M.B. Rubin

Cosserat Theories: Shells, Rods and Points

Preface. 1. Introduction. 2. Basic Tensor Operations in Curvilinear Coordinates. 3. Three-Dimensional Continua. 4. Cosserat Shells. 5. Cosserat Rods. 6. Cosserat Points. 7. Numerical Solutions using Cosserat Theories. Appendix A: Tensors, Tensor Products and Tensor Operations in Three Dimensions. Appendix B: Summary of Tensor Operations in Specific Coordinate Systems. Exercises. Acknowledgments. References. Index.

A three-dimensional nonlinear model for dissipative response of soft tissue

A set of three-dimensional constitutive equations is proposed for modeling the nonlinear dissipative response of soft tissue. These constitutive equations are phenomenological in nature and they model a number of physical features that have been observed in soft tissue. The equations model the tissue as a composite of a purely elastic component and a dissipative component, both of which experience the same total dilatation and distortion. The stress response of the purely elastic component depends on dilatation, distortion and the stretch of material fibers, whereas the stress response of the dissipative component depends on distortional deformation only. The equations are hyperelastic in the sense that the stress is obtained by derivatives of a strain energy function, and they are properly invariant under superposed rigid body motions. In contrast with standard viscoelastic models of tissues, the proposed constitutive model includes the total deformation rate in evolution equations that can reproduce the observed physical feature that the hysteresis loops of most biological soft tissues are nearly independent of strain rate (Biomechanics, Mechanical Properties of Living Tissues, second ed. (1993)). Material constants are determined which produce good agreement with uniaxial stress experiments on superficial musculoaponeurotic system and facial skin.

A new 3-D finite element for nonlinear elasticity using the theory of a Cosserat point

Abstract The theory of a Cosserat point has been used to formulate a new 3-D finite element for the numerical analysis of dynamic problems in nonlinear elasticity. The kinematics of this element are consistent with the standard tri-linear approximation in an eight node brick-element. Specifically, the Cosserat point is characterized by eight director vectors which are determined by balance laws and constitutive equations. For hyperelastic response, the constitutive equations for the director couples are determined by derivatives of a strain energy function. Restrictions are imposed on the strain energy function which ensure that the element satisfies a nonlinear version of the patch test. It is shown that the Cosserat balance laws are in one-to-one correspondence with those obtained using a Bubnov–Galerkin formulation. Nevertheless, there is an essential difference between the two approaches in the procedure for obtaining the strain energy function. Specifically, the Cosserat approach determines the constitutive coefficients for inhomogeneous deformations by comparison with exact solutions or experimental data. In contrast, the Bubnov–Galerkin approach determines these constitutive coefficients by integrating the 3-D strain energy function using the kinematic approximation. It is shown that the resulting Cosserat equations eliminate unphysical locking, and hourglassing in large compression without the need for using assumed enhanced strains or special weighting functions.

On the Theory of a Cosserat Point and Its Application to the Numerical Solution of Continuum Problems

The theory of a Cosserat point is developed to describe motion of a body that is essentially a material point surrounded by some small volume. The development of this theory is motivated mainly by its applicability to the numerical solution of continuum problems. Attention is confined to the purely mechanical theory and nonlinear balance laws are proposed for Cosserat points with arbitrary constitutive properties. The linearized theory is developed and constitutive equations for an elastic material are discussed within the context of both the nonlinear and linear theories. Explicit constitutive equations for a linear-elastic isotropic Cosserat point are developed to model a parallelepiped composed of a linear-elastic homogeneous isotropic material.

Hyperbolic heat conduction and the second law

Abstract It is well known that Cattaneos modification of the constitutive equation for heat flux yields a telegraph equation for determining temperature that models hyperbolic heat conduction with a finite thermal wave speed. By analyzing the solution of an example of thermal equilibration of an initially inhomogeneous state it is shown that Cattaneos model violates a notion of the second law of thermodynamics because it predicts that heat may flow from cold to hot regions during finite time periods. In this paper we discuss modified restrictions associated with the second law and we propose an alternative formulation of the equations for heat conduction which retains the usual Fourier equation for heat flux but modifies the constitutive equations for internal energy, Helmholtz free energy, and entropy to include dependence on temperature rate. This alternative formulation satisfies the restrictions associated with the second law and produces the same telegraph equation for hyperbolic heat conduction as the Cattaneo model. However, since Fourier heat conduction is retained the new model predicts that heat always flows from hot to cold regions. An additional example of uniform heating is considered to show that the solution for temperature of the model proposed here differs from the solution associated with the parabolic and Cattaneo models.

Continuum model of dispersion caused by an inherent material characteristic length

Modifications of the Helmholtz free energy and the stress associated with general constitutive equations of a simple continuum are proposed to model dispersive effects of an inherent material characteristic length. These modifications do not alter the usual restrictions on the unmodified constitutive equations imposed by the first and second laws of thermodynamics. The special case of a thermoelastic compressible Newtonian viscous fluid is considered with attention focused on uniaxial strain. Within this context, the linearized problems of wave propagation in an infinite media and free vibrations of a finite column are considered for the simple case of elastic response. It is shown that the proposed model predicts the dispersive effects observed in wave propagation through a chain of springs and masses as the wavelength decreases. Also, the nonlinear problems of steady wave propagation of a soliton in the absence of viscosity and of a shock wave in the presence of viscosity are discussed. In particular it...

Constrained theories of rods

Utilizing a nonlinear theory of rods, which is formulated on the basis of a Cosserat curve with two directors, a number of constrained theories of varios degrees of generality are developed. In addition to the nonlinear version of the Bernoulli-Euler beam theory (discussed for completeness and clarity), six other less restrictive nonlinear constrained theories are also discussed. A table is provided, which summarizes the degree of exclusion of certain modes of motion or deformation, and which indicates the system of differential equations to be used in applications.

An Elastic–Viscoplastic Model for Excised Facial Tissues

Unified constitutive equations for elastic-viscoplastic materials were modified and used to model the highly nonlinear elastic and rate-dependent inelastic response exhibited in recent experiments on excised facial tissues. These included the skin and the underlying supportive tissue SMAS (the Superficial Musculoaponeurotic System). This study indicates a number of relevant results: The skin is more strain rate dependent than the SMAS; the nonlinearity of the elasticity of the skin is greater than that of the SMAS; both tissues exhibit a hardening effect indicated by increased resistance to inelastic deformation due to stress acting over a time period; the hardening effect leads to a decrease in time dependence and an increased elastic range, which is more pronounced for SMAS. Consequently, the SMAS can be viewed as the firmer elastic foundation of the more viscous skin. Moreover, the relaxation time for the skin is fairly short so the skin would be expected to conform to the deformation of the SMAS if it remained attached to the SMAS during stretching. This is relevant when it is undesirable to separate the skin from the SMAS for physiological reasons.

On the significance of normal cross-sectional extension in beam theory with application to contact problems

Abstract The main features of contact problems of elastic beams are explored by considering a specific equilibrium problem of a beam in contact with a smooth rigid flat surface. Solutions of four separate linear theories, namely a general theory (G) and three others which are constructed as constrained theories are considered. These constrained theories differ from the general theory only by the degree of exclusion of one or both types of deformation usually referred to as (a) transverse normal strain and (b) transverse shear deformation. Thus, with both (a) and (b) absent, the constrained theory, corresponds to the Bernoulli-Euler beam theory (BE); with only (b) absent, the constrained theory (N) accounts for normal extensional deformation; and with only (a) absent, the constrained theory corresponds to the Timoshenko beam theory (T). Comparison of the predictions of the solutions of the three constrained theories with that of theory G shows that theory N contains the main physics of the contact problem and correctly predicts the conditions under which the beam loses contact. Also the contact force is continuous at the end points of the contact region. In contrast, neither of the other two constrained theories (BE or T) correctly predicts these features.

Mechanical and numerical modeling of a porous elastic- viscoplastic material with tensile failure

The objective of this paper is to develop simple but comprehensive constitutive equations that model a number of physical phenomena exhibited by dry porous geological materials and metals. For geological materials the equations model: porous compaction; porous dilation due to distortional deformation and tensile failure; shear enhanced compaction; pressure hardening of the yield strength; damage of the yield strength due to distortional deformation and porosity changes; and dependence of the yield strength on the Lode angle. For metals the equations model: hardening of the yield strength due to plastic deformation; pressure and temperature dependence of the yield strength, and damage due to nucleation of porosity during tensile failure. The equations are valid for large deformations and the elastic response is hyperelastic in the sense that the stress is related to a derivative of the Helmholtz free energy. Also, the equations are viscoplastic with rate dependence occurring in both the evolution equations of porosity and elastic distortional deformations. Moreover, formulas are presented for robust numerical integration of the evolution equations at the element level that can be easily implemented into standard computer programs for dynamic response of materials.