Journal of Elasticity最新文献

We solve the inverse problem in three-dimensional elasticity associated with the design of a harmonic ellipsoidal isotropic elastic solid or compressible liquid inclusion that does not disturb the first invariant of the stress tensor in the surrounding isotropic elastic matrix subjected to uniform remote normal stresses. In order to achieve the harmonic condition, the two ratios of the remote normal stresses are uniquely determined for given geometric and material parameters.

The material tailoring problem for a hollow circular cylinder composed of an isotropic, incompressible, and linearly elastic functionally graded material has been analytically analyzed. The cylinder is deformed by torques and axial loads on the end faces, and pressures on its inner and outer surfaces. The cylinder material has one elastic parameter, the shear modulus (mu left ( r right ) ). For the direct problem it is a known positive and continuously varying function in the radial direction, (r). For the inverse problem (mu left ( r right )) is a design variable and is found to provide the desired radial variation of either the strain energy density, (W^{def} (r)), or the von Mises stress, (sigma ^{VM} left ( r right )), for the given loads and the cylinder geometry. If the three loads are simultaneously varied by a factor (gamma ) then (W^{def} left ( r right ) ) and (sigma ^{VM} left ( r right ) ), respectively, change by (gamma ^{2} ) and (gamma ) for fixed (mu left ( r right ) ) in the direct problem and (mu (r)) by (gamma ^{2} ) and (gamma ) in the inverse problem for preassigned (W^{def} (r) = W_{cr} left ( r right ) ) and (sigma ^{VM} left ( r right ) = sigma _{cr}^{VM} left ( r right )). The (W_{cr} left ( r right ) ) and (sigma _{cr}^{VM} left ( r right )) are, respectively, values at failure of the strain energy density and the von Mises stress. For the cylinder material composed of two constituents having positive shear moduli as is often the case in experiments we use a homogenization technique to find the radial variations of their volume fractions and ensure (mu (r)) is positive. We review three manufacturing techniques and propose an experimental program to find (W_{cr} left ( r right )) and (sigma _{cr}^{VM} left ( r right )). The expression for (mu (r)) is derived from the solution of the direct problem that has a unique solution. It provides reference solutions for similar nonlinear problems and verification of numerical algorithms. It supports the optimal design of cylinders.

We recently laid down the theoretical basis for the consistent formulation of the collocation boundary element method, as it should have been conceived from the beginning. We proposed a convergence theorem for two- and three-dimensional problems of elasticity and potential, which applies to generally curved elements in the frame of an isoparametric analysis. We also showed that the code implementation leads to controllable, highly precise and accurate results for arbitrarily small source-field distances of two-dimensional problems – limited only by the machine’s capacity to represent numbers. On the other hand, there still is the cost-benefit question of how to adequately describe a real problem’s geometry without increasing the number of degrees of freedom (h- and p-mesh refinement). We are proposing that the isoparametric implementation – with the introduced elegance of a convergence theorem – be replaced with a formulation that preserves the problem’s idealized geometry but is not isoparametric, in general. We also introduce a homothetic approach – for nodes and elements adaptively generated according to the same pattern along a boundary patch –, which is highly cost-effective. We present conceptual formulation, code implementation, and numerical illustrations that go from the simple case of an infinite plate with a circular hole to very challenging – physically unrealistic and only mathematically conceivable – topological applications: a multi-connected domain with generally curved boundary patches and presenting cracks, cusp and reentrant angles of virtually zero magnitude, and a strip of material of zero width. This cannot be manufactured in the real world but can be nevertheless simulated provided we have the proper mathematical tools, as presently proposed.

Formation of ice layers during winter is a common natural phenomenon in high-latitude regions. To evaluate the impact of the ice layer on the seismic response of a poroelastic medium, we develop a novel model to describe the dynamic interaction among the ice layer, water layer, and transversely isotropic poroelastic rock under vertical P-wave excitation. First, the general solutions for the poroelastic rock and overlying water and ice layers are derived by applying the Laplace transform. Then the dual-variable and position (DVP) method is employed to obtain a semi-analytical solution of the layered media in the transform domain. By applying a numerical inverse Laplace transform scheme, the time response of free-field motion in the layered rock under P-wave excitation is obtained. Numerical results show that the ice layer causes more complex waveforms and amplifies the vertical displacement in deeper locations in the poroelastic medium with low permeability. A higher anisotropic modulus ratio leads to an earlier arrival of displacement peaks and troughs, with this effect strengthening over time. Stiff interlayers amplify the displacement and advance the waveform, while soft interlayers have the opposite effect.

This paper develops an axisymmetric boundary element method (BEM) for analyzing a transversely isotropic (TI) layered half-space with an internal cavity subjected to pressure. The BEM formulation utilizes the fundamental solution of a TI layered solid of infinite extent under body forces uniformly concentrated along a circular ring. Three types of isoparametric elements are used to discretize the core region surrounding the cavity and an infinite element is utilized to discretize the external boundary away from the cavity. A novel numerical quadrature scheme is introduced to calculate the regular and singular integrals in the BEM formulation. Numerical verifications are carried out to confirm the accuracy and computational efficiency of the proposed BEM. The numerical results demonstrate the influence of the heterogeneity and anisotropy of the TI layered solid on the elastic fields in the surrounding rocks around the cavity of either sphere or ellipsoid.

In nonlinear elasticity, the square root of a tensor arises in the polar decomposition of the deformation gradient, and in many other applications in other areas as well. In this work, given a positive integer (p), we derive an explicit expression for the principal (p)-th root of a real-valued second-order tensor, which is not necessarily diagonalizable, whose eigenvalues do not lie on the closed negative real axis, but which is otherwise arbitrary, for any underlying space dimension (n). We also present a method for the explicit evaluation of the derivative of the (p)-th root of a tensor.

Edges are abundant when elastic solids glide in guiding rails or fluids are contained in vessels. We here address induced displacements in elastic solids or small-scale flows in viscous fluids in the vicinity of one such edge. For this purpose, we solve the governing elasticity equations for linearly elastic, potentially compressible solids, as well as the low-Reynolds-number flow equations for incompressible fluids. Technically speaking, we derive the associated Green’s functions under confinement by two planar boundaries that meet at a straight edge. The two boundaries both feature no-slip or free-slip conditions, or one of these two conditions per boundary. Previously, we solved the simpler case of the force being oriented parallel to the straight edge. Here, we complement this solution by the more challenging case of the force pointing into a direction perpendicular to the edge. Together, these two cases provide the general solution. Specific situations in which our analysis may find application in terms of quantitative theoretical descriptions are particle motion in confined colloidal suspensions, dynamics of active microswimmers near edges, or actuated distortions of elastic materials due to activated contained functionalized particles.

This study presents an analytical solution for an exponentially-graded transversely isotropic half-space reinforced by a thin plate, subjected to frictionless contact conditions and buried static point loads. The adopted material model captures the behavior of anisotropic and functionally graded media with improved accuracy compared to the assumption of transverse isotropy. Expressions for displacement field are derived, integrating material anisotropy, functional gradation, and interface conditions. The results show that increasing the thin plate stiffness significantly reduces the normal displacement of the half-space, with displacement nearly vanishing under extreme conditions. Additionally, variations in material gradation initially lead to a substantial reduction in displacement, while specific reinforcement conditions may result in localized increases. These findings provide insights into the mechanical behavior of surface-reinforced anisotropic media and multi-layered structures in applications such as protective coatings, pavements, and ground reinforcements.

This study investigates the dispersion of Rayleigh waves propagating in fiber-reinforced medium over a compressible porous semi-infinite medium under the influence of micropolar elasticity. Perfect and slip boundary conditions are applied on the interface of stratum and substrate to yield the dispersion relation. The model incorporates one of the most effective scheme, the Finite Difference Scheme (FDS) to compute the phase velocities and group velocities of Rayleigh waves. Stability condition of FDS has been derived for the phase velocity and group velocity. The effects of porosity, reinforcement, and micropolarity on Rayleigh waves propagation are studied graphically. A higher mode of the dispersion curve is also analyzed with respect to welded (perfect) and slip boundary conditions.

This paper presents solutions for a transversely isotropic unsaturated half-space subjected to axisymmetric time-harmonic vertical and fluid pressure loading. The Biot’s coupled poroelastodynamic equations are modified to include the unsaturated case by adding the air phase into the balance equations. The resulting fully coupled field equations of the three-phase medium are analytically solved to obtain the frequency-domain general solutions by using Hankel integral transforms in the radial direction. A boundary value problem is then formulated to obtain explicit expressions for the solutions of a half-space subjected to internally applied time-harmonic vertical loading and applied fluid pressure. To validate the accuracy of the proposed formulation, a comparison is made with existing solutions for surface loading on an unsaturated half-space. Numerical results are presented to illustrate the influence of the degree of saturation and the frequency of excitation on the dynamic response of the unsaturated half-space under internal excitations. The analysis reveals significant differences in profiles of displacement, stress, and pore pressure for the unsaturated soils compared to fully saturated cases, highlighting the importance of incorporating partial saturation in dynamic poroelastic modelling for geotechnical and earthquake engineering applications.