A Framework for Modeling Large Deformations and Stress Wave Mechanics in Soft Biological Tissue
Abstract
An oblique, Cartesian, coordinate system arises from the geometry affiliated with a Gram-Schmidt (QR) factorization of the deformation gradient F, wherein Q is a proper orthogonal matrix and R is an upper-triangular matrix.
First, a cube deforms into a parallelepiped whose edges are oblique and serve as the base vectors for a convected coordinate system. Components for the metric tensor, its dual, and their rates, evaluated in this convected coordinate system, are established for any state of deformation. Strains and strain rates are defined and quantified in terms of these metrics and their rates. Quotient laws and their affiliated Jacobians are constructed that govern how vector and tensor fields transform between this oblique coordinate system, where constitutive equations are ideally cast, and the reference, rectangular, Cartesian, coordinate system described in terms of Lagrangian variables, where boundary value problems are solved.
Then, we derived two sets of thermodynamically admissible stress-strain pairs. They are quantified in terms of physical components extracted from a convected stress and a convected velocity gradient, with elastic models being presented for both sets. The first model supports two modes of deformation: elongation and shear. The second model supports three modes of deformation: dilatation, squeeze and shear. These models are distinguished by their pure-and simple-shear responses. They contain the coupling effects of Lord Kelvin [1], Poisson [2] and Poynting [3].
The Eulerian formulation, consists of a lower-triangular stretch postmultiplied by a different rotation tensor is studied. The corresponding stretch tensors is denoted as the Eulerian Laplace stretches. Kinematics (with physical interpretations) and work conjugate stress measures are analyzed. 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.
As the application of our work, we introduced a dodecahedron to model an alveolus. Its geometric properties are derived in detail with regard to its three geometric features: 1D septal chords, 2D septal membranes, and the 3D alveolar sac. The kinematics are derived for us to model a deforming dodecahedron, including the shape functions needed for interpolating each geometry. Constitutive models are derived that are suitable for describing the thermomechanical response for the structural constituents of an alveolus: its septal chords, its permeable membranes, and its volume. Numerical methods are advanced for solving first- and second-order ordinary differential equations (ODEs) and spatial integrations along a bar, across a pentagon, and throughout a tetrahedron using Gaussian quadrature schemes designed for each geometry. A variational formulation is used to create our structural modeling of an alveolus. Constitutive equations suitable for modeling biological tissues are derived from thermodynamics using the theory of implicit elasticity, presented in an appendix.
Citation
Zamani Mehrian, Shahla (2021). A Framework for Modeling Large Deformations and Stress Wave Mechanics in Soft Biological Tissue. Doctoral dissertation, Texas A&M University. Available electronically from https : / /hdl .handle .net /1969 .1 /197506.