David J. Steigmann

Tension-Field Theory

A general theory of the tension field is developed for application to the analysis of wrinkling in isotropic elastic membranes undergoing finite deformations. The principal contribution is a partial differential equation describing a geometrical property of tension trajectories. This is one of a system of two equations which describes the state of stress independently of the deformation. This system is strongly elliptic at any stable solution, whereas the deformation is described by a system of parabolic type. Controllable solutions, i. e. those states that can be maintained in any isotropic elastic material by application of edge tractions and lateral pressure alone, are obtained. The general axisymmetric problem is solved implicitly and the theory is applied to the solution of two representative examples. Existing small strain theories are shown to correspond to a singular limit of the general theory, at which the underlying system changes from elliptic to parabolic type.

ELASTIC SURFACE-SUBSTRATE INTERACTIONS

A theory for three–dimensional finite deformations of elastic solids with conforming elastic films attached to their bounding surfaces is described. The Gurtin–Murdoch theory incorporating elastic resistance of the film to strain is generalized to account for the effects of intrinsic flexural resistance. This modification yields a model that can be used to describe equilibrium deformations in the presence of compressive–surface stress fields. An associated variational theory is given and material symmetry considerations are discussed. The theory is illustrated by examples.

Plane deformations of elastic solids with intrinsic boundary elasticity

In this paper, a nonlinear theory of elastic boundary coating (or reinforcement) of an elastic solid is developed for plane strain deformations. The coating consists of a material curve endowed with intrinsic elastic properties associated with extensibility and bending stiffness bonded to part, or all, of the bounding curve of the elastic body. The equations describing the equilibrium of the coated body when subject to finite deformation are derived using a variational method. The incremental equations describing a small departure from an equilibrium configuration are then derived and used to investigate the stability of a deformed configuration and the possibility of bifurcation. The theory is applied to the analysis of the equilibrium of a finitely deformed half-plane consisting of compressible elastic material coated along its edge. The influence of the coating on the bifurcation behaviour of the half–plane is assessed against known results for an uncoated half–plane. Numerical results are used to illustrate the influence of certain material parameters on bifurcation.

Analysis of partly wrinkled membranes by the method of dynamic relaxation

A version of the method of dynamic relaxation is developed to analyze equilibrium configurations of partly wrinkled membranes. In this method equilibria are regarded as long time limits of a damped dynamical problem. The membrane theory considered is based on the concept of a relaxed strain energy function that automatically incorporates the effects of wrinkling. For neo-Hookean materials, existence theorems of nonlinear elasticity are used to show that the relaxed potential energy possesses minimizers in a certain function space. Moreover, solutions of the equilibrium equations furnish global minima of the energy, for certain classes of boundary data. Such deformations are automatically stable according to the minimum-energy criterion. This result motivates the search for solutions of the equilibrium equations, although the existence theory does not guarantee that energy minimizers possess the degree of regularity required by these equations. Several examples of two-and three-dimensional deformations are presented.

Synthesis of Fibrous Complex Structures: Designing Microstructure to Deliver Targeted Macroscale Response

In Mechanics, material properties are most often regarded as being given, and based on this, many technical solutions are usually conceived and constructed. However, nowadays manufacturing processes have advanced to the point that metamaterials having selected properties can be designed and fabricated. Three-dimensional printing, electrospinning, self-assembly, and many other advanced manufacturing techniques are raising a number of scientific questions which must be addressed if the potential of these new technologies is to be fully realized. In this work, we report on the status of modeling and analysis of metamaterials exhibiting a rich and varied macroscopic response conferred by complex microstructures and particularly focus on strongly interacting inextensible or nearly inextensible fibers. The principal aim is to furnish a framework in which the mechanics of 3D rapid prototyping of microstructured lattices and fabrics can be clearly understood and exploited. Moreover, several-related open questions will be identified and discussed, and some methodological considerations of general interest are provided.

On the variational theory of cell-membrane equilibria

The equivalence of two approaches to the variational theory of cell-membrane equilibria which have been proposed in the literature is demonstrated. Both assume a constraint on surface area, global in one formulation and local in the alternative, in accordance with measurements which reveal negligible surface dilation in the presence of membrane deformation. We thus address a potential controversy in the mathematical modeling of an important problem in biophysics.

Equilibrium of Elastic Nets

A general equilibrium theory for nets constructed from two families of perfectly flexible elastic fibres is presented. The fibres are assumed to be continuously distributed and to offer negligible resistance to shear distortion. Configurations of nets are shown to be minimizers of the potential energy of deformation only if the associated fibre stretches are points of convexity of the fibre strain energy functions and the stresses in the fibres are tensile. These results are used to construct a relaxed energy density that automatically accounts for wrinkling of the network. Universal solutions are obtained. These are the deformations that can be maintained in every elastic net by the application of edge tractions and lateral pressure alone. A detailed study of the differential geometry of nets is included to aid in their interpretation. The equilibrium theory for half-slack (wrinkled) nets is developed and applied to the solution of some representative examples.

Coupled deformations of elastic curves and surfaces

An equilibrium theory for the coupled finite deformations of elastic curves and surfaces is described. Possible wrinkling of the curve or surface is taken into account by using a relaxed version of the theory obtained from minimum energy considerations. The relaxed theory admits a dual formulation leading to extremum principles and uniqueness of the equilibrium stress distribution. A number of examples are treated using spatial finite differences together with a dynamic relaxation method in which equilibrium configurations are obtained in the long-time limit of a damped dynamical problem.

Variational theory for spatial rods

The simplest theory of spatial rods is presented in a variational setting and certain necessary conditions for minimizers of the potential energy are derived. These include the Weierstrass and Legendre inequalities, which require that the vector describing curvature and twist belong to a domain of convexity of the strain energy function.

Frame-Invariant Polyconvex Strain-Energy Functions for Some Anisotropic Solids

We present a set of simple sufficient conditions for the polyconvexity and coercivity of strainenergy functions for transversely isotropic and orthotropic elastic solids. The formulation is based on appropriate function bases for the right stretch tensor in the polar decomposition of the deformation gradient and furnishes numerical analysts with a priori existence criteria for boundary-value problems.