A Model for Predicting the Fatigue Life of Fiber-Reinforced 

Polymeric Composites in Offshore Applications 

Tinh Nguyen, Hai Tang, Tze-jer Chuang, Joannie Chin, Felix Wu, and Jack Lesko*

National Institute of Standards and Technology, Gaithersburg, MD 20899

*Virginia Polytechnic Institute and State University, Blacksburg, VA 24061

ABSTRACT

A model based on cumulative damage has been developed for predicting the fatigue life 

of fiber-reinforced polymeric composites used in offshore environments. The model incorporates 

applied stress, stress amplitude, loading frequency, residual tensile modulus, and material 

constants as parameters. The model is verified with experimental data from a glass fiber/vinyl 

ester composite fatigued under different environmental conditions. Specimens are subjected to 

tension-tension fatigue at four levels of applied maximum tensile stress at two different 

frequencies while exposed to air, fresh water, and sea water at 30 °C. Both the residual 

mechanical properties at specified loading cycles and the number of cycles at which the 

specimens fail are measured. For the material used in this study, the loss in mechanical 

properties (residual tensile strength and modulus) in salt water is approximately the same as that 

in fresh water, and the fatigue life of the composite in these aqueous environments is shorter than 

that in air. The S-N curves for specimens subjected to the three environments have 

approximately the same slope, suggesting that the failure mechanism does not change with these 

environments. Furthermore, specimens that are fatigued at a lower frequency failed at a lower 

number of cycles than those tested at a higher frequency. Numerical analysis is performed using 

the fatigue experimental data to determine the material constants of the composite. The model 

agrees well with the experimental data, and it can be used to predict the fatigue life of polymeric 

composites subjected to an applied load in different environments or the residual tensile modulus 

after a number of loading cycles.
2

INTRODUCTION

Over the last three decades, fiber-reinforced polymer (FRP) composites have been used 

extensively in a wide range of aerospace and military applications. There is a growing interest in 

extending the use of these materials into civil engineering structures such as offshore platforms 

[1,2]. An urgent need exists to develop quantitative analysis and methodology for assessing the 

safety and reliability of using polymer composites in these applications. One particularly critical 

issue in regard to using polymer composites in structural applications is their fatigue reliability in 

different environments and loading conditions. Fatigue damage in polymer composites for noncivil engineering applications has been extensively investigated [3-7]. Studies of the effects of 

water or salt water on the fatigue behavior of polymer composites have also been widely 

reported [8-14]. However, there is little quantitative research on the effects of civil engineering 

environments, namely water, sea water, concrete pore solution, temperature, ultraviolet radiation, 

and loading on the fatigue of polymeric composites. The main objective of this study is to 

develop experimentally verifiable models for predicting the fatigue life of polymer composites 

used in civil engineering structures.

This report presents the development of a fatigue model, results of the fatigue 

experiments for a glass fiber/vinyl ester composite subjected to different loading and 

environmental conditions, and the verification of the model. The model can be used to predict 

fatigue life of polymeric composites under an applied load and to estimate the residual tensile 

modulus after a certain number of loading cycles.

DERIVATION OF FATIGUE MODEL FOR FIBER-REINFORCED

POLYMER COMPOSITES FOR OFFSHORE APPLICATIONS

Fatigue Model

The basic failure mechanisms most commonly observed in fatigue of fiber-reinforced 

polymer composites are matrix cracking, fiber-matrix debonding, ply delamination, and fiber 

breakage [3,4,15-18]. Cumulative damage as a function of loading cycles may be generally 

described by the curve given in Figure 1, where D represents cumulative damage, N is the 
3

number of loading cycles, and Nf is the number of cycles to failure. The change in material 

modulus has been commonly used to express the state of damage in polymer composites 

[3,15,16,19]. In this work, cumulative damage, D, is defined as: 

0

1

E

E

D = − (1)

where E and E0 are the residual and initial moduli, respectively. For typical multidirectional 

laminate polymer composites, matrix cracks are initiated early in the fatigue process due to high 

stress concentrations. As the crack density increases, stress redistribution limits the initiation of 

new cracks. At this stage, the damage appears to grow at a constant rate as the cyclic loading 

continues. In the region of damage, fiber/matrix interfacial debonding, delamination of the 

laminates, and fiber breakage begin to occur and gradually increase. As the fiber breakage 

progresses and intensifies, the rate of fiber fracture increases rapidly and, finally, the composite 

ruptures.

The incremental damage per loading cycle, dD/dN, as a function of cumulative damage, 

D, which may be derived from Figure 1, is displayed in Figure 2. The initial damage growth per 

loading cycle may be mathematically expressed in either of the following two functions:

1

1

n D

C

dN

dD

= (2) 

 or

n D

B e

dN

dD 1

1

−

= (3) 

where N is the number of loading cycles, and C1 and B1 and n1 are constants with n1 > 1. 
4

Figure 1: Cumulative damage, D, as a function of the ratio of the number of loading cycles, N, 

to the number of cycles at failure, Nf.

Figure 2: Damage growth as a function of cumulative damage.
5

The damage growth rate near the end of the life cycle before failure may be described by:

2

(1 )

2

n D

C

dN

dD

−

= (4)

or 

n D

B e

dN

dD 2 = 2

(5)

where B2, C2 and n2 are constants and n2 > 1.

The general fatigue model as depicted in Figure 2 may take the form:

= +

1

1

n D

C

dN

dD

2

(1 )

2

n D

C

−

(6)

or

= +

−n D

B e

dN

dD 1

1

n D

B e

2

2

(7)

However, our experimental data on the cumulative damage, D, as a function of the 

number of loading cycles, N, for a glass fiber/vinyl ester composite do not show any initial 

strength degradation until the number of loading cycles exceeds 1000. Further, the initial damage 

does not reveal any evidence of an abrupt growth phenomenon (see, e.g. Figure 12). Instead, the 

normalized modulus data shows smooth, gradual decrease from the undamaged state. Therefore, 

for this composite material, C1 is negligible in comparison with C2. Consequently, the damage 

rate per loading cycle may be expressed by the following function:

n

D

C

dN

dD

(1 − )

= (8)

or

nD Be

dN

dD

= (9)
6

where B, C and n are again material constants. In Figure 3, the curves represented by Eq. 8 and 

Eq. 9 are compared with B = C = 1. From this figure, Eq. 8 appears to better represent the 

sudden rupture of the composite material as D approaches unity. Therefore, Eq. 8 will be 

expanded to describe the fatigue process for vinyl ester/E-glass fiber composites used in civil 

engineering applications.

Figure 3: Comparison between damage growth as a function of exponential damage and inverse 

exponential damage for different values of n.

Besides the state of damage, the maximum stress and stress amplitude may also have 

substantial effects on the damage growth. Accordingly, Eq. 8 may be rewritten as:

n

m

o

D

C

dN

dD

(1 )

( ) max amp

−

=

s s

(10)

where

Co, m and n are the material constants,

σmax is the maximum stress,
7

σmin is the minimum stress, and

σamp is the cyclic stress amplitude and is equal to (σmax - σmin)

We can express the maximum stress and stress amplitude in normalized terms with respect to the 

ultimate tensile strength (σult) and the minimum to maximum stress ratio R:

ult

max

max

s

s

S = (11)

(1 ) max (1 )

max

min

ult

max

ult

max min

ult

amp S amp = − = S − R

−

= =

s

s

s

s

s

s s

s

s

(12)

Substituting for σmax and σamp in Eq. 10 and then simplifying, the equation becomes:

n

m

D

S R

C

dN

dD

(1 )

( max (1 )) 2

−

−

= (13)

where Smax ≤ 1 and C = Co(σult)

2m

.

Integrating Eq. 13 and substituting the initial condition, i.e., D = 0 when N = 0, we 

obtain:

 C S R N

n

D

n

m

n

( (1 ))

1

(1 )

1

1

max

2

1

= −

+

−

−

+

+

(14)

where 0 ≤ D ≤ 1.

When the number of loading cycles approaches the maximum number of cycles Nf, D 

approaches 1. Under this condition, Eq. 14 becomes:

f

m C S R N

n

( (1 ))

1

1

max

2

= −

+

(15)
8

Eq. 15 is equivalent to the power formula often seen in the literature as:

max N = constant f

A

s (16)

which can also be expressed as:

 log constant

1

log max = − N +

A

s f

(17)

where A is a constant.

In the model validation section, this equation will be compared with the following 

expression: 

max = m log N + constant s f

(18)

which is commonly used for characterizing fatigue life, Nf, of materials.

Determination of the Material Constants

The constants m, n, and C in Eqs. 14 and 15 can be obtained if experimental fatigue data 

of polymer composites are available. This section describes the methods for determining these 

constants. 

Taking the logarithm of both sides of Eq. 15, we obtain:

log[(n + 1)C] + 2m log(S max ) + m log(1 − R) + log N f = 0 (19)

Therefore, the linear regression slope (-1/2m) of the Smax versus Nf plot in log-log scale can be 

used to compute the material constant m.

To determine the value of n, we need to use the solution in the intermediate state with 

partial damage D in Eq. 14. Since D is usually small until it approaches the maximum number of 

life cycles, we can expand the (1-D)n+1 term with a Taylor's series and omit the higher order 

terms. Accordingly, Eq. 14 becomes:

D C S R N

n n

n D

n

m

( (1 ))

2!

( 1)

1 1 ( 1)

1

1 2

max

2

= −













+

+

− + + −

+

K (20)
9

For the same applied load where the residual modulus is measured at two different numbers of 

cycles N1 and N2, the respective damages D1 and D2 can be computed, and Eq. 20 gives:

1

2

1 1 max ) ( (1 ))

2

(1 D C S R N

n

D

m

− = − (21)

2

2

2 2 max ) ( (1 ))

2

(1 D C S R N

n

D

m

− = − (22)

Dividing Eq. 21 by Eq. 22 and canceling the common terms, we have:

 

2

1

2 2

1 1

)

2

(1

)

2

(1

N

N

D

n

D

D

n

D

=

−

−

(23)

 or

 )

2

) (1

2

) (1

2

(1 2 12 2

1

2

2

1

1 D

n

D K

n

D

D

N

N

D

n

− = − = − (24)

where

1

2

2

1

12 D

D

N

N

K = (25)

Therefore,

( )

2(1 )

1 2 12

12

D D K

K

n

−

−

= (26)

The remaining parameter C can then be determined using the plot of maximum stress versus the 

number of cycles at failure (S-N curve) in Eq. 15 and the previously determined m and n values.

It should be noted that the material constants, C, m, and n could be dependent on the 

fatigue frequency. However, as will be shown in the following section, our experimental data for 
10

two frequencies, 2 Hz and 10 Hz, indicate that frequency has little effect on m or n, but it has a 

substantial effect on C.

EXPERIMENTAL PROCEDURE1

Materials

The specimens used for this study were obtained from a pultruded vinyl ester/E-glass 

fiber composite supplied by Strongwell (Bristol, VA). The composite is similar to the material 

used in the construction of the Tom's Creek composite bridge in Blacksburg, VA. The flat 

laminate was constructed from unidirectional roving and continuous strand mat. The fiber 

volume fraction was approximately 28 % to 30 %, as determined by combustion of the matrix. 

Coupons having dimensions of 22.9 cm long x 2.54 cm wide x 0.32 cm thick were cut from a 

40.6 cm x 122 cm plate. The edges of the specimen were sanded to ensure uniformity and were 

coated with a filled epoxy material to minimize the sorption of exposure solution into the 

composite from the edges. 

Preparation of Test Solution 

A solution containing 3.5 % mass fraction of NaCl in distilled water was used to simulate 

sea water. To ensure that all specimens were exposed to approximately the same salt

concentration, a fresh solution was used for each batch of specimens. In addition, the salinity of 

the solution was monitored by a salimeter and adjusted by adding fresh solution. A calibration

curve was generated for 1 % and 4 % mass fraction of NaCl, and the concentration was plotted 

against their corresponding volume displacements, as measured by the salimeter. Subsequent

measurements were checked against these established conditions.

 

1

 Certain commercial materials and equipment are identified in this paper in order to specify adequately the 

experimental procedure. In no case does such identification imply recommendation or endorsement by the National 

Institute of Standards and Technology (NIST), nor does it imply that they are necessarily the best available for the 

purpose.
11

Fluid Cell

A transparent fluid cell was designed to hold the salt solution and water around the 

specimens during the fatigue experiment. Figure 4 displays a typical fluid cell assembly with a 

test specimen in place. The fluid cell consisted of two poly(methylmethacrylate) (PMMA) plates. 

Each of the plates had a groove to hold the specimen and a central depression to contain the 

solution during cycling. Six screws were used to fasten the two plates together, enclosing the 

mid-section of the specimen. Silicone sealant was laid along the edges of the depression to 

prevent leaking during testing. In most cases, the fluid did not leak from the cell during cycling, 

but only when the specimen failed.

Figure 4: Fluid cell assembly used for fatigue testing of polymer composite in simulated sea 

water at a specified temperature.

Procedure for Saturation of Specimens in Simulated Sea Water at 65 °C 

All composite specimens used for fatigue measurement in water and simulated sea water 

were presaturated with the respective solutions at 65 °C. This elevated temperature was used to 

accelerate the diffusion of solutions into the composite. Equipment used for the saturation 

consisted of a 76-liter tank made of transparent glass to facilitate experimental monitoring 

without disturbing the setup. Four aquarium heaters were mounted at four corners of the tank. A 

temperature controller was used to maintain the temperature at 65 °C. Two power pumps were 

Central depression

Specimen
12

placed in the tank to ensure uniform heat distribution in the solution during aging. The tank was 

insulated to reduce evaporation and heat loss. 

Edge-coated specimens were individually weighed before placing them into the 

saturation tank. Specimens were carefully arranged on a rack to ensure a uniform exposure of 

each specimen to the solution. The mass uptake of each specimen was measured daily by 

removing it from the tank, blotting, weighing, and placing it back in the tank. The result was the 

average of seven specimens.

Percent mass uptake was computed as follows:

% 100

1

2 1

x

M

M M

M 













 −

= (27)

Where: % M = percent mass uptake

M1 = dry mass

M2 = mass at specified time of immersion in solution 

Specimen Preparation for Fatigue Test

Specimens were prepared by enclosing the central portion of the specimen within the 

fluid cell (see Figure 4). The remaining equal lengths on either side of the cell were used for 

gripping. A length of 12.5 mm was left between the cell and the grips of the testing machine on 

either side of the cell. Thus, a length of 50 mm of the specimen resided in the grips. Specimen 

ends were wrapped with fine wire mesh to prevent slippage within the grip areas during testing. 

As indicated earlier, six screws were used to fasten the two PMMA plates together and silicone 

sealant was used along the edges of the cell depression to seal the cell during fatiguing. Prepared 

specimens were stored for approximately 6 h to allow the silicone to cure before fatigue cycling 

began. 
13

Measurement of Mechanical Properties and Fatigue Testing

Quasi-static Tensile

The quasi-static properties of the composite material were obtained for each batch of preconditioned specimens. A screw driven Instron tensile testing machine was used to measure the 

quasi-static tensile strength and stiffness. Strain data was recorded with an extensometer, and 

transverse and axial strains were obtained using CEA-13-125WT-3501strain gages. Gages were 

attached to the wet specimens using a cynoacrylate adhesive (M-bond 200)1. Specimens were 

tested at a displacement rate of 2.5 mm/min.

Fatigue Testing

Fatigue experiments were conducted under the following loading and environmental 

conditions: 

1) Loading mode: tension/tension 

2) R (minimum stress to maximum stress ratio) = 0.1

3) Loading level (Smax): 30 %, 35 %, 40 %, and 45 % of ultimate tensile strength 

(UTS)

4) Test frequency: 2 Hz and 10 Hz 

5) Exposure environment: air, filtered tap water, and 3.5 % mass fraction NaCl in 

distilled water (simulated sea water).

6) Test temperature: 4 oC, 30 oC and 65 oC. 

All fatigue specimens were tested using a 4.5 x 104 N MTS servo-hydraulic machine; a 

schematic sketch of the experimental setup is displayed in Figure 5a, and a corresponding 

photograph is shown in Figure 5b. The fluid cell (with the specimen enclosed) was connected to 

the solution circulation bath using polyurethane tubing. A thermometer attached to the cell 

provided a means to continuously monitor the temperature of the solution in the cell. After the 

appropriate stress levels were entered into the data acquisition system and the desired frequency 

(fatigue) was selected, the circulating system was turned on. This same procedure was employed 

for fatigue testing at 2 Hz and 10 Hz frequencies. S-N curves were generated for the composite 
14

material by cycling the coupons to failure and recording the number of cycles-to-failure. The 

residual mechanical properties of the material were measured after fatiguing the specimens to a 

preset number of cycles. The tensile modulus and Poisson's ratio of these specimens were then 

determined using a screw-driven Instron testing machine. 

Figure 5a: A schematic of the experimental setup to measure fatigue life of polymer composite 

in simulated seawater at a desired temperature.

Figure 5b: Photograph of the experimental setup to measure fatigue life of polymer composite 

in simulated sea water at a specified temperature. 
15

Test Matrix for Fatigue 

Table 1 presents the test matrix used to study the fatigue life of E-glass fiber/vinyl ester 

composites under different loading conditions at each temperature in each environment, i.e., 

ambient air, fresh water, or simulated sea water. Each batch of the saturated specimens was 

allocated to a particular set of tests. For instance, the batch of specimens used to generate S-N 

curve at 10 Hz was also employed to measure the residual mechanical properties at 10 Hz as 

well. The numbers noted in Table 1 at a given stress level and load cycles represent the number 

of replicates. For example, at 2 Hz, 35 % UTS load level and 104 cycles, three specimens were 

usually tested.

Table 1: Test matrix showing number of replicates performed for fatigue test of polymer 

composite under different loading conditions 

2 Hz

Preset Cycles

10 Hz

Preset Cycles

Load levels

(Smax)

1031041051031041052x 1055x105106

2.5x1065x106

30 % 0 0 0 0 0 0 0 1 2 1 1

35 % 3 3 3 5 5 6 0 0 4 0 0

40 % 3 3 3 5 5 6 3 0 0 0 0

45 % 3 3 3 8 6 6 0 0 0 0 0
16

RESULTS AND DISCUSSION

Uptake of Salt Solution in Composite

As indicated earlier, all specimens were fully saturated with water or simulated sea water 

prior to fatigue testing. Further, to speed up the sorption process, the solution was heated at 

65 °C. Figure 6 presents the sorption behavior of the simulated seawater in the composite as a 

function of time at 65 °C. Each experimental data point is the average of seven specimens. After 

approximately 450 h, the simulated sea water nearly saturates the composite, reaching 0.9 % 

equilibrium mass fraction. This immersion time was used for pre-saturating all remaining 

specimens before fatigue cycling. The dashed line in Figure 6 represents a theoretical prediction 

using a Fickian model described in Reference 20. 

0

0.2

0.4

0.6

0.8

1.0

1.2

0 5 10 15 20 25

Time1/2 (h1/2)

Figure 6: Sorption behavior of simulated sea water at 65 oC in the test composite (t

experimental values, n Fickian prediction). Error bars represent one standard deviation.
17

The experimental data of Figure 6 were used to calculate the diffusion coefficient of

composite in simulated seawater solution using the following expression (20):

















−

−

=

2 1

2 1

2

16 t t

M M

M

h

D

m

z p

(28)

where: Dz: the diffusion coefficient 

h : thickness of composite

M1, M2 : two levels of mass uptake in the linear region

t1, t2 : times in the linear region corresponding to M1 and M2.

Dz of the vinyl ester/E-glass fiber composite used in this study was approximately 2.9 x 10-8

cm

2

/s in simulated sea water at 65 °C. A similar value was recorded for specimens immersed in 

fresh water at 65 °C, suggesting that water was the main component sorbed in the composite. 

The diffusion coefficients for a vinyl ester resin commonly used for polymer/fiber composites in 

water at 30 oC and 60 °C have been reported to be 0.9 x 10-8 cm

2

/s and 2.4 x 10-8 cm

2

/s, 

respectively (21). 

Mechanical Properties 

Before Exposure and After Saturation with Fresh Water and Simulated Sea Water

Mechanical properties of the composite were measured before exposure (as received) and 

following saturation with fresh water and simulated sea water at 65 oC. As indicated earlier, 

eleven specimens were tested in the as-received condition and seven specimens were tested after 

exposure. Table 2 provides the tensile strengths, tensile moduli, and Poisson's ratios and their 

corresponding standard deviations for the composite exposed to three environments. Decreases 

of 25 % and 32 % in the tensile strength are observed for samples saturated with fresh water and 

simulated seawater at 65 oC, respectively. Further, saturation in water and the same salt solution 

also resulted in a reduction in the tensile modulus of 15 % and 11 %, respectively. The 

differences between samples tested before exposure and after water and simulated sea water 

saturation were statistically significant at 95 % confidence level, as determined by the t-test 

analysis. However, no significant difference was observed between the water-saturated and 
18

simulated sea water-saturated specimens for the same mechanical properties. On the other hand, 

the Poisson's ratio was essentially unchanged for both fresh water and simulated sea water 

exposure. 

Table 2: Mechanical properties of E-glass/vinyl ester composite before exposure and 

after saturation with fresh water and simulated sea water 

Dry Fresh Water Simulated Sea 

Water

Mean Std dev Mean std dev Mean std dev

Ultimate tensile strength (MPa) 212 17.9 158 13.4 144 14.4

Modulus (GPa) 15.55 0.66 13.28 1.20 13.85 4.01

Poisson's Ratio 0.31 0.03 0.31 0.03 0.33 0.04

Fatigue Experiments

As indicated in the experimental section, the effects of ultimate tensile stress, cyclic frequency, 

cyclic temperature, and exposure environments on the fatigue behavior of the composite were 

investigated to provide inputs for the fatigue model. The fatigue experiments were conducted at 

four different levels of ultimate tensile stress (UTS) and at two frequencies in air, fresh water and 

simulated seawater and at three different temperatures (4 oC, 30 oC, and 65 oC). Some 

representative data on the effects of these parameters on fatigue properties are presented in this 

section. However, discussions of these results are treated in the model verification section. 

The effects of cycling frequency, exposure environments, and the temperature at which 

fatigue experiments were conducted are illustrated in Figures 7, 8 and 9, respectively. In these 

figures, the normalized % UTS (with respect to the value of the as-received material) is plotted 

against the number of cycles at failure. Except for fatigue testing in air, all specimens were 

completely saturated with their respective solutions (18 days immersed in solution) before 

testing. Further, all failures were observed to occur within the fluid cell. 
19

Figure 7 is a S-N plot for specimens fatigued at 2 Hz and 10 Hz in simulated sea water at 

30 oC. A slightly steeper slope for specimens tested at 2 Hz is observed compared to those at 10 

Hz, suggesting that the deterioration of the composite was greater at 2 Hz than at 10 Hz.

0

10

20

30

40

50

60

70

80

90

100

102

103104105106107108

Nf (cycles to failure)

Sult = 144 MPa

0.9 % Moisture Content

R = 0.1

freq = 2 & 10 Hz

10 Hz (18 days)

2 Hz (18 days)

Figure 7: S-N plot of a vinyl ester/glass composite fatigued at 2 Hz (◊) and 10 Hz (•)

in simulated sea water at 30 oC.

The influence of the exposure environment on fatigue behavior of the composite is 

displayed in Figure 8. These S-N plots were generated for specimens cycled at 10 Hz in air, fresh 

water, and simulated sea water at 30 oC. Although the exposure environment appears to have 

little effect on the slope of the curves, the presence of aqueous solutions caused a substantial 

decrease on the total life time. Figure 8 indicates that subjecting this composite material to a 

fresh water or simulated sea water at 30 °C can result in a reduction of approximately one decade 

of its fatigue life as compared to that in air under the same loading condition. 
20

0

34.5

69.0

103.5

138.0

172.5

207.0

241.5

276.0

10 102103104105106107

Nf (# cycles to failure)

MPa)

Sult,dry = 212 MPa

Sult,wet (fresh water) = 158 MPa

Sult,wet (simulated sea water) = 144 MPa

as-received

fresh water (0.9 %)

R = 0.1

freq = 10 Hz

simulated sea water

(0.9 %)

Figure 8: S-N plot of a vinyl ester/glass fiber composite fatigued at 10 Hz in air( ), fresh 

water(•), and simulated sea water (o) at 30 oC.

Figure 9 displays the S-N data for the composite cycling at 10 Hz in simulated sea water 

at three different temperatures (4 oC, 30 oC and 65 oC). These temperatures represent a wide 

range of environments to which the polymer composites for civil engineering applications may 

be subjected. The results show that the losses in both the amplitude and rate of fatigue were 

smaller at 4 oC than at 30 oC or 65 oC. Except for low cycles (103

) where little difference exists 

between the two temperatures, the reduction at 65 oC appears to be greater than that at 30 oC. 

Since the difference between 65 oC and 30 oC is relatively small, additional data is required to 

provide a more definite conclusion.
21

0

10

20

30

40

50

60

70

80

90

100

102

103104105106107

Nf (# cycles to failure)

Sult = 144 MPa

R = 0.1

freq = 10 Hz

Figure 9: S-N plot of a glass fiber/vinyl ester composite fatigued at 10 Hz in simulated sea water

at 4 oC (t), 30 oC (h), and 65 oC( ).

Residual Strength and Modulus

The residual strengths and moduli of composites fatigued at specified cycles were 

recorded for different loadings and frequencies in simulated sea water at 30 oC. The data were 

fitted to the following equation (Eq. 29) given by Broutman-Sahu [22] with an α = 1. 

a s

s

s

s

N

constant 1 1

















= − −

ult

a

ult

res (29)

where σres is the residual tensile strength, σult is the the ultimate tensile strength, σa is the applied 

stress, and N is the number of cycles. Figures 10a and 10b give the residual strength plots for 

loading levels at 2 Hz and 10 Hz, respectively, showing progressive damage with increased 

loading levels at either frequency. 
22

The effects of frequency on the residual strengths at a particular loading stress are 

presented in Figure 11, in which the results for 35 % UTS and 40 % UTS are shown. Plots of 

residual strengths vs number of cycles at other loading levels are similar to that of Figure 11. In 

general, the reduction in strength at 2 Hz is slightly greater than that at 10 Hz, similar to that 

observed for the S-N behavior.

The residual modulus characterizes changes in the stiffness of the composite when 

subjected to preset cyclic loading. The residual moduli for several loading levels at 2 Hz and 10 

Hz in simulated sea water are shown in Figure 12a and 12b, respectively (the lines in these 

figures are the fitted curves). From these results, the change in stiffness (damage) appears to be 

more prominent at 40 % UTS and 45 % UTS than at 35 % UTS, indicating that the applied load 

has an influence on the residual modulus of the glass fiber/vinyl ester composite. 
23

0

0.2

0.4

0.6

0.8

1.0

1.2

1 10 105

106

N (# cycles)

res/σult

35 % UTS - 2 Hz

40 % UTS - 2 Hz

45 % UTS - 2 Hz

103102104

(a)

0

0.2

0.4

0.6

0.8

1.0

1.2

1 10 102103104105106107108

N (# cycles)

35 % UTS - 10 Hz

40 % UTS - 10 Hz

45 % UTS - 10 Hz

res/σult

(b)

Figure 10: Residual strength as a function of loading cycles N for several load levels at (a) 2 Hz 

and (b) 10 Hz in simulated sea water at 30 oC (error bar represents one standard deviation). 
24

0

0.2

0.4

0.6

0.8

1.0

00

1.2

1 10 102 103

104104106107

N (# cycles)

35 % UTS –10 Hz

35 % UTS – 2 Hz

res

σ/ ult

(a)

0

0.2

0.4

0.6

0.8

1.0

1.2

1 10 102103

104105106

N (# cycles)

40 % UTS – 10 Hz

40 % UTS – 2 Hz

res

σ/ ult

(b)

Figure 11: Effects of cyclic frequency on residual strength of test composite fatigued at several 

loading conditions: (a) 35 % UTS and (b) 40 % UTS in simulated sea water at 30 oC (error bar 

represents one standard deviation). 
25

0

0.2

0.4

0.6

0.8

1.0

1.2

1 10 102

103

104105

106

107

N (# cycles)

35 %UTS-10Hz

40 %UTS-10Hz

45 %UTS-10Hz

res

/

Eult

(a)

0

0.2

0.4

0.6

0.8

1.0

1.2

1 10 102103104105

106

N (# cycles)

35 % UTS – 2 Hz

40 % UTS – 2 Hz

45 % UTS – 2 Hz

res/

Eult

(b)

Figure 12: Residual modulus as a function of loading cycles for three load levels

for 2 Hz (a) and 10 Hz (b) in simulated sea water at 30 oC (error bar represents one standard 

deviation). 
26

Comparative plots of residual modulus for test frequencies of 2 Hz and 10 Hz at a 

particular loading level are displayed in Figures 13a and 13b, respectively, for two loading 

levels. In these plots, the error bars overlap considerably, suggesting a negligible difference in 

modulus reduction between 2 Hz and 10 Hz experiments. 

0

0.2

0.4

0.6

0.8

1.0

1.2

1 10 102103104105

106

107

N (# cycles)

40 % UTS – 10 Hz

40 % UTS – 2 Hz

res

/

Eult

(a)

0

0.2

0.4

0.6

0.8

1.0

1.2

1 10 102

103104105106

 N (# cycles)

45 % UTS – 10 Hz

45 % UTS – 2 Hz

res

/

Eult

(b)

Figure 13: Effect of cyclic frequency on residual modulus at a particular loading level for 

(a) 40 %UTS and (b) 45 % UTS in simulated sea water at 30 oC (error bar represents one 

standard deviation). 
27

Residual Poisson’s Ratio

The Poisson’s ratios were obtained at 2 Hz and 10 Hz in simulated sea water at 30 oC. 

Normalized plots (with respect to as-received value) are presented in Figures 14a and 14b. No 

significant changes in the residual Poisson’s ratio with number of cycles were observed. The 

fairly even distribution of data on Poisson’s ratio may be attributed to the higher proportion of 

unidirectional fiber in the crosswise direction, resulting in a strong resistance to change in the 

transverse direction.

(a) (b)

Figure 14: Residual Poisson's ratio as a function of cycles N for several loading levels at (a) 2 

Hz and (b)10 Hz in simulated sea water at 30 oC.

MODEL VERIFICATION

Figure 15 displays the predicted S-N curves and experimental fatigue data for the vinyl 

ester composite tested in air, fresh water, and simulated sea water at 30 °C. In this figure, the 

maximum stress is plotted against the number of cycles at failure on a log-log scale. Values of 

the square of correlation coefficient (r2) between the experimental maximum stress and the 

¥

0

0.2

0.4

0.6

0.8

1.0

1.2

0 1 2 3 4 5 6 7

Log N

35 % UTS – 10 Hz

40 % UTS – 10 Hz

45 % UTS – 10 Hz

res/νult

0

0.2

0.4

0.6

0.8

1.0

1.2

0 1 2 3 4 5 6

Log N

35 % UTS – 2 Hz

40 % UTS – 2 Hz

45 % UTS – 2 Hz

res/νult
28

number of cycles at failure are 0.856 for ambient air, 0.803 for fresh water, and 0.840 for 

simulated seawater. The same fatigue data are also plotted in semilog scale (i.e., σmax = m log Nf

+ C, Equation 18); this result is shown in Figure 16. In this case, the r2 value for dry air, fresh 

water, and simulated seawater environments are 0.824, 0.794, and 0.863, respectively. The 

square of the linear correlation coefficient quantitatively indicates the linear correspondence 

between two variables [23], e.g., maximum stress and the number of cycles at failure in this case. 

An r2 value of 1.0 means a perfect linear relationship between the two quantities; a value less 

than 1.0 means a less good fit. Based on these r2 values, both formulas (Equations 8 and 18) can 

be used to predict fatigue life of the polymer composite used in this study. However, combining 

with additional tests at lower load levels that failed at a higher number of cycles, Eq. 8 gives a 

higher r2 value than Equation 18. Therefore, Equation 8 is better suited to describe the fatigue 

with a wider range of life time. 

In Figure 15, little difference in the slopes of the regression lines for ambient air, fresh 

water, and simulated seawater is observed, suggesting that the same mechanism of failure may

be operative in all three environments. The slopes of the regression lines are in the range (10 % 

to 11 %) UTS per decade, which are consistent with earlier results [24]. Also, although the

difference in fatigue behavior between fresh water and simulated sea water is small, the fatigue

life of the composite in fresh water or simulated salt water is shorter than that in ambient air.

Thus, the presence of water alone has a significant effect on the durability of this material; this

result is in agreement with previous observations [24,25]. The loss of fatigue life of polymer

composites has been explained by the stress corrosion mechanism [26]. Based on this concept, 

the strength of E-glass fiber is controlled by the chemical reaction of the glass with the surfacesorbed water. This reaction removes certain chemical elements from the fiber, resulting in

microflaws (stress concentration locations), which are the precursors to fiber fracture. This

process is irreversible. Thus, it appears that the stress concentration developed along the fiber

surface does not change the mechanism of fatigue failure, but rather, reduces the ultimate

strength.
29

Number of Life Cycles

1e+2 1e+3 1e+4 1e+5 1e+6

Maximum Stress, 

σmax (MPa)

40

50

60

70

80

90

200

100

Dry Air ------------------ 

Fresh Water ----- -----

 Salt Water --- --- ---

Linear Regression

Figure 15: Predicted S-N curves and experimental fatigue data in log-log scale of the test

composite fatigued in air, fresh water, and simulated seawater at 30 oC.

Number of Life Cycles

1e+2 1e+3 1e+4 1e+5 1e+6

Maximum Stress, 

σmax (MPa)

40

60

70

80

90

110

120

130

140

160

50

100

150

 Dry Air 

 Fresh Water 

 Salt Water 

Linear Regression

Figure 16: Predicted S-N curves and experimental fatigue data in semi-log scale of the test 

composite fatigued in air, fresh water, and simulated seawater at 30 oC.

Figure 17 shows the normalized predicted S-N curves and normalized experimental S-N 

data of the composites fatigued in air, fresh water and simulated sea water. The normalized 

values were obtained by dividing the maximum applied stress by its respective ultimate strength 
30

at a particular exposure and loading condition. The closeness of the regression curves in this 

figure suggests that a single normalized S-N curve can be used to predict the fatigue life of a 

vinyl ester/E-glass fiber composite exposed to all three environments. The good fit between the 

experimental data and Equation 8, as shown in Figure 15, suggests that the model is valid for 

predicting the fatigue life of the polymer composites used in this study under a wide range of 

loading and exposure conditions.

Number of Life Cycles

1e+3 1e+4 1e+5 1e+6

Normalized Stress, 

σmax / 

σult

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Dry Air -----------------

Fresh Water ----- -----

Salt Water --- --- ---

Linear Regression

Figure 17: Predicted normalized S-N curves and experimental fatigue data for the test composite 

in air, water, and simulated seawater at 30 oC. 

In order to determine the value of the material constant m, Eq. 19 is rewritten into a 

simple linear form (y=a+bx):

[ ] N

m

n C R

m

S f

m

log

2

1

log ( 1)( (1 ) )

2

1

log( max ) = − + − −

N

m n C R m

m f

log

2

1

( 1)( (1 ) )

1

log

2

1

−

+ −

= (30)

such that 

( 1)( (1 ) )

1

log

2

1

m m n C R

a

+ −

= (31)
31

m

b

2

1

= − (32)

Values of a and b are obtained from separate fatigue experimental data at 10 Hz in dry air, fresh 

water and salt water environments; and m can be obtained from -1/2b. Table 3 presents these 

values and their standard errors for the three environments.

Table 3: The linear constants and m values for three environments

a Std. Error b Std. Error m Std. Error

 Dry Air 0.0701 0.0272 -0.0882 0.0065 5.67 0.42 

 Fresh Water 0.0766 0.0340 -0.0859 0.0085 5.82 0.53 

 Salt Water 0.0729 0.0450 -0.0869 0.0095 5.75 0.64

Combined 0.0756 0.0181 -0.0877 0.0042 5.70 0.23

The small differences among the a and b values in the three environments confirms that a single 

normalized S-N curve can be used to predict the fatigue life in three environments.

Figure 18: Predicted normalized S-N curves and experimental fatigue data for the test 

composite in all three environments.
32

Figure 18 depicts the single linear regression that combines all experimental fatigue data 

of the three environments. The 95 % confidence interval and the 95 % prediction interval are 

also shown. Overlaying the plots in Figure 17 and Figure 18, the three S-N curves for three 

environments lie well within the 95 % confidence interval. Thus, the error induced by a single 

normalized S-N curve is well within the measurement uncertainty. Therefore, a single S-N curve 

(described by Eq. 33) can be used to predict the fatigue life of a E-glass fiber/vinyl ester 

composite exposed to air, fresh water, or simulated seawater. The corresponding combined 

values of a, b, m, and their standard errors are presented in Table 3.

When a and b values are substituted into Eq. 17, the S-N curve for a E-glass/vinyl ester 

composite at 10 Hz and Nf ≥ 10 may be expressed as:

 (Smax)

11.40Nf = 7.28 for the three environments combined together (33)

Figure 19: Partial damage at selected loading cycles for 2 Hz and 10 Hz.
33

The value of n may be determined from the partial damage data at different numbers of 

loading cycles under a given maximum load, which are given in Figure 19 for 2 Hz and 10 Hz. 

It is interesting to note that whenever severe damage has occurred, the lines become parallel. In 

other words, when the damage becomes substantial, it increases at a constant rate. Since we only 

measured the partial damage at the beginning of each order of magnitude, i.e., 103, 104, 105, and 

106 cycles, we may not accurately record the number of cycles where the substantial damage 

starts. Figure 19 suggests that the damage growth within an order of magnitude (dD/dlog N) is 

constant and equals approximately 0.12. Therefore, for a given maximum applied load with two 

partial damages D1 and D2 measured at two respective numbers of loading cycles N1 and N2, we 

have:

 10

1

2 =

N

N

(34)

and

2 1

1

2

2 1

2 1

2 1

log log log( ) ( )

D D

N

N

D D

N Log N

D D

d N

dD

= −

−

=

−

−

= = 0.12 (35)

From Figure 19, D1 starts from approximately 1 % to 5 % . Therefore,

1 (2.4 12)

0.12 1

0.12

1 1

1

1

2

to

D D

D

D

D

= + = +

+

= (36)

Consequently, from Eq. 25, we may have:

0.34 1.3

1

2

2

1

12 to

D

D

N

N

K = = (37)

However, from Eq. 6 and Figure 4, n must be ≥ 1:
34

1

( )

2(1 )

1 2 12

12 ≥

−

−

=

D D K

K

n (38)

1.065

2 0.12

2

2

2

1

1

2

1

12 ≈

− −

−

=

−

−

≥

D

D

D

D

K (39)

1.065 1.3 ≤ K12 ≤ (40)

( )

2( 1)

( )

2(1 )

12 2 1

12

1 2 12

12

K D D

K

D D K

K

n

−

−

≈

−

−

= (41)

Thus we have:

1.02 ≤ n ≤ 3.85

More experiments are required to determine the value of n. However, based on our 

limited experimental data, the n value should be closer to the lower bound. For now, we set n 

equal to 2.

To determine the C parameter, experimental data at two frequencies are used. Figure 20 

displays the linear-fit S-N lines for the simulated seawater environment at 2 Hz and 10 Hz

frequencies. From this figure, it appears that for a given maximum applied load, the number of

cycles at failure for 2 Hz is less than that for 10 Hz. It is also evident that, at lower cycles-tofailure, the two plots appear to converge. The effect of cyclic frequency on the fatigue behavior 

of polymer composites has been reported (27-30), for which the residual modulus and residual

strength of specimens that had been cycled at lower frequency were lower than those fatigued at 

higher frequency. 
35

Figure 20: Predicted S-N curves and experimental fatigue data for the test composite at 2 Hz 

and 10 Hz in simulated sea water at 30 oC.

The small difference between the slopes of the S-N curves for 2 Hz and 10 Hz in Figure 

20 could be due to an inability to statistically average out experimental variations by the limited

number of samples. The scattering of the experimental data from the S-N curve for 2 Hz is

obviously greater than that for 10 Hz. If we reconstruct the S-N curve for 2 Hz using the same 

slope of the S-N curve for 10 Hz as shown in the dot line, the newly constructed S-N curve

reasonably fits the experimental data for 2 Hz. From this analysis, we can conclude that the slope 

of S-N curve is not sensitive to frequency, and neither is the m value.

The parallel partial damage curves in Figure 19 for 2 Hz and 10 Hz indicates that, for the 

composite used in this study, n is not sensitive to frequency. Any sensitivity to frequency must 

lie in the C constant. If we assume that composites fatigued at a lower frequency fail at a lower 

number of cycles for a given maximum applied load, then the C constant will be higher at a 

lower frequency, i.e.:

f

C

C C

2 = 1 + (42)
36

where C1 and C2 are constants and f is the frequency. Substituting C into Eqs. 14 and 15, we 

obtain:

f

m

n

S R N

f

C

C

n

D

n

( )( (1 ))

1

(1 )

1

1

max

2 2

1

1

= + −

+

−

−

+

+

(43)

f

m

S R N

f

C

C

n

( )( (1 ))

1

1

max

2 2

= 1 + −

+

 (44)

By taking a logarithm of both sides of Eq. 44 and substituting σmax and Nf for 2 Hz and 10 Hz, 

respectively, into the equation, two separate equations for 2 Hz and 10 Hz can then be subtracted 

from each other. These two parallel S-N curves relate C2/f to C1, as expressed in Equations 45 

through 47:

) 2 (log log ) 0.445

10

) log(

2

log( max,10 max,2

2

1

2

1 + − + = m s − s =

C

C

C

C (45)

where σmax,10 and σmax,2 are the maximum stresses for 10 Hz and 2 Hz, respectively.

 2.78

10

2

2

1

2

1

=

+

+

C

C

C

C

(46)

8

1

2

=

C

C

(47)

Eq. 43 and Eq. 44 can then be rewritten as:

f

m

n

S R N

f

C

n

D

n

)( (1 )) 8(

1

(1 )

1

1

max

2

1

1

= + −

+

−

−

+

+

(48)

f

m

S R N

f

C

n

)( (1 )) 8(

1

1

max

2

= 1 + −

+

(49)
37

As for partial damage, Figure 19 shows that the residual modulus is higher at 10 Hz than at 2 Hz, 

which is consistent with previous studies cited earlier.

SUMMARY AND CONCLUSIONS

A fatigue model based on the cumulative damage due to cyclic loading is developed and 

verified with experimental data for a glass fiber/vinyl ester composite. The fatigue damage per 

cycle, residual modulus in terms of partial damage, and fatigue life of fiber-reinforced polymeric 

composites are described as functions of applied stress, stress amplitude, loading frequency, state 

of damage, and materials constants as follows:

S R N

f

C

C

n

S R N

f

C

C

n

D

n

D

S R

f

C

C

dN

dD

m

m

n

n

m

( )( (1 ))

1

1

)( (1 )) 1(

1

(1 )

1

1

(1 )

( (1 )) ( )

max

2 2

1

max

2 2

1

max

2

2

1

= + −

+

= + −

+

−

−

+

−

−

= +

+

The material constants used in the model are determined using the experimental data. 

This model can be used to predict the fatigue life of an E-glass/vinyl ester polymer composite at 

an applied load and to predict the residual strength modulus after a number of cycles at a given 

load in various civil engineering and offshore environments. Using the experimental data to 

determine the model constants, we have obtained the following S-N curve for a glass fiber 

reinforced vinyl ester composite exposed to the three environments:
38

 

f

S N f

8

1

11.40 13.11

max

+

= for the three environments combined

where Smax is the normalized stress to respective static ultimate strength ( Smax ≤ 1). 

The residual modulus for a given maximum applied load after N loading cycles can be predicted 

as:

1

1

11.40

max

13.11

8

1

1

+













































+

= −

n

o

f

S N

E

E

for the three environments combined

The value of n is not conclusive at this time, and more experiments are required to determine its 

value. 

The damage tolerance and durability of E-glass/vinyl ester composite is adversely

affected by exposure to the simulated saltwater environment. The presence of the moisture alone 

enhances the damage. A drop of about 11 % in modulus and 32 % in strength, respectively, was 

observed for specimens pre-conditioned in fresh water and simulated sea water at 65 °C and

tested quasi-statically. Stress corrosion is suspect in the degradation of this composite system. 

Initial strength loss observed under various conditions namely: salt, dry and water, exhibit a

similar trend indicating that failure mechanisms are similar and dominated by fiber damage

process. Normalized fatigue stress against percent ultimate tensile strength indicates a slope of

about 10 % UTS/decade [23]. 

ACKNOWLEDGEMENTS

This research is supported by the Advanced Technology Program, National Institute of Standards 

and Technology, U.S. Department of Commerce. 
39

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