Stress solution for a crack at an arbitrary angle to an interface

Two-dimensional elasticity solution and the stress intensity factors are determined for a finite crack in one of the materials of a bimaterial composite. The crack has an arbitrary orientation and distance from the straight interface. The solution for general stress boundary conditions on the crack surface is presented in the form of coupled Fredholm integral equations of the second kind. Numerical values of the stress intensity factors are computed for various crack orientations, distances from the interface, and different combinations of material properties when the boundary conditions are uniform pressure and uniform shear stress.     

Stress intensity factors for an interface crack along an elliptical inclusion

The problem of a crack along the interface of an elliptical elastic inclusion embedded in an infinite plate subjected to uniform stresses at infinity is analyzed by the body force method. The crack tip stress intensity factors are calculated for various inclusion geometries and material combinations. Based on numerical results, the effect of the inclusion geometry on the stress intensity factors is investigated. It is found that for small interface cracks the stress intensity factors are mainly determined by the stresses, occurring at the crack center point before the crack initiation, and interface curvature radius alone.     

The Stress Intensity Factor of an Edge Dislocation Near an Elliptically Blunted Crack Tip

The stress field around the tip of an elliptically blunted crack induced by an edge dislocation has been obtained in closed form, from which the mode I and mode II stress intensity factors induced by the edge dislocation are obtained. The solutions apply to the edge dislocation either emitted from crack-tip surface or originated elsewhere, and for the dislocation located anywhere around the crack tip. The effects of the crack length, the crack-tip bluntness, the origination and position of the dislocation on the stress intensity factors are examined.     

Antiplane interaction of an anisotropic elliptic inclusion with an arbitrarily oriented crack

The antiplane interaction problem for an anisotropic elastic inclusion embedded in an anisotropic elastic matrix with an arbitrarily oriented crack, located either in the matrix or in the inclusion, is considered in this paper. The proposed analysis is based upon the use of conformal mapping, analytical continuation and Laurent series expansion of the corresponding complex potentials. By applying the existing solutions for dislocation functions, the integral equations for a line crack are formulated and the mode-III stress intensity factors are obtained numerically. Several numerical examples are given to demonstrate the effects of geometrical parameters and material property combinations on the strength of the antiplane stress singularity.     

Thermoelastic analysis for an opening crack in an orthotropic material

The thermoelastic analysis of an opening crack embedded in an orthotropic material is made under applied uniform heat flux and mechanical loadings. To simulate the case of an opening crack filled with a medium, a thermal-medium crack model is proposed. The thermally permeable and impermeable cracks are the limiting ones of the proposed thermal-medium one. The crack-tip thermoelastic fields induced by a crack in an orthotropic material are determined in closed forms. The elastic T-stress can be also obtained explicitly. The effects of applied mechanical loadings and the thermal conductivity of crack interior on the heat flux at the crack surfaces and the mode-II stress intensity factor are investigated through numerical computations. The obtained results reveal that an increase of the thermal conductivity of crack interior decreases the mode-II stress intensity factor. And when an applied mechanical loading is increasing, the mode-II stress intensity factor is rising.     

Thermal stress intensities at an interface crack between two elastic layers

The thermal stress intensities (energy release rate and stress intensity factors) due to temperature changes are derived in closed-form for an interface crack between two elastic layers of dissimilar materials. The solutions are two-dimensional and tabulated over a wide range of material and layer thickness combinations. The tables serve as rapid evaluations of the thermal stress intensities for given temperature changes. A strain gauge technique is given for determining constraint coefficients which reflect the constraint conditions during the temperature changes. The solutions are compared with results from the literature. The stress intensities due to thermal and mechanical loads are generally superimposed. As an example of application, the solutions are utilized to obtain the complete thermal and mechanical stress intensities for a four-point bend specimen.     

Continuum structural topological optimizations with stress constraints based on an active constraint technique

Stress‐related problems have not been given the same attention as the minimum compliance topological optimization problem in the literature. Continuum structural topological optimization with stress constraints is of wide engineering application prospect, in which there still are many problems to solve, such as the stress concentration, an equivalent approximate optimization model and etc. A new and effective topological optimization method of continuum structures with the stress constraints and the objective function being the structural volume has been presented in this paper. To solve the stress concentration issue, an approximate stress gradient evaluation for any element is introduced, and a total aggregation normalized stress gradient constraint is constructed for the optimized structure under the r?th load case. To obtain stable convergent series solutions and enhance the control on the stress level, two p‐norm global stress constraint functions with different indexes are adopted, and some weighting p‐norm global stress constraint functions are introduced for any load case. And an equivalent topological optimization model with reduced stress constraints is constructed,being incorporated with the rational approximation for material properties, an active constraint technique, a trust region scheme, and an effective local stress approach like the qp approach to resolve the stress singularity phenomenon. Hence, a set of stress quadratic explicit approximations are constructed, based on stress sensitivities and the method of moving asymptotes. A set of algorithm for the one level optimization problem with artificial variables and many possible non‐active design variables is proposed by adopting an inequality constrained nonlinear programming method with simple trust regions, based on the primal‐dual theory, in which the non‐smooth expressions of the design variable solutions are reformulated as smoothing functions of the Lagrange multipliers by using a novel smoothing function. Finally, a two‐level optimization design scheme with active constraint technique, i.e. varied constraint limits, is proposed to deal with the aggregation constraints that always are of loose constraint (non active constraint) features in the conventional structural optimization method. A novel structural topological optimization method with stress constraints and its algorithm are formed, and examples are provided to demonstrate that the proposed method is feasible and very effective. © 2016 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.     

On an inverse thermo-elasticity problem for an infinite medium containing a cavity of unknown shape

The static inverse thermoelastic problem for an infinite elastic isotropic medium containing a cavity of unknown shape, under three-axes tension and given constant values of the pressure and the temperature on the cavity surface, is considered. The shape of a cavity is sought subject to the condition that certain stress components are uniform on the cavity surface. It is shown that ellipsoidal shapes furnish a solution of this inverse thermo-elasticity problem. Nonlinear equations for determining the geometric cavity parameters, which lead to an equal-stress state along the cavity surface, are obtained. Results of other authors for force loading only are obtained as special cases. Numerical investigations have been carried out and correlations between values of the force and temperature loadings and geometrical parameters for an equal-stress cavity surface are studied. The stress values on the equal-stress cavity surface under force and temperature loadings are investigated.     

Axisymmetric stress in an electrostrictive hollow cylinder under electric loading

Axisymmetric problems for an electrostrictive hollow cylinder under electric loading are studied based on the potential function method. First, in the Cartesian coordinate system, the general solutions for the displacement are presented. Then, explicit results of displacement and stress are derived. For two special cases of in-plane and off-plane electric loading, the stresses around the inner and outer boundaries of the cylinder are given in an analytic form and the effects of Maxwell stress are discussed. Numerical results are also presented and graphically shown. It is found that for a thick-walled hollow cylinder, the electrostrictive stress in the internal boundary of the cylinder would be significantly large in magnitude when in-plane electrical loading is applied.     

Scattering of elastic waves by an elastic sphere

The problem of scattering of plane compressional wave by an elastic sphere embedded in an isotropic elastic medium of different material properties is solved. Approximate formulas are derived for the displacement field, stress tensor, stress intensity factors, far-field amplitudes and the scattering cross-section. It is assumed that the wave length is large compared to the radius of the scatterer. Various elastostatic limits are also presented.     

The torsion of an infinite hollow cylinder with an external crack

This paper deals with the axially symmetric torsion of a hollow cylinder with an external crack. The mixed boundary value problem is reduced to a pair of dual series equations. These equations are shown to be equivalent to an infinite system of simultaneous equations. Numerical results are presented for stresses, displacement and stress intensity factors.     

Stress intensification due to an edge crack in an anisotropic elastic solid

Integral transform techniques are used to determine the stress intensity factors of a crack at the edge of an anisotropic elastic half space under generalized plane strain conditions. Numerical results are given for a carbon fibre reinforced epoxy in uniaxial tension.     

The interaction between a crack and an inclusion

A numerical method described recently[1] is used here to obtain the stress intensity factor for a crack near an inclusion. Results for the variation of the stress intensity factor with the distance of the crack tip from the inclusion, are shown graphically.     

The effect of an elliptical inclusion on a crack

The interaction between an elliptical inclusion and a crack is analyzed by body force method. The investigated stress field is simulated by superposing the fundamental solutions for a point force applied at a point in an infinite plate containing an elliptical inclusion. Based on numerical results, effects of the inclusion shape on the crack tip stress intensity factor are discussed. It is found that for small cracks emanating from a stress-higher point on the inclusion interface the stress intensity factors are mainly determined by the stresses, occurring at the crack starting point before the crack initiation, and the inclusion root radius, besides the crack length. However, for the cracks occurring in a stress-lower region around the inclusion, it is difficult to characterize the effect of the inclusion geometry on the stress intensity factors of small cracks by the inclusion root radius alone. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stress intensity factor analysis at an interfacial corner between anisotropic bimaterials under thermal stress

A numerical method using a path-independent H-integral based on the Betti reciprocal principle was developed to analyze the stress intensity factors of an interfacial corner between anisotropic bimaterials under thermal stress. According to the theory of linear elasticity, asymptotic stress near the tip of a sharp interfacial corner is generally singular as a result of a mismatch of the materials’ elastic constants. The eigenvalues and the eigenfunctions are obtained using the Williams eigenfunction method, which depends on the materials’ properties and the geometry of an interfacial corner. The order of the singularity related to the eigenvalue is real, complex or power-logarithmic. The amplitudes of the singular stress terms can be calculated using the H-integral. The stress and displacement fields around an interfacial corner for the H-integral are obtained using finite element analysis. A proposed definition of the stress intensity factors of an interfacial corner involves a smooth expansion of the stress intensity factors of an interfacial crack between dissimilar materials. The asymptotic solutions of stress and displacement around an interfacial corner are uniquely obtained using these stress intensity factors.     

An approximate analysis of an internally loaded elastic plate containing an infinite row of closely spaced parallel cracks

The stress transfer which occurs in an internally loaded infinite elastic plate containing an array of closely spaced parallel cracks of finite width is examined. The internal loading corresponds to a doublet of concentrated forces which act at finite distances from the cracked region. The solution presented is approximate to the extent that the state of stress in the strip regions contained between adjacent cracks is considered to be one-dimensional. Such a simplification enables the derivation of certain general results for the stress distribution in a strip region contained within internally loaded half-planes of differing elastic characteristics. These solutions are obtained by Fourier transform methods. Attention is particularly focussed on the estimation of the stress magnification which occurs in the strip region.     

Two coplanar Griffith cracks in an infinite elastic layer under torsion

Summary We consider the problem of determining the stress distribution in an infinitely long isotropic homogeneous elastic layer containing two coplanar Griffith cracks which are opened by internal shear stress acting along the lengths of the cracks. The faces of the layer are assumed to be stress free. The cracks are located in the middle plane of the layer parallel to its faces. By using Fourier transforms, we reduce the problem to the solution of a set of triple integral equations with a cosine kernel and a weight function. These equations are solved exactly by using finite Hilbert transform techniques. Finally we derive the closed form expressions for the stress intensity factors and the crack energy. Solutions to the following problems are derived as particular cases: (i) a single crack in an infinite layer under torsion, (ii) two coplanar cracks in an infinite space under torsion, (iii) a single crack in an infinite space under torsion.     

Kinking of an interface crack in an orthotropic bimaterial

We investigated the asymptotic problem of a kinked interface crack in an orthotropic bimaterial under in‐plane loading conditions. The stress intensity factors at the tip of the kinked interface crack are described in terms of the stress intensity factors of the interface crack prior to the kink combined with a dimensionless matrix function. Using a modified Stroh formalism and an orthotropy rescaling technique, the matrix function was obtained from the solutions of the corresponding problem in transformed bimaterial. The effects of orthotropic and bimaterial parameters on the matrix function were examined. A reduction in the number of dependent material parameters on the matrix function was made using the modified Stroh formalism. Moreover, the explicit dependence of one orthotropic parameter on the matrix function was determined using an orthotropic rescaling technique. The effects of the other material parameters on the matrix function were numerically examined. The energy release rate was obtained for a kinked interface crack in an orthotropic bimaterial.     

Multiple interacting planar cracks in an anisotropic multilayered medium under an antiplane shear stress: a hypersingular integral approach

The problem of determining the stress field around an arbitrary number of arbitrarily-located planar cracks in an anisotropic elastic half-space which adheres perfectly to an infinitely-long elastic strip is considered. The strip is made up of several layers of anisotropic materials which are perfectly bonded to one another. The multilayered medium is assumed to undergo an antiplane deformation. Suitable integral expressions are used to represent the displacement and the stress, leading to a system of hypersingular integral equations to be solved. For a specific example of the problem, which involves particular transversely-isotropic materials, the hypersingular integral equations are solved numerically, in order to calculate the relevant crack tip stress intensity factors.     

Numerical simulation of a two-dimensional turbulent wall jet in an external stream

Turbulent wall jets possess a region with negative production of turbulent kinetic energy between the points of maximum velocity and vanishing shear stress. This characteristic feature cannot be shown with many turbulence models. The use of an extended expression for the primary turbulent shear stress together with a k–? or an algebraic Reynolds stress model results in a model which can show this physical property. Computed results obtained with this concept are compared with measurements and results obtained with the standard k–? model and a full Reynolds stress closure. It is shown that the computed results with the present and the Reynolds stress model are of similar quality. However, the Reynolds stress solution is more costly in computing time.