research

This paper presents some developments related to the idea of covariance in elasticity. The geometric point of view in continuum mechanics is briefly reviewed. Building on this, regarding the reference configuration and the ambient space as Riemannian manifolds with their own metrics, a Lagrangian field theory of elastic bodies with evolving reference configurations is developed. It is shown that even in this general setting, the Euler-Lagrange equations resulting from horizontal (referential) variations are equivalent to those resulting from vertical (spatial) variations. The classical Green-Naghdi-Rivilin theorem is revisited and a material version of it is discussed. It is shown that energy balance, in general, cannot be invariant under isometries of the reference configuration, which in this case is identified with a subset of R^3. Transformation properties of balance of energy under rigid translations and rotations of the reference configuration is obtained. The spatial covariant theory of elasticity is also revisited. The transformation of balance of energy under an arbitrary diffeomorphism of the reference configuration is obtained and it is shown that some nonstandard terms appear in the transformed balance of energy. Then conditions under which energy balance is materially covariant are obtained. It is seen that material covariance of energy balance is equivalent to conservation of mass, isotropy, material Doyle-Ericksen formula and an extra condition that we call ‘configurational inviscidity’. In the last part of the paper, the connection between Noether’s theorem and covariance is investigated. It is shown that the Doyle-Ericksen formula can be obtained as a consequence of spatial covariance of Lagrangian density. Similarly, it is shown that the material Doyle-Ericksen formula can be obtained from material covariance of Lagrangian density.

I would like to share some of our more recent findings on nano-pillar compression, namely the role of the surface treatment in plastic deformation at the nano-scale. Recent advances in the 2-beam focused ion beams technology (FIB) have enabled researchers to not only perform high-precision nanolithography and micro-machining, but also to apply these novel fabrication techniques to investigating a broad range of materials' properties at the sub-micron and nano-scales. In our work, the FIB is utilized in manufacturing of sub-micron cylinders, or nano-pillars, as well as of TEM cross-sections to directly investigate plasticity of metals at these small length scales. Single crystal nano-pillars, ranging in diameter between 300 nm and 870 nm, were fabricated in the FIB from epitaxial gold films on MgO substrates and subsequently compressed using a Nanoindenter fitted with a custom-fabricated diamond flat punch. We show convincingly that flow stresses strongly depend on the sample size, as some of our smaller specimens were found to plastically deform in uniaxial compression at stresses as high as 600 MPa, a value ~25 times higher than for bulk gold. We believe that these high strengths are hardened by dislocation starvation. In this mechanism, once the sample is small enough, the mobile dislocations have a higher probability of annihilating at a nearby free surface than of multiplying and being pinned by other dislocations. Contrary to this, if the dislocations are trapped inside the specimen by a coating, the strengthening mechanism is expected to be different. Here we present for the first time the comparison of plastic deformation of passivated and unpassivated single crystal specimens at the sub-micron scale. The role of free surfaces is investigated by comparing stress results of both as-FIB'd, annealed, and alumina-passivated pillars. Preliminary results show that ALD-coated pillars exhibit much higher flow stresses at equivalent sizes and strains compared with the uncoated samples. We also found that while FIB damage during pillar fabrication might account for a small portion of the strength increase, it is not the major contributor.

Singular elastic stress fields are generally developed at sharp re-entrant corners and at the end of bonded interfaces between dissimilar elastic materials. This behaviour can present difficulties in both analytical and numerical solution of such problems. For example, excessive mesh refinement might be needed in a finite element solution.

This paper develops a theory of anharmonic lattice statics for the analysis of defective complex lattices. This theory differs from the classical treatments of defects in lattice statics in that it does not rely on harmonic and homogeneous force constants. Instead, it starts with an interatomic potential, possibly with in¯nite range as appropriate for situations with electrostatics, and calculates the equilibrium states of defects. In particular, the present theory accounts for the differences in the force constants near defects and in the bulk. The present formulation reduces the analysis of defective crystals to the solution of a system of nonlinear difference equations with appropriate boundary conditions. A harmonic problem is obtained by linearizing the nonlinear equations, and a method for obtaining analytical solutions is described in situations where one can exploit symmetry. It is then extended to the anharmonic problem using modified Newton-Raphson iteration. The method is demonstrated for model problems motivated by domain walls in ferroelectric materials.

J.R.Barber

The contact of rough surfaces

Surfaces are rough on the microscopic scale, so contact is restricted to a few `actual contact areas'. If a current flows between two contacting bodies, it has to pass through these areas, causing an electrical contact resistance. The problem can be seen as analogous to a large number of people trying to get out of a hall through a small number of doors.

Classical treatments of the problem are mostly based on the approximation of the surfaces as a set of `asperities' of idealized shape. The real surfaces are represented as a statistical distribution of such asperities with height above some datum surface. However, modern measurement techniques have shown surfaces have multiscale, quasi-fractal characteristics over a wide range of length scales. This makes it difficult to decide on what scale to define the asperities.

The Division of Civil, Mechanical and Manufacturing Innovation (CMMI) of NSF and the DOE Office of Freedom CAR and Vehicle Technologies intend to co-sponsor proposals addressing fundamental research issues in advanced high strength steels (AHSS). Specifically, proposals focused on

1. AHSS materials development and characterization,
2. predictive modeling that integrates AHSS material structure and product performance, and
3. fundamental research in the area of processing and manufacturing of AHSS, are of interest. This collaborative effort is a direct outcome of the Advanced High Strength Steel Workshop.

Interested PIs should consider submitting an unsolicited proposal to the core programs of the CMMI Division namely, (1) Materials Processing & Manufacturing (MPM), (2) Materials Design & Surface Engineering (MDSE), (3) Applications & Structural Mechanics, or (4) Mechanics & Structures of Materials (MSM), during the January 15, 2007 to February 15, 2007 submission window. Unsolicited proposals in response to this letter should have titles beginning with "AHSS:".  Proposals from the March-April 2007 panel review will be eligible for co-funding, pending availability of funds.

Electro-sensitive (ES) elastomers form a class of smart materials whose mechanical properties can be changed rapidly by the application of an electric field. These materials have attracted considerable interest recently because of their potential for providing relatively cheap and light replacements for mechanical devices, such as actuators, and also for the development of artificial muscles. In this paper we are concerned with a theoretical framework for the analysis of boundary-value problems that underpin the applications of the associated electromechanical interactions. We confine attention to the static situation and first summarize the governing equations for a solid material capable of large electroelastic deformations. The general constitutive laws for the Cauchy stress tensor and the electric field vectors for an isotropic electroelastic material are developed in a compact form following recent work by the authors. The equations are then applied, in the case of an incompressible material, to the solution of a number of representative boundary-value problems. Specifically, we consider the influence of a radial electric field on the azimuthal shear response of a thick-walled circular cylindrical tube, the extension and inflation characteristics of the same tube under either a radial or an axial electric field (or both fields combined), and the effect of a radial field on the deformation of an internally pressurized spherical shell.