Plane asymptotic crack-tip solutions in gradient elasticity

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 τ=τ(0)-ℓ2∇2τ(0)τ=τ(0)-ℓ2∇2τ(0) is considered, where (τ,ε)(τ,ε) are the stress and strain tensors, τ(0)=λεkkδ+2με,(λ,μ)τ(0)=λεkkδ+2με,(λ,μ) are the Lamé constants, and ℓ 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 u is of the formu1=Ax1+ℓrℓ3/2A1u˜11(θ,ν)+A2u˜12(θ,ν)+O(r2),u2=Bx2+ℓrℓ3/2A1u˜21(θ,ν)+A2u˜22(θ,ν)+O(r2),where (x1,x2)and(r,θ) are crack-tip Cartesian and polar coordinates, respectively, ν   is Poisson’s ratio, and (A,B,A1,A2)(A,B,A1,A2) 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 (ε11)(ε11) and normal (ε22)(ε22) 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 of order ℓ/10. The “energy release rate” is found to decrease with increasing ℓ.