Continuum mechanics and thermodynamics are foundational theories of many fields of science and engineering. This book presents a fresh perspective on these fundamental topics, connecting micro- and nanoscopic theories and emphasizing topics relevant to understanding solid-state thermo-mechanical behavior. Providing clear, in-depth coverage, the book gives a self-contained treatment of topics directly related to nonlinear materials modeling. It starts with vectors and tensors, finite deformation kinematics, the fundamental balance and conservation laws, and classical thermodynamics. It then discusses the principles of constitutive theory and examples of constitutive models, presents a foundational treatment of energy principles and stability theory, and concludes with example closed-form solutions and the essentials of finite elements. Together with its companion book, Modeling Materials, (Cambridge University Press, 2011), this work presents the fundamentals of multiscale materials modeling for graduate students and researchers in physics, materials science, chemistry and engineering.

Continuum Mechanics and Thermodynamics: From Fundamental Concepts to Governing Equations

This paper is concerned with investigation of the effects of strain-stiffening on the classical limit point instability that is well-known to occur in the inflation of internally pressurized rubber-like spherical thin shells (balloons) and circular cylindrical thin tubes composed of incompressible isotropic non-linearly elastic materials. For a variety of specific strain-energy densities that give rise to strain-stiffening in the stress-stretch response, the inflation pressure versus stretch relations are given explicitly and the non-monotonic character of the inflation curves is examined. While such results are known for constitutive models that exhibit a gradual stiffening (e.g. exponential and power-law models), our primary focus is on materials that undergo severe strain-stiffening in the stress-stretch response. In particular, we consider two phenomenological constitutive models that reflect limiting chain extensibility at the molecular level. It is shown that for materials with sufficiently low extensibility no limit point instability occurs and so stable inflation is then predicted for such materials. Potential applications of the results to the biomechanics of soft tissues are indicated.

Elastic instabilities for strain-stiffening rubber-like spherical and cylindrical thin shells under inflation

This paper is concerned with the effect of residual stress on the elastic behaviour of materials undergoing finite elastic deformations. The theory is based on a general constitutive framework for hyperelastic materials with residual stress. Several simple problems, whose solutions are known in the situation where there is no residual stress, are analyzed in order to elucidate the influence of residual stress, and the results are illustrated for two prototype constitutive laws.

The influence of residual stress on finite deformation elastic response

In this paper we analyze some aspects of the mechanics of anisotropic non-linearly elastic solids for which the anisotropy is associated with two preferred directions, thereby modelling the effect of two families of fiber reinforcement. Particular attention is focused on the role of the invariant I 8 , which couples the fiber kinematics in the constitutive law, and results are illustrated for a model strain–energy function that includes this invariant. The distinctive effects of this invariant on the stress–strain behaviour of the material and on the failure of ellipticity are discussed separately for pure homogeneous deformation and for a simple shear deformation. For pure homogeneous deformation it is shown that ellipticity fails with an associated weak surface of discontinuity that bisects the fiber directions. Moreover, ellipticity fails close to the reference configuration in either tension or compression. For simple shear it is found that loss of ellipticity occurs both for positive and negative shearing, and the orientation of the weak surface relative to the fiber directions is characterized in each case. Again ellipticity fails close to the reference configuration, and inclusion of the invariant I 8 in the constitutive law therefore has a pronounced destabilizing influence.

The influence of the invariant I8I8 on the stress deformation and ellipticity characteristics of doubly fiber-reinforced non-linearly elastic solids

We propose a new elastic constitutive law for arterial tissue in which the limiting polymeric chain extensibility of both collagen and elastin fibres is accounted for. The elastic strain-energy function is separated additively into two parts: an isotropic contribution associated with the matrix (incorporating the elastin fibre network) and an anisotropic one associated with the collagen fibres. Information on the limiting extensibility in each case provides some mesoscopic input into the model. The (logarithm-based) model is compared with the Fung–Demiray exponential model and certain other recently proposed models. Some aspects of the elastic response under extension and inflation of a thin-walled circular cylindrical tube (the artery) are then examined and compared with the corresponding response of a rubberlike tube. We point out that our model, when both isotropic and anisotropic terms are included, can be developed to accommodate changing mechanical properties associated with degradation of the elastin and collagen by considering the material constants that define the limit of chain extensibility to evolve in time.

G. Saccomandi

Papers

Nonlinear Elasticity: Universal Results in Finite Elasticity