Variational formulation of boundary-value problems

This chapter discusses a variational formulation of boundary value problems in small deformation solid mechanics. It begins by introducing the important principle of virtual power, and shows that it encapsulates Cauchy’s traction law, and the local form of the basic balance of forces (equation of equilibrium), and the local from of the balance of moments (symmetry of the stress). Since the principle of virtual power encapsulates both the equation of equilibrium and the Cauchy relation for tractions, it can be used to formulate and solve boundary-value problems in solid mechanics in a variational or weak sense. Specifically, it is shown how the displacement problem of linear elastostatics may be formulated variationally using the principle of virtual power.

Single-crystal elasticity of Ice at high pressures

<p>Recent astronomy studies observed the existence of hundreds of ice giants in the universe. Ice is one of the major components of these ice giants. Experimental studies on the physical properties of ice at high pressures are thus important for understanding the composition and evolution of these ice giants. Here, we have synthesized high-quality single-crystal H<sub>2</sub>O-ice with 0.5 m NaCl. Single-crystal elasticity of ice was measured by Brillouin spectroscopy combined with X-ray diffraction up to 93 GPa at 300 K using diamond anvil cells. All the elastic moduli of ice-VII exhibit a nearly linear increase with pressure up to 43 GPa at 300 K, although the off-diagonal modulus C<sub>12</sub> and shear modulus C<sub>44</sub> slightly deviate from the Cauchy relation between 10 and 20 GPa. The longitudinal modulus, C<sub>11</sub>, and C<sub>12</sub> show a clear softening when ice-VII changes to the pre-transition ice-VII state. Meanwhile, we also observed a weak drop in the unit cell volume across this change in our high-quality single-crystal X-ray diffraction measurements. We also present first experimental measurements on the single-crystal elasticity of ice-X. Our experimental results were also used to model the anisotropy of ice at high pressures.</p>

An Improved Diagonal Jacobian Approximation via a New Quasi-Cauchy Condition for Solving Large-Scale Systems of Nonlinear Equations

We present a new diagonal quasi-Newton update with an improved diagonal Jacobian approximation for solving large-scale systems of nonlinear equations. In this approach, the Jacobian approximation is derived based on the quasi-Cauchy condition. The anticipation has been to further improve the performance of diagonal updating, by modifying the quasi-Cauchy relation so as to carry some additional information from the functions. The effectiveness of our proposed scheme is appraised through numerical comparison with some well-known Newton-like methods.