2.5. Stress–Strain Curve Analysis

A general observation is that the stress–strain curves of carbon composites curve upwards [1,2], and this indicates an increasing tangent stiffness with increasing strain. Markussen et al. [2] reported an almost linear relation between tangent stiffness and strain for unidirectional carbon composites, implying a second-order polynomial relation between stress (σ) and strain (ε).

A linear relation between tangent stiffness and strain can be expressed as:

where E_t (= dσ/dε) is the tangent stiffness at a given value of ε, E₀ is the initial stiffness (at zero strain), and α is the stress–strain curvature coefficient, respectively.

The second-order polynomial relation between stress and strain is derived by integrating the above equation with respect to ε (for intercept at origin):

Between any two points on the second-order polynomial stress–strain curve, the relationship between relative increment in stress σ %=σ2−σ1σ1·100 and relative increment in strain ε %=ε2−ε1ε1·100 can be expressed as:

For a straight line (α = 0), Equation (6) reduces toσ %ε %=1, i.e., the relative increment in stress and strain is the same. Equation (6) will be used in Section 3.2. to calculate relative increment in failure stress based on the determined relative increments in strain to failure.

In the present study, the experimental stress–strain curves were fitted with this second-order polynomial equation, and the fitted curves were used as practical operational stress–strain curves. Initial stiffness (E₀) at zero strain was determined based on the fitted stress–strain curves using Equation (5). This method of stiffness determination takes into account the shape of the whole stress–strain curve. It overcomes the challenges of not well-defined stress–strain curves at the start of test, and the non-linear stress–strain behavior of carbon composites.

As exemplified in Figure 5, some of the experimental stress–strain curves show serrations at the end of the curves, observed for both unprotected and protected test specimens. The reason of such small serrations could be due to various reasons like local failures in the specimen, slight slippage, gauge separation, or delamination. Therefore, due to these cases, a practical operational definition of failure stress and strain to failure is needed. As shown in Figure 5, failure stress was obtained as the measured ultimate stress (at maximum load). Then, strain to failure was obtained by extrapolating the fitted line to the ultimate stress. Based on the results of 36 test specimens (unprotected and protected carbon composites), the average corrected strain to failure (εcorrectedfailure) was lowered by 0.02% compared to measured strain to failure (εmeasuredfailure); εmeasuredfailure−εcorrectedfailure ≤0.02 %. Therefore, this correction method had an extremely small influence on the determined strain to failure value.

Example of experimental stress–strain curve and second-order polynomial fit showing the approach used to determine strain to failure.

During tensile testing of the protected carbon composite (G/C/G), the strain in each layer can be assumed equal to the measured strain for the whole composite due to the parallel arrangement of layers. Therefore, the corresponding stresses in each layer follow the rule of mixture relationships with the measured stress of the composite.

The stress in the carbon composite layer in the protected carbon composite (G/C/G), at a given value of strain, can be back-calculated as follows:

where σ_C, σ_G/C/G, σ_G, and σ_A are stresses in carbon composite layer, protected carbon composite, glass composite layer, and adhesive layer, respectively. Moreover, V_G, V_A, and V_C are volume fractions of glass composite, adhesive, and carbon composite in the protected carbon composite, respectively.

The stress values in the glass composite layers (σ_G) were assumed similar to stress values in the tested unprotected glass composites (G). The experimental stress–strain curves of unprotected glass composites (G) were fitted with the second-order polynomial (Equation (5)).

The stress values in the adhesive layers (σ_A) were found from an established stress–strain curve of epoxy adhesive, see Figure 6. The curve was established using the second-order polynomial (Equation (5)) based on values of failure stress, strain to failure, and stiffness found in the manufacturer’s datasheet. The approach is summarized in Appendix A.

Established stress–strain curves of epoxy adhesive and vinyl ester matrix, shown in the relevant strain range. See approach in Appendix A.

For the carbon composite, using the assumption of equal strain, the stress of the carbon fibres, at a given value of strain, was back-calculated as follows:

where σ_f, σ_C, and σ_m are stresses in carbon fibres, carbon composite and matrix, respectively. Moreover, V_m and V_f are volume fractions of matrix and fibres, respectively. The stress values in the matrix (σ_m) were found with a similar approach as used above for the adhesive layer.