Mechanics and Materials Center

TEXAS A&M UNIVERSITY

College Station, Texas

• (NASA-CB-177233) INTEGBATID EESEAECH IN

CONSTITUTIVE MODELLING, ft! ELEVATED

TEMPEBATUBES* PABT 2 Final Beport (Texas

: ASM Univ.) 224. p HC A10/MI A01 CSCL 20K

N86-26455

THEU

N86-28460

Unclas

^

. G3/39 431 19

INTEGRATED RESEARCH IN CONSTITUTIVE MODELLING

AT ELEVATED TEMPERATURES

FINAL REPORT

by

W.E. Haisler

and

D.H. Allen

Part 2

MM 4998-86-13 June 1986
AN EXPERIMENTAL COMPARISON OF CURRENT VISCOPLASTIC CONSTITUTIVE MODELS

AT ELEVATED TEMPERATURE

A Thesis

by

GEORGE H. JAMES III

Submitted to the Graduate College of

Texas A&M University

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

August 1986

Major Subject: Aerospace Engineeing
ABSTRACT

Four current viscoplastic models are compared experimentally with

Inconel 718 at 1100°F. This material system responds with apparent

negative strain rate sensitivity, undergoes cyclic work softening, and

mis susceptible to low cycle fatigue. The models used include Bodner's

anisotropic model, Krieg, Swearengen, and Rhode's model, Schmidt and

Miller's model, and Walker's exponential model. Schmidt and Miller's

model and Walker's model correct for negative strain rate sensitivity

response. A correction similar to Schmidt's is applied to the models of

Bodner and Krieg, et al.

A series of tests have been performed to create a sufficient data

base from which to evaluate material constants. A method to evaluate

the constants is developed which draws on common assumptions for this

type of material, recent advances by other researchers, and iterative

techniques. A complex history test, not used in calculating the

constants, is then used to compare to the predictive capabilities of the

models.

The combination of exponentially based inelastic strain rate

equations and dynamic recovery is shown to model this material system

with the greatest success. The method of constant calculation developed

in this work was successfully applied to the complex material response

encountered and suggestions for improvement are provided. Back-stress

measuring tests were found to be invaluable and warrant further

development.
LIST OF TABLES

1. Test Programs 37

2. Inconel 718 Composition 49

3. Example Iterations 61

4. Computer Iteration Summary of Krieg, Swearengen, and Rhode's

Model 65

5. Constants for Krieg, Swearengen, and Rhode's Model 68

6. Computer Iteration Summary of Bodner's Anisotropic Model 73

7. Constants for Bodner's Anisotropic Model 74

8. Computer Iteration Summary of Schmidt and Miller's Model 79

9. Constants for Schmidt and Miller's Model 80

10. Computer Iteration Summary of Walker's Exponential Model 85

11. Constants for Walker's Exponential Model 86

12. Complex History Test Input 100
List of Figures

1 A Strain Transient Test 29

2 A Stress Transient Test 30

3 Cyclic Hysteresis Loop with Hold Times 33

4 Creep Hold Response 34

5 Stress Relaxation Hold Response 35

6 Final Creep Frame Configuration 40

7 MTS Configuration 44

8 Furnace and Extensometer for MTS 45

9 Method of Thermocouple Attachment for MTS 47

10 Creep Frame Sample Configuration 50

11 MTS Sample Configuration 52

12 Test 65 Back Stress Determination Data 54

13 Model Response as Compared to Test 70 88

14 Model Response as Compared to Test 71 89

15 Stress-Strain Response at +.8% for Cycle 1 91

16 Stress-Strain Response at -.8% for Cycle 1 92

17 Stress-Strain Response at +.8% Saturated Cycle 93

18 10th Cycle Hysteresis Loops for Test 86 95

19 Test 86 +.8% Response at Each Cycle 96

20 Test 80 +.8% Response at Each Cycle 97

21 Test 72 +.8% Response at Each Cycle 98

22 Back Stress Plot 99

23 Complex History-Bodner's Uncorrected Model 101

24 Complex History-Bodner's Corrected Model 102
List of Figures Continued

25 Complex History-Uncorrected Model of Krieg, et al. 103

26 Complex History-Corrected Model of Krieg, et al. 104

27 Complex History-Schmidt and Miller's Model 105

28 Complex History-Walker's Model 106
TABLE OF CONTENTS

INTRODUCTION ' 1

LITERATURE REVIEW 4

Historical Review 4

Classical plasticity 4

Microphenominological based plasticity 5

Nonlinear viscoelasticity 5

Internal state variables 6

Selected Models For Further Comparison 7

Bodner's model 7

Krieg, Swearengen, and Rhode's model 8

Miller's model 9

Walker's model 11

Other Current Models 11

Krempl, et. al.'s model 11

Robinson's model , 12

Hart, et. al.'s model 12

Cescotto and Leckie's model 12

Previous Comparative Studies 12

OVERVIEW OF MODELS 15

Thermodynamic Framework 15

General Viscoplastic Framework 18

Bodner's Anisotropic Model 20

Krieg, Swearengen, and Rhode's Model 21

Miller's Model 22

Walker's Exponential Model 23

EXPERIMENTAL REQUIREMENTS AND PROGRAM 25

Creep Tests 25

Stress Relaxation Tests 26

Monotonic Strain Rate Tests 26

Fully Reversed Cyclic Tests 27

Transient Tests During Monotonic Loading 27

Transient Tests During Cyclic Loading 31

Experimental Program 36

EXPERIMENTAL APPARATUS AND PROCEDURES 38
Creep Frame Set-up 38

Table of Contents Continued

Load frame 38

Data acquisition 39

Temperature measurement and control 41

Test procedures 42

Computer Controlled Test Set-up 43

Load Frame 43

Data acquisition 43

Temperature measurement and control 43

Test procedures 46

Basic materials characteristics 48

Sample Configurations 49

Heat treatment 51

Back Stress Measuring Tests 51

CALCULATION OF MATERIAL CONSTANTS 56

Generic Summary Of Constant Calculation Methods 57

Generic viscoplastic model 57

Initial assumptions 57

Computer iteration 59

Krieg, Swearengen, and Rhode's Model 61

Initial calculations 62

Computer iterations 64

Bodner's Anisotropic Model 67

Initial calculations 68

Computer iterations 72

Miller's Model 73

Initial calculations 74

Computer iterations 77

Walker's Exponential Model . 78

Initial calculations 78

Computer iterations 84
MODEL RESULTS 87

Table of Contents Continued

Reproduction Of Test Data 87

Predictive Capabilities 100

CONCLUSIONS AND RECOMMENDATIONS 108

Conclusions and Recommendations Based On Models 108

Conclusions and Recommendations Based On Calculation of Constants 109

Conclusions and Recommendations Based On The Experimental Work 113

REFERENCES 115

APPENDIX A - DATA SET 125

APPENDIX B - PROPOSED INITIAL CALCULATION METHODS 136

Generic Summary Of Constant Calculation Methods 136

Generic viscoplastic model 136

Initial assumptions 137

Initial calculations 137

Krieg, Swearengen, And Rhode's Model 138

Bodner's Anisotropic Model 142

Miller's Model 143

Walker's Exponential Model 146

APPENDIX C - AUXILIARY MODEL RESULTS 149

APPENDIX D - EXPERIMENTAL COMPUTER PROGRAMS 175
INTRODUCTION

This thesis experimentally compares four current viscoplastic

models at elevated temperature. The primary objective of this work is

to uncover the mathematical forms which model reality most successfully

and to develop basic understanding of the models. A secondary objective

is to develop methods of constant calculation which are systematic and

repeatable. A final objective is,to develop experimental tests and test

software to support viscoplastic modeling.

Current technological applications such as jet engines, rocket

engines, and nuclear reactors operate at high temperature. The

components of these mechanisms are also subjected to complex loading and

temperature histories. Similar conditions will be found in future

technologies such as the skins and engines of hypersonic transports,

laser technologies, and alternative energy conversion systems.

Hypersonic technology holds promise for revolutionizing intercontinental

and orbital transportation. The proposed anti-ballistic missile shield

may use laser technologies. Alternative energy conversion systems such

as nuclear fusion reactors and solar energy conversion devices will be

necessary to continue the advancement of modern civilization through the

21st century.

The materials considered for the technologies listed above have a

thermoviscoplastic thermomechanical constitution. Thermomechanical

constitution describes the relation between the load applied to the
material (measured as stress), the change in the size of the material

(measured as strain), and the temperature. Thermoviscoplastic

mechanical response is dependent on time, temperature, and loading

history. The constitutive behavior of the material must be modelled

before any of the devices mentioned above can be designed or built.

This is especially critical in such expensive and potentially dangerous

applications. Thermoviscoplastic models are mathematical

representations which allow engineers to predict the behavior of these

materials. Major improvements are needed in these current models, as

will be demonstrated in this thesis.

Further development and evaluation of these models requires that

the models be compared to the real response of a sample of material at

elevated temperature and subjected to a complex loading history.

Several problems must be overcome in order to carry out such a

comparison. The models to be considered must be studied to determine

the types of inputs needed to use each model. These inputs are material

parameters which are unique to each model and must be determined from

specialized tests on the material being studied. The hardware and software for these specialized tests must be collected or developed. These

tests must be perfomed and methods developed to calculate the material

parameters for each model being considered in the comparison. Tests

with a very general loading history must be performed for comparison, so

that the actual material response observed in these general tests can be

compared to the response predicted by the models.

This research produces many positive results. First, the aspects of
each model which need further development are uncovered. Also, the most

accurate mathematical forms of the models are determined. Third, basic

understanding of the models is generated. Such understanding is

necessary for actual engineering application of the models and for

expanding the capabilities of the models. Fourth, systematic methods of

material parameter evaluation are developed which draw on advances by

all the modelers. Systematic constant calculation methods make the

models much easier to use by researchers and engineers in the field and

advance the technology toward automation and standardization. Finally,

experimental techniques and needs are developed or reported which can

either lead or support theoretical advances.

This thesis will begin with a review of the pertinent literature.

An overview of the selected models will follow. Separate chapters

dealing with experimental requirements, and experimental procedures will

be presented in that order. The methods used to calculate material

constants will be presented next. A chapter detailing model results

will follow. Conclusions of the current research and recommendations

for future research will conclude this thesis.

The intent of this thesis is to provide the interested reader with

the necessary information to repeat and improve on the work presented.

Four appendices are provided to support this intent. The first appendix

provides necessary experimental data to recalculate the material

constants. The second appendix presents proposed upgrades to the

constant calculation procedures. A complete set of experimental and

model results is contained in the next appendix. The final appendix
provides listings of the computer programs developed to perform the

experimental and data reduction portions of this work.
LITERATURE REVIEW

The following literature review begins with the historical

development of viscoplastic modelling. The next section traces the

development of specific models chosen for comparison in this work.

Current literature pertaining to the use, expected improvements, and

attempted transient temperature extensions of the chosen models are also

reviewed. Sections listing other current viscoplastic models and

previous comparative studies are also provided.

Historical Review

Three primary areas of research motivate current viscoplastic

models. The first area is rate-independent classical plasticity. This

is a macrophenomenologically based theory which attempts modelling by

observing the locally averaged response of the material. The second

area of research is microphenomenological modelling. This area attempts

to predict general mechanical response by studying microphysical

response. The third area is nonlinear viscoelasticity. This study is

inherently rate-dependent and draws heavily on thermodynamics.

Thermodynamic considerations provide a framework for any constitutive

model which can limit its functional forms.

Classical Plasticity. Tresca stated a possible means of predicting

the onset of yield in 1864 and thus began the study of inelastic

behavior [1]. Levy in 1870 [2] and Von Mises in 1913 [3] formed the

basis of classical plasticity by independently forming the Levy-Mises

equations. Von Mises also proposed his own yield criterion in the same

paper. Hencky [4], Prandtl [5], and Reuss [6] also made important
advances in rate-independent plasticity. Prager [71, Ziegler [81, and

Drucker[9J made advances which allowed classical plasticity to model

work-hardening materials more accurately.

Bingham [10] and Hohenemser and Prager [111 made initial attempts

to add rate-dependency to plasticity in 1922 and 1932. Freudenthal

[12], Malvern [13,14], Lubliner [151, Perzyna [16-20], Cristescu [21],

and Naghdi and Murch [22] attempted to model rate-dependent

plasticity. Howeve, these models did not unify rate-dependent and rateindependent plasticity. Zienkiewicz and Cormeau [23] presented a

classical plasticity based model which was unified and drew heavily on

viscoelasticity. Bodner [24] and Robinson [25] proposed models which

are still actively under development. Allen and Haisler [26] proposed a

model which attempts to model transient temperature effects.

Microphenomenologically Based Plasticity. Coble [27], Nabarro

[28], and Herring [29] produced early works on diffusion controlled

creep. Sherby, et al. [30-32], Garofalo [33], Weertman [34], Alden

[35], Mukherjee, Bird, and Dorn [36], Gibbs [37], Argon [38], Kocks

[39], and Hart [40] are some other important contributes to the

foundations of microphenomenologically based theory. Hart [40], Krieg,

et al. [41], and Miller [42] have produced models which are still under

development.

Nonlinear Viscoelasticity. Biot [43] produced a linear

viscoelastic model in 1954 which was later expanded by Shapery to model

nonlinearity [44]. A reduced time scale was used by Shapery to model

nonlinear behavior. Valanis later used the same reduced time technique
to model plasticity [45,46]. Some other viscoelasticity based theories

with thermodynamic considerations include Green and Rivlin [47], Coleman

and Noll [48,49], and Green and Naghdi [50]. The models of Krempl [51]

and Walker [52] are viscoelasticity based models which are still under

development.

Internal State Variables. Rational thermodynamics has its basis in

the work of Truesdell and Toupin [53]. Other works in the area include

Coleman and Noll [49], Muller [54,55], Green and Naghdi [50], Green and

Law [56], Day [57], and Kratochvil and Dillon [58,59]. The application

of thermodynamics to constitutive modelling of history-dependent

materials can be accomplished by a functional approach as shown by

Coleman [60] or by internal state variables as shown by Coleman and

Gurtin [61]. These two methods were shown to be equivalent in most

cases by Lubliner [62].

The internal state variable approach is used extensively in current

literature [63] and will be discussed further in the next chapter. An

internal state variable maintains a material's memory by integrating an

associated growth law or evolution equation through the history of the

loading. The evolution equations for internal state variables usually

follow a standard pattern of competing hardening and softening terms as

proposed by Bailey 164] and Orowan 165].

Inelastic strain can be considered to be an internal state

variable. In addition, several micro-structural phenomena may be

represented by internal state variables. Dislocation arrangement, back

stress, equilibrium stress, or kinematic hardening may be considered to
be an internal state variable. Dislocation density, drag stress, or

isotropic hardening is another example. Damage mechanisms may also be

represented as internal state variables [66].

Selected Models for Further Comparison

Four current viscoplastic models have been chosen for experimental

comparison in this research. A review of these models follows. The

models chosen are those of Bodner [67], Krieg, et al., [41], Miller

[42], and Walker [52]. These models have been chosen because they are

under active development, they have been put in a common thermodynamic

framework [68], methods of determination of constants has been reported

[41,42,52,69,70], the models have been studied in the past [68-73], and

some attempt has been made or is being made to expand them to transient

temperature modelling [69,72,74].

Bodner's Model. Bodner first proposed his model in 1975 [24J. The

model assumed isotropic and isothermal conditions and required no yield

function. One internal state variable was utilized to model isotropic

hardening [75]. The model was later expanded to account for anisotropy

and hence kinematic hardening [76-78]. A damage parameter has also been

added [78,79]. The forms of the internal state variable growth laws are

based on the earlier work of Onat [67,80,81].

Bodner's model is a unified phenomenologically based model which

has a scalar variable for isotropic hardening and a symmetric second

order tensor variable for anisotropic hardening. An effective scalar

variable is produced from the anisotropic variable and added to the

isotropic variable to produce a single hardness variable [78].
Transient temperature modelling has been attempted with Hastelloy X

[74]. A coupled heat conduction equation has been produced which can be

used in conjunction with the model [82].

The i.sotropic form of the model has been used to predict the

response of Rene 95 at 650° F [83], Inconel 100 at 732° C [84], and

Inconel 718 at 650° C [85,86,87]. Methods for the determination of

material constants have also been reported [84,87,88]. The anisotropic

form of the model has been used to predict the response of Hastelloy X

at room temperature [89]. A method for the determination of the

material constants for the anisotropic form has been reported [90].

Material constants for several other materials have also been reported

167].

Krieg. Swearengen, and Rhode's Model. Krieg, Swearengen, and Rhode

presented a microphenomenologically based model in 1978 [41]. The model

is equipped with an internal state variable for isotropic hardening and

one for kinematic hardening. The hardening functions are given as

constants. Krieg has suggested a modification for the hardening

functions which has not been incorporated into this model [91,92]. The

recovery terms in the internal state variable growth law follow the work

of Friedel [92].

A method for the determination of the constants has been presented

for the case where the isotropic hardening variable is constant. This

method makes extensive use of a stress drop or strain transient dip test

[41]. The authors have engaged in subsequent work to further develop

this test for constitutive modelling [93,94]. The kinematic form of the
model has been used to model aluminum at room temperature [41] and

Inconel 100 at 1350 F [71]. Walker has provided a method for evaluating

the constants and has modelled Hastelloy X at several temperatures

[72]. Transient temperature modelling has been attempted with Hastelloy

X [72].

Miller's Model. Miller first proposed his microphenomenologically

based model in 1976 [42]. The model relies heavily on microphysical

considerations. Both a kinematic hardening internal state variable and

an isotropic hardening variable are provided. Annealing is simulated,

but recovery terms in the growth laws depend only on temperature. The

hardening functions in the growth laws are given as constants [42].

A simplified form of the model was presented in 1977 [95]. This

form has eliminated the kinematic hardening back stress variable and

allows only isotropic hardening to be modelled. Initial attempts were

made to model solute strengthening in the drag stress variable. The

hardening function in the back stress growth law was altered in 1978 to

produce a more realistic transition from elastic to inelastic behavior

[96]. The form of this improvement was first proposed by Krieg [91].

Schmidt and Miller further improved the capability to model solute

strengthening [97,98]. These modifications were designed to allow the

model to be operated over a broader range of stress and temperature when

solute effects can be expected. Kagawa and Asada cast Miller's model in

multiaxial form in 1983 [99]. Miller has developed a new form of the

model with four structurally based internal state variables for

modelling long range back stresses, short range back stresses, long

10
range drag stresses, and short range drag stresses. This form of the

model holds promise for predicting transient temperature effects

[69,100] and work softening phenomena.

Miller has also presented a series of papers devoted solely to

microstructural topics which give rise to various functions found in his

models. Sherby and Miller explained in 1979 the form of the inelastic

strain rate equation and its basis in steady state creep. They also

explained the initial attempts to model deformation strengthening

[101]. Miller and Ziaai-Moayyed pointed out and confirmed with further

experimental work several results of Miller's model pertaining to steady

state back stress, strain softening, and peak back stress [102]. Ruano,

Miller, and Sherby emphasized the need to properly take into account

diffusional properties in modelling creep at various stress levels

[103]. Schmidt and Miller presented a paper dealing with the effect

solutes have on material behavior [104]. Miller, Kasner, Rubin, and

Sherby produced work correlating isotropic hardening transients to

dislocation strengthening [105,106].

Miller first presented a method for calculating the constants for

his model in 1976 [42], This method used steady state creep, cyclic

tests, and tensile tests. Miller presented a method for calculating the

constants in his simplified model in 1977 [95]. This method used steady

state creep tests and tensile tests. Schmidt presented a method of

constant evaluation for the model incorporating solute strengthening

effects in 1979 [97,98]. This method used microstructural data, creep

tests, tensile tests, and cyclic tests. Walker presented a method of

11
constant evaluation for Miller's-model based solely on cyclic tests in

1980 [721. Miller's model has been used at several temperatures to

model stainless steel 304 [42], aluminum [95], stainless steel 316

[97,98] and Hastelloy X. Transient temperature modelling has been

attempted with Miller's model for Hastelloy X [72]. The newest form of

the model is intended to have transient temperature capability [69,100].

Walker's Model. Walker proposed his theory in 1980 [52]. It is a

viscoelasticity based model with microphenomenological considerations.

Two microstructural internal state variables, back stress and drag

stress, are provided. The model has been proposed with a term in the

back stress growth law to allow for nonisothermal elastic excursions

[72]. The model was presented in a different form and coupled with a

damage internal state variable in 1984 [107],

Walker has provided a method of determination of constants for a

kinematically hardening material (drag stress constant). The method

relies totally on cyclic testing [52]. Walker's model has been used to

model Hastelloy X at several constant as well as transient temperatures

[72]. The latest form of the model has been used to model Inconel 718

at 650° C [107].

Other Current Models.

Krempl, et al.'s Model. Krempl's model is a viscoelasticity based

model in which an equilibrium stress is subtracted from the applied

stress. This fills the function of the back stress in a

phenomenological model. A drag stress parameter is not utilized in the

roost recent presentation of the model. Recovery is not provided. The

12
equilibrium stress hardening term is dependent on total strain rather

than inelastic strain [51,108-113].

Robinson's Model. Robinson's model is a classical plasticity based

model with microphenomenological considerations and draws on work by

Onat, Lagneborg, and Rice. Auxiliary equations are provided to better

model separate material points [25,114-118]. It has .been modified to

model anisotropic behavior [119]. The model has also been used to model

nonisothermal conditions [69,114-118,120-122]. Robinson has presented a

series of papers outlining development of his model [123-130]. The

constants are determined in torsion [131].

Hart, et al.'s Model. Hart presented his microphenomenologically

based model in 1976. It includes a microstructural internal state

variable for back stress, a constant for drag stress, and an internal

state variable called hardness which affects the back stress growth law

[40]. A method for the calculation of constants has been given [132].

The latest form of the model has four internal state variables. Two

back stress and two hardness internal state variables are used [69,133].

Cescotto and Leckie's Model. Cescotto and Leckie presented a

microphenomenologically based model in 1983 [134]. The model includes

an internal state variable for back stress and one for drag stress. The

evolution equations for these variables are not given in a specific

form. They must be evaluated after a series of appropriate tests.

Previous Comparative Studies

Delph performed a comparative study between Robinson-'s [25] and

Hart's [40] models in 1980 [115]. This was a quantitative study based

13
on constants derived from 2 1/4 Cr - 1 Mo steel at 1050 F. Cernocky

produced a study in 1981 comparing the models of Bodner and Partom [24],

Cernocky and Krempl [51,135] and two versions of a viscoelasticity based

model by Wu and Yip [136,137]. This was an analytical and numerical

study [73]. Walker completed a study in 1981 [72] which included a

qualitative examination of several viscoplastic models. The study also

performed an experimental comparison of Walker's theory [52], Miller's

theory [42], and Krieg, Swearengen, and Rhode's theory [41]. This study

utilized Hastelloy X at several temperatures and included extensions to

thermomechanical testing.

Milly and Allen reviewed Bodner and Partom1s theory [24], Krieg, et

al.'s theory [41], Walker's theory [52], Krempl, et al.'s theory [135],

\

and Zienkiewicz and Cormeau's theory [23]. A comparison of Bodner and

Partom's model and Krieg, et al.'s model for Inconel 100 at 1350 F was

also performed [71]. Beek, Allen, and Milly did a comparative study in

1983 of all the listed in the previous section plus the models of

Valanis [45,46] and Allen and Haisler [26,138]. These models were all

put in a common internal state variable framework to allow easier

comparison [68]. Lindholm, et al. produced a review of several

viscoplastic constitutive theories in 1984 [69]. The models covered

included those of Walker [72], Bodner and Partom [75,78], Miller

[42,97,98], Krieg, et al. [41], Chaboche [139], Robinson [25], Hart

[40,133], Bodner's anisotropic model [67], Lee and Zaverl [140], and

Ghosh [141]. Follow-up research to this project will utilize the models

of Walker and Bodner's anisotropic form for thermomechanical testing.

14
Beek [142] used aluminum 5086 at room temperature to compare the models

of Krieg, et al. [41], Bodner's isotropic model [75], and Miller's

simplified model [95].

15
N86-28456

OVERVIEW OF MODELS

A brief description of the thermodynamic framework ofinternal state

variables will open this chapter. The general form of current

viscoplastic models will be discussed next. Individual sections will

follow which deal with the specific forms of the models selected for

this work.

Thermodynamic Framework

The material in this section is an abbreviated form of discussions

found elsewhere [68,143]. A state variable is a parameter whose

magnitude can be determined by looking at the current time only. The

following state variables are necessary to fully characterize the

thermomechanical state of a body at all points x,- and at all times t,

while undergoing infinitesimal deformations:

1) The displacement field i^- ui(xk,t) (1)

2) the stress tensor o. .= o. .(x.,t) (2) I J 1J K

3) the body force per unit mass f..= fi(x(<,t) (3)

4) the internal energy per unit mass u = u(xk,t) (4)

5) the heat supply per unit mass r = r(x.,t) (5)

6) the entropy per unit mass s = s(xk,t) (6)

16
7) the absolute temperature T = T(x. ,t) (7)

8) the heat flux vector q..= qi(xk,t) (8)

and

9) a

m..= afj(xk,t) , m = 1, 2, ...n (9)

where aTM, are internal state variables (ISV) which are necessary to

account for the nonlinear effects of rate, history, and temperature.

ISV are not necessarily limited to second rank tensors.

The variables u^ (displacements) and T(temperature) are

hypothesized to be specifiable variables. The following variables are

then defined from the spatial derivatives of u^ and T:

e^= % ( u. , + u. , ) ; the strain

g = T ;the spatial temperature gradient (11)

III 9^"

The following constitutive equations of state are then postulated:

Oij(xk,t) - o..[ emn(xk,t), T (xk.t). gn(xk.t). aPn(xk,t) ] (12)

17
')'

 T

-

 s I *)' T (*) - 8(x.t). "(x,t) ] (14)

The quantities o. . , u, s, and qn- are called observable state variables

' J

because they can be determined from the constitutive equations of state

by observing at the current position(xk) and current time(t) only. The

magnitudes of e , T,gm, and ajj|n and the relationships between these

independent variables will give the magnitudes of the dependent

variables. The current magnitudes of the ISV a'3 are arrived at through

growth laws or evolution equations of the form:

c^.(xm,t) = Q !f-(e , T, g , 0|J ) (16)

Equations (16) must be integrated through all time to arrive at the

current magnitudes of the ISV:

•«• J" "ii'v*1 "'• <17>

Where t1 is a dummy variable of integration. The magnitudes cannot be

determined by looking at the current time only and these quantities are

18
therefore called hidden or internal state variables. The distribution of

f .j (body force) and r(heat supply) may be determined by the method of

Coleman and Noll [49] with equations (12) through (16), conservation of

linear momentum, and conservation of energy. Analysis of the mechanical

response of materials will require that the precise forms of equations

(12) and (16) be produced. A simplification is performed here in which

the temperature gradient is assumed to be zero in the body of interest

and in equations (12) and (16). This leaves the independent variables

of strain ( e ) and temperature (T). These conditions are those

needed to model thermoviscoplastic response. The mechanical equations

of state for viscoplastic materials take the following form when T is

assumed to be constant in time and space:

<

18>

(19)

General Viscoplastic Framework

This research deals only with uniaxial response and the specific

models will be presented in uniaxial form only. A general outline will

be presented which defines the various parts of a typical, unified,

internal state variable, viscoplastic model. The notation used will be

that of Imbrie, et al. [70]. The general form of equation (18) for the

uniaxial case is as follows:

19
o - E (e - e1). (20)

where E is given to be Young's modulus, e is the total strain, and e is

the inelastic strain. The thermodynamic framework presented in the

preceding section requires .that e be an ISV. Labelling inelastic

strain as the first ISV is a controversial issue and the reader is

referred elsewhere for clarification [66].

A growth law for the inelastic strain, also called the flow law or

inelastic strain rate equation, must be presented to fulfill equations

(19):

l

l= f ( e, e1. B, D,...am) (21)

where B ( a ) and D ( a ) are the second and third ISV, and am are

other internal state variables not used in the specific model forms

presented here. Equation (21) commonly takes the following form:

i

l= ^(-^--) (22)

where a = E ( e - e ) . The function f^ can take several forms

including exponential, power law, and hyperbolic functions. B is called

back stress or equilibrium stress and is an ISV responsible for

kinematic or directional hardening, the Bauschinger effect, or lower reyield in compression. B may be thought of as a measure of the

arrangement of dislocations in the crystalline lattice. D is called drag

20
stress and is responsible for isotropic work hardening, such as cyclic

hardening or softening. Physically, this variable measures dislocation

density. am are variables which may be used to account for the effects

of damage, solution strengthening, short range dislocation effects, or

temperature [79,97,98,1001.

The growth laws for B and D commonly take the following forms:

B = h( M ) - r°( ft. B ) - r( B ) (23)

0 = hD( M ) - rjj( M, D ) - rj( D ) (24)

where h are the hardening functions, rd are dynamic recovery terms, and

_ \

r are static thermal recovery terms. Various models may not have both

recovery terms [42,41]. M is the measure of work hardening and is

commonly inelastic strain ( e ), total strain ( e ), or plastic work

( W ). Equations (20-23) are specific forms of equations (18) and

(19). Individual models will now be presented.

Bodner's Anisotropic Model

This model has the following form [67]:

= -^- D0exp{ - 4- hf]2n} sgn a (25)

.1

e

3

Z = Il+ ZA= ZJ+ B sgn a (26)

Z^mJ Zr Z1] Wp- A.ZJ— p (27)

21
B = ma[ Z3sgn „ - ZA] W- A^J- 2 sgn ZA (28)

where DQ, n, m^, Zj, Z£, A^, r^, n^, Zg, A£, and T^ are material

constants.

The function f required for the flow law is exponentially based in

equation (25). The model gives a limiting strain rate in shear of DQ

[70]. The term -rrijZ ft is a dynamic recovery term for Z in the

isotropic growth law (27) and - KlZl[( I - Z2) Z"1]

1"1 is a static

thermal recovery term. 8 is a uni axial representation of a second order

tensor in the multiaxial state which handles directional or anisotropic

hardening. B is assumed to act as an isotropic variable on an

incremental basis [78]. The growth law for B (28) has the same

components as the growth law for D (27).

Bodner's model is seen to use the rate of plastic work instead of

inelastic strain rate as the measure of work hardening (27,28). This is

designed to allow for better modelling of strain rate jump tests [67].

Krieg, Swearengen. and Rhode's Model

Krieg, et al.'s growth laws have the following form [70]:

8I. C p--)n sgn o (29)

B = A^1- A2B

2

[ e(A3

B

 )- 1 ] sgn B (30)

0 = A,*1- As( 0 - 00)

n

 (31)

22
where C, n, Aj_, A2» A3, A4, and Ag are material constants.

Krieg, et al.'s flow law is seen to be a power law based

equation. The back stress and drag stress growth laws (30,31) contain

static thermal recovery terms but no dynamic recovery terms. The

recovery term in (30) is based on a dislocation climb model by

Friedel. The recovery term in (31) is based on a special case of the

same climb recovery model used in (30) [41,92].

Miller's Model

Miller's growth laws have the following form [42].

i

l= B'{ sinh( ° " B j1'5}" sgn( a - B ) (32)

B = H^1- H^'f sinh( AjB| ) }n sgn( B ) (33)

6 = Hje'KC^ |B| ---D 3) - H2C2B'{ sinh( A/) }n (34)

where B1, n, H^, Aj_, h^. C2» and A£ are material constants.

Miller's flow law has the form of a hyperbolic sine. This form was

chosen to model creep response better [42]. This same form is found in

the static thermal recovery terms of the back stress and drag stress

growth laws (33,34). The drag stress hardening term contains a

hardening term, a dynamic recovery term, and a term which couples drag

stress hardening to back stress magnitude. These three terms provide

the proper cyclic, hardening, softening and saturation behavior [421.

23
Walker's Exponential Model

Walker's growth laws have the following form [90,107]:

e

!= exp t ° D ^ " * sgn( o - B ) (35)

B = n2- B { [ n3+ n,,exp(-ns| log(JJli) |) ] R + n6} (36)

RO

D = Dj+ D2exp(-n7R) (37)

R = (I1! (38)

•

where B, n£, ng, n4, n^, R0, ng, Dj_, D2» and n7 are material constants.

This version of Walker's flow law (35) is based on an exponential

function. The term n2e is seen to be a work hardening term in the back

stress growth law. The term B [ n3+ n,texp(-n5|log(|R|/R0) |) ] R is a

dynamic recovery term. Negative strain rate sensitivity effects can be

modelled with the term nltexp(-ns |log(|R|/R0) |) . Back stress thermal

recovery is handled by the B n6 term. Drag stress hardening is modelled

through the D2exp(-n7R) term. No provision is made for drag stress

recovery in this model.

24
EXPERIMENTAL REQUIREMENTS AND PROGRAM

This chapter contains sections covering the tests used as input to

calculate material constants for viscoplastic models. These standard

tests include creep tests, stress relaxation tests, monotonic strain

rate tests, fully reversed cyclic tests, transient tests during

monotonic loading, and transient tests during cyclic loading. These

sections cover the input and expected output for each test, the

information sought from each test, and the experimental requirements for

each test. The final section covers the experimental program actually

perfomed for this thesis.

Creep Tests

These tests are load controlled tests in which a constant stress,

usually simulated by a constant load, is applied to a material sample.

The resulting strain versus time response is monitored. The information

sought in this test is the linear slope during secondary or steady-state

creep. This region of material response is characterized by a stable

and saturated microstructure [41,42]. A saturated microstructure in

this case is a condition where the back stress and drag stress are

unchanging due to a balance between the work hardening and the static

thermal recovery terms. The saturated back stress is commonly assumed

to have a linear relationship to applied stress [42]. Miller assumes

the saturated drag stress values have a power law relation to applied

stress [42,95]. Some model evaluation methods utilize points on the

primary or transient creep curve [41].

The requirements for this test include a constant stress or a

25
constant load with small changes in cross-sectional area. A stable

temperature over a long period of time and good control over initial

load-up are also necessary. The time required for initial load-up

should be short enough to eliminate thermal recovery effects and long

enough to eliminate dynamic effects. The usual test length requires

long term stability of the extensometry or strain measuring equipment.

The strain rates encountered may be slow and good resolution is needed

with the extensometry. Creep tests are usually run at several initial

loads to arrive at a table of initial load versus steady state strain

rate results. These are simple tests to run. However, they are prone to

give scattered results [131].

Stress Relaxation Tests

These tests are strain controlled tests in which a constant strain

is applied to a material sample. The resulting stress (or load) versus

time response is monitored. The information sought in this test is the

constant stress achieved following a hold at constant strain. The

requirements for this test include a constant strain and a constant

temperature [131].

Monotonic Strain Rate Tests

These tests are strain controlled tests in which a constant strain

rate is applied to the sample. The stress versus strain response is

then monitored. The information sought in this test includes the yield

point or onset of substantial inelastic strain, the initial elastic

modulus, the final saturated stress or final constant stress rate, and

the strain rate sensitivity of the material. These tests are generally

26
run rapidly enough to neglect static thermal recovery terms and

therefore provide data on work hardening and dynamic recovery constants

[67). This type of test requires the ability to input a constant strain

rate and tests using several different strain rates are run.

Fully Reversed Cyclic Tests

These tests are strain controlled tests which ramp between the same

value of strain in tension and compression. The resulting stress versus

strain response is monitored. The data sought in this test include the

yield point, initial elastic modulus, the cyclic hardening or softening

characteristics, and the final saturated hysteresis loop. The drag

stress usually changes in magnitude with each cycle until saturation to

*

control the cyclic hardening, softening, and saturation characteristics

[42]. This saturated state is often used as a reference condition from

which transient tests are run [52,131]. Tests of this type are usually

run at several different strain rates and strain limits [42,52]. The

most difficult requirement for this test is usually machine

alignment,si nee samples with straight gauge sections are prone to

buckling when loaded compressively.

Transient Tests During Monotonic Loading

The purpose of these tests is to arrive at magnitudes for back

stresses. The strain transient test, also called the stress drop test,

during secondary creep and the stress transient test during saturated

monotonic strain rate loading are the two tests mentioned here. These

tests require the independent or controlled variable (stress for creep

and strain for monotonic strain rate) to be reduced to a lower value and

27
held constant. The dependent variable (strain for creep or stress for

monotonic strain rate) is then monitored versus time. The direction and

magnitude of the change in the dependent variable for different

reductions in the independent variable is sought. The stress on the

sample is equal to the back stress whenever the dependent variable

remains at a constant value following a drop [144]. Figure 1 shows the

typical stress-time and strain-time response of a strain transient or

stress drop test. Figure 2 shows the typical strain-time and stresstime response of a stress transient test. Several transient tests are

usually run with one sample when the independent variable is returned to

the reference condition before each drop.

These tests are both highly controversial. However, the strain

i

transient test is generally better accepted. The stress transient test

allows the microstructure to change (back stress magnitude changes)

during the transient period [144]. The strain transient test during

creep requires a great deal of understanding to properly interpret the

results. The following paragraph will present a brief review of

pertinent literature concerning the strain transient test.

Some of the earliest reports on strain transient testing were by

Ahlquist and Nix [145-146]. They reported some early techniques and

results. Poirier presented a paper in 1977 which gave the expected

results of strain transient tests and possible means of analysis

[147]. Gibeling, Nix, and Fuchs explained the resulting transient in

terms of the microstructure [148-149]. These papers also discussed the

28
L±U

A02

H 1-

e

Z

>0

4

X

<0

•H fC

l

 C2

Figure 1 - A Strain Transient Test
fc

I—I—I—\-

Figure 2 - A Stress Transient Test
physical nature of the back stress and when it was microphysically

appropriate to include a back stress variable in a constitutive

equation. It was also mentioned that hand smoothing of the data was

usually necessary. Jones, Rhode, and Swearengen presented a machine for

use in strain transient testing [93]. The typical method of analyzing

strain transient data is to produce a zero strain rate following a

stress drop. Blum and Finkel proposed an alternate means of analyzing

strain transient data. This technique involves measuring initial and

maximum changes in strain following a stress drop. Plotting these

changes against the stress decrement allows the back stress to be

calculated. This method assumes the ratio of back stress to applied

stress is constant and independent of stress [150].

Strain transient tests during creep require varying amounts of

stress to be quickly removed and reapplied. The strain must be measured

with a resolution of a few microstrain for an indefinite amount of

time. The data acquisition system must be able to take in data slowly

during the creep portion of the test and rapidly during the transient

portion of the test. These requirements make this a difficult test to

carry out.

Transient Tests During Cyclic Loading

The purpose of these tests is to determine values of back stress

for cyclically saturated conditions. A cyclically saturated condition

is characterized by each successive cycle repeating the previous cycle

when subjected to the same maximum and minimum strain limits. The creep

test or the stress relaxation test is used during constant strain rate

31
cyclic conditions to produce these tests. The general assumption behind

these tests is that the back stress changes only with inelastic

strain. This assumption requires that elastic unload take place rapidly

enough to ignore static thermal recovery [52].

These tests are performed by cycling a material until saturated

conditions are reached. Hold times are then inserted at various points

in the unloading region as shown in Figure 3. The stress value is

greater than the back stress value if positive creep (strain increasing)

or negative strain relaxation (stress decreasing) is observed. The

stress value is less than the back stress value if negative creep

(strain decreasing) or positive stress relaxation (stress increasing) is

observed. The applied stress value equals the back stress value when no

creep or stress relaxation is observed immediately following a hold.

Figure 4 summarizes this typical response if creep tests are used and

Figure 5 shows the typical response if stress relaxation tests are

used. The material is cycled between each hold time to return the

material to saturated conditions [52,131].

The requirements for these types of tests include the ability to

cycle the material into compression, measure strain from 10~^ to

microstrain, and to hold the test at various points. Creep transients

also require the ability to switch between load and strain control.

Ellis and Robinson have described an extensometer and associated

calibration fixture for carrying out these transient tests [131]. They

also report the merits of carrying out such tests under conditions of

torsional cycling while using stress relaxation response.

32
Hold Time #1

a >Back Stress

Hold Time //3

a = Back Stress

Hold Time #2

a < Back Stress

Figure 3 - Cyclic Hysteresis Loop With Hold Times
Hold Time #1

e

OBack Stress

Hold Time #2

£

0<Back Stress

Hold Time #3

e

o=Back Stress

Figure 4 - Creep Hold Response .
Hold Time #1 a>Back Stress

Hold Time #2 o<Back Stress

Hold Time #3 o=Back Stress

-o

Figure 5 - Stress Relaxation Hold Response
Experimental Program

The basic experimental program consisted of the following tests:

(1) 2 monotonic tension tests to 1.5% strain ( strain rates of

3.15X10'3 sec'1 and 7.25X1CT6 sec'1);

(2) 5 fully reversed cyclic tests to + .8% strain ( strain rates

between 1.00X10'3 sec"1 and 7.63X10'6 sec'1 );

(3) 5 constant load creep tests ( applied stresses between 119 ksi

and 139 ksi );

(4) 4 back stress measuring tests during cyclic loading and 4 during

secondary creep; and

(5) 1 complex history test. An auxiliary set of data was generated

from a second series of material and was utilized only in specific

instances when a broader data base was necessary. The second series

was presumably from a different lot or heat treatment. The primary

series of material was denoted as D2, whereas the auxiliary series

was denoted as D8. The auxiliary data included the following:

(1) 8 monotonic tension tests ( strain limits of 1.2% to 4% and

strain rates between 9.25X10'4 sec'1 and 7.03X10'6 sec'1 );

(2) 5 constant load creep tests (applied stresses between 110 ksi

and 125 ksi ); and

(3) 1 additional complex history test from the D2 series.

Table 1 provides more specific information on the test program.

Column 1 provides the test number and column 2 provides the material

series. The type of test is given in column 3. The strain rate and

strain limits are given in columns 4 and 5. The applied stresses for

36
the creep tests are given in column 6. A complete data set in tabular

form is provided in Appendix A.

Table 1 - Test Program

Test

70

71

86

56

65

83

80

72

64

63

62

61

60

84

88

81

*65

63

62

61

60

89

35

37

39

34

36

42

38

40

19

11

17

12

10

85

Series

D2

D2

D2

D2

D2

D2

D2

D2

D2

D2

D2

D2

D2

D2

D2

D2

D2

D2

D2

02

02

02

08

08

08

08

08

08

08

08

08

08

08

08

08

02

Type

tension

tension

cyclic

cyclic

cyclic

cyclic

Cyclic

cyclic

creep

creep

creep

creep

creep

back

back

back

back

back

back

back

back

complex

tension

tension

tension

tension

tension

tension

tension

tension

creep

creep

creep

creep

creep

complex

eapPi

sec"

3.151E-3

7.253E-6

1.002E-3

9.966E-4

3.127E-4

9.926E-5

3.054E-5

7.626E-6

2.812E-3

9.272E-4

8.635E-4

3.127E-4

9.253E-4

6.048E-4

1.793E-4

7.637E-5

5.703E-5

1.914E-5

1.410E-5

7.029E-6

e l im

1.5%

1.5%

+/-.8%

+/-.8X

+/-.8%

+/-.8%

+/-.8%

+/-.8%

+/-.8%

+/-.8%

+/-.8%

+/-.8%

4.0%

1.2%

1.2%

2.4%

1.2%

3.5%

1.2%

1.2%

°app

ksi

138.8

133.9

127.0

124.0

119.0

133.9

127.0

124.0

119.0

125.0

120.0

120.0

115.0

110.0

37
EXPERIMENTAL APPARATUS AND PROCEDURES

This chapter will describe the test equipment used to carry out the

tests described in the previous chapter. The first section will

describe the test set-up used for the creep.and strain transient tests

based on a dead weight creep frame. Topics to be covered include the

load frame, data acquisition, temperature measurement and control, and

test procedures. The next section will describe the test set-up used

for the monotonic strain rate, fully reversed cyclic tests, and

transient tests during cyclic loading as well as creep and strain

transient tests based on a computer controlled test set-up. The same

topics will be covered. The next section covers material considerations

of this work. Topics to be covered include basic material

characteristics, sample configuration, and heat treatment. The final

section will be devoted to a discussion of the procedures used to

perform back-stress measuring tests on the automated load frame.

Creep Frame Test Set-up

Load Frame. The load frame utilized in this thesis was a creep

frame produced at Texas A&M University. The initial configuration

included a constant load cam and a constant stress cam as described by

Garofalo, et al. [151]. The lift for the pan was provided by an ATS

(Applied Test Systems) 2081 cyclic load module. Several modifications

were made to this set-up. The constant stress cam was removed and a

second constant load cam installed. This increased the maximum weight

limit from 400 Ibs. to 800 Ibs. This, also increased the stability of the

load-pan during load-up and provided a seven to one load magnification.

38
The load pan was increased in size and supports were added to further

aid stability during load-up. The cyclic load table was removed and

replaced with a Century-Fox model CF-59 5 ton capacity hydraulic jack.

This provided more lift capacity and gave more room for machine

deflection during load-up. The final configuration of the machine is

shown in Figure 6.

Further improvements needed include a more advanced load pan which

would provide automated removal and replacement of portions of the

load. Jones, et al. [93] have described such an apparatus. Similar

modifications are needed to improve the results of the strain transient

tests. Another improvement needed is a more advanced method of loadup. An apparatus such as an electric or pneumatic jack which could

provide a calculable and repeatable load-up rate is needed.

Data Acquisition. The load was measured with a Strainsert TLN20-

256K Tension Link driven by a Hewlett Packard 8805A Carrier Preamp. The

strain was measured with an ATS model 4112 LVDT (Linear Variable

Displacement Transducer) and extensometer driven by an ATS model 6974

signal conditioner. Two Hewlett Packard 8803A Low Level Preamps and two

low pass filters were necessary to achieve the necessary resolution.

Load and strain measurements were recorded on a Hewlett Packard series

7700 strip chart recorder. These components can be seen in Figure 6

also.

This data acquisition system was very prone to drift, noise, and

non-linearity. All strain amplification, filtration, and data

39
PAGE is

POOR QUALITY

Figure 6 - Final Creep Frame Configuration
acquisition equipment were eventually replaced with a Kiethly model 197

logging digital multimeter. This set-up proved much more stable.

However, it lacked the ability to take in a large number of data

points. Much improvement is needed in strain data acquisition to

produce a system which is stable, can take in large numbers of points

and has sufficient resolution.

Temperature Measurement and Control. An ATS 2961 Clamshell oven

and an LFE series 230 temperature controller were used for temperature

control. Temperature measurement was handled by two 20 gauge K-type

thermocouples. These were placed in contact with the sample in the

middle of the gauge section on diametrically opposing sides. One

thermocouple was used as input to the temperature controller. The other

thermocouple was input to a Fluke 2176A digital thermometer for readout.

The oven, controller, and digital thermometer can be seen in Figure 6

also.

Improvements in this system would include the ability to monitor

the temperature at several points simultaneously along the gauge

section. The temperature should also be input to a data acquisition

system. The method of thermocouple attachment should also be

upgraded. The optimum method of thermocouple attachment would be

individual welding of the leads to the sample surface. This is

described in the ASTM Thermocouple Handbook [152]. Such a procedure was

attempted with 20 gauge thermocouple wire and a Duracom thermocouple

welder. The samples tended to fail prematurely at the thermocouple

welds and this method of thermocouple attachment was abandoned. Another

41
possible method using smaller 28 gauge thermocouples was used with the

computer-controlled set-up and will be discussed in that section.

Test Procedures. The initial step was to measure the gauge length

and gauge diameter. The gauge diameter was measured at the top, middle,

and bottom of the gauge section. These measurements were averaged and

an area calculated. The weight needed to produce the desired stress was

calculated and loaded on the load pan. The weight of the pan alone was

45 Ibs. and the load magnification of the frame was seven to one. The

sample was then loaded in the extensometer and load train.

The sample temperature was stabilized at 1100 F using small

changes in set point and zone power. This usually took 45 minutes to

one hour. Zeroing and calibration of the LVDT followed a warm-up period

of 30 minutes. A micrometer was provided on the extensometer for

calibration. The LVDT was calibrated between 0 and 5 volts (.25 inch

displacement).. The digital voltmeter was set to take in one point every

minute for the first 100 minutes and one point every 10 minutes for the

remainder of the test. The weight was applied to the sample by

releasing the pressure in the jack over a period of about 30 seconds.

Stress drop tests were carried out after three to six hours of

creep testing. The digital voltmeter was set to take in a point every

second and an increment of load was removed. The load was reapplied one

to 15 minutes later after a creep rate was determined. The sample was

allowed to creep at least 30 minutes before the next stress drop test.

Following a creep test the stress change during the test could be

approximated by assuming a Poisson's ratio of 1/2 throughout the test.

42
This gave an upper limit as to the expected stress variation during the

test.

Computer Controlled Test Set-up

i

Load Frame. The load frame utilized in these tests was an MTS

(Materials Test System) model 880 electrohydraulic testing machine shown

in Figure 7. MTS 652.01 Water-cooled hydraulic grips allowed fully

reversed cyclic tests to be carried out at high temperature. The frame

was controlled by a Digital Micro PDP-11 computer. Computer programs

were written to run monotonic tension tests, cyclic tests, cyclic tests

with hold times, creep tests, and creep stress drop tests. These

programs are presented in Appendix D.

Data Acquisition. The Micro PDP-11 also handled data acquisition

functions. An MTS 661.21A-02 10 kip load cell was the load

transducer. An MTS 632.418-02 axial extensometer was the strain

transducer. This device had quartz extension rods which contacted the

sample at two 120 punch holes. The extensometer can be seen in Figure

8. All data were stored on 5.25 inch floppy diskettes and could be

retrieved in hard copy or graphical form.

Temperature Measurement and Control. An MTS 652 three-zone

clamshell furnace and three Research Incorporated 63911 Process

Temperature and Power Controllers were used for temperature control.

The furnace is shown in Figure 8. Temperature Measurement was handled

by six 28 gauge K-type thermocouples. These were placed in contact with

the sample. Three thermocouples were fed into a Fluke 2176A Digital

Thermometer for readout. These were placed with one each at the top,

43
ORIGINAL PAGE IS

OF POOR QUALITY

Figure 7 - MTS Configuration
ORIGINAL PAGE IS

OF POOR QUALITY

- - ' -, '•• '

Figure 8 - Furnace and Extensometer for MTS
middle, and bottom of the gauge section. The other three thermocouples

were fed into the temperature controllers. These were placed in the

center of the furnace zone each was to sense. One thermocouple was

placed in the center of the gauge section and one on each grip.

The thermocouples were fastened to the grips by fiberglass thread

attached to the sample by self-supporting means. The thermocouples at

the top and bottom of the gauge section were wound around the sample.

The thermocouples used in the center of the gauge section were brought

into the oven from different directions and tied to each other. These

thermocouples were then wound around the sample for contact. This

arrangement is shown in Figure 9. Welding the thermocouples to the

sample would have produced harder contacts with more reliable

temperature measurement. However, as mentioned earlier, premature

failure occurred at the welds.

Test Procedures. The sample gauge length and gauge diameter were

measured as mentioned earlier. The sample was then installed in the

grip inserts using Nikal Nickel Special anti-seize lubricating compound

to prevent freeze-up to the grips from high-temperatures. The sample

was installed in the bottom grip and then the top grip. The hydraulic

pressure was applied to the grips. Thermocouples, furnace, and

extensometer were then mounted, respectively. The water and air for

cooling of the grips and extensometer were then turned on, followed by

the furnace. The machine was left on low pressure and in load control

to compensate for thermal expansion. The temperature and temperature

gradient were then stabilized using the set points of the temperature

46
ORIGINAL PAGE IS

OF POOR QUALITY

Figure 9 - Method of Thermocouple Attachment

for MTS
controllers. This usually took two to three hours to achieve an

acceptable temperature gradient. The most efficient method was to

gradually work the temperature up to the desired level.

Calibration of the extensometer was carried out with a Del-tron

calibrater approximately every 5 tests. Grip alignment was checked with

an Ames 212.5 .0001 inch dial gauge. The dial gauge was attached to the

lower grip and the entire grip and fixture were rotated. A dial gauge

reading of the relative displacement of the upper grip was taken at four

places. Adjustments were made until the relative displacement was less

than .0015 inches. Alignment was carried out as needed.

Material Considerations

Basic Material Characteristics. The material used in this work was

Inconel 718 and was provided by NASA Lewis Research Center in Cleveland,

Ohio. The temperature used was 593° C. (1100° F.). The typical

chemical composition as reported by Thakker and Cowles [153] is shown in

Table 2. The alloy is precipitation-hardenable and maintains high

strength and creep resistance to 700° C. [153,154]. The material is

used extensively in the aerospace and nuclear industries [154]. Young's

modulus at 593° C. is reported by Korth and Smolik to be 24,230 ksi

[154] and the average value produced in this work was 24,658 ksi. The

same researchers report .2% yield stress values between 128 ksi and 140

ksi, ultimate tensile strengths between 147 ksi and 174 ksi, and a mean

coefficient of thermal expansion of 14.56X106 °C ~

l [154]. The material

used in this work produced .2% yield stress values between 115 and 131

ksi. Inconel 718 is reported to cyclically work soften by several

48
authors [153-158]. Such response was also experienced during this

research.

Table 2 - Chemical Composition of Inconel 718

Chemical Component

Ni

Cr

Fe

Cb+Ta

Mo

Ti

Co

Al

Mn

Si

Cu

C

P

S

B

Required Weight Percent

50.0-55.0

17.0-21.0

15.0-21.0

4.75-5.50

2.80-3.30

0.75-1.15

1.0 Max

0.3-0.7

0.35 Max

0.35 Max

0.30 Max

0.02-0.08

0.015 Max

0.015 Max

0.006 Max

The material underwent strain aging and therefore responded with

apparent negative strain rate sensitivity between the strain rates of

1X10~5 sec"1 and 1X10'3 sec'1. A fatigue life of 5 to 30 cycles

resulted specimens were cycled at strain limits over +1% strain. Lower

strain rates and the inclusion of creep hold times also adversly

affected the fatigue life. These observations are supported by several

other researchers in the field [153-158].

Sample Configurations. Sample design for the creep frame was

driven by machine requirements. Figure 10 shows the creep frame sample

configuration. The shoulder grooves were necessary to maintain seating

by the extensometer knives. The gauge diameter was reduced to .20

inches to bring the applied load values into the range of the load

49
,75"

3.75"

10 threads/in.

1.00"

V

r

J

t

.5"

.50"

groove -.025" wide

.20"

groove -.025" wide

Figure 10 - Creep Sample Configuration
frame. The MTS .880 samples designed to ASTM E606-77T specifications

for low cycle fatigue specimens. The sample configuration is shown in

Figure 11.

Heat Treatment. All samples were subjected to the same heat

treatment prior to being tested. The heat treatment used was given by

the Metals Handbook [159]. The material was annealed at 954° C. for one

hour followed by oil quench. Ageing at 718 C. for eight hours was next

with a furnace cool. The furnace used was a Hevi Duty Electric Co. type

66-P. Temperature was monitored with a Keithly 871 Digital

Thermometer. The resulting material state was found to be the easiest

to machine. Therefore, the heat treatment was carried out before

machining and again before testing.

Back Stress Measuring Tests

The values for back stress values reported in this work are at best

estimates of the true values of and should be treated as such. Three

reasons exist for this uncertainty. First, the transient tests during

cyclic loading require saturated conditions. However, the fatigue

lifetime problems for this material mentioned earlier did not permit

complete saturation of the microstructure for fear of sample fracture.

The criterion used to define saturation in this work was a cycle to

cycle variation of the maximum stress of less than one ksi. These

conditions were met after 10 to 15 cycles for this material. Second,

the resulting strain measurements following a cyclic hold time or a

creep stress drop were on the order of the digital resolution of the

data aquisition system ( = 5.0X10" in ). This problem was lessened by

51
3.75"

T

.95"

1

.43"

T

1.00"

I

A

.43"

t

.95"

\

\

.50"

1.00" radius

2 holes -120°

.25"

20 threads/in.

Figure 11 - MTS Sample Configuration
taking in large numbers of data points and using least squares

regression analysis to calculate strain rates. Third, the determination

of when recovery effects became significant following a cyclic hold time

or creep stress drop was a very subjective decision. Each transient

test had to be individually scrutinized to decide how many points to

consider in the regression analysis.

The cyclic back stress numbers were obtained by holding a cyclic

test at various points on the unloading curve, switching to load control

and monitoring the strain rate following the hold. The material was

recycled and a hold time at another stress value was carried out. A

linear least squares fit to the strain versus time data provided a

strain rate at each hold time. Another linear least squares fit to the

strain rate versus hold stress data provided stress at which a zero

strain rate would have been expected. The back stress was assumed to be

equal to the stress at this point.

Figure 12 shows the data obtained from test 65 cyclic hold times.

Test 65 was a cyclic test run at a strain rate of 3.127X10"4 sec'1 with

strain limits of + .8% strain. The x axis provides values of the strain

rate calculated with the least squares regression after each cycle. The

y axis provides the associated value of the stress at the hold time.

The average maximum applied stress over the range of these tests was 108

ksi. A bilinear relationship can be seen in the data. A linear least

squares fit to the data of the lower slope provided the back stress

value used in this work of 48 ksi. It was assumed that the upper slope

represented a breakdown in the linear assumption and was not

53
o

o

o

•

C\Jo

o

d

o.

>— .o

CO

CO

o

IE

8

d-|

OJ

-10.00 0.00 10.00 20.00 30.00

STRAIN RATE <IN/IN/SEC)*E-7

40.00

Figure 12 - Test 65 Back Stress Determination Data
considered. This analysis should be reconsidered. The upper slope may

actually be considered and a back stress value of 80 ksi reported.

Poirier [147] presents some models which would support a zero strain

rate for a cyclic hold or stress drop which is below the back stress.

The lower slope could be a viewed as a region of approximately zero

strain rate and the associated stress values would be below the back

stress value.

The back stress numbers were invaluable in estimating some material

constants as will be seen in the next, chapter. The results were also

promising enough to warrant further study. The procedures used here

could be greatly enhanced by equipment with greater resolution such as

used by Jones, et al. [93] and less subjective methods of data reduction

such as the method of Blum and Finkel [150]. Other techniques such as

torsional cycling could also be considered [131]. The stress transient

test [144] might also provide information for a material such a Inconel

718 which suffers from a short fatigue life when cycled. It should also

be noted that cyclic hold times were attempted with stress relaxation

holds rather than creep holds. The machine was found to be less stable

in strain control. A smaller load frame might allow stress relaxation

holds to be more stable and eliminate the need to switch between load

and strain control. The back stress data presented graphically in

Figure 12 are presented in tabular form in Appendix A as well as data

from the other back stress measuring tests.

55
CALCULATION OF MATERIAL CONSTANTS

The complex response of Inconel 718 at 1100 F prompted flexible

methods of constant calculation to be developed. The method for

calculating constants for the models began by making a series of

judicious assumptions which allowed commonly used constant calculation

schemes to produce initial estimates of the constants. Some

nonlinearity was avoided in this step and was reintroduced by a series

of repeatable iterations to the final constants. The iterative step

numerically integrated the models to predict the stress-strain response

at a certain point. One material constant was then changed to match the

prediction to the experimental value at this point. Another material

constant was then changed to match another material point.

Physical insight, familiarity with the uncertainty in the data set,

and engineering intuition guided the organization of the calculation

process. However, the actual process was carried out as systematically

as possible. The eventual creation of systematic and automatable

methods to calculate constants has been a major driver in this phase of

the work. This chapter is organized to allow an interested reader to

recalculate the constants and to improve on the process used here.

Table 1 should be consulted throughout this chapter as tests will be

referred to by number. Appendix A presents all experimental data needed

to repeat these calculations. Appendix B provides proposed upgrades to

these procedures based on experience developed in this "work.

56
The method used to calculate the material constants will be

summarized using a generic viscoplastic model in the first section of

this chapter. This is presented to make the actual calculations more

understandable to the reader. The generic model used as an example will

be presented first followed by a subsection outlining the general method

of initial calculations and a subsection outlining the iterative step.

Individual sections covering the four models used in this work will

follow. Each section will contain one subsection describing the initial

calculations and one describing the computer iterations.

Generic Summary of Constant Calculation Methods

Generic viscoplastic model. The growth laws for the example model

are presented below:

gI. (_2_=_B_)n (39)

B = C^l+ C2B il+ C3B (40)

D = Cjl1! + C5D (41)

n is a constant measuring strain rate sensitivity. Cj is a constant

measuring back stress hardening. C,£ is handling back stress dynamic

recovery and Cg measures back stress thermal recovery. C4 produces drag

stress hardening and C5 models drag stress recovery.

Initial assumptions. The following initial assumptions were made

in this work:

57
(1) back stress was assumed responsible for hardening in

monotonic tension;

(2) drag stress was assumed responsible for cyclic softening;

(3) thermal recovery was assumed negligible for rapid

tests (e!> LOXlO'^ec"1) ;

(4) drag stress thermal recovery was present in low

strain rate saturated cyclic tests; and

(5) back stress thermal recovery was present in creep tests.

These assumptions allowed the constants for the inelastic strain rate

equations, back stress hardening, drag stress hardening, drag stress

recovery, and back stress recovery to be calculated in that general

order. These assumptions also allowed much of the constant calculation

schemes reported in the literature to be utilized with this material

[41,42,52,70,72,90,97,142]. A new set of initial assumptions was found

to be more appropriate upon completion of this work and proposed methods

based on these new assumptions can be found in Appendix B.

The first step was to estimate the constants in the back stress

growth law assuming thermal recovery was negligible. The back stress

growth law took on the following form:

B = [C1+B C2] e1 (42)

Differential techniques for calculating work hardening such as seen in

Chan's gamma and theta plot concepts [90] were useful. Experimental

estimations of back stress values such as used by Krieg, et al. [41] and

58
Walker [52] were usually necessary. Relationships between saturated

stresses and saturated back stresses as used by Miller [42] have also

been used.

The next step was to calculate the strain rate sensitivity constant

n and the initial value of drag stress denoted by DQ. Rewriting the

inelastic strain rate equation in the following form was useful:

o - B ) ---ln ( e) + ln( D0) (43)

A linear fit to several data points typically provided 1/n as the slope

and ln( DQ ) as the intercept. This is a technique commonly used with

Bodner's model [84]. The ability to estimate saturated stresses and

back stresses using techniques such as the gamma or theta plot [90] and

relations between saturated stress and back stress are useful [42,90].

Initial determination of the drag stress parameter C4 was carried

out by assuming that thermal recovery could be neglected for rapid

tests. The cumulative inelastic strain was calculated to a point on the

cyclic curve where B and D could be estimated. The drag stress recovery

parameter C5 was then calculated by assuming the drag stress growth law

was equal to zero for the saturated cycle of a low strain rate test.

The back stress recovery parameter Cj was calculated by assuming the

back stress growth law was equal to zero for creep tests.

Computer iterations. The computer iterations began by pairing each

material constant with an experimental stress-strain point which the

constant should intuitively have the greatest effect in predicting in a

59
sequential fashion. The constants in the generic model were paired in

the following fashion for this work:

(1) DQ was paired with a stress-strain point at .8% strain on test

70 ( i = 3.151X10~3 sec"1);

(2) G£ was paired with a stress-strain point at 1.3% strain on test

70;

(3) GI was used to assure that the theoretical back stress values

were in the same range expected from experimental values;

(4) n was paired with a stress-strain point at .8% strain on test 71

( e = 7.253X10"6 sec'1);

(5) Cg was paired with a point at 1.3% strain on test 71;

(6) C4 was paired with a point at .8% on the 10th cycle of test 86

( I = 1.002X10'3 sec'1);

(7) C5 was paired with a point at .8% ot the 4th cycle of test 72

( e = 7.626X10'6 sec"1);

The iterative procedure then progressed by numerically integrating

the model to predict the experimental stress-strain value for a specific

constant. The constant was altered to match this point while the others

were held constant. Then another constant was altered to produce the

proper prediction at its paired experimental point. The expected order

with which these steps were to be carried out is shown in Table 3. The

x marks indicate which constant is being altered during the step

indicated in column 1. Steps 1 through 5 are setting the back stress

hardening characteristics. Steps 6 through 9 are setting the strain

rate sensitivity of the model. Steps 10 through 12 are setting the drag

60
stress hardening constant. The back- stress recovery constants are being

set in steps 13 through 15. Drag stress recovery is set in steps 16

through 18.

Table 3 - An Example Set of Iterations

Step

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

D0

X

X

X

C2

X

X

X

X

X

Cl

X

n

X

X

C4

X

X

C3

X

X

X

C5

X

X

This method allows the entire process to be recorded. Automation

of such a method is also possible if the initial calculations produce

values which are close to the final constants. A systematic set of

iterations may also allow a standard method for calculating constants to

be produced. The lack of correction for strain aging effects in the

initial calculations caused problems in implementing this iterative

scheme. Appendix B provides some suggestions to avoid this.

61
Krieg, Swearengen and Rhode's Model

The growth laws for the model of Krieg, et al. are repeated below

for completeness:

i

1- CHH^-)n

 sgn( o - B ) (44)

2

B = Aje1- A2B

2[ e( A3B ) - 1 ] sgn( B ) (45)

D - AJ il\ - A5( 0 - D0)

n (46)

Initial calculations. The initial calculations began by rewriting

the inelastic strain rate equation in the following form:

— ) + n-ln(o-B) (47)

Assuming a constant D a linear fit between ln(o - B) and ln(e )

produces a slope of n and intercept ln(C/Dn) . Experimental values of

o, B, and e from tests 60, 61, 63, 65, 81, 84, and 88 produced an n

value of 4.41257. Using a DQ value of 1.00, C is estimated at

3.177X10~H sec . Now, assuming monotonic hardening is due to back

stress, no thermal recovery is present, and the initial value of back

stress is zero, then back stress is written as:

B = Aje (48)

62
The constant A^ is calculated from the following:

(.1, r^l/n

I

Experimental values of o, e, and I at .B% total strain from tests 70,

86, 56 provide an average value of Aj = 20,533.82 ksi.

The constant A4 is calculated by assuming that B and D grow without

recovery for the first three half cycles. The following formula then

provides an estimate of A4 :

f « ( I, I. 111( 6 ^~l/n n I o - Aj( e,+ e2+ e3J [-*-J - D

A,= £ °— (49)

where e1$ e2, and e3 are the values of inelastic strain produced

during each half cycle. Using values of a, e|, e2, e3, and e from

tests 86 and 56, the average estimate of A4 = -8.784 ksi was produced.

Cyclic saturation is assumed to be a balance between drag stress

growth and recovery. Rewriting the drag stress growth law provides the

following equation:

• I

A = Aj^l-Ko - B).(-H~1/n- Dj"n (50)

C

Test 84 provides experimental values of o, B, e and produces A5 =

-13.8768 ksi sec'1. DQ has been set to 1.000.

Creep saturation is assumed to rely heavily on directional

recovery. The back stress growth law is written as follows:

63
2',( A3B

2

A,r= A2B [ev "3" i. i] (51)

Using two tests, the following equation results for A3 :

2

A3- { ln[ K( e( A3

B2)_i) + i]} B"2= 0 (52)

where K =

?

! R 2

e2 Bj

All possible combinations of tests 60, 61, and 63 provided

experimental values of e1§ B1§ e2, and B2 . The above equation was

solved iteratively for A3 and the resulting estimates averaged to

produce A3 = 1.134X10~^ ksi~ . The following equation then produced

values for A2:

2

A2= A1e

IB-2[e( A3B )- I]'1 (53)

Averaging the values produced by tests 60, 61, and 63 provided A2 =

1.452XKT14 ksi sec"1.

Computer Iterations. Table 4 summarizes the computer iterations

carried out to obtain the final constants. The first column lists the

step number of these sequential iterations. Each step consisted of

changes in the constant provided in column two. The model was

integrated at the same strain rate as the experimental test given in

column three to match the experimental point at the total strain value

64
Table 4 - Computer Iteration Summary of Krieg, Swearengen, and Rhode's

Model

Step

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

Const.

Al

C

Al

C

Al

C

A4

A5

Al

C

Al

C

A4

C

A2

C

Al

C

C

C

C

C

C

C

DO

A4

A5

8

J

F

8

J

F

8

J

F

8

J

F

DO

n

DO

C

Al

8

J

Test

70

70

70

70

70

86

86

86

70

70

70

86

86

71

71

70

70

86

71

72

65

83

80

86

86

86

70

70

70

71

71

71

70

70

70

71

70

70

70

70

70

70

Strain(%)

1.28

.8

1.28

.8

1.28

.8

.8

.8

1.28

.8

1.28

.8

.8

.8

1.28

.8

1.28

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

Cycle

1

1

1

1

1

1

10

10

1

1

1

1

10

1

1

1

1

1

1

1

L

1

1

1

10

10

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

Old Value

20t533

3E-11

5,000

3E-12

3,000

2E-12

-8.78

-13.876

2,500

9E-13

750

1.75E-12

-3.0

7E-13

1.45E-14

3.7E-15

1,000

1.75E-12

7E-13

3.7E-15

3.7E-15

1.8E-13

1.8E-13

1

-3.5

-6.0

2.15

7E-6

14,029

2

1E-7

1,500

5

1E-9

3,000

568.34

4.41257

475

1

1,200

1.5

7E-6

New Value

5,000

3E-12

3,000

2E-12

2,500

9E-13

-3.0

-6.0

750

1.75E-12

1,000

7E-13

-3.5

3.7E-15

1E-4

1.75E-12

1,200

7E-13

3.7E-15

3.7E-15

1.8E-13

5E-14

1.375E-14

1

568.34

-500

l-E-19

2.1549

7E-6

1402.9

2

1E-1

1,500

5

1E-9

3,000

1.5

7E-6

1,300

475

15

100

2E-4

1,000

3.5

1E-9

65
Fable 4

Step

47

48

49

50

continuec

Const.

F

6

F

Al

1

Test

70

71

71

70

Stra1n(X)

.8

.8

.8

1.28

Cycle

1

1

1

1

Old Value

1,300

3.5

100

1,000

New Value

100

4.0

120

1,400

given in columns four and five. The old value of the constant is given

in column six and the new value is given in column seven. Several

iterations were required within each step to arrive at the proper new

value. The steps were carried out sequentially. The constant being

changed was chosen as the one which could be expected to have the

greatest influence on the stress-strain point considered.

Steps 1 through 17 of Table 4 are tuning constants Aj, A2, A4,

A5. The true impact of the negative strain rate sensitivity material

response showed itself at this point. The model could not match both

tests 70 (3.015X10'3 sec'1) and test 71 (7.626X10'6 sec'1) in the form

presented here. Steps 18 through 23 were used to locate individual

values of the constant C to. fit each strain rate. The values were then

used to calculate a correction for the negative strain rate sensitivity

problem. The modification used in this work came from Schmidt and

Miller's non-interactive solute stregthening correction [981. The

inelastic strain rate equation is written in the following form:

sol

sgn(o - B (54)

66
Fsol= F.exp( .(I ion (It'll > I*, f|J|H )., (S5)

Steps 24 through 27 of Table 4 show iterations in which all scaling is

transferred from the C constant to the D variable. An initial

assumption was that the maximum effect of the solutes was at

I 6 1

e = 7.000X10 sec . The effect was assumed to be non-existent for

i = 1.000X10" sec" . The constant F is the maximum value of the solute

stregthening. This constant was set by differencing DQ values for tests

71 and 86 needed to provide proper scaling. The constant J is the

strain rate at the peak effect and was set at 7.000X10~6 sec'1. The

constant 6 is the width of the correction and was set by the following

equation:

- log( J ) (56)

e

1 was set at 1.000X10"3 sec"1 and 3 = 2.1549 resulted.

Steps 31 through 40 tuned these constants. The correction was

still not adequate. Steps 41 through 50 increased the constant n to

lessen the strain rate sensitivity of the basic model. Table 5 shows

the values of the constants after the initial calculations and the

values after the computer iterations.

Bodner's Am'sotropic Model

The growth laws for Bodner's anisotropic model are repeated below

67
for completeness:

D0exp[-(-^-)2n] sgn o (57)

Z = l+ Z= Z+ B sgn a

• T T • 2

Z

1- BI( Zr Z1

) Wp - A^t—

(58)

(59)

B = m2( Z3sgn a - Z )• W - A2ZJ sgn ZA (60)

' .1

Wp= a e (61)

Table 5 - Constants for Krieg, Swerengen, and Rhode's Model

Constant

n

C

Al

A2

A3

A4

A5

BO

DO

F

8

J

Hand Calculations

4.412

3.176E-11

20,530

1.451E-14

1.133E-3

-8.784

-13.88

0

1.000

Computer Iterations

15.00

2.000E-4

1,400

0.000

1.133E-3

-500.0

-l.OOOE-19

0.00

100.0

55

1,000

7.000E-6

Initial calculations. The initial calculations began by rewriting

the inelastic strain rate equation in the following form:

68
•>

 Do 2-ln(— )] = -2n-ln(o) + 2n-ln(Z) = ln(S) (62)

'3 il

A linear fit of 1n(o) versus ln(S) gave -2n as the slope and 2n[ln(Z)]

as the intercept for stationary values of Z. n = .81316 was calculated

in this fashion using tests 60, 61, 62, 63, 64, 71, 72, 80, 40, 38, 42,

36, and 34. These tests were low strain rate tests

T "4 1 (e < 1.0X10" sec ) and exhibited positive strain rate sensitivity.

The values of a from tests 60, 61, 62, 63, and 64 were applied stresses

and e the strain rates during secondary creep. The a values for the

remaining tests were saturated stresses from the gamma plot and

the e values were values of applied strain rate. The gamma plot will

be discussed later in this section. The value for On was set at \

 u

1.000X104 sec'1.

At this point it is assumed that Z2 = Zg at this point and ZQ was

estimated. The value of ZQ is given by the following equation:

Z0=o0[ 2.ln(—A— J]1/

2" (63)

/ , .1 3 e0

The values of o0 were the values of stress at the onset of visible

plastic flow. e0 was the associated plastic strain rate and was

typically 1X10'6 sec'1. Test 72 produced a Z0 value of 1063.23 ksi.

The gamma plot as described by Chan in reference [90] is a proposed

method of analyzing the work hardening characteristics of a monotonic

test assuming dynamic recovery and a measure of work hardening based on

plastic work. The following assumptions are also made:

69
(1) Only values of o and e after .2% offset will be used;

(2) e is constant and approximately equal to applied

strain rate; and

(3) thermal recovery is negligible.

The assumption that Z1 is constant during a monotonic cycle was also

made for this material system. The following equation is derived:

. / , .1 -l/2n

Y = -fir = [ -2-ln(-^.-S—)] m2(Z3+ Z0) - m2a (64)

P 2 D0

Values of y were calculated by differentiating a polynomial curve fit of

a stress-inelastic strain curve and dividing by the stress value at each

point. A linear fit to the first ten points of a o versus Y plot

provided a slope of -n^ and an x intercept of saturated stress.

Saturated stress is defined to be the stress at which the internal state

variables are unchanging. Values of Zg were calculated from the

saturated stress using the following formula:

-7 r 0 i_r 3 ° ^il/2n 7 /cc\ Z3= a[ 2-ln( J] - Z0 (65)

2 e1

The ZQ values used in this step were individual values for each test and

not the single value reported earlier. Averaging tests 71, 72, 80, 65,

56, 86, and 70 the following values were obtained:

Z3 = 317.01 ksi

m2 = 3.3068 ksi'1

An initial estimate of Z is obtained by assuming the gamma plot

70
works analogously for the positive side of the saturated cycle (assumed

to be the tenth cycle). The value of ZQ has changed to Z^ at this cycle

and the following equation results from the x intercept:

Zi=o[ 2 . ln[ _]_ Z 3 (66)

2 e*

Tests 65, 56, and 86 were averaged to produce the following value for

zl!

Zt = 724.35 ksi

The gamma plot was used to obtain values of Z1 and W_ for every

half cycle of 10 cycle cyclic tests. Z1 is calculated as Z} had been

calculated above. Wp was calculated by integrating the polynomial

producing Y. These values are then combined in the following equation

to produce an estimate for m:

ln( Z1- Z,) = -mlWp+ ln(Z0- ZJ (67)

Tests 86 and 56 produced an average value of m± = .4850 ksi"1.

Calculation of the constants A^ and r^ required assuming that

recovery of B was negligible and the gamma plot was still operative at

the saturated cyclic loops of low strain rate tests. Values of Z1 were

calculated as above and W values were calculated from the last two data

points of each positive 10th cycle load-up. The following equation then

allowed a linear fit to produce values for

A and r:

71
T • - 2 ln{ mj Z1

- Zj Wp} = riln[ - ] + ln( -A^) (68

Tests 72 ( W = 8.454X10~"ksi/sec ) 80 ( W = 3.138X10~-3ksi/sec ), 83

( W = 1.014XKT2ksi/sec ), and 65 ( W = 3.044X10"2ksi/sec ), produced

the following values:

A! = -.001 sec'1

rj_ = .49258

Creep tests were used to determine values for A2 and r2. The

assumption that Z1 = ZQ was used at this point. Values of B were

• •

calculated from the inelastic strain rate equation. W values were also

calculated from a i . The following equation then provided r2 and A2

after a linear fit:

m2(Z3- B) Wp] = r 2 ln[ ] + ln( -A2ZJ (69)

Tests 60, 61, 62, 63, and 64 provided the following values:

A2 = -6.9935 sec'1

r2 = 21.9558

Computer iterations. Table 6 summarizes the computer iterations

carried out to obtain the final constants for Bodner's model. It should

be interpreted in the same manner as in the section describing the model

of Krieg, et al. Step 1 removes the directional recovery term from

consideration. Steps 2 through 7 are setting the constants ZQ and m2

for test 70. Steps 8, 9, and 10 are setting the cyclic response through

72
the constants Z^ and Aj_. A correction for solute strengthening was

found to be necessary for this model also. The model response was

determined as a function of ZQ in steps 11 through 15. This information

was used to determine a solute strengthening correction. Schmidt and

Miller's correction was again used to produce an inelastic strain rate

equation of the following form:

Z + Fsol

0

2n] sgn a (70)

(71)

Table 6 - Computer Iteration Summary for Bodner's Anisotropic Model

Step

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

Const.

A2

20/12

M2

ZO/Z2

Ml

ZO/Z2

M2

Zl

Al

Zl

ZO/Z2

ZO/Z2

20/12

20/22

20/22

F

6

J

20/22

J

20/22

Test

70

70

70

70

70

70

70

86

71

86

71

72

65

83

80

70

70

71

Strain(%)

.8

.8

1.25

.8

1.25

.8

1.25

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

Cycle

1

1

1

1

1

1

1

10

1

10

1

1

1

1

1

1

1

1

Old Value

-6.99

1062.23

3.3068

825

.485

1050

.5

72435

-.001

600

975

975

975

975

975

7E-6

975

New Value

0.0

825

.5

1050

.05

975

.4

600

-.0001

700

1240

1250

1050

1150

1225

350

3

7E-6

975

1E-6

900

73
The constants F, 6, and J, were calculated in the same manner as with

the model of Krieg, et al. These parameters were introduced and tuned

in steps 16 through 21. Table 7 compares the constants before and after

the computer iteration process.

Miller's Model

The growth laws for Miller's model are repeated below for

completeness:

e = B1{ sinh[

- B ,1.L55 ]n sgn(o - B) (72)

B = H^1- HjB1 sinh( AjB| )n sgn(B)

\ -

D = H2| e C2+

U5n

- H2C,B' sinh( A 2 D )

(73)

(74)

Table 7 - Constants for Bodner's Anisotropic Model

Constant

n

Al

A2

Ml

M2

rl

r2

ZO

Zl

Z2

Z3

F

8

J

Hand Calculations

.8132

-.0010

-6.994

.4850

3.307

.4926

21.958

1063.23

724.35

1063.23

317.01

Computer Iterations

.8132

-.0010

0.000

.0500

. 4000

.4926

.4926

900.0

700.0

900.0

317.0

-350.0

3.0

1E-6

74
Initial Calculations. The first step in the calculation of

Miller's constants is to write an auxiliary equation based on the

inelastic strain rate equation in the following form:

sinh(A o)J + In(B') (75)

e and o are values of inelastic strain rate during secondary creep

and the associated applied stress. The constant A is to be determined

from the following equation:

ln[sinh(A o,)] - ln[sinh(A a'

(76)

ln[sinh(A o3)] - ln[sinh(A Oi;

The e and o values are again secondary creep rate and applied stress

from 3 separate tests. This equation was solved iteratively for various

combinations of tests 60, 61, 62, 63, and 64. The results were averaged

for an A value of .0917092 ksi"1. The same tests were then applied to

the previous linear equations to produce n and B' through a linear

fit. The following values were produced:

n = 1.813

B' = 2.206X10"16 sec"1

DO was calculated from the following equation assuming B = 0.000:

1 .1

D0- oJslnhK-M ] }- (77)

B

1

o0 and e were the values at the appearance of plastic deformation. A

75
DO value of 16.264 ksi was produced by averaging the results of tests

71, 72, 80, 83, 65, 56, 86, and 70.

The constant H^ was calculated from the following equation assuming

drag stress had not grown and recovery was negligible:

H1= (a - D0{ sinh-'U-M"] }}(*r (78)

B

1

Utilizing points at .8% total strain the average of tests 70, 86, and 56

was H1 = 4128.4085 ksi.

Estimates of A^ and A2 will be obtained from the following

equations:

A^A-^S- (79)

Bss

A2= { A [ 1 - —^- I"1 }3 (80)

Bss

Values of o and B from tests 65, 81, 84, and 88 were used to

calculate the following values:

A! = .170673 ksi"1

A2 = .0095544 ksi'3

An estimate of C2 was obtained from fast saturated cyclic tests

with back stress recovery assumed negligible, drag stress saturated, and

back stress value known. The following formulation results if such

conditions can be met:

76
C2= (— D- B)( 1 - [ B'{ Sinh( Ald)-) - (81)

D = (a - B){ sinh^K-M17"}'273 (82)

B

1

The results of tests 65, 81, 84, and 88 are averaged to produce C2 =

-5.335XKT2 ksi.

The constant H2 was estimated from a fast cyclic test assuming no

back stress recovery and no drag stress recovery. Back stress is

estimated by calculating the cumulative inelastic strain for each of the

first three half cycles. The following equation is written for H2:

H2= (C.e o - B) (sinh-!!--)"] }-2/3 (83)

where e= |.| + |e| + \t\

The values ej, e2, and e3 are the maximum inelastic strains from the

first three half cycles. Tests 56,86, and 65 produced an average value

of H2 = -20189.385.

Computer iterations. Experience with the models of Bodner and

Krieg, et al. prompted a change at this point to an upgraded version of

this model which accounted for solute strengthening. The inelastic

strain rate equation of the Schmidt and Miller model is as follows:

sinh[ ( °/E -

 B j1-5]" } sgn( 0/E - B ) (84)

D + Fsol

77
exp( - | - q , (85)

The noninteractive solute strengthening parameter was the only one used

in this work (interactive solute strengthening modifies the growth of

the drag stress). This allowed a more meaningful comparison to the

other models since they were all corrected in a similar fashion. The

calculation of the material constants was also simplified since only one

parameter was utilized. The scale factor of Young's modulus in the

inelastic strain rate equation should also be noted.

Table 8 summarizes the computer iterations carried out. Steps 1

through 14 have no order other than the intuitive guesses of the user.

These iterations were correcting major deficiencies of the initial

calculations. The solute strengthening parameters were set in steps 15

through 19. The basic strain rate sensitivity of the model is still too

positive and steps 20 through 30 were increasing n to eliminate this

problem. The solute strengthening parameters were being tuned in steps

31 through 36. The work hardening parameters were being set in steps 37

through 50. The constants after initial calculations and after computer

iterations are shown in Table 9.

Walker's Exponential Model

The growth laws for Walker's model are repeated below for

completeness:

(86)

78
Table 8 - Computer Iteration Summary of Schmidt and Miller's Model

Step

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

Const.

H2

C2

DO

HI

Al

A2

A2

C2

Al

DO

C2

H2

Al

A2

F

6

J

A2

B

1

C2

DO

HI

H2

n

n

DO

n

DO

n

B

1

J

F

8

J

F

F

HI

C2

H2

HI

HI

F

8

B

1

HI

B

1

C2

Test

86

86

86

86

86

86

86

86

86

86

86

86

86

86

70

70

70

71

71

70

70

71

70

71

71

70

71

71

71

70

86

86

70

71

70

70

70

70

71

86

Stra1n(X)

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

Cycle

1

1

1

1

1

1

10

10

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

10

Old Value

-20189

-.053353

16.264

4128.41

.17087

.0095544

.0001879

20189

.183445

0.0

40

.053353

.2714418

.02

.027

2.206E-16

l.OE-11

1.81

2.5

1.0

5.0

.01

6.0

1.0E6

l.OE-6

.001

5

7.0E-6

.002

3.0

0

0

0

l.OE-5

.15

.005

4.0

l.OE-7

.1

3.0E6

-1,000

New Value

.053353

20189

10.0

12000

.183445

.0061674

.02

40

.2714418

11.0

1E-11

2.5E-1

.3

.027

.001

5

1E-6

.023

1.0E6

0

1.0

0

0

2.5

5.0

.01

6.0

.001

7.0

1.0E7

7.0E-6

.002

3

l.OE-9

.005

4.0

l.OE-5

-1,000

l.OE-7

.15

.1

.007

3.5

3.0E6

.07

1.5E6

-3.0E4

79
fable 8

Step

48

49

50

Continuec

Const.

A2

Al

Al

1

Test

72

72

86

StrainW

.8

.8

.8

Cycle

1

1

1

Old Value

.023

.001

1

New Value

10,000

1

.001

Table 9 - Constants for Schmidt and Miller's Model

Constant

n

Al

A2

B

1

C2

DO

HI

H2

F

B

J

Hand Calculations

1.813

.1707

.0096

2.206E-16

-5.335E-2

16.26

4128

-20190

Computer Iterations

7.0

.001

10.000

1.5E6

-3E4

.001

.07

1E-7

.007

3.5

1E-9

B = n 2 e - B { [ n3+ ^-) |)] R + n6} (87)

D = Dj+ D2exp(-n7R) (88)

(89)

Initial calculations. The initial step in evaluating Walker's

constants was to calculate provisional values of C and D from

experimental back stress measuring tests. The inelastic strain rate

equation was rewritten in the following form:

80
a - B = D ln( e1) + 0 ln( C ) (90)

A linear fit to the data from tests 84, 88, 81, 65, 63, 61, and 60

produced the following values:

D' = 4.372 ksi

C

1

 = 7.211X107

The constant RQ was set at the strain rate at which the strain rate

sensitivity switched sign. The value used here was 3.05X10"^

sec . Chan [90] has proposed a method similar to the gamma plot for

analyzing the work hardening characteristics of a back stress type model

with dynamic recovery. The growth law for the back stress is first

rewritten in a form similar to Bodner's model:

B = ( n2- B-N ) e1 (91)

N = n3+ n4f (92)

.1

f = exp(-ns| log[H— |] |) (93)

The same assumptions are made as with the gamma plot: only values of

o and e past .2% offset will be used, e is approximately constant and

equal -to applied strain rate, exp( °I ) 1.00 , D is constant during

the monotonic portions of a cyclic test, and thermal recovery is

negligible. The following equation results:

81
e = -^j- = -N o + [ n2+ N 0 ln( C i) ] (94)

de1

A linear fit to the first ten points produced a line of slope -N and an

x intercept of saturated stress. A saturated stress condition would

produce the following relationship:

n2

B - -fl— (95)

This is also reported in [90]. The following equation was then used to

produce the constant nt

n2= Ns[ a - D' ln( ^C1 +1) ] (96)

The values of NS and o are the slope and x intercept, respectively, of

the theta plot for a specific test, e is the total strain rate for the

test if saturated conditions are assumed. The results of tests 70, 86,

56, and 65 produced an average r\2 of 22,523 ksi. The strain rate

sensitivity appeared to be returning to positive after 1.0X10"^ sec .

The assumption that N = n^ was made for tests 70, 86, 56, and the above

equation produced values of n^. Averaging the results of these tests

gave n3 = 324.897.

The constant n^ was chosen by setting the f function equal to a

I 3 small number (.1) for e = 1.0X10 . The following equation resulted:

= .6566 (97)

82
This equation was incorrect since-the In function in the denominator was

supposed to be a log function.

The theta plot for test 80 was analyzed assuming N = n^ + n4. This

produced the following value for n4:

n4 = -116.67

N is rewritten in the following form if recovery is present:

N = n3+ n,f + -^f- (98)

The following equation then results for n:

na+ n.,exp(-n5|log(

where B = o - D' ln( C1 + 1 )

The x intercept of the theta plots for tests 71 and 72 produced an

average value of ng = -5.081X10-4.

The next step was to calculate a new value for C to replace the

provisional value. New values for B-jin) = n2/N (where N = nj + n4f

+ng/ e ) were calculated. Theta plots for the third half cycle then

produced values of a,, for the following equation:

o - B = D ln( e1) + D ln( C ) (100)

Tests 86, 56, 65, 83, 80, and 72 produced a linear fit with C =

83
1.132X108 sec value produced.

The next step was to calculate the drag stress growth law

constants. Theta plots from the positive half cycles 3, 5, and 7

produced values of ol im for tests 86, 56, 65, 83, 80, and 72. Values of

Blim = n2/N were Ca1culated for each test- Equation (100) and the

saturated values just calculated were used to calculate a value for D at

each half cycle. The cumulative inelastic strain was estimated for each

half cycle based on the stress values at .8% total strain. The

following formula was used for this purpose:

R,= z I e| ; j = 3, 5, 7 ' (101) J 1-1

 n

where o.. is the stress value at +.8% strain for each cycle. Three

equations in three unknowns could then be written in the following form:

Dj= D,+ D2exp(-n7Rj) ; J = 3, 5. 7 (102)

where D,- and R,- were calculated above and D2, D^, and n7 were the

constants to be determined. These equations were solved iteratively to

produce the following values:

D! = 2.794 ksi

D2 = 8.150 ksi

n7 = 1.688

Computer iterations. Table 10 summarizes steps used to finalize

the constants for Walker's model. The immediate realization with

84
Table 10 - Computer Iteration Summary of Walker's Exponential Model

Step

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

Const.

D2

n4

n2

n3

Dl

8

n2

n3

n2

n3

n3

n3

n4

n5

n2

n4

n3

Dl

C

n2

n2

n3

n3

n3

n3

n5

n4

n5

n3

n4

n5

n4

n3

n5

n3

n2

n2

n6

PI

02

n7

Dl

02

n7

Test

70

70

70

71

80

70

70

70

71

70

70

80

71

86

70

86

70

80

71

71

71

70

70

71

70

71

86

86

86

86

Strain(%)

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

.8

1.26

1.28

1.28

.8

.8

.8

.8

Cycle

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

10

10

10

10

Old Value

1.13E8

0

22,523

r,ooo

325

0

.6566

1.0E6

0

1,000

0

700

450

700

0

.66

750

-300

2.5

-350

775

1.7

700

31,000

35^000

0

0.0

.2

-.8

3.5

New Value

0

0

0

0

11.0

1.0E6

22523.623

1,000

41,000

325

95

IjOOO

-900

1.7

0

0

0

1.0

1.0E40

1,000

31,000

700

450

700

750

.66

-300

2.5

775

-350

1.7

-250

700

.66

750

35,000

33,000

2.5E-4

.2

-.8

3.5

.7

-.3

18.0

85
Walker's model was that the negative strain rate-sensitivity correction

was not strong enough. This could have been a result of the error

mentioned previously in calculating ng. Steps 1 through 14 recalculated

the work hardening parameters to correct for negative strain rate

sensitivity. These changes still did not correct the problem and steps

15 through 35 changed the basic strain rate sensitivity of the model and

recalculated the work hardening parameters. The high strain work

hardening characteristics were set in steps 36 through 37. Thermal

recovery was set in step 38. The cyclic response was tuned in steps 39

through 44. Table 11 presents the constants before and after the

computer iterations.

Table 11 - Constants for Walker's Exponential Model

Constant

C

Dl

D2

n2

n3

n4

n5

n6

n7

RO

Hand Calculations

1.132E8

2.794

-8.150

22,520

324.9

-116.7

.6567

-5.081E-4

1.688

3.050E-5

Computer Iterations

1.000E40

.7

-.3

33,000

750.0

-250.0

.6600

2.5E-4

18.00

3.050E-5

86
MODEL RESULTS

This chapter will be divided into two sections. The first section

will discuss the capabilities of the models to reproduce input data.

The second section will cover the predictive abilities of the models.

The forms of the models to be covered include Bodner's model without a

correction for solute strengthening, Bodner's model with a solute

strengthening correction, the model of Krieg, et al. with and without a

correction for solute strengthening, Schmidt and Miller's model, and

Walker's exponential model. The models were numerically integrated with

an Euler forward integration scheme on a Perkin-lilmer 32/10 computer.

The time steps used ranged from 5.0X10"4 sec for test 70 to 5.0X10"^

sec for test 71.

Reproduction of Test Data

Figure 13 shows the response of the models as compared to test 70 (

= 3.151X10"^ sec"*). The models are all oversquare except for

Walker's. Walker's model is showing adverse effects from its dynamic

recovery term as the stress is decreasing at higher strain levels. This

is more of a problem with the method of constant calculation than the

model itself. The iterative portion of the constant calculation process

was performed with access to only two points on this curve. Using three

points or interactive graphics would have solved this problem.

Figure 14 compares the model outputs to test 71 ( e = 7.253X10"6

sec"1). Walker's model is still following the shape of the curve

best. The dynamic recovery problem still exists with the Walker

87
CO

CO

cu

CO

TEST 70 --- STRfllN RRTE = 3.151E-3/SEC

BOONER MODEL (WITHOUT FSOL CORRECTION)

BODNEr? MODEL I WITH FSOL CORRECTION)

KRIE D MODEL (WITHOUT FSOL CORRECTION)

KRIEC MODEL (WITH FSOL CORRECTION)

MILLER MODEL

WfiLKER MODEL

150.0

125.0

100.0

75.0

50.0

25.0

0.0

I I I

I

0.0 .002 .004 .006 .008 .010 .012

Strain (in/in)

Figure 13 - Model Response as Compared to Test 70

.014
TEST 71 -- STRR1N RRTE = 7.253E-6/SEC

BODNER MODEL 1H1THOUT FSOL CORRECTION)

BODNER MODEL IWIT H FOL CORRECTION)

KRIEO nODEL (WITHOUT FSOL CORRECTION)

KRIEC MODEL (WITH FSOL CORRECTION)

HILLER HODEL

HflLKER MODEL

150.0

125.0

100.0

CO . 75.0

CO

50.0

25.0

0.0

0.0

I I I

I

.002 .004 .006 .008 .010 .01 2

Strain (in/in)

Figure 14 - Model Response as Compared to Test 71

.014
model. The Krieg, et al. model is showing some numerical instability

due to the presence of the solute strengthening parameters. The

uncorrected versions of Bodner and Krieg, et al. are much lower than the

other models. The FSQ-| parameter was simply set to zero in these

versions. The other constants remained the same as in the corrected

versions. Therefore, the reponse of the uncorrected versions could have

been averaged over the strain range better. However, the basic strain

rate sensitivity would have remained the same.

Figure 15 interpolates the model response and experimental response

between these two strain rates presented above by picking off stress

values at .8% total strain for tests of intermediate strain rates and

plotting these values versus the log of the applied strain rate. The

tests used in Figures 13 and 14 are shown on this figure also.

Individual plots of the intermediate strain rate tests are provided in

Appendix C. Walker's model is exhibiting negative strain rate

sensitivity and the corrected Bodner model is showing no strain rate

sensitivity. The other models clearly produce positive strain rate

sensitivity. Figure 16 shows these results with greater clarity.

Presented in this Figure are the stress values at -.8% for the first

compressive cycle.

Figure 17 shows the stress values at +.8% strain for the saturated

cycle response. The slowest strain rate provides data from the fourth

cycle and the other points are from the 10th cycle. The trend has

changed and all the models with correction for solute strengthening are

exhibiting negative strain rate sensitivity. This is probably an effect

90
a

CT: CO — • z + o

l_> O •- Z

LU UJ — <_> O

_i o: •- uj — (_> a: LJ a: >— >- o uj ce LJ LJ i_> cc o uj o; <_) te. »— _i o a: co o o _i o — to o o — u. _ J co o u. _i CO H- CO O CO Z> U. H- CO UJ O D U. Ct I X O CL LU UJ _J _l U

J UJ

•— o o uj uj o o

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of the drag stress thermal recovery parameters and the cyclic work

softening response.

Figure 18 provides the complete 10th cycle hysteresis loops for

test 86 ( e = 1.002X1CT3 sec'1 ) as compared to the model output.

Bodner's model is seen to show some numerical instability due to

interaction with the solute strengthening parameter. Figure 19 allows

interpolation between the first cycle and the 10th cycle by presenting

the values at +.3% strain for each cycle. The corrected Krieg, et al.

model and the uncorrected Bodner model reproduce this data closest.

Figure 20 provides the same data for test 80 ( e = 9.926X10'5

sec" ). This strain rate shows Walker's model following the experiment

the closest. The peak value at the second cycle is reproduced with this

model only. Figure 21 presents the cyclic data for test 72 ( e =

7.626X10"^ sec'1 ). The corrected Bodner model is following the data

closest. The Walker model is clearly suffering from the lack of a drag

stress thermal recovery term. Appendix C contains the cyclic data for

the remaining tests as well as the compressive cyclic data.

Figure 22 shows experimental values of back stress compared to the

values of back stress compared to the values at the tenth cycle of tests

86 and 65 and the first cycle of test 70 for the models of Walker,

Schmidt and Miller, corrected Krieg, et al., and uncorrected Krieg, et

al. This figure points to the fact that the constants produced to fit

the models' of Krieg, et al. and Schmidt and Miller to the input data do

not produce physically realizable results. This can probably be traced

to the abnormally high values of the inelastic strain rate equation

94
CO

CO

CD

72

tEST 86 (CYCLE 10) -- STRRIN RfiTE : 1 .002E-3/SEC

BODNER MODEL (WITHOUT FSOL CORRECTION)

BODNER MODEL (WITH FSOL CORRECTION)

— — KRIEO MODEL (WITHOUT FSOL CORRECTION)

KRIEG MODEL (WITH FSOL CORRECTION)

MILLER MODEL

WflLKER MODEL

150.0

125.0 -

100.0

75.0 -

50.0

25.0

0.0

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-50.0

-75.0 -

-100.0 -

-125.0 -

-150.0

-.010 -.008 -.006 -.004 -.002 0.0 .002 .004 .006 .008 .010

Strain (in/in)

Figure 18 - 10th Cycle Hysteresis Loops Test 86
0

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exponent needed to match the strain ageing effects. This may also

create the oversquareness of these models.

Predictive Capabilities

The predictive capabilities of the models were explored by the use

a complex history test. This experimental test was not used in the

calculation of the material constants. Table 12 gives the input history

of this test. Figures 23 through 28 show the comparison of the models

to this complex history test. The corrected Bodner model in Figure 24

is the least affected by strain rate jumps. Bodner attributes this to

the use of plastic work as the measure of work hardening [67]. The

interaction of the solute strengthening corrections of all the models

may be having an effect on this aspect of all the models. The

uncorrected versions in Figures 23 and 25 are very suseptible to these

jumps. Yao and Krempl report that the overshoots and undershoots

observed during the strain rate jumps are a transient effect of the

behavior of a system of coupled nonlinear differential equations [160].

Table 12 - Complex History Test Input

Interval

1

2

3

4

5

6

7

8

9

10

11

12

13

Beginning Strain

0.0

.004

.006

.008

0

0

-.004

-.009

.006

.008

.01

.01

.015

Endinq Strain

.004

.006

.008

0

0

-.004

-.009

.006

008

.01

.01

.015

0

Strain Rate

9.991E-5

4.784E-4

9.762E-4

-5.0E-3

0.0

-9.878E-4

-9.795E-5

9.933E-4

9.532E-6

5.0E-3

0

4.95E-4

-1.4925E-3

100
•TEST 89 -- COMPLEX INPUT HISTORY

•SOONER MODEL (HITHOUT FSOL CORRECTION)

150.0

125.0 -

100.0 -

75.0

50.0 -

25.0

CO

to

cu

-150.0

-25. OJ-

-50.0 -

-75.0

-100.0 -

-125.0 -

-.012 -.008 -.004 0.0 .004 .008 .012 .016

Strain (in/in)

Figure 23 - Complex History - Bodner's Uncorrected Model
•TEST 89 -- COMPLEX INPUT HISTORY

BODNER MODEL (WITH FSOL CORRECTION)

150.0

125.0 -

100.0

75.0

50.0

25.0

CO

en

0>

-150.0

A . r~. - \.

-25. OK-

-50.0 -

-75.0 -

-100.0 -

-125.0 -

-.012 -.008 -.004- 0.0 .004 .008 .012 .016

Strain (in/in)

Figure 24 - Complex History - Bodner's Corrected Model
-TEST 89 -- COMPLEX INPUT HISTORY

KRIEG MODEL (WITHOUT FSOL CORRECTION)

V)

CO

CO

0)

150.0

125.0 -

100.0

75.0 -

50.0

25.0

• ui f _j '

p . r-—• '•

,1 vi TM-

-25. OH

-.012 -.008 -.004 0.0 .004- .008 .012 .016

Strain (in/in)

Figure 25 - Complex History - Uncorrected Model of Krieg, et al.
•TEST 89 -- COMPLEX INPUT HISTORY

•KRIEG MODEL (WITH FSOL CORRECTION)

OT

150.0

125.0 -

100.0

75.0 -

50.0 -

25.0 -

0.0

-25.01-

-50.0

-75.0

-100.0

-125.0

-150.0

-.012 -.008 -.004 0.0 .004 .008 .012 .016

Strain (in/in)

Figure 26 - Corrected Model of Kreig, et al. - Complex History
-TEST 89 -- COMPLEX INPUT HISTORY

•MILLER MODEL

W

73

CO

CD

150.0

125.0

100.0

75.0

50.0

25.0

0..0

•*> -2.5.0

-50.0

-75.0

-100.0

-125.0

-150.0

T

500.0

0-01

-.012 -.008 -.004 0.0 .004 .008 .012 .016

Strain (in/in)

Figure 27 - Complex History - Schmidt and Miller's Model
-TEST 89 -- COMPLEX INPUT HISTORY

•HRLKER MODEL

150.0

125.0 -

100.0

75.0 -

50.0 -

C/3

CO

0>

£-,

. r:— .

-.012 -.008 -.004- 0.0 .004 .008 .012 .016

Strain (in/in)

Figure 28 - Complex History - Walker's Model •
A comparison of the response of the corrected and uncorrected

versions of the Bodner and Krieg, et al. models at the zero strain hold

time shows that the FSO-| correction negates the effects of thermal

recovery in such instances. This could be a result of the low value of

J or the inelastic strain rate of maximum correction used in these

models. The corrected Bodner model had J = 1.0X10 sec , the model of

Krieg, et al had J = 7.0X10"6 sec'1, and Schmidt and Miller had J =

1.0X10 sec" . Schmidt and Miller's model showed no thermal recovery at

this hold either. The small inelastic strain rates produced by thermal

recovery terms would meet increasing hardness if their magnitude was

below J. Increasing hardness would tend to drive the stresses up and

oppose the action of the thermal recovery terms.

The Walker model follows the shape of the stress strain curve

better than the other models. This could be a result of the better

modelling of the back stress growth and the lack of an inelastic strain

rate exponent. The model of Krieg, et al. had n = 15.0 and Schmidt and

Miller had n = 7.0. The constant values of work hardening have also

been reported as reasons for this [41,96]. Bodner's model may be

suffering from the lack of a back stress or the effects of the plastic

work measure of strain hardening. Further study would be required to

show this however. The corrected model of Krieg, et al. reproduces the

actual stress levels best after initial yield. No explanation can be

given for this at this time.

107
CONCLUSIONS AND RECOMMENDATIONS

This chapter will be divided into three sections. The first

section will cover the conclusions and recommendations of this thesis

pertaining to the models themselves. The second section will cover the

methods of calculating the material constants. The final section will

cover the experimental portion of this work. Conclusions reached and

recommendations for further study will be presented simultaneously as

these topics are very closely related in many cases.

Conclusions and Recommendations based on the Models

The theories of Walker and Bodner with exponentially based

inelastic strain rate equations and dynamic recovery terms handle the

strain rate sensitivity the best. Bodner's model shows less sensitivity

to strain rate jumps possibly due to the plastic work rate measure of

strain hardening. The reproduction of the general shape using Walker's

model may be aided by better modelling of the back stress term and by

the exponentially based inelastic strain rate equation. The drag stress

growth law of the Walker model provided the closest fit to data over

several cycles at higher strain rates. The second cycle peak seen at

the lower strain rates was modelled only by Walker. Bodner's model

handled cyclic response best over several cycles at the lower strain

rates due to the thermal recovery term. The solute strengthening

correction caused numerical instability, negating the effect of thermal

recovery during hold times and may have lessened the sensitivity to

strain rate jumps.

108
Future study of these models could take two directions. First a

comparison to a material which does not exhibit strain aging effects

would be beneficial. The corrections necessary to account for this

phenomenon masked some of the information which could have been obtained

in this work. An example of this is information about the effect of

strain jumps on the predictive capabilities of the models. The thermal

recovery capabilities of the models were also adversely affected by the

strain ageing corrections. The methods for calculating constants should

be checked with a positve strain rate sensitive material.

Second, further study which concentrates on the specific model form

should be carried out by the use of extended models. These would be

models extended from the existing ones. An example of this would be to

replace the inelastic strain measure of work hardening in the model of

Krieg, et al. with a measure based on plastic work. The inelastic work

measure in Bodner's model could be replaced with an inelastic strain

measure. The extended models could then provide true insight into the

ramifications of using a measure of plastic work. The effect of using

an inelastic strain rate equation based on exponential, power law, and

hyperbolic sine functions could be studied. The advantages and

disadvantages of providing a model with a back stress term could be

studied by providing the Bodner model with one as Moreno and Jordan have

done [74].

Conclusions and Recommendations based on the Calculation of Constants

The initial assumptions of back stress being responsible for

hardening in monotonic tension and drag stress being responsible for

109
cyclic hardening/softening appears to be a good assumption for this

material system. These assumptions were used for every model in the

hand calculations and the computer iterations with success. Krempl,

McMahon, and Yao report that a changing drag stress parameter alters the

strain rate sensitivity of the model [108]. This effect was not

considered in this work and might warrant further study. The initial

assumptions of thermal recovery being negligible for rapid tests ( e >

1.0X10 sec ), drag stress recovery being dominate in low strain

rate cyclic tests, and back stress recovery being dominate in creep

tests appear difficult to apply in the presence of solute strengthening

effects. This material system requires that a correction for solute

strengthening be employed before recovery effects can be calculated.

The recovery effects were much smaller than the original hand

calculations for the models of Krieg, et al., Bodner, and Walker

produced. This observation leads to the conclusion that the recovery

effects are largely insignificant for £ > l.OXKT5 sec'1. Miller's

model requires the recovery terms to be much more active than the other

models. This inflexibility gave some problems in the calculation of

Miller's constants.

The solute strengthening effects also masked the true strain rate

sensitivity of the material. Information on the strain rate sensitivity

needs to be obtained outside the region of solute strengthening

effects. The following initial assumptions would have been more

appropriate based on these observations:

(1) back stress was assumed responsible for hardening in

110
monotom'c tension;

(2) drag stress was assumed responsible for cyclic softening;

(3) thermal recovery effects were small and masked by solute

strengthening effects;

(4) solute strengthening or strain aging effects masked the

basic positive strain rate sensitivity of the material.

The model of Krieg et, al. was an easy model to work with since

each term in the growth laws could be scaled somewhat seperately of the

others. An interesting observation of this model was that the constants

of the inelastic strain rate equation could be swept over a broad range

but the monotom'c hardening stayed relatively constant. There was also

a mathematical ambiguity between the constant C and the scaling of the

drag stress. The scaling could be transferred from one parameter to the

other without any visible change in model response.

Bodner's model "converged" to the final constants with fewer

iterations than the other models using the iterative scheme developed in

this work. This was probably due to the lack of a back stress

parameter. A mathematical ambiguity existed between n and the scaling

of the internal state variables when information was not available to

calculate n. This is why a value for n can often be picked and still

produce a workable model..

Miller's model was highly coupled in that the recovery terms were

not were not separated from the hardening terms. The recovery terms can

therefore change on the same order of magnitude as the hardening

terms. Miller readily admits that this model is designed for materials

111
which have a very active drag stress parameter [42]. He states that

this model may not be applicable for this type of material system.

However, a reevaluation of the constants for Miller's model might prove

fruitful. A majority of the constants should be calculated outside the

region of solute strengthening effects and without artificially

seperating the hardening and recovery terms. A solute strengthening

parameter would then be added to fit the response to the negative strain

rate sensitive region.

Miller's model also maintains control over the saturated states of

the internal state variables B and D with the A^ and Ag constants. A

correction for strain ageing as well as cyclic work softening might be

possible by controlling these saturated states. A recommendation for

further study based on this model would contain an expanded study of

back stress magnitudes over the entire strain rate region considered. A

possible method for this will be disscussed in the next section.

Suggestions can also be found in Appendix B. The latest form of

Miller's model [100] should also be studied as it may be used with

material systems similar to this.

Walker's model holds promise for automating the calculation

procedure for this type of material. Walker's model has fewer

constants, appears to be tailored for this type of material, and can

utilize the theta plot concept [90]. The drag stress scaling performs

the same strain rate sensivity functions as the n in the power law

related models. Expanding knowledge of the back stress values would

also be useful for this model.

112
Conclusions and Recommendations Based on the Experimental Work

The back stress measuring tests both in creep and in cyclic loading

were very subjective and uncertain. However, their extreme usefulness

and relative success in application with an automated test set-up

warrant further study. It appears possible that these tests can be

developed into useful inputs to the constant calculation process. More

sensitive data acquisition devises with greater resolution and a smaller

less massive load frame for more precise control would greatly enhance

the usefulness of these tests. The subjectivity could be lessened by

using a method such as proposed by Blum and Finkel [150] to analyze the

data.

The back stress measuring tests during creep were useful for strain

rates less than 1.0X10 sec . The cyclic back stress measuring tests

were useful for strain rates greater than 1.0X10 sec . The region

between these two tests could be filled by performing tests during

monotonic tension as the stress transient test mentioned by Soloman,

Alhquist and Nix [144]. This type of test takes on greater usefulness

for a material such as Inconel 718 which exhibits good ductility in

tension and high susceptibility to low cycle fatigue.

An automated load frame was invaluable in this work for the complex

tests. A smaller load frame might provide more stability during highly

sensitive and mode-switching tests. A dead weight load frame would also

have worth for the creep and creep-stress drop tests. A more advanced

and controllable method of load-up would be a necessity. It would also

be useful to utilize the same grips, furnace, extensometer, and data

113
aquisition equipment as with automated load frame. This would remove

•some relative errors between the two systems.

114
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N86 -28 457

/

APPENDIX A - DATA SET

The major purpose of this appendix is allow the calculations of the

chapter IV to be repeated and upgrades to the methods used developed.

The data will be presented in tabular fornu The average value of Young's

modulus for this material was 24,658 ksi.Table A.I contains data from

the initial tensile portion of the cyclic tests and monotonic tensile

tests. The first column provides the test number and the second column

provides the applied total strain rate. The third column is the stress

at which the first visible sign of inelastic flow is seen and the fourth

column provides the inelastic strain rate at this stress. The fifth

column is the .2% offset value and the sixth column is the stress at .8%

total strain.

Table A.2 provides auxiliary information similar to Table A.I.

These data were from a second series of material and utilized only when

a broader data base was necessary. The material series labeled D2 was

the series studied in this work. The material series D8 was presumably

from a different lot or heat treatment than the D2 series. Columns one

through five contain the same types of information as the respective

columns in Table A.I. Column six is the saturated stress from the gamma

plot and column seven is the slope of the gamma plot. The gamma plot

was described in chapter V under the section covering Bodner's model.

Table A.3 gives data from high strain monotonic tensile tests.

This table contains data from both the D2 and D8 series of the

126
material. Column three provides the value of total strain and column

four provides the associated stress value. Column six and seven are the

slopes from the gamma and theta plots, respectively. The theta plot was

described in chapter V under the section covering Walker's model.

Table A.4 presents the creep data produced in this work. Mixed

series data are again presented. Column three contains the applied

stress and column four the associated secondary creep rate. The D8

series data were obtained using the dead weight load frame discussed in

the previous chapter. The D2 series data were produced on the computer

controlled MTS 880 load frame discussed in the previous chapter.

Table A.5 contains the back stress data from this work. The values

for back stress are at best estimates of the true value and should be

treated as such. Column three of Table A.5 contains the inelastic

strain rate and column four contains the stress value associated with

the back stress value of column five. Table A.6 contains the creep

stress drop data used to arrive at the back stress values in Table

A.5. The stress associated with the drop and the associated inelastic

strain rate are provided under each test. A linear regression curve fit

provided the stress at which a zero strain rate was expected for each

test. Table A.7 provides the same data for the cyclic hold time data.

Table A.8 through Table A.13 provide cyclic data from individual

tests. The first column provides the half cycle currently being

considered. The second column provides the stress value at .8% strain

for that half cycle. Columns three, four, and five contain the slope of

the gamma plot, saturated stress of the gamma plot, and plastic work

127
from the gamma plot for each half cycle. Columns six and seven contain

the slope and saturated.stress from the theta plot for each half cycle.

Table A.I - First Half Cycle Material Data - 02 Series

Test

70

86

56

65

83

80

72

71

•T . e (sec'1)

3.151E-2

1.002E-3

9.966E-3

3.127E-4

9.926E-5

3.054E-5

7.626E-6

7.253E-6

°°(ksi)

97.83

96.68

95.56

97.84

94.56

100.94

101.71

97.67

•I . e° (sec'1)

9.144E-4

2.304E-3

2.684E-4

7.599E-5

1.319E-5

7.213E-6

1.550E-6

1.402E-6

°-2(ksi)

122.17

117.66

115.17

121.41

123.56

128.86

126.78

124.67

°-8(ksi)

124.64

120.19

117.52

123.56

127.52

130.51

128.42

126.95

Table A.2 - Auxilary Tensile Data - 08 Series

Test

35

37

39

34

36

42

38

40

'

T

 (sec'1)

9.253E-4

6.048E-4

1.793E-4

7.637E-5

5.703E-5

1.914E-5

1.410E-5

7.029E-6

°.z (ksi)

125.29

125.30

119.82

131.36

126.82

127.16

127.17

128.72

°Y (ksi)

136.8

136.3

130.5

147.4

137.2

147.2

144.3

137.5

Y(ksi)

2.037

2.202

1.772

1.4004

2.237

.6772

1.2867

2.917

°Y (ksi)

136.3

137.5

130.6

142.1

138.5

137.6

140.7

138.3

e

2.772

240.9

206.6

319.8

241.6

291.8

212.4

328.5

Table A.3 - High Strain Data

Test

70

35

34

42

71

35

34

42

Series

D2

D8

08

D8

D2

08

08

08

T

e

.0128

.0130

.0130

.0129

.0126

.0399

.0243

.0347

° (ksi)

130.7

134.43

140.38

135.96

134.4

143.0

146.08

144.71

Y (ksi)

2.037

1.4004

.6772

2.929

6 (ksi)

287.29

342.25

128
Table A.4 .- Creep Data

Test

60

61

62

63

64

10

12

17

11

19

Series

02

D2

D2

02

02

08

08

08

08

08

°a (ksi)

119.0

124.0

127.0

133.9

138.8

110.0

115.0 •

120.0

120.0

125.0

•I . es (sec'1)

1.792E-8

1.012E-7

1.125E-7

2.42E-7

7.117E-7

5.3E-8

9.5E-9

9.55E-8

8.88E-8

3.03E-4

Table A.5 - Back Stress Data - 02 Series

Test

84

88

81

65

63

61

60

Type

Cyclic

Cycl ic

Cycl ic

Cyclic

Creep

Creep

Creep

•I . e (sec'1)

2.812E-3

9.272E-4

8.635E-4

2.751E-4

2.420E-7

1.012E-7

1.792E-8

° (ksi)

116.48

100.98

116.45

107.84

133.91

124.00

119.04

B(ksi)

68.30

62.47

64.65

48.58

128.22

116.78

113.85

129
Table A.6 - Creep Stress Drop Data

Test

60

61

62

63

o (ksi)

117.6

115.7

114.2

111.7

95.53

85.27

77.96

121.8

119.4

117.1

113.9

107.4

126.2

122.3

118.8

115.5

112.9

131.5

129.5

129.3

120.5

117.3

•I

e

5.616E-8

-2.292E-8

-1.194E-7

0.00

-2.753E-7

-4.800E-7

-8.809E-7

1.635E-7

8.545E-8

-1.194E-7

2.415E-8

-3.190E-7

-1.313E-7

3.166E-7

-2.197E-7

1.137E-7

-4.288E-8

-3.07E-7

.1839E-7

.06405E-7

-.4958E-7

-.3820E-7

130
Table A.7 - Cyclic Back Stress Hold Data

Test

65

81

88

84

o

104.51

99.38

94.49

89.71

84.61

79.32

74.46

74.13

69.29

64.28

59.23

54.34

49.36

44.58

39.24

78.56

73.91

73.38

68.64

68.33

58.94

55.85

42.28

79.03

73.61

68.71

58.37

54.24

38.66

85.56

74.76

65.01

56.11

46.28

•I

e

2.658E-6

1.785E-6

1.340E-6

9.102E-7

3.973E-7

2.268E-7

7.293E-7

5.788E-7

4.171E-7

3.576E-7

1.386E-7

3.034E-7

7.509E-8

-1.784E-7

-1.182E-7

3.128E-7

-3.852E-7

1.535E-7

1.616E-7

1.201E-7

-6.898E-8

-3.633E-7

-4.009E-7

1.606E-7

8.755E-8

6.099E-8

-2.448E-7

-7.279E-8

-6.285E-7

1.756E-7

3.912E-7

-2.256E-7

-3.135E-7

-3.837E-7

131
Table A.8 - Cyclic Data - Test 86

Half

Cycle

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

°.8

(ksi)

120.33

-124.10

114.49

719.05

111.42

-115.98

109.4-7

-113.04

108.01

-112.42

106.86

-111.27

105.91

-110.26

105.09

-109.44

104.43

-108.74

103.88

-108.14

103.42

-107.65

Y

(ksi)

3.620

3.843

2.223

3.824

4.068

4.124

4.325

4.079

4.192

4.173

4.318

4.193

4.357

2.363

4.282

4.255

4.234

4.128

4.354

2.426

2.500

a

(ksi)

-128.094

116.80

-131.73

113.00

-115.94

109.78

-113.04

108.43

-112.04

106.91

-110.53

106.02

-109.49

115.18

-109.00

104.36

-108.50

104.24

-107.50

112.99

-117.50

Wp

(ksi)

.12852

.4643

.4560

.4796

.4568

.5016

.4873

.5111

.4846

.5098

.4964

.5196

.4947

.5190

.4930

.5170

.5052

.5166

.5044

.5276

.5036

.5271

e

(ksi)

307.03

-307.10

295.03

-269.30

310.49

-306.22

322.62

-302.41

310.81

-302.19

312.05

-303.10

315.07

-296.55

308.22

-299.03

303.44

-288.42

303.96

-292.83

298.35

°e

(ksi)

-133.47

120.93

-125.83

116.98

-120.93

113.91

-117.84

112.28

-116.75

110.91

-115.45

109.84

-114.13

109.24

-133.62

108.38

-113.10

108.36

-112.50

107.56

-112.31

132
Table A.9 - Cyclic Data - Test 56

Half

Cycle

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

°.8

(ksi)

117.64

-120.33

111.95

-115.33

108.88

-112.41

106.87

-110.44

105.50

-109.00

104.50

-107.94

103.59

-107.03

102.90

-106.25

102.24

-105.05

101.71

-105.00

101.18

-104.59

Y

(ksi)

3.523

3.863

3.860

4.016

4.013

4.068

4.182

4.073

4.164

4.182

4.163

4.200

4.142

4.263

4.099

4.249

4.149

4.255

4.077

4.297

4.000

o

(ksi)

-124.67

114.40

-116.89

110.14

-112.84

107.82

-110.54

106.40

-109.23

105.01

-108.10

104.11

-107.31

103.15

-106.83

102.61

-105.98

102.10

-105.68

101.52

-105.50

Wp

(ksi)

.1282

.4224

.4350

.4651

.4549

.4729

.4640

.4820

.4621

.4799

.4736

.4923

.4711

.4920

.4836

.4896

.4826

.5010

.4814

.5010

.4921

.4977

e

(ksi)

319.63

-305.44

298.85

-300.78

304.61

-299.00

298.64

-304.87

297.19

-298.61

291.06

-269.18

287.98

-293.68

283.14

-294.02

276.97

-294.58

273.76

-288.11

264.87

°9

(ksi)

-127.92

118.22

-122.04

114.15

-117.71

111.74

-115.77

109.78

-114.08

108.86

-113.18

107.96

-112.30

107.15

-111.83

106.53

-111.54

105.90

-111.07

105.82

-111.14

133
Table A.10 - Cyclic Data - Test 65

Half

Cycle

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

°.B

(ksi)

123.91

-129.76

122.55

-126.17

119.95

-123.33

117.96

-121.22

116.38

-119.65

115.12

-118.35

114.15

-117.32

113.27

-116.48

112.55

-115.72

111.94

-115.12

111.34

-114.54

Y

(ksi)

3.257

2.599

3.022

2.950

2.926

2.219

30.43

2.954

2.169

2.189

2.163

2.968

3.398

3.148

3.366

3.298

3.415

3.205

a

(ksi)

130.37

-140.77

129.55

-133.43

127.15

-137.56

124.15

-124.98

127.76

-131.27

126.76

-122.75

116.10

-120.63

115.58

-118.99

114.80

-118.86

Wp

(ksi)

.1064

.4727

.3921

.4140

.3959

.4336

.4020

.4116

.4081

.4099

.4068

.4217

.4525

.4540

.4514

.4656

.4506

.4648

9

(ksi)

314.80

268.17

261.29

243.38

253.43

241.85

221.41

224.23

223.58

229.72

214.72

239.09

220.90

248.02

217.93

247.65

221.88

°e

(ksi)

-139.93

134.14

-138.99

132.85

-136.24

130.32

-133.19

124.76

-131.23

125.08

-131.39

123.14

-129.41

121.43

-128.92

120.78

-127.60

Table A.11 - Cyclic Data - Test 83

Half

Cycle

1

2

3

4

5

6

7

8

9

10

11

°.8

(ksi)

128.66

-138.82

130.07

-137.35

129.29

-135.64

128.06

-134.01

126.86

-132.60

125.75

o

(ksi)

2.912

3.399

2.885

3.292

0

(ksi)

134.30

-148.09

135.50

-145.28

135.20

Wp

(ksi)

. 14978

.4249

.3829

.40093

.3755

e

(ksi)

395.79

334.40

299.96

319.92

320.43

285.42

290.42

290.33

280.31

267.18

°e

(ksi)

134.57

-146.84

138.81

-149.29

138.81

-146.09

139.79

-146.64

137.94

-146.01

138.76

134
Table A.12 - Cyclic Data - Test 80

Half

Cycle

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

°.8

(ksi)

130.59

-139.24

130.89

-137.85

129.92

-136.07

128.54

-134.32

127.25

-132.82

126.04

-131.52

124.99

-130.35

123.93

-129.35

123.18

-128.42

122.39

-127.70

121.58

Y

(ksi)

2.684

3.075

3.140

a

(ksi)

-149.25

138.46

137.30

Wp

(ksi)

.06833

.3866

.3661

.3936

.3734

9

(ksi)

368.98

310.78

321.97

303.02

333.99

292.52

304.11

265.97

295.35

241.28

285.46

251.77

287.86

254.05

272.44

233.61

253.32

255.53

224.54

209.09

Og

(ksi)

-148.07

141.82

-149.40

141.50

-146.98

140.72

-147.28

142.03

-146.24

142.88

-145.43

140.31

-143.56

138.75

-143.51

139.68

-144.32

135.90

-146.89

140.72

Table A.13 - Cyclic Data - Test 72

Half

Cycle

1

2

3

4

5

6

7

8

°.B

(ksi)

128.79

-137.83

130.04

-138.25

130.66

-138.10

130.69

-137.12

Y

(ksi)

2.711

2.713

2.818

3.243

o

(ksi)

-146.44

139.66

-147.44

137.86

Wp

(ksi)

.0674

.3479

.3212

.3396

.3185

e

(ksi)

355.88

358.55

309.59

334.58

293.64

-375.04

341.27

°e

(ksi)

-146.28

138.16

-150.57

140.69

-152.15

139.38

-148.97

135
N86-28458

APPENDIX B - PROPOSED INITIAL CALCULATION METHODS

The proposed methods will be summarized using a generic

viscoplastic modelin the first section of this chapter. This is

presented to make the actual calculations more transparent to the

reader. The generic model used as an example will be presented first

followed by the generic method of initial calculations. Individual

sections covering the four models used in this work will follow. The

material in Table 1 of the main text and Appendix A should be referred

to as needed.

Generic Summary of Constant Calculation Methods

Generic viscoplastic model. The growth laws for the example model

are presented below:

B = C^1* C2B e!+ C3B (104)

l! + C5D (105)

F = F( e1) (106)

n is a constant measuring strain rate sensitivity. Cj is a constant

measuring back stress hardening. C2 1s handeling back stress dynamic

recovery and C3 measures back stress thermal recovery. C4 produces drag

stress hardening and C5 models drag stress recovery. F is a parameter

136
inserted to handle strain aging effects. This parameter can be removed

from any model in this appendix and the method still used.

Initial assumptions. The following initial assumptions were made

for these methods:

(1) Back stress was assumed responsible for hardening in

monotonic tension;

(2) drag stress was assumed responsible for cyclic softening;

(3) thermal recovery effects were small and masked by solute

strengthening effects

(4) solute strengthening or strain aging masked the basic positive

strain rate sensitivity of the material.

\

Initial calculations. The first step would be to estimate the

constants in the back stress growth law. The back stress growth law is

written in the following form to allow the thermal recovery constant C^

to be included in the calculations:

C3 T

B = [C1+B ( C2+— )] e1 (107)

Differential descriptions of work hardening, experimental estimations of

back stress values, relationships between saturated stresses and

saturated back stresses, linear regression analysis will all be useful

in this step.

The next step was to calculate the strain rate sensitivity constant

n and the initial value of drag stress denoted by DQ. The function F

137
will need to be set to assure a positive value of n. An estimation of n

from literature or a region outside of the strain aging region will be

helpful. The inelastic strain rate equation is rewritten in the

following form:

ln( a - B ) = -£- ln( e1) + ln( D0+ F ) (108)

The ability to estimate saturated stresses and back stresses, remove

drag stress scaling to another parameter, or place F in the back stress

growth law are useful at this stage.

Initial determination of the drag stress parameters C4 and Cg can

be initiated by rewriting the drag stress growth law in the following

form:

cs T

= ( C^--D ) e1 (109)

The constants can be evaluated by repeated application of a differential

technique such as the gamma or theta plot at each cycle and/or by

monitoring the growth of D versus the cumulative inelastic strain.

Krieg. Swearenqen and Rhode's Model

The growth laws for the model of Krieg, et al are repeated below

for completeness:

e

j= C( ° » B )" sgn( a - B ) (110)

138
B = A,!1- A2B

2[ e( A3B ^ - 1 ] sgn( B ) (111)

D - AJ e!| - A5( D -D0)

n (112)

This method assumes the solute strengthening corrected version of the

model is being used. The first step would be to utilize the assumption

that saturated back stress is proportional to applied stress as reported

by Miller [42]. Tests 84,88, 81. and 65 would be used to approximate this

proportionality and the results averaged. An equation of the following

form would result:

Blinf

where a is the constant of proportionality.

The constant A^ will be set by assuming that no thermal recovery is

present and B can be written in the following form for saturated

conditions:

B = Ai e

X= a o (114)

Tests 80,83,65,56,86,70 would be used to produce an average value of A^

based on e and o at .8515 total strain.

The initial value of D is again set to 1.000 and B is assumed equal

to Aje . The inelastic strain rate equation is written in the

following form:

139
I

1n(

 l£ ) = i ln( e1) - i ln( C ) (115)

1 + Fsol

The Fsol constants F, B , and J will be set such that a linear fit of

test data at .8% total strain will produce a physically realizable n

value. The constants 6 and J could be set as previously mentioned to

eliminate some of the "curve-fitting" aspects of this step. The initial

value of n = 4.412 from the previous initial calculations could also be

used as an estimate of n. Tests with little thermal recovery such as

80,83,65,56,86,and 70 should be used in this step.

The inelastic strain rate equation will then be rewritten in the

following form:

.1

Values of B will be calculated based on experimental values of

o and e at .8% strain for tests 71,72,80,83. The following

approximation for back stress is then made:

B = [ A,- A2exp( A3B

2)( e1

)'1] e1 (117)

The following linearized equation results:

i

l(— - Aj] = ln( A2) + A3B2 (118)

e

140
A linear least squares fit to the data of tests 71,72,80,and 83 would

produce estimations for A£ and A^.

Data at .B% total strain from the saturated cyclic loops of tests

72,80,83,65,56,86 would be input into the following equation to produce

saturated values of D:

n

D - (o - B) (-H-n- F (119)

The following approximation for drag stress is then made:

D - [ AH- As( D - D0)

n

( I1)'1] R (120)

T N T o.

where R = z |e}| = z I ej- — - 1 1-1

 n E

and D.J, E.J are the maximum and minimum stress and strain for each half

cycle. N is the number of half cycles associated with each current

value of D in a cyclic test. The D values and associated e values are

those calculated above and DQ = 1.0000. A least squares curve fit could

provide estimates of A4 and Ag.

The constants produced with this proposed procedure should be tuned

with a systematic procedure such as described in the computer iteration

section. A possible automation of the constant calculation procedure

would consist of good initial approximations of the constants which

ignores some inherent nonlinearity in the model coupled with a

systematic and repeatable iterative method to return the nonlinearity to

141
the constants. The true test of the above scheme is in its application to

a material which responds with positive strain rate sensitivity. The

FSQ-| terms could simply be dropped from the above discussionin such a

case.

Bodner's Anisotropic Model

The growth laws for Bodner's anisotropic model are repeated below

for completeness:

i

l= j^- D0exp[-(-?-)2n] sgn o (121)

Z = Z1* ZA= Zl+ B sgn o (122)

Z

1-^ Zt- Z1) Wp - A^f ^ 2 )ri (123)

B = m2( Z3sgn a - ZA) Wp - A^J-^-J-)^ sgn ZA (124)

Wp= o e (125)

Bodner's model is amenable to a correction for negative strain rate

sensitivity similar to Walker's model. The Z3 variable would be

replaced with the following expression:

.1

Z3= Bt+ B2exP(-B3log|-Y|) (126)

£o

The value of n will be calculated from a region of positive strain

142
rate sensitive material response as presented earlier. The constants ZQ

and n\2

 wil1 a1so be calculated as before. The following equation will

then result for 1^:

.1 9 D0 1/9

Z3= B!+ B2exp(-B3logHy|) = o[ 21^-^ p)]1'^- Z0 (127)

e0 3 e

The constants 63 and eQ wiould be set in a manner as dscribed in the

section discussing Walker's model. The saturated stresses of the gamma

plots would be used from tests 71, 72, 80, 83, 65, 56, 86, and 70 to

produce least squares estimations of B^ and 62- The constants Zj, m^,

Aj_t and r^ would be calculated in the same manner as before. The

approximations commonly used by Bodner of A2 = A^ and r2 = r^ could be

employed.

Miller's Model

The growth laws for Miller's model are repeated below for

completeness:

i

l= B'{ sinh[ ( ° p B )1>5 ]n sgn(o - B) (128)

B = H^1- HjB1 sinh( AjB| )n sgn(B) (129)

D = H2| el|( C2+ |B| - -^- D3) - H2C2B' sinh( A2D

L5)n (130)

"i

Miller's model is a highly "coupled" model. The back stress recovery

and drag stress recovery terms are designed to reproduce specific

143
saturated states. The recovery terms are not seperated by individual

constants from the hardening terms. Instananeous changes on the order

of the changes in hardening are possible. A complete calculation of

the constants for Miller's model will probably involve iterative

procedures since hardening and recovery terms can not be artificially

seperated.

Most of the constants for Miller's model should be calculated

outside the region of solute strengthening effects. The proposed method

brgins by calculating the constants A, B', n, DQ, Aj_, A2, Cj, C2, and Hj

from tests 42, 38, 40, 34, 80, 72, 71, 60, 61, 62, 63, and 64. The

constants A, B', n, and DQ would be calculated as before. The constants

AI and A2

 could De calculated using the experimental back stress values

of tests 60, 61, and 63. The constant H2 should be set to an

artificially small number to simplify the drag stress hardening term and

the calculations.

The constants H^ and C2 should be set from the strain controlled

tests 42, 38, 34, 80, 72, and 71. These constants must be evaluated

iteratively unless a differential method of evaluating the hardening

constants such as the gamma plot can be developed. The following

equation can be explored as a possible starting point if the same

assumtions made for the gamma plot are made and if back stress is the

main contributor to monotonic hardening:

e = -1- = Htf (131)

de

144
f = 1 - --{ sinh( At[ o - {sinh-^-H}])} (132)

A linear curve fit to f versus e data might give an estimate of H^ as

the slope when monotonic tensile data is used.

Utilizing values of stress and cumulative inelastic strain at .8%

total strain from cyclic tests, making the usual gamma plot assumtions

and assuming B = CjO ( Cj_ 1s the experimentally determined ratio of

saturated stress to back stress) the following equation results:

-4°_ = (i _ Cl)-1H2C2f3a

3 (133)

de

.1

f3= 1 - —j—[ sinh ( A2{ a[l - Cj[sinh~ ( g, )]" ' } )] (134)

e

A linear fit of such an equation may give H2C2 as the slope.

The response of the model to the higher strain rates of tests 80,

83, 65, 86, 56, and 70 could be tuned using the Fsol parameters. A

possible alternative to the Fsol correction would be to make the

auxiliary constant C} dependent on strain rate. C^ is the constant of

proportionality between saturated stress and saturated back stress:

ss ,ss (135)

The constants Aj and A2 depend on C^ as follow:

A,-4- (136)
A2= [ A / (1-CJ ]3 (137)

The GI value for the creep tests is approximately .99. The C^ value for

cyclic tests is approximately .5. This is somewhat to be expected since

stress can not reach full saturation in a cyclic test. The suggestion

here is to evaluate each cyclic test in terms of C^ and produce an

appropriate functional form for G£. It is possible that such a method

might handle the negative strain rate sensitivity response, the cyclic

work softening response, and produce correct values of saturated back

stresses.

Walker's Exponential Model

The growth laws for Walker's model are repeated below for

completeness:

.x= CAHL p ; (138)

C

•

• r O *

B = n2e - B { [ n3+ nltexp(-ns| log(-^-) |)] R + n6} (139)

D = Dj+ D2exp(-n7R) (140)

R= I e!| (141)

The constants R0and ns would be calculated in the same manner as

previously detailed. The negative strain rate sensitivity correction

146
term could then be calculated for each test using the applied strain

rate and the following formula:

.1

f.= exp(-n5abs{ ln[ abs(——)]}) (142) i •

where the i index denotes each individual test being considered and abs

denotes absolute value.

The constants n3, n4, and ng will be determined with a linear least

squares regression scheme applied to the following equations:

n3 + n4f.j + n6/e. = N1

where N is the slope of the theta plot and the i index denotes each test

considered. The tests considered would be 70, 86, 56, 65, 83, 80, 72,

and 71.

The constant n2 would be estimated by averaging values calculated

from the following equation:

where C2 is the ratio of saturated stress to saturated back stress as

estimated from tests 65, 81, 84, and 88. Tests 70, 86, 56, and 65 would

be used in this calculation.

The next step would be to calculate C and D = D^ + D2 from the

following equation:

147
n2

n3+ v f.

A linear fit using the saturated stress from the theta plot, fi§ and the

applied strain rate for tests 70, 86, 56, 65, 83, and 80 gives D as the

slope and allows C to be calculated from the intercept. The constants

Dj, D2, and n7 would be calculated in the same manner as presented

earlier. This method of calculation has been applied by Imbrie, et al.

[161].

148
2>

N86-28459

APPENDIX C - AUXILIARY MODEL RESULTS

This appendix presents model predictions not presented in the text

and are included here for completeness. Figure C.I presents the

experimental results for the first half cycle of the cyclic tests and

the long term tension tests. Figures C.2 through C.6 present the model

predictions of the first half cycle of the cyclic tests 86, 65, 83, 80,

and 72. Numerical instabilities can be seen which result from the Fsol

parameters.

Figure C.7 provides the experimental hystenesis loops. Figure C.8

through C.ll compare the model results to tests 65, 83, 80, and 72. The

values at -.8% strain for each test and the associated model predictions

are shown in Figure C.12.

The values at +.8% of each cycle are shown for test 65 and 83 in

Figure C.13 and C.14 respectively. The values at -.8% for tests 86, 65,

83, 80, and 72 are shown in Figures C.15 through C.19.

The model's predictions are shown compared to a complex history

test of the same input and material series as the test 89 mentioned in

Chapter VII in Figures C.20 through C.26. A region of discontinuous

yielding or the Portvin - Le Chatelier effect can be seen in the region

between .005 and .014 strain. Bodner and Rosen discuss this phenomena

in reference [162].

149
CO

co

CO

o>

CO

TEST 70

TEST 86

TEST 65

TEST 83

TEST 80

TEST 72

TEST 71

STRfllN RRTE

STRfllN ROTE

STRfllN RflTE

STRfllN RRTE

STRfllN RflTE

STRfllN ROTE

STRfllN RflTE

151E-3/SEC

002E-3/SEC

127E-4/SEC

926E-S/SEC

054E-5/SEC

626E-6/SEC

2S3E-6/SEC

150.0

125.0

100.0

75.0

50.0

25.0

0.0

I I

I i I 1 1 .

0.0 .002 .004 .006 .008 .010 .012

Strain (in/in)

Figure C.I - Experimental Monotonic Tests

.014
CO

CO

CO

86 -- STRR1N RflTE s \ .002E-3/SEC

BOONER HOOEL (WITHOUT FSOL CORRECTION)

BOONER MODEL [WITH FSOL CORRECTION)

— — KRIEC tlODEL (WITHOUT FSOL CORRECTION)

KRIEC MODEL (WITH FSOL CORRECTION)

MILLER MODEL

WRLKER MODEL

150.0

125.0

100.0

75.0 -

50.0

25.0 -

.002 .004 .006 .008 .010 .012

Strain (in/in)

Figure C.2 - Model Response As Compared to Test 86

.014
co

<i>

CO

TEST 65 -- STRRIN ROTE = 3.127E-4/SEC

BOONER MODEL (WITHOUT FSOL CORRECTION)

BODNER MODEL (WITH FSOL CORRECTION)

— — KRIEC MODEL (WITHOUT FSOL CORRECTION)

KRIEO MODEL (WITH FSOL CORRECTION)

MILLER MODEL

WRLKER MODEL

150.0

1 25.0

100.0

75.0

50.0

25.0

0.0

0.0 .002 .004 .006 .008 .010 .012

Strain (in/in)

Figure C.3 - Model Response as Compared to Test 65

.014
CO

V)

0)

&H

+J

CO

TEST 83 -- STRflIN RflTE = 9-926E-5/SEC

BOOMER MODEL (WITHOUT FSOL CORRECTION)

BODNER MODEL IWIT H FSOL CORRECTION)

— — KRIEC MODEL (WITHOUT FSOL CORRECTION)

KRIEO MODEL (WITH FSOL CORRECTION)

MILLER MODEL

WfiLKER MODEL

150.0

125.0

100.0

75.0

50.0

25.0

0.0

T T

0.0 .002 .004- .006 .008 .010 .012 .014

Strain (in/in)

Figure C.4 - Model Response as Compared to Test 83
TEST 80 -- STRRIN RflTE r 3.054E-5/SEC

BOONER MODEL (WITHOUT FSOL CORRECTION)

BODNER MODEL (WITH FSOL CORRECTION)

— — KRIEC MODEL (WITHOUT FSOL CORRECTION)

KRIEC MODEL (WITH FSOL CORRECTION)

MILLER MODEL

HflLKER MODEL

150.0

125.0

100.0

CO

CO 75.0

CO

CO

50.0

25.0

0.0

r i i

j

0:0 .002 .004 .006 .008 .010 .012

Strain (in/in)

Figure C.5 - Model Response as Compared to Test 80

.014
to

1/3

(7)

TEST 72 -- STRRIN RRTE = 7.626E-6/SEC

BOONER MODEL (WITHOUT FSOL CORRECTION)

BOONER rtOOEL (WITH FSOL CORRECTION)

— — KR1EO MODEL (WITHOUT FSOL CORRECTION)

KRIEC MODEL (WITH FSOL CORRECTION)

MILLER MODEL

WRLKER MODEL

150.0

125.0

100.0

75.0

50.0

25.0

0.0

0.0 .002 .004 .006 .008 .010 .012

Strain (in/in)

Figure C.6 - Model Response as Compared to Test 72

.014
OT

CO

CO

CO

TEST BB ICYCLE 10) -- STRR1N RRTE = 1 .002E-3/SEC

TEST 65 (CYCLE 10) -- STRRIN RRTE = 3.127E-4/SEC

TEST 83 ICYCUE 10) -- STRRIN ROTE = 9.926E-5/SEC

— — TEST BO (CYCLE 10) -- STRRIN ROTE = 3.0S4E-S/SEC

TEST 72 (CYCLE 4) -- STRRIN RRTE = 7.626E-6/SEC

150.0

125.0

100.0

75.0

50.0

25.0

0.0

-25.0

-50.0

-75.0

-100.0

-125.0

-150.0

I I

I I I

I

I I

-.01 0-.008 -.006 -.004 -.002 0.0 .002 .004 .006 .008 .010

Strain (in/in)

Figure C.7 - Experimental Hysteresis Loops
CO

CO

0>

(H

TEST 63 (CYCLE 10) -- STRflIN RRTE = 9.926E-5/5EC

BOOMER nOOEL (WITHOUT FSOL CORRECTION)

BODNER MODEL (WITH FSOL CORRECTION)

— — KRIEC MOOEL (WITHOUT FSOL CORRECTION)

KRIEG MODEL (WI.TH FSOL CORRECTION)

MILLER MODEL

WRLKER MODEL

150.0

125.0

1 00.0

75.0

50.0

25.0

0.0

-25.0

-50.0

-75.0

-100.0

-125.0

-150.0

I T I

-.010 -.008 -.006 -.004 -.002 0.0 .002 .004 .006 .008 .010

Strain (in/in)

Figure C.9 - 10th Cycle. Hysteresis Loops for Test 83
TEST 65 (CTCLE 10) -- STRRIN RRTE = 3.I27E-4/SEC

BOONER MODEL (WITHOUT FSOL CORRECTION)

BOONER MODEL (WITH FSOL CORRECTION)

— — KRIEC MODEL (WITHOUT FSOL CORRECTION)

KRIEC MODEL (WITH FSOL CORRECTION)

MILLER MODEL

HflLKER MODEL

CO

cn

o>

CO

150.0

125.0

100.0

75.0

50.0 -

25.0

0.0

-25.0 -

-50.0 -

-75.0

-100.0

-125.0 -

-150.0

-.010-.008 -.006-.004 -.002 0.0 .002 .004 .006 .008 .010

Strain (in/in)

Figure C.8 10th Cycle Hysteresis Loops for Test 65
CO

CO

CL>

CO

TEST 80 (CYCLE 10) -- STRfilN RRTE = 3.054E-5/SEC

SOONER MODEL (WITHOUT FSOL CORRECTION)

BODNER MODEL IHIT H FSOL CORRECTION)

— — KRIED nODEL (WITHOUT FSOL CORRECTION)

KRIED nODEL (WITH FSOL CORRECTION)

MILLER MODEL

WRLKER MODEL

150.0

125.0 -

1 00.0

75.0 -

50.0 -

25.0 -

0.0

-25.0

-50.0

-75.0 -

-100.0 -

-125.0 -

-150.0

-.010-.008 -.006 -.004 -.002 0.0 .002 .004 .006 .008 .010

Strain (in/in)

Figure C.10 - 10th Cycle Hysteresis Loops for Test 80
CO

CO

•TEST 72 (CYCLE 4) -- STRPIN RRTE = 7.626E-6/SEC

•BDDNER MODEL 1H1THOUT FSOL CORRECTION)

BODNER MODEL (WITH FSOL CORRECTION)

KRIEG MODEL (WITHOUT FSOL CORRECTION)

•KRIEC MODEL (WITH FSOL CORRECTION)

•MILLER MODEL

•HRLKER MODEL

150.0

125.0

i 00.0

75.0

50.0

25.0

0.0

-25.0

-50.0

-75.0

-100.0

-125.0

-150.0

// •• />•' /;•//

!!/if'/ i.- >'

f/f

T

-.010-.008 -.006-.004 -.002 0.0 .002 .004 .006 .008 .010

Strain (in/in)

Figure C.ll - 4th Cycle Hyeteresis Loops for Test 72
EXPERIMENTAL - STRESS MOTH CYCLE o -.87. STRfllNI

BODNER MODEL (WITHOU T FSOL CORRECTION)

BODNER MODEL (WIT H FSOL CORRECTION )

KRIEG MODEL (WITHOU T FSOL CORRECTION)

KRIEG MODEL (WIT H FSOL CORRECTION)

t— i -.--4>

CO -».»

Jv^ 7O D

c

— • -80.0

4->

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°° -100.0

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(3, -110.0

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P -140.0

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XH 150 0

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5

- MILLER MODEL

— - HRLKER MODEL

1 1 1 1 1

-K. ^_

~~ "~ -K _

m, _ ~^ -^4>

*

~~ -••—•'"£.'-.-.^-—~ '~'* ' * ^-^iS^tZZr:-^ -

x — -•_-•-.•.—— — "a^"""^"^ A

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)!(••" •• — —••••" *• ~ ^^^

l i i i i

.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2

Log(Strain Rate)

Figure C.I2 - Stress-Strain Response at -.8% Saturated Cycle
TEST 65 -- STRRIN RflTE = 3.127E-4/SEC

t— 1

CO

^

£•<

•fH

ctf

£

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--A- BODNER MODEL (WIT H FSOL CORRECTION)

—4- KRIEG MODEL (WITHOU T FSOL CORRECTION)

--*- KRIEG MODEL (WIT H FSOL CORRECTION)

-..-«>—• MILLER MODEL

— -3K-— WflLKER MODEL

1 O U .U

140.0

130.0

I

120.0

1

1 r\ r\

I 0.0

I

1I V nJn U . U n

90.0

80.0

7rv r»

1 II 1 1 1 1 1

-

—

.. ..-. — *•—...._

t

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k-

-- -&• ^....

_ "" ^ & ^ ^

M " —

1 1 1 1 1 II 1

1 234567 , 8 9 K

Cycle Number

Figure C.13 - Test 65 +.8% Response at Each Cycle
TEST 83 -- STRfllN ROTE r 9.926E-5/SEC

• — G>

- -A

— 4-

- -X

-..-o

-•••*

i

1

 c DU. . nU r>

tf) 140.0

6

 130-°«

•i— i

rtl

_fj 1 20.0

fex 1 1 0.0

CO

4- 100.0'

<§>

OT 90-0

0>

£ 80.0

r/3

•7 f\ r\

/ U.U

i

— BODNER MODEL (WITHOU T FSOL CORRECTION)

BODNER MODEL (WIT H FSOL CORRECTION )

KR1EG HOOEL (WITHOUT FSOL CORRECTION)

— KRIEG MODEL (WIT H FSOL CORRECTION)

— • MILLER MODEL

— WRLKER MODEL

1 1 1 1 1 1 1 1

P^^«-«^T'

7

^=TnT-7r-e — pi_ ""

r^ • *— — ..^..

 H g m

A A --— Jlt-T^— . ^ - A _ _ _ -— •• QJ '-• — ^

~---x ' """* •<. *"-"-"*•» * ;

'^-^ H *" *•"""" """"* *

\ "^ ^ ^ ©- ^. ^

i i l i i i i i

2 3 4 5 6 7 8 91 <

Cycle Number

Figure C.14 - Test 83 +.8% Response at Each Cycle
C/3

CO

O

I

<§>

CO

CO

—S— TEST 06 -- STRfilN ROTE = I .002E-3/SEC

—-G>— BODNER MODEL (WITHOUT KSOL CORRECTION)

- -A - BOONER MODEL (WITH FSOL CORRECTION)

— 4- KRIEG MODEL (WITHOUT FSOL CORECTION)

—X— KRIEG MODEL (WITH FSOL CORRECTION)

—-O—• MILLER MODEL

WRLKER MODEL

-70.0

C/} -80.0

90.0

-100.0

-1 10.0i ^

-120.0

-i 30.0

-140.0

).0

1 1

_—-e —*~— --*::r::---3 ::^::——-

1 1

3456 7

Cycle Number

Figure C.15 - Test 86 -.8% Response at Each Cycle

8 10
....0....

_ -A -

— 4-

_._*._.

TEST 65 -EXPERIMENTS- STRESS --8X STRBIN FOR EfiCH CYCLE

BODNER MODEL (WITHOUT FSOL CORRECTION)

BOONER MODEL (WITH FSOL CORRECTION)

KRIEG MODEL (WITHOUT FSOL CORRECTION)

KRIEG MODEL (WITH FSOL CORRECTION)

MILLER MODEL

WRLKER MODEL

m -

fi

f-t

-100.0 -

CO

d.

I

<3>

en

<r>

o>

-1 10.0 -

-120.0 ^

-1 30.0

-140.0 -

3456 7

Cycle Number

Figure C.16 - Test 65 -.8% Response at Each Cycle

8 10
—B— TEST 83 -- STRRIN RRTE - 9.926E-5/SEC

....0.... BOONER MODEL (WITHOUT FSOL CORRECTION)

- -A. - OODNER MODEL (WITH FSOL CORRECTION)

—4- KRIEG MODEL (WITHOUT FSOL CORRECTION)

—H— KRIEG MODEL (WITH FSOL CORRECTION)

- -O- • MILLER MODEL

1— 1

g

jw

NO

00

1

<3>

V)

V)

fl

in

— X

7(1 n

-80.0

-90.0

-100.0

i

1 1 O O

-1 20.0'

-1 30.0

-140.01

150 0

i

•— WflLKER MODEL

1 1 1 1 1 1 1 1

i 4 • 1 . 1 1 1 1 1

- €>• '<*• *'

^* ^ W *^£ *W V xx j^ ^, „ — i - -• • • — ' " ~^^

._.--•'" ^

_'-':* ^ — "* *

_ - - A- - - f -_**.&• *~ At

_- - A ' "" ~ _.--3l^""""""""

- .-.••'••'-*""""' B— — 0 ' B B :-

_— e— — Q " °~

t — *

1 1 1 1 1 II I

2345 . 678 9 1 C

Cycle Number

Figure C.I7 - Test 83 -.8% Response at Each Cycle
1— (

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CO

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1

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(s>

V)

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— B

— -©

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-80.0

-90.0

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- 1• £ 2, V/0 • V 0

-1 30.0

-140.0'

i r. r\ /-)

1 J U . U

1

— TEST 80 -- STRRIN RflTE = 3-054E-5/SEC

— BODNER MODEL (WITHOU T FSOL CORRECTION)

80DNER MODEL (WIT H FSOL CORRECTION)

KRIEG MODEL (HITHOU T FSOL CORRECTION)

— KRiEG MODEL (M1TH FSOL CORRECTION)

— • MILLER MODEL

— HRLKER MODEL

1 1 1 1 1 1 1 1

^ ^ & •©• €> €>• ©• •&— -T •<

-—••-'-

x— -"" '"~K K £ * «..•-••-"•"""*" *" ^'"

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- ^...-*-'-'-~" ^_^-^ ^ — — *

r"'" g • ~Q

•

l l I I l I l l

2345 6 7 8 9 K

Cycle Number

Figure C.I8 - Test 80 -.8% Response at Each Cycle.
t— <

• r— 1

cfl

W

00

d

1

(3) vy

to

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0)

£

w

— B

--©

- -A

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7/ \0j ,\ nj

-80.0

-90.0

-100.0

-1 10.0

-120.0

-1 30.0

i

-140.0

1 so n

1

— TEST 72 -- STRflIN RflTE = 7.626E-6/SEC

— BOOMER MODEL (WITHOUT FSOL CORRECTION)

BOONER MODEL (WIT H FSOL CORRECTION)

KR1EG MODEL (WITHOUT FSOL CORRECTION)

— KR1EG MODEL (WIT H FSOL CORRECTION)

— • MILLER MODEL

— HRLKER MODEL

1 1 1 1 1 1 1 1

1 i i ^ "1 —— — — — | ' ' — — *~ ^~ T

-

- - —

"

.,-'•'' •"*

^ ~" ^ •*•• "" -

..--*""""""

. ,- — ' ^ „__ ^ — A "•

m m PI

£3 . • gj

1 1 1 1 1 1 1 1

234 5 6 7 8 9 1 C

Cycle Number

Figure C.I9 - Test 72 -.8% Response at Each Cycle
•TEST 85 — COMPLEX INPUT HISTORY

•BODNER MODEL (WITHOUT FSOL CORRECTION)

W

CD

IH

150.0

125.0 -

100.0

75.0

50.0

25.0

-25.0

-.012 -.008 -.004 0.0 .004 .008 .012 .016

Strain (in/in)

Figure C.20 - Test 85 Bodner's Uncorrected Model
•TEST 85 -- COMPLEX INPUT HISTORY

BOONER flODEL (WITH FSOL CORRECTION)

m

CO

CO

o

CO

150.0

125.0

100.Oh

75.0

50.0 -

25.0

0.0

-25.0

-50.0 -

-75.0 -

-100.0

-125.0 -

-150.0

•ui r j U . r:—. I.

1 W1 ^i

.01 t -

-.012 -.008 -.004 0.0 .004 .008 .012 .016

Strain (in/in)

Figure C. 21 - Test 85 Bodner's Corrected Model
-TEST 85 -- COMPLEX INPUT HISTORY

•KRIEG MODEL (WITHOUT FSOL CORRECTION)

C/3

(73

(/)

CD

E*

•^

OT

150.0

125.0 -

100.0

75.0

50.0

25.0

0.0

-25. Oh-

-50.0 -

•75.0 -

-100.0 -

-125.0

-150.0

-.012 .008 -.004 0.0 .004 .008 .012 .016

Strain (in/in)

Figure C.22 - Test 85 Uncorrected Model of Krieg, et al.
•TEST 85 -- COMPLEX INPUT HISTORY

•KR1EC MODEL (WITH FSOL CORRECTION)

to

CO

CO

150.0

125.0

100.0

75.0

50.0

25.0

0.0

-25.0

-50.0

-75.0

-100.0

-125.0

-150.0

I I

-.012 -.008 -.004 0.0 .004 .008 .012 .016

Strain (in/in)

Figure C.23 - Test 85 Corrected Model of Krieg, et al.
•TEST 85 -- COMPLEX INPUT HISTORY

•MILLER MODEL

CO

CO

o>

72

150.0

125.0 -

100.0 -

75.0 -

50.0

25.0 -

0.0

-25.01-

-50.0 -

-75.0 -

-100.0

-125.0

-150.0

i f _r-—\

u . r~.—. \.

• W »•

-.012 008 -.004 0.0 .004 .008 .012

Strain (in/in)

Figure C.24 - Test 85 Miller's Model

.016
-TEST 85 -- COMPLEX INPUT HISTORY

•HHLKER MODEL

Cfl

CO

0

150.0

125.0 -

I 00.0

75.0 -

50.0

25.0

0.0

*» -25.0 -

-50.0

-75.0

-100.0

-125.0

-150.0

-.012 .008 -.004 0.0 .004 .008

Strain "(in/in)

Figure C.25 - Test 85 Walker's Model

.012 .016
N86 -28460

litAPPENDIX 0 - EXPERIMENTAL COMPUTER PROGRAMS

The appendix presents the test software developed for this work

which would be necessary to repeat or expand on this work. All programs

are written in MTS Basic. Program VCYC6 allows the user to run cyclic

tests, monotonic tension tests, and cyclic tests with creep hold

times. The test parameters can be changed on the fly. Data acquisition

for both the cyclic test and the hold times as provided for. All data

is stored in binary form on floppy diskettes. The program CRP3 performs

creep tests and creep stress drop tests. The program BACK2 allows the

user to analyze cyclic creep hold time data on creep stress drop data.

The program does a linear regression curve fit through the data. The

number of points used for the curve fit is chosen by the user for each

step.

6MPLT8 analyzes each test with the gamma plot. The stress

inelastic strain history is calculated and a stress-inelastic strain

polynomial curve fit performed. This curve fit is differentiated at each

data point and divided by the value of stress at that point to create a

-Tg- versus o plot. Linear regression curve fits for the upper and dwp

lower slopes are performed based on the number of points input by the

user. This version is configured to automatically step through a cyclic

test and produce gamma plot for each half cycle. The program WALK2

performs the same functions with the theta plot.

VCYC6

175
1 REM CYCLIC TEST - STRAIN CONTROL

8 CLRTXWIN

10 REM WRITTEN BY GEORGE JAMES AND PAUL SEAN HILL - JUNE 1985

20 PRINT 'THIS PROGRAM PERFORMS A CYCLIC STRAIN CONTROLLED TEST WHICH

IS1

24 PRINT 'IS CAPABLE OF SWITCHING STRAIN LIMITS DURING THE TEST.1

28 PRINT

30 PRINT 'THE PROGRAM SWITCHES FROM LOAD TO STRAIN CONTROL AND IT IS IMi

40 PRINT 'PORTANT THAT THE DISPLAY #2 SHOWS A POSITIVE VALUE FOR THIS'

45 PRINT 'SWITCH.'

47 PRINT

70 PRINT 'THE TEST DATA IS STORED ON 5.25" FLEXIBLE DISC ON DU1: IN

BINARY1

100 PRINT 'FORM. RELAXATION DATA IS STORED IN HARD COPY FORM ONLY.' \

PRINT

110 PRINT 'THE CAPABILITY TO PRINT AND PLOT THE DATA IS PROVIDED FOR

THE'

120 PRINT 'CURRENT TEST OR FOR EXISTING FILES.1

125 PRINT \ PRINT

126 PRINT \ PRINT

130 DIM A(64,3),B(64,3),M(2,2),MO(2,2),M1(2,2),M2(2,2)

140 REM

150 REM PRINT ONLY INPUT

160 REM

170 PRINT

180 PRINT 'TYPE'; \ TXTBOLD \ PRINT ' TEST';

182 TXTNORMAL \ PRINT ' IF YOU WANT TO RUN A NEW TEST'

184 PRINT \ PRINT ' OR'

186 PRINT

190 PRINT 'TYPE'; \ TXTBOLD \ PRINT ' PRINT'; \ TXTNORMAL

195 PRINT ' IF YOU WANT TO PRINT EXISTING FILES'

196 PRINT

200 INPUT R$

210 IF R$='TEST' THEN GO TO 330

220 IF R$='PRINT' THEN GO TO 240

225 TXTINVERS

230 PRINT 'INVALID ENTRY, TYPE TEST OR PRINT' \ TXTNORMAL \ GO TO 170

240 PRINT 'ENTER FILE NAME (<=5 CHARACTERS)'; \ INPUT F$

250 IF LEN(F$)<=5 THEN GO TO 270

255 TXTINVERS

260 PRINT 'INVALID FILE NAME' \ TXTNORMAL \ GO TO 240

270 F1$='DU1:'+F$+'C.DAT'

280 F2$='DU1:'+F$+'.DAT1

285 P0=0

290 GO TO 6900

300 REM

310 REM TEST INPUT

320 REM

176
330 GO TO 350

340 CLOSE #1,2,3

350 PRINT

360 PRINT 'ENTER FILE NAME (<=5 CHARACTERS)1; \ INPUT F$

370 IF LEN(F$)<=5 THEN GO TO 390

375 TXTINVERS

380 PRINT 'INVALID FILE NAME ' \ TXTNORMAL \ GO TO 350

390 F1$='DU1:

400 F2$='DU1:

+F$+'C.DAT'

+F$+'.DAT'

410 OPEN Fl$ FOR OUTPUT AS FILE 1, FILESIZE 2

420 OPEN F2$ FOR OUTPUT AS FILE 2, FILESIZE 30

FOR OUTPUT AS FILE 3

ENTER THE SAMPLE NUMBER1; \ INPUT B$

ENTER THE SAMPLE TEMPERATURE IN DEGREES CELSIUS1; \ INPUT A$

ENTER LOAD CALIBRATION (LBS/VOLT)1; \ INPUT LO

ENTER EXTENSOMETER CALIBRATION (IN/VOLT)'; \ INPUT XO

ENTER GAUGE LENGTH (IN)'; \ INPUT GO

ENTER GAUGE AREA (IN*IN)'; \ INPUT AO

430 OPEN 'LP

431 PRINT

432 PRINT

440 PRINT

460 PRINT

470 PRINT

480 PRINT

490 PRINT ENTER STRAIN RATE (IN/IN/SEC)1; \ INPUT RO

500 R1=(RO*GO)/(XO*10)

510 PRINT 'ENTER MAXIMUM STRAIN (IN/IN)1; \ INPUT SO

520 S1=(SO*GO)/(XO*10)

530 PRINT 'ENTER MINIMUM STRAIN (IN/IN)'; \ INPUT S2

540 S3=(S2*GO)/(XO*10)

550 PRINT 'ENTER TIME/POINT (SEC)1; \ INPUT TO

560 PRINT 'ENTER LENGTH OF DATA ACQUISITION (CYCLES)1; \ INPUT CO

570 C1=2*CO

580 T1=((2*CO)*((SO-S2)/(RO*TO)))

590 PRINT 'NUMBER OF POINTS EXPECTED ';T1

620 PRINT 'ENTER LOAD RANGE DURING HOLD TIMES (1,2,3, OR 4)'; \ INPUT LI

625 PRINT 'ENTER LOAD CALIBRATION FOR HOLD TIMES (LBS/VOLT)'; \ INPUT L2

630 PRINT 'ENTER TIME/POINT DURING HOLD TIMES (MIN)1; \ INPUT T2

632 CLRTXWIN

SAMPLE NUMBER

SAMPLE TEMPERATURE

LOAD CALIBRATION

EXTENSOMETER CAL.

GAUGE LENGTH

GAUGE AREA

STRAIN RATE

MAXIMUM STRAIN

MINIMUM STRAIN

633 PRINT

634 PRINT

635 PRINT

640 PRINT

650 PRINT

660 PRINT

680 PRINT

690 PRINT

700 PRINT

710 PRINT

720 PRINT

730 PRINT

735 PRINT

740 PRINT

745 PRINT

750 PRINT

•TIME/POINT

'LENGTH OF D.A.

'NUMBER OF POINTS

'HOLD-LOAD RANGE

'HOLD-LOAD CAL.

'HOLD-TIME/POINT

\ PRINT 'ARE ALL PARAMETERS

(DEG C)

(LBS/VOLT)

(IN/VOLT)

(IN)

(IN*IN)

(IN/IN/SEC)

(IN/IN)

(IN/IN)

(SEC)

(CYCLES)

(LBS/VOLT)

(MIN)

DAUCTCDC r\V

;B$

;A$

;LO

;XO

;GO

;AO

;RO

;SO

;S2

;TO

;CO

;L2

;T2

752 TXTBOLD \ PRINT '(Y/N)'; \ TXTNORMAL \ INPUT R$

177
760 IF R$='N' THEN GO TO 340

780 PRINT #1,LO;',';XO;',';GO;',';AO;',';RO;',';R1;',';SO;',';S1

790 PRINT #1,S2;',';S3;',';TO;',';CO;',';T1;',';L1;',';L2;',';T2

800 REM

810 REM DATA ACQUISITION FUNCTIONS

820 REM

830 CLINIT

880 CKSTOP

890 CKTIME(1,.01,K)

900 I1=INT(TO/K)

910 ADTIMED(1SASBS1SI1)

920 ADSLAVE(1,2,6)

940 ADRMAX(3,M,1)

950 ADRMIN(3,MO,1)

960 ADRMAX(3,M1,2)

970 ADRMIN(3,M2,2)

980 REM

990 REM REAL TIME PLOT INPUT

1000 REM

1010 PRINT \ PRINT 'DO YOU WANT A REAL TIME PLOT?'

1015 PRINT ' (Y/N)'; \ TXTNORMAL \ INPUT R$

1030 IF R$='N' THEN GO TO 1060

1050 G=l \ GO TO 1070

1060 G=0 \ GO TO 1209

 }

1070 X$='IN/IN'

1080 PRINT \ PRINT 'REAL TIME PLOT PARAMETERS'

1090 PRINT 'ENTER MAXIMUM STRAIN1

\ TXTBOLD

1100 PRINT

1110 PRINT

'ENTER MINIMUM STRAIN1

\

\

INPUT X2

INPUT XI

\ PRINT

'ENTER STRAIN STEP SIZE'; \ INPUT X3

1120 Y$='PSI' \ PRINT

1130 PRINT 'ENTER MAXIMUM STRESS1; \

1140 PRINT 'ENTER MINIMUM STRESS1; \

1150 PRINT 'ENTER STRESS STEP SIZE1;

1160 PRINT 'ARE THESE PARAMETERS OK?1;

1165 TXTNORMAL \ INPUT R$

1180 IF R$='N' THEN GO TO 1090

1200 REM

1203 REM TEST FUNCTIONS

1206 REM

1209 CLRTXWIN

1212 TXTNORMAL \ PRINT \ TXTINVERS

1215 PRINT 'KEY INTERUPT SUMMARY1

INPUT Y2

INPUT Yl

\ INPUT Y3

\ TXTBOLD \ PRINT '(Y/N)';

1218 TXTSPOS(1,8,1) \ PRINT

1221 PRINT

1224 PRINT

1227 PRINT

\ TXTSPOS(1,5,10)

\ TXTNORMAL

'<ESC>E - END TEST'

'<ESC>H - HOLD TEST FOR RELAXATION'

'<ESOR - RESUME TEST AFTER HOLD1

'<ESC>P - CHANGE CYCLE PARAMETERS'

1228 PRINT '<ESC>A - HALT DATA ACQUISITION AND CONTINUE TEST1

1230 PRINT \ TXTFLASH \ PRINT \ PRINT \ PRINT

1232 PRINT 'BRING UP THE HYDRAULIC PRESSURE '

1233 PRINT 'ZERO LOAD WITH SET POINT CONTROL AND STRAIN WITH ZERO

178
ADJUST1

1234 PRINT 'PLACE PRINTER ON SWITCH "A" '

1235 TXTNORMAL \ PRINT \ PRINT

1236 PRINT 'HIT RETURN TO SWITCH TO STRAIN CONTROL.1

1240 INPUT R$

1242 PRINT 'ZERO STRAIN AGAIN WITH SET POINT CONTROLLER' \ TXTNORMAL

1245 KBINT('E',3, LINE 4600 )

1248 KBINT('H',4, LINE 1266 )

1251 KBINT('R',4, LINE 1269 )

1254 KBINT('P',5, LINE 12000 )

1255 KBINT('A'.3S LINE 13000 )

1257 KBENB

1260 Gl=0

1263 GO TO 1272

1266 Gl=l \ RETURN

1269 Gl=2 \ RETURN

1272 ADINIT

1275 CLRANGESENSE(1,2,X5)

1300 IF X5=4 THEN X5=10

1310 IF X5=3 THEN X5=5

1320 ADIMMED(2,S4)

1330 S4=-(S4/X5)

1340 CLSWITCH(1,2,,S4)

1341 PRINT \ PRINT \ TXTFLASH

1345 PRINT \ PRINT 'HIT RETURN TO START TEST.' \ INPUT R$

1346 GTIME(Z3,Z2,Z1) \ GDATE(Z6,Z5,Z4)

1350 IF G=0 THEN GO TO 1420

1360 TEKMODE(l.l)

1370 PHYL(1,10,85,10,85)

1380 SCALE(1,0,X1,X2,Y1,Y2)

1390 AXES(1,0,0)

1400 LABEL(1,X$,Y$,X3,Y3,1)

1410 INVEC

1425 T6=0

1430 ADGO \ ETIME

1500 FGGO

1505 B0=0 \ Fl=0

1550 FGREPT(1,'SINE', RATE R1,C1,S1,S3)

1600 FGSTATUS(l.Hl)

1605 IF G=0 THEN GO TO 1635

1610 ADIMMED(2,X4)

1615 ADIMMED(1,Y4)

1620 X4=(X4*((10*XO)/GO))

1625 Y4=(Y4*((10*LO)/AO))

1630 PLOT(1,X4,Y4)

1635 IF Hl=0 THEN GO TO 4650

1640 CLHYDSENSE(1,H2)

1645 IF H2=0 THEN GO TO 5100

1650 IF Gl=l THEN GO TO 2200

1653 IF Al=l THEN GO TO 1600

179
1655 IF A=64 THEN B=0

1660 IF A<64 THEN GO TO 1600 \ A=-l

1665 Fl=l

1670 AOUT(#2,A(1,0),64*4,BO,E)

1675 P=INT(E/4) \ PO=PO+P

1680 BO=BO+1

1685 FGSTATUS(l.Hl)

1690 IF G=0 THEN GO TO 1710

1695 ADIMMED(2,X4) \ ADIMMED(1,Y4)

1700 X4=(X4*((10*XO)/GO)) \ Y4=(Y4*((10*LO)/AO))

1705 PLOT(1,X4,Y4)

1710 IF Hl=0 THEN GO TO 4650

1715 CLHYDSENSE(1,H2)

1720 IF H2=0 THEN GO TO 5100

1725 IF Gl=l THEN GO TO 2200

1730 IF B=64 THEN A=0

1735 IF B<64 THEN GO TO 1685 \ B=-l

1740 Fl=0

1745 AOUT(#2,B(1,0),64*4,BO,E)

1750 P=INT(E/4) \ PO=PO+P

1755 BO=BO+1

1760 GO TO 1600

2100 REM

2150 REM HOLD FUNCTIONS

2200 REM

2300 IF G=0 THEN GO TO 2450

2350 HOME

2400 VTMODE(l.O)

2450 ADIMMED(2,X5)

2500 ADIMMED(1,L6)

2505 IF S8=0 THEN GO TO 2525

2510 X9=L6*L2*10/AO

2515 IF X9<=S9 THEN GO TO 2525

2520 GO TO 2450

2525 FGHOLD \ ADHOLD(l)

2550 PRINT 'HOLD AT1 ;((X5*XO*10)/GO);'(IN/IN)' \ PRINT

2600 PRINT 'INITIAL STRESS =';((L6*L2*10)/AO);' PSI' \ PRINT

2650 PRINT #3,'HOLD AT';((X5*XO*10)/GO);'(IN/IN)' \ PRINT #3

2700 PRINT #3,'INITIAL STRESS =';((L6*L2*10)/AO);' PSI'

2750 PRINT #3

2800 PRINT #3,'STRESS (PSI)';TAB(30)'TIME (MIN)'

2850 ETIME(T3)

2900 CLOFFSENSE(1,1,L3,F)

2950 CLRANGESENSE(1,1,L4)

3000 ADIMMED(1,L5)

3050 IF L4=4 THEN L5=-(L5/10)

3100 IF L4=3 THEN L5=-(L5/5)

3150 IF L4=2 THEN L5=-(L5/2)

3200 IF L4=l THEN L5=-L5

3250 CLRANGESWITCH(1,1,L1,L5)

180
3300 ADIMMED(1,L5) \ ETIME(T4)

3350 SLEEP(T4+(T2*60))

3400 L5=((L5*L2*10)/AO)

3450 ETIME(T4) \ T4=((T4-T3)/60)

3500 PRINT 'STRESS =';L5;'(PSI)';TAB(30)'TIME =';T4;'(MIN)' \ PRINT

3550 PRINT #3,L5;TAB(30)T4

3560 CLHYDSENSE(1,H2)

3570 IF H2=0 THEN GO TO 5100

3600 IF Gl=2 THEN GO TO 3850

3650 GO TO 3300

3700 REM

3750 REM RESUME FUNCTIONS

3800 REM

3850 PRINT

3855 PRINT 'DO YOU WANT TO SPECIFY THE INITIAL STRESS FOR THE NEXT

HOLD1;

3857 TXTBOLD \ PRINT '(Y/N)'; \ TXTNORMAL

3860 INPUT R$ \ IF R$='N' THEN GO TO 3875

3865 PRINT \ PRINT 'UHAT INITIAL STRESS DO YOU WANT TO HOLD AT NEXT1; \

INPUT S9

3870 S8=l \ GO TO 3899

3875 S8=0

3899 CLRANGESWITCH(1,1,L4,L3)

3900 ETIME(T5) \ T5=T5-T3 \ T6=T6+T5

3950 FGRESUME \ ADRESUME(l)

4000 IF G=0 THEN GO TO 4400

4050 TEKMODE(l.l)

4100 PHYL(1,10,85,10,85)

4150 SCALE(1,0,X1,X2,Y1,Y2)

4200 AXES(1,0,0)

4250 LABEL(1,X$,Y$,X3,Y3,1)

4300 INVEC

4350 Gl=0

4400 IF Fl=0 THEN GO TO 1655

4410 IF Fl=l THEN GO TO 1730

4450 ^

4500 REM END OF TEST FUNCTIONS

4550

4600 FGREMOVE(l.l)

4650 CLSWITCH(1,1,,0)

4700 FGGO

4750 FGARB(1,'RAMP', TIME 5,0)

4800 IF G=l THEN GO TO 4900

4850 PRINT 'SWITCHING TO LOAD CONTROL AND RAMPING DOWN...1

4900 FGSTATUS(l.Hl)

4950 IF Hl=0 THEN GO TO 5350

5000 GO TO 4900

5100 FGREMOVE(l)

5150 CLSWITCH(1,1,,0)

5250 IF 6=1 THEN GO TO 5350

181
5300 PRINT 'HYDRAULIC SHUT-DOWN DETECTED1

5350 FGTOTALSEG(1,C2)

5400 FGSTOP \ ADSTOP

5450 ETIME(T7) \ T7=T7-T6

5500 IF G=0 THEN GO TO 5700

5550 INVEC

5600 HOME

5650 VTMODE(l.O)

5700 PRINT 'END OF TEST' \ PRINT

5750 CLRANGESWITCH(1,2,,0)

5800 REM

5850 REM STORING DATA POINTS

5900 REM

5950 IF Fl=l THEN GO TO 5970

5960 N=A \ AOUT(#2,A(1,0),N*4,BO,E) \ GO TO 5980

5970 N=B \ AOUT(#2,B(1,0),N*4,BO,E)

5980 BO=BO+1

5990 P=INT(E/4)

6000 PO=PO+P

6500 PRINT 'TOTAL POINTS TRANSFERRED = ';PO

6550 C2=C2/2

6600 M3=((ELEVEL(M(1,1))*LO*10)/AO)

6650 M4=((ELEVEL(MO(1,1))*LO*10)/AO)

6700 M5=((ELEVEL(M1(1,1))*XO*10)/GO)

6750 M6=((ELEVEL(M2(1,1))*XO*10)/GO)

6800 PRINT #1,C2;1,

I;M3;',1;M4;',';M5;',';M6;',';T7;',';PO

6810 PRINT #1,Z1;V;Z2;V;Z3;',';Z4;',';Z5;V;Z6

6811 PRINT #1,A$

6812 PRINT #1,B$

6850 CLOSE #1,2,3

6900 PRINT \ PRINT 'DO YOU WANT TO PRINT THE RESULTS? ';

6905 TXTBOLD \ PRINT '(Y/N)'; \ TXTNORMAL \ INPUT R$

7000 IF R$='N' THEN GO TO 7250

7100 PRINT 'PLACE PRINTER SWITCH TO POSITION "A" '

7150 PRINT 'HIT RETURN TO CONTINUE' \ INPUT R$

7200 GOSUB 7450

7205 IF Z9=l THEN GO TO 7270

7250 PRINT 'DO YOU WANT TO PLOT THE RESULTS (DEFAULT ANS. IS NO THIS

TIME)1;

7252 TXTBOLD \ PRINT '(Y/N)'; \ TXTNORMAL \ INPUT R$

7260 IF R$='Y' THEN Z9=l

7265 IF R$='Y' THEN GO TO 7100

7270 CLRTXWIN \ STOP

7300 REM

7350 REM SUBROUTINE FOR PRINTING RESULTS

7400 REM

7450 OPEN Fl$ FOR INPUT AS FILE 1

7500 OPEN F2$ FOR INPUT AS FILE 2

7550 OPEN 'LP:' FOR OUTPUT AS FILE 3

7600 INPUT #1,LO,XO,GO,AO,RO,R1,SO,S1,S2,S4,TO,CO,T1,L1,L2J2

182
7650 INPUT 4U,C2,M3,M4,M5,M6J7,PO

7755

7760

7765

7700

7750

7800

7810

7811

7812

7850

7855

7860

7900

7950

8000

8050

8100

8150

8200

8250

8300

8350

8400

8450

8500

8550

8600 PRINT #3,'HOLD-TIME/POINT

8610 IF Z9=l THEN GO TO 10100

REM

REM READING STORED DATA

REM

8800 PRINT #3,'TIME (SEC)';TAB(20)'STRESS (PSI)';TAB(40)'STRAIN (IN/IN)1

8805 A(0,0)=0 \ A(0,1)=(A(0,1)*LO)/AO

8810 A(0,2)=(A(0,2)*LO)/GO

8815 PRINT #3,A(0,0);TAB(20)A(0,1);TAB(40)A(0,2)

8850 B0=0 \ N1=PO \ T=0

8900 Nl=Nl-64 \ N2=64 \ IF Nl<=0 THEN N2=Nl+64

8950 AINP(#2,A(1,0),N2*4,BO,P)

9050 IF Nl<:=0 THEN GO TO 9200

9100 BO=BO+1

9200 ELEVAR(A,1,2,10)

9300 FOR J=l TO N2

9350 T=T+TO \ A(J,0)=T

9400 A(J,1)=(A(J,1)*LO)/AO

9450 A(J,2)=(A(J,2)*XO)/GO

9500 PRINT #3,A(J,0);TAB(20)A(J,1);TAB(40)A(J,2)

9550 NEXT J

9552 IF Nl<=0 THEN GO TO 9600

9554 GO TO 8900

INPUT I1,Z1.Z2,Z3,Z4,Z5,Z6

INPUT #1,A$

INPUT #1,B$

PRINT #3

Z$=' FILENAME

PRINT #3, USING Z$,F$

PRINT #3

PRINT #3, 'TIME: ' ;Z1; ' : ' ;Z2; '

PRINT #3

PRINT #3, 'SAMPLE NUMBER

PRINT #3, 'SAMPLE TEMPERATURE

PRINT #3, 'LOAD CALIBRATION

PRINT #3,'EXTENSOMETER CAL.

PRINT #3, 'GAUGE LENGTH

PRINT #3, 'GAUGE AREA

PRINT #3, 'STRAIN RATE

PRINT #3, 'CYCLES COMPLETED

PRINT #3, 'MAXIMUM STRESS

PRINT #3, 'MINIMUM STRESS

PRINT #3,' MAXIMUM STRAIN

PRINT #3, 'MINIMUM STRAIN

PRINT #3, 'POINTS TAKEN

PRINT #3, 'TIME/POINT

PRINT #3, 'HOLD-LOAD RANGE

PRINT #3, 'HOLD- LOAD CAL.

<####

1 ' 6 '

':';Z3

(DEG C)

(LBS/VOLT)

(IN/ VOLT)

(IN)

(IN*IN)

(IN/IN/SEC)

(PSI)

(PSI)

(IN/IN)

(IN/IN)

(SEC)

(LBS/SEC)

(SEC)

;B$

;A$

;LO

;XO

;GO

;AO

;RO

;C2

;M3

;M4

;M5

;M6

;PO

;TO

IL2

;T2

8650

8700

8750

183
9600 PRINT \ PRINT 'DO YOU WANT TO PLOT THE RESULTS? ';

9605 TXTBOLD \ PRINT '(Y/N)1; \ TXTNORMAL \ INPUT R$

9700 IF R$='N' THEN GO TO 9850

9800 GOSUB 10100

9850 CLOSE #1,2,3

9900 RETURN

9950 REM

10000 REM SUBROUTINE TO PLOT DATA

10050 REM

10100 PRINT 'PLACE PRINTER SWITCH IN POSITION "B" '

10150 PRINT 'HIT RETURN TO CONTINUE1; \ INPUT R$ \ PRINT \ PRINT

10200 PRINT 'MAXIMUM STRAIN (IN/IN) ';M5

10250 PRINT 'MIMIMUM STRAIN (IN/IN) ';M6

10300 X$='IN/IN'

10350 PRINT 'ENTER MAXIMUM STRAIN'; \ INPUT X2

10400 PRINT 'ENTER MINIMUM STRAIN'; \ INPUT XI

10450 PRINT 'ENTER STRAIN STEP SIZE'; \ INPUT X3

10500 PRINT 'MAXIMUM STRESS (PSI) ';M3

10550 PRINT 'MINIMUM STRESS (PSI) ';M4

10600 Y$='PSI'

10650 PRINT 'ENTER MAXIMUM STRESS'; \ INPUT Y2

10700 PRINT 'ENTER MINIMUM STRESS'; \ INPUT Yl

10750 PRINT 'ENTER STRESS STEP SIZE1; \ INPUT Y3

10800 PRINT #3 \ PRINT #3 \ PRINT #3

10850 PRINT 'ARE THESE PARAMETERS OK? ';

10860 TXTBOLD \ PRINT '(Y/N)'; \ TXTNORMAL \ INPUT R$

10950 IF R$='N' THEN GO TO 10200

11050 REM

11100 REM GRAPHICS FUNCTIONS

11150 REM

11200 TEKMODE(l.l) \ Z9=l

11250 PHYL(1,10,85,10,85)

11300 SCALE(1,0,X1,X2,Y1,Y2)

11350 AXES(1,0,0)

11400 LABEL(1,X$,Y$,X3,Y3,1)

11450 INVEC

11451 B0=0 \ N1=PO \ T=0

11452 Nl=Nl-64 \ N2=64 \ IF Nl<=0 THEN N2=Nl+64

11453 AINP(#2,A(1,0),N2*4,BO,P)

11454 IF Nl<=0 THEN GO TO 11456

11455 BO=BO+1

11456 ELEVAR(A,1,2,10)

11457 FOR J=l TO N2

11458 T=T+TO \ A(J,0)=T

11459 A(J,1)=(A(J,1)*LO)/AO

11460 A(J,2)=(A(J,2)*XO)/GO

11461 X4=A(J,2)

11462 Y4=A(J,1)

11463 PLOT(1,X4,Y4)

11464 NEXT J

184
11465 IF Nl<=0 THEN GO TO 11750

11466 GO TO 11452

11750 INVEC

11800 HOME

11850 COPY

11900 VTMODE(l.O)

11950 RETURN

11960 REM

11970 REM PARAMETER CHANGE FUNCTIONS

11980 REM

12000 VTMODE \ CLRTXWIN

12050 PRINT 'WHAT IS THE NEW MAXIMUM STRAIN1;

12100 INPUT SO

12150 PRINT 'WHAT IS THE NEW MINIMUM STRAIN';

12200 INPUT S2

12205 PRINT 'DO YOU WANT TO CHANGE THE STRAIN RATE';

12206 TXTBOLD \ PRINT '(Y/N)'; \ TXTNORMAL \ INPUT R$ .

12210 IF R$='N' THEN GO TO 12230

12220 PRINT 'WHAT IS THE NEW STRAIN RATE (IN/IN/SEC)'; \ INPUT RO

12225 R1=(RO*GO)/(XO*10)

12230 PRINT 'DO YOU WANT TO CHANGE THE NUMBER OF TOTAL CYCLES';

12232 TXTBOLD \ PRINT '(Y/N)1; \ TXTNORMAL \ INPUT R$

12235 IF R$='N' THEN GO TO 12250

12245'PRINT 'WHAT is THE NEW TOTAL NUMBER OF CYCLES'; \ INPUT co \

C1=CO*2

12250 S1=(SO*GO)/(XO*10)

12300 S3=(S2*GO)/(XO*10)

12350 FGREMOVE(l.l)

12400 FGTOTALSEG(1,C2)

12450 C1=C1-C2

12500 FGREPT(1,'SINE', RATE R1.C1.S1.S3)

12505 IF G=0 THEN GO TO 12550

12510 TEKMODE(l.l)

12515 PHYL(1,10,85,10,85)

12520 SCALE(1,0,X1,X2,Y1,Y2)

12525 AXES(l.O.O)

12530 LABEL(1,X$,Y$,X3,Y3,1)

12535 INVEC

12550 RETURN

13000 REM SET FLAGS TO HALT DATA ACQUISTION

13010 Al=l

13020 ADREMOVE(l)

13500 RETURN

1 REM CREEP/STRESS DROP - LOAD CONTROL

8 CLRTXWIN

9 U=SYS(7)

10 REM WRITTEN BY GEORGE JAMES AND PAUL SEAN HILL - JULY 1985

20 PRINT 'THIS PROGRAM PERFORMS A CREEP TEST AND IS LOADED AT A'

185
24 PRINT

28 PRINT

30 PRINT

40 PRINT

'CONSTANT LOAD RATE.1

'THE PROGRAM ALLOWS A SPECIFIED STRESS INCREMENT TO BE1

'INSTANTANEOUSLY REMOVED AND THEN REPLACED ON COMMAND.'

45 PRINT 'AFTER A PERIOD OF CREEP, THE LOAD CAN BE RAMPED UP AGAIN,'

46 PRINT 'OR THE TEST CAN BE ENDED BY KEYBOARD INTERUPTS.'

47 PRINT

70 PRINT

BINARY'

100 PRINT 'FORM. STRESS DROP DATA IS ALSO STORED ON 5.25" DISC.' \

PRINT

110 PRINT

THE'

120 PRINT

125 PRINT

126 PRINT

'THE TEST DATA IS STORED ON 5.25" FLEXIBLE DISC ON DU1: IN

'THE CAPABILITY TO PRINT AND PLOT THE DATA IS PROVIDED FOR

CURRENT TEST OR FOR EXISTING FILES.'

\ PRINT

\ PRINT

130 DIM A(64,3),B(64,3),M(2,2),MO(2,2),M1(2,2),M2(2,2)

140 REM

150 REM PRINT ONLY INPUT

160 REM

170 PRINT

180 PRINT 'TYPE1; \ TXTBOLD \ PRINT ' TEST1;

182 TXTNORMAL \ PRINT ' IF YOU WANT TO RUN A NEW TEST1

184 PRINT \ PRINT ' OR1

186 PRINT

190 PRINT 'TYPE'; \ TXTBOLD \ PRINT ' PRINT'; \ TXTNORMAL

195 PRINT ' IF YOU WANT TO PRINT EXISTING FILES'

196 PRINT

200 INPUT R$

210 IF R$='TEST' THEN GO TO 330

220 IF R$='PRINT1 THEN GO TO 240

225 TXTINVERS

230 PRINT 'INVALID ENTRY, TYPE TEST OR PRINT

240 PRINT 'ENTER FILE NAME (<=5 CHARACTERS)'

250 IF LEN(F$)<=5 THEN GO TO 270

255 TXTINVERS

260 PRINT 'INVALID FILE NAME

270 F1$='DU1:'+F$+'C.DAT'

280 F2$='DU1:'+F$+'.DAT'

285 P0=0

290 GO TO 6900

300 REM

310 REM TEST INPUT

320 REM

330 GO TO 350

340 CLOSE #1,2,3

350 PRINT

360 PRINT 'ENTER FILE NAME (<=5 CHARACTERS)1; \ INPUT F$

370 IF LEN(F$)<=5 THEN GO TO 390

375 TXTINVERS

\ TXTNORMAL \ GO TO 170

\ INPUT F$

\ TXTNORMAL \ GO TO 240

186
380 PRINT 'INVALID FILE NAME \ TXTNORMAL \ GO TO 350

390 Fl$='OUl:'+F$-4-lC.DAT1

400 F2$='DU1:'+F$+'.DAT'

410 OPEN Fl$ FOR OUTPUT AS FILE 1,

420 OPEN F2$ FOR OUTPUT AS FILE 2,

430 OPEN 'LP:' FOR OUTPUT AS FILE 3

431 PRINT 'ENTER THE SAMPLE NUMBER1; \ INPUT B$

ENTER THE SAMPLE TEMPERATURE IN DEGREES CELSIUS1

ENTER LOAD CALIBRATION (LBS/VOLT)1; \ INPUT LO

FILESIZE 1

FILESIZE 30

435 PRINT

440 PRINT

460 PRINT

470 PRINT

480 PRINT

490 PRINT

\ INPUT A$

ENTER EXTENSOMETER CALIBRATION (IN/VOLT) \ INPUT XO

1

ENTER GAUGE LENGTH (IN)': \ INPUT GO

ENTER GAUGE AREA (IN*IN)'; \ INPUT AO

ENTER LOAD RATE (PSI/SEC)1; \ INPUT RO

500 R1=(RO*LO)/(XO*10)*AO

510 PRINT 'BEGIN CREEP AT WHAT STRESS (PSI)'; \ INPUT SO

520 S1=(SO*LO)/(XO*10)*AO

525 T1=S1/R1

550 PRINT 'ENTER TIME/POINT FOR DATA ACQUISITION (SEC)1; \ INPUT TO

560 T3=T1/TO

620 PRINT 'ENTER LOAD RANGE DURING STRESS DROP (1,2,3, OR 4)

LI

625 PRINT 'ENTER LOAD CALIBRATION FOR STRESS DROP (LBS/VOLT)

L2

630 PRINT 'ENTER TIME/POINT DURING STRESS DROP (MIN)'; \ INPUT T2

632 CLRTXWIN

633 PRINT 'SAMPLE NUMBER

635 PRINT 'SAMPLE TEMPERATURE

636 PRINT 'LOAD CALIBRATION

640 PRINT 'EXTENSOMETER CAL.

650 PRINT 'GAUGE LENGTH

660 PRINT 'GAUGE AREA

680 PRINT 'LOAD RATE

690 PRINT 'INIT. CREEP STRESS

710 PRINT 'TIME/POINT

715 PRINT 'TOT. POINTS IN RAMP UP

735 PRINT 'STRESS DROP RANGE

740 PRINT 'STR DROP LOAD CAL

745 PRINT 'DROP-TIME/POINT

750 PRINT \ PRINT 'ARE ALL PARAME

752 TXTBOLD \ PRINT '(Y/N)'; \ TXTNORMAL \ INPUT R$

760 IF R$='N' THEN GO TO 340

780 PRINT #1,LO;',';XO;',';GO;',';AO;',';RO;',';R1;',';SO;',';S1

790 PRINT #1,TO;V;L1;V;L2;V;T2;',';T3

800 REM

810 REM DATA ACQUISITION FUNCTIONS

820 REM

830 CLINIT

880 CKSTOP

890 CKTIME(1,.01,K)

900 I1=INT(TO/K)

\ INPUT

\ INPUT

(DEG C)

(LBS/VOLT)

(IN/VOLT)

(IN)

(IN*IN)

(PSI/SEC)

(PSI)

(SEC)

(SEC)

(LBS/VOLT)

(MIN)

TERS OK ';

;B$

;A$

;LO

;XO

;GO

;AO

;RO

;SO

;TO

;T3

;L2

;T2

187
910 ADTIMED(1,A.B,1,I1)

920 ADSLAVE(1,2,6)

940 ADRMAX(3,M,1)

950 ADRMIN(3,MO,1)

960 ADRMAX(3,M1,2)

970 ADRMIN(3,M2,2)

980 REM

990 REM REAL TIME PLOT INPUT

1000 REM

1010 PRINT \ PRINT 'DO YOU WANT A REAL TIME PLOT?1; \ TXTBOLD

1015 PRINT ' (Y/N)1; \ TXTNORMAL \ INPUT R$

1030 IF R$='N' THEN GO TO 1060

1050 G=l \ GO TO 1070

1060 G=0 \ GO TO 1209

1070 X$='IN/IN1

1080 PRINT \ PRINT 'REAL TIME PLOT PARAMETERS' \ PRINT

1090 PRINT 'ENTER MAXIMUM STRAIN'; \ INPUT X2

1100 PRINT 'ENTER MINIMUM STRAIN1; \ INPUT XI

1110 PRINT 'ENTER STRAIN STEP SIZE'; \ INPUT X3

1120 Y$='PSI' \ PRINT

1130 PRINT 'ENTER MAXIMUM STRESS1; \ INPUT Y2

1140 PRINT 'ENTER MINIMUM STRESS'; \ INPUT Yl

1150 PRINT 'ENTER STRESS STEP SIZE'; \ INPUT Y3

1160 PRINT 'ARE THESE PARAMETERS OK?1; \ TXTBOLD \ PRINT '(Y/N)';

1165 TXTNORMAL \ INPUT R$

1180 IF R$='N' THEN GO TO 1090

1200 REM

1203 REM TEST FUNCTIONS

1206 REM

1209 CLRTXWIN

1212 TXTNORMAL \ PRINT \ TXTINVERS \ TXTSPOS(1,5,10)

1215 PRINT 'KEY INTERUPT SUMMARY' \ TXTNORMAL

1218 TXTSPOS( 1,8,1) \ PRINT '<ESOE - END TEST'

1221 PRINT '<ESOD - DROP STRESS INCREMENT FOR STRESS DROP TEST1

1222 PRINT '<ESC>S - STEP UP TO ORIGINAL LOAD.'

1224 PRINT '<ESC>R - RAMP UP TO ANOTHER STRESS LEVEL.'

1227 PRINT '<ESOE - END TEST, RAMP TO ZERO LOAD.'

1228 PRINT '<ESC>A - HALT DATA ACQUISITION AND CONTINUE TEST1

1229 ETIME(TS) \ SLEEP(T3+5)

1230 PRINT \ TXTFLASH \ PRINT \ PRINT \ PRINT

1232 PRINT 'BRING UP THE HYDRAULIC PRESSURE '

1233 PRINT 'ZERO LOAD WITH SET POINT CONTROL AND STRAIN WITH ZERO

ADJUST1

1234 ETIME(T3) \ SLEEP(T3+3) \ PRINT \ PRINT 'PLACE PRINTER ON SWITCH

"A" '

1235 PRINT \ PRINT

1240 ETIME(T3) \ SLEEP(T3+3)

1245 KBINT('E',3, LINE 4500 )

1248 KBINT('D',4, LINE 1266 )

1251 KBINT('R',4, LINE 1270 )

188
1255 KBINT('A',3, LINE 13000 )

1256 KBINT('S',4, LINE 1269 )

1257 KBENB

1260 61=0

1263 GO TO 1272

1266 61=1 \ RETURN

1269 Gl=2 \ RETURN

1270 G1=3\RETURN

1272 ADINIT

1275 CLRAN6ESENSE(1,2,X5)

1300 IF X5=4 THEN X5=10

1310 IF X5=3 THEN X5=5

1341 PRINT \ PRINT

1345 PRINT \ PRINT 'HIT RETURN TO START TEST.1 \ INPUT R$ \ TXTNORMAL

1346 GTIME(Z3,Z2,Z1) \ GDATE(Z6,Z5,Z4)

1350 IF G=0 THEN GO TO 1420

1360 TEKMODE(l.l)

1370 PHYL(1,10,85,10,85)

1380 SCALE(1,0,X1,X2,Y1,Y2)

1390 AXES(1,0,0)

1400 LABEL(1,X$,Y$,X3,Y3,1)

1410 INVEC

1420 REM

1425 T6=0

1430 ADGO \ ETIME

1500 FGGO

1505 B0=0 \ Fl=0

1550 FGRAMP(l.Tl.Sl)

1600 REM

1605 IF G=0 THEN GO TO 1640

1610 ADIMMED(2,X4)

1615 ADIMMED(1,Y4)

1620 X4=(X4*((10*XO)/GO))

1625 Y4=(Y4*((10*LO)/AO))

1630 PLOT(1,X4,Y4)

1640 CLHYDSENSE(1,H2)

1645 IF H2=0 THEN GO TO 5100

1650 IF 61=1 THEN GO TO 2200

1652 IF Gl=3 THEN GOTO 3700

1655 IF A=64 THEN B=0

1660 IF A<64 THEN GO TO 1600 \ A=-l

1665 Fl=l

1670 AOUT(#2,A(1,0),64*4,BO,E)

1675 P=INT(E/4) \ PO=PO+P

1680 BO=BO+1

1685 REM

1690 IF G=0 THEN GO TO 1710

1695 ADIMMED(2,X4) \ ADIMMED(1,Y4)

1700 X4=(X4*((10*XO)/GO)) \ Y4=(Y4*((10*LO)/AO))

1705 PLOT(1,X4,Y4)

189
1715 CLHYDSENSE(1,H2)

1720 IF H2=0 THEN GO TO 5100

1725 IF Gl=l THEN GO TO 2200

1727 IF Gl=3 THEN GOTO 3700

1730 IF B=64 THEN A=0

1735 IF B<64 THEN GO TO 1685 \ B=-l

1740 Fl=0

1745 AOUT(#2,B(1,0),64*4,BO,E)

1750 P=INT(E/4) \ PO=PO+P

1755 BO=BO+1

1760 GO TO 1600

2100 REM

2150 REM DROP FUNCTIONS

2200 REM

2300 IF G=0 THEN GO TO 2450

2350 HOME

2400 VTMODE(l.O)

2450 ADIMMED(2,X5)

2500 ADIMMED(1,L6)

2525 ADHOLD(l)

2550 PRINT 'HOW MUCH STRESS DO YOU WANT TO DROP1; \ INPUT DO

2555 D1=(SO-DO)*LO*AO/XO/10

2560 FGSTEP(l.Dl)

2850 ETIME(T3)

3300 ADIMMED(1,L5) \ ETIME(T4)

3350 SLEEP(T4+(T2*60))

3400 L5=((L5*L2*10)/AO)

3450 ETIME(T4) \ T4=((T4-T3)/60)

3560 CLHYDSENSE(1,H2)

3570 IF H2=0 THEN GO TO 5100

3600 IF 61=2 THEN GO TO 12000

3650 GO TO 3300

3700 REM

3750 REM RAMP UP FUNCTIONS

3800 REM

3850 PRINT

3855 PRINT 'ENTER LOAD RATE (PSI/SEC)1; \ INPUT RO

3860 R1=(RO*LO)/(XO*10)*AO

3865 PRINT 'BEGIN NEXT CREEP AT WHAT STRESS (PSI)1; \ INPUT SO

3870 S1=(SO*LO)/(XO*10)*AO

3871 ADIMMED(1,S5)

3875 T1=(S1-S5)/R1

3899 CLRANGESWITCH(1,1,L4,L3)

4000 IF G=0 THEN GO TO 4400

4050 TEKMODE(l.l)

4100 PHYL(1,10,85,10,85)

4150 SCALE(1,0,X1,X2,Y1,Y2)

4200 AXES(1,0,0)

4250 LABEL(1,X$,Y$,X3,Y3,1)

4300 INVEC

190
4310 FGRAMP(1,T1,S1)

4350 Gl=0

4400 IF Fl=0 THEN GO TO 1655

4410 GO TO 1730

4450

4500 REM END OF TEST FUNCTIONS

4550 -—

4700 FGGO

4750 FGRAMP(1,5,0)

4800 IF G=l THEN GO TO 4900

4850 PRINT 'TEST OVER. RAMPING DOWN...1

4900 FGSTATUS(l.Hl)

4950 IF Hl=0 THEN GO TO 5350

5000 GO TO 4900

5100 REM

5250 IF G=l THEN GO TO 5400

5300 PRINT 'HYDRAULIC SHUT-DOWN DETECTED'

5400 FGSTOP \ ADSTOP

5450 ETIME(T7) \ T7=T7-T6

5500 IF 6=0 THEN GO TO 5700

5550 INVEC

5600 HOME

5650 VTMODE(l.O)

5700 PRINT 'END OF TEST' \ PRINT

5750 CLRANGESWITCH(1,2,,0)

5800 REM

5850 REM STORING DATA POINTS

5900 REM

5950 IF Fl=l THEN GO TO 5970

5960 N=A \ AOUT(#2,A(1,0),N*4,BO,E) \ GO TO 5980

5970 N=B \ AOUT(#2,B(1,0),N*4,BO,E)

5980 BO=BO+1

5990 P=INT(E/4)

6000 PO=PO+P

6500 PRINT 'TOTAL POINTS TRANSFERRED = ' ;PO

6550 C2=C2/2

6600 M3=((ELEVEL(M(1,1))*LO*10)/AO)

6650 M4=((ELEVEL(MO(1,1))*LO*10)/AO)

6700 M5=((ELEVEL(M1(1,1))*XO*10)/GO)

6750 M6=((ELEVEL(M2(1,1))*XO*10)/GO)

6800 PRINT #1,C2;I,';M3;',';M4;',I;M5;',I;M6;I,';T7;',';PO

6810 PRINT #1,Z1;',';Z2;V;Z3;V;Z4;',';Z5;',';Z6

6811 PRINT ll.AS

6812 PRINT #1,B$

6850 CLOSE #1,2,3

6900 PRINT \ PRINT 'DO YOU WANT TO PRINT THE RESULTS? ';

6905 TXTBOLD \ PRINT '(Y/N)'; \ TXTNORMAL \ INPUT R$

7000 IF R$='N' THEN GO TO 7250

7100 PRINT 'PLACE PRINTER SWITCH TO POSITION "A" '

7150 PRINT 'HIT RETURN TO CONTINUE1 \ INPUT R$

191
7200 GOSUB 7450

7205 IF Z9=l THEN GO TO 7270

7250 PRINT 'DO YOU WANT TO PLOT THE RESULTS (DEFAULT ANS. IS NO THIS

TIME)1;

7252 TXTBOLD \ PRINT '(Y/N)'; \ TXTNORMAL \ INPUT R$

7260 IF R$='Y' THEN Z9=l

7265 IF R$='Y' THEN GO TO 7100

7270 CLRTXWIN \ STOP

/ ouu

7350

7450

7500

7550

7600

7650

7655

7660

7665

7700

7750

7800

7810

7811

7812

7850

7855

7860

7900

7950

8000

8050

8100

8150

8200

8250

8300

8350

8400

8450

8500

8550

8600

8610

Rfi^ OU9Un

8700

ft7«;n

REM SUBROUTINE FOR PRINTING RESULTS

OPEN

OPEN

OPEN

INPUT

INPUT

INPUT

INPUT

INPUT

PRINT

Fl$

F2$

•LP

#1

#1

#1

#1

#1

13

*

,

,

,

9

9

FOR INPUT AS FILE 1

FOR INPUT AS FILE 2

' FOR OUTPUT AS FILE

LO,XO,GO,AO,RO,R1,SO

C2,M3,M4,M5,M6,T7,PO

Z1,Z2,Z3,Z4,Z5,Z6

A$

B$

Z$=' FILENAME

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

#3

#3

#3

#3

#3

#3

#3

#3

#3

#3

#3

#3

#3

#3

#3

#3

#3

#3

#3

#3

#3

#3

f

9

,

,

,

9

9

9

9

9

9

9

9

1

9

,

,

9

9

9

USING Z$,F$

•DATE: ';Z5;'-';Z4;'

'TIME: ';Z1;':';Z2;'

•SAMPLE NUMBER

•SAMPLE TEMPERATURE

•LOAD CALIBRATION

'EXTENSOMETER CAL.

•GAUGE LENGTH

•GAUGE AREA

•STRAIN RATE

•CYCLES COMPLETED

•MAXIMUM STRESS

'MINIMUM STRESS

•MAXIMUM STRAIN

•MINIMUM STRAIN

•POINTS TAKEN

'TIME/POINT

'HOLD- LOAD RANGE

'HOLD- LOAD CAL.

'HOLD-TIME/POINT

IF Z9=l THEN GO TO 10100

REM

3

,S1,S2,S4,TO,CO,T1

<####'

-' ;Z6;'

:';Z3

(DEG C)

(LBS/VOLT)

(IN/ VOLT)

(IN)

(IN*IN)

(IN/IN/SEC)

(PSI)

(PSI)

(IN/IN)

(IN/IN)

(SEC)

(LBS/SEC)

;B$

;A$

;LO

;XO

;GO

;AO

;RO

;C2

;M3

;M4

;M5

;M6

;PO

;TO

;L1

;L2

(SEC) ;T2

REM READING STORED DATA

RFM -

L1,L2,T2

8800 PRINT #3,'TIME.(SEC)';TAB(20)'STRESS (PSI)1;TAB(40)'STRAIN (IN/IN)'

8805 A(0,0)=0 \ A(0,1)=(A(0,1)*LO)/AO

8810 A(0,2)=(A(0,2)*LO)/GO

192
8815 PRINT #3,A(0,0);TAB(20)A(0,1);TAB(40)A(0,2)

8850 60=0 \ N1=PO \ T=0

8900 Nl=Nl-64 \ N2=64 \ IF Nl<=0 THEN N2=Nl+64

8950 AINP(l2,A(lfO),N2*4,BO,P)

9050 IF Nl<=0 THEN GO TO 9200

9100 BO=BO+1

9200 ELEVAR(A,1,2,10)

9300 FOR J=l TO N2

9350 T=T+TO \ A(J,0)=T

9400 A(J,1)=(A(J,1)*LO)/AO

9450 A(J,2)=(A(J,2)*XO)/GO

9500 PRINT #3,A(J,0);TAB(20)A(,J,1);TAB(40)A(J,2)

9550 NEXT J

9552 IF Nl<=0 THEN GO TO 9600

9554 GO TO 8900

9600 PRINT \ PRINT 'DO YOU WANT TO PLOT THE RESULTS? ';

9605 TXTBOLD \ PRINT '(Y/N)1; \ TXTNORMAL \ INPUT R$

9700 IF R$='N' THEN GO TO 9850

9800 GOSUB 10100

9850 CLOSE #1,2,3

9900 RETURN

9950 REM

10000 REM SUBROUTINE TO PLOT DATA

10050 REM

10100 PRINT 'PLACE PRINTER SWITCH IN POSITION "B" '

10150 PRINT 'HIT RETURN TO CONTINUE1; \ INPUT R$ \ PRINT \ PRINT

10200 PRINT 'MAXIMUM STRAIN (IN/IN) ';M5

10250 PRINT 'MIMIMUM STRAIN (IN/IN) ';M6

10300 X$='IN/IN'

10350 PRINT 'ENTER MAXIMUM STRAIN1; \ INPUT X2

10400 PRINT 'ENTER MINIMUM STRAIN'; \ INPUT XI

10450 PRINT 'ENTER STRAIN STEP SIZE'; \ INPUT X3

10500 PRINT 'MAXIMUM STRESS (PSI) ';M3

10550 PRINT 'MINIMUM STRESS (PSI) ';M4

10600 Y$='PSI'

10650 PRINT 'ENTER MAXIMUM STRESS'; \ INPUT Y2

10700 PRINT 'ENTER MINIMUM STRESS'; \ INPUT Yl

10750 PRINT 'ENTER STRESS STEP SIZE1; \ INPUT Y3

10800 PRINT #3 \ PRINT #3 \ PRINT #3

10850 PRINT 'ARE THESE PARAMETERS OK? ';

10860 TXTBOLD \ PRINT '(Y/N)1; \ TXTNORMAL \ INPUT R$

10950 IF R$='N' THEN GO TO 10200

11050 REM

11100 REM GRAPHICS FUNCTIONS

11150 REM

11200 TEKMODE(l.l) \ Z9=l

11250 PHYL(1,10,85,10,85)

11300 SCALE(1,0,X1,X2,Y1,Y2)

11350 AXES(1,0,0)

11400 LABEL(1,X$,Y$,X3,Y3,1)

193
11450 INVEC

11451 80=0 \ N1=PO \ T=0

11452 Nl=Nl-64 \ N2=64 \ IF Nl<=0 THEN N2=Nl+64

11453 AINP(#2,A(1,0),N2*4,BO,P)

11454 IF Nl<=0 THEN GO TO 11456

11455 BO=BO+1

11456 ELEVAR(A,1,2,10)

11457 FOR J=l TO N2

11458 T=T+TO \ A(J,0)=T

11459 A(J,1)=(A(J,1)*LO)/AO

11460 A(J,2)=(A(J,2)*XO)/GO

11461 X4=A(J,2)

11462 Y4=A(J,1)

11463 PLOT(1,X4,Y4)

11464 NEXT J

11465 IF Nl<=0 THEN GO TO 11750

11466 GO TO 11452

11750 INVEC

11800 HOME

11850 COPY

11900 VTMODE(l.O)

11950 RETURN

12000 REM

12010 REM STEP BACK UP AFTER A STRESS DROP

12020 REM

12030 FGSTEP(1,S1)

12040 IF G=0 THEN GO TO 4400

12050 TEKMODE(l.l)

12060 PHYL(1,10,85,10,85)

12070 SCALE(1,0,X1,X2,Y1,Y2)

12080 AXES(1,0,0)

12090 LABEL(1,X$,Y$,X3,Y3,1)

12100 INVEC

12110 ADRESUME(l)

12550 IF Fl=0 THEN GOTO 1655

12560 GOTO 1730

12900 REM

13000 REM SET FLAGS TO HALT DATA ACQUISTION

13001 REM

13020 ADREMOVE(1)

13500 RETURN

BACK2

1 REM THIS PROGRAM FITS A LINEAR CURVE TO DATA TAKEN FROM

2 REM HOLD TIME FROM DATA STORED IN BINARY FORM STORED

3 REM ON DISK DU1

194
4 REM WRITTEN BY GEORGE JAMES & EVA SMITH, AUGUST 3, 1984

20 DIM Cl(321,3),0(32,3),01(50,6),02(2,2),03(2.2),04(2,2).05(2,2)

25 DIM N(321,3),X(321),Y(321),C(10),A(10,11),X1(321).8(10,11),Al(32,2)

30 PRINT 'ENTER FILE NAME (<=5 CHARACTERS)1; \ INPUT F$

40 IF LEN(F$)<=5 THEN GO TO 70

50 TXTINVERS

60 PRINT 'INVALID FILE NAME-' \ TXTNORMAL \ GO TO 30

70 F1$='DU1:'+F$+'C.DAT'

80 F2$='DU1:'+F$+'.DAT'

90 Kl=l

100 REM

110 REM SUBROUTINE TO PRINT RELAXATION DATA

120 REM

125 PRINT \ PRINT 'SWITCH PRINTER TO "A" AND HIT RETURN';

126 INPUT R$

127 PRINT

130 OPEN 'LP:1 FOR OUTPUT AS FILE 3

140 F4$='DU1:'+F$+'S.DAT'

150 OPEN F4$ FOR INPUT AS FILE #4

160 INPUT #4,DO,T2,L2,AO,XO,GO

170 PRINT #3,'HOLD #';TAB(30)'STRESS(PSI)';TAB(60)'# OF POINTS'

180 FOR 1=1 TO DO

190 INPUT 14,01(1,1),D1(I,2),D1(I,3),D1(I,4),D1(I,5),D1(I,6)

200 PRINT #3,I;TAB(30)D1(I,1);TAB(60)D1(I,2)

210 NEXT I

220 CLOSE #4

230 PRINT 'INPUT # OF HOLD TIME DESIRED'; \ INPUT DO

240 P4=D1(DO,2)

250 F5$='DU1:'+F$+'S.D'+STR$(DO)

260 OPEN F5$ FOR INPUT AS FILE 5

270 81=0 \ N1=D1(DO,2) \ T=0 \ N3=l

280 Nl=Nl-32 \ N2=32 \ IF Nl<=0 THEN N2=Nl+32

290 FINP(#5,C1(N3,0),N2*4,B1,P3)

300 B1=B1+1 \ N3=N3+32

320 IF Nl<=0 THEN GO TO 335

330 GO TO 280

335 ELEVAR(C1,1,2,10)

340 CLOSE #5

342 PRINT ,'INPUT LENGTH OF TIME DESIRED FOR CURVE FIT (SEC)';

344 INPUT T4

346 P1=T4/T2

348 PRINT \ PRINT 'NUMBER OF POINTS USED =';P1

360 REM CALCULATING E

370 REM

380 REM FORMING A MATRIX

390 REM

395 T=-T2

400 FOR J=l TO PI

405 T=T+T2

410 X(J)=T

195
420 Y(J)=(C1(J,2)*XO)/GO

430 X1(J)=1

440 NEXT J

450 FOR J=l TO 2

460 A(J,1)=0

470 A(J,3)=0

480 FOR K=l TO PI

490 A(J,1)=A(J,1)+X1(K)

500 A(J,3)=A(J,3)+Y(K)*X1(K)

510 X1(K)=X1(K)*X(K)

520 NEXT K

530 NEXT J

540 A(2,2)=0

550 FOR J=l TO PI

560 A(2,2)=A(2,2)+X1(J)

570 X1(J)=X1(J)*X(J)

580 NEXT J

590 A(1,2)=A(2,1)

600 PRINT #3,'A MATRIX' \ PRINT #3

610 PRINT #3,A(1,1);A(1,2);A(1,3)

620 PRINT #3,A(2,1);A(2,2);A(2,3)

630 PRINT #3

640 REM

650 REM LU DECOMPOSITION

660 REM

670 FOR 1=1 TO 2

680 FOR J=2 TO 2

690 S5=0

700 IF J>I THEN 60 TO 770

710 J1=J-1

720 FOR K=l TO Jl

730 S5=S5+A(I,K)*A(K,J)

740 NEXT K

750 A(I,J)=A(I,J)-S5

760 GO TO 870

770 11=1-1

780 IF 1=0 THEN GO TO 820

790 FOR K=l TO II

800 S5=S5+A(I,K)*A(K,J)

810 NEXT K

820 IF ABS(A(I,!))<!.OOOOOE-10 THEN GO TO 840

830 GO TO 860

840 PRINT 'SMALL VALUE ON DIAGONAL DETECTED'

850 STOP

860 A(I,J)=(A(I,J)-S5)/A(I,I)

870 NEXT J

880 NEXT I

890 PRINT #3 \ PRINT #3,'LU DECOMPOSITION OF A MATRIX'

900 PRINT #3

910 PRINT #3,A(1,1);A(1,2);A(1,3)

196
920 PRINT #3,A(2,1);A(2,2);A(2,3)

930 PRINT 13

940 REM

950 REM SOLVING MATRIX EQUATIONS

960 REM

970 C(1)=A(1,3)/A(1,1)

980 C(2HA(2,3)-A(2,1)*C(1))/A(2,2)

990 C(1K(1)-A(1,2)*C(2)

1000 PRINT #3

1010 PRINT #3,'COEFFFICIENTS OF LEAST SQUARES FIT1

1020 PRINT #3,C(1);C(2)

1030 PRINT #3

1040 Bl=0

1050 FOR 1=1 TO PI

1060 S5=C(2)*X(I)+C(1)

1070 B1=B1+(Y(I)-S5)*2

1080 NEXT I

1090 B1=B1/(P1-1)

1100 PRINT #3 \ PRINT #3,'E=';C(2)

1110 PRINT #3,'BETA=';B1

1120 PRINT #3,'T2=';T2

1130 PRINT #3,'FILENAME^ ;F$

1140 PRINT #3,'HOLD NUMBER=';DO

1142 PRINT

#3 " ^^**VH*^**-iHt**;IH***1Ht**^*Ht-iHt^ I

1145 PRINT #3 \ PRINT #3 \ PRINT #3 \ PRINT #3

1150 Q1=0 \ Q2=P1*T2

1160 Q3=Q2/10

1170 Q$='TIME(SEC)'

1180 Y1=((C1(1,2)*XO)/GO)-1.00000E-04 \ Y2=Y1+2.00000E-04

1190 Y3=(Y2-Y1)/10

1200 Y$='IN/IN'

1420 REM

1430 REM GRAPHICS FUNCTIONS

1440 REM

1445 PRINT \ PRINT 'SWITCH PRINTER TO "B" & HIT RETURN1

1446 PRINT 'WHEN PROMPTED HIT RETURN AGAIN'; \ INPUT R$

1450 TEKMODE(l.l)

1460 PHYL(1,10,85,10,85)

1470 SCALE(1,0,Q1,Q2,Y1,Y2)

1480 AXES(1,0,0)

1490 LABEL(1,Q$,Y$,Q3,Y3,1)

1505 T=-T2

1510 FOR 1=1 TO PI

1515 T=T+T2

1520 Q4=T

1530 Y4=(C1(I,2)*XO)/GO

1550 MARK(1,4,Q4,Y4)

1570 NEXT I

1575 INVEC

197
1580 T=-T2

1590 FOR 1=1 TO PI

1600 T=T+T2

1610 Q4=T

1620. W4=(T*C(2))+C(1)

1630 PLOT(1,Q4,W4)

1640 NEXT I

1650 INVEC

1660 HOME

1665 INPUT R$

1670 COPY

1680 VTMODE(l.O)

1685 A1(K1,2)=C(2) \ A1(K1,1)=DO \ K1=K1+1

1690 PRINT \ PRINT 'DO YOU WISH TO RUN ANOTHER HOLD TIME (Y/N)1;

1700 INPUT R$

1710 IF R$='N' THEN 60 TO 1750

1720 PRINT \ PRINT 'SWITCH PRINTER TO "A" & HIT RETURN1; \ INPUT R$

1730 PRINT

1740 GO TO 230

1750 PRINT \ PRINT '00 YOU WANT TO FIT THE RATE VALUES (Y/N)1; \ INPUT

R$

1760 IF R$='N' THEN GO TO 12008

1770 FOR 1=1 TO Kl-1

1775 PRINT

1780 PRINT 'FOR HOLD TIME #';A1(I,1);' DO YOU WANT TO USE THIS RATE

(Y/N)';

1790 PRINT A1(I,2) \ INPUT R$

1800 IF R$='N' THEN GO TO 1830

1810 PRINT \ PRINT 'INPUT THE ASSOCIATED STRESS VALUE' \ INPUT A1(I,0)

1820 GO TO 1840 \ PRINT

1830 A1(I,0)=0

1840 NEXT I

1850 PRINT \ PRINT 'SWITCH PRINTER TO "A" AND HIT RETURN1; \ INPUT R$

2652 REM RATE CURVE FIT

2654 REM —

2655 REM FORMING A MATRIX

2656 REM

2658 J=0 \ PRINT \ PRINT #3

2660 FOR Jl=l TO Kl-1

2665 IF A1(J,0)=0 THEN GO TO 2700 \ J=J+1

2670 X(J)=A1(J1,0)

2680 Y(0)=A1(J1,2)*1-OOOOOE+06

2690 X1(J)=1

2692 PRINT 'HOLD TIME=';A1(J1.1) \ PRINT #3,'HOLD TIME=';A1(J1,1)

2694 PRINT 'STRAIN RATE=';A1(J1,2) \ PRINT #3,'STRAIN RATE=';A1(J1,2)

2696 PRINT 'STRESS=';A1(J1,0) \ PRINT #3,'STRESS=';A1(J1,0)

2698 PRINT \ PRINT #3

2700 NEXT Jl

2702 P2=J

2704 PRINT 'ARE THESE VALUES OK (Y/N)1; \ INPUT R$

198
2706 IF R$='N' THEN GO TO 1775

2708 PRINT

2710 FOR J=l TO 9

2720 A(J,1)=0

2730 A(0,ll)=0

2740 FOR K=l TO P2

2750 A(0,1)=A(0,1)+X1(K)

2760 A(J,11)=A(J,11)+Y(K)*X1(K)

2770 X1(K)=X1(K)*X(K)

2780 NEXT K

2790 NEXT J

2795 FOR 1=2 TO 10

2800 A(10,I)=0

2810 FOR J=l TO P2

2820 A(10,I)=A(10,I)+X1(0)

2830 X1(0)=X1(0)*X(0)

2840 NEXT 0

2845 NEXT I

2847 FOR 0=2 TO 10

2849 FOR 1=1 TO 9

2850 A(I,0)=A(I+1,0-1)

2855 PRINT #3,'A(';!;',';0;')= ';A(I,J)

2857 NEXT I

2859 NEXT J

2860 PRINT #3

2900 REM

2910 REM LU DECOMPOSITION

2920 REM

2930 FOR 1=1 TO 10

2940 FOR J=2 TO 10

2950 S5=0

2960 IF 0>I THEN GO TO 3006 .

2970 J1=J-1

2980 FOR K=l TO 01

2990 S5=S5+A(I,K)*A(K,0)

3000 NEXT K

3002 A(I,0)=A(I,0)-S5

3004 GO TO 3060

3006 11=1-1

3008 IF 1=0 THEN GO TO 3016

3010 FOR K=l TO II

3012 S5=S5+A(I,K)*A(K,0)

3014 NEXT K

3016 IF ABS(A(I,I))<1.00000E-10 THEN GO TO 3030

3020 GO TO 3050

3030 PRINT 'SMALL VALUE ON DIAGONAL DETECTED FOR REDUCTION I1;I

3040 STOP

3050 A(I,0)=(A(I,0)-S5)/A(I,I)

3060 NEXT 0

3070 NEXT I

199
3130 REM

3140 REM SOLVING MATRIX EQUATIONS

3150 REM

3155 FOR 1=2 TO 10

3160 FOR J=l TO I

3165 C(J)=A(J,11)

3170 NEXT J

3175 GOSUB 4000

3180 11=1-1

3190 PRINT #3

3200 PRINT #3,'COEFFFICIENTS OF LEAST SQUARES FIT OF DEGREE1:I1

3210 FOR 0=1 TO I

3220 PRINT «,C(J)

3230 NEXT J

3235 81=0

324.0 FOR 12=1 TO P2

3245 S5=0

3250 FOR 13=2 TO I

3255 J3=I-l3+2

3260 S5=(S5+C(J3))*X(I2)

3265 NEXT 13

3267 S5=S5+C(1)

3270 B1=B1+(Y(I2)-S5)'2

3280 NEXT 12

3290 B1=B1/(P2-1)

3300 PRINT #3

3400 PRINT #3,'BETA=';B1

3405 B(I,0)=B1

3410 FOR J=l TO I

3420 B(I,J)=C(J)

3430 NEXT J

3440 NEXT I

3450 PRINT #3

12000 PRINT \ PRINT 'DO YOU WANT TO DO ANOTHER RATE FIT (Y/N)';

12002 INPUT R$

12004 IF R$='N' THEN GO TO 12008

12006 GO TO 1690

12008 STOP

12010 END

GMPLT8

10 REM PLOT WORK RATE VS STRESS

20 X$='* * *'

30 X=SYS(7)

40 CLRTXWIN

50 REM

60 REM This program was written by Paul Sean Hill and George

200
65 REM James Nov 1985 in

70 REM order to calculate the inelastic work rate vs stress

80 REM for use with the Bodner constant code in determining

85 REM saturation stresses for various cases.

90 REM

100 DIM N(250,4),X(250),Y(250),C(10),A(10,11),X1(250),B(10,11)

110 DIM S(10,5),E(10,5),Y1(250),E1(250,3),M(2)

120 PRINT 'What is Youngs modulus for this test1; \ INPUT C(2) \ PRINT

130 C9=C(2)

135 PRINT \ PRINT ,'Input block number to start with1; \ INPUT BO

140 PRINT \ PRINT ,'Input point to start with1; \ INPUT PI

145 PRINT \ PRINT ,'Input point to end with1; \ INPUT P3

150 PRINT \ PRINT ,'input beginning half cycle'; \ INPUT F

210 PRINT

270 PRINT 'Enter the file name for experimental data (<=5 CHARACTERS)1;

280 INPUT F$

290 IF LEN(F$)<=5 THEN GO TO 305

300 PRINT 'INVALID FILE NAME' \ GO TO 270

305 OPEN 'LP:' FOR OUTPUT AS FILE 3

310 F2$='DU1:'+F$+'.DAT'

320 F1$='DU1:'+F$+'C.DAT'

330 REM FILE INPUT

340 OPEN F2$ FOR INPUT AS FILE 2

350 OPEN Fl$ FOR INPUT AS FILE 1

370 Z$='FILENAME <####'

380 INPUT #l,LO,XO,GO,AO,Ul,U2,U3,U4,TO,U5fU6,U7,U8,PO

390 PRINT #3,USING Z$,F$

400 PRINT #3

410 PRINT #3,'BLOCK NUMBER = ';BO

420 PRINT #3,'STARTING POINT = ';P1

430 PRINT #3,'ENDING POINT = ';P3

440 PRINT #3,'HALF CYCLE = ' ;F

480 PO=P3

500 P2=P3

510 REM

520 REM READING STORED DATA

530 REM

540 N4=P3 \ N3=l \ T=0-T0

550 N4=N4-64 \ N2=64 \ IF N4<=0 THEN N2=N4+64

560 AINP(#2,E1(N3,0),N2*4,BO,P)

570 IF N4<=0 THEN GO TO 600

580 BO=BO+1 \ N3=N3+64

590 GO TO 550

600 ELEVAR(E1,1,2,10)

610 FOR J=P1 TO P3

620 T=T+TO \ N(J,0)=T

630 N(J,1)=E1(J,2)*LO/AO/1000

640 N(J,2)=E1(J,1)*XO/GO

650 NEXT J

660 CLOSE #1,2

201
970 PRINT #3 \ PRINT #3 \ PRINT #3,'For Youngs modulus = ';C9;

1370 REM CALCULATING PLASTIC STRAINS AND RATE OF PLASTIC WORK1375 N(P1,3)=0

1380 FOR I=P1+1 TO P3

1390 N(I,3)=N(I-1,3)+(N(I,2)-N(I-1,2))-((N(I,1)-N(I-1,1))/C9)

1410 NEXT I

1412 P6=P1

1420 IF ABS(N(P6,3))<2.00000E-03 THEN GO TO 1424

1422 GO TO 1430

1424 P6=P6+1

1426 GO TO 1420

1430 P4=P3-1

1440 P5=P6+1

1450 P2=P3-P6+1

1480 REM RATE CURVE FIT

1490 REM

1500 REM FORMING A MATRIX

1510 REM

1520 FOR J1=P6 TO P3

1530 J=J1-P6+1

1540 X(J)=N(J1,3)

1550 Y(J)=N(J1,1)

1560 X1(J)=1

1570 NEXT Jl

1580 FOR J=l TO 9

1590 A(J,1)=0

1600 A(J,11)=0

1610 FOR K=l TO P2

1620 A(J,1)=A(J,1)+X1(K)

1630 A(J,11)=A(J,11)+Y(K)*X1(K)

1640 X1(K)=X1(K)*X(K)

1650 NEXT K

1660 NEXT J

1670 FOR 1=2 TO 10

1680 A(10,I)=0

1690 FOR J=l TO P2

1700 A(10,I)=A(10,I)+X1(J)

1710 X1(J)=X1(J)*X(J)

1720 NEXT J

1730 NEXT I

1740 FOR J=2 TO 10

1750 FOR 1=1 TO 9

1760 A(I,J)=A(I+1,J-1)

1770 NEXT I

1780 NEXT J

1790 REM

1800 REM LU DECOMPOSITION

1810 REM

1820 FOR 1=1 TO 10

1830 FOR J=2 TO 10

202
1840 S5=0

1850 IF J>I THEN GO TO 1920

1860 J1=J-1

1870 FOR K=l TO 01

1880 S5=S5+A(I,K)*A(K,J)

1890 NEXT K

1900 A(I,J)=A(I,J)-S5

1910 GO TO 2020

1920 11=1-1

1930 IF 1=0 THEN GO TO 1970

1940 FOR K=l TO II

1950 S5=S5+A(I,K)*A(K,J)

1960 NEXT K

1970 IF ABS(A(I,!))<!.OOOOOE-20 THEN GO TO 1990

1980 GO TO 2010

1990 PRINT 'SMALL VALUE ON DIAGONAL DETECTED FOR REDUCTION #';!

2010 A(I,J)=(A(I,J)-S5)/A(I,I)

2020 NEXT J

2030 NEXT I

2040 REM

2050 REM SOLVING MATRIX EQUATIONS

2060 REM

2070 FOR 1=2 TO 10

2080 FOR J=l TO I

2090 C(J)=A(J,11)

2100 NEXT J

2110 GOSUB 2590

2120 11=1-1

2130 PRINT #3

2140 PRINT #3,'COEFFFICIENTS OF LEAST SQUARES FIT OF DEGREE1;I1

2150 FOR J=l TO I

2160 PRINT #3,C(J)

2170 NEXT J

2180 81=0

2190 FOR 12=1 TO P2

2200 S5=0

2210 FOR 13=2 TO I

2220 J3=I-I3+2

2230 S5=(S5+C(J3))*X(I2)

2240 NEXT 13

2250 S5=S5+C(lj

2260 B1=B1+(Y(I2)-S5)"2

2270 NEXT 12

2280 B1=B1/(P2-1)

2290 PRINT #3,'BETA=';B1

2300 B(I,0)=B1

2310 FOR J=l TO I

2320 B(I,J)=C(J)

2330 NEXT J

2340 NEXT I

203
2350 PRINT #3

2360 PRINT ,'INPUT DEGREE OF POLYNOMIAL DESIRED1; \ INPUT I

2370 19=1

2380 FOR J=P6 TO P3

2390 N(J,2)=0

2400 N(J,4)=0

2405 N(J,0)=0

2410 FOR K=l TO 19+1

2420 N(J,4)=N(J,4)+B(I,K)*N(J,3)"(K-1)

2430 IF K<2 THEN GO TO 2445

2440 N(J,2)=N(J.2)+B(I,K)*(K-l)*(N(J.3)"(K-2))

2445 N(J,0)=N(J»0)+(B(I,K)/K)*(N(J,3) K)

2450 NEXT K

2455 N(J,2)=N(J,2)/N(J,1)

2460 NEXT J

2470 FOR J=l TO P6-1

2480 N(J,2)=0

2490 N(J,4)=0

2500 NEXT J

2502 W1=N(P3,0)-N(P6,0)

2504 PRINT #3 \ PRINT #3,'PLASTIC WORK =';W1

2510 PRINT #3,'STRESS(KSI)'TAB(15)'INEL STRN';

2520 PRINT #3,TAB(30)'STRESS-FIT';TAB(45)'dS/dWP-FIT1

2540 FOR I=P6 TO P3

2550 PRINT #3,N(I,1);TAB(15)N(I,3);TAB(30)N(I,4);TAB(45)N(I,2)

2570 NEXT I

2580 GO TO 2790

2590 C(1)=C(1)/A(1,1)

2600 FOR 12=2 TO I

2610 11=12-1

2620 S5=0

2630 FOR K=l TO II

2640 S5=S5+A(I2,K)*C(K)

2650 NEXT K

2660 C(I2)=(C(I2)-S5)/A(I2,I2)

2670 NEXT 12

2680 FOR J=2 TO I

2690 N2=I-J+2

2700 N1=I-J+1

2710 S5=0

2720 FOR K=N2 TO I

2730 S5=S5+A(N1,K)*C(K)

2740 NEXT K

2750 C(N1)=C(N1)-S5

2760 NEXT J

2770 RETURN

2780 REM PLOT DATA

2790 PRINT 'PLACE PRINTER SWITCH IN POSITION "B" '

2800 PRINT 'HIT RETURN TO CONTINUE1; \ INPUT R$ \ PRINT \ PRINT

2810 Y$='STRESS (KSI)1

204
2820 PRINT 'ENTER MAXIMUM, MINIMUM STRESS1; \ INPUT Y2.Y1

2830 PRINT 'ENTER STRESS STEP SIZE'; \ INPUT Y3

2840 X$='INEL STRAIN'

2850 PRINT 'ENTER MAXIMUM, MINIMUM INELASTIC WORK'; \ INPUT X2,X1

2860 PRINT 'ENTER INELASTIC WORK STEP SIZE'; \ INPUT X3

2870 PRINT #3 \ PRINT #3 \ PRINT #3

2880 PRINT #3,Y$,I9;'th ORDER1

2890 PRINT #3,'* *'

2900 PRINT #3,'* *'

2910 PRINT 'ARE THESE PARAMETERS OK? ';

2920 INPUT R$

2930 IF R$='N' THEN GO TO 2820

2940 TEKMODE(l.l)

2950 PHYL(1,10,85,10,85)

2960 SCALE(1,0,X1,X2,Y1,Y2)

2970 AXES(1,0,0)

2980 LABEL(1,X$,Y$,X3,Y3,1)

2990 INVEC

3000 FOR I=P5 TO P4

3010 X4=N(I,3)

3020 Y4=N(I,1)

3030 MARK(1,4,X4,Y4)

3040 INVEC

3050 NEXT I

3060 FOR I=P5 TO P4

3070 X4=N(I,3)

3080 Y4=N(I,4)

3090 PLOT(1,X4,Y4)

3100 NEXT I

3110 HOME

3120 COPY

3130 VTMODE(l.O)

3140 PRINT 'Is this fit close enough1.; \ INPUT A$

3150 PRINT \ PRINT

3160 IF A$='N' THEN GO TO 2360

3170 PRINT 'Do you wish to continue the calculations'; \ INPUT A$

3180 IF A$='N' THEN GO TO 4580

3185 X$=Y$ \ X1=Y1 \ X2=Y2 \ X3=Y3 \ X4=Y4

3190 Y$='INEL WORK RATE'

3200 PRINT 'ENTER MAXIMUM RATE'; \ INPUT Y2

3210 PRINT 'ENTER MINIMUM RATE'; \ INPUT Yl

3220 PRINT 'ENTER RATE STEP SIZE1; \ INPUT Y3

3230 PRINT #3 \ PRINT #3 \ PRINT #3

3240 PRINT 'ARE THESE PARAMETERS OK? ';

3250 TXTBOLD \ PRINT '(Y/N)'; \ TXTNORMAL \ INPUT R$

3260 IF R$='N' THEN GO TO 3200

3270 TEKMODE(l.l)

3280 PHYL(1,10,85,10,85)

3290 SCALE(1,0,X1,X2,Y1,Y2)

3300 AXES(1,0,0)

205
3310 LABEL(1.X$,Y$,X3,Y3,1)

3320 INVEC

3330 FOR I=P5 TO P4

3340 X4=N(I,1)

3350 Y4=N(I,2)

3360 MARK(1,4,X4,Y4)

3370 INVEC

3380 NEXT I

3440 HOME

3510 PRINT 'How many points do you want for the lower portion';

3520 INPUT T2

3530 PRINT 'How many points do you want for the upper portion';

3540 INPUT Tl

3545 INPUT R$

3560 P7=P6+T1

3565 P=T1 \ 11=1

3566 FOR I=P6 TO P7

3567 X(I1)=N(I,1)

3568 Y(I1)=N(I,2) \ PRINT I3,X(I1),Y(I1) \ 11=11+1

3569 NEXT I

3570 GOSUB 15000

3580 M2=C2

3590 M4=C1

3595 M5=-M4/M2

3600 P7=P3

3601 P=T2 \ 11=1

3602 P8=P7-T2

3603 FOR I=P8 TO P7

3604 X(I1)=N(I,1) \ Y(I1)=N(I,2) \ PRINT *3.X(I1).Y(I1) \ 11=11+1

3605 NEXT I

3607 GOSUB 15000

3608 M1=C2

3609 M6=C1

3610 M7=-M6/M1

3670 FOR I=P5 TO P4

3680 E1(I,1)=M1*N(I,1)+M6

3690 E1(I,2)=M2*N(I,1)+M4

3710 NEXT I

3720 PRINT #3 \ PRINT #3

3730 PRINT #3,'M2 = ';M2

3740 PRINT #3,'UPPER Y INTERCEPT = ';M4

3750 PRINT #3,'BACK STRESS = ';M5

3752 PRINT #3,'Ml = ';M1

3754 PRINT #3,'LOWER Y INTERCEPT = ';M6

3755 PRINT #3,'SATURATED STRESS = ';M7

3760 PRINT #3 \ PRINT #3

3790 PRINT #3

3800 PRINT #3 \ PRINT #3,Y$

3900 INVEC

3940 FOR I=P6 TO P3

206
3950 X4=N(I,1)

3960 Y4=E1(I,1)

3970 PLOT(1,X4,Y4)

3980 NEXT I

3990 INVEC

4000 INVEC

4100 FOR I=P6 TO P3

4110 X4=N(I,1)

4120 Y4=E1(I,2)

4130 PLOT(1,X4,Y4)

4140 NEXT I

4145 INVEC

4150 HOME

4160 COPY

4170 VTMODE(l.l)

4180 PRINT ,'DO YOU WANT TO CONTINUE'; \ INPUT R$

4190 IF R$='N' THEN GO TO 4580

4200 Pl=P3-INT(P3/64)*64

4210 P3=Pl+79

4220 F=F+1

4230 PRINT #3,' -.

4240 GO TO 340

4580 CLOSE (#3)

4585 STOP i

4590 END

15000 REM Sl= SUM OF X(J)

15010 REM S2 = SUM OF X(J)"2

15020 REM S3 = SUM OF Y(J)

15030 REM S4 = SUM OF X(J) * Y(J)

15040 Sl=0 \ S2=0 \ S3=0 \ S4=0

15050 FOR J=l TO P

15060 S1=S1+X(J) \ S2=S2+X(J)*2 \ S3=S3+Y(J) \ S4=S4-t-(X(J)*Y(J))

15070 NEXT J

15080 C1=(S3*S2-S1*S4)/(P*S2-S1*S1)

15090 C2=(P*S4-S1*S3)/(P*S2-S1*S1)

15100 RETURN

15110 END

15200 REM FOR J=l TO P

15210 REM X(J)=LOG(X(J))

15220 REM Y(J)=LOG(Y(J))

15230 REM NEXT J

15250 REM C1=EXP(C1)

15300 REMFOR J=l TO P

15400 REM Y(J)=LOG(Y(J))

15410 REM NEXT J

15500 REM C1=EXP(C1)

207
WALK2

10 REM PLOT WORK RATE VS STRESS

20 X$='* * *'

30 X=SYS(7)

40 CLRTXWIN

50 REM

60 REM This program was written by Paul Sean Hill and George

65 REM James Nov 1985 in

70 REM order to calculate the inelastic strain rate vs stress

80 REM for use in determining

85 REM saturation stresses for the calculation of Walker's model.

90 REM

100 DIM N(250,4),X(250),Y(250),C(10),A(10,11),Xl(250),8(10,11)

110 DIM S(10,5),E(10,5),Y1(250),E1(250,3),M(2)

120 PRINT 'What is Youngs modulus for this test'; \ INPUT C(2) \ PRINT

130 C9=C(2)

135 PRINT \ PRINT ,'Input block number to start with'; \ INPUT BO

140 PRINT \ PRINT ,'Input point to start with1; \ INPUT PI

145 PRINT \ PRINT ,'Input point to end with'; \ INPUT P3

150 PRINT \ PRINT ,'input beginning half cycle'; \ INPUT F

210 PRINT

270 PRINT 'Enter the file name for experimental data (<=5 CHARACTERS)';

280 INPUT F$

290 IF LEN(F$)<=5 THEN GO TO 305

300 PRINT 'INVALID FILE NAME1 \ GO TO 270

305 OPEN 'LP:1 FOR OUTPUT AS FILE 3

310 F2$='DU1:'+F$+'.DAT'

320 F1$='DU1:'+F$+'C.DAT'

330 REM FILE INPUT

340 OPEN F2$ FOR INPUT AS FILE 2

350 OPEN Fl$ FOR INPUT AS FILE 1

370 Z$='FILENAME <####'

380 INPUT #1,LO,XO,GO,AO,U1,U2,U3,U4,TO,U5,U6,U7,U8,PO

390. PRINT #3,USING Z$,F$

400 PRINT #3

410 PRINT #3,'BLOCK NUMBER = ';BO

420 PRINT #3,'STARTING POINT = ';P1

430 PRINT #3,'ENDING POINT = ';P3

440 PRINT #3,'HALF CYCLE = ';F

480 PO=P3

500 P2=P3

510 REM

520 REM READING STORED DATA

530 REM

540 N4=P3 \ N3=l \ T=0-T0

550 N4=N4-64 \ N2=64 \ IF N4<=0 THEN N2=N4+64

560 AINP(#2,E1(N3,0),N2*4,BO,P)

570 IF N4<=0 THEN GO TO 600

208
580 BO=BO+1 \ N3=N3+64

590 GO TO 550

600 ELEVAR(E1,1,2,10)

610 FOR J=P1 TO P3

620 T=T+TO \ N(J,0)=T

630 N(J,1)=E1(J,2)*LO/AO/1000

640 N(J,2)=E1(J,1)*XO/GO

650 NEXT J

660 CLOSE #1,2

970 REM PRINT #3 \ PRINT #3 \ PRINT #3,'For Youngs modulus = ';C9;

1370 REM CALCULATING PLASTIC STRAINS AND RATE OF PLASTIC WORK—

1375 N(P1,3)=0

1380 FOR I=P1+1 TO P3

1390 N(I,3)=N(I-1,3)+(N(I,2)-N(I-1,2))-((N(I,1)-N(I-1,1))/C9)

1410 NEXT I

1412 P6=P1

1420 IF ABS(N(P6,3))<2.00000E-03 THEN GO TO 1424

1422 GO TO 1430

1424 P6=P6+1

1426 GO TO 1420

1430 P4=P3-1

1440 P5=P6+1

1450 P2=P3-P6+1

1480 REM RATE CURVE FIT

1490 REM

1500 REM FORMING A MATRIX

1510 REM

1520 FOR J1=P6 TO P3

1530 J=J1-P6+1

1540 X(J)=N(J1,3)

1550 Y(J)=N(J1,1)

1560 X1(J)=1

1570 NEXT Jl

1580 FOR J=l TO 9

1590 A(J,1)=0

1600 A(J,11)=0

1610 FOR K=l TO P2

1620 A(J,1)=A(J,1)+X1(K)

1630 A(J,11)=A(J,11)+Y(K)*X1(K)

1640 X1(K)=X1(K)*X(K)

1650 NEXT K

1660 NEXT J

1670 FOR 1=2 TO 10

1680 A(10,I)=0

1690 FOR J=l TO P2

1700 A(10,I)=A(10,I)+X1(J)

1710 X1(J)=X1(J)*X(J)

1720 NEXT J

1730 NEXT I

1740 FOR J=2 TO 10

209
1750 FOR 1=1 TO 9

1760 A(I,J)=A(I+1,J-1)

1770 NEXT I

1780 NEXT J

1790 REM

1800 REM LU DECOMPOSITION

1810 REM

1820 FOR 1=1 TO 10

1830 FOR J=2 TO 10

1840 S5=0

1850 IF J>I THEN GO TO 1920

1860 J1=0-1

1870 FOR K=l TO Jl

1880 S5=S5+A(I,K)*A(K,J)

1890 NEXT K

1900 A(I,J)=A(I,J)-S5

1910 GO TO 2020

1920 11=1-1

1930 IF 1=0 THEN GO TO 1970

1940 FOR K=l TO II

1950 S5=S5+A(I,K)*A(K,J)

1960 NEXT K

1970 IF ABS(A(I,!))<!.OOOOOE-20 THEN GO TO 1990

1980 GO TO 2010

1990 PRINT 'SMALL VALUE ON DIAGONAL DETECTED FOR REDUCTION *';!

2010 A(I.J)=(A(I,J)-S5)/A(I.I)

2020 NEXT J

2030 NEXT I

2040 REM

2050 REM SOLVING MATRIX EQUATIONS

2060 REM

2070 FOR 1=2 TO 10

2080 FOR J=l TO I

2090 C(J)=A(J,11)

2100 NEXT J

2110 GOSUB 2590

2120 11=1-1

2130 PRINT

2140 PRINT ,'COEFFFICIENTS OF LEAST SQUARES FIT OF DEGREE';I1

2150 FOR J=l TO I

2160 REM PRINT ,C(J)

2170 NEXT J

2180 Bl=0

2190 FOR 12=1 TO P2

2200 S5=0

2210 FOR 13=2 TO I

2220 J3=I-I3+2

2230 S5=(S5+C(J3))*X(I2)

2240 NEXT 13

2250 S5=S5+C(1)

210
2260 B1=B1+(Y(I2)-S5)"2

2270 NEXT 12

2280 B1=B1/(P2-1)

2290 PRINT ,'BETA=';B1

2300 B(I,0)=B1

2310 FOR J=l TO I

2320 B(I,J)=C(J)

2330 NEXT J

2340 NEXT I

2350 PRINT

2360 PRINT .'INPUT DEGREE OF POLYNOMIAL DESIRED1; \ 1=4

2370 19=1

2380 FOR J=P6 TO P3

2390 N(J,2)=0

2400 N(J,4)=0

2405 N(J,0)=0

2410 FOR K=l TO 19+1

2420 N(J,4)=N(J,4)+B(I,K)*N(J,3r(K-l)

2430 IF K<2 THEN 60 TO 2445

2440 N(J,2)=N(J,2)+B(I,K)*(K-l)*(N(J,3)*(K-2))

2445 N(J,0)=N(J,0)+(B(I,K)/K)*(N(J,3)*K)

2450 NEXT K

2460 NEXT J

2470 FOR J=l TO P6-1

2490 N(J,4)=0

2500 NEXT J

2502 W1=N(P3,0)-N(P6,0)

2504 PRINT #3 \ PRINT #3,'PLASTIC STRAIN = ';W1

2510 REM PRINT #3,'STRESS(KSI)'TAB(15)'INEL STRN';

2520 REM PRINT #3,TAB(30)'STRESS-FIT';TAB(45)'dS/dEP-FIT'

2540 FOR I=P6 TO P3

2550 REM PRINT #3,N(I,1);TAB(15)N(I,3);TAB(30)N(I,4);TAB(45)N(I,2)

2570 NEXT I

2580 GO TO 3510

2590 C(1)=C(1)/A(1,1)

2600 FOR 12=2 TO I

2610 11=12-1

2620 S5=0

2630 FOR K=l TO II

2640 S5=S5+A(I2,K)*C(K)

2650 NEXT K

2660 C(I2)=(C(I2)-S5)/A(I2,I2)

2670 NEXT 12

2680 FOR J=2 TO I

2690 N2=I-J+2

2700 N1=I-J+1

2710 S5=0

2720 FOR K=N2 TO I

2730 S5=S5+A(N1,K)*C(K)

2740 NEXT K

211
2750 C(N1)=C(N1)-S5

2760 NEXT J

2770 RETURN

2780 REM PLOT DATA

2790 PRINT 'PLACE PRINTER SWITCH IN POSITION "B" '

2800 PRINT 'HIT RETURN TO CONTINUE'; \ INPUT R$ \ PRINT \ PRINT

2810 Y$='STRESS (KSI)1

2820 PRINT 'ENTER MAXIMUM, MINIMUM STRESS'; \ INPUT Y2.Y1

2830 PRINT 'ENTER STRESS STEP SIZE'; \ INPUT Y3

2840 X$='INEL STRAIN'

2850 PRINT 'ENTER MAXIMUM, MINIMUM INELASTIC STRAIN'; \ INPUT X2.X1

2860 PRINT 'ENTER INELASTIC STRAIN STEP SIZE'; \ INPUT X3

2870 PRINT #3 \ PRINT #3 \ PRINT #3

2880 PRINT #3,Y$,I9;'th ORDER1

2890 PRINT #3,'* *'

2900 PRINT #3,'* *'

2910 PRINT 'ARE THESE PARAMETERS OK? ';

2920 INPUT R$

2930 IF R$='N' THEN GO TO 2820

2940 TEKMODE(1,1)

2950 PHYL(1,10,85,10,85)

2960 SCALE(1,0,X1,X2,Y1,Y2)

2970 AXES(1,0,0)

2980 LABEL(1,X$,Y$,X3,Y3,1)

2990 INVEC

3000 FOR I=P5 TO P4

3010 X4=N(I,3)

3020 Y4=N(I,1)

3030 MARK(1,4,X4,Y4)

3040 INVEC

3050 NEXT I

3060 FOR I=P5 TO P4

3070 X4=N(I,3)

3080 Y4=N(I,4)

3090 PLOT(1,X4,Y4)

3100 NEXT I

3110 HOME

3120 COPY

3130 VTMODE(l.O)

3140 PRINT 'Is this fit close enough'; \ INPUT A$

3150 PRINT \ PRINT

3160 IF A$='N' THEN GO TO 2360

3170 PRINT 'Do you wish to continue the calculations'; \ INPUT A$

3180 IF A$='N' THEN GO TO 4580

3185 X$=Y$ \ X1-Y1 \ X2=Y2 \ X3=Y3 \ X4=Y4

3190 Y$='INEL WORK RATE1

3200 PRINT 'ENTER MAXIMUM RATE1; \ INPUT Y2

3210 PRINT 'ENTER MINIMUM RATE'; \ INPUT Yl

3220 PRINT 'ENTER RATE STEP SIZE1; \ INPUT Y3

3230 PRINT #3 \ PRINT #3 \ PRINT #3

212
3240 PRINT 'ARE THESE PARAMETERS OK? ';

3250 TXTBOLD \ PRINT '(Y/N)1; \ TXTNORMAL \ INPUT R$

3260 IF R$='N' THEN GO TO 3200

3270 TEKMODE(l.l)

3280 PHYL(1,10,85,10,85)

3290 SCALE(1,0,X1,X2,Y1,Y2)

3300 AXES(1,0,0)

3310 LABEL(1.X$,Y$.X3,Y3,1)

3320 INVEC

3330 FOR I=P5 TO P4

3340 X4=N(I.l)

3350 Y4=N(I,2)

3360 MARK(1,4,X4,Y4)

3370 INVEC

3380 NEXT I

3440 HOME

3510 PRINT 'How many points do you want for the lower portion1;

3520 T2*5

3530 PRINT 'How many points do you want for the upper portion1;

3540 T>10

3545 REM INPUT R$

3560 P7=P6+T1

3565 P=T1 \ 11=1

3566 FOR I=P6 TO P7

3567 X(I1)=N(I,1)

3568 Y(I1)=N(I:.2, ) \ PRINT #3,X(I1),Y(I1) \ 11=11+1

3569 NEXT I

3570 GOSUB 15000

3580 M2=C2

3590 M4=C1

3595 M5=-M4/M2

3600 P7=P3

3601 P=T2 \ 11=1

3602 P8*P7-T2

3603 FOR I=P8 TO P7

3604 X(I1)=N(I,1) \ Y(I1)=N(I,2) \ PRINT #3,X(I1),Y(I1) \ 11=11+1

3605 NEXT I

3607 GOSUB 15000

3608 M1=C2

3609 M6=C1

3610 M7=-M6/M1

3670 FOR I=P5 TO P4

3680 E1(I,1)=M1*N(I,1)+M6

3690 E1(I,2)=M2*N(I,1)+M4

3710 NEXT I

3720 PRINT #3 \ PRINT #3

3730 PRINT #3,'M2 = ';M2

3740 PRINT #3,'UPPER Y INTERCEPT = ';M4

3750 PRINT #3,'BACK STRESS = ';M5

3752 PRINT #3,'Ml = ';M1

213
3754 PRINT #3,'LOWER Y INTERCEPT = ';M6

3755 PRINT #3,'SATURATED STRESS = ';M7

3760 PRINT #3 \ PRINT #3

3790 PRINT #3

3800 PRINT #3 \ PRINT #3,Y$

3810 GO TO 4200

3900 INVEC

3940 FOR I=P6 TO P3

3950 X4=N(I,1)

3960 Y4=E1(I,1)

3970 PLOT(1,X4,Y4)

3980 NEXT I

3990 INVEC

4000 INVEC

4100 FOR I=P6 TO P3

4110 X4=N(I,1)

4120 Y4=E1(I,2)

4130 PLOT(1,X4,Y4)

4140 NEXT I

4145 INVEC

4150 HOME

4160 COPY

4170 VTMODE(1,1)

4180 PRINT ,'DO YOU WANT TO CONTINUE1; \ INPUT R$

4190 IF R$='N' THEN GO TO 4580

4200 Pl=P3-INT(P3/64)*64

4210 P3=P1+80

4220 F=F+1

4230 PRINT #3,' —

4240 GO TO 340

4580 CLOSE (#3)

4585 STOP

4590 END

15000 REM Sl= SUM OF X(J)

15010 REM S2 = SUM OF X(J)"2

15020 REM S3 = SUM OF Y(J)

15030 REM S4 = SUM OF X(J) * Y(J)

15040 Sl=0 \ S2=0 \ S3=0 \ S4=0

15050 FOR J=l TO P

15060 S1=S1+X(J) \ S2=S2+X(J)A2 \ S3=S3+Y(J) \ S4=S4+(X(J)*Y(J))

15070 NEXT J

15080 C1=(S3*S2-S1*S4)/(P*S2-S1*S1)

15090 C2=(P*S4-S1*S3)/(P*S2-S1*S1)

15100 RETURN

15110 END

15200 REM FOR J=l TO P

15210 REM X(J)=LOG(X(J))

15220 REM Y(J)=LOG(Y(J))

15230 REM NEXT J

15250 REM C1=EXP(C1)

214
15300 REMFOR J=l TO P

15400 REM Y(J)=LOG(Y(J))

15410 REM NEXT J

15500 REM C1=EXP(C1)

215