Solution of two-parameter cohesive law using Chebyshev polynomials for singular integral equation

The inapplicability of variational methods in the cohesive crack problem for initially rigid traction‐separation relation and its solution using integral equations

“…An initial crack size of 2𝑎 is taken and hence the substitutions 𝑏 = −𝑎 and 𝑐 = 𝑎 are made. Equation ( 13) will now be numerically solved using Chebyshev polynomials [64,71]. Integrating Equation ( 13) from −𝑎 to 𝑥 yields…”

“…An initial crack size of 2 a is taken and hence the substitutions $$b=-a$$ and $$c =a$$ are made. Equation () will now be numerically solved using Chebyshev polynomials [64, 71]. Integrating Equation () from $$-a$$ to x yields $$$$\begin{equation} u_{2}(x) = \frac{-1}{2(1 - k^{2}) G} {\left\lbrace \frac{1}{\pi } \int _{-a}^{x} \frac{1}{\sqrt {a^{2}-x^{2}}} \left [\int _{-a}^{a} \frac{t_{2}(\xi ) \sqrt {a^{2} - \xi ^{2}}}{\xi - x} \mathrm{d}\xi \right ] \mathrm{d}x \right\rbrace} + \frac{-1}{2(1 - k^{2}) G} \int _{-a}^{x} \frac{\sigma _{yy}^{\infty } x}{\sqrt {a^{2} - x^{2}}} \mathrm{d}x \end{equation}$$$$…”

“…The initial crack opening is assumed to be almost zero throughout. Brief [70] and detailed [71] derivations may be found in literature. The cohesive tractions are taken only between the opposite crack faces and not in the bulk.…”

“…The cohesive tractions are taken only between the opposite crack faces and not in the bulk. This singular integral equation has been numerically solved (and verified using perturbation method) for a linear hardening traction‐separation cohesive relation [71]. In the present work, Broberg's equation of displacement gradient and normal stress will be numerically solved for initially rigid traction‐separation cohesive relation.…”