Surface phenomena of gradient materials

The behavior of third gradient materials is analyzed. They possess stress tensor fields of second, third and fourth order. Starting from the principle of virtual power, we derive the admissible boundary conditions. Those on free surfaces can only be obtained by the application of the divergence theorem of surfaces. On the other hand, such an application to fictitious internal cuts makes no sense although it is usually practiced. We prove that some of the boundary conditions on a free surface may be interpreted as the equilibrium conditions of a shell. So a crust shell exists on such a surface and a beam exists where patches of the surface meet. On the other hand, no such shells or beams can be found with fictitious surfaces in the interior of a continuum. Our finding does not depend on any specific constitutive assumption.

The Influence of Dimensionality on the Rate of Diffusive Escape From an Energy Well

A commonly used idealization when describing separation of a chemical bond between molecules is that of an energy well which prescribes the dependence of energy of interaction between the molecules in terms of a reaction coordinate. The energy difference between the peak to be overcome and the root of the well is the so-called activation energy, and the overall shape of the well dictates the kinetics of separation through a constitutive assumption concerning transport. An assumption tacit in this description is that the state of the bond evolves with only a single degree of freedom—the reaction coordinate—as the system explores its energy environment under random thermal excitation. In this discussion we will consider several bonds described by one and the same energy profile. The cases differ in that the energy profile varies along a line extending from the root of the well in the first case, along any radial line in a plane extending from the root of the well in a second case, and along any radial line in space extending from the root of the well in a third case. To focus the discussion we determine the statistical rate of escape of states from the well in each case, requiring that the profile of the well is the same in all three cases. It is found that the rates of escape each depend exponentially on the depth of the well but that the coefficients of the exponential vary with depth of the well differently in the three cases considered.

A direct method for the determination of the mean orientation-dependent elastic strains and stresses in polycrystalline materials from strain pole figures

A salient manifestation of anisotropy in the mechanical response of polycrystalline materials is the inhomogeneous partitioning of elastic strains over the aggregate. For bulk samples, the distributions of these intergranular strains are expected to have a strong functional dependence on grain orientations. It is then useful to formulate a mean lattice strain distribution function (LSDF) over the orientation space, which serves to characterize the micromechanical state of the aggregate. Orientation-dependent intergranular stresses may be recovered from the LSDFviaa constitutive assumption, such as anisotropic linear elasticity. While the LSDF may be determined directly from simulation data, its experimental determination relies on solving an inverse problem that is similar in character to the fundamental problem of texture analysis. In this paper, a versatile and robust direct method for determining an LSDF from strain pole figures is presented. The effectiveness of this method is demonstrated using synthetic strain pole figures from a model LSDF obtained from the simulated uniaxial deformation of a 1000-crystal aggregate.

Forum: The hierarchicalization of political community

To respond to Andrew Linklater's The Transformation of Political Community is to walk a very fine line between admiration and perplexity. I have admiration especially for his willingness to confront some of the most important and difficult questions of contemporary political theory. To pursue the claim that we need to rethink the accounts of political community that anyone with credibility as a political analyst would much prefer to leave as a constitutive assumption of almost everything they want to say, takes some courage. And this is certainly a bold book, both in its sweep and its ambitions. It is one of the very few books in the contemporary theory of international relations that provokes more than passive surrender to the tediously familiar and the three-fold typology. Even so, I remain perplexed at the way Linklater manages to keep retying himself up in all the knots he tries so vigorously to disentangle.

On homotopy conditions and the existence of multiple equilibria in finite elasticity

In this paper we study homotopy classes of deformations and their properties under weak convergence. As an application, we give an analytic proof (in two and three dimensions) of the existence of infinitely many local minimisers for a displacement boundary-value problem from finite elasticity, posed on a nonconvex domain, under the constitutive assumption of polyconvexity.

A continuum approach for modelling induced anisotropy in glaciers and ice sheets

This paper presents a formulation of a continuum model for so-called (stress or deformation) induced anisotropy in natural ice which, unlike computer-based Taylor-type models, can be incorporated in numerical simulations of large ice masses to account for the effects of this process on the flow of these bodies in a physical fashion. To do this, we treat natural ice as a rigid-elastic, non-linear inelastic material which can develop transverse isotropic behaviour (accounting for the simplest kind of induced anisotropy in natural ice masses), where the degree of such anisotropy at each point is controlled by the distribution of crystal glide-plane orientations there. This distribution is described by a so-called orientation-distribution function, for which an evolution relation can be derived. The central constitutive assumption of this formulation relates this distribution to the “structure” tensor representing the transverse isotropy of the material. On the basis of this relation, the model predicts in particular isotropic (e.g. classical Glen’s flow-law type) behaviour at a given point when the distribution of crystal glide-plane orientations is uniform there.

A continuum approach for modelling induced anisotropy in glaciers and ice sheets

This paper presents a formulation of a continuum model for so-called (stress or deformation) induced anisotropy in natural ice which, unlike computer-based Taylor-type models, can be incorporated in numerical simulations of large ice masses to account for the effects of this process on the flow of these bodies in a physical fashion. To do this, we treat natural ice as a rigid-elastic, non-linear inelastic material which can develop transverse isotropic behaviour (accounting for the simplest kind of induced anisotropy in natural ice masses), where the degree of such anisotropy at each point is controlled by the distribution of crystal glide-plane orientations there. This distribution is described by a so-called orientation-distribution function, for which an evolution relation can be derived. The central constitutive assumption of this formulation relates this distribution to the “structure” tensor representing the transverse isotropy of the material. On the basis of this relation, the model predicts in particular isotropic (e.g. classical Glen’s flow-law type) behaviour at a given point when the distribution of crystal glide-plane orientations is uniform there.

The overall constitutive behaviour of nonlinearly elastic composites

This work uses the general framework of Hill for the definition of overall mechanical properties of composite materials undergoing large deformations, together with Ball’s constitutive assumption of polyconvexity, and ideas based on a Hashin–Shtrikman variational structure developed recently by Talbot & Willis for nonlinear problems, to obtain rigorous first-order (depending only on the initial volume fractions of the phases) and second-order (depending also on the overall isotropy of the composite) bounds on the overall properties of composite materials with nonlinearly elastic phases. Although the proposed first-order upper bound, or Voigt bound, agrees with a previous result, the corresponding lower bound, or Reuss bound, is significantly tighter than the previously available result. The proposed second-order bounds are completely new, and are of course tighter than the first-order bounds. Additionally, a ‘self-consistent’ estimate is given for the case when the phases are neo-hookean.