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This paper discusses the stress fields when a spheroidal inclusion, free to slip along its interface, is subjected to a constant nonshear eigenstrain, and when an elastic body containing the inhomogeneity is under all-around tension or uniaxial tension at infinity. In each case the stress field in the inclusion or the inhomogeneity is not constant, contrary to Eshelby's solution. When sliding takes place, the stress increases locally compared with the perfect bonding case, but the elastic energy decreases due to the relaxation. The relative displacement (slip) along the interface is also evaluated.

A new approach to estimate the overall behavior of an inhomogeneous body is applied to investigate the average elastic-plastic state of composite materials in small deformation. The method is based on the concept of the average field of an infinite body with inhomogeneities, which is replaced by an equivalent body with homogeneous inclusions having appropriate eigenstrains that are composed of the actual plastic misfit strains and the properly determined eigenstrains by the equivalent replacement of the inhomogeneities. A simple example is given for a pure shear state of the elastic-perfectly plastic composite materials with spherical inhomogeneities. The overall hardening rate, quantitative estimate of ductility improvement, and unloading behavior are discussed.

We propose the extended Hamilton's principle to investigate the dynamic plastic behavior of a beam or a plate under an impact loading. Material is assumed to be rigid perfectly plastic. The impact loading is given in the form of initial velocity. Good agreement between our numerical solutions and experimental results done by Parkes and Jones indicates that the proposed variational method is a powerful approximation method for the dynamic plasticity analysis. The effect of strain-rate sensitivity on the permanent deflection of a plate is investigated in some detail. Instead of nonlinear differential equations, nonlinear algebraic equations are solved in the proposed method.

An attempt is made to correlate some morphological characteristics of pitting cracks with a simple asperity interaction model. The stress field created by a semispherical pressure is combined with the macroscopic Hertzian stress field. For the initiation of fatigue cracks, emphasis is given to two stresses: the maximum range of the superimposed average shear stress over a semicircular area inclined at an angle α to the rolling surface; and the maximum of the superimposed average normal stress over the same semicircular area. Possible mechanisms for initiation of surface pitting are proposed.

The three-dimensional stress distribution for a periodic array of penny-shaped cracks in an infinite isotropic elastic solid under arbitrary uniform loadings is considered by means of the Somigliana dislocation method. The displacement discontinuity of the Somigliana dislocations, by which the penny-shaped cracks are simulated mathematically, is assumed to be in the form p(x'1, x'2)(1 − x'12− x'22)½. The unknown polynomial p(x'1, x'2) is determined from the boundary conditions on the surfaces of the penny-shaped cracks. It is shown that the singular behavior of the three-dimensional stresses in the vicinity of the crack border is easily examined and the stress intensity factors can be expressed explicitly as functions of position along the crack border. Numerical results for the stress intensity factors are shown for the case of the periodic array of coplanar penny-shaped cracks.

Stresses for the top and bottom of an expansive cylinder in a half-space are investigated. The maximum hoop stress is used for controlled cracking of concrete in demolitions. A design formula is given for determination of spacing between holes necessary to fracture when many holes are lying in a straight line.

Transform methods are used to solve the problem of an elastic quarter space subjected to tractions on its surface. The present analysis uses the reflection principle to formulate problem as the solution of unknown residual surface tractions. A Fourier transform is taken in the direction of the edge of the quarter space. The problem is thereby reduced to decouplable integral equations, which are similar to those for the elastic quarter plane. Field quantities are obtained by performing the appropriate inverse transforms and the present results are compared with those of Hetenyi for the normally loaded case. Additional results are given for the cases of tangential surface loading.

The stress relaxation caused by diffusion round a second-phase particle has been measured by internal friction in Al-Si alloys. Reproducible clear internal-friction peaks were found at ∼420 K at a frequency of ∼ 1 Hz. These were unambiguously assigned to the presence of Si particles in Al. In accordance with a theory of the relaxation caused by diffusion along the matrix - particle interface, the relaxation time is proportional to the third power of the particle size. The activation energy and the pre-exponential factor of this diffusion are ∼ 0·92 eV and ∼ 3·9 × 10-5 m2/s, respectively. The magnitude of the relaxation strength agrees with the theory.

The elastic field produced by a straight dislocation dipole in an unbounded anisotropic elastic medium is investigated in the framework of the elastic continuum approximation. The problem is formulated on the basis of the solution of eigen-strain problem of anisotropic media. The result is expressed by line integrals along a contour on a unit sphere.

When plastic damage regions are accumulated in a material, there exist residual displacements on the surface of the material after all the loadings are removed. The residual displacements are defined as the difference between before and after loading, and can be measured experimentally without destruction of the material. This paper addresses the problem of evaluating the residual stress field caused by the accumulation of the plastic damage regions in a subdomain of the material. The problem is formulated as a system of integral equations relating the surface displacements to the unknown plastic strains. The damage domain, which appears as the domain of integration of the integral equations, is also unknown. Determination of the shape of the damage domain, together with the plastic strains, is a very complicated nonlinear problem. In addition to the residual surface displacement data, it requires more information about the loading history or other restrictive assumptions. However, the residual stress field in the vicinity of the damage domain is obtained after the equivalent damage domain and the equivalent plastic strains are introduced. The problem is an inverse problem, which is substantially different from the conventional forward analysis of structural mechanics. Special attention is given to the uniqueness and stability of the solution.

The elastic fields due to a spheroidal inclusion, which undergoes constant shear eigenstrains and a spheroidal inhomogeneity under uniform shear stress applied at infinity are investigated. The interface between the material and the spheroid allows free sliding (cannot sustain shear tractions). The Papkovich-Neuber displacement potentials in the form of infinite series are used in the analysis. The dependence of stresses and displacements on the shear moduli ratio, Poisson's ratio and the shape of the subdomain is analyzed. It is discovered that stresses in the sliding spheroidal subdomain are not uniform, unlike the perfectly bonded inclusion solution where Eshelby's result holds. The applications to the evaluation of average elastic properties of the composites with sliding interfaces are also investigated and it is observed that sliding causes relaxation of the elastic moduli.

The elastic field outside an ellipsoidal inclusion is investigated. No particular restrictions are imposed to eigenstrain distributions throughout the inclusion and to the anisotropy of elastic media for matrices. The solution is expressed in the form of integrals defined on a subspace of a unit sphere. Since the integrands of these integrals have no singularities, the numerical calculation on a computer can easily be performed.

By using the solution of eigenstrain problems in anisotropic media, flat ellipsoidal inclusions and crack problems are investigated. When a dilatational misfit strain is defined in a flat ellipsoidal inclusion (i.e., disk-shaped precipitate), the elastic strain energy associated with the misfit becomes minimum when the plane of the inclusion coincides with one of the crystalline planes, where the matrix and the inclusion are assumed as two different cubic crystals having the same crystalline directions. The minimum value, however surprisingly, depends only on the elastic moduli of the inclusion. When crystalline directions of the inclusion are parallel to the principal axes of the ellipsoid, the elastic strain energy is independent of the orientation of the inclusion with respect to the crystalline directions of the matrix and the constant value is equal to the minimum value in the first case. If an inclusion is simply defined by giving a uniform dilatational misfit strain in an ellipsoidal domain in a homogeneous material (i.e., matrix and inclusion are same material), the condition for the minimum elastic strain energy is the same as that of the first case, namely the minimum occurs when the plane of the inclusion is parallel to one of the crystalline directions. The method employed here can be applied equally to generally anisotropic materials. The method is also applicable to fracture problems by taking the elastic moduli of the inclusion as zero. As examples, an elliptical crack is considered for a simple tension and a pure shear. The interaction energy between the applied stress and a crack is calculated. By using this result, the Griffith criterion for fracture is derived for the penny-shaped crack. Some numerical results for cubic crystals are shown.

The high stress concentration near the interface of fiber-matrix in composite materials causes the accumulation of the interfacial damages and the degradation of mechanical properties of the materials. In this paper, by using the residual surface displacement data, we evaluate the microscopic damages in terms of Somigliano's dislocations in composites near the interface caused by a series of unknown loadings. The goal of this research is to monitor the processes of the damage evolution near the interface and to relate this microscopic damage to the degradation of the macroscopic mechanical properties of the materials.

A general method of numerical solution in the non-Hertzian elastic contact problem is developed using rectangular subdivisions. The major application of the method given is in the sliding contact of a short rough cylinder with an elastic half space; the contact region is the side of the cylinder, whose axis may not be exactly aligned to the surface of the half space. The stress singularity at the ends of the cylinder is taken as either square root (to determine stress intensification factor) or is approximated by compact elements to reduce computing time. An iterative scheme is used and the convergence to the unknown shape of the contact region is rapid.

The problem of the contact between a smooth, rigid indenter and an elastic quarter space is solved using integral transform techniques. The indenter can have an arbitrary plan form and can extend to the edge of the quarter space. Numerical results are given when a rigid, right-circular body is pressed into the top surface of the quarter space, while keeping its axis parallel to the surface and perpendicular to the edge.

Based upon the Mori-Tanaka method, the constitutive equations of power-law materials and the failure criteria of multiple cracks materials are investigated. The piecewise linear incremental approach is also employed to analyze the effective stress and strain of the power-law materials. Results are presented for the case of pure shear where the matrix is a power-law material with rigid or void inhomogeneities. For the multiple cracked materials, the Griffith fracture criterion is applied to determine the critical volume fraction which causes the catastrophic failure of a material. The failure criteria of penny shaped, flat ellipsoidal, and slit-like cracked materials are examined and it is found that the volume fraction of cracks and critical applied stress are in linear relation.

The stress field produced by a dislocation segment in an unbounded anisotropic elastic medium is derived. The solution, which does not require solving one sextic equation, is expressed by a line integral of rational functions along a unit circle.

Based on morphological tests in rolling and sliding contact, the effect of surface film, the subsurface heavily dislocated layer, crack initiation, crack propagation, and micropitting have been investigated by using a two-disk rig. A micro-macro contact model is presented for surface pitting in rolling and sliding contacts. In this model the pressures related to asperity interaction and the subsurface stress map along whole contact area have been analyzed by superimposing the asperity contacting loads on the Hertzian load for a given specimen surface. Compared to the experimental results, this model is capable of predicting the onset of micropits for given mechanical and lubrication conditions. Cracking is assumed to occur when the accumulated strain energy of the dislocation reaches a critical value. A three-dimensional crack propagating model has also been used for pitting life prediction in rolling and sliding contact.

A series solution is presented for a hemispheroidal elastic inhomogeneity at the free surface of an elastic half space. The loading is either all around tension at infinity, perpendicular to the axis of symmetry of the inhomogeneity, or uniform, nonshear type eigenstrains sustained by the inhomogeneity. The displacement potentials of Boussinesq are used to represent the solution and several numerical calculations are performed to illustrate the results.

One of the experimental findings on short-fiber reinforced composite materials is that the fiber-ends act as a crack initiator. The effect of the fiber-end crack on the overall stiffness and the strength of the composite are investigated here. A particular emphasis is placed upon the weakening longitudinal Young’s modulus by the fiber-end crack which is assumed to be penny-shaped. The energy release rate of the penny-shaped crack at the fiber-end under a uniaxial applied stress is also calculated for a fracture criterion. It is assumed in our theoretical model that short-fibers are all aligned in the loading direction and the penny-shaped crack at the fiber-end extends in the direction perpendicular to the fiber axis. Our analytical technique is a combination of Eshelby’s equivalent inclusion method and Mori-Tanaka’s back stress analysis so that our results are valid even for large volume fraction of fibers.

The slip band formed in a grain on the material surface is a preferential site for crack initiation during low strain fatigue of polycrystalline metals. The forward and reverse plastic flow within the slip band is modeled in the present study by dislocations with different signs moving on two closely located layers, and it is assumed that their movement is irreversible. Based on the model, the monotonic buildup of dislocation dipoles piled up at the grain boundary is systematically derived using the theory of continuously distributed dislocations. This buildup is associated with the progress of extrusion or intrusion. The number of stress cycles up to the initiation of a crack of the grain size order is defined as the cycle when the stored strain energy of accumulated dislocations reaches a critical value. The relation between the initiation life and the plastic strain range derived theoretically is in agreement with a Coffin-Manson type law, and that between the fatigue strength and the grain size is expressed in an equation of the Petch type.

The paper analyzes the elastic fields caused by an elliptic inclusion which undergoes a uniform expansion. The interface between the inclusion and the matrix cannot sustain shear tractions and is free to slip. Papkovich–Neuber displacement potentials are used to solve the problem. In contrast to the perfectly bonded interface, the solution cannot be expressed in closed form and involves infinite series. The results are illustrated by numerical examples.

Analytical solutions are proposed for the stress and displacement fields in a quasi three-dimensional elastic anisotropic body containing an elliptic cavity or rigid inclusion. The directions of the principal elastic axes are allowed to be inclined arbitrarily with respect to the axes of the elliptic cavity. As an application, expressions for the stress intensity factors are formulated when the cavity reduces to a colinear crack.

Analytical closed-form solutions are proposed in a rather compact form for the stress and displacement fields induced by out-of-plane loading of a semi-infinite anisotropic material with inclined strata. The solutions are then extended to include the case of a bimaterial with a planar interface. Several boundary conditions are considered for the interface which may be between two anisotropic half-planes with different elastic properties, or two different orientations of the strata in the same material.

Analytical solutions are proposed for the stress and displacement fields induced by in-plane loading of a bimaterial under several boundary conditions. The two joined orthotropic layers forming the bimaterial are allowed to possess different types of anisotropy with their planes of elastic symmetry arbitrarily inclined with respect to the horizontal.

Summary The buckling of thin elastic rectangular plates due to two arrays of edge dislocations symmetrically placed with respect to the longitudinal axis of the plate is studied for two different dislocation orientations. The analysis is based on the governing equation of plate buckling. Through a standard eigenvalue analysis the buckling criterion is analytically-numerically computed and numerical results are presented pertaining to cases of practical engineering interest such as welding.

Stress distributions are evaluated for a semi-infinite elastic body having a spheroidal inhomogeneity under an all-around tension (axisymmetric uniform tension) at infinity. Boussinesq’s potential functions are used for the analysis. A special technique is employed for the traction-free condition on the free surface, where the ordinary Legendre functions are expressed in terms of the Bessel functions in cylindrical coordinates. A closed-form expression for the stress concentration factor is found for a special case of a needlelike inhomogeneity.

The utility of surface displacement measurements as a source of quantitative information about subsurface plastic deformation is illustrated by analysis of the transformed zone in magnesia partially stabilized zirconia (Mg-PSZ). The surface displacements are derived from partial fringe analysis of optical interference micrographs. Their interpretation is based on analytical and numerical calculations of the surface displacements generated by a transforming semi-ellipsoidal in-homogeneity at the surface of a half-space. Inhomogeneous transformed zones, i.e., those possessing nonuniform distributions of transformation strain, are modeled as linear sums of nested, uniformly transforming semi-ellipsoids. It is found that the transformed zones around either through cracks or small, surface-breaking cracks in Mg-PSZ are indeed inhomogeneous. The degree of transformation is near the maximum possible only within a few microns of the crack plane. It falls away linearly or faster as the distance from the crack plane increases. Qualitative evidence suggests furthermore that significant net shear transformation persists near the crack plane. These conclusions are consistent with phase analysis using Raman spectroscopy.

The paper analyzes thermal stresses and effective thermal expansion coefficients of the composites in which the fiber-matrix interface is allowed to slip. Thermal stresses are evaluated by introducing an eigenstrain in the fibers, which corresponds to the strain due to the mismatch of thermal expansion coefficients. For simplicity, the effect of friction is neglected. Boussinesq-Papkovich displacement potential method is used in the analysis. Then, the results for a single sliding fiber are used to predict the average thermal expansion coefficients of the composite containing finite concentration of fibers. It is observed that sliding at the fiber-matrix interface causes higher stress concentrations and affects significantly the average coefficients of thermal expansion of the composite.

Elastic fields due to a sliding inclusion and inhomogeneity with the frictional interface are investigated. The exact solution in closed form is presented for three cases: (i) a spherical inclusion which undergoes constant eigenstrains, (ii) a nondegenerate ellipsoidal inclusion with uniform shear eigenstrains and (iii) a nondegenerate ellipsoidal inhomogeneity subjected to an applied shear stress at infinity. Moreover, the existence of solution for a sliding inclusion and inhomogeneity with frictional interface is also demonstrated.

The plastic flow at the crack tip is characterized by a model compatible with slip line theory. From this model it is shown that a continuous distribution of dislocations may be derived. Then assuming that these dislocations are emitted from the crack tip and move along slip lines to their final position, the Peach-Koehler force is used to calculate the plastic work involved. Since the plastic zone size is dependent on crack length, two plastic effects are present upon propagation. Primarily the distribution of dislocations present moves along with the crack tip, secondarily new dislocations are emitted to fill the larger plastic zone. These effects yield plastic work which is dependent on both σ2 and σ4, with σ being the applied stress. This dependancy yields a critical stress relationship different from that proposed by either Irwin or Orowan. It also leads to the determination of a subcritical flaw size, i.e., one which will never become unstable.

The elastic stress field caused by an ellipsoidal inclusion with uniform dilatational eigenstrains is investigated for a semi-infinite solid. The result is expressed in terms of derivatives of elliptic integrals. Numerical examples are given to show the effect of the free surface with parameters of depth from the surface and shapes of the inclusion.

The problem of adhesively bonded sheets, one of which is cracked, is formulated by the utilization of integral transform methods. The objective of the investigation is to calculate the stress-intensity factor at the crack tip for the cracked sheets. Results are obtained when the cracked sheet has a single crack and an array of identical, equally spaced, coplanar cracks. Results tend to indicate that the growth of crack implies a reduction in the stress-intensity factor.

The problem of two ellipsoidal inhomogeneities in an infinitely extended isotropic matrix is treated by the equivalent inclusion method. The matrix is subjected to an applied strain field in the form of a polynomial of degree M in the position coordinates xi. The final stress and strain states are calculated for two isotropic ellipsoidal inhomogeneities both in the interior and the exterior (in the matrix) by using a computer program developed. The method can be extended to more than two inhomogeneities.

A solution for an elastic quarter plane containing an arbitrarily oriented crack is presented. The problem is formulated by means of Mellin integral transforms and reduced to a system of two coupled singular integral equations where the unknown quantities are the dislocation densities that characterize the crack. Numerical solutions are investigated for various orientations of the cracks. In each case the stress intensity factors are computed for the different parameters.

Exact elasticity solutions are used to discuss the interaction between a circular inclusion and an edge dislocation in the matrix. The force on the dislocation is given, and it is shown that for certain combinations of material constants, the dislocation has a stable equilibrium position in the matrix near the interface. Moreover, the interaction is very much affected by the orientation of the Burgers vector with respect to the inclusion. The results also exhibit a strong dependency on the difference between the Poisson's ratios of the matrix and the inclusion.

A variational method for calculating the lower and upper bounds on the safety factor of perfectly plastic simply supported conical sandwich shells subjected to a uniform internal pressure and obeying Mises yield criteria is presented. The solutions obtained by the proposed method are compared with those obtained by means of the classical method using the two-moment limited interaction yield condition based on Tresca yield criterion.

The elastic field around a long, straight edge dislocation moving with an arbitrary velocity is derived in the form of time integrals. An electromagnetic analogy, which is known to apply to a screw dislocation, is found to be absent. The results are applied to an edge dislocation oscillating with a small amplitude. A brief discussion of the scattering properties shows no significant difference between screw and edge dislocations.

A deformation field due to a moving single dislocation is expressed by line integrals along the dislocation line. Proper definitions for a continuous distribution of moving dislocations and its velocities enable us to extend the expression for the deformation field due to a single dislocation, to the deformation field due to a continuous distribution of dislocations. The fundamental relationship between plastic strains and dislocation density tensors, and the relationship between plastic strain rates and velocities of a continuous distribution of dislocations are determined from a formulation similar to the law of conservation of matter. Also it is found that a continuous distribution of dislocations must satisfy a certain boundary condition on the free boundary of the material. The relationship between the total strain, and the elastic and plastic strains, is discussed by referring to their definitions in the mathematical theory of plasticity.