Asymptotic Justification of Models of Plates Containing Inside Hard Thin Inclusions

An equilibrium problem of the Kirchhoff–Love plate containing a nonhomogeneous inclusion is considered. It is assumed that elastic properties of the inclusion depend on a small parameter characterizing the width of the inclusion ε as εN with N<1. The passage to the limit as the parameter ε tends to zero is justified, and an asymptotic model of a plate containing a thin inhomogeneous hard inclusion is constructed. It is shown that there exists two types of thin inclusions: rigid inclusion (N<−1) and elastic inclusion (N=−1). The inhomogeneity disappears in the case of N∈(−1,1).

Asymptotic Justification of Models of Plates Containing Inside Hard Thin Inclusions

An equilibrium problem of the Kirchhoff-Love plate containing a nonhomogeneous inclusion is considered. It is assumed that elastic properties of the inclusion depend on a small parameter characterizing width of the inclusion $\varepsilon$ as $\varepsilon^N$ with $N&lt;1$. The passage to the limit as the parameter $\varepsilon$ tends to zero is justified, and an asymptotic model of a plate containing a thin inhomogeneous hard inclusion is constructed. It is shown that there exists two types of thin inclusions: rigid inclusion ($N&lt;-1$) and elastic inclusion ($N=-1$). The inhomogeneity disappears in the case of $N\in (-1,1)$.

Static elastic cloaking, low-frequency elastic wave transparency and neutral inclusions

New connections between static elastic cloaking, low-frequency elastic wave scattering and neutral inclusions (NIs) are established in the context of two-dimensional elasticity. A cylindrical core surrounded by a cylindrical shell is embedded in a uniform elastic matrix. Given the core and matrix properties, we answer the questions of how to select the shell material such that (i) it acts as a static elastic cloak, and (ii) it eliminates low-frequency scattering of incident elastic waves. It is shown that static cloaking (i) requires an anisotropic shell, whereas scattering reduction (ii) can be satisfied more simply with isotropic materials. Implicit solutions for the shell material are obtained by considering the core–shell composite cylinder as a neutral elastic inclusion. Two types of NI are distinguished, weak and strong with the former equivalent to low-frequency transparency and the classical Christensen and Lo generalized self-consistent result for in-plane shear from 1979. Our introduction of the strong NI is an important extension of this result in that we show that standard anisotropic shells can act as perfect static cloaks, contrasting previous work that has employed ‘unphysical’ materials. The relationships between low-frequency transparency, static cloaking and NIs provide the material designer with options for achieving elastic cloaking in the quasi-static limit.

Inverse problem for elastic body with thin elastic inclusion

An inverse problem for an elastic body with a thin elastic inclusion is investigated. It is assumed that the inclusion crosses the external boundary of the elastic body. A connection between the inclusion and the elastic body is characterized by the damage parameter. We study a dependence of the solutions on the damage parameter. In particular, passages to infinity and to zero of the damage parameter are investigated. Limit models are analyzed. Assuming that the damage and rigidity parameters of the model are unknown, inverse problems are formulated. Sufficient conditions for the inverse problems to have solutions are found. Estimates concerning solutions of the inverse problem are established.

Determination of the Stress State of the Layer with a Cylindrical Elastic Inclusion

A spatial problem of the theory of elasticity for the layer with an infinite round cylindrical inclusion is investigated. At the boundaries of the layer, displacements are given. The cylindrical elastic inclusion is rigidly coupled with the layer and their boundary surfaces do not intersect. The solution to the spatial problem is obtained by the generalized Fourier method, with regard to the Lamé system of equations. The obtained infinite systems of linear algebraic equations are solved by a reduction method. As a result, the values ​​of displacements and stresses in the elastic body are determined. A comparative analysis of the stress state for different geometrical parameters is carried out, and a comparison is made with the stress state in the layer with a cylindrical cavity.

Residual Stress Analysis of an Orthotropic Composite Cylinder under Thermal Loading and Unloading

Elastoplastic analysis of a composite cylinder, consisting of an isotropic elastic inclusion surrounded by orthotropic matrix, is conducted via numerical parametric studies for examining its residual stress under thermal cycles. The matrix is assumed to be elastically and plastically orthotropic, and all of its material properties are temperature-dependent (TD). The Hill’s anisotropic plasticity material model is adopted. The interface between the inclusion and matrix is perfectly bonded, and the outer boundary of the cylinder is fully constrained. A quasi-static, uniform temperature field is applied to the cylinder, which is analyzed under the plane-strain assumption. The mechanical responses of the composite cylinder are strongly affected by the material symmetry and temperature-dependent material properties. When the temperature-independent material properties are assumed, larger internal stresses at the loading phase are predicted. Furthermore, considering only yield stress being temperature dependent may be insufficient since other TD material parameters may also affect the stress distributions. In addition, plastic orthotropy inducing preferential yielding along certain directions leads to complex residual stress distributions when material properties are temperature-dependent.