Prestressed Microstretch Elastic Solids

An Eulerian formulation has been developed for the constitutive response of a group of materials that includes anisotropic elastic and viscoelastic solids and viscous fluids. The material is considered to be a composite of an elastic solid and a viscous fluid. Evolution equations are proposed for a triad of vectors mi that represent the stretches and orientations of material line elements in the solid component. Evolution equations for an orthonormal triad of vectors si are also proposed to characterize anisotropy of the fluid component. In particular, for an elastic solid it is shown that the material response is totally characterized by the functional form of the strain energy and by the current values of mi, which are measurable in the current state of the material. Moreover, it is shown that the proposed Eulerian formulation removes unphysical arbitrariness of the choice of the reference configuration in the standard formulation of constitutive equations for anisotropic elastic solids.

Chapter Four - Elastic solid with spherical inhomogeneities

We develop a new operator splitting formulation for the simulation of corotated linearly elastic solids with Smoothed Particle Hydrodynamics (SPH). Based on the technique of Kugelstadt et al. [2018] originally developed for the Finite Element Method (FEM), we split the elastic energy into two separate terms corresponding to stretching and volume conservation, and based on this principle, we design a splitting scheme compatible with SPH. The operator splitting scheme enables us to treat the two terms separately, and because the stretching forces lead to a stiffness matrix that is constant in time, we are able to prefactor the system matrix for the implicit integration step. Solid-solid contact and fluid-solid interaction is achieved through a unified pressure solve. We demonstrate more than an order of magnitude improvement in computation time compared to a state-of-the-art SPH simulator for elastic solids. We further improve the stability and reliability of the simulation through several additional contributions. We introduce a new implicit penalty mechanism that suppresses zero-energy modes inherent in the SPH formulation for elastic solids, and present a new, physics-inspired sampling algorithm for generating high-quality particle distributions for the rest shape of an elastic solid. We finally also devise an efficient method for interpolating vertex positions of a high-resolution surface mesh based on the SPH particle positions for use in high-fidelity visualization.

The simpler linear rheological bodies—such as the elastic and Kelvin's solids, Maxwell's and Newton's viscous fluids—obey a generalized linear differential equation with constant coefficients between the strain and stress deviators and their time rates. The various coefficients are correlated to the asymptotic rigidity or static elastic modulusG, elastic firmness or dynamic modulusH, solid viscosity µ, times of relaxationR and of lagging or retardationL, and the endosity η.‡ Expressing the time and stress in non-dimensional forms, a universal equation is obtained, dependent on a single non-dimensional parameter, the ‘time factor’ τ = R/L = G/H. This defines a principle of similitude for all bodies of identical τ. The mechanical and thermodynamic study of periodic, impulsive, and transient stresses leads towards a new classification of the linear bodies, based on the value of τ: The exothermal or dissipative bodies (τ < 1) are less rigid than firm, less strained than elastic solids, dissipate energy in a cycle. They warm up adiabatically. Here belong also Maxwell's elastico-viscous fluids and Kelvin's firmo-viscous solids (t = 0). The endows solids (τ > 1) are more rigid than firm, more strained than elastic solids, may show a' negative' dissipation in a cycle, which is compensated by internal structural changes. This is not in contradiction with the second law of thermodynamics and may explain the fatigue of such solids. The homothermal solids (τ = 1) are as rigid as firm, show no apparent dissipation in a cycle. A particular case is that of elastic solids. The stress-strain diagram under periodio stresses tends towards an ellipse, which shows accommodation of the body. The smaller axis passes through a maximum for a critical frequency, and the body has two independent elastic moduli—a static and a dynamic one. These facts were experimentally confirmed for certain plastics.

High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: Viscous heat-conducting fluids and elastic solids

Part 1 Overall properties of heterogeneous solids: aggregate properties and averaging methods aggregate properties, averaging methods elastic solids with microcavities and microcracks linearly elastic solids, elastic solids with traction-free defects, elastic solids with micrcavities, elastic solids with microcracks elastic solids with micro-inclusions overall elastic modulus and compliance tensors, examples o elastic solids with elastic micro-inclusions, upper and lower bounds for overall elastic moduli, self-consistent differential and related averaging methods, Eshelby's tensor and related topics solids with periodic microstructure general properties and field equations, overall properties of solids with periodic microstructure, mirror-image decomposition of periodic fields. Part 2 Introduction to basic elements of elasticity theory: foundations geometric foundations, kinematic foundations, dynamic foundations, constitutive relations elastostatic problems of linear elasticity boundary-value problems and extremum principles three-dimensional problems solution of singular problems. Appendix: references.

In the context of wave propagation in damaged (elastic) solids, an analytical approach for oblique penetration of a plane wave through a doubly periodic array of cracks is developed. By using a uniform approximation in one-mode range previously obtained, we give explicit representations for the wave field throughout the structure and the relevant parameters. Two figures show the peculiarity of such results.

Elastic waves and microstructure: Oxford University, UK, 16–17 December 1980

Multi-inclusion method for finite deformations: exact results and applications

In a standard T‐matrix calculation, the scattered radiation field is evaluated by assuming that both the source and receiver are a large distance from the scatterer. These conditions may not be satisfied in practice. We consider the effects of an incident spherical wave and of relaxing the farfield assumptions for the scattered wave amplitude for both rigid and solid elastic prolate spheroidal solids in water. The T‐matrix formalism is used to calculate the scattered form functions and radiation patterns at a variety of distances. Special attention is given to nearfield effects on the measured widths and strengths of the resonances generated in the elastic solid.

Harmonic decomposition analysis of contact mechanics of bonded layered elastic solids

At previous meetings of the Acoustical Society of America [L. H. Green, D. H. Trivett, and L. Flax, J. Acoust. Soc. Am. Suppl. 1 77, S79 (1985)], it was demonstrated that, for the analysis of the elastic excitations in the acoustic scattering from elastic targets, the proper background to subtract is a rigid background. This choice was utilized to analyze the low‐frequency scattering from spherical and infinite cylindrical shells over a broad range of frequencies and shell thicknesses. However, in a recent paper on the acoustic scattering from elastic spheroidal solids [M. F. Werby and G. J. Tango, J. Acoust. Soc. Am. 79, 1260–1268 (1986)], it is concluded that a rigid background is inappropriate for elastic targets that have either low shear speeds or low densities. A complete reexamination of the scattering from elastic solids is presented to demonstrate that the rigid background is the appropriate choice in this case, too. In the course of this analysis, it is also demonstrated that Werby and Tungo have misinterpreted the nature of the elastic response of the scatterer.

On the SPH tensile instability in forming viscous liquid drops

In the context of wave propagation in damaged (elastic) solids, an analytical method previously introduced for scalar problems, is now applied to study the (vector) problem for normal penetration of a longitudinal plane wave into a periodic array of collinear cracks. Reduced the problem to an integral equation holding over the openings, an approximation of one-mode type leads to analytical solutions and then to explicit representations for the wave fields and the scattering parameters. Some graphs will finally compare our results with the numerical ones by other authors.

Hydrostatic loading causes an isotropic elastic solid to be in a state of pure dilatation with no distortion relative to its unstressed reference configuration. Similarly, hydrostatic loading causes a general orthotropic solid to be distorted relative to its unstressed reference configuration. This paper introduces physically based invariants for orthotropic nonlinear elastic solids which are measures of distortions that cause deviatoric Cauchy stress. Specifically, these invariants allow for the modeling of the distortion in a hydrostatic state of stress independently of the form of the strain energy function. Consequently, use of these invariants may lead to simpler forms of the strain energy function which adequately model specific orthotropic solids.