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The notion of material symmetry has been formalised by W. Noll and Truesdell’s school of continuum mechanics, and is now explained in many classical textbooks. Mostly, attention has been confined to simple materials. Applying the notion to shells requires to consider second-grade continua and is more concealed. Attempts have been done among others by Murdoch and Cohen, Steigmann and Ogden, Steigmann. Here I present an independent approach for hyper-elastic shells with second-gradient elasticity. The equation that characterizes the material symmetries is deduced and, by way of illustration, the case of a shell with unstressed reference configuration is discussed.

Recent studies suggest that cells routinely probe their mechanical environments and that this mechanosensitive behavior regulates some of their cellular activities. The finite elasticity theory of small-on-large deformation (SoL) has been shown to be effective in interpreting the mechanosensitive behavior of cells on a substrate that has been subjected to a prior large static stretch before the culturing of the cells. Small on large deformation is the superposition of a small deformation onto a prior large deformation that serves as the new reference configuration. This article aims to refine SoL theory to develop a theoretical framework for improved physical interpretation of mechanosensing. Given the initial large deformation in SoL, the stress generated by the small deformation is linearized, and the linearized elasticity tensor is taken to be a significant factor facilitating prediction of cellular behavior. The pre-stretch is shown to produce direction-based, effective elastic moduli for cellular mechanosensing. The utility of this SoL theory is illustrated by culturing of two different cell types grown on uniaxially pre-stretched surfaces that induce changes to the cell orientation and behavior.

A chemically reacting mixture of elastic solids is considered. As a constitutive assumption, the peculiar functions (such as the free energy, the entropy, and the stress) of a constituent are taken to be functions of a set of variables pertaining to that constituent. The interaction terms, namely the growth of mass, linear momentum, and energy, are allowed to depend on the set of variables pertaining to all of the constituents. While the dependence on the mass density is usually disregarded, the paper accounts also for such a dependence, which seems to be in order especially in connection with reacting mixtures where the mass densities change also in the reference configuration. The thermodynamic restrictions are derived by starting from the non-negative value of the sum of entropy growths and involving the properties of the peculiar functions. The results so obtained for stresses and chemical potentials are examined in connection with similar schemes (swelling solids). While the correct relations for the mass diffusion flux arise from balance equations, an analysis is given of whether and how Fick-type models are acceptable possibly depending on the fluid or solid character of the mixture.

The conventional theory of linearized elastodynamics addresses the case of motions that have small displacement gradients with respect to a reference configuration of the elastic body that is unstressed and at rest. Here, we develop a theory of much wider applicability in which the linearization is with respect to a reference configuration that is in rigid motion.

In this note we study the condition of strong ellipticity under changes in the current and reference configuration for the finite hyperelastostatic case. The outcome is that strong ellipticity is preserved provided one adjusts the vectors used in the definition of this condition accordingly.

Restrictions imposed on stress response functions by objectivity and material symmetry are usually obtained by appeal to both a reference configuration and a response function common to all observers. Here these restrictions are obtained by regarding the reference configuration to be observer-dependent in a natural manner, and without the need to assume the existence of a common response function. Remarks are made on the distinction which should be drawn between the notions of observer transformation and rigid deformation, the consequences of this distinction for the correct interpretation of ‘invariance under superposed rigid motions’, and on the domains of response functions.

We show a necessary and sufficient condition for the undistorted reference configuration y(x)=x, with ∇y=F=1, to be a minimizer of the total stored energy for an isotropic elastic body. Polyconvexity of the stored energy function is not sufficient, and we give an example which possesses two distinct natural (i.e., unstressed) states to illustrate this point.

A body is composed of a homogeneous elastic material if there is a configuration of the body relative to which the Cauchy stress-response function does not depend upon position in that configuration. In this paper, necessary and sufficient conditions for the existence of such a configuration are deduced. These are expressed as restrictions on the stress-response function associated with a single reference configuration. In particular, it is shown that a body that possesses a stress-free reference configuration and has an associatedinvertible (but position-dependent) elasticity four tensor that governs the response of this material at infinitesimal deformations is comprised of anin homogeneous material.

Non-homogeneous conformal deformations are shown to be possible in an isotropic elastic material in the absence of body forces if and only if the material satisfies a certain condition which renders it incapable of obeying the classical pressure-compression inequality. The undistorted states of materials in this class (which are obtained by subjecting an undistorted state called reference configuration to all possible conformal deformations) are shown to be at best neutrally stable when subject to hydrostatic loading everywhere on the boundary.

Let ℰ be an ellipsoid in ℝ3 contained in a region Ω. Suppose one body occupies the region Ω−ℰ in a certain stress-free reference configuration while a second body, the inclusion, occupies the region ℰ in a stress-free reference configuration. Assume the inclusion is free to slip at ∂ℰ. Now suppose that by changing some variable such as the temperature, pressure, humidity, etc., we cause the trivial deformation y(x)=x of the inclusion to become unstable relative to some other deformation. For example, the inclusion may be made out of such a material that if it were removed from the body, it would suddenly change shape to another stress-free configuration specified by a deformation y=Fx, FΥF=C, C being a fixed tensor characteristic of the material, at a certain temperature. However, with an appropriate material model for the surrounding body, we expect it will resist the transformation, and both body and inclusion will end up stressed.