A theoretical framework underlying an accelerated testing method and its application to composites under constant strain rates and fatigue loading

1 Introduction

Fibre-reinforced composite structures, which have a better stiffness-to-weight ratio, strength-to-weight ratio and significantly higher tolerance to fatigue loading than metallic materials, have been used widely in aerospace engineering to reduce the weight of structures. Composite structures started to be used as fairings and radomes in the 1970s on the Airbus A300-B2. By the time of writing, more than 45% of composites by weight are used in the A350, including composite main wing and composite fuselage. The past design method was based on a no-damage growth criterion (it is assumed that no damage occurs during the lifetime) and this was achieved by conservative strain limitations and verification testing on components [1]. Fatigue was not an issue at that stage since the operational strain levels were kept safely below the fatigue threshold for graphite/epoxy composite materials.

As polymer composite materials began to be employed in the primary structures of aircraft and, therefore, were subjected to higher strain levels, the improvement in these properties and a greater understanding of the static properties of composite materials have raised the allowable design strain to the critical point at which zero growth of defects due to fatigue cannot be guaranteed. Furthermore, the fatigue properties of helicopter rotor blades, ship propellers and wind turbines are a major concern of the designers. Thus, the fatigue of composites has now become an issue that requires further investigation and predictive tools.

To achieve the greatest possible weight saving of aircraft based on designing the aircraft structure using rigorous analysis procedures, we need a good understanding of the fatigue properties of composite materials and composite structures. Because of the lack of fatigue life prediction methodologies for composite materials, a large factor of safety is required. Consequently, composite structures used in high cycle fatigue applications are often over-designed and are somewhat heavier and more costly than necessary. Improved life prediction methodologies for composite materials are essential and would result in more efficient use of these materials and hence would enable the expected benefits of lower weight and lower cost structures to be realised.

Due to the complex damage mechanisms for fibre-reinforced composite materials during fatigue, such as matrix cracking, matrix crazing, debonding, fibre–matrix interface failure, delamination, and fibre fracture, and their interactions and their different growth rates, it is difficult to construct a fatigue damage model, which can cover all of the damage modes. However, some efforts have been made by a number of researchers to establish a fatigue model for composite materials. Existing fatigue models for composites can be classified into three major categories [2]. The fatigue life models [3,4,5,6] expressed fatigue life in terms of stress levels. Whilst this was in a desirable format in terms of applications, their applicability was confined within a limited scope in terms of the types of structures and the loading conditions because these models rest heavily on the test data from which curves were fitted. It is therefore impractical to envisage the fatigue life prediction of composite materials and structures, such as aircraft using these models. The constant-life models [7,8,9,10,11] shared much of the curve-fitting nature of the fatigue life models and hence their limitations. The semi-empirical models, such as residual strength models and residual stiffness models [12,13,14,15,16,17,18], allowed damage accumulation but still relied on the fitting of curves to experimental data. The progressive damage model directly predicts the growth of damage (e.g. transverse matrix crack density, size of the delaminated area) and correlates the damage growth with the residual mechanical properties (stiffness and strength) [19,20] using one or more damage variables, which describe the deterioration of the composite materials. Its applicability is limited to the specific damage types incorporated, such as matrix cracks and delamination.

Most of the fatigue models mentioned above are only directly applicable to composite coupons, and significant difficulties have to be overcome when modelling composite structure fatigue under general loading conditions for the following reasons. First, most experiments are carried out under uniaxial stress conditions, although this stress state is not typical for real structures. Second, the governing damage mechanism varies with stress/strain level states [21,22], and failure patterns vary with cyclic stress levels and even with the number of cycles to failure. The load history, in the form of different block loading sequences, will result in differences in damage growth [23]. Third, the fatigue properties of composites, such as residual strength and fatigue life, have been observed to decrease more rapidly when the loading sequence is repeatedly changed [24]. This is the so-called “cycle-mix effect,” which refers to the observations, that laminates experiencing small cycle blocks tend to have reduced average fatigue lives when compared with those subjected to large cycle blocks, even though the blocks are of similar structure but involve fewer cycles. Thus, in the authors’ opinion, the aim of formulating a comprehensive fatigue model for use in aircraft design is still too ambitious to achieve at the present time.

Experimental data from fatigue tests performed directly on composite materials and structures remain the primary means of evaluating their fatigue behaviour under a specific set of test loading conditions. However, airframes have long service lives (up to 50 years) during which they will be subjected to complex loading cycles. Direct replication of the load spectrum for fatigue tests is clearly not feasible due to the timescales involved. In practice, tests have to be accelerated and relatively established procedures are available for fatigue tests of metallic materials, typically by increasing the loading frequency. However, in the case of polymeric composite materials, an increase in the loading frequency generates a significant amount of heat owing to their high hysteretic losses and poor thermal conductivity, which lead to a temperature rise within the composite specimen. Moreover, the properties of the matrix or the interface of composites are typically sensitive to temperature, leading to a physically coupled problem amongst damage, viscoelasticity, temperature, frequency and fatigue life. Consequently, a change in frequency will result in a different fatigue life from that obtained from testing at the normal loading frequency. Any reasonable acceleration scheme for composites will have to incorporate the effects of temperature change caused by the change in frequency.

In the previous work, the authors presented a continuum damage model for transverse cracking in a unidirectional (UD) composite of linear viscoelastic behaviour. In that study, it was demonstrated that the properties of the matrix or the properties in the transverse direction of UD composites were affected by the combination of the damage and viscoelasticity [25].

The present article describes a theoretical framework for accelerated testing of UD composite, derived by modelling the mechanical behaviour of the composite based on a full understanding of properties of its constituent fibres and resin. The temperature effect and frequency effect on the fatigue life and the time–temperature superposition principle are explained in this article. This will lay a rational basis for the development of accelerated testing methods for fatigue testing for a specific UD composite. This article will also present some simple demonstrations of the theory involving tests under different constant strain rates and simple fatigue loading conditions.

The article is structured as follows. The damage model for the matrix of UD composites is introduced in Section 2. By taking advantage of the Wiechert model, the mechanical behaviour model of UD composites is built in Section 3. Based on the mechanical behaviour model, the test acceleration methods for the constant strain rate test and fatigue test are built in Sections 4 and 5, respectively. Then, the applications of these two methods are elaborated and discussed in Section 6. Finally, Section 7 ends the article with some conclusions and perspectives.

2 The damage model for the matrix of UD composite material

The mechanical behaviour of cured polymeric resins exhibiting viscoelastic characteristics shows a degree of time and temperature dependence under certain conditions. This will further affect the mechanical properties of composites when the temperature of the specimen rises due to the heat generated by the high-frequency fatigue test.

When this occurs, neither the effects of frequency nor the temperature on the composites can validly be ignored. From the authors’ perspective, the effects of frequency and temperature on the fatigue life of the composites are mainly caused by the viscoelastic behaviour of the matrix and its tendency to undergo damage. To accelerate testing of the composites both under fatigue and constant strain rate tests, a damage model for the viscoelastic matrix material of the composites will need to be constructed to obtain a full understanding of their viscoelasticity, damage and their interaction.

2.2 Damage evolution law for viscoelastic materials

A real polymer does not relax with a single relaxation time since molecular segments of varying lengths contribute to the relaxation, that is, the relaxation with the simpler and shorter segments is much quicker than that with the long ones. This leads to a distribution of relaxation times, which in turn produces a relaxation spreading over a much longer time and is thusly modelled much more accurately than that with a single relaxation time. The most proper mathematical model to describe the viscoelasticity of polymer composites in a one-dimensional form is the Wiechert model [27] (Figure 1), which represents viscoelasticity using a series of springs (elasticity) and dashpots (viscosity) connected in parallel.

Figure 1

The Wiechert model.

The relaxation modulus for this model is

(3) E rel ( t ) = k e + ∑ j k j exp − t τ j ,

where k _e is the elastic modulus of the main spring, k _j is the elastic modulus of a spring connected with a dashpot and τ _j is their relaxation time and

(4) τ j = η j k j ,

η _j is the viscosity of the dashpot.

In the continuum damage mechanics (CDM) theory, the effects of damage are often represented in terms of a reduction in stiffness modulus [28,29,30]. For viscoelastic materials represented by the Wiechert model, it will be assumed that the damage affects all of its parts (springs k _j and dashpots η _j ) equally.

The relaxation modulus with damage can then be defined as

(5) E D ( t ) = ( 1 − D ) k e + ∑ j k j exp − t τ j .

To describe the cracking of the viscoelastic materials in terms of the CDM, a damage evolution law will be formulated here, based on the assumption that the damage evolution is controlled by the tensile strain in line with Miyano and Nakada’s work [31]. The Weibull distribution will be employed to characterise the defects in the viscoelastic material, in which damage (cracking) is expressed as a function of the strain. Tensile tests will be conducted on polymer resin to obtain the necessary material properties.

The concept of a representative volume element (RVE) in a viscoelastic material will be employed to introduce damage. It will be assumed that the RVE contains a large number of microscopic defects of different sizes as shown in Figure 2(a). These defects develop into cracks as the strain increases as illustrated in Figure 2(b). These discrete cracks are the physical manifestation of the idealised, continuous damage. The effect of these cracks is expressed via the damage parameter D, the value of which is in the range from 0 to 1. When D is equal to 0, the material is undamaged. When D becomes equal to 1, the material has completely failed. The probability density of these defects can be described by the Weibull distribution, which is defined here as a function of strain.

(6) ρ ( ϵ ; λ , h ) = h λ ϵ λ h − 1 e − ϵ λ h , σ ≥ 0 , 0 , σ < 0 ,

where ϵ is the strain under uniaxial stress conditions and h and λ are material constants. Then,

(7) D = ∫ 0 ϵ ρ d x = 1 − e − ϵ λ h .

Figure 2

The representative volume element showing (a) microscopic defects and (b) the cracks into which they develop.

There are various possible definitions or physical interpretations for the damage parameter D. Here, it is defined as a ratio, specifically, the relative reduction in stiffness E due to damage as

(8) D = ( E 0 − E ) / E 0 = 1 − E / E 0 .

Once the damage D is obtained from the experimental data as a function of ϵ , parameters, such as h and λ, can be obtained by fitting the data to function (7).

Using the viscoelastic constitutive relations in equations (1) or (2) and treating the relaxation modulus as a function of damage as expressed in equation (5), the constitutive equation for the damaged viscoelastic material is written as

(9) σ   ( t ) = E D ( t ) ϵ 0 + ∫ 0 t E D ( t − τ ) ∂ ϵ ( τ ) ∂ τ d τ = ( 1 − D ) E rel ( t ) ϵ 0 + ( 1 − D ) ∫ 0 t E rel ( t − τ ) ∂ ϵ ( τ ) ∂ τ d τ = e − ϵ λ h E rel ( t ) ϵ 0 + e − ϵ λ h ∫ 0 t E rel ( t − τ ) ∂ ϵ ( τ ) ∂ τ d τ = e − ϵ λ h k e + ∑ j k j e − t τ j ϵ 0 + e − ϵ λ h ∫ 0 t k e + ∑ j k j e − t − τ τ j ∂ ϵ ( τ ) ∂ τ d τ ,

where ϵ = ϵ 0 + ∫ 0 t ∂ ϵ ( τ ) ∂ τ d τ and ϵ = ϵ 0 when t = 0 . Therefore, equation (9) presents the damage evolution law for the viscoelastic material.

In the case of a constant strain rate test at a strain rate a, equation (9) will transform into

(10) σ   ( t ) = a e − a t λ h ∫ 0 t k e + ∑ j k j e − t − τ τ j d τ = a e − a t λ h k e t + ∑ j k j τ j − k j τ j e − t τ j .

In Kumar and Talreja [26] and Koyanagi et al. [32], the purely elastic part of the Wiechert model is absent for the cases of pure resin or the transverse properties of the UD composite, so that equation (10) can be further simplified as

(11) σ   ( t ) = a e − a t λ h ⋅ ∑ j k j τ j − k j τ j e − t τ j .

The extremum values of a function can be determined when its first derivative vanishes. Then, the strength under constant strain rate can be determined as the maximum value of equation (11) corresponding to its first derivative with respect to time becoming zero as

(12) d σ ( t ) d t = a e − a t λ h ∑ j k j e − t τ j − a h λ τ j a t λ h − 1 1 − e − t τ j = 0 .

3 Mechanical behaviour model of UD composites

The Wiechert model is used to describe the properties of UD composite materials as illustrated in Figure 3. As mentioned above, there is no purely elastic part k _e in the matrix (resin). Thus, part A is used to describe the properties of the fibre and part B is used to describe the properties of the resin.

Figure 3

The mathematic model for UD composites in 1 dimension.

The relaxation modulus including the consideration of damage evolution of the matrix for UD composites is obtained as

As listed previously, there are several damage modes during the whole process of failure of UD composites, such as matrix cracks, debonding and fibre breakage. If it is desired to predict the properties of UD composite during deformation, the damage model should be formulated to take account of all the damage modes as well as the interactions between them.

It is noted that a damage evolution law for the fibre has not been proposed. Due to the randomness of the fibre strength and the nonprogressive fibre breakage mechanism mentioned in Talreja’s work [21,33,34], the fibre failure of UD composites during fatigue cannot be predicted. Also, to the best of the authors’ knowledge, the no-damage model is currently available to describe the modes of the damage, such as debonding (interface damage), which commonly occurs during the failure of UD composites. Whilst waiting for an acceptable predictive theoretical model to become available, the determination of fatigue strength of composites will have to rely mainly on experimental means. Fatigue experiments are notoriously time-consuming and an appropriate method to accelerate such experiments is essential. Even in the presence of an applicable theoretical model, to support the model in terms of necessary input data to the model and the validation of the model will still require experimental data. This underlines the objective of the present article.

4 The acceleration of constant strain rate tests for UD composites

If constant strain rate loading is applied to the UD composite in the axial direction, the stress can be expressed as

where a e − a t λ h ⋅ ∑ j k j τ j − k j τ j e − t τ j is the stress applied to the matrix, atk _e is the stress applied to the fibre and k _e is used for representing the elastic modulus of the fibre.

As shown in equation (14), the properties of a UD composite can be described by the Wiechert model combined with a damage evolution law for the matrix. Either a different loading rate or a different testing temperature can lead to a different result. This is due to the fact that the distribution of stresses on the fibre and resin during deformation varies with the loading rate or the temperature. It is reasonable to assume that the same stress distribution in fibre and matrix during deformation should lead to the same testing result. Thus, the result of a CSR test at a low loading rate can be obtained through the test at a high loading rate with an appropriate temperature shift. This process is defined in the following paragraphs.

The stress in the composite is the average of those in the fibres and matrix microscopically and the ratio between the stress in the fibres and that in the matrix is broadly determined by the stiffness ratio between these two constituents. However, this ratio can be upset by the loading rate because of the different viscosities of the two constituents. To counterbalance the effects of the disparity in the changes of viscosities on the microscopic stresses in the fibres and matrix, the temperature can be introduced to moderate the viscosities. The theory proposed in the present article is based on the assumption that temperature within the range involved does not affect the fatigue strength. With this, it is then possible to increase the loading rate without upsetting the stress ratio between the fibres and matrix by adjusting the temperature appropriately. ϵ = a t By substituting equation (4) into equation (14), one obtains

It can be seen from equation (15) that the loading rate a and viscosity η _j always occur in pairs, meaning that the effect of a change in one of them can be counterbalanced by an opposite change in the other, provided that the product of the two remains unchanged. The change in the value of viscosity η _j can be controlled by adjusting the temperature T after appropriate calibration.

It is known that the viscosity η _j is temperature dependent. From Ojovan’s work [35,36], if the temperature is significantly lower than the glass transition temperature, T ≪ T g , then the viscosity can be expressed via an Arrhenius relationship as

with

where H _d is the enthalpy of formation of broken bonds, H _m is the enthalpy of their motion and R is the gas constant, 8.314 × 10⁻³ [kJ/s(K × mol)].

If the loading rate is changed from a ₀ to a higher rate a, the testing time will be reduced to time t from t′, and t t ' becomes equal to a 0 a by assuming that the specimen failed at the same strain [31]. According to equation (15), if it is desired to accelerate a test by increasing the loading rate from a ₀ to a, it will be necessary to change the viscosity of the matrix from η ₀ to η for the accelerated test to be representative of the true test. Then, the time–temperature shift factor b _T is defined as the ratio of t to t′ and can be further described by the viscosity changing at different temperatures as

According to equations (15) and (18), when the loading rate has been changed from a ₀ to a to accelerate the test, the temperature should be shifted from T ₀ to T to counteract the difference on the stress distribution caused by the change in the loading rate. This will be further discussed and with the outcome supported by experimental data as shown in Figure 5.

Figure 4

Specimen drawing.

Figure 5

Strength comparison of different loading conditions.

5 The fatigue acceleration of UD composites

The ultimate objective of this article is the acceleration of fatigue tests for UD composite materials. To achieve this objective, it is not necessary to build a damage model for composite fatigue to predict fatigue life, which would be an unrealistic aim when considering the complex damage mechanisms, which occur during fatigue failure. Instead, the aim is to propose a rational method to accelerate fatigue tests through the mechanical behaviour model since the traditional fatigue acceleration method, such as increasing the frequency, cannot be used directly due to the viscoelasticity of the resin.

As illustrated in the previous section, the loading condition affects the process and result of monotonic loading tests, such as creep tests, stress relaxation tests and constant strain rate tests. To accelerate fatigue tests, the viscoelastic behaviour under dynamic loading needs to be characterised. When a viscoelastic material is subjected to a sinusoidally varying stress, a steady state will eventually be reached in which the resulting strain is also sinusoidal, having the same angular frequency but retarded in phase by an angle θ [37]. This is also true even if the strain is the controlled variable.

Suppose a sinusoidal load is applied to this model as follows

The strain function should be

Advantages can be taken of the complex modulus to calculate the strain of the UD composite. This calculation makes use of the composite mechanical behaviour model (13) proposed in Section 3, which is the relaxation modulus described by the Wiechert model with the consideration of damage evolution of the resin controlled by the strain.

where

Reference [38],

and

Then, the strain ϵ ( t ) is obtained as the real part of equation (21)

Substituting equation (4) into equation (26), the maximum strain ϵ max and the phase angle θ are obtained as

and

As shown in equation (27), the maximum strain is affected by the frequency. If the frequency is increased to accelerate the fatigue test, the maximum strain will decrease. According to Talreja’s theory [21], the fatigue life of a UD composite is dominated by the maximum strain. When the maximum strain is decreased, the fatigue life will increase. Thus, if the frequency is increased directly to accelerate the fatigue test, an over-estimated fatigue life will be obtained. Furthermore, the phase angle will be changed (the changes in the phase angle can be calculated from the value of relaxation modulus) along with changes in frequency according to equation (28). The stress distributed on the fibre can be expressed by

It can be inferred from equation (29) that the load applied on the fibre at the same phase position will change when the frequency is changed since the phase angle θ is affected by frequency. This means that the stress distribution on fibre and resin will be changed if the frequency is changed.

Directly increasing the frequency to accelerate the fatigue test of UD composite will not only cause the maximum strain to decrease but will also cause the stress distribution to change at a given phase position within the loading cycle. In other words, directly increasing the frequency does not validly accelerate the fatigue test on UD composites. The over-estimate of fatigue life will pose a hazard when using the results to evaluate a design.

Fortunately, the viscosity η _j and frequency ω always come in pairs. The viscosity η _j can be reduced by increasing the temperature (16) by a suitable value to counteract the effect of increasing frequency. Then, the maximum strain and the stress state in the fibres remain unchanged during each cycle and this means that the stress state in the resin also remains unchanged. Thus, the fatigue test can be accelerated by increasing the frequency and the temperature at the same time. The viscosity–temperature shift factor b _T is defined as

which is the same as equation (18) and the value of Q can be obtained from stress relaxation tests.

6 Application

The methods of accelerating both constant strain rate tests and fatigue tests have been demonstrated experimentally. The specimens are made from T700 carbon fibre and VTM 264-1 resin produced by Cytec Industries Inc. The volume fractions of the fibre and resin are 65% and 35%, respectively. The curing condition is 120°C for 1 h followed by 80°C for 5 h. The specimen design for the tests is illustrated in Figure 4.

In the previous work [25], the properties of the resin VTM 264-1 described by the Wiechert model in terms of values of parameters k _j and τ _j were obtained at the reference temperature of 50°C (Table 1).

Meanwhile, the activation energy Q was 298.4 kJ/mol and the damage evolution parameters h and λ were 2.7381 and 0.0117, respectively [25].

6.2 The acceleration of fatigue tests

The tensile fatigue tests were carried out under three different loading conditions using an Instron 8801 servohydraulic fatigue testing machine. The stress ratio R is 0.1. The original fatigue tests were carried out at 70°C and 0.5 Hz. If the frequency needs to be increased to 5 Hz to accelerate the tests, the temperature should be shifted to 78°C according to equation (32), which is a rearrangement of equation (30) noting that ω = f 2 π .

(32) T T 0 e Q R 1 T − 1 T 0 − f 0 f = 0 .

Tests carried out at 70°C and 5 Hz were also undertaken to provide a comparison. Testing results are shown in Figure 6.

Figure 6

The comparison of the S–N curves.

As shown in Figure 6, when the original fatigue test (black line) is accelerated by increasing the frequency from 0.5 to 5 Hz whilst maintaining the temperature at 70°C, the fatigue life increases significantly (red line). This result is as would be expected, as simply increasing the frequency will reduce the maximum strain, thus leading to longer fatigue life. According to the calculation based on the present theory, the fatigue properties when the frequency is 0.5 Hz and the temperature is 70°C are equivalent to the fatigue properties when the frequency is 5 Hz and the temperature is 78°C (blue line). Then, the S–N curve of 0.5 Hz at 70°C should overlap on the S–N curve for 5 Hz at 78°C. However, the fatigue life at 5 Hz and 78°C is less than the fatigue life at 0.5 Hz and 70°C. There are several reasons that might cause this result, which will be discussed later. Although the present fatigue acceleration method leads to a shorter fatigue life compared with the fatigue life without acceleration, this method still can be used to accelerate the fatigue test since the accelerated result is conservative and therefore safe compared with directly increasing the frequency, for which the resulting fatigue lives are overestimated and unsafe.

The fatigue acceleration method is based on the assumption that the fatigue life is dominated by the maximum strain and the stress distribution between fibre and resin during fatigue tests. Actually, there is another factor, the interfacial shear strength, which also affects the strength and fatigue life of UD composite materials. According to Koyanagi’s theory [32], the interfacial shear strength is affected by the initial interfacial compressive stress, which consists of thermal residual stress and will be affected by the temperature. When the temperature is increased, the interfacial compressive stress decreases. This will lead to a decrease in the interfacial shear strength, which will result in a reduction in the strength or the fatigue life.

The described methods for accelerated constant strain rate and fatigue testing methods for UD composites were demonstrated on a proof-of-concept basis and were verified through a limited number of experiments, which were carried out at representative values of loading rate and reference temperature. It is demonstrated that the present theoretical model is consistent with the experimental results well and the present acceleration method provides a much more accurate prediction for the constant strain rate test and fatigue test of composite materials. These accelerated testing methods can of course be used for other loading rates, frequencies, loading ratios, temperatures or materials as long as the material exhibits linear viscoelasticity under that testing condition. The key parameter of these accelerated methods is the time–temperature shift factor, which can be obtained from stress relaxation tests at different temperatures. To apply these methods to the acceleration of constant strain rate test or fatigue tests of other composites at a different temperature, loading rate or frequency, the time–temperature shift factor for the relevant material at that temperature range will need to be determined. Whilst the results presented here are promising and appear to be slightly conservative, the limited validation conducted within this proof-of-concept study means that further validation of the newly proposed methods will be needed before they can be used for accelerating safety-critical tests.

7 Conclusion and perspectives

A CSR test acceleration method and a fatigue acceleration method for UD composite materials have been proposed and verified in this article. A damage evolution law for viscoelastic materials (the composite matrix) has been proposed based on CDM. Then, a mechanical behaviour model for UD composites has been formulated using the Wiechert model and the damage evolution law for the matrix. Based on this mechanical behaviour model, a rational method with a physical basis for the acceleration of CSR tests and fatigue tests on a UD composite has been proposed and demonstrated on a proof-of-concept basis.

Both of the testing acceleration methods have been verified through tests. Compared with increasing loading rate alone for CSR tests or increasing frequency only for fatigue tests, which lead to an over-estimate of the UD composite properties, the testing acceleration method provides more accurate and conservative predictions.

The present testing acceleration method can be applied to a greater extent. First, it can improve the testing efficiency of the durability verification of aircraft composite structures. Moreover, it can be extended to a high-frequency fatigue test and, then, helps evaluate the high cycle fatigue performance of composite blades used in aeroengine, wind turbine, etc.