Robert L. Lowe

Numerical Solution by the CESE Method of a First-Order Hyperbolic Form of the Equations of Dynamic Nonlinear Elasticity

This paper reports the application of the space-time conservation element and solution element (CESE) method to the numerical solution of nonlinear waves in elastic solids. The governing equations consist of a pair of coupled first-order nonlinear hyperbolic partial differential equations, formulated in the Eulerian frame. We report their derivations and present conservative, nonconservative, and diagonal forms. The conservative form is solved numerically by the CESE method; the other forms are used to study the eigenstructure of the hyperbolic system (which reveals the underlying wave physics) and deduce the Riemann invariants. The proposed theoretical/numerical approach is demonstrated by directly solving two benchmark elastic wave problems: one involving linear propagating extensional waves, the other involving nonlinear resonant standing waves. For the extensional wave problem, the CESE method accurately captures the sharp propagating wavefront without excessive numerical diffusion or spurious oscillations, and predicts correct reflection characteristics at the boundaries. For the resonant vibrations problem, the CESE method captures the linear-to-nonlinear evolution of the resonant waves and the distribution of wave energy among multiple modes in the nonlinear regime.

Modeling of Thermo-Electro-Magneto-Mechanical Behavior, with Application to Smart Materials

A key feature of smart materials is their ability to convert energy from one form into another. For instance, piezoelectric materials deform when exposed to an electric field, thus converting electrical energy to mechanical energy. Other common smart materials include magnetostrictives, magnetorheological fluids, shape memory alloys, and electroactive polymers. These materials couple different physical effects, e.g., thermal, electrical, magnetic, and/or mechanical. In this chapter, we present a continuum framework that lays the groundwork for modeling a broad range of smart materials exhibiting coupled thermal, electrical, magnetic, and/or mechanical behavior. This framework has the breadth to accommodate large deformations (i.e., geometric nonlinearities), anisotropy, and nonlinear constitutive response (i.e., material nonlinearity). We devote special attention to developing the fundamental laws of continuum electrodynamics and presenting key aspects of thermodynamic constitutive modeling. For instance, we show how the thermodynamic formalism that produced constitutive models for classical elastic solids and viscous fluids can also be used to facilitate the constitutive modeling of smart materials with coupled thermo-electro-magneto-mechanical (TEMM) behavior. Finally, we illustrate that our modeling approach provides an overarching framework that encompasses many well-known types of smart material behavior. For instance, we explicitly demonstrate that the linear theory of piezoelectricity falls out as a special case of our more general finite-deformation TEMM framework, much the same way linear elasticity falls out of finite-deformation elasticity.

Chapter 7 – Fluid Mechanics

This chapter presents constitutive equations appropriate for modeling the flow of several technologically important classes of fluids, namely viscous fluids and inviscid fluids. Viscous fluids are sensitive to the rate at which they are deformed, whereas inviscid fluids are insensitive to the rate at which they are deformed. Of course, almost no fluids of practical importance are truly “inviscid” (i.e., have zero viscosity). Nevertheless, from a modeling perspective, the notion of an inviscid fluid is quite useful. In particular, it can be used as an idealization for modeling flows where viscous effects only weakly influence the flow physics. For instance, in most applications, water is modeled as a viscous fluid. However, in modeling the high-speed flow of water far from a bounding surface (e.g., a pipeline), it is more appropriate to model water as an inviscid fluid. Hence, it is the fluid and physical application together that dictate if neglecting viscous effects is an appropriate assumption. In this chapter, we discuss viscous fluids and inviscid fluids in the context of the mechanical theory and the thermomechanical theory.

Chapter 6 – Nonlinear Elasticity

This chapter presents constitutive equations appropriate for a broad class of engineering materials known as nonlinear elastic solids. Common examples include rubber, elastomers (rubber-like polymers), and soft biological tissues. Nonlinear elastic solids are characterized by their ability to undergo large recoverable deformations and their highly nonlinear stress-strain response. Hence, they exhibit geometric nonlinearity (i.e., strain-displacement nonlinearity) due to finite elastic deformations, and material nonlinearity (i.e., stress-strain nonlinearity) due to nonlinear constitutive response. We examine nonlinear elastic materials in the context of the mechanical (isothermal) theory as well as the thermomechanical theory. In the latter case, we explicitly illustrate how the constitutive equations must satisfy the second law of thermodynamics, invariance, conservation of angular momentum, and material symmetry (isotropy).

Chapter 1 – What Is a Continuum?

This chapter provides a brief overview of mathematical modeling in mechanics. Special attention is devoted to contrasting discrete approaches with continuum approaches, both of which play an important and unique role in the modern mechanics-of-materials landscape. Discrete models underpin state-of-the-art molecular dynamics simulation tools. Continuum models, on the other hand, underpin modern finite element and computational fluid dynamics codes, which are widely used in industry and engineering practice.

Chapter 4 – The Fundamental Laws of Thermomechanics

This chapter discusses the kinetics and thermodynamics of a continuum, introducing concepts such as force, moment, momentum, stress, energy, work, heat, and entropy. Special attention is devoted to developing the fundamental laws (or first principles) of thermomechanics. These include the mechanical conservation laws of mass, linear momentum, and angular momentum, as well as the first law of thermodynamics (or conservation of energy). Each of the fundamental laws is first postulated as a primitive statement (in words), from which we carefully progress to material, integral, and pointwise forms. Both Eulerian (present configuration) and Lagrangian (reference configuration) representations of the fundamental laws are discussed. The chapter concludes with a presentation of the second law of thermodynamics in the form of the Clausius-Duhem inequality.

Chapter 3 – Kinematics: Motion and Deformation

This chapter discusses the kinematics of a continuum. The reader is first introduced to the fundamental concepts of body, configuration, and motion, with the motion taking the body from its reference configuration to its present configuration. Special attention is devoted to developing the Lagrangian (referential) and Eulerian (spatial) descriptions of a quantity and showing that they are nothing more than labels: in the Lagrangian description, each particle is labeled by its position in the reference configuration, whereas in the Eulerian description, each particle is labeled by its position in the present configuration. The material derivative is introduced, from which fundamental definitions of velocity and acceleration follow. Finally, various deformation and strain measures are introduced and their physical meanings interpreted.

Chapter 2 – Our Mathematical Playground

The purpose of this chapter is to enable the reader to become fluent in the language of this textbook: tensor algebra and tensor calculus. In particular, special attention is devoted to rigorously developing the mathematical foundations underlying tensor algebra and tensor calculus from the ground up, starting with the fundamental concepts of a vector space and an inner product space. Throughout, we favor a direct (or coordinate-free) presentation of the mathematics. Although direct notation requires some effort to master, it ultimately lends itself to a more transparent presentation of the physical concepts. Care is taken to provide the reader with sufficient background to specialize the coordinate-free results to Cartesian or curvilinear coordinate systems. Almost all examples are worked by the authors to facilitate self-study and to ensure the reader has a firm mathematical foundation.

Chapter 5 – Constitutive Modeling in Mechanics and Thermomechanics

The constitutive equations for a particular material, together with the conservation laws of mass, linear momentum, angular momentum, and energy, govern the response of that material. Unlike the conservation laws, however, the constitutive equations vary from material to material. In other words, they depend on the physical behavior of the particular material being modeled. In this chapter, we provide a broad overview of constitutive modeling in mechanics and thermomechanics. We describe how the constitutive equations must satisfy the second law of thermodynamics, conservation of angular momentum, material symmetry requirements, and invariance under superposed rigid body motions. Special attention is devoted to determining how various kinematic, kinetic, and thermodynamic quantities transform under a superposed rigid body motion and, in turn, how these transformations can be used to develop appropriate invariance requirements. We also examine useful thermodynamic concepts in constitutive modeling, such as thermomechanical processes and Legendre transformations.

Incompressibility and Thermal Expansion

In this chapter, we develop thermomechanical models for (i) incompressible viscous fluids, (ii) incompressible nonlinear elastic solids, and (iii) viscous fluids that thermally expand and contract. A material is referred to as incompressible if, within a certain class of loadings, only isochoric or volume-preserving motions are possible. In other words, the volume of the material cannot be changed appreciably by any means, be they thermal or mechanical. In the same vein, for certain classes of loadings, some materials are mechanically incompressible, yet experience significant thermal expansion, i.e., thermally induced volume change. For instance, polymer melts are relatively insensitive to changes in pressure, but can experience significant volume shrinkage (sometimes 30 percent or more) as they cool from their melt-processing temperature to room temperature.