Crack opening displacement for two unequal straight cracks with coalesced plastic zones – A modified Dugdale model

The problem investigated is of an infinite plate weakened by two collinear unequal hairline straight quasi-static cracks. Uniform constant tension is applied at infinity in a direction perpendicular to the rims of the cracks. Consequently the rims of the cracks open in Mode I type deformation. The tension at infinity is increased to the limit such that the plastic zones developed at the two adjacent interior tips of cracks get coalesced. To arrest the crack from further opening normal cohesive variable stress distribution is applied on the rims of the plastic zones. Closed form analytic expressions are obtained for load bearing capacity and crack opening displacement (COD). An illustrative case is discussed to study the behavior of load bearing capacity and crack opening displacement with respect to affecting parameters viz. crack length, plastic zone length and inter crack distance between the two cracks. Results obtained are reported graphically and analyzed.     

Crack-tip-opening displacement for four symmetrically situated cracks with coalesced interior yield zones

Crack-tip opening displacements are obtained for four collinear straight cracks, weakening an unbounded homogeneous and isotropic elastic-perfectly plastic plate. The cracks are so configured that two symmetrically situated and interiorly lying cracks are of equal-lengths. Other two exteriorly lying, collinear straight cracks (surrounding the interiorly lying straight cracks) are of mutually equal-lengths. Thus an exterior and an interior crack-set are symmetrically oriented with respect to the other interior–exterior collinear cracks-set configuration. Uniform constant load prescribed at remote boundary of the plate, opens the crack in self-similar fashion developing a strip-yield zone ahead each tip of the cracks. It is assumed that the strip-yield zone developed at each of interior tips of an exteriorly and interiorly lying crack-set configuration gets coalesced. The developed yield zones are subjected to normal cohesive yield stress to arrest the crack from further opening. The solution of the problem is obtained by superposing the solutions of the two auxiliary problems, appropriately derived from the given problem. Each of the auxiliary problems, in turn, is solved using complex variable technique. Expressions are derived for quantities of interest viz. crack-tip opening displacement (CTOD), length of each developed yield zone. The effect of applied load and closing load on the parameters CTOD and strip yield zone affecting the crack arrest is presented graphically and concluded.     

偏心裂纹板在裂纹面受两对反平面点力的弹塑性解析解

采用线场分析方法对理想弹塑性材料偏心裂纹板在裂纹面受两对反平面点力的情形进行弹塑性分析，分析不受小范围屈服条件的限制，求得了裂纹线附近应力场和位移场的弹塑性解析解、裂纹线上的塑性区长度随外荷载的变化规律及有限宽板具有偏心裂纹的承载力．     

有限宽板在裂纹面受两对反平面集中力时裂纹线场的弹塑性分析

本文采用线场分析方法对理想弹塑性材料有限宽板中心裂纹在裂纹面上受两对反平面集中力的情形进行弹塑性分析，求得了裂纹线附近的弹塑性解析解、裂纹线上的塑性区长度随外荷载的变化规律及有限宽板具有中心裂纹的承载力·本文的分析不受小范围屈服假设的限制，并且不附加其他假设条件，其结果在裂纹线附近足够精确·     

Crack Arrest Model for a Plate Weakened by Internal and External Cracks - a Modified Dugdale Approach

A modified Dugdale model solution is obtained for an elastic-perfectly-plastic plate weakened by one internal and two external straight collinear hairline cracks. The tension applied to the infinite boundary of the plate opens the rims of cracks with forming a plastic zone ahead of each tip of the internal crack and also at each finitely distant tip of the two external cracks. The developed plastic zones are closed by normal cohesive linearly varying yield-point stress distributions applied to their rims. The problem is solved using the complex-variable technique. A case study is carried out to find the load required to prevent the cracks from further growing with respect to affecting parameters. The results obtained are reported graphically and analyzed.     

Arrest of Opening of Two Circular-Arc Cracks with Coalesced Plastic Zones

The closure of plastic zones developed ahead of the tips of two unequal hairline arc cracks in an unbounded elastic-perfectly plastic plate is studied. The cracks lie along the circumference of one and the same circle. The rims of the cracks are opened in mode I type deformation by biaxial tension applied at infinity, and consequently plastic zones develop ahead of the tips of the cracks. The tension is increased to such an extent that the plastic zones of both cracks, lying adjacent to each other, are coalesced. To prevent the cracks from further opening, the rim of the plastic zone is subjected to a uniform, constant compressive yield-point stress. The problem is solved using the complex variable technique and the principle of superimposition of the stress intensity factors. The Dugdale hypothesis is used to determined the length of the plastic zones developed. The behavior of each of the parameters, viz. the length of the plastic zone, the crack length, and the intercrack distance effecting the crack closure, is investigated and reported graphically.     

Biaxial tension of a homogeneous isotropic plate with two equal coaxial cracks with regard for plastic zones near their tips

We investigate the problem of determination of the stress-strain state of an isotropic plate with two equal cracks at a set homogeneous field of forces at infinity. It is assumed that the lips of the cracks are free of load and that, near their tips, plastic zones are formed. Using Kolosov–Muskhelishvili complex potentials, we seek a solution of the problem in the class of functions bounded at the tips of the cracks and reduce it to problems of linear conjugation. Relations for the determination of the values of plastic zones and crack tip opening displacements are obtained. We perform a numerical analysis of the problem and construct graphs of dependences of the lengths of plastic zones and crack tip opening displacements on the distance between the centers of the cracks.     

Influence of initial stresses on fracture of composite materials containing interacting cracks

Considered in this study are the axially-symmetric problems of fracture of composite materials with interacting cracks, which are subjected to initial (residual) stresses acting along the cracks planes. An analytical approach within the framework of three-dimensional linearized mechanics of solids is used. Two geometric schemes of cracks location are studied: a circular crack is located parallel to the surface of a semi-infinite composite with initial stresses, and two parallel co-axial penny-shaped cracks are contained in an infinite composite material with initial stresses. The cracks are assumed to be under a normal or a radial shear load. Analysis involves reducing the problems to systems of second-kind Fredholm integral equations, where the solutions are identified with harmonic potential functions. Representations of the stress intensity factors near the cracks edges are obtained. These stress intensity factors are influenced by the initial stresses. The presence of the free boundary and the interaction between cracks has a significant effect on the stress intensity factors as well. The parameters of fracture for two types of composites (a laminar composite made of aluminum/boron/silicate glass with epoxy-maleic resin and a carbon/plastic composite with stochastic reinforcement by short ellipsoidal carbon fibers) are analyzed numerically. The dependence of the stress intensity factors on the initial stresses, physical-mechanical parameters of the composites, and the geometric parameters of the problem are investigated.     

Interfacial fracture of layered magnetoelectroelastic materials

A problem for an interface crack located in a layered magnetoelectroelastic material strip of semi-infinite length is solved. A closed-form solution is obtained for anti-plane mechanical and in-plane electric and magnetic fields. Explicit expressions for stresses and electric and magnetic fields, together with their intensity factors and the energy release rate, are obtained. The extreme cases of impermeable and permeable cracks are discussed. Using the basic solution for a single crack, solutions for two collinear interface cracks in an infinitely long layered magnetoelectroelastic medium, an interface crack in an infinitely long layered magnetoelectroelastic medium, and an edge crack at the interface of a semi-infinitely long layered magnetoelectroelastic medium are also obtained. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 145–164, March–April, 2008.     

The problem of an interface crack with a rigid patch plate on part of its edge

A mixed boundary-value problem is solved for a piecewise-homogeneous elastic body with a rectilinear semi-infinite crack on the line where the materials are joined. A rigid patch plate (a reinforcing plate) of specified shape is attached to the upper edge of the crack on a finite interval adjacent to the crack tip. The edges of the crack are loaded with specified stresses. The body is stretched at infinity by a specified longitudinal stress. External forces with a given principal vector and moment act on the patch plate. The problem reduces to a Riemann-Hilbert boundary-value matrix problem with a piecewise-constant coefficient, the solution of which is explicitly constructed using a Gaussian hypergeometric function. The angle of rotation of the patch plate and the complex potentials describing the stress state of the body are found and the stress state of the body close to the ends of the patch plate, one of which is also simultaneously the crack tip, is investigated. Numerical examples are presented that illustrate the effect of the initial force parameters, the length of the patch plate and other parameters of the body on the angle of rotation of the patch plate and the stress state of the body.