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
The effects of tension and compression along a crack on the plastic zone in a finite anisotropic body under plane strain are studied. The formation pattern for the plastic zone with increasing load is established by numerically solving a boundary-value problem for each of the cases. In particular, a new plastic zone is revealed. It occurs at the crack face under a compressive load of certain magnitude. How this plastic zone interacts with that at the crack tip is established Introduction. Loads acting along a crack have always attracted particular interest of experts because analytic solutions obtained within the framework of classical fracture mechanics lead to paradoxes. For example, the calculation of ultimate loads for the Griffiths-Irwin, Dugdale, Barenblatt, Leonov-Panasyuk cracks has revealed that loads acting along a crack do not affect the fracture characteristics. This fact contradicts experimental data [4,10]. Therefore, several nonclassical approaches have been developed to solve the problem. One approach, based on the concept of local buckling at the crack periphery, is outlined in [4,8]. The other approaches are analyzed in [10]. Recently various models of the plastic zone at the crack front have been actively developed [10,11], which urgently require solutions of the corresponding boundary-value problems. A great many boundary-value problems on plastic zones near a crack in an isotropic body were solved in [5, 6, etc.]. The plastic zone near a crack in an anisotropic body, however, has been studied inadequately [7]. It was the subject of few studies of which [12] is noteworthy.The object of study here is an elastoplastic anisotropic body of finite dimensions with a crack regarded as a cut of zero width. We will use governing equations written in terms of the components of the displacement vector and examine, by way of a specific example, the influence of tensile and compressive loads along a crack on the formation of a plastic zone (the compressive load is assumed to be less than the critical load that causes local buckling at the crack periphery).1. 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. The following tensor-linear constitutive equations [2] relating the contravariant stress tensor S and the covariant strain tensor D are used in [12]:
The influence of the length of a mode I crack on the plastic zone in an anisotropic body under hard loading is studied. The case of a generalized plane stress state is examined. A boundary-value problem is solved numerically to study the behavior of the main plastic zone at the crack tip, the additional plastic zone on the lateral face of the body, and the merged plastic zone Introduction. Various crack models are widely used in elastoplastic fracture mechanics [9-11, 13-16, 19]. To justify these models, it is necessary to know the size and shape of the plastic zone at a crack and to solve the corresponding boundary-value problems. The papers [4-6, etc.] solved (both analytically and numerically) many boundary-value problems for plane and antiplane strains and generalized plane stress state. However, they are all concerned with the plastic zone in an isotropic body. The plastic zone in an anisotropic body is still inadequately understood. There are just a few studies on the subject [17,18], where several boundary-value problems have been solved (numerically) for the case of plane strain and the effect of anisotropy and loads along the crack on the size and shape of the plastic zone has been established.This paper studies the plastic zone at a crack in an anisotropic body in the case of a generalized plane stress state. Strains are assumed small. The body is rectangular and thin and has a mode I crack at the center. The governing equations are written for the components of the displacement vector. By numerically solving the boundary-value problem, we can describe how the plastic zone forms and, in particular, can establish the effect of the crack length on the size and shape of the plastic zone.1. Preliminaries. We assume that Poynting's effect is absent when a body is deformed. Hence, tensor-linear constitutive equations may be used to derive the governing equations.1.1. Tensor-Linear Constitutive Equations. The following equations are derived in [2] to relate the components of the stress tensor S with the components of the strain tensor D:
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