Rate-independent plasticity and statistically stored dislocations

We present here a continuum model for the evolution of the total dislocation density in the framework of rate-independent plasticity. Three basic physical features are taken into account: (i) the role of dislocation densities on hardening; (ii) the relations between the slip velocity and the mobility of gliding dislocations; (iii) the energetics of self and mutual interactions between dislocations. We restrict attention to plastic processes corresponding to single slip. Numerical simulations showing the formation of bands are also presented.     

Some problems in nonlinear magnetoelasticity

In this paper we examine the influence of magnetic fields on the static response of magnetoelastic materials, such as magneto-sensitive elastomers, that are capable of large deformations. The analysis is based on a simple formulation of the mechanical equilibrium equations and constitutive law for such materials developed recently by the authors, coupled with the governing magnetic field equations. The equations are applied in the solution of some simple representative and illustrative problems, with the focus on incompressible materials. First, we consider the pure homogeneous deformation of a slab of material in the presence of a magnetic field normal to its faces. This is followed by a review of the problem of simple shear of the slab in the presence of the same magnetic field. Next we examine a problem involving non-homogeneous deformations, namely the extension and inflation of a circular cylindrical tube. In this problem the magnetic field is taken to be either axial (a uniform field) or circumferential. For each problem we give a general formulation for the case of an isotropic magnetoelastic constitutive law, and then, for illustration, specific results are derived for a prototype constitutive law. We emphasize that in general there are significant differences in the results for formulations in which the magnetic field or the magnetic induction is taken as the independent magnetic variable. This is demonstrated for one particular problem, in which restrictions are placed on the admissible class of constitutive laws if the magnetic induction is the independent variable but no restrictions if the magnetic field is the independent variable.Received: May 17, 2004     

Resonant hydraulic sloshing in a tank of variable depth

The forced resonant oscillations of a fluid in a tank of variable depth are considered within the hydraulic approximation. It is shown that for certain bottom topographies a continuous periodic output dominated by the first normal mode is possible. This contrasts with the case of a tank of constant depth, where hydraulic jumps are a feature of the motion. The amplitude and frequency of the output are connected by a cubic equation. The fluid response can act like that of a hard or soft spring, depending on the bottom topography. There is also a critical bottom topography that yields a higher order response amplitude.     

End effects for pre-stressed and pre-polarized piezoelectric solids in anti-plane shear

This paper is concerned with investigation of the influence of pre-stress and pre-polarization on the decay of Saint-Venant end effects in piezoelectric solids. This issue is examined within the framework of incremental anti-plane shear states for prestressed and prepolarized solids. Attention is confined to a specific class of materials, namely those of hexagonal 6mm or 4mm symmetry. It is shown that the mechanical and electrical fields uncouple in this case, and the governing system of coupled second-order linear partial differential equations uncouple to a single homogeneous second-order equation with constant coefficients for the displacement and a Poisson equation for the electric potential. The exact decay rate for mechanical and electrical end effects is determined and the influence of pre-stress and pre-polarization is assessed. Motivation for this work arises from the current rapid developments in smart materials technology.     

Formulation and analysis of two energy-consistent methods for nonlinear elastodynamic frictional contact problems

Energy-conserving algorithms are necessary to solve nonlinear elastodynamic problems in order to recover long term time integration accuracy and stability. Furthermore, some physical phenomena (such as friction) can generate dissipation; then in this work, we present and analyse two energy-consistent algorithms for hyperelastodynamic frictional contact problems which are characterised by a conserving behaviour for frictionless impacts but also by an admissible frictional dissipation phenomenon. The first approach permits one to enforce, respectively, the Kuhn–Tucker and persistency conditions during each time step by combining an adapted continuation of the Newton method and a Lagrangean formulation. In addition the second method which is based on the work in [P. Hauret, P. Le Tallec, Energy-controlling time integration methods for nonlinear elastodynamics and low-velocity impact, Comput. Methods Appl. Mech. Eng. 195 (2006) 4890–4916] represents a specific penalisation of the unilateral contact conditions. Some numerical simulations are presented to underscore the conservative or dissipative behaviour of the proposed methods.     

Effective transverse elastic moduli of composites at non-dilute concentration of a random field of aligned fibers

We consider a transversal loading of a linearly elastic isotropic media containing the identical isotropic aligned circular fibers at non-dilute concentration c. By the use of solution obtained by the Kolosov–Muskhelishvili complex potential method for two interacting circles subjected to three different applied stresses at infinity, and exact integral representations for both the stress and strain distributions in a microinhomogeneous medium, one estimates the effective moduli of the composite accurately to order c². Received: March 4, 2003; revised: August 8, 2003     

Effective elastic moduli and stress concentrator factors in random structure aligned fiber composites

We consider a linearly elastic composite medium, which consists of a homogeneous matrix containing a statistically uniform random set of aligned fibers. Effective elastic moduli as well as the stress concentrator factors in the components are estimated. The micromechanical approach is based on the Green’s function technique as well as on the generalization of the “multiparticle effective field method” (MEFM, see for references, Buryachenko [1]). The refined version of the MEFM takes into account the variation of the effective fields acting on each pair of fibers. The dependence of effective elastic moduli and stress concentrator factors on the radial distribution function of the fiber locations is analyzed. Received: October 20, 2004     

Numerical analysis of a frictional contact problem for viscoelastic materials with long-term memory

We consider a mathematical model which describes the bilateral contact between a deformable body and an obstacle. The process is quasistatic, the material is assumed to be viscoelastic with long memory and the friction is modeled with Tresca’s law. The problem has a unique weak solution. Here we study spatially semi-discrete and fully discrete schemes using finite differences and finite elements. We show the convergence of the schemes under the basic solution regularity and we derive order error estimates. Finally, we present an algorithm for the numerical realization and simulations for a two-dimensional test problem.     

Steady-state vibrations of an unbounded linear piezoelectric medium

We consider fundamental (Dirichlet and Neumann-type) boundary value problems in a theory of generalized plane strain for the steady-state vibrations of an infinite piezoelectric medium with transversely isotropic symmetry (6 mm). Using integral equation methods with the appropriate Sommerfeld-type radiation conditions, we prove existence and uniqueness results for the corresponding exterior boundary value problems. Exact solutions are obtained in the form of integral potentials. (Received: September 27, 2005)     

Group invariant solution for a pre-existing fluid-driven fracture in impermeable rock

The propagation of a two-dimensional fluid-driven fracture in impermeable rock is considered. The fluid flow in the fracture is laminar. By applying lubrication theory a partial differential equation relating the half-width of the fracture to the fluid pressure is derived. To close the model the PKN formulation is adopted in which the fluid pressure is proportional to the half-width of the fracture. By considering a linear combination of the Lie point symmetries of the resulting non-linear diffusion equation the boundary value problem is expressed in a form appropriate for a similarity solution. The boundary value problem is reformulated as two initial value problems which are readily solved numerically. The similarity solution describes a preexisting fracture since both the total volume and length of the fracture are initially finite and non-zero. Applications in which the rate of fluid injection into the fracture and the pressure at the fracture entry are independent of time are considered.     

Dynamic bilateral contact problem for viscoelastic piezoelectric materials with adhesion

A model of a dynamic viscoelastic adhesive contact between a piezoelectric body and a deformable foundation is described. The model consists of a system of the hemivariational inequality of hyperbolic type for the displacement, the time dependent elliptic equation for the electric potential and the ordinary differential equation for the adhesion field. In the hemivariational inequality the friction forces are derived from a nonconvex superpotential through the generalized Clarke subdifferential. The existence of a weak solution is proved by embedding the problem into a class of second-order evolution inclusions and by applying a surjectivity result for multivalued operators.     

Dynamic frictionless contact with adhesion

A model for the dynamic, adhesive, frictionless contact between a viscoelastic body and a deformable foundation is described. The adhesion process is modeled by a bonding field on the contact surface. The contact is described by a modified normal compliance condition. The tangential shear due to the bonding field is included. The problem is formulated as a coupled system of a variational equality for the displacements and a differential equation for the bonding field. The existence of a unique weak solution for the problem is established, together with a partial regularity result. The existence proof proceeds by construction of an appropriate mapping which is shown to be a contraction on a Hilbert space.     

A Chebyshev spectral collocation method for solving Burgers’-type equations

In this paper, we elaborated a spectral collocation method based on differentiated Chebyshev polynomials to obtain numerical solutions for some different kinds of nonlinear partial differential equations. The problem is reduced to a system of ordinary differential equations that are solved by Runge–Kutta method of order four. Numerical results for the nonlinear evolution equations such as 1D Burgers’, KdV–Burgers’, coupled Burgers’, 2D Burgers’ and system of 2D Burgers’ equations are obtained. The numerical results are found to be in good agreement with the exact solutions. Numerical computations for a wide range of values of Reynolds’ number, show that the present method offers better accuracy in comparison with other previous methods. Moreover the method can be applied to a wide class of nonlinear partial differential equations.     

Analysis of a Class of Frictional Contact Problems for the Bingham Fluid

We consider a mathematical model which describes the stationary flow of a Bingham fluid with friction. The frictional contact is modeled by a general velocity dependent dissipation functional. We derive a weak formulation of the model which consists in a variational inequality for the velocity field. We establish the existence and uniqueness of the weak solution as well as its continuous dependence with respect to the contact condition. Finally, we describe a number of concrete friction conditions which may be set in this general framework and for which our results apply.     

Optimal shape of a strongest inverted column

By using Pontryagin's maximum principle we determine the shape of the strongest column positioned in a constant gravity field, simply supported at the lower end and clamped at upper end (with the possibility of axial sliding). It is shown that the cross-sectional area function is determined from the solution of a nonlinear boundary value problem. A variational principle for this boundary value problem is formulated and two first integrals are constructed. These integrals lead to an a priori estimate of the value of one the missing initial condition and to the reduction of the order of the system. The optimal shape of a column is determined by numerical integration.     

Thermodynamics and analysis of rate-independent adhesive contact at small strains

We address a model for adhesive unilateral frictionless Signorini-type contact between bodies of heat-conductive viscoelastic material, in the linear Kelvin-Voigt rheology, undergoing thermal expansion. The flow rule for debonding the adhesion is considered rate-independent and unidirectional, and a thermodynamically consistent model is derived and analysed as far as the existence of a weak solution is concerned.     

The solution methods for the largest eigenvalue (singular value) of nonnegative tensors and convergence analysis

In this paper we study two solution methods for finding the largest eigenvalue (singular value) of general square (rectangular) nonnegative tensors. For a positive tensor, one can find the largest eigenvalue (singular value) based on the properties of the positive tensor and the power-type method. While for a general nonnegative tensor, we use a series of decreasing positive perturbations of the original tensor and repeatedly recall power-type method for finding the largest eigenvalue (singular value) of a positive tensor with an inexact strategy. We prove the convergence of the method for the general nonnegative tensor. Under a certain assumption, the computing complexity of the method is established. Motivated by the interior-point method for the convex optimization, we put forward a one-step inner iteration power-type method, whose convergence is also established under certain assumption. Additionally, by using embedding technique, we show the relationship between the singular values of the rectangular tensor and the eigenvalues of related square tensor, which suggests another way for finding the largest singular value of nonnegative rectangular tensor besides direct power-type method for this problem. Finally, numerical examples of our algorithms are reported, which demonstrate the convergence behaviors of our methods and show that the algorithms presented are promising.     

DFT modal analysis of spectral element methods for the 2D elastic wave equation

The DFT modal analysis is a dispersion analysis technique that transforms the equations of a numerical scheme to the discrete Fourier transform domain sampled in the mesh nodes. This technique provides a natural matching of exact and approximate modes of propagation. We extend this technique to spectral element methods for the 2D isotropic elastic wave equation, by using a Rayleigh quotient approximation of the eigenvalue problem that characterizes the dispersion relation, taking full advantage of the tensor product representation of the spectral element matrices. Numerical experiments illustrate the dependence of dispersion errors on the grid resolution, polynomial degree, and discretization in time. We consider spectral element methods with Chebyshev and Legendre collocation points.     

Analysis of a dynamic contact problem for electro-viscoelastic cylinders

We consider a mathematical model which describes the antiplane shear deformations of a piezoelectric cylinder in frictional contact with a foundation. The process is mechanically dynamic and electrically static, the material behavior is described with a linearly electro-viscoelastic constitutive law, the contact is frictional and the foundation is assumed to be electrically conductive. Both the friction and the electrical conductivity condition on the contact surface are described with subdifferential boundary conditions. We derive a variational formulation of the problem which is of the form of a system coupling a second order hemivariational inequality for the displacement field with a time-dependent hemivariational inequality for the electric potential field. Then we prove the existence of a unique weak solution to the model. The proof is based on abstract results for second order evolutionary inclusions in Banach spaces. Finally, we present concrete examples of friction laws and electrical conductivity conditions for which our result is valid.