Conditions for the onset of elastic and material instabilities in hyperelastic materials

Several constitutive inequalities have been proposed in the literature to quantify the notion that ‘stress increases with strain’ in an elastic material. Due to some inherent shortcomings in them, which we discuss, we propose a new tensorial criterion for isotropic materials. We also present necessary conditions in terms of elasticity tensors for the onset of elastic instabilities.

Note on modelling directionality in piezoresistivity

When computing the change in electrical resistivity of a piezoresistive cubic material embedded in a deforming structure, the piezoresistive and the stress tensors should be in the same coordinate system. While the stress tensor is usually calculated in a coordinate system aligned with the principal axes of a regular structure, the specified piezoresistive coefficients may not be in that coordinate system. For instance, piezoresistive coefficients are usually given in an orthogonal cartesian coordinate system aligned with the <100> crystallographic directions and designers sometimes deliberately orient a crystallographic direction other than <100> along the principal directions of the structure to increase the gauge factor. In such structures, it is advantageous to calculate the piezoresistivity tensor in the coordinate system along which the stress tensors are known rather than the other way around. This is because the transformation of stress will have to be done at every point in the structure but piezoresistivity tensor needs to be transformed only once. Here, using tensor transformation relations, we show how to calculate the piezoresistive tensor along any arbitrary Cartesian coordinate system from the piezoresistive coefficients for the <100> coordinate system. Some of the software packages that simulate the piezoresistive effect do not have interfaces for calculation of the entire piezoresistive tensor for arbitrary directions. This warrants additional work for the user because not considering the complete piezoresisitive tensor can lead to large errors. This is illustrated with an example where the error is as high as 33%. Additionally, for elastic analysis, we used hybrid finite element formulation that estimates stresses more accurately than displacement-based formulation. Therefore, as shown in an example where the change in resistance can be calculated analytically, the percentage error of our piezoresistive program is an order of magnitude lower relative to displacement-based finite element method.

Holographic stress tensor at finite coupling

We calculate one, two and three point functions of the holographic stress tensor for any bulk Lagrangian of the form L (g(ab), R-abcd, del(e) R-abcd). Using the first law of entanglement, a simple method has recently been proposed to compute the holographic stress tensor arising from a higher derivative gravity dual. The stress tensor is proportional to a dimension dependent factor which depends on the higher derivative couplings. In this paper, we identify this proportionality constant with a B-type trace anomaly in even dimensions for any bulk Lagrangian of the above form. This in turn relates to C-T, the coefficient appearing in the two point function of stress tensors. We use a background field method to compute the two and three point function of stress tensors for any bulk Lagrangian of the above form in arbitrary dimensions. As an application we consider general situations where eta/s for holographic plasmas is less than the KSS bound.

Dielectric elastomers: Asymptotically-correct three-dimensional displacement field

Asymptotically-accurate dimensional reduction from three to two dimensions and recovery of 3-D displacement field of non-prestretched dielectric hyperelastic membranes are carried out using the Variational Asymptotic Method (VAM) with moderate strains and very small ratio of the membrane thickness to its shortest wavelength of the deformation along the plate reference surface chosen as the small parameters for asymptotic expansion. Present work incorporates large deformations (displacements and rotations), material nonlinearity (hyperelasticity), and electrical effects. It begins with 3-D nonlinear electroelastic energy and mathematically splits the analysis into a one-dimensional (1-D) through-the-thickness analysis and a 2-D nonlinear plate analysis. Major contribution of this paper is a comprehensive nonlinear through-the-thickness analysis which provides a 2-D energy asymptotically equivalent of the 3-D energy, a 2-D constitutive relation between the 2-D generalized strain and stress tensors for the plate analysis and a set of recovery relations to express the 3-D displacement field. Analytical expressions are derived for warping functions and stiffness coefficients. This is the first attempt to integrate an analytical work on asymptotically-accurate nonlinear electro-elastic constitutive relation for compressible dielectric hyperelastic model with a generalized finite element analysis of plates to provide 3-D displacement fields using VAM. A unified software package `VAMNLM' (Variational Asymptotic Method applied to Non-Linear Material models) was developed to carry out 1-D non-linear analysis (analytical), 2-D non-linear finite element analysis and 3-D recovery analysis. The applicability of the current theory is demonstrated through an actuation test case, for which distribution of 3-D displacements are provided. (C) 2014 Elsevier Ltd. All rights reserved.

Nonlinear constraints on gravity from entanglement

Using the positivity of relative entropy arising from the Ryu-Takayanagi formula for spherical entangling surfaces, we obtain constraints at the nonlinear level for the gravitational dual. We calculate the Green's function necessary to compute the first order correction to the entangling surface and use this to find the relative entropy for non-constant stress tensors in a derivative expansion. We show that the Einstein value satisfies the positivity condition, while the multidimensional parameter space away from it gets constrained.