Darboux classification of cauchy elastic bodies and some of its consequences

The Darboux theorem on the classification and canonical forms of first degree differential forms is used to obtain a classification and canonical representation of Cauchy elastic solids. If the stress is assumed to be determined uniquely by the Green deformation tensor C, the double Piola-Kirchhoff stress tensor is shown to belong to one of six possible classes, each of which possesses a unique representation in terms of linear combinations of gradients with respect to C of scalar-valued functions with scalar-valued coefficients. There are three distinct classes for isotropic materials, each of which has a unique representation. It is shown that these three classes are distinguishable in terms of the existence or nonexistence of cycles with nonzero total work and by the accessibility or inaccessibility of any deformation state from any neighboring deformation state by a path of zero work.     

Reflection of plane waves at the boundary of a pre-stressed elastic half-space subject to biaxial stretch and simple shear

This paper is concerned with the effect of finite pure homogeneous biaxial stretch together with simple shear deformation on the reflection from a plane boundary of elastic waves propagating in a half-space of incompressible isotropic elastic material. This generalizes the previous work in which, separately, pure homogeneous strain and simple shear were considered. For a special class of constitutive law, it is shown that an incident plane harmonic wave propagating in the considered plane gives rise to a surface wave in addition to a reflected wave for every angle of incidence. The amplitude of the surface wave may vanish at certain discrete angles depending on the state of stress, biaxial stretch and simple shear deformation and then specialized to recover results obtained previously. The amplitude of the reflected wave is independent of the pre-stress but does depend upon the magnitude of deformation under consideration. The dependence of the reflected and surface wave behavior on the angle of incidence, amount of shear strain, biaxial stretch and the state of stress is illustrated graphically.     

Radiance Theorem with Partially Coherent Light

The transmission of a generalized radiance across a planar boundary separating two homogeneous media is considered. It is assumed that the optical field remains continuous at the interface and reflection is neglected. A result is obtained which may be regarded as a generalization of the conventional radiance theorem for fields of any state of coherence. This result differs from the conventional theorem by a factor that depends, in general, both on the optical intensity and on the degree of coherence of the field. However, over a wide range of circumstances the generalized radiance theorem is shown to be in good agreement with the conventional theorem.     

Symplectic analysis of plane problems of functionally graded piezoelectric materials

This paper develops an analytical method to investigate the plane problems of functionally graded piezoelectric materials within the symplectic framework. The material constants, including the elastic, piezoelectric and dielectric constants are assumed to vary along the length in an identical exponential form. A matrix state equation is derived by introducing new stress and electric displacement components, and is solved using the method of separation of variables. The operator matrix in the state equation is found to have similar properties as Hamiltonian matrix for homogeneous materials. Its eigenvectors (and hence eigensolutions) corresponding to particular eigenvalues (0 and −α) are derived; while the former also present in the homogeneous materials, the latter bear complete different forms. A detailed analysis shows, however, that the −α-group eigensolutions can degenerate to the ones for the homogeneous materials after eliminating certain rigid motions. Numerical results are given to show the effect of material inhomogeneity on these eigensolutions.     

Characteristics of mechanical properties of materials in studies of conditions of attainment of marginal states

It is shown that conditions of attainment of marginal states in the material of structural components are formulated on the basis of analysis of force and deformation criteria of formation and development of destruction. The parameters of these criteria are data on types of structural materials, standard and special characteristics of their mechanical properties, structural forms of considered elements of the equipment, and their operating stress loading. In this case, the construction of a system of state equations for describing interrelation of current damage, external actions, and responses to them is based on the concept of calculating variations in properties of materials at all stages of the design life cycle.     

Computer-Aided Design of High-Strength Gradient Materials Operating under Dynamic Loads

We consider a procedure of computer-aided design of materials with high strength characteristics based on using multicomponent systems with a gradient distribution of the concentration of components. It is shown that, in accordance with the synergetic definition of an impact-deformable material as an open system, with the help of gradient materials one can exercise control of the behavior of a loaded material in nonequilibrium state. The use of gradient materials under intense dynamic loads (impact, explosion) allows improving the stability of the system due to a decrease in the total increment of entropy that is confirmed by the Prigozhin–Klimontovich theorem.     

Study on cavitated bifurcation problems for spheres composed of hyper-elastic materials

In this paper, spherical cavitated bifurcation problems are examined for incompressible hyper-elastic materials and compressible hyper-elastic materials, respectively. For incompressible hyper-elastic materials, a cavitated bifurcation equation that describes cavity formation and growth for a solid sphere, composed of a class of transversely isotropic incompressible hyper-elastic materials, is obtained. Some qualitative properties of the solutions of the cavitated bifurcation equation are discussed in the different regions of the plane partitioned by material parameters indicating the degree of radial anisotropy in detail. It is shown that the cavitated bifurcation equation is equivalent, by use of singularity theory, to a class of normal forms with single-sided constraint conditions at the critical point. Stability and catastrophe of the solutions of the cavitated bifurcation equation are discussed by using the minimal potential-energy principle. For compressible hyper-elastic materials, a group of parameter-type solutions for the cavitated deformation for a solid sphere, composed of a class of isotropic compressible hyper-elastic materials, is obtained. Stability of the solutions is also discussed.     

Disorder is good for you: the influence of local disorder on strain localization and ductility of strain softening materials

We formulate a generic concept model for the deformation of a locally disordered, macroscopically homogeneous material which undergoes irreversible strain softening during plastic deformation. We investigate the influence of the degree of microstructural heterogeneity and disorder on strain localization (formation of a macroscopic shear band) in such materials. It is shown that increased microstructural heterogeneity delays strain localization and leads to an increase of the plastic regime in the macroscopic stress–strain curves. The evolving strain localization patterns are characterized.     

An internal state variable approach to constitutive theories for granular materials with snow as an example

In this paper, a microphysical constitutive theory is developed for a class of rate dependent granular materials under finite deformation. The theory is based on non-equilibrium thermodynamics with internal state variables. The state variables may be thought of as representing the current pattern of microstructural arrangemenp and hence characterize the plastic state of the material. A significant feature of this theory is that the state variables are identified at the granular level, as opposed to the crystalline level. This allows one to develop a microdynamical theory in terms of experimentally observable quantities and is a unique feature of granular materials.The theory is used to describe the mechanical properties of snow under high rate multiaxial deformation. Snow is a highly nonlinear, rate dependent material which exhibits significant microstructural alternations under finite strain. These alternations are tracked mathematically by temporal evolution equations governing the internal state variables. The change in the state variables is directly related to the plastic strain of the material.     

A class of universal relations for constrained,isotropic elastic materials

Summary A class of universal relations for all kinematically constrained, isotropic, elastic materials is described by the equationSB=BS relating the symmetric extra stress and the Cauchy-Green deformation tensors. This rule generates easily at most three universal relations for all kinematically admissible deformations of any constrained, isotropic body for which these tensors are nondiagonal. New universal formulae for homogeneous, compressible and incompressible materials reinforced by inextensible fibers in a variety of arrangements are presented for several kinds of homogeneous and nonhomogeneous, controllable universal deformations.     

Numerical simulation of brittle fracture of thin-walled structures1

The model of brittle material, allowing for the determination of initiation and growth of cracks in thin-walled structures up to their destruction, is offered. Solutions of some problems are submitted: destruction of a plate at a homogeneous stress state; destruction of a beam at a pure bending; dynamic deformation and destruction of a plate with the concentrated mass having various initial speeds. Destruction of part of the material of the plate is shown to result in dynamic loss of stability of the solution.     

Numerical simulation of brittle fracture of thin-walled structures1

The model of brittle material, allowing for the determination of initiation and growth of cracks in thin-walled structures up to their destruction, is offered. Solutions of some problems are submitted: destruction of a plate at a homogeneous stress state; destruction of a beam at a pure bending; dynamic deformation and destruction of a plate with the concentrated mass having various initial speeds. Destruction of part of the material of the plate is shown to result in dynamic loss of stability of the solution.     

Modeling the progressive collapse behavior of metal foams

Quasi-static axial compression behavior of cellular materials can be explained by two ideal deformation scenarios: homogeneous deformation and progressive collapse. An elastic–plastic–rigid (E–P–R) foam model is used to derive the energy absorbed under these two deformation scenarios. It has been identified that the progressive collapse mode of deformation can absorb more energy than homogeneous deformation prior to full densification. The additional energy is shown to be proportional to the magnitude of the tangent modulus in the plateau region of the stress–strain curve. The model also shows that energy absorption is equivalent for all deformation states of progressive and homogenous collapse for materials that exhibit elastic–perfectly-plastic–rigid behavior.     

An entropy model for shear deformation of granular materials

A statistical mechanical analogy for characterization of granular materials is discussed by using such notions as the state of the material, the density of states, entropy, canonical distribution and the partition function. The transition law of states during shear deformations of the material is microscopically investigated in the case of two-dimensional model granular materials. The assumption of entropy growth is shown to characterize the dilatancy of the material. A rough proof is given by assuming the measure preserving property of the transition and showing its ergodicity.     

Optical momentum in negative-index media

The correct form of the energy-momentum tensor in a dielectric has been the subject of a controversy spanning the last hundred years. Three principal forms for the momentum, in particular, have been identified for dispersive media. These are the Abraham, Minkowski and canonical momenta. We investigate whether any of these forms can be distinguished by considering pulses of light incident on materials with a negative refractive index and find that, for such materials, the canonical momentum has a qualitatively different behaviour to the other two.     

Achieving homogeneity in a two-phase Cu–Ag composite during high-pressure torsion

A Cu–8 wt%Ag alloy was processed at room temperature either by high-pressure torsion (HPT) or using a two-step deformation mode of equal-channel angular pressing (ECAP) and HPT. After HPT deformation, different flow patterns were observed on the disk surface without any post-deformation treatment thereby indicating an inhomogeneous shearing deformation. The microhardness distributions throughout the disks were compared after the two different processing routes. It is shown that the microhardness remains very low near the center of the disk although saturated in the outer regions after 10 revolutions. By contrast, an intrinsically homogeneous microhardness may be attained throughout the disk after the two-step deformation of ECAP and HPT. This study suggests a convenient procedure for achieving full homogeneity in high-strength materials during HPT processing.     

Water permeability of engineered cementitious composites

The water permeability of a unique class of high performance fiber reinforced cementitious composites (HPFRCC) called engineered cementitious composites (ECC) is investigated. These composites are deliberately tailored using microcmechanical design principles to exhibit pseudo-strain-hardening characteristics in uniaxial tension, up to greater than 4% strain. While undergoing tensile deformation, microcracks are designed to saturate the specimen rather than localize into large cracks. This tendency to form microcracks, which are experimentally shown to be approximately 60 μm in width, allows ECC material in the cracked state to maintain water permeability similar to that of uncracked concrete or mortar, and magnitudes lower than cracked reinforced mortar or concrete. It is also shown that the self-healing properties of cracks within ECC material significantly aids in reducing the coefficient of permeability of cracked ECC.     

Optimal heating control of thermosensitive bodies under plastic deformation of material

A mathematical statement has been formulated and algorithms for solving one- and two-dimensional problems on fast operation optimal heating control of homogeneous thermosensitive canonical form bodies has been developed. We have determined the control (the surrounding temperature at one of the boundary surfaces or heat flux at one of the boundary surfaces) which in a minimal time carries the body from the initial thermal state to the final one, characterized by the given mean-integral temperature. In addition, restrictions on the control function and the maximal tangential stress intensity or accumulated plastic shear strain have been considered. The elastoplastic deformation of the material has been studied within the framework of general plasticity theory proposed by Il’yushin.     

Periodic dynamics of coupled cell networks II: cyclic symmetry

A coupled cell network is a directed graph whose nodes represent dynamical systems and whose directed edges specify how those systems are coupled to each other. The typical dynamic behaviour of a network is strongly constrained by its topology. Especially important constraints arise from global (group) symmetries and local (groupoid) symmetries. The H/K theorem of Buono and Golubitsky characterises the possible spatio-temporal symmetries of time-periodic states of group-equivariant dynamical systems. A version of this theorem for group-symmetric networks has been proved by Josi? and Török. In networks, spatial symmetries correspond to synchrony of cells, and spatio-temporal symmetries correspond to phase relations between cells. Associated with any coupled cell network is a canonical class of admissible ODEs that respect the network topology. A pattern of synchrony or phase relations in a hyperbolic time-periodic state of such an ODE is rigid if the pattern persists under small admissible perturbations. We characterise rigid patterns of synchrony and rigid phase patterns in coupled cell networks, on the assumption that the periodic state is fully oscillatory (no cell is in equilibrium) and the network has a basic property, the rigid phase property. We conjecture that all networks have the rigid phase property, and that in any path-connected network an admissible ODE with a hyperbolic periodic state can always be perturbed to make the perturbed periodic state fully oscillatory. Our main result states that in any path-connected network with the rigid phase property, every rigid pattern of phase relations can be characterised in two stages. First, sets of cells form synchronous clumps according to a balanced equivalence relation. Second, the corresponding quotient network has a cyclic group of automorphisms, and the phase relations are induced by associating a fixed phase shift with a generator of this group. Thus the clumps of synchronous cells form a discrete rotating wave. As a corollary, we prove an analogue of the H/K theorem for any path-connected network. We also discuss the non-path-connected case.