A Geometric Theory of Growth Mechanics

Abstract

In this paper we formulate a geometric theory of the mechanics of growing solids. Bulk growth is modeled by a material manifold with an evolving metric. Time dependence of metric represents the evolution of the stress-free (natural) configuration of the body in response to changes in mass density and "shape". We show that time dependency of material metric will affect the energy balance and the entropy production inequality; both the energy balance and the entropy production inequality have to be modified. We then obtain the governing equations covariantly by postulating invariance of energy balance under time-dependent spatial diffeomorphisms. We use the principle of maximum entropy production in deriving an evolution equation for the material metric. In the case of isotropic growth, we find those growth distributions that do not result in residual stresses. We then look at Lagrangian field theory of growing elastic solids. We will use the Lagrange-d'Alembert's principle with Rayleigh's dissipation functions to derive all the governing equations. We make an explicit connection between our geometric theory and the conventional multiplicative decomposition of deformation gradient $\mathbf{F}=\mathbf{F}_e\mathbf{F}_g$ into growth and elastic parts. We linearize the nonlinear theory and derive a linearized theory of growth mechanics. Finally, we obtain the stress-free growth distributions in the linearized theory

A short introduction to morphoelasticity: the mechanics of growing elastic tissues

“…We will not go further into differential geometric aspects here. Detailed treatments can be found in (Marsden and Hughes, 1994) and (Yavari, 2010).…”

An Approach to Modeling of Additive Manufacturing Technologies

“…Investigation of the stress-strain state for growing solids has been carried out in numerous works [1,2,6,11,12,20]. One may highlight a number of trends in generalization of the classical continuum mechanics that has been utilized in this papers.…”

“…At the same time we have to use the stressed configuration as a reference which complicates the formulation of constitutive relations. In particular, they have one more tensor argument -"implant" [6,14,20] which characterizes the initial local deformation. However, the geometric meaning of the implant becomes clear if we consider it as an initial ("assembly") local deformation of the element in natural state which leads directly to the notion of local transformation of the natural frame used in the geometry of the space with absolute parallelism and, thus, introduces the concept of non-Euclidean geometry.…”

On Spatial Covariance, Second Law of Thermodynamics and Configurational Forces in Continua

“…x which satisfies the balance of energy, consider an arbitrary superposed spatial diffeomorphism : S S. It is noted that, unlike the standard approaches to bodies with internal structure (see, e.g., Mariano [27]; Yavari and Marsden [13]; Yavari [15]) the thermodynamic force work-conjugate to c, that is , e  c does not enter the expression of working. Accordingly, the proposed scheme resembles the standard internal variables approaches, in the sense that the material metric c in the spatial configuration does not enter explicitly the balance of energy equation.…”

“…Moreover, he derived the standard splitting of the total energy into internal and kinetic energies and the transformation law for external forces. Since then a lot of effort has been placed and a series of papers have appeared in the literature dealing with the concept of covariant energy balance in both a conservative (see, e.g., Yavari et al [10]; Kanso et al [11]; Yavari and Ozakin [12]; Yavari and Marsden [13]; Panoskaltsis and Soldatos [14]) as well as a dissipative (see, e.g., Yavari [15]; Panoskaltsis et al [16,17]) setting.…”