NASA Technical Memorandum 102761

FINITE ELEMENT THERMO-VISCOPLASTIC ANALYSIS OF

AEROSPACE STRUCTURES

AJAY K. PANDEY

PRAMOTE DECHAUMPHAI

EARL A. THORNTON

(NASA-/u-I02761) FINITF FL_M_NT

THE_N_-VI_COPLA3TIC ANALY_[S OF A_ROSPACE

STRUCTUPE_ (NASA) 23 O CSCL ?OK

G3139

Ngl-I6Q07

Unclds

03265sI

NOVEMBER 1990

NASA

Nalional Aeronautics and

Space Administration

Langley Research Center

Hampton, Virginia 23665



FINITE-ELEMENT T|IERMO-VISCOPLASTIC ANALYSIS

OF AEROSPACE STRUCTURES

A jay K. Pandey

Lockheed Engineering and Sciences Company

ttampton, Virginia

Pramote Dechaumphai

NASA 'Langley Research Center

llampton, Virginia

and

Earl A. Thornton

University of Virginia

Charlottesville, Virginia

The time-dependent thermo-viscoplastic response of aerospace structures

subjected to intense aerothermal loads is predicted using the finite-element

method. The finite-element analysis uses the Bodner-Partom unified

viscoplastic constitutive relations to determine rate-dependent nonlinear material

behavior. The methodology is verified by comparison with experimental data

and other numerical results fo¢ a uniaxially-loaded "bar. The method is then used

(1) to predict the structural response of a rectangular plate subjected to line

heating along a centerline, and (2) to predict the themr, d-structural response of a

convectively-cooled engine cowl leading edge subjected to aerodynamic shockshock interference heating. Compared to linear elastic analysis, the viscoplastic

analysis results in lower peak stresses and regions of plastic dcformadons.

INI"RODUC'FION

As hypersonic vehicles accelerate or decelerate, aerodynamic shock waves sweep across the vehicle and

interact with local shocks and boundary layers. These shock interactions expose the vehicle surfaces to severe

pressures and heating rates in l_al regions where the interactions occur. Leading edges present a particularly difficult

design problem because of such intense, highly-localized loads. TILe thermal-structural response of hydrogen-cooled

super "thermal" conducting leading edge subjected to intense aerodynamic heating was studied in [1]. A thermalstructural elasto/plastic analysis with experimental verification [21 of a cowl lip design demonstrated that plastic

effects occur and may be significant.

Until recent years, the study of structural response at elevated temperatures due to transient thermal and

pressure loads was not possible because of an inability to model the viscoplastic material behavior. Traditionally,

tile thermal-structural analayis has been performed using elasto/plastic theories. These theories lack the capabilities

to accurately model the load rate effect, cyclic material behavior, stress relaxation and other such phenomena. Over

the last twenty years unified viscoplastic constitutive models have evolved to meet these needs. These unified

constitutive models provide a means for analytically representing a material's response from the elastic through the

plastic range including strain-rate dependent plastic flow, creep, stress relaxation, and temperature effects. The first

multi-dimensional formulation of viseoplastic constitutive model was due to Bodner and Partom [3]. Since then a

number of constitutive models have appeared; many of these models and the related theories are suinmarized in

review articles that appear in [4].

When unified constitutive models are implemented in finite element programs, it enables viscoplastic
analysis of complex structures. Finite-element methods with unified constitutive models have been under

development for the past 15 years. A thermo-viscop "lastic finite-element computational approach for hypersonic

structures was presented in [5]. Analysis of convectively-cooled hypersonic structures illustrated the effectiveness of

the approach and provided insight into Uansient plastic structural behavior at elevated temperatures. Further

application of the approach to high temperature structures and improvements in finite-element modeling techniques

are required to: (1) assist in planning thermal-structural experiments to validate the computational approach, and (2)

aid in und_standing the high-temperature belmvior of difficult design problems such as leading edges of hypersonic

vehicles.

The purpose of this paper is to present the thermo-viscoplastic analysis methodology, investigate its

application to representative aerospace structures of current interest, and to investigate the need for such analysis.

The Bodner-Partom unified constitutive relations and the thermal-structural equations are described first. Next, the

finite-element formulation and the computational steps are presented for the thermo-viscoplastic analysis

methodology. The methodiogy is evaluated by comparison of results for a uniaxially loaded 'bar with experimental

data and other finite-element results. The thermo-viscoplastic response of a plate subjecttal to line heating along the

centerline is then obtained. The last application example presents the transient thermal-structural response of

convectively-cooled engine cowl leading edge subjected to intense aerodynamic heating and pressure.

BODNER-PARTOM UNIFIED VISCOPLASTIC MODEL

The structural analysis of aerospace structures at high temperature has traditionally been carried out by

using either elastic or classic',d elasto/plastic analysis methodology [6]. The elastic analysis solution can't be relied

upon once the material has yielded because elastic constitutive relations are not valid. In classical time independent

elasto/plastic theory, traditionally used for the post-yield analysis, the structural behavior is predicted using a yield

condition (such as the von-Mises yield condition), a time independent flow rule (such as the Prandtl-Reuss equation)

and a "hardening rule. Hardening rules are used to determine reverse yielding under cyclic loading. A separate creep

law is required to predict creep behavior. This classical elasto/plastic theory can't predict the effect of varying strain

rate [71 which is significant if the strain rate and temperature vary over a wide range.

Several unified viscoplastic models have been developed in recent years, which combine the time

independent and time dependent response into a single inelastic component. These models utilize internal state

variables as the repositories for the effect of past deformation history and do not rely on the existence of a

conventional yield condition. A unified viscoplastic model [3,81 is used in the present analysis because of its

widespread use and its capability to closely simulate the time dependent material response under various kinds of

loading [5,7].

The Bodner-Partom unified constitutive model represents plastic strain rate in terms of stresses and internal

state variables. The internal slate variable rates, in turn, are given as functions of stresses, plastic work, and the

current value of internal state variables. These equations, briefly described below, include certain physical concepts

based on dislocation dynamics. Details of these equations are given in [5,8] and are briefly described herein.

The strain-displacement relation written in the rate form is,

(1)

where e,j denotes the total strain components and superscripts E and P denote elastic and plastic strain components,

respectively. The displacement components are denoted by ui and the dot on top of the variable represents derivative

with time. The subscripts i,j and j,i denote differentiation with respect to the spati',d coordinates. The plastic strain

rate is described by the evolution equation (also known as flow rule) and is given by 181,

'i_ _ Do exp [ - 21(Z2/3J2) * ] (2)

where Do is the limiting shear strain rate, and n is the temperature dependent material parameter known as the strain

rate sensitivity parameter. In the above equation, the deviatorie stress components, S_j, are related to the stress

components by,
where _j is the Dirac delta function.

components by,

(_)

The deviatoric stress invariant J2 is given in terms of deviatoric stress

h = I S_ s_j (2b)

2

Tile variable Z appearing in Eq. (2) is interpreted as a k_ad history dependent parmneter (known as internal state

variable) that represents hardness of the material with respect to plastic flow. This internal state variable consists of

isotropic ( ZI) and directional (Z°) "hardening components,

Z=Zl+Z °

Tim rate of change of the internal state variable isotropic hardening component, :_, is given by,

(3)

• i 1 [21-22] rl

Z = mdZvZ 1Wp - AIZ_ 1 Zl J

(4)

whereml is the isotropic hardening rate; Zl and Z2 are the maximum and minimum values of ZI, respectively;

constants Atand rtare temperature dependent material co.rants, known as thermal recovery parameter and thermal

recovery exponent, respectively; and the plastic work rate,Wp 0 which is taken as measure of hardening, is given by,

_Vp ffi Oijl_

The directional hardening component of the internal state variable, Zn , is given by,

Z°=i_a_

(4a)

where aij are the direction cosines of the current stress state,

a,j = ou I 1o_1o_11[ (Sa)

and the rate of change of the parameter 13ijhas the stone general fonn as that for isotropic hardening component (Eq.

(4)) but has tensorial character,

=m21Z3aij-l]ij] /p -A2Z, Ill t l]tz!r2 (5b)

wherem2 is the directional hardening rate; Z3 is the maximum value of ZD; and A2 and r2 are the temperature

dependent material cons_ts, known as thermal recovery parameter and thermal recovery exponent, respectively.

In Eqs. (2-5), the parmnetersDo, Z_, Zs, rh r2, mh and m2 are temperature-independent material constants;

the parameters n, Z2, Z0, Ah and A2 are temperature-dependent material constants and all other parameters are

problem variables. The material constants for different materials are given in [8-10] along with the detailed

procedures for determining these constants. The particular values of these constants for the nickel-based superalloy

(B 1900+Hf), used in the present study, are given in the Appendix.
T|IERMAL-STRUCTLIRAL FINITE-ELEMENT FORMULATION

Governing Equatio.s

The thermal response of the structure is governed by the energy equation in the form,

pc _----+qi,i = Q

_t

(6)

where p is the density; c is the specific heat; T is the temperature; Q is the internal heat generation rate; and qi,i is

tire spatial derivative of the heat flux qi in the ith direction. The conduction heat flux components, qi, are related

to the temperature gradients by Fourier's law,

qi = - kij _q-----J"

_xj (6a)

where kij is the temperature-dependent thermal conductivity tensor.

The structural response of die structure is goverend by the equilibrium equations, written in rate form [5]

as,

where fi are die body force components per unit volume. The stress rate compenents are related to the elastic strain

rate components and the rate of change of temlx:rature by the generalized llooke's law,

c_q= Ei_,_( *_ - akt a'i') (8)

where Eda represents Hooke's tensor of elasticity parameters, and otj,t are components of a tensor of therm'.d

expansion coefficients.

Finite-Element Equations and Solution Algorithm

The flux-based finite-element approach [1] is used to derive the finite-element equations for the thermal

analysis. The flux-based finite-element approach is used because it reduces the computational time and increases the

solution accuracy [1,11]. The finite-element equations used to solve for rate of change of nodal temperatures, are in

the form

[MI ('F} = {R}I + {Rlz (9)

where [M} is the eapaci_ce matrix. The two vectors on the right hand side are associated with d_e fluxes within the

element and across the element boundary, respectively. These vectors are delifw.d by,

t

{R}_ = | _-{_}-INIdg_ {q,}

J,j _Xi

(ga)

{R}_ = f= IN) INI ds [qS} (gb)

where {N} are the interpolation functions; vector (q] represents die nodal heat flux in the ith direction; (qS}vector

represents the nodal heat flux on the boundary and includes specified heating, surface convection and surface radiation;

symbols 1_ and s on the integral sign, respectively, denote domain and boundary integrals. An explicit time

marching scheme (with lumped capacitance matrix) is used to obtain the transient temperature response. Details o[

the matrices in Eq. (9) are given in IlL

+

4
The flux-based solution algorithm is also used to derive the finite-element equationsfor the structural

analysis [11]. The finite-element equations are given by,

[Ki {fi} ,, {l_)r+ {l_}'r+ {1_}o+ {I_},,

0o)

where [K] is the element stiffness matrix. Tile load vectors on the right hand side of the above equation are

associated with the rate of change of plastic strains, prescribed temperatures, surface tractions, and body forces,

respectively. These load voctors arc dcfineJJby,

{i_}v = I_IBITIE} {_p} d_

0o0

{I_)T = f,, [B]T IEI {cx}A'FdH Oob)

{1_}o = [s [NIT{_} ds (lOc)

f

{i_}a = -LINIT {i} d[_ (10d)

where [E] is the elasticity matrix; [N] is the matrix of element interpolationfunctions;[B] is the strain-displacement

matrix; f_ denotesthe clement domain; and s denotesthe boundaryof the element where tractionsarc specified.

Details of these matrices can b_ rou,d in 112].

To predict the structural response, the solution algorithm proceeds through the following steps:

1. At time t = 0, initialize stresses ( oO and internal state variable components (ZI and Zn).

2. Compute plastic strain rates using Eq. (2).

3. Form finite-clement matrices of Eq. (10) and assemble thcJn to obtain global matrix equations.

4. Solve global matrix equations [Ki {fi} = {I_] for displacement rates.

5. Compute strain rates from the displacement rates using Eq. (1).

6. Compute stress rates from the generalized llooke's hw using Eq. (8).

7. Compute the internal state variable rates using Eqs. (4-5).

8. Integrate stress rates and internal state variable rates to obtain the solutions of the stress componenLs and

internal state variables at time I+AI.

9. If t+At. < h-,.d go to step 2, otherwise stop.

The above solution algorithm has been developed into a finite-element code, which includes eight-node

three-dimensional hexahcdral elements, four-node two-di,nensional quadrilateral elements and three-node twodimensional triangular elements. The internal state variable rates, plastic strain rates and other variable rate terms arc

integrated using Euler forward time integration scheme with a small time-step. Sm.'dl time-step is required because

the viscoplastic differential equations are very stiff in nature. The time-step for viscoplastic analysis is chosen so

that the computed solutions are smooth in time and don't diverge. The time-step for the viscoplastic analysis

depends on the rate of applied thermo-mechanical loads. The elastic analysis solutions are computed at selected

times, without any consideration of ti,ne-step.

VERIFICATION OF METHODOLOGY AND APPLICATIONS

The application of this viscoplastic analysis approach is presented for three different problems. The first

application highlights simple ,naterial behavior by obtaining the thcrmo-viscoplastic response of a bar at uniform

5
temperaturesubjccte, d to uniaxial strain. The thcrrao-viscoplastic response of a rectangular plate under intense line

heating is presented as the second application. A convectivcly-cooled leading edge structure is then analyzed for

transient thcrra',dand viscoplastic response. A high tcrape,raturc nickel-based supcralloy (B 19(X)+LIf) is used as the

structural raatcrial in all the example probleras. The raatcrial properties arc listed in the Appendix.

Uniaxial Loading of a Bar

The viscoplastic behavior of a nickel supcrailoy bar subjected to different strain rates and temperatures were

reported for the NASA Lewis HOST (HOt Section Technology) program [9,10]. The authors compared the bar's

experimental viscoplastic re.spons¢ with a finite element solution to verify the capability of the Bodner-Partom

constitutive model in predicting the effects of strain rate and temperatures on the stress-strain rcspon_ of the

material. In the present example, the finitc-cleraent algorithm, described in previous sections, is used to prcdic!

response of the bar and results arc compared with numerical and experimental results from [9,10]. The numerical

results in [9,10] were obtained using the MARC finilc-clerncnt code, which was modified to include a subroutine for

Bodner-Partom unified constitutive model. The finitc-eleraent model of the bar consists of one thrcc-diracnsional

eight-node hexahedral eleraent. The b_ is fixed at one end, while the other end is subjcccd to loading.

In the f'u'sl cxperiraent, the end of the bar is subjected to tensile strain at a talc of 100 p inJin.-s at three

different tcraperatures of 2060 °R, 2260 °R and 2460 °R. Computed sucss-strain curves compare well with the strain

controlled experimental data and finite element results from [9] as shown in Fig. 1. The stress-strain curve is linear

for sin',all values of strain, then yielding starts, and a saturation stress level is reached, above which strain increases

without any further increase in stress. The saturation stress refers to the stress value, which corresponds to the

material plastic flow. As the temperature of the bar is increased (se.¢ Fig. 1), the materi',d sohcns and a lower value

of saturation stress is obtained. The elastic modulus is reduced as temperature increases, as shown in Fig. 1.

150 o Ref [9], Experiment I/_ = 100 g in./in.-s I

I ...... Present, Ref [9], F.E. F. E.Analysis Analysis

,oo 2o?o°.

50 _iii1460.i" "'"" 2460 °R

I J I J

•0 5000 10000 15000 20000

Strain, p. in./in.

Fig. 1 Effect of tcmpcrartur¢on thestress-strain curves of a nickel-based supcralloy (B 19(X)+H0

bar at a uniaxiai strain rate of 100 _ inJin.-s.
The effect of different applied strain rates on the bar is presented in the second expcrimenL In this second

experiment, the bar is kept at a constant temperature of 2060 °R. The stress-strain curves, computed using the

present methodology, compare well with experimental data and another numerical solution [9] for three different

strain rates as shown in Fig. 2. The material behavior is unique and linear for small strain. At higher strain levels,

the material response is not unique at different strain rates because of the viseoplasfic effecL The material becomes

harder when it is pulled at a higher strain rate (e.g. at 4000 p in./in.-s), resulting in a higher value of saturation

stress.

Creep phenomena is demostrated by applying a constant load to an isothermal bar. The strain Increases

with time without any increa_ in applied load, which is the basic characteristic of creep loading. Strain history

compares well with the available experimental results and numerical solution [10] as shown in Fig. 3. The figure

shows that an isothermal bar at 2060 °R and at a constant applied stress of 74 ksi results in a larger creep strain rate

than the bar at 2460 °R and at a constant applied stress of 12 ksi. The effect of cyclic load at two different

temperatures, 2060 °R and 2260 °R, is shown in Fig. 4. The bar is subjected to a maximum of +!.5% strain at a

rate of :LI00 p in./in.-s. The virgin i_thcrmal bar is subjected to the cyclic strain loading and two stress-strain

curves are obtained cocresponding to two different temperatures. At higher temperature, the saturation stress level is

lower in both tension and compression. Significant plastic strain is observed at zero stress level with relatively

more plastic strain for the bar at higher temperature.

The above example problems highlight some of the material rc._x)nses, which are predicted using the

Bodner-Partom viscoplastic model applied to a unlaxially-loaded bar. These examples also validate the present

methodology and analysis code.

150

100

Stress, ksi

5O

[ T = 2060 °R I

- o Ref [9], Experiment '

...... Ref [9], F.E. Analysis

Pre_4000.. .... .. p in./in.-s

100 g in./in.-s

-o .... 0--_'.4 p in./in.-s

I I I I

•0 5000 10000 15000 20000

Strain, p in./in.

Fig. 2 Effect of varying uniaxial strain rate on the st/ess-strain curves of a nickel-based superalloy

(B 1900+H0 bar at a uniform temperature of 2060 °R.
30000 [-

20000

Strain, p in./in.

I

/

/

I

/

/

i t

l

0 Ref [10], Experiment

Ref [10], F.E. Analysis

Present, F.E. Analysis

ksi 2060 °R

Fig. 3

10000

-12 ksi at 2460 °R

I 1

0 3 6

Time, s

Time dependent strain in a nickel-bend superalioy (B 1900+H0 bar subjected to a constant

load at two different temperatures.

200 r -- 2060o°R

Stress, ksi / .... 2260 R

-2oooo,../ o /,.." .2oooo

c_'/. ................. .-//'" Strain, p ,n./in.

..................

Fig. 4

-200 _

Predictedstress-straincurvesdueto uniaxial cyclicloadingof a nickel-ba_l superalloy

(B 1900+Hf) bar at two different temperatures and at a strain rate of 100 p in./in.-s.

8
Plate Subjected to Line Heating

The thermo-elastic behavior of a flat plate subjected to local heating was investigated in the early 1950s

[13]. The investigation consisted of thcoretical study "bascd on approximate analytical solutions and an experimental

study of rectangular aluminum plates. In the experimental study, simple "tent like" steady-state temperature

distributions were introduced by heating a rectangular plate along a centerline with a heating wire and maintaining

constant temperatures along parallel edges by water flow through coolant tubes. Top and bottom surfaces of the

plate were insulated to produce uniform one-dimensional, linear temperature variations between the heated centedine

and cooled parallel edges. In the experimental study [13], in-plane plate displacemcnts were permitted to occur freely,

but out-of-plane displacement were constrained by restraints that forced the plate to remain flat.

In this paper, the thermo-viscoplasdc behavior of a rectangular plate is studied assuming the plate is

pezfectly flat and a state of plane stress occurs. Tbe rectangular plate subjected to line heating along the ccnterline is

shown schematically in Fig. 5. The plate is subjected to a uniform heating rate, Q, 0.73 Btu/ft-s and the line

heating width on the plate is 0.125 in. The top and bottom surfaces of the plate are assumed to be perfectly

insulated, and with the assumption of constant thermal properties, file transient temperature distribution can be

determined from a one-dimcnsional analytical solution [14]. For this study, the temperature T(y,t) was computed

from,

T(y,t) Q(I- y) 4 QI _-' _ .-K(2n+l)2n2t- (2n+l)n( /- Z) = ' " 2., _ expl _ I sin

2k kn 2 ,,.o (2n+ I)" 4 l" 2 I

(11)

where l = 5.0 in. is the plate half-width; k is the thermal conductivity; I¢ is the thermal diffusivity; and t is time.

The initial temperature of the plate is assumed to be at room temperature (530 °R). Transient temperature

dislxibudons, calculated using eq.(l 1), are shown in Fig. 6 at four instants of time. For times up to 200 s, steep

temperature gradients occur only over the central one-half (-2.5 < y < 2.5) of the plate. The highly localized

temperature change is characteristic of low thermal conductivity nickel-based materials. Because of symmetry, only

one-fourtlt of the plate was modeled in the structural analysis. A reference temperature of 530 °R was assumed as the

stress-free temperature. The finite-element model and the structural boun "dary conditions are shown in Fig. 7. A

stretched mesh was used with small elements along the centerline and near the edge in the region of the steep

temperature gradients. The plate structural response was first computed assuming elastic behavior and then

viscoplastic behavior.

Coolant

Applied _

heating.

Coolant

J 15 '/--Computational

model

P,as,,c,ube / \

Test panel--/

j''"

Fig. 5 Schcmatic diagram of a rectangular plate subjected to line heating along a centerline.
5

4

3 j lx

y, in. ', _-- 200 s

2 \ 100 s

0 1000 2000 3000

Temperature, °R

Fig. 6 Plate transient temperature distribution in the direction perpendicular to the centerline.

546 Nodes

500 Quads

B(.3, .075)

A(.3, .025)

1t

. r

• _- _ _

I

•lli I

| • :

I ! " L "

e5 II

Fig. 7 Finite-clement model and structural boundary conditions of one-quarter of the plate

heated along centerlinc at y--0.

X

.

10
Temperature histories and comparisons of the elastic and viscoplasfic strc_s (Ox) histories at two locations

(A and B as marked in Fig. 7) near the center of the plate are shown in Fig. 8. For elastic behavior, the stresses, Ox,

increase monotonically as temperature increases with time (Fig. 8). In contrast to the elastic model, the vi_coplasfic

model predicts yielding early in the response with the stres_s remaining nearly constant at 100 ksi for over 100

_onds. The stress at point A decree, s after time t = 120 seconds due to the higher temperature at point A and the

softening of the ma_rial (_e Appendix). A further underslanding of the plate response is provided by Figs. 9-10,

which show the variation of Ox (x,0,t) anti oy (7.5",y,t) for the elastic and vi_oplastic analysis respectively. As

might be expected the elastic analysis grossly over-estimates the stress levels. The vi_oplastic predictions for O'x

show that the plate yields along the center line at y = 0 in a region -6 in. < x < 6 in. very early in Ihe respon_ at t

= 10 s. In contrast, the viscoplastic predictions for Oy along the edge at x = 7.5 in. (scc Fig. 10) show that the

behavior is elastic early in the respond, but yielding occurs perpendicular to the plate centerline at about t ,, 30 s.

Contour plots of Ox stres_s on the deformed shape of the plate, given in Fig. ! 1, show regions of maximum stress,

Ox, for elastic and vi_oplastic analysis. The viscoplastic analysis shows that the region near the centerline has

reached saturation stress and stresses remain almost constant at about -110 ksi in this region with increasing

temperature.

3000 [- __ Location A 0 F. -- Location A

,,]_"-..... Location B

.....

2000 I \ ""'. _.-Vnscoplastic

T,°R -200 _- _ "'-.. ._- Elastic

1000

0 I "1 l l

0 100 200 0 100 200

Time, s Time, s

(a) Temperature variation with time (b) Stress variation with time

Fig. 8 Temperature histories and compari_n of elastic versus vi_oplastic o, stress histories

at two locations in a plate heated along its centerlinc.

A further measure of the thermally-induced plastic behavior was derived by calculating an effective plastic

strain, which is ba,_d on the normal and shearing plastic strain components. The effective plastic strain component

is defined by [15],

f-Pecd_--'-- _ _V/(E_-E_)2+(£_'I_2+(E["d_ 2+ _(._p2 +,p2 +.,(_2) (12)

Contours of the effective plastic strain are pk)tted at four different of times on the deformed plate (see Fig. 12).

ll
0

-100

ax, ksi

-200

-300

0 4 8 0

x, in.

(a) Elastic analysis

Yt

X /--

-- 0 // /

ksi

........ 1.//xj -,oo , oOo s

/- 100 s ./-" ,., " '-180 s

=._/_._._._.7._......,. I

l,_y.s. -aoo

>" J I

4

x, In,

(b) Viscoplastic analysis

Fig. 9 Variation of o, sUcsses along the, centerlinc of the plate.

8

t

5- t x 5 il1

4 4-

y'in 3 - !li 3 /

,I- _] //-100 s ,.,n. 2 1il O0 s

2/ t'.,//F3os __.__._3os

",,_4///- 10 s 10 s

/ \'-,/7"--,.."'---

_. ,",,:/'-.,-"-_ '-, o_ '/6 ' '

0 ,, 0 100 200 300 0 200 300

6y, ksi Oy, ksi

(a) Elastic analysis (b) Viscoplastic analysis

Fig. 10 Variation of oy stresses along the free cdge of the plate.

12 '
-290 ksl \--110 ksi

S

-360 ksi -110 ksi

(a) Elastic analysis (b) Viscoplastic analysis

Fig. 1! Elastic and vi_oplastic ox stress contours on deformed plate (plate heated along centerline, y = 0).

30 s

y

5O s

_-5000 3000 -/ x 22000 32000 -/

Fig. 12 Effective plastic strain contours, indicating the plastic region and the maximum plastic

strain at different times.

33 _
These plots indicate the evolution of the plastically deformed regions near the centerline, The plots show that early

in the response/or 0 s < t < 50 s the largest inelastic behavior occurs near the plate's center and along the plate's

ccntcrline, which are primarily due toG, stress, llowever, later in the response for 150 s < t < 180 s, the largest

plastic strain occurs on the plate centerline at the free edges at x = :t:7.5 in. which is induced by the dominant Oy

stress at that location. These results indicate that after the plate returns to room temperature permanent contractions

can be expected along the plate ccntcrline at y = 0, and permanent elongations can be expected along the edges x =

:t:7.5 in.

Computational results for this example problem indicate that large, local temperature gradients induce

significant plastic behavior over a narrow band along the plate centerline. The viscoplastic analysis of the plate was

performed using a time step of 0.1 s. The analysi._ required two hours of CPU time on CRAY-2 to obtain the plate

viscoplastic response. Compulation of elastic solutions at the selected times (one _cond interval) required 800

_conds of CPU time.

Engine Cowl Leading _lge

Aerodynamically heated leading edges are a critical design problem for scramjet engine structures of

hypersonic vehicles because of the highly localized, intense aerothermal loads induced by shock-shock interactions.

The flow interaction at the scramjet engine inlet for the aerospace plane cowl leading edge is schematically shown in

Fig. 13. The problem has been the subject of a number of studies [1,2]. A flow-thermal-structural study Ill

describes a finite-element approach used in prediction of the aerothermal loads, the structural temperatures, and the

structural rcspon_ of a convectively-cooled leading edge. In this study, super-thermal conductivity materials were

evaluated, relative to their ability to alleviate elevated leading edge temperatures, temperature gradients, and attendant

stress levels. Another study [16] evaluated an adaptive, unstructured finite-element approach for integrated fluidthermal-structural analysis by analyzing a convcctively-cooled leading edge made of copper. A recent study [17]

describes a three-dimensional thermal-suuctural analysis of a swept cowl leading edge subjected to interference

heating. In the latter study [171, the computed peak stresses were beyond the elastic range, and a need for the plastic

analysis was indicated.

Vehicle nose

bow shock

 Shock-shock

ctlon

 nal

 °0,ne inlet -

Cowl C bo

shock

Fig. 13 Schematic diagram of flow interaction at _ramjet engine inlet for aerospace plane cowl leading edge.

14
The purpo_ of this example problem is to evaluate the significance of vi_oplastic behavior on the thermal

structural response of a leading edge made from the low thermal conductivity, high ductility, B1900+Hf nickel

superalloy material. This material was selected because the material constants for vi_oplastic model are available

for a wide range of temperatures. The leading edge geometry and boundary conditions are shown schematically in

Fig. 14. The outer edge is subject to intense, highly localized, transient aerodynamic heating and emits radiant

energy to space. The inner surface is convectively cooled by the direct impingement of a hydrogen flow at a constant

temperature of 50 °R. The coolant convective Coefficient is 7.8 Btu/fl2-s-°R. The coolant flow occurs at an internal

pressure of 1000 psi which is assumed to bc uniform over the internal surface. The leading edge has an external

radius of 0.125 in. and a wall thickness of 0.015 in. The assumed struclural boundary conditions, shown in Fig. 14,

prohibit horizontal displacement at the supports, but permit the leading edge to expand or contract in the vertical

direction.

The specified aerodynamic heating on the outer surface of the leading edge is shown in Fig. 15. The heat

flux shown is a consequence of a shock interaction that occurs a.sthe vehicle accelerates through Mach 16. The

interaction results from the vehicle's no_ bow shock sweeping across the leading edge and intersecting the leading

edge bow shock (see Fig. 13). The distribution shown in Fig 15a is based on experimental and computational

studies described in [1] and the references contained therein. The a._.sumedtime history of the aerodynamic heating is

shown in Fig. 15b. The leading edge is subjected to the intcn_ heat flux pulses each of 6 ms duration. The shock

on the leading is assumed stationary during this 6 ms duration, which repre_nts the time required for the shock to

sweep across the leading edge. In addition to the aerodynamic heating, an external surface pressure with a peak value

of 1000 psi is also impend. The spatial and temporal distributions of the external surface pressure are the same as

the heat flux (.¢¢eFig. !5a and 15b).

Aero. Heating

390 Nodes _._

626 Triangles

Radia_o_ 0_ 0.25 in.

Internal pressure I

__ 1000 psi uniform) [ Aero. I

pressure

(1000 psi peak) ---_0.015 in. _1,

Fig. 14 Finite-element model of the convecfively-cooled leading edge structure with applied loads

and boundary conditions.

15
Qo- 30000 Btu/ft2-s

Po - 1200 psi

1

0-90- -45 L 0 L 45 9C) O0

O, degrees

w

F24ms

/ 6ms _/_ 48ms-_,

25 50

Time, ms

Fig. 15

(a) Aerodynamic heating and (b) Peak loading histrocy

pressureprofile

Aerodynamic heating and external pressure profiles along the outer surface and the peak load history.

The finite-element two-dimensional thermal and structural analyses were performed using the unstructured

mesh containing 390 nodes and 626 triangles which is shown in Fig. 14. This mesh was developed in reference [16]

using an adaptive procedure that clustered small elements in regions of steep temperature gradients. Plane strain

condition is assumed for the structural analysis. A time step of 0.01 ms was chosen for the viscoplasdc analysis.

The computations were performed on CRAY-2 computer and the viseoplastic analysis required 3.0 hours of cpu

time. The analysis required 1000 cpu seconds to obtain elastic solutions at desired times.

Temperature histories predicted at two locations on the leading edge are shown in Fig 16. The temperat_e

histories correspond to two points direcdy under the most intense heat flux, point A on the outer surface and point B

the inner surface (Fig. 16a). The "saw-tooth" response at point A corresponds to the heat pulses shown in Fig.

15b. The initial heat pulse causes the surface temperature to rise rapidly to a peak of 2550 °R at 6 ms. After the

peak, the temperature decays to about 900 ORbefore the second heat pulse increases the temperature at 24 ms. The

temperature rises again to a second peak of about 2550 °R at 30 ms. The temperature response at point B on the

inner surface shows a relatively small increase of about 200 °R corresponding to the external heating. This small

temperature change shows the predominant role of the intem'al convective cooling. Temperature contours at 6 ms are

shown in Fig. 16b. These contours show the highly localized heat-affected region, and the very steep temperature

drop from the outer to the inner surface.

Using these prescribed temperatures, the structural response was computed quasi-staticaUy assuming: (1)

elastic material behavior, and (2) inelastic material behavior according to the Bodner-Partom model.

Circumferential stress histories at points A and B, as predicted by the elastic and viscoplastic analyses, are

shown in Fig. 17. At time t = 0 when the exter,ml pressure, i,_ternal pressure, and external heating are

instantaneously applied, local bending introduces compressive initial stress on the outer surface and tensile stress on

the inner surface. "lhereafter, the stress histories reflect a complex inteq_lay between thermal stress and temperaturedependent inelastic material behavior. During the first heat pulse, temperature on the outer surface rises very rapidly

introducing high compression stresses at point A. The viscoplastic analysis predicts yielding within 1 ms. Point B

at this time experiences small compressive stresses. Temperature on the outer surface rises to the peak value of

about 2550 °R in only 6 ms during the first heat pulse. On die outer surface, the material softens with a steep

reduction in the elastic modulus. At die same time, the steep temperature gradient introduces a significant thermal

bending moment. After the heat pulse ends at 6 ms, the bending moment persists, causing increasing tensile

16
stresses at point B until a peak is reached at about 8 ms. The vi_oplastic analysis predictions suggest slight tensile

yielding for point B at this time. From 6 ms until the second heat pulse occurs at 24 ms, the stress at point A

remains roughly constant, but the stress at point B decre&_es as the wall temperature gradient diminishes. At 24 ms,

the second heat pul_ begins, temperature incre&ses rapidly and the stress histories tend to repeat their first cycles.

The axial stre.qs_s (perpendicular to analysis plane) for the elastic analysis are much higher than the in-plane stresses

(circumferential and radialst_sses) due to plane-strain constraint, but the vi_oplastic axial stres_s are of comparable

value to the vi_oplastic in-plane stresses.

During the two heat pul_s, the predicted elastic and viscoplasdc stresses follow each other closely. The

most significant difference in stresses occur at 25 ms. Compari_n of stress contours for the elastic and viscoplastic

analy_s at t = 25 ms is shown in Fig. 18. The stre_s contours show steep gradients from the outer to inner surface,

and stres_s are confined locally to the heat affected region as might be expected. Although the viscoplastic and

elastic stresses are at similar levels over much of the response, consideration of the stress levels alone does not fully

explain the behavior. At saturation stress level, significant plastic strain occurs. This is particularly true at point A

where the highest temperature is experienced. At very high temperature.s, the saturation stress level is reduced (see

Fig. !) and considerable plastic strain may be caused by relatively low stresses.

Contours of effective plastic strains at four times in the response are presented in Fig. 19 (for reference,

yielding occurs at a strain of about 2000 11 in./in, at room temperature). The figure shows that point A sustains a

plastic strain of 12000 It inJin, over the period of 24 ms due to the first heat pulse. The second heat pulse causes

less plastic strain due to built-in residual stresses, but a total of 17000 It inJin, accumulates after 48 ms. The

contours suggest that the largest plastic strains occur near the outer surface although small plastic strains are

experienced through the wall thickness. These plastic strains will result in permanent deformations and residual

stresses after the leading edge temperatures have returned to steady-slate conditions.

The elastic and viscoplastic analyses of a two-dimensional model of the leading edge provide valuable

insight into the local thermal-structural behavior at elevated temperatures. For the nickel superalloy material studied,

significant permanent plastic strain re.qulLs. To understand the three-dimensional viscoplastic behavior of the leading

edge, further study is required.

3000 -

2000 A

T,°R

1000 -

ms 25 200 °R

- -,7-.- .__ LS--_..f .... __._

--"F -Tr" ,,,:1

0 25 50

Time, ms

(a) Temperature variation with time (b) Temperature contours at 6 ms

Fig. 16 Temperature history of two locations and the leading edge temperature contours at 6 ms.

17
200 I- /_.Zone for external heating & pressures L,/ _v,sco , st,c

P'-_ __ ___ ..... Elastic

O[ //." _.." _-- B

-100_/....: i_ / \\ ..............

I'.V/ 6 ms i....-'"" _ A

illmsi 24ims 48ms

-2oo_ _ _5 5'o

Time, ms

Fig. 17 Comparison of circumferential stresses at two locations for elastic and vi_oplastic analysis.

-140 ksi

Elastic analysis

25 ms

20 ksi

-95 ksi

Viscoplastic

analysis

25 ms

30 ksi

Fig. 18 Circumferenlial slre,_qcontours for elastic and viscoplastic analysis.

18
Fig.19 Effectiveplasticstraincontoursindicatingthepropagationofplasticregionand maximum plastic strain.

CONCLUDING REMARKS

A thermo-vi_oplaslic computational method for structures subjected to severe local heating is used to

investigate the transient thermal-structural behavior of .severalproblems of current intere_st.The method employs the

Bodner-Partom unified vi_oplastic constitutive model implemented in a finite-element approach for quasi-static

thermai-sU'uctural respond.

The analysis method was verified by comparing with previous uniaxial experimental and numerical results

using B1900+Hf nickel superalloy material. The two application problems considered were: (I) a flat plate in plane

sa'ess condition subjected to line heating along the plate centerline, and (2) a convectively-cooled leading edge in

plane-strain condition subjected to localize! heating representing a shock-shock interaction. Elastic and viscoplastic

analyses were compared for each of the problems. The elastic analysis tended to over-estimate stresses and, of

course, did not predict plastic deformations. The viscoplastic analysis predicted local yielding and provided insight

into the history of multi-dimensional inelastic behavior induced by transicnt heating. The viscoplastic analy_s are

relatively expensive in comparison to elastic analysis and lack experimental validation. Yet this study shows that

such analyses can have a valuable role in understanding complex transient structural behavior at elevated

I_.mperalures.

19
.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12,

13.

14.

15.

16.

17.

REFERENCES

P. Dechaumphai, E. A. Thorntonand A. R. Wieting, "Flow Thermal-Structural Study of Aerodynamically

lleated Leading Edge," Journal of Spacecraft and Rockets, Vol. 26, No. 4, July 1989, pp. 201-209.

M. W. Melis and tt. J. Gladden, "Thermostructural Analysis with Experimental Verification in a High

Heat Flux Facility of a Simulated Cowl Lip," AIAA Paper No. 88-2222, April 1988.

S. R. Bodner and Y. Partom, "Constitutive Equations for Elasfc-Vi_oplastic Strain-Hardening Materials,"

Journal of Applied Mechanics, Vol. 42, 1975, pp. 385-389.

A.K. Miller (edito0, _tiltllive Equations for Creep and Plasticity, El_vier Applied Science

Publishers, 1987.

E. A. Thornton, J. T. Oden, W. W. Tworzydlo and S. K. Youn, "Thermo-Vi_oplastic Analysis of

Hypersonic Structures Subjected to Severe Aerodynamic Heating," AIAA Paper No. 89-1226, April 1989.

H. J. Gladden, M. E. Melis and T. T. Mockler, "Thermal/Structural Analyses of Several Hydrogen-Cooled

Leading-FAge Concepts for Hypersonic Flight Vehicles," AIAA Paper No. 90-0053, January 1990.

V. Moreno and E. H.Jordan, "Prediction of Material Thermo-Mechanical Response with a Unified

Vi_oplasfic Constitutive Model," AIAA Paper No. 85-0622, April 1985.

S. R. Bodner, "Review of a Unified Elastic-Vi_oplastic Theory," Unified Constitutive F..__uationsfor

Creep and Plasticity, A. K. Miller, Editor, El_vier Applied Science Publishers, 1987.

U. S. Lindholm, K. S. Chan, S. R. Bodner, R. M. Weber, K. P. Walker and B. N. Cassenti, "Constitutive

Modeling for Isotropic Materials (HOST)," NASA CR- i 74980, July !985.

K. S. Chan, U. S. Lindholm, S. R. Bodner, J. T. Hill, R. M. Weber, T. G. Meyer, "Constitutive

Modeling for Isotropic Materials (HOST)," NASA CR-179522, July 1986.

A. K. Pandey, P. Dechaumphai and A. R. Wieting, "Thermal-Structural Finite Element Analysis Using

Linear Flux Formulation," AIAA Paper No. 89-1224, April 1989.

K. H. Huebner and E. A. Thornton, The Finite Element Method for Engineers, Second Edition, John

Wiley and Sons, 1982.

R. R. Heldenfels and W. M. Roberts, "Experimental and Theoretical Determination of Thermal Stress in a

Plate," NACA TN-2769, 1952.

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford University Press, 1980, p. 113.

J. O. Smith and O. M. Sidebottom, Inelastic Behavior of Load Carryin. " g Members, John Wiley and Sons,

1965, p. 86.

P. Dechaumphai, "Evaluation of an Adaptive Unstructured Remeshing Technique forlntegrated FluidThermal-Structural Analysis," AIAA Paper No. 90-0556, January 1990.

S. P. Polesky, P. Dechanmphai, C. E. Glass and A. K. Pandey, "Three-dimensional Thermal-Structural

Analysis of a Swept Cowl Leading Edge Subjected to Skewed Shock-Shock Interference Healing," AIAA

Paper No. 90-1710, June 1990.

APPENDIX

Material properties for nickel-ba,_d supemlloy (B 1900+II0

Density (p) = 0.283 Ibm/in3

Thermal conductivity (It) and Specific tteat (c)

Temperature (°R) k (Btu/in-s-°R) c (Btu/lbm-°R)

0 ! .23x 10-4 0.12

500 1.80x 10-4 0.135

1500 3.25x i0 "4 0. !77

3000 6,00x10-4 0.26

2O
StructuralConstants

1. Elastic Constants

Temperature (°R)

492.0

762.0

1032.0

1302.0

1572.0

1842.0

2112.0

2382.0

2652.0

2. Thermal Expansion Coefficient (ix)

Temperature (°R)

492.0

852.0

1212.0

!572.0

1932.0

2292.0

2652.0

3. Viscoplastic constants

Temperature-independent constanLq

ml= 1.8616x 10-3 psi-l;

ZI = 43.51 x 104 psi;

1"I=I"2= 2;

E (psi x 106)

28.88

28.91

28.31

27.09

25.30

22.97

20.13

16.82

13.07

ct (in/in-°R)

0.638 x 10-5

0.744 x 10-5

0.755x 10-5

0.783x 10-5

0.855x 10-5

0.898 x 10-5

0.922 x 10-5

m2-- 1.0480 x 10.2 psi-I

Z2 = 16.67 x 104 psi

1)_ = I0000 s-!

O(psi x 106)

! 2.54

12.22

! ! .94

11.62

!!.14

10.40

9.31

7.76

5.65

Temperature-dependent conslanL_

Temperature(°R) 1860 20(_0 2260 2460

n 1.055 1.03 0.85 0.70

Zo= Z2 (ksi) 391.6 34g.0 275.5 174.0

Ai = A2(s-!) 0.0 0.0055 0.02 0.25

w =

21
Report Documentation Page

#

1. Repo_ No 2 Government Accession No

NASA TM 102761

4, Titleand Subtitle

Finite Element Thermo-Viscopiastic Analysis of

Aerospace Structures

7. Author(s)

A jay K.

Pramote

Earl A.

Pandey, Lockheed Engineering & Sciences Co.

Dechaumphai, NASA Langley Research Center

Thornton, University of Virginia

9. Performing Organization Name and Address

Langley Research Center

Hampton, VA 23665-5225

12Spon_ring AgencyNameandAddress

National Aeronautics and Space Administration

Washington, D.C. 20546-0001

3. Recipient's Catalog No.

5. Repo_ Date

November ]990

6. Performing Organization Code

8. Performing Organization Report No.

10. Work Unit No.

506-40-21

1t. Contract or Grant No.

13. Type of Report and Period Covered

Technical Memorandum

,1_4. Sponsoring ,_gency Code

15. Supplementaw Notes

Presented at the First Thermal Structures Conference, Charlottesville,

November 13-15, 1990.

Virginia,

16. Abstract

The time-dependent thermo-viscoplastic response of aerospace structures

subjected to intense aerothermal loads is predicted using the finite-element

method. The finite-element analqsis uses the Bodner-Partom unified viscoplastic

constitutive relations to determine rate-dependent nonlinear material behavior_

The methodology is verified by comparison with experimental data and other

numerical results for a uniaxially-loaded bar. The method is then used

i) to predict the structural response of a rectangular plate sub]ected to line

heating along a centerline, and 2) to predict the thermal-structural response of

a convectively-cooled engine cowl leading edge subjected to aerodynamic shockshock interference heating. Compared to linear elastic analqsis, the viscoplastic

analysis results in lower peak stresses and regions of plastic deformations.

/

i

s

/

17. Key Words(Sug_stedbyAuthor(s))

Viscoplastic Analysis, Elastic Analysis

Aerospace Structures, Finite Element

Methods

Thermal-Structural Analqsis

19. SecuriW Cla_if.(ofthisreport)

Unclassified

NASA FORM 1626 OCT 86

18. D_tribut_nStatement

Unclassified - Unlimited

Subject Cateqory - 39

_, S_urityCla_if.(ofthispa_)

Unclassified

21. No. of pages 22. Price

22 A03

i