4. Large Deformation Framework
4.1. Introduction
In this chapter and the next several chapters we extend our discussion of the linear elastic problem to accommodate two categories of important nonlinearities: potentially large motions and deformations, and nonlinear material response. We will do this by introducing a more general notational framework. While the equations governing large deformation initial/boundary value problems are similar in form to their counterparts from the small deformation theory just discussed, a rigorous prescription and understanding of large deformation problems can only be achieved through a careful examination of the concepts of nonlinear continuum mechanics, which will be the concern of the next several chapters.
The organization of this material is as follows. This chapter establishes a notational framework for the generic specification of a nonlinear solid mechanics problem. Section 5 and Section 6 discuss large deformation kinematics in a general context. Section 7 will then discuss the various measures of stress that are frequently encountered in large deformation analysis. Then, with these preliminaries in hand, we will be in a position to state relevant balance laws in notation appropriate for large deformation problems in Section 8. Finally, in Section 9, we will discuss the important concept of material frame indifference, which demands that material laws be unaltered by rigid body motions. We will see that this concept places important restrictions on the kinematic and stress measures that are suitable for prescription of constitutive laws, providing important background information for the chapter on material models.
4.2. Notational Framework
The system we wish to consider is depicted schematically in Fig. 4.1. We consider a body, initially in a location denoted by \(\Omega\), undergoing a time dependent motion \(\varphi\) that describes its trajectory through space (assumed here to be \(\mathbb{R}^3\)).
Fig. 4.1 Notation for large deformation initial/boundary value problems.
The set \(\Omega\) is called the reference configuration and can be thought of as consisting of points \(\mathbf{X}\) that serve as labels for the material points existing at their respective locations. For this reason, the coordinates \(\mathbf{X}\) are often called reference or material coordinates.
We assume, as before, that the surface \(\partial \Omega\) of \(\Omega\) can be decomposed into subsets \(\Gamma _{\sigma}\) and \(\Gamma _u\) obeying restrictions in (2.1). The general interpretation of these surfaces remains the same. Traction boundary conditions will be imposed on \(\Gamma _{\sigma}\) and displacement boundary conditions will be imposed on \(\Gamma _u\). Full specification of these boundary conditions must be deferred, however, until some continuum mechanical preliminaries are discussed.
We have mentioned that the motion \(\varphi\) is in general time dependent. In fact, we could write this fact in mathematical terms as \(\varphi : \bar{\Omega} \times (0,T) \rightarrow \mathbb{R}^3\). If we fix the time argument of \(\varphi\), we obtain a configuration mapping \(\varphi_t\), summarized as \(\varphi_t : \bar{\Omega} \rightarrow \mathbb{R}^3\), which gives us the location of the body at time \(t\) given the reference configuration \(\Omega\). Coordinates in the current location \(\varphi (\Omega)\) of the body will be denoted by \(\mathbf{x}\).
The current location is often called the spatial configuration and the coordinates, \(\mathbf{x}\) spatial coordinates. Given a material point \(\mathbf{X} \in \Omega\) and a configuration mapping \(\varphi_t\), we may write
A key decision in writing the equations of motion for this system is whether to express the equations in terms of \(\mathbf{X} \in \Omega\) or \(\mathbf{x} \in \varphi_t(\Omega)\).
4.3. Lagrangian and Eulerian Descriptions
The choice of whether to use the reference coordinates \(\mathbf{X}\) or the spatial coordinates \(\mathbf{x}\) in the problem description is generally highly dependent on the physical system to be studied.
For example, suppose we wish to write the equations of motion for a gas flowing through a duct, or for a fluid flowing through a nozzle. In these cases the physical region of interest (the control volume bounded by the duct or nozzle) is fixed, and does not depend on the solution or time. It could also be observed that identification of individual particle trajectories in such problems is probably not of primary interest, with such quantities as pressure, velocity, and temperature at particular locations in the flow field being more desirable. In such problems, it is generally most appropriate to associate field variables and equations with spatial points, or in the current notation, \(\mathbf{x}\). A system described in this manner is said to be utilizing the Eulerian description and implicitly associates all field variables and equations with spatial points \(\mathbf{x}\) without specific regard for the material points \(\mathbf{X}\) involved in the flow of the problem. Most fluid and gas dynamics problems are written in this way, as are problems in hydrodynamics and some problems in solid mechanics involving fully developed plastic flow.
When thinking of Eulerian coordinate systems, it is sometimes useful to invoke the analogy of watching an event through a window; the window represents the Eulerian frame and has our coordinate system attached to it. Particles pass through our field of view, thereby defining a flow, but we describe this flow from the frame of reference of our window without specific reference to the particles undergoing the motion we observe.
In most solid mechanics applications, by contrast, the identity of specific material particles is of central interest in modeling a system. For example, the plastic response of metals is history dependent, meaning that the current relationship between stress and strain (the material model) at a point in the body depends on the deformation history associated with that material point. To construct and use such models effectively requires knowledge of the history of individual particles, or material points, throughout a deformation process. Furthermore, many physical processes we wish to describe do not lend themselves to an invariant Eulerian frame. In a forging process, for example, the metal at the end of the procedure occupies a very different region in space than it did at the outset. In addition, there may be periods of time over which boundary conditions are applied requiring precise knowledge of the boundary of the region of interest. For these reasons, as well as others, the predominant approach to solid mechanics systems is to write all equations in terms of the material coordinates, or to use the Lagrangian frame of reference.
Returning to the notation summarized in Fig. 4.1, for a Lagrangian description we associate all field variables and equations with points \(\mathbf{X} \in \Omega\), and keep track of these reference particles throughout the process. One may note in the last subsection a bias toward this approach already. We have written the primary unknown in the problem, \(\varphi\), as a function of \(\mathbf{X} \in \Omega\) and \(t \in (0,T)\). Sierra/SM uses the Lagrangian frame of reference though as we will see next, the spatial frame is also of great interest to us.
4.4. Governing Equations in the Spatial Frame
We turn now to the equations governing the motion of a medium. If we adopt for the moment the spatial frame as our frame of reference, the form of these equations is largely unchanged from the linear elastic case presented previously (where we explicitly took advantage of the fact that for linear problems there is no difference between material and spatial descriptions). We fix our attention on some time \(t \in (0,T)\) and consider the current (unknown) location of the body \(\Omega\). Over this region \(\varphi_t (\Omega)\), the following conditions must hold:
and
subject to initial conditions at \(t=0\). Some explanation of these equations is necessary. The operator \(\nabla\) in (4.2) is with respect to spatial coordinates \(\mathbf{x}\). The acceleration \(\mathbf{a}\) is the acceleration of the particle currently at \(\mathbf{x}\) written with respect to spatial coordinates, and \(\bar{\varphi_t}\) is the prescribed location for the particles on the Dirichlet boundary. We leave the constitutive law governing \(\mathbf{T}\) unspecified at this point but remark that in general the stress must depend on \(\varphi_t\) through appropriate strain/displacement and stress/strain relations.
We see from (4.2) through (4.4) that the equations of motion are easily written in the form inherited from the kinematically linear case, but that the frame in which this is done, the spatial frame, is not independent of the unknown field \(\varphi_t\) but relies upon it for its own definition. Thus, although the equations we now consider are essentially identical in form to those from linear elasticity, they posses a considerably more complex relationship to the dependent variable. Rigorous specification of this general boundary value problem requires an in-depth treatment of the continuum mechanics of large deformation, as will be provided in the next chapters.
Before leaving this topic, we address an item which frequently causes confusion. Although we have written the governing equations in (4.2) through (4.4) in terms of the spatial domain, this does not imply an Eulerian statement of the problem. In fact, if we choose (as we have done) to consider our dependent variable (in this case \(\varphi_t\)) to be a function of reference coordinates, the framework we have chosen is inherently Lagrangian. Another way of saying this is that (4.2) through (4.4) are the Lagrangian equations of motion which have been converted through a change-of-variables so that they are written in terms of \(\mathbf{x}\). In the remainder of this text, the reader should assume a Lagrangian framework unless otherwise noted.