Plane strain crack problems for elastic materials with variable properties

Distributions of stress, strain and displacement occurring at the tip of a crack in a material with properties dependent on the type of loading are investigated for the conditions of plane strain in both far-field tensile and shear loads. The causes of the dependence of material properties on the type of external forces are the various inhomogeneities such as microcracks, pores, inclusions or reinforcing components in a material. The behaviour of these inhomogeneities depends substantially on the conditions of loading or deformation. Hence, the deformation properties of a material are not fixed intrinsic material characteristics that are invariant to the loading conditions, but rather the macroproperties of such materials are stress-state-dependent ones, and this effect becomes more noticeable as the volume content of the inhomogeneities increases. The asymptotic solutions of crack problems are obtained on the basis of proposed stress-strain relations describing not only the stress-state dependence of material properties, but the interrelation between the characteristics of volume and shear deformation as well. In a non-uniform stress state the primary macrohomogeneous material becomes an heterogeneous one. The use of the stress function is not effective for the solution of plane strain crack problems for the materials under consideration. Therefore, an approach based on the corresponding representation for the strains is used. It is shown that the commonly used suppositions of the symmetry or anti-symmetry in the stress distribution relative to the crack plane can not be accepted, since they do not allow all the boundary conditions to be satisfied. The opening of the crack surfaces in the case of far shear field is observed. The influence of stress-state sensitivity of material properties on the values of the stress intensity factor is more significant for tensile crack than for the crack in far shear field.     

On a generalised plane strain crack problem for inhomogeneous anisotropic elastic materials

A generalised plane strain crack problem is considered for a class of inhomogeneous anisotropic elastic materials. The problem is reduced to a boundary integral equation involving hypersingular integrals. The boundary integral equation may be solved numerically using standard procedures. Some crack problems for a particular inhomogeneous material are considered in detail and the stress intensity factors are obtained in order to assess the effect of the anisotropy and inhomogeneity on the stress field near the crack tips.     

Plane strain problems in piezoelectricity

The paper is concerned with some boundary-value problems in the static theory of linear piezoelectricity. First, fundamental solutions are established. Then, the boundary-value problems are reduced to integral equations for which Fredholm's basic theorems are valid. Existence theorems are derived.     

Interface crack problems with strain gradient effects

In this paper, the strain gradient theory proposed by Chen and Wang (2001a, 2002b) is used to analyze an interface crack tip field at micron scales. Numerical results show that at a distance much larger than the dislocation spacing the classical continuum plasticity is applicable; but the stress level with the strain gradient effect is significantly higher than that in classical plasticity immediately ahead of the crack tip. The singularity of stresses in the strain gradient theory is higher than that in HRR field and it slightly exceeds or equals to the square root singularity and has no relation with the material hardening exponents. Several kinds of interface crack fields are calculated and compared. The interface crack tip field between an elastic-plastic material and a rigid substrate is different from that between two elastic-plastic solids. This study provides explanations for the crack growth in materials by decohesion at the atomic scale.     

Plane strain crack closure and cyclic hardening

This study is focused on the understanding of the mechanical effects of cyclic hardening on crack tip plasticity and on plasticity-induced crack closure. Various finite element analyses were conducted using abaqus. Cyclic hardening is found to affect both crack closure and the shape of the plastic zone at the crack tip. Crack growth modelling in plane strain conditions in a cyclically hardening material is discussed. An empirical formula is provided which allows the calculation of the crack tip plastic zone size under plane strain conditions in a cyclically hardening material. The effects of overloads are also examined.     

Plane strain asymptotic fields for cracks terminating at the interface between elastic and pressure-sensitive dilatant materials

The plane strain asymptotic fields for cracks terminating at the interface between elastic and pressure-sensitive dilatant material are investigated in this paper. Applying the stress-strain relation for the pressure-sensitive dilatant material, we have obtained an exact asymptotic solution for the plane strain tip fields for two types of cracks, one of which lies in the pressure-sensitive dilatant material and the other in the elastic material and their tips touch both the bimaterial interface. In cases, numerical results show that the singularity and the angular variations of the fields obtained depend on the material hardening exponent n, the pressure sensitivity parameter μ and geometrical parameter λ. This revised version was published online in August 2006 with corrections to the Cover Date.     

Contact and crack problems for an elastic wedge

In this paper the contact and the crack problems for an elastic wedge of arbitrary angle are considered. The problem is reduced to a singular integral equation which, in the general case, may have a generalized Cauchy kernel. The singularities under the stamp as well as at the wedge apex are studied and the relevant stress intensity factors are defined. The problem is solved for various wedge geometries and loading conditions. The results may be applicable to certain foundation problems and to crack problems in symmetrically loaded wedges in which cracks may initiate from the apex.     

Plane strain problem in microstretch elastic solid

The eigenvalue approach is developed for the two-dimensional plane strain problem in a microstretch elastic medium. Applying Laplace and Fourier transforms, an infinite space subjected to a concentrated force is studied. The integral transforms are inverted using a numerical technique to get displacement, force stress, couple stress and first moment, which are also shown graphically. The results of micropolar elasticity are deduced as a special case from the present formulation.     

On interface crack propagation with variable velocity for longitudinal shear problems

The antiplane strain problem of straight interface crack propagation between two elastic half-spaces under arbitrary variable loading is considered. The crack edge is specified as an arbitrary smooth function of time. It is assumed that the crack speed is less than the smaller of the shear wave velocities of two media. An integral transform method and factorization technique are used to solve the problem. The solutions are worked out for semi-infinite crack and finite crack problems. The dynamic stress intensity factors at the crack tip of the moving interface crack are given and it is found that the stress intensity factor of the interface crack is slightly higher than that in the homogeneous medium with slower shear wave velocity.     

A hypersingular boundary integral equation for antiplane crack problems for a class of inhomogeneous anisotropic elastic materials

An antiplane multiple crack problem is considered for a class of inhomogeneous anisotropic elastic materials. The problem is reduced to a boundary integral equation involving hypersingular integrals. The boundary integral equation may be solved numerically using standard procedures. Some crack problems for a particular inhomogeneous material is considered in detail and the stress intensity factors are obtained in order to assess the effect of the anisotropy and inhomogeneity on the stress field near the crack tips.     

Plane strain crack instability under small-scale yielding conditions

A theoretical analysis shows that if a plane strain crack becomes unstable under small-scale yielding conditions after only a very small increment of stable crack extension, there being no change in fracture mode, then the difference between the crack tip stress intensity factors for the onset of crack extension and instability is also small. Implications of this prediction are discussed in relation to the fracture toughness testing of materials.     

Tailoring materials with prescribed elastic properties

This paper describes a method to design the periodic microstructure of a material to obtain prescribed constitutive properties. The microstructure is modelled as a truss or thin frame structure in 2 and 3 dimensions. The problem of finding the simplest possible microstructure with the prescribed elastic properties can be called an inverse homogenization problem, and is formulated as an optimization problem of finding a microstructure with the lowest possible weight which fulfils the specified behavioral requirements. A full ground structure known from topology optimization of trusses is used as starting guess for the optimization algorithm. This implies that the optimal microstructure of a base cell is found from a truss or frame structure with 120 possible members in the 2-dimensional case and 2016 possible members in the 3-dimensional case. The material parameters are found by a numerical homogenization method, using Finite-Elements to model the representative base cell, and the optimization problem is solved by an optimality criteria method.

Numerical examples in two and three dimensions show that it is possible to design materials with many different properties using base cells modelled as truss or frame works. Hereunder is shown that it is possible to tailor extreme materials, such as isotropic materials with Poisson's ratio close to − 1, 0 and 0.5, by the proposed method. Some of the proposed materials have been tested as macro models which demonstrate the expected behaviour.     

Crack front elastic stress state for three-dimensional crack problems

The problems inherent in the estimation of fracture mechanics parameters for three-dimensional crack problems are reviewed. A novel parameter is introduced to quantify the loss of constraint along that portion of a crack front adjacent to a free surface. Finite element analyses of a compact tension specimen with a straight crack front indicate the usefulness of this parameter in studying the influence of Poisson's ratio and plate thickness on the extent of the free surface boundary layer effect. A method for predicting the shape of a natural crack in this specimen is also presented.

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Résumé On passe en revue les problèmes inhérents à l'estimation des paramètres de mécanique de rupture dans le cas des fissures tridimensionnelles. Afin de quantifier l'absence de bridage qui caractérise la portion du front de fissure qui est adjacente à une surface libre, on a introduit un nouveau paramè00E8;tre. L'analyse par éléments finis d'une éprouvette de traction compacte présentant un front de fissure droit indique l'utilité de ce paramètre pour l'étude de l'influence du module de Poisson et de l'épaisseur d'une tôle sur l'effet de couche de bord caractérisant la surface libre. On présente également une méthode de prédiction de la forme d'une fissure naturelle dans ce type d'éprouvette.

     

Plane stress crack trajectories in low modulus materials

Crack trajectory studies in propellant and liner materials under combined loading conditions have been conducted using plane stress sheets. Linear strain finite element modeling was used to determine the direction of maximum principal stress, which was hypothesized as the direction of crack propagation. However, it was found experimentally that contrary to many other engineering materials, this was not the nature of the behavior of cracks in these low modulus materials. Instead it was found that in order to predict the trajectory it was necessary to model the crack with a finite tip radius which in reality was produced by the deformation of the soft materials before the crack propagated. Experimental results were accurately predicted in this manner, and it was not necessary to use nonlinear strain measures nor nonlinear constitutive laws.     

Plane stress crack-line fields for crack growth in an elastic perfectly-plastic material

Mode-I crack growth in an elastic perfectly-plastic material under conditions of generalized plane stress has been investigated. In the plastic loading zone, near the plane of the crack, the stresses and strains have been expanded in powers of the distance, $\text{y}$, to the crack line. Substitution of the expansions in the equilibrium equations, the yield condition and the constitutive equations yields a system of simple ordinary differential equations for the coefficients of the expansions. This system is solvable if it is assumed that the cleavage stress is uniform on the crack line. By matching the relevant stress components and particle velocities to the dominant terms of appropriate elastic fields at the elastic-plastic boundary, a complete solution has been obtained for ?_y in the plane of the crack. The solution depends on crack-line position and time, and applies from the propagating crack tip up to the moving elastic-plastic boundary. Numerical results are presented for the edge crack geometry.     

A fatigue crack propagation model for strain hardening materials

In this paper a crack propagation model based on Tomkins concept (dl/dN ∝ Δε_p · ω) has been developed using the theoretically developed cyclic plastic zone sizes. The crack propagation rates are found to be functions of stress intensity factor, Elber's effective stress range ratio, cyclic yield strength of material, crack length, specimen width and cyclic strain hardening exponent. Suitably grouped to give the crack growth rate in terms of five constants termed as Loading Constant, Material constant, Crack size constant, specimen Width Constant and Stress Intensity Exponent. The crack growth rates found by theory are compared with the experimental results available in literature and a good agreement is found.     

Benchmark problems for wave propagation in elastic materials

The application of the new numerical approach for elastodynamics problems developed in our previous paper and based on the new solution strategy and the new time-integration methods is considered for 1D and 2D axisymmetric impact problems. It is not easy to solve these problems accurately because the exact solutions of the corresponding semi-discrete elastodynamics problems contain a large number of spurious high-frequency oscillations. We use the 1D impact problem for the calibration of a new analytical expression describing the minimum amount of numerical dissipation necessary for the new time-integration method used for filtering spurious oscillations. Then, we show that the new numerical approach for elastodynamics along with the new expression for numerical dissipation for the first time yield accurate and non-oscillatory solutions of the considered impact problems. The comparison of effectiveness of linear and quadratic elements as well as rectangular and triangular finite elements for elastodynamics problems is also considered.     

Nondestructive characterization of materials with variable properties

Summary Materials with continuous variable properties in time and space are considered. The possibility of nondestructive characterization of these materials on the basis of nonlinear longitudinal wave propagation data is discussed on the basis of recent results. The model direct and inverse problems are solved. The utilization of obtained results enables one to solve several nondestructive material characterization problems.