Detailed derivations of the Legendre-Hadamard necessary conditions for energy-minimizing states of fiber-reinforced three-dimensional solids and two-dimensional shells are presented. The underlying conceptual framework is Cosserat elasticity theory in which the Cosserat rotation field controls the orientation of the embedded fibers. This is partially coupled to the continuum deformation gradient by the requirement that the fibers convect as material curves with respect to the matrix material in which they are embedded. The conditions obtained combine the effects of deformation and rotation and subsume previously obtained decoupled inequalities involving these effects separately.

This chapter surveys fundamental aspects of the modern theory of finite elastoplasticity. We emphasize the now-ubiquitous decomposition of the deformation into elastic and plastic parts, the central roles played by dissipation and material symmetry, and a framework for the modeling of scale-dependent work hardening in crystalline and isotropic materials.

We present a model for incremental deformations of an elastic solid reinforced by a single family offibers that offer resistance to extension, flexure, and torsion. The theory is cast in the setting of small-on-large deformations and provides a framework for the multiscale analysis of bifurcation of equilibria in fi brous composites. The model is based on a theory of three-dimensional Cosserat elasticity in which fi ber kinematics are controlled by a rotation fi eld that is weakly coupled to the bulk deformation through a pointwise fiber-materiality constraint. Fiber-matrix interaction forces are explicitly accounted for via the attendant Lagrange multipliers. We demonstrate the utility of the model by investigating the onset of bifurcation in an incompressible fiber-reinforced elastic half-plane. In particular, we study the influence of axial fiber stiffness, flexural stiffness, and fiber - matrix interaction forces on planar buckling modes. We envisage a model for the study of buckling problems of biological and industrial relevance with a view to gaining better insight into the roles offiber bending, twisting, and fiber - matrix interaction forces in regulating the buckling of fibrous composites.

We present an ordinary state-based formulation of the equation of motion of an elastic peridynamic solid, where the response function state is derived from a free energy functional that depends nonlinearly on measures of both stretch and volume changes and contains two peridynamic coefficients. To determine these coefficients, we assume that the quadratic form of this functional near the undistorted natural state of the peridynamic material corresponds to the quadratic strain energy of the classical linear elasticity at an interior point of the peridynamic solid. Next, we consider a particular class of homogeneous peridynamic cylindrical solids and expand the displacement field asymptotically in terms of the thickness coordinate, following classical arguments of thin plate theory. We then reduce the equation of motion to a system of equations for the determination of the expansion coefficients.

The relatively new field of “gradient plasticity” is concerned with the modeling of length scale effects observed in plastically deformed solids. The older subject of inhomogeneity in materially uniform bodies, inaugurated by Noll, offers the possibility of constructing such models in terms of inhomogeneity fields. In the case of materially uniform crystalline solids, for example, inhomogeneity is described by the torsion tensor—equivalently, the “geometrically necessary” dislocation density—induced by plastic deformation. This is the unique descriptor of length-scale effects modeled by constitutive functions involving both the plastic deformation and its gradient. In isotropic solids, inhomogeneity is characterized by the Riemann tensor induced by the plastic metric. This involves the plastic deformation together with its first and second gradients. However, in the case of isotropy, it is an open question as to whether or not inhomogeneity furnishes a complete model of length scale effects described by functions of the plastic deformation and its gradients. In this note, we show that such functions are ultimately expressible in terms of the Riemann tensor alone, and hence, that scale effects are modeled by these functions via the inhomogeneity field.

We present a model of the equilibrium response of a pressurized circular cylindrical elastic tube, reinforced by two families of continuously distributed spatial rods arranged as right- and left-handed helices. The framework underlying our analysis is a modified version of Cosserat elasticity in which the basic kinematic descriptors are a single deformation field and two rotation fields, one for each fiber family, subject to the constraint that the fibers convect with the deformation as material curves. The effect of the intrinsic and torsional stiffnesses of the fibers on the overall force-extension response of the cylinder is quantified.

A concise derivation of Kirchhoff's theory for naturally curved and twisted rods is presented as a prelude to the derivation of the theory for the elastic response of rods that have undergone prior plastic deformation.

Convexity conditions of the Legendre–Hadamard and quasiconvexity type are derived in the context of a theory for fiber-reinforced shells based on Cosserat elasticity. These furnish two-dimensional shell-theoretic versions of their three-dimensional counterparts.

Necessary conditions of the Legendre–Hadamard type for energy minimizers are derived in the setting of a special Cosserat elasticity theory for fiber-reinforced solids. These extend and generalize earlier results pertaining to fiber flexure-twist strain vectors formed from the Cosserat wryness tensors. New necessary conditions involving the associated Cosserat deformation tensors are also derived.

Abstract In this chapter the contents of Chapter 3 are applied to the description of continuum mechanics using convected coordinates to identify material points. This unifies various concepts in continuum mechanics with other branches of the physical sciences, and facilitates understanding of the crucial concept of strain compatibility.

Abstract The decomposition of deformation into elastic and plastic parts is introduced in terms of transformations of the material symmetry group, due to plastic evolution, pertaining to a local undistorted state of the material. This leads naturally to the notion of dislocation density, interpreted as the torsion of the underlying material manifold. Torsion, in turn, implies that the elastic strain is incompatible, and hence that the material is residually stressed when unloaded.

In this chapter, we describe the deformation of continua and define the strain and stress tensors. Then, we review the main results of the differential geometry of surfaces in the Euclidean space.

We present a theory for thin elastic plates reinforced by a single family of continuously distributed fibers. The novelty of the model is the assignment of intrinsic flexural and torsional elasticity to the embedded fibers, regarded as material curves. Our derivation is based on a dimension reduction from the three-dimensional theory to obtain a plate model valid to leading order in the small plate thickness.

Abstract The small-deformation theory of elastic-plastic response, due to Prandtl and Reuss, is the basis of most analytical treatments of elastic-plastic behavior. This model is examined from a modern perspective. The shifter concept is combined with the convected coordinate formalism to cast the theory in a manner that ensures its invariance under a change of frame, in contrast to conventional expositions of the small-deformation theories of elasticity and elasto-plastic response

A continuum theory of pantographic lattices, based on second-grade elasticity, is presented. The proposed model is able to describe the mechanical behavior of a type of material structure made up of multiple layers of pantographic sheets connected with a third family of fibers. Thus, these materials are characterized by an orthogonal pattern of fibers that can bend, stretch and twist. Numerical experiments illustrate the predictive potential of the model when the material is subjected to different types of mechanical loads, including compression, torsion and two kinds of bending. Analyzing the material responses for these various tests makes it possible to reveal unusual deformation patterns characteristic of such “pantographic blocks.” Numerical simulations using the finite element method are intended to assist in designing an experimental program using 3D-printed specimens made of different materials.