Stresses between 3D fractures in infinite and layered elastic solids

Periodically distributed opening mode fractures are often found in layered sedimentary rocks. The stress analysis related to opening mode fractures in layered solids is solved by a new numerical approach combining (3D) fast Fourier transform with the theory of periodic eigenstrain and the conjugate gradient method. Results show that the fracture spacing to layer thickness ratio for embedded opening mode fractures, using a three-dimensional (3D) model, is in good comparison with that of the plane strain case (two-dimensional model). The critical value of the fracture spacing to layer thickness ratio increases for a stiff layer case and decreases for a stiff surrounding solid case. Out-of-plane fracture length is also studied as a parameter in the 3D modeling. Opening mode surface fractures in a layered half-space were also studied. The results show that a critical fracture saturation ratio exists for this case and occurs when the normal surface stress transitions from tensile to compressive. This stress state is shown to be caused by a bending effect in the layer.     

Stress intensity factors of two diametrically opposed edge cracks in a thick-walled functionally graded material cylinder

A method is developed to evaluate stress intensity factors for two diametrically-opposed edge cracks emanating from the inner surface of a thick-walled functionally graded material (FGM) cylinder. The crack and the cylinder inner surfaces are subjected to an internal pressure. The thermal eigenstrain induced in the cylinder material due to nonuniform coefficient of thermal expansion after cooling from the sintering temperature is taken into account. First, the FGM cylinder is homogenized by simulating its nonhomogeneous material properties by an equivalent eigenstrain, whereby the problem is reduced to the solution of a cracked homogenized cylinder with an induced thermal and an equivalent eigenstrains and under an internal pressure. Then, representing the cracks by a continuous distribution of edge dislocations and using their complex potential functions, generalized formulations are developed to calculate stress intensity factors for the cracks in the homogenized cylinder. The stress intensity factors calculated for the cracks in homogenized cylinder represents the stress intensity factors for the same cracks in the FGM cylinder. The application of the formulations are demonstrated for a thick-walled TiC/Al₂O₃ FGM cylinder and some numerical results of stress intensity factors are presented for different profiles of material distribution in the FGM cylinder.     

Relations between compliances of inhomogeneities having the same shape but different elastic constants

The contributions of inhomogeneities having the same shape but different elastic constants, to the overall elastic properties are interrelated. The utility of these relations lies, in particular, in the possibility to extend available results for pores or rigid inclusions to inhomogeneities of arbitrary elastic properties. The relations are exact for ellipsoids and approximate for non-ellipsoidal shapes. The constructed approximation also constitutes approximate connection between the first Eshelby’s problem (the eigenstrain problem) and the second one (the inhomogeneity problem), for non-ellipsoidal shapes. It also yields approximate formulas for the contribution of a non-ellipsoidal inhomogeneity to effective elastic properties.     

Effect of FGM coating thickness on apparent fracture toughness of a thick-walled cylinder

This study describes the effect of FGM coating thickness on apparent fracture toughness (AFT) of a thick-walled cylinder containing two diametrically-opposed edge cracks emanating from the inner surface of the cylinder. The incompatible eigenstrain developed in the cylinder due to nonuniform coefficient of thermal expansion because of cooling from sintering temperature is taken into account. Based on a generalized method of evaluating stress intensity factor (SIF) addressed in a previous study, an approach is developed to calculate AFT. Some numerical results of AFT are presented for a TiC/Al₂O₃ FGM coating at the inner surface of an Al₂O₃ thick-walled cylinder.     

Simulation of ellipsoidal particle-reinforced materials with eigenstrain formulation of 3D BIE

Based on the concepts of eigenstrain and equivalent inclusion of Eshelby for inhomogeneity problems, a computational model and its solution procedure are presented using the proposed three-dimensional (3D) eigenstrain formulation of boundary integral equations (BIE) for simulating ellipsoidal particle-reinforced (and/or void-weakened) inhomogeneous materials. In the model, the eigenstrains characterizing deformation behaviors of each particle embedded in the matrix are determined using an iterative scheme with the aid of the corresponding Eshelby tensors, which can be obtained beforehand either analytically or numerically. With the proposed numerical model, the unknowns of the problem appear only on the boundary of the solution domain, since the interface condition between particles/voids and the matrix is satisfied naturally. The solution scale of the inhomogeneity problem can thus be significantly reduced. Using the algorithm, the stress distribution and the overall elastic properties are identified for ellipsoidal particle-reinforced/void-weakened inhomogeneous materials over a representative volume element (RVE). The effects of a variety of factors on the overall properties of the materials as well as the convergence behavior of the algorithm are studied numerically, showing the validity and efficiency of the proposed algorithm.     

Surface deformation of an elastic half-space with attenuating eigenstrain in a striped region

Surface deformation of an elastically isotropic half-space with eigenstrain in a striped region perpendicular to the surface is studied. The eigenstrain attenuates exponentially from the surface, representing some typical cases in technological applications. By utilizing the Fourier transform technique, the surface displacement components on the surface of the half-space are obtained analytically. The effects of such factors as the width of the striped region, the Poisson’s ratio of the material, and the state of the eigenstrain are discussed. The result maps the eigenstrain to the surface profile, and may be used to either explore related phenomena or create desired topography on the surface of photosensitive solids.     

Thermal Stress Analysis for an Inclusion with Nonuniform Temperature Distribution in an Infinite Kirchhoff Plate

Abstract

An elliptic inclusion with prescribed polynomial eigenstrains in an infinite Kirchhoff plate is analyzed. The integral type general solutions for the in-plane and out-of-plane displacements on the mid-plane of the plate were derived. The integrals were simplified by using Green's function for the Kirchhoff plate. The integrals could be explicitly expressed by calculating two potential functions defined in this work. After some manipulation of Ferrers and Dyson's formula related to the integration of the harmonic potential for the three-dimensional ellipsoid, we evaluated the potential functions, which can be algebraically expressed by the I-integrals. The results were applied to the analysts of the thermal stress for an inclusion with non uniform temperature distribution that might be approximated by a polynomial. For mathematical convenience, we consider an inclusion with a linear temperature distribution. The expressions for the displacements were decomposed in order to separately investigate the effects of the constant and the first-order term of the temperature distribution. The elastic fields caused by an elliptic inhomogeneity with polynomial eigenstrains, which is called the inhomogeneous inclusion, were also determined by the equivalent eigenstrain method.     

Analysis of angular distortion in line-heating

A simplified analytical model for prediction of deformations of metal plates due to line-heating process is presented. In the present work, a formula for plate deformation produced by line-heating in terms of process parameters such as heat input and plate dimensions is developed analytically using an eigenstrain concept. The residual deformation due to thermal processing was depends on the magnitude and region of plastic strains at heating zone. The magnitude of plastic strains was determined by disk model and its region was calculated using the Rosenthal's solution. The vertical displacement of the plate was analysed by using an infinite laminated plate theory to consider a cuboidal inclusion with an eigenstrain. Comparison of calculated results and experimental data shows the accuracy and validity of proposed model.     

Gauss integration applied to a Green''s function formulation for cylindrical fiber composites

The stress in an elastic two phase material is formulated in terms of eigenstrains and Green's functions leading to a singular integral equation. The singularity is dealt with by a subtraction technique and contour integrations, one of which corresponds to the Eshelby solution. The remaining non-singular integration is evaluated numerically using a Gauss rule. The calculated local stresses are within 1% of the analytical solution for an isolated inclusion problem. Also, the effective properties as a function of volume fraction are in good agreement with Christensen's analytical result. The method is then used to study rectangular versus square packing. The volume averaged stiffness for rectangular packing may be either higher or lower than that for square packing, depending on the loading direction. The stress concentration factor is shown to increase with fiber volume fraction. The stress concentration factor also increases at a decreasing rate as the fiber to matrix stiffness ratio is increased. All changes examined that an increase in the stiffness, whether by increased fiber volume fraction, increased fiber stiffness or an alternate packing arrangement, also increase the fiber stress concentration factor.     

Effect of residual stresses on ductile crack growth resistance

It is well known that residual stresses influence the ductile fracture behaviour. In this paper, a numerical study was performed to assess the effect of residual stresses on ductile crack growth resistance of a typical pipeline steel. A modified boundary layer model was employed for the analysis under plane strain, Mode I loading condition. The residual stress fields were introduced into the finite element model by the eigenstrain method. A sharp crack was embedded in the center of the weld region. The complete Gurson model has been applied to simulate the ductile fracture by microvoid nucleation, growth and coalescence. Results show that tensile residual stresses can significantly reduce the crack growth resistance when the crack growth is small compared with the length scale of the tensile residual stress field. With the crack growth, the effect of residual stresses on the crack growth resistance tends to diminish. The effect of residual stress on ductile crack growth resistance seems independent of the size of geometrically similar welds. When normalized by the weld zone size, the ductile crack growth resistance collapses into one curve, which can be used to assess the structural integrity and evaluate the effect of residual stresses. It has also been found that the effect of residual stresses on crack growth resistance depends on the initial void volume fraction f₀, hardening exponent n and T-stress.