Laplace stretch: Eulerian and Lagrangian formulations
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Laplace stretch: Eulerian and Lagrangian formulations Alan D. Freed , Shahla Zamani, L´ aszl´o Szab´ o and John D. Clayton
Abstract. Two triangular factorizations of the deformation gradient tensor are studied. The first, termed the Lagrangian formulation, consists of an upper-triangular stretch premultiplied by a rotation tensor. The second, termed the Eulerian formulation, consists of a lower-triangular stretch postmultiplied by a different rotation tensor. The corresponding stretch tensors are denoted as the Lagrangian and Eulerian Laplace stretches, respectively. Kinematics (with physical interpretations) and work-conjugate stress measures are analyzed and compared for each formulation. While the Lagrangian formulation has been used in prior work for constitutive modeling of anisotropic and hyperelastic materials, the Eulerian formulation, which may be advantageous for modeling isotropic solids and fluids with no physically identifiable reference configuration, does not seem to have been used elsewhere in a continuum mechanical setting for the purpose of constitutive development, though it has been introduced before in a purely kinematic setting. Mathematics Subject Classification. Primary 74A05; Secondary 15A90. Keywords. Continuum mechanics, Kinematics, Finite strain, Gram–Schmidt factorization.
1. Introduction Lagrangian formulations (i.e., constitutive models based on Lagrangian measures of strain) are typically preferred for modeling anisotropic solids, as well as certain isotropic solids that have a clearly defined initial, stress-free, or ‘reference’ state. This is readily apparent for single crystals, e.g., where a reference state is identified with a regular lattice geometry occupied by atoms in their minimum energy (ground) state. Hyperelasticity [1,2] is typically invoked in this context, where energy potentials that depend upon a Lagrangian strain are employed. Eulerian formulations (i.e., constitutive models based on Eulerian measures of strain), in contrast, are often preferred for modeling isotropic solids (and fluids) that have no obvious initial or reference state. Hypoelasticity [3,4] is often invoked in this context for the purpose of solving initial-boundary value problems numerically. Motivation for this study is a continued need to develop constitutive models for biological tissues that can be understood and used by those who work in the medical profession. In vivo, soft tissues are under tension perpetually, and a stress-free reference state is never physically realized. In such cases, it becomes advantageous to choose a ‘reference’ state with clinical relevance, e.g., at max systole for cardiac analyses, or at total lung capacity for pulmonary analyses, etc. Consequently, an Eulerian formulation would be optimal in such cases. The intent of this paper is to create a theoretical framework suitable for such constitutive developments. It is not the intent of this paper to create said models nor to apply them. With regard to creating a frame
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