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.     

Developing a Compressive Failure Theory for Nanocomposites

The paper addresses a compressive-failure theory for polymer-matrix nanocomposites in the case where failure onset is due to microbuckling. Two approaches based on the three-dimensional linearized theory of stability of deformable bodies are applied to laminated and fibrous nanocomposites. According to the first approach (continuum compressive-failure theory), nanocomposites are modeled by a homogeneous anisotropic medium with effective constants, including microstructural parameters. The second approach uses the piecewise-homogeneous model, three-dimensional relations for fibers (CNT) and matrix, and continuity conditions at the fiber-matrix interface. The compressive-failure theory is used to solve specific problems for laminated and fibrous nanocomposites. Some approximate failure theories based on the one- and two-dimensional applied theories of stability of rods, plates, and shells are analyzed__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 3, pp. 3–37, March 2005.     

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.     

Continuum approximation in three-dimensional nonaxisymmetric problems of the stability theory of laminar compressible composite materials

Conclusions In the present work it has been rigorously proven that, for three-dimensional nonaxisymmetric stability problems of laminar compressible composite materials, forms of stability loss with a period along the axis ox₃ larger than the period of the structure (the second and fourth forms) in the continuum approximation do not give results corresponding to internal instability; forms of stability loss with the period along the axis ox₃ equal to the period of the structure (the first and third forms) in the continuum approximation give results corresponding to internal instability; the continuum theory of internal instability [2, 3] follows in the long-wave approximation, from the results corresponding to the first form of stability loss within the framework of the model of a piecewise-homogeneous medium (accurate formulation), and hence the continuum theory [2] is asymptotically accurate.The above conclusions corresponding to a three-dimensional nonaxisymmetric problem coincide completely with the conclusions of [5] obtained for plane problems.Kiev University. Translated from Prikladnaya Mekhanika, Vol. 26, No. 3, pp. 23–27, March, 1990.     

Stability of layered materials with zero in-plane strains

The equilibrium state of layered bodies under biaxial loading is analyzed for stability. It is assumed that the in-plane strains in the layers are zero. The three-dimensional linearized theory of stability and the piecewise-homogeneous material model are used. Two models of layered bodies are considered. Specific problems for layered bodies of different structure are solved. The critical load and wave number that cause instability of layered bodies are found     

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     

Dynamic interaction of a prestressed nonlinear elastic layer and a half-plane

Generalized Rayleigh waves propagating in a prestressed layer on a prestressed half plane under plane-strain conditions are studied using the piecewise-homogeneous model and the three-dimensional linearized theory of elasticity. The constitutive relations of the layer and half-plane include the Murnaghan potential. Numerical results have been obtained for two materials: bronze for the layer and steel for the half-plane.Published in Prikladnaya Mekhanika, Vol. 40, No. 9, pp. 137–144, September 2004.     

On the formation of linear folds in regularly layered rock masses under biaxial loading

The piecewise-homogeneous model and the three-dimensional theory of stability for small and finite subcritical strains are used to study the formation of folds in layered rock masses of the Earth’s crust under biaxial loading. The general statement of the problem is given, and the governing characteristic equations are derived. Numerical results are presented for a layered rock mass composed of two alternating layers __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 12, pp. 26–34, December 2005.     

Strain gradient plasticity modeling of the cyclic behavior of laminate microstructures

Two recently proposed Helmholtz free energy potentials including the full dislocation density tensor as an argument within the framework of strain gradient plasticity are used to predict the cyclic elastoplastic response of periodic laminate microstructures. First, a rank-one defect energy is considered, allowing for a size-effect on the overall yield strength of micro-heterogeneous materials. As a second candidate, a logarithmic defect energy is investigated, which is motivated by the work of Groma et al. (2003). The properties of the back-stress arising from both energies are investigated in the case of a laminate microstructure for which analytical as well as numerical solutions are derived. In this context, a new regularization technique for the numerical treatment of the rank-one potential is presented based on an incremental potential involving Lagrange multipliers. The results illustrate the effect of the two energies on the macroscopic size-dependent stress–strain response in monotonic and cyclic shear loading, as well as the arising pile-up distributions. Under cyclic loading, stress–strain hysteresis loops with inflections are predicted by both models. The logarithmic potential is shown to provide a continuum formulation of Asaro's type III kinematic hardening model. Experimental evidence in the literature of such loops with inflections in two-phased FFC alloys is provided, showing that the proposed strain gradient models reflect the occurrence of reversible plasticity phenomena under reverse loading.     

Stability of layered coatings with outer layer made of ceramics

The stability of layered coatings with a ceramic layer under biaxial loading is studied. The three-dimensional linearized theory of stability and piecewise-homogeneous material model are used. Specific problems for layered bodies of various structure are solved. The critical loads and wave numbers responsible for loss of stability of layered bodies are determined     

含液颗粒材料液固耦合分析的离散颗粒模型及特征线SPH法

对含液颗粒材料流固耦合分析建议了一个基于离散颗粒模型与特征线SPH法的显式拉格朗日-欧拉无网格方案。在已有的用以模拟固体颗粒集合体的离散颗粒模型[1]基础上,将颗粒间间隙内的流体模型化为连续介质,对其提出并推导了基于特征线的SPH法。数值例题显示了所建议方案在模拟颗粒材料与间隙流相互作用的能力和性能以及间隙流体对颗粒结构承载能力及变形的影响。     

Axisymmetric longitudinal wave propagation in a finite prestrained circular cylinder embedded in a finite prestrained compressible medium

Axisymmetric longitudinal wave propagation in a finite prestrained circular cylinder contained in a finite prestrained infinite body is investigated within the scope of the piecewise-homogeneous body model by employing the three-dimensional linearized theory of elastic waves in a prestressed body. It is assumed that the materials of the cylinder and infinite body are compressible and that their elastic relations are described by a harmonic potential. Numerical results are presented and discussed for the case where the elastic constants of the cylinder are greater than those of the surrounding infinite body     

Stability of an Elastic Layer Stack Between Two Half-Spaces Under Compressive Loads

A stability problem is solved for a multicomponent medium consisting of a layer stack and two structurally homogeneous half-spaces. The analysis is made within the framework of the piecewise-homogeneous model and the three-dimensional linearized theory of stability for small subcritical strains. The properties of isolated elements of the medium are determined within the framework of the orthotropic linearly elastic model. Specific numerical results are obtained for the case of plane strain     

A BEM solution to transverse shear loading of composite beams

In this paper a boundary element method is developed for the solution of the general transverse shear loading problem of composite beams of arbitrary constant cross-section. The composite beam consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli with same Poisson’s ratio and are firmly bonded together. The analysis of the beam is accomplished with respect to a coordinate system that has its origin at the centroid of the cross-section, while its axes are not necessarily the principal ones. The transverse shear loading is applied at the shear centre of the cross-section, avoiding in this way the induction of a twisting moment. Two boundary value problems that take into account the effect of Poisson’s ratio are formulated with respect to stress functions and solved employing a pure BEM approach, that is only boundary discretization is used. The evaluation of the transverse shear stresses is accomplished by direct differentiation of these stress functions, while both the coordinates of the shear center and the shear deformation coefficients are obtained from these functions using only boundary integration. Numerical examples with great practical interest are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. The accuracy of the proposed shear deformation coefficients compared with those obtained from a 3-D FEM solution of the ‘exact’ elastic beam theory is remarkable.     

A continuum approach to the analysis of the stress field in a fiber reinforced composite with a transverse crack

The stress field in a periodically layered composite with an embedded crack oriented in the normal direction to the layering and subjected to a tensile far-field loading is obtained based on the continuum equations of elasticity. This geometry models the 2D problem of fiber reinforced materials with a transverse crack. The analysis is based on the combination of the representative cell method and the higher-order theory. The representative cell method is employed for the construction of Green’s functions for the displacements jumps along the crack line. The problem of the infinite domain is reduced, in conjunction with the discrete Fourier transform, to a finite domain (representative cell) on which the Born–von Karman type boundary conditions are applied. In the framework of the higher-order theory, the transformed elastic field is determined by a second-order expansion of the displacement vector in terms of local coordinates, in conjunction with the equilibrium equations and these boundary conditions. The accuracy of the proposed approach is verified by a comparison with the analytical solution for a crack embedded in a homogeneous plane.Results show the effects of crack lengths, fiber volume fractions, ratios of fiber to matrix Young’s moduli and matrix Poisson’s ratio on the resulting elastic field at various locations of interest. Comparisons with the predictions obtained from the shear lag theory are presented.     

On the simulation of cohesive fatigue effects in grain boundaries of a piezoelectric mesostructure

Ferroelectric materials offer a variety of new applications in the field of smart structures and intelligent systems. Accordingly, the modelling of these materials constitutes an active field of research. A critical limitation of the performance of such materials is given when electrical, mechanical, or mixed loading fatigue occurs, combined with, for instance, microcracking phenomena. In this contribution, fatigue effects in ferroelectric materials are numerically investigated by utilisation of a cohesive-type approach. In view of finite element-based simulations, the geometry of a natural grain structure, as observed on the so-called meso-level, is represented by an appropriate mesh. While the response of the grains themselves is approximated by coupled continuum elements, grain boundaries are numerically incorporated via so-called cohesive-type or interface elements. These offer a great potential for numerical simulations: as an advantage, they do not result in bad-conditioned systems of equations as compared with the application of standard continuum elements inhering a very high ratio of length and height. The grain boundary behaviour is modelled by cohesive-type constitutive laws, designed to capture fatigue phenomena. Being a first attempt, switching effects are planned to be added to the grain model in the future. Two differently motivated fatigue evolution techniques are applied, the first being appropriate for low-cycle-fatigue, and a second one adequate to simulate high-cycle-fatigue. Subsequent to a demonstration of the theoretical and numerical framework, studies of benchmark boundary value problems with fatigue-motivated boundary conditions are presented.     

On the influence of singular-type finite elements on the critical force in studying the buckling of a circular plate with a crack

The axisymmetric buckling (delamination) of a circular disk (plate) with a penny-shaped crack is analyzed using a continuum model, piecewise-homogeneous model, and the three-dimensional linearized theory of stability. The FEM is used. The analysis is carried out using various singular and ordinary finite elements. The numerical results obtained indicate that it is not necessary to use singular finite elements to solve the problem Published in Prikladnaya Mekhanika, Vol. 43, No. 9, pp. 120–129, September 2007.     

Statistical continuum theory for inelastic behavior of a two-phase medium

A statistical continuum mechanics formulation is presented to predict the inelastic behavior of a medium consisting of two isotropic phases. The phase distribution and morphology are represented by a two-point probability function. The isotropic behavior of the single phase medium is represented by a power law relationship between the strain rate and the resolved local shear stress. It is assumed that the elastic contribution to deformation is negligible. A Green’s function solution to the equations of stress equilibrium is used to obtain the constitutive law for the heterogeneous medium. This relationship links the local velocity gradient to the macroscopic velocity gradient and local viscoplastic modulus. The statistical continuum theory is introduced into the localization relation to obtain a closed form solution. Using a Taylor series expansion an approximate solution is obtained and compared to the Taylor’s upper-bound for the inelastic effective modulus. The model is applied for the two classical cases of spherical and unidirectional discontinuous fiber-reinforced two-phase media with varying size and orientation.     

On the determination of loads upon the ground support in underground mine workings

The piecewise-homogeneous model and the three-dimensional linearized theory of stability for small subcritical strains are used to study the surface instability of a regularly layered rock mass under biaxial loading. A plane problem is formulated. Basic characteristic equations are derived. A specific problem is solved as an example to demonstrate the selection of loads and the interaction of support elements with the wall rock __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 11, pp. 38–46, November 2005.