Abstract
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.
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Financial support by the Austrian Federal Government and the Styrian Provincial Government within the research activities of the K2 Competence Center on “Integrated Research in Materials, Processing and Product Engineering”, under the frame of the Austrian COMET Competence Center Programme, is gratefully acknowledged (Strategic project A4.20-WP1).
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Appendix: Integrals
Appendix: Integrals
1.1 Integral \(I_\mathrm{u} \)
The integral \(I_\mathrm{u} \left( {{E_\mathrm{A} }/{E_0 }, {X_0 }/\lambda , R/\lambda } \right) \), Eq. (14b), is re-formulated with \(E\left( x \right) \) from Eq. (3) in the form
This integral can be solved both numerically and analytically, see e.g. http://www.wolfram.com/mathematica. Analytical solutions are given for \(R/{\lambda =1}\) and two different values for the relative phase shift, \({X_0 }/\lambda =1/2\) (corresponding to the location of the maxima in \(\text {J}_{\mathrm{tip}})\) and \({X_0 }/\lambda =0\) or 1 (corresponding to the location of the minima in \(\text {J}_{\mathrm{tip}})\) as
The quantity \(\hbox {J}_1 \) symbolizes the Bessel function of first kind of order 1.
1.2 Integral \(I_\mathrm{t} \)
The integral \(I_\mathrm{t} \left( {{E_\mathrm{A} }/{E_0 }, {X_0 }/\lambda , R/\lambda } \right) \), Eq. (14c), follows with Eq. (12b) as
This integral must be solved numerically. Note that the integrand becomes an indefinite form \(0/0\) for \({\theta =\pi }/2\), identified as \(\left( {3-\nu } \right) /4\cdot 2\pi R/\lambda \times E_\mathrm{A} \cos \left( {2\pi {X_0 }/\lambda } \right) /E\left( {x=0} \right) \).
To avoid numerical problems \({E_\mathrm{A} }/{E_0 }\) should be kept smaller than 0.999.
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Fischer, F.D., Predan, J., Müller, R. et al. On problems with the determination of the fracture resistance for materials with spatial variations of the Young’s modulus. Int J Fract 190, 23–38 (2014). https://doi.org/10.1007/s10704-014-9972-2
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DOI: https://doi.org/10.1007/s10704-014-9972-2