 Award ID(s):
 1829447
 NSFPAR ID:
 10247794
 Date Published:
 Journal Name:
 Geophysical Journal International
 ISSN:
 0956540X
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract An elastic map $\mathbf {T}$ T describes the strainstress relation at a particular point $\mathbf {p}$ p in some material. A symmetry of $\mathbf {T}$ T is a rotation of the material, about $\mathbf {p}$ p , that does not change $\mathbf {T}$ T . We describe two ways of inferring the group $\mathcal {S} _{ \mathbf {T} }$ S T of symmetries of any elastic map $\mathbf {T}$ T ; one way is qualitative and visual, the other is quantitative. In the first method, we associate to each $\mathbf {T}$ T its “monoclinic distance function” "Equation missing" on the unit sphere. The function "Equation missing" is invariant under all of the symmetries of $\mathbf {T}$ T , so the group $\mathcal {S} _{ \mathbf {T} }$ S T is seen, approximately, in a contour plot of "Equation missing" . The second method is harder to summarize, but it complements the first by providing an algorithm to compute the symmetry group $\mathcal {S} _{ \mathbf {T} }$ S T . In addition to $\mathcal {S} _{ \mathbf {T} }$ S T , the algorithm gives a quantitative description of the overall approximate symmetry of $\mathbf {T}$ T . Mathematica codes are provided for implementing both the visual and the quantitative approaches.more » « less

Abstract In linear elasticity, universal displacements for a given symmetry class are those displacements that can be maintained by only applying boundary tractions (no body forces) and for arbitrary elastic constants in the symmetry class. In a previous work, we showed that the larger the symmetry group, the larger the space of universal displacements. Here, we generalize these ideas to the case of anelasticity. In linear anelasticity, the total strain is additively decomposed into elastic strain and anelastic strain, often referred to as an eigenstrain. We show that the
universality constraints (equilibrium equations and arbitrariness of the elastic constants) completely specify theuniversal elastic strains for each of the eight anisotropy symmetry classes. The corresponding universal eigenstrains are the set of solutions to a system of secondorder linear PDEs that ensure compatibility of the total strains. We show that for three symmetry classes, namely triclinic, monoclinic, and trigonal, only compatible (impotent) eigenstrains are universal. For the remaining five classes universal eigenstrains (up to the impotent ones) are the set of solutions to a system of linear secondorder PDEs with certain arbitrary forcing terms that depend on the symmetry class. 
A comparative study is presented to solve the inverse problem in elasticity for the shear modulus (stiffness) distribution utilizing two constitutive equations: (1) linear elasticity assuming small strain theory, and (2) finite elasticity with a hyperelastic neoHookean material model. Assuming that a material undergoes large deformations and material nonlinearity is assumed negligible, the inverse solution using (2) is anticipated to yield better results than (1). Given the fact that solving a linear elastic model is significantly faster than a nonlinear model and more robust numerically, we posed the following question: How accurately could we map the shear modulus distribution with a linear elastic model using small strain theory for a specimen undergoing large deformations? To this end, experimental displacement data of a silicone composite sample containing two stiff inclusions of different sizes under uniaxial displacement controlled extension were acquired using a digital image correlation system. The silicone based composite was modeled both as a linear elastic solid under infinitesimal strains and as a neoHookean hyperelastic solid that takes into account geometrically nonlinear finite deformations. We observed that the mapped shear modulus contrast, determined by solving an inverse problem, between inclusion and background was higher for the linear elastic model as compared to that of the hyperelastic one. A similar trend was observed for simulated experiments, where synthetically computed displacement data were produced and the inverse problem solved using both, the linear elastic model and the neoHookean material model. In addition, it was observed that the inverse problem solution was inclusion sizesensitive. Consequently, an 1D model was introduced to broaden our understanding of this issue. This 1D analysis revealed that by using a linear elastic approach, the overestimation of the shear modulus contrast between inclusion and background increases with the increase of external loads and target shear modulus contrast. Finally, this investigation provides valuable information on the validity of the assumption for utilizing linear elasticity in solving inverse problems for the spatial distribution of shear modulus associated with soft solids undergoing large deformations. Thus, this work could be of importance to characterize mechanical property variations of polymer based materials such as rubbers or in elasticity imaging of tissues for pathology.more » « less

There seems to be a basic misconception in several recent papers concerning the material symmetry of bodies in configurations that are prestressed. In this short paper we point to the source of the error and show that the material symmetry that is possible depends on the nature of the prestress. We also extend the results for material symmetry which have been well known within the context of simple elastic solids to the general class of simple materials. This generalization has relevance to the material symmetry of biological solids that are viscoelastic.more » « less

Abstract Numerical simulations using nonlinear hyperelastic material models to describe interactions between brain white matter (axons and extra cellular matrix (ECM)) have enabled highfidelity characterization of stressstrain response. In this paper, a novel finite element model (FEM) has been developed to study mechanical response of axons embedded in ECM when subjected to tensile loads under purely nonaffine kinematic boundary conditions. FEM leveraging Ogden hyperelastic material model is deployed to understand impact of parametrically varying oligodendrocyteaxon tethering and analyze influence of aging material characteristics on stress propagation. In proposed FEM, oligodendrocyte connections to axons are represented via springdashpot model, such tethering technique facilitates contact definition at various locations, parameterize connection points and vary stiffness of connection hubs. Two FE submodels are discussed: 1) multiple oligodendrocytes arbitrarily tethered to the nearest axons, and 2) single oligodendrocyte tethered to all axons at various locations. Root mean square deviation (RMSD) were computed between stressstrain plots to depict trends in mechanical response. Axonal stiffness was found to rise with increasing tethering, indicating role of oligodendrocytes in stress redistribution. Finally, stress state results for aging axon material, with varying stiffnesses and number of connections in FEM ensemble have also been discussed to demonstrate gradual softening of tissues.