Analytical and numerical analyses of the micromechanics of soft fibrous connective tissues

State of the art research and treatment of biological tissues require accurate and efficient methods for describing their mechanical properties. Indeed, micromechanics motivated approaches provide a systematic method for elevating relevant data from the microscopic level to the macroscopic one. In this work the mechanical responses of hyperelastic tissues with one and two families of collagen fibers are analyzed by application of a new variational estimate accounting for their histology and the behaviors of their constituents. The resulting, close form expressions, are used to determine the overall response of the wall of a healthy human coronary artery. To demonstrate the accuracy of the proposed method these predictions are compared with corresponding 3-D finite element simulations of a periodic unit cell of the tissue with two families of fibers. Throughout, the analytical predictions for the highly nonlinear and anisotropic tissue are in agreement with the numerical simulations

Basic equations of continuum mechanics

This first chapter of the volume on Constitutive Modelling of Solid Continua sets out briefly the main concepts of general continuum mechanics without reference to specific material behaviour as a backdrop for the detailed descriptions of different types of material behaviour that are contained in the remaining chapters. The focus is therefore on the fundamental ideas of kinematics of a continuum (deformation and motion), the global balance equations governing the motion of a general continuum, stress and the energy balance equation and the derivation of the local governing equations, including mass conservation and the equation of motion. For the most part, detailed proofs are not provided and reference is made to standard texts for more details since this volume is aimed at researchers who have a reasonable background in continuum mechanics and wish to widen the scope of their knowledge in different areas of solid mechanics

Finite deformation elasticity theory

This chapter provides the framework for the development of constitutive theories of solids by focusing on constitutive laws for nonlinearly elastic solids. These exemplify the general principles of constitutive theory that should be applied to all types of material behaviour, in particular, the notions of objectivity and material symmetry, including the important symmetries of isotropy, transverse isotropy and orthotropy based in part on deformation invariants. Details are given for the various general stress–deformation relations for each case of symmetry in respect of hyperelastic materials (which are characterized by a strain-energy function), with or without an internal constraint such as incompressibility, and these are illustrated by particular prototype models. The notion of residual stress (in an unloaded configuration) is discussed and the form of strain-energy function required to accommodate residual stress in the material response is developed

Physics of growing biological tissues: the complex cross-talk between cell activity, growth and resistance

For many organisms, shapes emerge from growth, which generates stresses, which in turn can feedback on growth. In this review, theoretical methods to analyse various aspects of morphogenesis are discussed with the aim to determine the most adapted method for tissue mechanics. We discuss the need to work at scales intermediate between cells and tissues and emphasize the use of finite elasticity for this. We detail the application of these ideas to four systems: active cells embedded in tissues, brain cortical convolutions, the cortex of Caenorhabditis elegans during elongation and finally the proliferation of epithelia on extracellular matrix. Numerical models well adapted to inhomogeneities are also presented