A novel biaxial specimen for inducing residual stresses in thermoset polymers and fibre composite material

Abstract

A new type of specimen configuration with the purpose of introducing a well-defined biaxial residual (axisymmetric) stress field in a neat thermoset or a fibre composite material is presented. The ability to experimentally validate residual stress predictions is an increasing need for design engineers when they challenge the material limits in present and future thermoset and composite component. In addition to the new specimen configuration, this paper presents an analytical solution for the residual stress state in the specimen. The analytical solution assumes linear elastic and isotropic material behaviour. Experimental strain release measurements and the analytical solution determine the residual stress state present in the material. A demonstration on neat epoxy is conducted and residual stress predictions of high accuracy and repeatability have been achieved. The precise determination of the biaxial stress state in the specimen after cure makes it suitable for calibrating residual stress models.

Keywords

Polymers curing residual stress digital image correlation

Introduction

Polymers and especially polymer matrix fibre composite material is used extensively in commercial parts today and it is widely recognised that its utilisation will increase in the future. Some of the more well-known present applications are commercial aeroplanes, wind turbine blades, bicycle frames and rims. Due to an increased acceptance in the industry for this material class, increased attention is given to optimise manufacturing techniques and to ensure sufficient quality. The importance of the manufacturing process is further supported by the properties being dependent on the process conditions such as temperature and pressure.

Warpage may be seen on produced fibre composite parts and is a result of the mismatch in thermal and chemical strain between the material constituents. Furthermore, the discrepancy in thermal strain may rise from the difference in coefficient of thermal expansion between the polymer matrix and the fibrous reinforcement and/or a temperature difference between them. A chemical shrinkage takes place in the matrix (and not in the fibre) during cure which further leads to a difference strain behaviour between the two constituents. Yet another important parameter to cause warpage of fibre composite parts is the laminate architecture, i.e. laminate stacking sequence. Shape distortion and residual stresses in fibre composite parts as a consequence of the processing history have been given a vast attention and predictive models have been developed during the past three decades. The field of modelling stresses caused by the curing process was pioneered decades ago^1,2 and later on authors like White,^3,4 Filiou⁶ and Bogetti⁵ have refined and expanded the models. Studies showing the effect of different manufacturing techniques and enhanced models describing the material behaviour have more recently been presented by different research groups.^7–11 More specific attention on model characterisation of warpage in composite parts and comparison with experimental data has been provided by Svanberg et al.^12–14 In the continuation of the work presented by Svanberg et al., other authors have been studying the details of the spring-in effect on various configurations of curved and flanged panels^15–20 and obtained good correlations with experimental data of the measured spring-in angle. However, none of the just mentioned works have attempted to validate the residual stress state in test configurations without shape distortions, i.e. in specimen with a pure in-plane residual stress state.

Predicting the residual stress state in fibre composite material is complex due to the multiple processes generating the stress and deformation state in the material as reviewed elsewhere.^21–23 The experimental techniques for determination of the residual stress state can be divided into three general categories: (1) nondestructive testing, which includes X-ray diffraction^24,25 and photoelastic stress analysis or measurement of shape distortions, (2) semi-destructive testing including the hole-drilling technique^26–28 and (3) destructive testing such as the contour method.²⁹

At present, experimental techniques for characterisation of residual stress are available; however, there still is a gap between model predictions and experimental measurements. As the complexity in studied processes and material behaviour increases together with the associated uncertainties introduced from various sources (e.g. statistic variation in material properties, process history and boundary conditions), model validation is still a key issue. The present paper presents a new simple specimen configuration that creates an in-plane biaxial, and furthermore axisymmetric stress state and meanwhile minimises the aforementioned uncertainties. This latter feature of the experimental setup makes the specimen suitable for calibrating numerical residual stress models by comparing experimental data from the proposed specimen with model predictions.

Sample structure and geometry

The proposed residual stress specimen comprises a reinforcing steel plate with a circular cut out and a test material, which in this work is neat epoxy. However, due to the versatility of the sample production technique, other test materials such as fibre-reinforced composite materials can be examined as well. The circular cut out forms the gauge area of the specimen (cf. Figure 1).

As the test material bonds to the reinforcing plate during the curing process, a well-defined stress state develops in the gauge area. The developed stress state depends on how the test material has been processed and its material characteristics. Using a thermoset material such as neat epoxy, the chemical shrinkage and/or thermal expansion will start to introduce a stress state in the gauge area as soon as the epoxy bonds to the reinforcing plate. In most cases, models for prediction of induced residual stresses are sophisticated and rely on advanced material models that originate from a vast experimental program and process history. Model uncertainties arising from e.g. material properties and boundary condition are often unavoidable when predicting the residual stress state in complex parts. The simplicity of this specimen configuration limits the uncertainties in the boundary conditions and process history and can be used to validate model predictions and asses the quality of the employed material models.

Theory

In this work an elastic solution for the stress and strain state in the entire specimen have been developed. The solution is derived using Muskhelishvili³⁰ potential theory as explained in detail in the Appendix. Figure 2 shows the dimensional parameters and regions in the test specimen; #1: the test material and #2: the reinforcing plate.

The radius of the gauge area and major radius of the reinforcing plate are denoted r and R, respectively. The diameter of the drilled hole is ρ. ξ is the radial coordinate with origin in the centre of the drilled hole.

The elastic solution is derived assuming that a misfit (δ) between region #1 and #2 exists (cf. Figure 3(a)) because of the difference in the straining behaviour for the test material and the reinforcing plate. The misfit is balanced by an eigenstress such that continuity between region #1 and #2 is established (cf. Figure 3(b)). Drilling a hole in the centre of the gauge area will create a strain release and similarly reduce the magnitude of the eigenstress (cf. Figure 3(c)).

The radial and tangential strain decay from a hole with radius ρ in the gauge area (ξ < r) is expressed in equation (1)

ɛ_{rr} (ξ) = \frac{σ_{0}}{4 μ_{1}} (\frac{1 - k_{1} + \frac{2 ρ^{2}}{ξ^{2}}}{Γ} + (1 - k_{1})) ɛ_{θ θ} (ξ) = \frac{σ_{0}}{4 μ_{1}} (\frac{1 - k_{1} + \frac{2 ρ^{2}}{ξ^{2}}}{Γ} + (1 - k_{1}))

(1)

The shear moduli for the gauge area and reinforcing region are denoted μ₁ and μ₂, respectively. The Kolosov parameters for plane stress are κ _i = (3 − ν _i )/(1 + ν _i ) with i taking the values 1 and 2 for the two regions and ν is Poisson’s ratio. Γ is a constant that depends on geometry and stiffness through Dundur’s parameters α and β,³¹ and presented in equation (2).

Γ = \frac{ρ^{2}}{r^{2}} - 1 + \frac{(α - 1) (1 - \frac{r^{2}}{R^{2}}) ρ^{2}}{r^{2} (α + 1 - 2 β (1 - \frac{r^{2}}{R^{2}}))}

(2)

As can be seen from equation (1), the radial and tangential strains depend linearly on the biaxial stress state σ₀, which is the unreleased stress state in the gauge area (prior to the hole drilling). In addition, equation (1) shows that close to the hole the strain decays with the distance from the hole with the power of −2.

The unreleased stress state can be described through geometrical properties, Dundurs stiffness parameters and a misfit (δ) between the gauge area and the reinforcing region as expressed in equation (3).

σ_{0} = \frac{4 (α - β) (1 - \frac{r^{2}}{R^{2}}) μ_{1} μ_{2} δ (t, T, c)}{(α + 1 - 2 β (1 - \frac{r^{2}}{R^{2}})) (μ_{1} - μ_{2}) r}

(3)

This may further be characterised by the straining rising from the thermal expansion (ɛ_therm), chemical shrinkage (ɛ_che) and creep (ɛ_vis) multiplied by the radius of the gauge area (r). It is proposed that the misfit can be expressed as in equation (4).

δ (t, T, c) = r (Δ ɛ_{therm} (T) = ɛ_{che} (c) - ɛ_{vis} (t, T, c)) = r (ϑ_{1} - ϑ_{2}) Δ T + r ϕ_{1} (c_{end} - c_{gel}) + r ɛ_{vis} (t, T, c)

(4)

In equation (4), the coefficients of thermal expansion for the epoxy and steel plate region are denoted θ₁ and θ₂, respectively. The temperature change is denoted ΔT and the coefficient of linear shrinkage of region 1 is denoted ϕ₁. The chemical conversion at gelation and after completion of the curing cycle is denoted as c_gel and c_end, respectively. The viscoelastic strain (ɛ_vis) contribution will tend to relax the induced residual stress state through material creep and for this reason it is subtracted from the thermal and chemical strain contributions. The viscoelastic strain for the polymer will depend on not only time (t) but also on temperature (T) and the chemical conversion (c) and may be of the same magnitude as the sum of thermal and chemical strain for material temperatures above its instantaneous glass transition temperature (T_g). This highlights the importance of the viscoelastic contribution to the residual stress build up in the specimen.

Assuming that all the residual stress induced in the viscoelastic phase will relax entirely, the residual stress state introduced will then only originate from the temperature changes in the glassy phase (ΔT_glassy). This will rewrite the equation (4) in a simplified form (cf. equation (5)).

δ (t, T, c) = r (ϑ_{1} - ϑ_{2}) Δ T_{glassy}

(5)

In this paper, the biaxial stress state will be determined as used in equations (3) and (5) since extensive knowledge in the viscoelastic behaviour and chemical shrinkage of the material is needed in order to apply the ‘more correct’ expression in equation (4). In addition, the residual stress state of the test material due to the curing process will be determined using strain data and regression analysis. The regression analysis done to estimate (σ₀) in equation (1) is essential when solving the optimisation problem and thereby achieve a best match to the measure strain release data recorded by applying the a ‘hole-drilling’ method.

Materials

The epoxy used for fabricating the samples is made from a diglycidyl ether bisphenol-A with mono reactive diluents resin and an amine-based curing agent containing 3-aminomethyl 3,5,5-trimethylcyclohexane (isophorone diamine) and monoethyleneglycol, polybutylenoxide-diamine from Momentive. The epoxy system is slowly curing and with a relatively low curing exotherm.

The E modulus of the epoxy has been measured by means of dynamic mechanical analysis in a bending experiments to E_Epoxy = 2.9 GPa and the Poisson’s ratio is set to ν_Epoxy = 0.35. The properties for the reinforcing material are set to typically accepted values of steel E_steel = 210 GPa and ν_steel = 0.30. The coefficient of linear thermal expansion of the epoxy and steel has been measured to θ₁ = 72 µm/(m℃) and θ₂ = 12 µm/(m℃), respectively.

Experimental method

Neat epoxy specimens were fabricated as a single batch of five specimens using a ‘resin bath’ technique. The ‘resin bath’ technique is carried out by having a liquid resin bath of 2–3 mm in depth on a slip surface wherein the reinforcing plates were submerged. Spacers were used between the slip surface and the reinforcing plate in order to obtain an equal thickness of resin on each side of the plate and thereby minimise the out of plane bending.

Digital image correlation (DIC) was used to capture the strain release caused by the hole drilling. The procedure is to capture two images before a 5-mm hole is drilled in the centre of the gauge area. The first image served as the reference image and the second image was used to evaluate the resolution (noise level) that was done by determining the standard deviation of the second image (cf. Table 1). A 5-mm hole was drilled in the cured specimen using a column drilling machine and a drilling speed of 300 r/min was chosen to minimise heating. This drilling speed was verified not to introduce any residual stresses of a significant magnitude by practise drills on a stress free epoxy disc and utilising photoelasticity to confirm that no fringes have been introduced.

Table 1.

DIC settings used in the evaluation of the strain field.

Technique used	Mono image correlation
Subset size	80 × 80 pixel
Shift	75 pixel (6% overlap)
Camera	8 bit, 2048 × 2048 ARAMIS 4M system
Field of view	34 × 34 mm
Measurement points	27 × 27
Strain
Calculation method	Numerical (3 × 3 subsets)
Smoothing method	No smoothing
Spatial resolution	1.37 mm
Resolution (noise)	61 µm/m

Note: DIC: digital image correlation.

After drilling the hole, three new images were recorded at a frequency of 1 Hz. The average strain field of the three images was used in the evaluation in order to average any dynamic effects in the test setup. The strain components (ɛ_rr) and (ɛ_θθ) were evaluated along the horizontal and vertical direction. The technical settings of the DIC used in the work are listed in Table 1.

Results/discussion

The residual stress state is estimated using equations (3) and (5). The temperature change in the glassy phase (ΔT_glassy) is determined using the onset temperature between the glassy and the viscoelastic phase (cf. Figure 4). The progress in elastic modulus is modelled using the proposed method suggested in Jakobsen et al.³² scaled to meet the magnitude of epoxy.

From Figure 4, the temperature change of the material in the glassy phase is determined to 26.7℃, which lead to σ₀ = 6.8 MPa using equations (3) and (5).

The strain decay from the drilled hole is evaluated using DIC and a typical image of the (ɛ_xx) is shown in Figure 5. The released strain exhibits a symmetric pattern as expected which supports that an equal biaxial stress state was present in the epoxy material prior to drilling the hole.

The radial and tangential strains are evaluated in the horizontal and vertical directions away from the hole. Using the measured strain data and the expression for the two strain components in equation (1), the initial residual stress state (σ₀) can be determined from regression analysis as exemplified in Figure 6.

The measured tangential and absolute radial strain components are similar in magnitude and the inclination of the data points fits well with the power of −2 decay predicted from equation (1). Only the data with the distance interval from 5 mm to 10 mm has been used in the evaluation of (σ₀). Strain data outside the 10 mm bound have been omitted since these values are of the magnitude of the resolution of the DIC equipment. Data between the hole edge and the 5-mm bound has been left out of the analysis due to the large strain gradient near the hole, which is not captured at a sufficient accuracy with the chosen subset size. A smaller subset size is an alternative, but will resolve in a higher resolution (more noise).

The initial residual stress states are summarised in Table 2 and it can be seen that the average stress state is 4.39 MPa and 5.58 MPa using the radial or tangential strain data, respectively. A discrepancy in the residual stress prediction using either the radial or tangential data is found. This discrepancy cannot be explained by pure experimental uncertainty as a high regression coefficient is achieved for each of the data sets. In addition, the 95% confidence intervals for the radial or tangential data sets are both narrow and significantly different. The discrepancy between the stress predicted from the radial and tangential stress is mainly a result of the stress difference for the first and third sample as the predicted difference between the two methods for these two samples is around 2 MPa. A possible explanation for this difference can be related to out of plane movement of the specimen when repositioning the specimen in front of the camera after hole drilling. In cases where the distance between the camera and specimen is unintendedly reduced between the acquisition of the reference image and image after the hole drilling, an artificial positive strain field is imposed. Due to the different sign of the radial and tangential strain component, this artificial strain field causes a deviation in the magnitude of the two measured strain fields. A method to circumvent this issue would be to average the two predictions, which will give a value of 4.99 MPa or one could use a stereo camera setup.

Table 2.

Residual stress determination from equation (1) using the measured strain release for the five specimens. R² values for the obtained fit are indicated in parentheses. The mean value and standard deviation for the five specimens is denoted ( $\bar{x}$ ) and (s_σ₀), respectively.

	Radial data		Tangential data
Specimen	σ₀ (MPa)	95% confidence interval	σ₀ (MPa)	95% confidence interval
Epoxy 1	4.1 (0.91)	[3.9–4.4]	6.0 (0.94)	[5.7–6.3]
Epoxy 2	4.5 (0.85)	[4.2–4.8]	5.5 (0.97)	[5.3–5.7]
Epoxy 3	3.8 (0.93)	[3.5–4.0]	5.8 (0.93)	[5.5–6.2]
Epoxy 4	4.6 (0.95)	[4.4–4.8]	5.7 (0.89)	[5.3–6.2]
Epoxy 5	5.0 (0.84)	[4.7–5.4]	4.8 (0.94)	[4.5–5.2]
$\bar{x}$	4.39	[4.2–4.5]	5.58	[5.4–5.7]
s_σ ₀	0.46		0.46

The standard deviation of the five tested specimens is around 10% of the average values, which indicates good reproducibility of the determined residual stress state.

The coefficient of determination for each specimen has been calculated to assess the quality of the obtained residual stress state and in general a good correlation between the model is presented in equation (1) and the measured strain release is obtained. The 95% confidence interval for each of the predicted stress state is determined and it is seen that these intervals are less than 1 MPa. This confirms that when using the model developed in this work, the biaxial stress state in the samples can be predicted with a high accuracy. However, care must be taken to avoid out-of-plane movement in the experimental setup when using a single camera setup.

The presented approach using equation (1) together with experiment data yields a lower residual stress state compared to the pure analytical approach (equations (3) and (5)). The deviation between the two proposed methods is significant (1.8 MPa) but within the same order of magnitude. It is believed that an improved agreement can be obtained by using equation (4) instead of using the simplified representation of δ given in equation (5).

Conclusion

A new residual stress specimen configuration that can be used for advanced model validation is proposed. The specimen configuration aims to be used in conjunction with thermoset resin and fibre composite material. The specimen consists of a gauge area where an equal biaxial residual stress state is formed as the material cures.

The evaluation of the residual stress state in the gauge area involves utilisation of the hole-drilling technique and measurements of the associated released strains. An elastic solution describing the strain decay from the hole has been derived using Muskhelishvili complex potentials. Regression analysis is used to obtain values for the induced residual stress state.

The calculated residual stress state was 4.39 MPa and 5.58 MPa using the measured radial and tangential strain data, respectively. The difference is believed to originate from out-of-plane movement in the experimental setup. An average of the two calculated stresses is proposed to give a value for the equal biaxial residual stress state in the material. The standard deviation of the obtained residual stress values for the five specimens is within 10% of the average obtained stress state that confirms a good reproducibility of the results.

The residual stress state has similarly been calculated using the proposed expression for the (σ₀) through the misfit parameter (δ) in equation (5). The stress state is determined to 6.8 MPa having a temperature change in the glassy phase of 26.7℃. This prediction is 1.8 MPa higher compared to the prediction made using equation (1) and the experimental data.

Footnotes

Conflict of interest

None declared.

Funding

The work presented has been sponsored by the Danish Council for Independent Research|Technology and Production Sciences (FTP, Grant Agreement 10-093339, ‘Mechanical Property Characterisation of Fibre Composites with Focus on Thermal Cure Conditions’. In addition, the test materials were sponsored by Siemens Wind Power. The received support is gratefully acknowledged.

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