Marcelo Epstein

Thermomechanics of volumetric growth in uniform bodies

Abstract A theory of material growth (mass creation and resorption) is presented in which growth is viewed as a local rearrangement of material inhomogeneities described by means of first- and second-order uniformity “transplants”. An essential role is played by the balance of canonical (material) momentum where the flux is none other than the so-called Eshelby material stress tensor. The corresponding irreversible thermodynamics is expanded. If the constitutive theory of basically elastic materials is only first-order in gradients, diffusion of mass growth cannot be accommodated, and volumetric growth then is essentially governed by the inhomogeneity velocity “gradient” (first-order transplant theory) while the driving force of irreversible growth is the Eshelby stress or, more precisely, the “Mandel” stress, although the possible influence of “elastic” strain and its time rate is not ruled out. The application of various invariance requirements leads to a rather simple and reasonable evolution law for the transplant. In the second-order theory which allows for growth diffusion, a second-order inhomogeneity tensor needs to be introduced but a special theory can be devised where the time evolution of the second-order transplant can be entirely dictated by that of the first-order one, thus avoiding insuperable complications. Differential geometric aspects are developed where needed.

Evaluation of the finite element software ABAQUS for biomechanical modelling of biphasic tissues

The biphasic cartilage model proposed by Mow et al. (1980) has proven successful to capture the essential mechanical features of articular cartilage. In order to analyse the joint contact mechanics in real, anatomical joints, the cartilage model needs to be implemented into a suitable finite element code to approximate the irregular surface geometries of such joints. However, systematic and extensive evaluation of the capacity of commercial software for modelling the contact mechanics with biphasic cartilage layers has not been made. This research was aimed at evaluating the commercial finite element software ABAQUS for analysing biphasic soft tissues. The solutions obtained using ABAQUS were compared with those obtained using other finite element models and analytical solutions for three numerical tests: an unconfined indentation test, a test with the contact of a spherical cartilage surface with a rigid plate, and an axi-symmetric joint contact test. It was concluded that the biphasic cartilage model can be implemented into the commercial finite element software ABAQUS to analyse practical joint contact problems with biphasic articular cartilage layers.

The energy=momentum tensor and material uniformity in finite elasticity

SummaryEshelbys elastic energy-momentum tensor is shown to satisfy a differential identity which, in the general case of a uniform elastic body with inhomogeneities, is expressible in terms of the torsion of the material connection.

Material and functional properties of articular cartilage and patellofemoral contact mechanics in an experimental model of osteoarthritis

The purposes of this study were to determine the in situ functional and material properties of articular cartilage in an experimental model of joint injury, and to quantify the corresponding in situ joint contact mechanics. Experiments were performed in the anterior cruciate ligament (ACL) transected knee of the cat and the corresponding, intact contralateral knee, 16 weeks following intervention. Cartilage thickness, stiffness, effective Youngs modulus, and permeability were measured and derived from six locations of the knee. The total contact area and peak pressures in the patellofemoral joint were obtained in situ using Fuji Pressensor film, and comparisons between experimental and contralateral joint were made for corresponding loading conditions. Total joint contact area and peak pressure were increased and decreased significantly (alpha=0.01), respectively, in the experimental compared to the contralateral joint. Articular cartilage thickness and stiffness were increased and decreased significantly (alpha=0.01), respectively, in the experimental compared to the contralateral joint in the four femoral and patellar test locations. Articular cartilage material properties (effective Youngs modulus and permeability) were the same in the ACL-transected and intact joints. These results demonstrate for the first time the effect of changes in articular cartilage properties on the load transmission across a joint. They further demonstrate a substantial change in the joint contact mechanics within 16 weeks of ACL transection. The results were corroborated by theoretical analysis of the contact mechanics in the intact and ACL-transected knee using biphasic contact analysis and direct input of cartilage properties and joint surface geometry from the experimental animals. We conclude that the joint contact mechanics in the ACL-transected cat change within 16 weeks of experimental intervention.

Modelling of location- and time-dependent deformation of chondrocytes during cartilage loading.

Experimental evidence suggests that the biosynthetic activity of chondrocytes is regulated primarily by the mechanical environment. In order to study the mechanisms underlying remodeling, adaptation, and degeneration of articular cartilage in a joint subjected to changing loads, it is important to know the time-dependent fluid pressure and stress-strain state in chondrocytes. The purpose of the present study was to develop a theoretical model to simulate the mechanical behaviour of articular cartilage and to describe the time-dependent stress-strain state and fluid pressure distribution in chondrocytes during cartilage deformation. It was assumed that the volume occupied by the chondrocytes is small and that cartilage can be treated as a macroscopically homogenized material with effective material properties which depend on the material properties of the cells and matrix and the volumetric fraction of the cells. Model predictions on the time-dependent distribution of fluid pressure and stress and on the time-dependent cell deformation during confined and unconfined compression tests agree with previous theoretical predictions and experimental observations. The proposed model supplies the tools to study the mechanisms of degeneration, adaptation and remodelling of cartilage associated with cell loading and deformation.

Electric-field induced interfacial cracking in multilayer electrostrictive actuators

Abstract Electric-field induced interfacial cracking in multilayer electrostrictive actuators is studied for two typical cases : (1) an interface crack lying between an electrode layer and ceramic matrix ; and (2) an interface crack with one tip at an embedded electrode-edge. Based on the small-scale saturation solutions, a direct method is proposed to calculate the electric-field induced stress intensity factors that avoids calculating the stress field. The effectiveness of the direct method is demonstrated by comparing the results derived with the known numerical solutions. For either case, the explicit condition is given that prohibits interfacial crack growth by restricting the thickness of ceramic layers and the intensity of the applied electric field. Especially, the maximum stress intensity factor derived for the above case (2) is found to be significantly larger than that obtained in the existing works for the electrode-tip matrix cracks. This result agrees with the experimental fact that interfacial cracking is the dominant failure mechanism in electrostrictive multilayer devices. Hence, the reliability design should be based on the conditions that prohibit interfacial cracking.

A finite element formulation for multilayered and thick shells

Abstract A bilinear isoparametric finite element concept is used for the numerical analysis of multilayered plates. The underlying theory used allows for transverse shear and normal strains in each layer, thus extending the analysis to very thick plates and laminates. To illustrate the versatility of the multilayered element, three examples are presented and the results are compared with available exact solutions.

Can a rheological muscle model predict force depression/enhancement?

A new phenomenological model of activated muscle is presented. The model is based on a combination of a contractile element, an elastic element that engages upon activation, a linear dashpot and a linear spring. Analytical solutions for a few selected experiments are provided. This model is able to reproduce the response of cat soleus muscle to ramp shortening and stretching and, unlike standard Hill-type models, computations are stable on the descending limb of the force-length relation and force enhancement (depression) following stretching (shortening) is predicted correctly. In its linear version, the model is consistent with a linear force-velocity law, which in this model is a consequence rather than a fundamental characteristic of the material. Results show that the mechanical response of activated muscle can be mimicked by a viscoelastic system. Conceptual differences between this model and standard Hill-type models are analyzed and the advantages of the present model are discussed.

Nonlinear analysis of multilayered shells

Abstract A large deformation theory for layered shells of arbitrarily varying thickness and using a piecewise smooth displacement field is developed. A system of layer coordinates is introduced which allows the results to be presented in a simple compact form analogous to the theory of monocoque shells.

On the geometrical material structure of anelasticity

SummaryG-structures are the geometric backbone of the theory of material uniformity in continuum mechanics. Within this geometric framework, anelasticity is seen as a result of evolving distributions of inhomogeneity reflected as material nonintegrability. Constitutive principles governing thetime evolution of the G-structure underlying the finite-strain theory of anelasticity (e.g., plasticity) are proposed. The material Eshelby stress tensor is shown to be thedriving force behind this evolution. This should allow for a thermodynamically admissible formulation of anelasticity viewed as a G-structure evolution.