On problems with the determination of the fracture resistance for materials with spatial variations of the Young’s modulus

The paper considers the near-tip J-integral $$\text {J}_{\mathrm{tip}}$$ , the far-field J-integral $$\text {J}_{\mathrm{far}} $$ , and the experimental J-integral $$\text {J}^{\mathrm{exp}}$$ in a material with sinusoidal variation of the Young’s modulus $$E$$ . The evaluations of $$\text {J}_{\mathrm{tip}} $$ and $$\text {J}_{\mathrm{far}} $$ are based on the concept of configurational forces; $$\text {J}^{\mathrm{exp}}$$ is evaluated from the area below the load point displacement curve, as prescribed by the standard testing procedures. Analytic expressions and/or approximation formulae are derived, how $$\text {J}_{\mathrm{tip}} ,\,\text {J}_{\mathrm{far}} $$ and $$\text {J}^{\mathrm{exp}}$$ depend on the system parameters of the material, i.e. wavelength and amplitude of the $$E$$ -variation and its phase shift with respect to the crack tip position, and the global specimen dimensions. The analyses show that $$\text {J}_{\mathrm{tip}} $$ and $$\text {J}_{\mathrm{far}} $$ exhibit a strong dependency on the phase shift, but not the experimental J-integral $$\text {J}^{\mathrm{exp}}$$ . This is the reason why the current procedures for fracture mechanics testing are not suitable to determine the true values of the fracture initiation toughness $$\text {J}_\mathrm{i} $$ or the crack growth resistance $$R$$ of a material, if the material properties exhibit a spatial variation in the direction of crack propagation. Relations are given to estimate the possible errors.