Ben Nadler

The Transverse Isotropy of Spinal Cord White Matter Under Dynamic Load

The rostral-caudally aligned fiber-reinforced structure of spinal cord white matter (WM) gives rise to transverse isotropy in the material. Stress and strain patterns generated in the spinal cord parenchyma following spinal cord injury (SCI) are multidirectional and dependent on the mechanism of the injury. Our objective was to develop a WM constitutive model that captures the material transverse isotropy under dynamic loading. The WM mechanical behavior was extracted from the published tensile and compressive experiments. Combinations of isotropic and fiber-reinforcing models were examined in a conditional quasi-linear viscoelastic (QLV) formulation to capture the WM mechanical behavior. The effect of WM transverse isotropy on SCI model outcomes was evaluated by simulating a nonhuman primate (NHP) contusion injury experiment. A second-order reduced polynomial hyperelastic energy potential conditionally combined with a quadratic reinforcing function in a four-term Prony series QLV model best captured the WM mechanical behavior (0.89 < R2 < 0.99). WM isotropic and transversely isotropic material models combined with discrete modeling of the pia mater resulted in peak impact forces that matched the experimental outcomes. The transversely isotropic WM with discrete pia mater resulted in maximum principal strain (MPS) distributions which effectively captured the combination of ipsilateral peripheral WM sparing, ipsilateral injury and contralateral sparing, and the rostral/caudal spread of damage observed in in vivo injuries. The results suggest that the WM transverse isotropy could have an important role in correlating tissue damage with mechanical measures and explaining the directional sensitivity of the spinal cord to injury.

A Theory of the Mechanics of Two Coupled Surfaces

In this work the mechanics of two coupled membrane-like surfaces are considered. It will be shown that under special restrictions the finite deformation of the two coupled surfaces can be described using only five field parameters. This is accomplished by introducing a pairing between the two surfaces. The surfaces are permitted to slip with respect to each other subjected to frictional slipping constitutive law, but restricted to maintain full contact at all time. Such a model can be used to model frictional slip in woven fabrics. The weak form of the equations are formulated to be used with the finite element method. This theory furnishes equations of motion and boundary conditions which have clear physical meaning.

Modeling of cell adhesion and deformation mediated by receptor-ligand interactions.

The current work is devoted to studying adhesion and deformation of biological cells mediated by receptors and ligands in order to enhance the existing models. Due to the sufficient in-plane continuity and fluidity of the phospholipid molecules, an isotropic continuum fluid membrane is proposed for modeling the cell membrane. The developed constitutive model accounts for the influence of the presence of receptors on the deformation and adhesion of the cell membrane through the introduction of spontaneous area dilation. Motivated by physics, a nonlinear receptor–ligand binding force is introduced based on charge-induced dipole interaction. Diffusion of the receptors on the membrane is governed by the receptor–ligand interaction via Fick’s Law and receptor-ligand interaction. The developed model is then applied to study the deformation and adhesion of a biological cell. The proposed model is used to study the role of the material, binding, spontaneous area dilation and environmental properties on the deformation and adhesion of the cell.

Slip-plane plasticity using the theory of material evolution

The theory of material evolution is used to model anisotropic plasticity. As suggested by the physics of plasticity, whereby the motion of dislocations yields relative slipping along a particular material plane, a shear-like isochoric material deformation is assumed as the admissible kinematic evolution variable, which is constitutively driven by the thermodynamically dual Mandel stress. A simple and intuitive anisotropic evolution law is considered, where slipping is permitted along a fixed material plane which is prescribed as part of the evolution law. An additional yield criterion is proposed to complete the model. It is demonstrated that the anisotropy of the evolution law renders perfect plasticity, hardening or softening and is rate dependent.

A general density-preserving remodeling law for isotropic materials

The theory of material evolution is specialized to accommodate density-preserving remodeling of isotropic materials. It is assumed that in the pure mechanical case, the rate of evolution depends on the stress, deformation and the evolution. The dissipation inequality and the conservation of density indicate that the driving force for the evolution is the symmetric deviatoric part of the Mandel stress. The isotropic tensor-valued function representation theorem is used to show that there are 18 different admissible evolution modes. Assuming that the dissipation inequality takes a quadratic form, each evolution mode is driven by an associated configurational force. In the proposed evolution model each mode is governed by a single material constant corresponding to viscosity. Moreover, consistent evolution criteria are developed such that evolution arises only if a certain threshold is reached.