Antiplane shear crack in a prestressed elastic medium based on the couple stress theory

A prestressed elastic medium containing a mode-Ⅲcrack is studied by means of the couple stress theory(CST).Based on the CST under initial stresses,a governing differential equation along with a mixed boundary value problem is established.The singularities of the couple stress and force stress near the crack tips are analyzed through the asymptotic crack-tip fields resulting from the characteristic expansion method.To determine their intensity,a hypersingular integral equation is derived and numerically solved with the help of the Chebyshev polynomial.The obtained results show a strong size-dependence of the out-of-plane displacement on the crack and the couple stress intensity factor(CSIF)and the force stress intensity factor(FSIF)around the crack tips.The symmetric part of the shear stress has no singularity,and the skew-symmetric part related to the couple stress exhibits an r^(-3/2)singularity,in which r is the distance from the crack tip.The initial stresses also affect the crack tearing displacement and the CSIF and FSIF.

EXACT SOLUTION FOR ORTHOTROPIC MATERIALS WEAKENED BY DOUBLY PERIODIC CRACKS OF UNEQUAL SIZE UNDER ANTIPLANE SHEAR

Orthotropic materials weakened by a doubly periodic array of cracks under far-field antiplane shear are investigated, where the fundamental cell contains four cracks of unequal size. By applying the mapping technique, the elliptical function theory and the theory of analytical function boundary value problems, a closed form solution of the whole-field stress is obtained. The exact formulae for the stress intensity factor at the crack tip and the effective antiplane shear modulus of the cracked orthotropic material are derived. A comparison with the finite element method shows the efficiency and accuracy of the present method. Several illustrative examples are provided, and an interesting phenomenon is observed, that is, the stress intensity factor and the dimensionless effective modulus are independent of the material property for a doubly periodic cracked isotropic material, but depend strongly on the material property for the doubly periodic cracked orthotropic material. Such a phenomenon for antiplane problems is similar to that for in-plane problems. The present solution can provide benchmark results for other numerical and approximate methods.

ANTIPLANE SHEAR FIELDS NEAR A CRACK-TIP IN A BRITTLE DAMAGED MATERIAL

In this paper,a simplified brittle damage model is proposed according to the Mazars-Lemaitre damage model for concrete.A closed-form solution for a mode Ⅲ crack is obtained based on the simplified model under small scale damage conditions,which allows for discontinuities of displacement-gradient and tangential stress on the damage boundary.It is pointed out that the discontinuities of field variables near the tip region exist for the brittle damaged material induced by the softening effect of the material.

ANTIPLANE PROBLEM OF CIRCULAR ARC INTERFACIAL RIGID LINE INCLUSIONS

The antiplane problem of circular arc rigid line inclusions under antiplane concentrated force and longitudinal shear loading was dealt with. By using Riemann-Schwarz's symmetry principle integrated with the singularity analysis of complex functions, the general solution of the problem and the closed form solutions for some important practical problems were presented. The stress distribution in the immediate vicinity of circular arc rigid line end was examined in detail. The results show that the singular stress fields near the rigid inclusion tip possess a square-root singularity similar to that for the corresponding crack problem under antiplane shear loading, but no oscillatory character. Furthermore, the stresses are found to depend on geometrical dimension, loading conditions and materials parameters. Some practical results concluded are in agreement with the previous solutions.

SHEAR BEAM MODEL FOR INTERFACE FAILURE UNDER ANTIPLANE SHEAR(Ⅱ)—INSTABILITY

Based on the (Ⅰ) of the present work, the behavior of shear beam model at crack initiation stage and at instable propagation stage was studied. The prime results include: 1) discriminant equation which clarifies the mode of instability, snap_back or snap_through, was established; 2) analytical solution was given out for the double shear beam and the load_displacement diagram for monotonic loading was presented for a full process; and 3) the problem of the energy release induced by instability was discussed.

Elastic-plastic analysis of antiplane crack near crack surface region

The elastic-plastic stress distribution and the elastic-plastic boundary con- figuration near a crack surface region are significant but hard to obtain by means of the conventional analysis. A crack line analysis method is developed in this paper by consid- ering the crack surface as an extension of the crack line. The stresses in the plastic zone, the length, and the unit normal vector of the elastic-plastic boundary near a crack surface region are obtained for an antiplane crack in an elastic-perfectly plastic solid. The usual small scale yielding assumptions are not needed in the analysis.

ANTIPLANE PROBLEMS OF PERIODICAL RIGID LINE INCLUSIONS BETWEEN DISSIMILAR ANISOTROPIC MATERIALS

The antiplane shear problems of periodical rigid line inclusions between dissimilar anisotropic materials are dealt with. By using the complex variable method, the closed form solutions are obtained. The stress distribution in the immediate vicinity of the rigid line is examined. The corresponding formulation between dissimilar isotropic materials and in homogeneous anisotropic medium can be derived from the special cases of those in the present paper, and the limit conditions are in agreement with the previously known results.

NEAR CRACK LINE ELASTIC－PLASTIC ANALYSIS FOR A CRACK LOADED BY ANTIPLANE POINT FORCES

In this paper, the improved near crack line analysis method proposed in Refs. [1]and [2] is used to investigate a mode Ⅲ crack loaded by antiplane point forces in aninfinite plate in an elastic-perfectly plastic solid. The solutions of this paper aresufficiently precise near the crack line region because. the assumptions of the smallscale yielding theory have not been used and no other assumptions have been taken.

Stress singularity analysis of anisotropic multi-material wedges under antiplane shear deformation using the symplectic approach

Symplectic approach has emerged a popular tool in dealing with elasticity problems especially for those with stress singularities. However, anisotropic material problem under polar coordinate system is still a bottleneck. This paper presents a subfield method coupled with the symplectic approach to study the anisotropic material under antiplane shear deformation. Anisotropic material around wedge tip is considered to be consisted of many subfields with constant material properties which can be handled by the symplectic approach individually. In this way, approximate solutions of the stress and displacement can be obtained. Numerical examples show that the present method is very accurate and efficient for such wedge problems. Besides, this paper has extended the application of the symplectic approach and provides a new idea for wedge problems of anisotropic material.

Interface end stress field of antiplane of orthotropic bimaterials

The orthotropic bimaterial antiplane interface end of a flat lap is studied by constructing new stress functions and using the composite complex function method of material fracture. The expressions of stress fields, displacements fields and the stress intensity factor around the flat lap interface end are derived by solving a class of generalized bi-harmonic equations. The result shows that this type of problem has one singularity, the stress field has no singularity when two materials have constant ratio F 〉 0, the stress field has power singularity, and the singularity index has a trend to -1/2 as F increases. The derived equation is verified with FEM analysis.

ANTIPLANE MULTIPLE CURVED CRACK PROBLEM IN CIR-CULAR INCLUSION AND MATRIX MATERIAL

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SOLUTION OF MULTIPLE CURVED RIGID LINE ANDANTIPLANE CIRCULAR INCLUSION PROBLEM BYWEAKLY SINGULAR INTEGRAL EQUATION

Interaction between multiple curved rigid line and circular inclusion in antiplane loading condition was considered. Two kinds of elementary solutions corresponding to a concentrated force applying at inclusion and matrix material respectively were presented. Utilizing the elementary solutions and taking density function of traction difference along curved rigid line, a group of weakly singular integral equations with log kernels can be obtained. After the numerical solution of the integral equations, the discrete values of density functions of traction difference are obtainable. So stress singularity coefficients at rigid line tips can be calculated, and several numerical examples are given.

HYPERSINGULAR INTEGRAL EQUATIONS FOR PERIODIC ARRAYS OF PLANAR CRACKS IN A PERIODICALLY-LAYERED ANISOTROPIC ELASTIC SPACE UNDER ANTIPLANE SHEAR STRESS

Hypersingular integral equations are derived for the problem of determining the antiplane shear stress around periodic arrays of planar cracks in a periodically-layered anisotropic elastic space. The unknown functions are directly related to the jump in the displacements across opposite crack faces. Once the integral equations are solved, crack parameters of interest, such as the clack tip stress intensity factors, may be readily computed.For some specific examples of the problem, the integral equations are solved numerically by using a collocation technique, in order to compute the relevant stress intensity factors.

ANTIPLANE CIRCULAR INCLUSION WITH A CURVED CRACK CROSSING THE BOUNDARY

The weakly singular integral equation sued to solve the problem ofthe curved crack crossing the boundary of the antiplane circularinclusion is presented. Using the principal part analysis method ofthe Cauchy type integral equation, the singular stress index at theintersection and the singular stress of angular Regions near theintersection are obtained. By using the singular stress obtained, thestress intensity factor at The intersection is defined. After thenumerical solution of the integral equation, the stress intensityfactors at The end points of the crack and intersection areobtainable.

Degenerate scale problem in antiplane elasticity or Laplace equation for contour shapes of triangles or quadrilaterals

This paper provides several solutions to the degenerate scale for the shapes of triangles or quadrilaterals in an exterior boundary value problem （BVP） of the antiplane elasticity or the Laplace equation. The Schwarz-Christoffel mapping is used throughout. It is found that a complex potential with a simple form in the mapping plane satisfies the vanishing displacement condition （or w ---- 0） along the boundary of the unit circle when the dimension R reaches its critical value 1. This means that the degenerate size in the physical plane is also achieved. The degenerate scales can be evaluated from the partic- ular integrals depending on certain parameters in the mapping function. The numerical results of degenerate sizes for the shapes of triangles or quadrilaterals are provided.

THE INTERACTION BETWEEN MULTIPLE CURVED RIGID LINE AND ANTIPLANE CIRCULAR INCLUSION

The interaction between multiple curved rigid line and circular inclusion in antiplane loading condition is considered in this paper. By utilizing the point force elementary solutions and taking density function of traction difference along curved rigid lines, a group of weakly singular integral equations with logarithmic kernels can be obtained. After the numerical solution of the integral equations, the discrete values of density functions of traction difference are obtainable. So the stress singularity coefficient at rigid line tips can be calculated, and two numerical examples are given.

BASIC SOLUTION FOR THE THREE-PHASE COMPOSITE CONSTITUTIVE MODEL IN ANTIPLANE PIEZOELECTRICITY

A basic solution in series form for the three-phase composite cylindrical model in antiplane piezoelectricity subjected to the action of a singularity in the intermediate matrix region is presented. The solution is obtained through the complex potential approach in conjunction with the techniques of analytical continuation, singularity analysis, Laurent series expansion in an annular region and Cauchy integral formulae, etc. Based on the complex potentials obtained, explicit expressions for the distribution of stress and electric displacement in the three regions are also derived.

COPLANAR CRACKS IN A FINITE TRANSVERSELY ISOTROPIC ELASTIC SLAB UNDER ANTIPLANE SHEARL STRESSES

An antiplane crack problem concerning a pair of coplanar cracks in a finite transversely isotropic elastic slab is considered. Using Fourier integral transform together with singular integral equation which can be solvel numerically by suing a collocation technique. Once the integral equation is solved, the relevant crack energy and stress intensity factors of the problem are given. The analysis present can be easily extended to include cases where there are two or more pairs of coplanar cracks in the slab.

TWO COPLANAR CRACKS IN A TRANSVERSELY ISOTROPIC ELASTIC SLAB SUBJECTED TO ANTIPLANE SHEAR STRESSES

The problem of a transversely isotropic elastic slab containing two coplanar cracks subjected to an antiplane deformation is considered. With the aid of an integral transform technique, we formulate the problem in terms of a finite-part singular integral equation which can be solved numerically, Once the integral equation is solved, relevant quantities such as the crack energy can be readily computed.

NEAR CRACK LINE ELASTIC-PLASTIC ANALYSIS FOR A FINITE PLATE LOADED BY TWO PAIRS OF ANTIPLANE POINT FORCES

In this paper, the improved near crack line analysis method proposed in Refs. [1] and [2] is used to investigate a center crack loaded by two pairs of antiplane point forces in a finite plate in an elastic-perfectly plastic solid. And the analytical solutions are obtained, that is elastic-plastic stress fields near the crack line and the law that the length of the plastic zone along the crack line is varied with an external loads tr,ld the bearing capacity of a finite plate with a center crack. The results of this paper are sufficiently precise near the crack line, because the assumptions of the small scale yielding theory have not been used and no other assumptions have been taken.