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This work develops a semi-analytic solution for multiple inhomogeneous inclusions of arbitrary shape and cracks in an isotropic infinite space. The solution is capable of fully taking into account the interactions among any number of inhomogeneous inclusions and cracks which no reported analytic or semi-analytic solution can handle. In the solution development, a novel method combining the equivalent inclusion method (EIM) and the distributed dislocation technique (DDT) is proposed. Each inhomogeneous inclusion is modeled as a homogenous inclusion with initial eigenstrain plus unknown equivalent eigenstrain using the EIM, and each crack of mixed modes I and II is modeled as a distribution of edge climb and glide dislocations with unknown densities. All the unknown equivalent eigenstrains and dislocation densities are solved simultaneously by means of iteration using the conjugate gradient method (CGM). The fast Fourier transform algorithm is also employed to greatly improve computational efficiency. The solution is verified by the finite element method (FEM) and its capability and generality are demonstrated through the study of a few sample cases. This work has potential applications in reliability analysis of heterogeneous materials.

This paper proposes a simple method based on analytical continuation and conformal mapping to obtain an analytic solution for a two-dimensional arbitrarily shaped Eshelby inclusion with uniform main plane eigenstrains and eigencurvatures in an infinite or semi-infinite isotropic laminated plate. The main plane of the plate is chosen in such a way that the in-plane displacements and out-of-plane deflection on the main plane are decoupled in the equilibrium equations. Consequently, the complex potential formalism for the isotropic laminate can be readily and elegantly established. One remarkable feature of the present method is that simple elementary expressions can be obtained for the internal elastic field within the inclusion of any shape in an infinite laminated plate. Several examples are presented to illustrate the general method.

A semi-analytic solution is presented for multiple inhomogeneous inclusions and cracks in a half-space under elastohydrodynamic lubrication contact. In formulating the governing equations, each inhomogeneous inclusion embedded under the contacting surfaces is modeled as a homogeneous inclusion with initial eigenstrains plus unknown equivalent eigenstrains by employing Eshelby's equivalent inclusion method, while each crack of mixed modes I and II is treated as a distribution of climb and glide dislocations with unknown densities according to the dislocation distribution technique. Such a treatment converts the problem into a homogeneous lubricated contact with disturbed deformation due to the inclusions and cracks. The unknowns in the governing equations are integrated by a numerical algorithm and determined iteratively by utilizing a modified conjugate gradient method. The iterative process is performed until the convergence of the half-space surface displacements, which involve the displacements due to the inhomogeneous inclusions and cracks as well as the fluid pressure. Samples are presented to demonstrate the generality of the solution.

In order to assure structural integrity for operating welded structures, it is necessary to evaluate crack growth rate and crack propagation direction for each observed crack non-destructively. Here, three dimensional (3D) welding residual stresses must be evaluated to predict crack propagation. Today, X-ray diffraction is used and the ultrasonic method has been proposed as non-destructive method to measure residual stresses. However, it is impossible to determine residual stress distributions in the thickness direction. Although residual stresses through a depth of several tens of millimeters can be evaluated non-destructively by neutron diffraction, it cannot be used as an on-site measurement technique. This is because neutron diffraction is only available in special irradiation facilities. Author pays attention to the bead flush method based on the eigenstrain methodology. In this method, 3D welding residual stresses are calculated by an elastic Finite Element Method (FEM) analysis from eigenstrains which are evaluated by an inverse analysis from released strains by strain gauges in the removal of the reinforcement of the weld. Here, the removal of the excess metal can be regarded as non-destructive treatment because toe of weld which may become crack starters can be eliminated. The effectiveness of the method has been proven for welded plates and pipes even with relatively lower bead height. In actual measurements, stress evaluation accuracy becomes poorer because measured values of strain gauges are affected by processing strains on the machined surface. In the previous studies, the author has developed the bead flush method that is free from the influence of the affecting strains by using residual strains on surface by X-ray diffraction. However, stress evaluation accuracy is not good enough because of relatively poor measurement accuracy of X-ray diffraction. In this study, a method to improve the estimation accuracy of residual stresses in this method is formulated, and it is shown numerically that inner welding residual stresses can be estimated accurately from the residual strains measured by X-ray diffraction.

In this paper, the recent literature of finite eignestrains in nonlinear elastic solids is reviewed, and Eshelby’s inclusion problem at finite strains is revisited. The subtleties of the analysis of combinations of finite eigenstrains for the example of combined finite radial, azimuthal, axial and twist eigenstrains in a finite circular cylindrical bar are discussed. The stress field of a spherical inclusion with uniform pure dilatational eigenstrain in a radially-inhomogeneous spherical ball made of arbitrary incompressible isotropic solids is analyzed. The same problem for a finite circular cylindrical bar is revisited. The stress and deformation fields of an orthotropic incompressible solid circular cylinder with distributed eigentwists are analyzed.

An inclusion under uniform eigenstrains is of fundamental importance in composite mechanics. Although it is well known that an ellipse is the only inclusion shape which achieves a uniform internal stress field under anti-plane shear, it remains unclear what inclusion shapes achieve a polynomial internal stress field. The present paper studies an anti-plane inhomogeneous or homogeneous inclusion in an infinite elastic plane under uniform stress-free eigenstrains. The inclusion shape which gives a polynomial internal stress field is determined and the relation between the degree of the non-uniformity of the internal polynomial stress field and the inclusion’s geometrical shape is derived.