Study of the plastic zone near a crack in an anisotropic body

Abstract

The plastic zone at a crack tip in a finite anisotropic body is studied. A boundary-value problem is formulated in terms of the components of the covariant displacement vector for small strains. Particular attention is given to the case of plain strain. In this case, a numerical solution is found for a long rectangular body with a central crack under tension. As a result, conditions for the occurrence and development of a plastic zone at the crack tip are established. A plastic zone on the lateral surface of the body is discovered. How both zones extend and coalesce is elucidated. The effect of anisotropy on the occurrence of a plastic zone is evaluated

Formation of a plastic zone in an anisotropic body under loads acting along a crack

“…Let us use the relations for the components of the displacement vector u derived in [12]. First, note that the invariants E and X (see (1.2)) can be expressed in terms of the components of the vector u as follows [12]:…”

“…Basics. The discussion below is based on the results obtained in [12] in formulating and solving a boundary-value problem for an elastoplastic anisotropic body described by nonorthogonal curvilinear coordinates x 1 , x 2 , x 3 .1.1. Tensor-Linear Constitutive Equations.…”

“…Tensor-Linear Constitutive Equations. The following tensor-linear constitutive equations [2] relating the contravariant stress tensor S and the covariant strain tensor D are used in [12]: …”

Influence of tension along a crack on the plastic zone in an anisotropic body

“…Plastic zones in anisotropic bodies are still poorly understood. They were studied in few publications [16][17][18], where several boundary-value problems for plane strain and generalized plane stress state were solved numerically and the influence of anisotropy and crack length on the size and shape of plastic zones was analyzed.Note that elastoplastic fracture under loads acting along a crack is studied by nonclassical fracture mechanics because the classical Griffith-Irwin-Barenblatt and Leonov-Panasyuk-Dugdale approaches fail to describe such processes [11,17]. It is therefore of current importance to thoroughly study this anomalous case to complete the overall picture of elastoplastic fracture.…”

“…Plastic zones in anisotropic bodies are still poorly understood. They were studied in few publications [16][17][18], where several boundary-value problems for plane strain and generalized plane stress state were solved numerically and the influence of anisotropy and crack length on the size and shape of plastic zones was analyzed.…”