## Plane asymptotic crack-tip solutions in gradient elasticity.(English)Zbl 1176.74155

Summary: An asymptotic crack-tip solution under conditions of plane strain is developed for a material that obeys a special form of linear isotropic strain gradient elasticity. In particular, an elastic constitutive equation of the form $$\tau =\tau ^{(0)} - \ell ^{2}\nabla ^{2}\tau ^{(0)}$$ is considered, where $$(\tau ,\mathbf {\epsilon })$$ are the stress and strain tensors, $$\tau ^{(0)}=\lambda \epsilon _{kk}\delta +2\mu \epsilon, (\lambda ,\mu )$$ are the Lamé constants, and $$\ell$$ is a material length. Both symmetric (mode-I) and antisymmetric (mode-II) solutions are developed. The asymptotic solution predicts finite strains at the crack-tip. The mode-I crack-tip displacement field $$\mathbf u$$ is of the form
\begin{aligned} u_1 = Ax_1 & + \ell\left(\frac{r}{\ell}\right)^{3/2}[A_1 \tilde u_{11} (\theta, \nu ) + A_2 \tilde u_{12} (\theta, \nu )] +O(r^2),\\ u_2 = Bx_2 & + \ell\left(\frac{r}{\ell}\right)^{3/2}[A_1 \tilde u_{21} (\theta , \nu ) + A_2 \tilde u_{22} (\theta, \nu )] +O(r^2), \end{aligned} where $$(x_1,x_2)$$ and $$(r,\theta )$$ are crack-tip Cartesian and polar coordinates, respectively, $$\nu$$ is Poisson’s ratio, and $$(A,B,A_1,A_2)$$ are dimensionless constants determined by the complete solution of a boundary value problem. The $$A$$- and $$B$$- terms above correspond to uniform normal strains parallel $$(\epsilon _{11})$$ and normal $$(\epsilon _{22})$$ to the crack line, which do not contribute to the crack-tip “energy release rate” ($$J$$-integral). Detailed finite element calculations are carried out for an edge-cracked-panel (ECP) loaded by point forces and the asymptotic solution is verified. The region of dominance of the asymptotic solution for the ECP geometry analyzed is found to be order $$\ell /10$$. The “energy release rate” is found to decrease with increasing $$\ell$$.

### MSC:

 74R10 Brittle fracture 74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics

Mathematica
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