Minimal surfaces and conservation laws for bidimensional structures

We discuss conservation laws for thin structures which could be modeled as a material minimal surface, i.e., a surface with zero mean curvatures. The models of an elastic membrane and micropolar (six-parameter) shell undergoing finite deformations are considered. We show that for a minimal surface, it is possible to formulate a conservation law similar to three-dimensional non-linear elasticity. It brings us a path-independent J-integral which could be used in mechanics of fracture. So, the class of minimal surfaces extends significantly a possible geometry of two-dimensional structures which possess conservation laws

A Micropolar Peridynamic Theory in Linear Elasticity

A state-based micropolar peridynamic theory for linear elastic solids is proposed. The main motivation is to introduce additional micro-rotational degrees of freedom to each material point and thus naturally bring in the physically relevant material length scale parameters into peridynamics. Non-ordinary type modeling via constitutive correspondence is adopted here to define the micropolar peridynamic material. Along with a general three dimensional model, homogenized one dimensional Timoshenko type beam models for both the proposed micropolar and the standard non-polar peridynamic variants are derived. The efficacy of the proposed models in analyzing continua with length scale effects is established via numerical simulations of a few beam and plane-stress problems

A MORE COMPLETE THERMODYNAMIC FRAMEWORK FOR SOLID CONTINUA

The Jacobian of deformation at a material point can be decomposed into the stretch tensor and the rotation tensor. Thus, varying Jacobians of deformation at the neighboring material points in the deforming volume of solid continua would yield varying stretch and rotation tensors at the material points. Measures of strain, such as Green’s strain, at a material point are purely a function of the stretch tensor, i.e. the rotation tensor plays no role in these measures. Alternatively, we could also consider decomposition of displacement gradient tensor into symmetric and skew symmetric tensors. Skew symmetric tensor is also a measure of pure rotations whereas symmetric tensor is a measure of strains, i.e. stretches. The measures of rotations in these two approaches describe the same physics but are in different forms. Polar decomposition gives the rotation matrix and not the rotation angles whereas the skew symmetric part of the displacement gradient tensor yields rotation angles that are explicitly and conveniently defined in terms of the displacement gradients. The varying rotations and rotation rates arise in all deforming solid continua due to varying deformation of the continua at neighboring material points, hence are internal to the volume of solid continua and are explicitly defined by the deformation, therefore do not require additional degrees of freedom to define them. If the internal varying rotations and their rates are resisted by the continua, then there must exist internal moments corresponding to these. The internal rotations and their rates and the corresponding moments can result in additional energy storage and dissipation. This physics is all internal to the deforming continua (hence does not require consideration of additional external degrees of freedom and associated external moments) and is neglected in the presently used continuum theories for isotropic, homogeneous solid continua. The continuum theory presented in this paper considers internal varying rotations and associated conjugate moments in the derivation of the conservation and balance laws, thus the theory presented in this paper is “a polar theory for solid continua” but is different than the micropolar theories published currently in which material points have six external degrees of freedom i.e. rotations are additional external degrees of freedom. This polar continuum theory only accounts for internal rotations and associated moments that exist as a consequence of deformation but are neglected in the present theories. We call this theory “a polar continuum theory” as it considers rotations and moments as conjugate pairs in a deforming solid continua though these are internal, hence are purely related to the deformation of the solid. It is shown that the polar continuum theory presented in this paper is not the same as the strain gradient theories reported in the literature. The differences are obviously in terms of the physics described by them and the mathematical details associated with conservation and balance laws. In this paper, we only consider polar continuum theory for small deformation and small strain. This polar continuum theory presented here is a more complete thermodynamic framework as it accounts for additional physics of internally varying rotations that is neglected in the currently used thermodynamic framework. This thermodynamic framework is suitable for isotropic, homogeneous solid matter such as thermoelastic and thermoviscoelastic solid continua with and without memory when the deformation is small. The paper also presents preliminary material helpful in consideration of the constitutive theories for polar continua

Mode III interfacial crack in the presence of couple stress elastic materials

In this paper we are concerned with the problem of a crack lying at the interface between dissimilar materials with microstructure undergoing antiplane deformations. The micropolar behaviour of the materials is described by the theory of couple stress elasticity developed by Koiter (1964). This constitutive model includes the characteristic lengths in bending and torsion and thus it is able to account for the underlying microstructure of the two materials. We perform an asymptotic analysis to investigate the behaviour of the solution near the crack tip. It turns out that the stress singularity at the crack tip is strongly influenced by the microstructural parameters and it may or may not show oscillatory behaviour depending on the ratio between the characteristic lengths.Comment: 19 pages, 3 figur

Fractional Calculus for Continuum Mechanics - anisotropic non-locality

In this paper the generalisation of previous author's formulation of fractional continuum mechanics to the case of anisotropic non-locality is presented. The considerations include the review of competitive formulations available in literature. The overall concept bases on the fractional deformation gradient which is non-local, as a consequence of fractional derivative definition. The main advantage of the proposed formulation is its analogical structure to the general framework of classical continuum mechanics. In this sense, it allows, to give similar physical and geometrical meaning of introduced objects

Nonlinear Elasticity, Fluctuations and Heterogeneity of Nematic Elastomers

Liquid crystal elastomers realize a fascinating new form of soft matter that is a composite of a conventional crosslinked polymer gel (rubber) and a liquid crystal. These {\em solid} liquid crystal amalgams, quite similarly to their (conventional, fluid) liquid crystal counterparts, can spontaneously partially break translational and/or orientational symmetries, accompanied by novel soft Goldstone modes. As a consequence, these materials can exhibit unconventional elasticity characterized by symmetry-enforced vanishing of some elastic moduli. Thus, a proper description of such solids requires an essential modification of the classical elasticity theory. In this work, we develop a {\em rotationally invariant}, {\em nonlinear} theory of elasticity for the nematic phase of ideal liquid crystal elastomers. We show that it is characterized by soft modes, corresponding to a combination of long wavelength shear deformations of the solid network and rotations of the nematic director field. We study thermal fluctuations of these soft modes in the presence of network heterogeneities and show that they lead to a large variety of anomalous elastic properties, such as singular length-scale dependent shear elastic moduli, a divergent elastic constant for splay distortion of the nematic director, long-scale incompressibility, universal Poisson ratios and a nonlinear stress-strain relation fo arbitrary small strains. These long-scale elastic properties are {\em universal}, controlled by a nontrivial zero-temperature fixed point and constitute a qualitative breakdown of the classical elasticity theory in nematic elastomers. Thus, nematic elastomers realize a stable ``critical phase'', characterized by universal power-law correlations, akin to a critical point of a continuous phase transition, but extending over an entire phase.Comment: 61 pages, 24 eps pages, submitted to Annals of Physic