Keywords: multi-scale, biological tissue, representative volume element, virial stress.

In this paper we present a micro-macro method for the computation of large elastic deformations of plant tissue, based on the concept of representative volume elements (RVEs) and similar to the approach described by Kouznetsova et al. [1]. At the microscopic level, we use a mass-spring model to describe the geometric structure and basic properties of the onion epidermis tissue [2]. The macroscopic domain is discretized using standard finite elements, in which the unknown material properties (the stress-strain relations) are computed using the microscopic model in small subdomains (the RVEs). To these RVEs, a macroscopic deformation is applied via appropriate boundary conditions, and from the equilibrium solution, the corresponding Cauchy stress tensor is computed via a virial stress formula. The spatial elasticity tensor is estimated via forward differencing of the Truesdell rate of the Cauchy stress.

We demonstrate via numerical experiments that the resulting computational multi-scale method converges to the solution of the full microscopic solution for successively refined meshes. We observed quadratic convergence for the Newton method for solving the nonlinear finite element equation, which ensures that the spatial elasticity tensor was correctly approximated. We also give an in-depth discussion on the computational efficiency of the method. We observe that by decoupling the full microscopic domain into many small microscopic simulations inside the RVEs, we can achieve a large speedup compared to a full microscopic simulation, this even when the separation in spatial scales is rather small.

While, in the present paper, we have restricted ourselves to a finite element formulation of elastic deformation at the macroscopic level, we emphasize that the proposed multi-scale approach can also be gradually refined to deal with visco-elasto-plastic deformation.

1

V. Kouznetsova, W.A.M. Brekelmans, F.P.T. Baaijens, "An approach to micro-macro modeling of heterogeneous materials", Computational Mechanics, vol. 27, pp. 37-48, 2001. doi:10.1007/s004660000212

2

J. Loodts, E. Tijskens, C. Wei, E. Vanstreels, B. Nicolai, H. Ramon, "Micromechanics: Simulating the elastic behavior of onion epidermis tissue", Journal of Texture Studies, vol. 37, pp. 16-34, February 2006. doi:10.1111/j.1745-4603.2006.00036.x

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