Mechanics of Growth and Growth of Mechanics

Abstract 

As an active research area in biomechanics, mechanics of growth studies the mechanical aspects of growth of biological tissues [1]-[4]. While the classical theories of mechanics have provided powerful tools in studying the growth of biological tissues, the physiological process of growth, in turn, presents special challenges to the theory of mechanics, and brings forth the growth of mechanics itself. In this talk we discuss some issues that have emerged in this process. 

The existing mechanics theory of growth is based on the decomposition of the defor- mation gradient tensor into a growth tensor and an accommodation elastic tensor. This decomposition requires a fixed reference configuration from which the deformation gradi- ent can be properly defined. It is built on a fundamental premise in continuum mechanics that there exists a one-to-one correspondence between the material particles in the reference configuration and in the current configuration. The assumption of the one-to-one corre- spondence is, however, inconsistent with a physical process of growth in which the newly deposited materials particles did not exist in a fixed reference configuration. In particular, the existing growth theory is incapable of modeling the surface growth in which changes in topology can occur as a material surface grows into a volume. In this talk, we present a theory that does not require the existence of a fixed reference configuration. 

To set the stage for the new growth theory, we first discuss a theory of elasticity that does not require a fixed reference configuration. In this theory, the movements of the material particles are described by the velocity field in the current configuration. The constitutive theory does not rely on the concept of a natural state and a fixed reference configuration. Instead, it uses the current configuration as reference. As the elastic body deforms, the response function, which gives the stress tensor in terms of the deformation gradient, is constantly updated. We derive an evolution equation for the response function. 

The proposed growth theory is then developed by incorporating a growth rate field into the above elasticity theory. The growth rate field, defined on the current configuration, de- scribes how new material particles are added to a growing elastic body in its current state. For volumetric growth, the growth rate field has its support on the growing region. In con- trast, for surface growth, the growth rate field has support on a surface and is conveniently described by singular distribution functions. The velocity field may suffer jump disconti- nuities across the surface. The evolution equation for the response function of a growing elastic body is derived that describes how the stress changes with growth and deformation. 

An example of internal surface growth is presented.

 About the Speaker

 Yi-chao Chen is a Professor at University of Houston. He received his Ph.D. from the University of Minnesota, and M.S. from Johns Hopkins University. His research interests include continuum mechanics, biomechanics, stability analysis, bifurcation theory, and multifunctional materials.

Host: Professor Callan Luetkemeyer