Piecewise Homogeneous Medium Research Articles

Problems of folding in the earth’s stratified crust

The problem of folding in the Earth’s stratified crust is addressed within the framework of solid mechanics. The history and formulation of the problem and approaches to its solution are briefly outlined. The initial stage of folding in strata with different mechanical properties is studied in detail using the piecewise-homogeneous medium model and the three-dimensional theory of stability. Conclusions of applied and fundamental importance are drawn from an analysis of solutions to a wide range of specific problems

Electromagnetic momentum and radiation pressure derived from the Fresnel relations

Using the Fresnel relations as axioms, we derive a generalized electromagnetic momentum for a piecewise homogeneous medium and a different generalized momentum for a medium with a spatially varying refractive index in the Wentzel-Kramers-Brillouin (WKB) limit. Both generalized momenta depend linearly on the field, but the refractive index appears to different powers due to the difference in translational symmetry. For the case of the slowly varying index, it is demonstrated that there is negligible transfer of momentum from the electromagnetic field to the material. Such a transfer occurs at the interface between the vacuum and a homogeneous material allowing us to derive the radiation pressure from the Fresnel reflection formula. The Lorentz volume force is shown to be nil.

Fracture of layered composites by internal fibre instability: effect of interfacial adhesion

AbstractIn this paper, a mechanism of compressive fracture for elastic and elastic-plastic composite materials with interfacial adhesion defects is investigated. A classification of different approaches in modelling compressive response of layered materials is given. The analysis finds the upper and the lower bounds for the critical load. In order to achieve this, the problem of the internal fibre (layer) instability is considered within the scope of the exact statement based on the application of the model of a piecewise-homogeneous medium and the equations of the three-dimensional (3D) stability theory. The solution of the 3D problem is found for the most general case accounting for the bi-axiality of compressive loads. The characteristic determinants are derived for the first four fibre instability (microbuckling) modes, which are more commonly observed. Special attention is given to the calculation of critical loads for practical elastic and elastic-plastic layered materials.

Numerical simulation of nonlinear elastic wave propagation in piecewise homogeneous media

Systematic experimental work [S. Zhuang, G. Ravichandran, D. Grady, J. Mech. Phys. Solids 51 (2003) 245–265] on laminated composites subjected to high velocity impact loading exhibits the dispersed wave field and the oscillatory behavior of waves with respect to a mean value. Such a behavior is absent in homogeneous solids. An approximate solution to the plate impact in layered heterogeneous solids has been developed in [X. Chen, N. Chandra, A.M. Rajendran, Int. J. Solids Struct. 41 (2004) 4635–4659]. The influence of the particle velocity on many process characteristics was demonstrated. Based on earlier results [A. Berezovski, J. Engelbrecht, G.A. Maugin, Arch. Appl. Mech. 70 (2000) 694–706], numerical simulations of one-dimensional wave propagation in layered nonlinear heterogeneous materials have been performed. The formulated problem follows a conventional experimental configuration of a plate impact. An extension of the high-resolution finite volume wave-propagation algorithm [R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002] is used. The speed of sound depends nonlinearly on a current stress value in each layer but also on the mismatch properties of layers. Results of numerical simulations capture the experimental data rather well.

Parallel Octree-Based Finite Element Method for Large-Scale Earthquake Ground Motion Simulation

We present a parallel octree-based finite element method for large-scale earthquake ground motion simulation in realistic basins. The octree representation combines the low memory per node and good cache performance of finite difference methods with the spatial adaptivity to local seismic wavelengths characteristic of unstructured finite element methods. Several tests are provided to verify the numerical performance of the method against Green’s function solutions for homogeneous and piecewise homogeneous media, both with and without anelastic attenuation. A comparison is also provided against a finite difference code and an unstructured tetrahedral finite element code for a simulation of the 1994 Northridge Earthquake. The numerical tests all show very good agreement with analytical solutions and other codes. Finally, performance evaluation indicates excellent single-processor performance and parallel scalability over a range of 1 to 2048 processors for Northridge simulations with up to 300 million degrees of freedom. keyword: Earthquake ground motion modeling, octree, parallel computing, finite element method, elastic wave propagation

Planar Stability Problem of Composite Weakly Reinforced by Short Fibers

A stability planar problem of composite material weakly reinforced by short fibers is formulated and a method for its solution is suggested. The problem is formulated with application of equations of linearized three-dimensional theory of stability within the frame of the model of the piecewise-homogeneous medium. The technique for solving the formulated problems is developed on the basis of the finite difference method using the base scheme concept. The results of solutions of problems for the specific composite are presented. The influence of geometric and mechanical parameters of the filling material on the critical strain value and on the form of stability loss is investigated.

Numerical modelling of interface scattering of seismic wavefield from a random rough interface in an acoustic medium: comparison between 2D and 3D cases

ABSTRACTSeismic wavefield scattering from a statistically randomly rough interface in a multilayered piecewise homogeneous medium is studied in 3D. The influence of the surface roughness on the scattered wavefield is analysed numerically by using a finite‐difference operator in the acoustic domain. Since interface scattering in the real practical sense is a 3D physical phenomenon, we show in this work that the scattering response of a randomly rough interface is not the same in 3D situations as in the 2D cases described in some earlier works. For a given interface roughness height in 3D, an interface roughness height at least three times greater is required to produce an equivalent phase scattering effect in 2D situations, for a given correlation length of the interface roughness scale. Based on observations from spectral analysis, we show that scattering results principally in de‐phasing and frequency band‐limiting of the incident wavefront, the frequency band‐limiting properties being comparable to cases reported in the literature for absorption and thin‐layer filtering. The interface scattering phenomenon should be critically considered when using amplitude and phase information from seismic signal during inversion processes.

The effect of the multi-axiality of compressive loading on the accuracy of a continuum model for layered materials

Two methods of analysis of the internal instability of layered materials are discussed: the continuum approach and the piecewise-homogeneous medium model. Based on the results obtained within the scope of the model of a piecewise-homogeneous medium and the 3-D stability theory, the accuracy of a continuum theory is examined for incompressible non-linear materials undergoing large deformations. Two different loading conditions are compared: biaxial and uniaxial compression. The effect of the multi-axiality of loading on the accuracy of the continuum theory is determined for the particular model of hyperelastic layers described by the Treloar’s potential (i.e. by a neo-Hookean type potential), which is a simplified version of the Mooney’s elastic potential.

Parameters of the Edge Effect as a Function of the Mechanical Characteristics of a Composite with an Interfacial Crack

The paper studies the dependence of edge effects in a laminated composite of regular structure with a periodic system of symmetric interfacial cracks on the mechanical characteristics of the composite components. Loading the composite induces constant strain in the reinforcement direction. The problem is solved in exact formulation using the linear elastic equations for piecewise-homogeneous media and a criterion for quantitative evaluation of edge effects. An approximate solution of the problem is found by the mesh approach. A difference scheme for the mixed problem under consideration is derived. The edge-effect zones for normal stresses are constructed. The maximum values of the normal stress concentration factors are determined

On the lower bounds for critical loads under large deformations in non-linear hyperelastic composites with imperfect interlaminar adhesion

The present paper investigates a mechanism of compressive fracture for heterogeneous incompressible non-linear materials with special kinds of defects of interfacial adhesion under large deformations. The analysis finds the lower bounds for the critical load. In order to calculate the bounds, the problem of the internal instability is considered within the scope of the exact statement based on the application of the model of a piecewise-homogeneous medium and the equations of the 3-D stability theory. The solution of the 3-D problem is found for the most general case accounting for large deformations and the biaxiality of compressive loads. The characteristic determinants are derived for the first four modes, which are more commonly observed. Special attention is given to the calculation of critical loads for hyperelastic layers described by a simplified version of Mooney's potential, namely the neo-Hookean potential.

Influence of Young's Moduli of a Laminated Composite with a Periodic System of Cracks on Edge Effects

A problem on edge effects in a laminated composite with a periodic system of symmetric contact cracks is studied. Loading of the composite induces constant deformation in the reinforcement direction beyond the edge effect. The problem is solved in an exact formulation under the linear theory of elasticity for piecewise-homogeneous media. An approximate solution is found by the mesh approach using the concept of a base scheme. For stresses σ ij , the shape and size of the edge effect zone are studied depending on the ratio of Young's moduli of the composite components, which varies over a wide range

On the Question of Calculation of Temperature Fields in Piecewise-Homogeneous Orthotropic Media

Based on the theory of R-functions, we propose structures of a solution of the boundary-value problem of heat conduction in piecewise-homogeneous orthotropic media, satisfying the conditions of convective heat exchange on the boundary of an orthotropic body and the conditions of ideal thermal contact on the boundary between media. We present the results of a computational experiment confirming the correctness and computational fitness of the proposed structural models.

An Analysis of the Decay of Saint-Venant End Effects in a Transversely Isotropic Matrix with an Isotropic Covering

A calculation model is proposed for a numerical analysis of the decay of Saint-Venant end effects in a laminate material of irregular structure to which corresponds a transversely isotropic matrix with an isotropic covering. The elastic properties of the matrix correspond to those of a unidirectional glass-fiber-reinforced plastic. The problem is investigated within the framework of the concept of a representative element of the material. The decay of the end effect in the direction perpendicular to the isotropy plane of a transversely isotropic matrix for the case of symmetric deformation of the material is considered. The source of the end effect is simulated by a piecewise-constant periodic surface load. This load is local for the calculated region and changes within a part of the boundary comparable with the typical size of the structural heterogeneity of the material. The equations of the linear elasticity theory of orthotropic bodies, a model of piecewise-homogeneous media, and quantitative criteria of decay of the end effect are used. Starting from the base system of equations, a discrete problem is constructed and solved. The results of a computing experiment for the choice of a representative element and the results of determining the maximum extent of the end-effect zone for this element are presented.

Mathematical and Numerical Methods in Inverse Acoustic Scattering Theory

A survey is given of the inverse acoustic scattering problem of determining the support of an inhomogeneous obstacle from a knowledge of the incident and scattered time-harmonic fields where the obstacle is situated in a homogeneous or piecewise homogeneous medium. Numerical examples are given for the determination of the support of buried obstacles in the lower half-space from near field data on a bounded region in the upper half-space.

Predicting fracture of layered composites caused by internal instability

The present paper estimates the critical strain at which ply instability occurs in compressible layered composites under uniaxial or biaxial compression. This is achieved by using two methods: the continuum approach and the model of piecewise-homogeneous medium in conjunction with the three-dimensional stability theory. The accuracy of the continuum theory is examined by taking into account the influence of layer thickness, stiffness properties and biaxiality of loads. Also, upper and lower bounds for the critical buckling strain of laminates with interfacial defects (cracks with connected edges) are determined using the results for perfectly bonded and sliding layers.

Compressive fracture of non-linear composites undergoing large deformations

Based on the results obtained within the scope of the model of piecewise-homogeneous medium and three-dimensional stability theory, the accuracy of the continuum theory is examined for laminated incompressible materials undergoing large deformations. Estimation of the accuracy of the continuum theory is illustrated by numerical results for the particular models of composites when the layers are hyperelastic materials with the elastic potential of the neo-Hookean type (Treloar’s potential). Based on this the influence of the layers’ thickness and their stiffness on the accuracy of the continuum theory is determined.

Fast semi-analytic computation of elastic edge singularities

The singularities that we consider are the characteristic non-smooth solutions of the equations of linear elasticity in piecewise homogeneous media near two-dimensional corners or three-dimensional edges. We describe here a method to compute their singularity exponents and the associated angular singular functions. We present the implementation of this method in a program whose input data are geometrical data, the elasticity coefficients of each material involved and the type of boundary conditions (Dirichlet, Neumann or mixed conditions). Our method is particularly useful with anisotropic materials and allows to “follow” the dependency of singularity exponents along a curved edge.

NUMERICAL SOLUTION OF THREE-DIMENSIONAL STABILITY PROBLEMS FOR ELASTIC BODIES

A solution in Cartesian coordinates to plane and spatial stability problems for composites is obtained within the framework of the second variant of the three-dimensional linearized theory of stability of deformable bodies. Two mechanical models are used: a homogeneous anisotropic medium with averaged mechanical characteristics and a piecewise-homogeneous medium with orthotropic linearly elastic components. To solve the problems, a mesh approach is applied. Discrete models are constructed using the concept of a base scheme. The calculated results are analyzed

A 3-D stability theory applied to layered rocks undergoing finite deformations in biaxial compression

Based on the results obtained within the scope of the model of piecewise-homogeneous medium and three-dimensional stability theory, the asymptotic accuracy of the continuum theory is examined for layered compressible rocks undergoing finite deformations in biaxial compression. The particular mode of stability loss, that corresponds to the continuum approximation is determined. The investigation is carried out for the cases of uniaxial and biaxial compression and is illustrated by several numerical examples for the particular models of rocks. At that the influence of the layers' thickness and their stiffness, as well as the biaxiality of loading, on the accuracy of the continuum theory is determined.

The Edge Effect in a Laminated Composite with a Transversally Isotropic Filler

The edge effects in a laminated composite with a regular structure are considered. The equations of the linear theory of elasticity of piecewise-homogeneous media are used as a mathematical and mechanical model. An approximate solution is obtained on the basis of the mesh approach. An example of solving a specific problem is given and the results are analyzed