Kinematics and Mechanics of Large Elastic Deformations - Continuum Mechanics of Anisotropic Materials - page 289

Chapter 11
Kinematics and Mechanics of Large
Elastic Deformations

“The perfectly elastic material is a particular kind of ideal material. It has a single
preferred or natural configuration. We think a portion of this material in a certain
configuration with material coordinates X , which we choose as specifying the
reference configuration. However this material is deformed, into whatever configu-
ration it is brought, it always remembers precisely its preferred or natural configu-
ration and attempts to get back to it. When the forces that maintain it its present
configuration are released, it will return precisely to initial configuration. It is a
material, in other words, which has perfect memory for one state and no memory
whatever for any other state. The forces required to maintain it in the configuration
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are completely independent of the matter in which it is brought from its original
configuration to the configuration , the time it has taken to get there, and all of its

intermediate history. This is a highly idealized kind of behavior, but it is one that
may be observed in a remarkably good approximation and rubber, for example.”
Truesdell ( 1960 )

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11.1 Large Deformations

Large deformations are more difficult to mathematically model than either small
deformations or fluid motions. The difficulty stems from the fact that, for the
analysis of large deformations, the knowledge or data associated with at least two
different configurations must be maintained. In the case of fluid motions, only the
knowledge of the present or instantaneous configuration is necessary and, in
the case of small deformations of solids, the difference between the initial reference
configurations and the present configuration is a small higher order quantity, and it
is neglected. In fact, this neglect of higher order terms between the two
configurations is the definition of “small” deformations. These difficulties may be
illustrated using the concept of stress in large deformations. For small deformations
the only definition of stress employed is force per unit of instantaneous cross-
sectional area. This is adequate since there is a negligible difference between the