This work investigates the asymptotic structure of a boundary-value problem proposed recently in connection with in-plane instabilities of spinning disks. Assuming an orthotropic elastic material with cylindrical symmetry we consider a perturbation with respect to the constitutive behavior. The material is assumed to be very stiff in the azimuthal direction, a situation which is commonly encountered in the case of composite flywheels based on hoop-wound carbon fibers in a flexible polyurethane resin. The accuracy of the asymptotic strategy is confirmed by a number of direct computer simulations of the original problem.

This paper is concerned with the asymptotic analysis of plates with periodically rapidly varying heterogeneities. The formal asymptotic procedure is performed when both the periods of changes of the material properties and the thickness of the plate are of the same orders of magnitude. Our approach is based on a sequence of recursive minimization problems. We consider a plate made of Ciarlet–Geymonat type materials. Depending on the order of magnitude of the applied loads, we obtain a nonlinear membrane model and a nonlinear membrane inextensional bending model as announced by Pruchnicki.

An approach is described for constructing (using molecular dynamics simulations at the atomic scale) a mathematical model for the constitutive behavior of a carbon nanotube. The method is based on applying homogenization theory to a hexagonal array of carbon atoms with a specific chirality vector. The molecular dynamics simulations generate a set of periodic, rapidly varying, elastic constants for the nanotube. An example is presented to illustrate the technique for a specific array on a nanotube surface.

In many practical situations, uncertainties affect the mechanical behavior that is given by a family of graphs instead of a single graph. In this paper, we show how the bipotential method is able to capture such blurred constitutive laws, using bipotential convex covers.

In this paper we consider a crack of arbitrary shape in a homogeneous elastic media in the absence of body forces, formulate variational Dirichlet and Neumann crack problems in a linear three-dimensional elasticity in Sobolev spaces and prove the existence and uniqueness of the corresponding (weak) solutions.

We present a mathematical framework for the theory of limit analysis of rigid, perfectly plastic bodies where the equality of the static multiplier and kinematic multiplier for incompressible fields is formulated and proved in a compact form. Assuming that the failure criterion is a norm on the space of deviatoric stress fields, we use standard properties of linear operators on Banach spaces.

In this paper we prove the relaxation theorem in micropolar elasticity and use it, together with the semicontinuity theorem, to justify lower-dimensional models of rods (and plates) by means of -convergence starting from general energy functionals. The internal energy density is assumed to be continuous and satisf–ies some growth and coercivity conditions. In particular, we apply these results to derive a rod model starting from quadratic isotropic energy density function of a cylindrical three-dimensional micropolar body.

Existence and uniqueness for infinitesimal dislocation based rate-independent gradient plasticity with linear isotropic hardening and plastic spin are shown using convex analysis and variational inequality methods. The dissipation potential is extended non-uniquely from symmetric plastic rates to non-symmetric plastic rates and three qualitatively different formats for the dissipation potential are distinguished.

Zimmer and collaborators have recently proposed an alternative approach to Landau's for phase transformations, in which they replace the energy expression in terms of polynomials, a stiff family of functions, with one in terms of more f lexible functions, for instance certain splines. One of the arguments in favor of this more f lexible choice is the inability of an energy proposed earlier by Ericksen and James to f it reasonably well all the elasticities of an InTl alloy, in particular the wide difference in the measured \hat{L}₄₄ and \hat{L}₆₆ elasticities of the InTl martensite. While the proposal of Zimmer is theoretically very interesting and f it for numerical applications, and shows potential for modelling materials with complex internal structure, here I show that the pessimism about the standard Landau approach is indeed excessive as far as the weak points of the Ericksen-James energy are concerned. I propose a fourth-degree polynomial energy which extends the one of Ericksen and provides a fairly accurate f it of the equilibria as well as the elasticities for an In15.5\%Tl alloy; and perhaps also for different compositions in the InTl family. The starting point is the version of Ericksen's theory [5] given in [12].

Elasticity is viewed here as a starting point in the description of inelastic behavior. Structured deformations and a field theory of elastic bodies undergoing disarrangements (non-smooth geometrical changes) and dissipation are used to formulate and illustrate a concept of "submacroscopically stable equilibrium configuration". A body in a submacroscopically stable equilibrium configuration resists additional submacroscopic geometrical changes such as the occurrence of microslips, the formation of microvoids, and the appearance of localized distortions that, together, leave the macroscopic configuration of the body unchanged. Submacroscopically stable configurations represent energetically preferred phases for bodies in equilibrium, and a procedure is described here for determining the submacroscopically stable equilibria of a body. The procedure is carried out in detail here for two classes of bodies that may undergo disarrangements and experience internal dissipation. One class is characterized by its bi-quadratic free energy response function, and the requirement of submacroscopic stability reduces from five to one the number of phases available to a body that is in equilibrium under mixed boundary conditions. Boundary-value problems for the macroscopic deformation corresponding to a submacroscopically stable equilibrium configuration in a body of this class are formulated. A second class of bodies, the "near-sighted fluids", has both a prolate and a spherical phase that may occur in equilibrium; the submacroscopically stable equilibria of a near-sighted fluid must be stress-free, without regard to the particular phase that appears. In all considerations in this article, the term "equilibrium" is synonymous with satisfaction of balance of forces and moments in a given environment.

The finite deformation kinematics of micromorphic plasticity is discussed in the framework of multiplicative decomposition of the macro- and microdeformation gradient tensor, suggesting the introduction of a so-called plastic intermediate configuration for the micromorphic continuum. The geometrical structure of the plastic intermediate configuration and the micromorphic curvature tensors are elucidated by invoking the differential operator of the relative covariant derivative with respect to the plastic intermediate configuration. Micromorphic curvature tensors arise in a natural way by considering scalar-valued differences. The latter measure the deformation process and are required to be form-invariant with respect to the chosen configuration.

In the recent work of H.-D. Alber and K. Chelminski (Mathematical Models and Methods in the Applied Sciences, 17, 189–213, 2007) the existence of the solutions to a model of inelastic (viscoplastic) behavior of materials at small strain is derived. In this work we show that the conditions of the existence theorem of Alber and Chelminski can be relaxed and the same result can be proved under less restrictive assumptions. The relaxation of the conditions of the existence theorem of Alber and Chelminski (2007) makes it possible to answer the question raised by them concerning the solvability of the model of nonlinear kinematic hardening without assuming a higher exponent in the constitutive law for one of the internal variables than the exponent in the constitutive law for the other one.

The transition from Neumann (traction-free) to Dirichlet (fixed-face) boundary conditions is investigated in respect of wave propagation in a linear isotropic elastic layer. Attention is focused on the implications of such a transition on the dispersion curve branches within the long-wave region. The formation of low-frequency band gap that is expected to exist in layers with Dirichlet boundary condition is shown to be caused by different mechanisms in anti-symmetric and symmetric cases. Certain implications to short-wave propagation in the layer are also investigated. The study includes both a numerical investigation and a multi-parameter asymptotic analysis.

A nested piecewise homogeneous elastic scatterer is embedded in a homogeneous elastic environment. The scatterer's core may be rigid, cavity, Robin, or lossy penetrable. A 2D or 3D incident elastic field, generated by a point-source located in the homogeneous environment, impinges on the scatterer. The scattering problem is formulated in a dyadic form. The main purpose of this paper is to establish scattering relations for the elastic point-source excitation of a nested piecewise homogeneous scatterer. To this direction, we establish reciprocity principles and general scattering theorems relating the scattered fields with the corresponding far-field patterns. Furthermore, for a scatterer excited by a point-source and a plane wave, mixed scattering relations are derived. The optical theorem, relating the scattering cross-section with the field at the point-source's location a is recovered as a corollary of the general scattering theorem. We present a detailed investigation for the 2D case and summarize the results for the 3D case, pointing out the main differences in the analysis.

The solution for the elastic three-phase circular inclusion problem plays a fundamental role in many practical and theoretical applications. In particular, it offers the fundamental solution for the generalized self-consistent method in the mechanics of composites materials. In this work, a semi-analytical method is presented for the problem of a pre-existing radial Griff ith crack embedded within the interphase layer surrounding a circular inclusion. Novel to this work is that the bonding at the inclusion–interphase interface and the interphase-matrix interface is considered to be imperfect with the assumption that the interface imperfections are constant. Employing complex variable techniques, we derive series representations for the corresponding stress functions inside the inclusion, in the interphase layer and the surrounding matrix. The governing boundary value problem is then formulated in such a way that these stress distributions simultaneously satisfy the traction-free condition along the crack face, the imperfect interface conditions and the prescribed asymptotic loading conditions. The advantage of the series method over other methods, such as the dislocation density method, is that in the former case the resulting expressions are linear and can be solved readily whereas in the latter case the method leads to cumbersome integral equations which are often numerically diff icult to solve.

Stress intensity factor (SIF) calculations are performed at the crack tips for different material property combinations, imperfect interface conditions and crack locations under mode I loading. The results not only provide for a quantitative description of the interaction between a radial interphase crack and a three-phase inclusion with imperfect interfaces but the results clearly demonstrate the signif icance of how two imperfect boundaries can inf luence crack behavior.

This short note introduces a methodology for sequential multiscale modeling of autonomous, microscopic, systems of Ordinary Differential Equations through a redef inition of the original dynamics as an augmented system with an explicit separation of time scales arising merely from the def inition of time-averaging. Associated mathematical questions are stated and discussed.

The global and local conditions of uniqueness and the criteria excluding a possibility of bifurcation of the equilibrium state for large and small strains are derived. The conditions and criteria are derived analyzing the problem of uniqueness of solution of the basic incremental boundary problem of coupled generalized thermoplasticity. This paper is a continuation of some previous work, but contains new derivation of the global and local criteria excluding a possibility of bifurcation of the equilibrium state for a comparison body dependent on statically admissible fields of stress velocity. All the thermomechanical coupling effects, non-associated laws of plastic flow and influence of plastic strains on thermoplastic properties of a body were taken into account in this work. Thus, the mathematical problem considered here is not a self-conjugated problem.

This paper is the second part of our work on generalized coupled thermoplasticity taking into consideration the large strains. In the first part of the work global uniqueness conditions for solution of fundamental incremental boundary-value problem were derived. In this part a model of non-compressible elasto-plastic body with non-associated plastic flow law is analyzed. The assumptions and constitutive equations for this model are presented and suitable comparison bodies dependent on kinematically admissible fields of velocity strain and statically admissible fields of velocity stress are derived. An attempt to derive local sufficiwent uniqueness condition for comparison bodies is also presented. This attempt was unsuccessful since it lead to contradictory results, which means that the derivation of local sufficient uniqueness condition for comparison bodies of non-compressible elasto-plastic body is still unsolved.

Cerebral aneurysms occur in weakened areas of artery walls resulting in a ballooning out of the wall filled with blood. A major catalyst for mathematical modeling of intracranial saccular aneurysms has been the axisymmetric membrane derivations in Shah and Humphrey (1999, Journal of Biomechanics, 32, 593–599) and David and Humphrey (2003, Journal of Biomechanics, 36, 1143–1150). We expand on the foundational membrane dynamics to develop a blood-aneurysm-cerebrospinal fluid model from the fully three-dimensional nonlinear elastic equations of motion with system coupling at both inner and outer fluid–aneurysm boundaries consistent with Navier-Stokes. We derive the 3D elastodynamics and determine subsequent governing nonlinear ordinary differential equations for the three general material symmetries possible under the classic initial assumption of axisymmetry. We employ biologically motivated strain-energy functions to numerically solve the equations and observe resulting aneurysm cyclic stretches, thickness changes, effects of material and geometric parameters, and through-the-thickness stresses due to biological forcing for each type of material symmetry and constitutive model.

This paper describes a stochastic analysis in the one-dimensional uncoupled linear thermoelasticity. The autocorrelation functions of transient temperature and thermal stresses are analytically obtained in seven simple geometries: infinite plate, infinite strip, hollow sphere, infinite body with a spherical cavity, infinite hollow cylinder, annular disk and infinite body with a cylindrical hole, which have a random initial temperature distribution. Numerical calculations are performed for some geometries under the condition that the randomness in the initial temperature is given as a homogeneous white noise. The transient behavior of the mean square temperature and thermal stresses is illustrated and it is observed in objects confined to a finite region that whereas the maximum value of the mean square thermal stresses occurs within the objects shortly after the heat flow begins, it appears at a lateral surface after a length of time has elapsed.

The possible role of the Eshelby stress in the general context of theories of material evolution is examined with particular emphasis on evolution laws that are not necessarily of the anelastic type.

In this paper, material and spatial motion problems of the coupled nonlinear problem of electro- and magneto-elastostatics are discussed in the context of non-potential loading where mechanical loads are not assumed to be derived explicitly from some potential. A virtual work approach is used to derive the corresponding balance equations and boundary conditions of the material motion problems.

In this paper we explore the consequences of prescribing constitutive relations for elastic bodies wherein deformations are given as functions of stresses. For this class of constitutive relations in, the particular case of small deformations, we study boundary value problems for plane strain and plane stresses, and we develop a weak formulation that can be considered as the starting point for numerical computations.

In this paper, we solve interior and exterior Dirichlet and Neumann problems of anti-plane micropolar elasticity in Sobolev space setting using boundary integral equation method. Corresponding weak solutions are derived in terms of modified single and double layer potentials with distributional densities.

We study the yield conditions of phase transformation initiation for shape memory alloys exhibiting asymmetry between tension and compression. An extension of the choice of the classical invariant parameters such as those of Lode is proposed. A necessary and sufficient condition of convexity of these surfaces representing the elastic domain of austenite in the stress space, is established. Moreover the transport of these surfaces in the space of effective transformations strains of martensite is done. Hence, the duality between these two spaces is built. Some applications involving Cu-Al-Be and NiTi shape memory alloys end the purpose.

With the aim of exploring extreme elastic properties, decompositions of the compliance tensor are established for various crystal classes. Representations are deduced that lead directly to well-structured forms for analyzing extreme properties. The properties of primary interest are the Lamé compliance and the axial compliance (reciprocal Young's modulus). These exteme values and the corresponding directions play significant role in the study of those for the Poisson's ratio, in particular the search for a negative minimum.

We prove that the first-order nonlinear differential system governing the plastic spin response in simple shear (Dafalias's equations) is reduced to an equivalent equation of the Abel normal form by means of admissible functional transformations. In similar Abel equations result also the original and the generalized Volterra differential systems, describing the problem of two populations conflicting with one another. The above reduced Abel equations do not admit exact analytic solutions in terms of known (tabulated) functions, since only very special cases of these types of equations can be analytically solved in parametric form. We provide a mathematical solution methodology leading to the construction of exact implicit analytic solutions of the above-mentioned type of equations. Since the plastic spin nonlinear differential system results in a special unsolvable form of Abel's equation of the normal form, we perform the exact implicit analytic solution of this system too.

The solutions of initial-boundary value problems for bending of a thermoelastic plate with a crack in the case of Dirichlet and Neumann boundary conditions prescribed on the crack edges are represented as a sum of thermoelastic single-layer and double-layer potentials with unknown densities, and the unique solvability of the corresponding boundary integral equations is discussed in a distributional setting.

We investigate the weakest possible constitutive assumptions on the curvature energy in linear Cosserat models still providing for existence, uniqueness and stability. The assumed curvature energy is µL _c ²||dev sym axl A||² where axl A is the axial vector of the skewsymmetric microrotation A (3) and dev is the orthogonal projection on the Lie-algebra (3) of trace free matrices. The proposed Cosserat parameter values coincide with values adopted in the experimental literature by R. S. Lakes. It is observed that unphysical stiffening for small samples is avoided in torsion and bending while size effects are still present. The number of Cosserat parameters is reduced from six to four. One Cosserat coupling parameter µ _c > 0 and only one length scale parameter L_c > 0. Use is made of a new coercive inequality for conformal Killing vectorfields. An interesting point is that no (controversial) essential boundary conditions on the microrotations need to be specified; thus avoiding boundary layer effects. Since the curvature energy is the weakest possible consistent with non-negativity of the energy, it seems that the Cosserat couple modulus µ _c > 0 remains a material parameter independent of the sample size which is impossible for stronger curvature expressions.

One of the possible effects of volumetric growth in elastic materials is the creation of residual stresses. These stresses are known to change many of the classical properties of the material and have been studied extensively in the context of volumetric growth in biomechanics. Here we consider the problem of elastic cavitation in a growing compressible elastic membrane. Growth is taken to be homogeneous but anisotropic, and the membrane is assumed to remain axisymmetric during growth and deformation. We prove that neo-Hookean membranes cannot cavitate, but for Varga elastic materials we find conditions under which the material exhibits spontaneous cavitation in the absence of external loads, in marked distinction from the cavitation problem without growth.

The main aim of this paper is to prove the existence and uniqueness of solutions to an initial-boundary value problem corresponding to the Biot model. The existence theorem is proved by Galerkin method and the passage to the limit in the approximation process is shown in a standard way. Assuming that the given data satisfy some natural regularity requirements a better regularity of solutions is obtained than it could be found in the literature.

Despite the prevalent use of continuum mechanics for the modeling of granular materials, the controversy surrounding the relationship between the properties of the discrete medium and those of its equivalent continuum has far from abated. The concept of strain is especially problematic. In a continuum body, the strain represents the deformation of an infinitesimal region about a material point. In a discrete granular assembly, however, deformation is governed by the relative motions of the constituent grains. Herein, we introduce a new microstructural definition for the deformation of a granular material within the framework of Micropolar Continuum Theory. The advantages of the new strain definition over existing formulations are: it accounts for particle rotations, it is relatively straightforward to calculate, and its global average matches the macroscopic strain of the assembly. The new definition leads to a patchwork strain field, the existence of which is linked to the nonaffine strain at the particle scale. A key aspect of this study is the construction of a set of local micropolar strain and curvature measures on the scale of a particle and its first ring of neighbors. We dissect these local continuum quantities and, with the aid of discrete element simulations, examine them for a specimen under biaxial compression. New insights are gained on the contributions of the relative particle motions for specific types of contacts at different stages in the deformation history. Results are discussed in light of past experimental findings on shear banding, as well as Oda�s hypothesis on force chain buckling.