Hybrid 3D mass-spring system for simulation of isotropic materials with any Poisson’s ratio

Abstract

Mass-spring systems (MSS) simulating elastic materials obey constraints known in elasticity as the Cauchy relations, restricting the Poisson ratio of isotropic systems to be exactly $\nu =1/4$. We remind that this limitation is intrinsic to centrosymmetric spring systems (where each node is a center of symmetry), forbidding them for instance to simulate incompressible materials (with $\nu =1/2$). To overcome this restriction, we propose to supplement the spring deformation energy with an energy depending on the volume only, insensitive to change of shape, permitting MSS to simulate any real isotropic materials. In addition, the freedom in choosing the spring constants realizing a given elastic behavior allows to manage instabilities. The proposed hybrid model is evaluated by comparing its response to various deformation geometries with analytical model and/or finite element model. The results show that the hybrid MSS model allows to simulate any compressible isotropic elastic material and in particular the nearly incompressible (Poisson ratio $\nu \simeq 1/2$) biological soft tissues to which it is dedicated.

Appendix A. Theoretical notions of linear elasticity

1.1 The multi-constant theory of elasticity

Consider a deformation that brings a material point from its rest position $\mathbf {x}\in \mathbb {R}^3$ to a new position $\mathbf {x}+\mathbf {u}(\mathbf {x})$. Obviously, a homogeneous displacement $\mathbf {u}(\mathbf {x})=cst$ amounts to a uniform translation, generating no stress in the material, so the lowest-order quantity of interest in elasticity is the displacement gradient ${\varvec{\nabla }}\mathbf {u}$, the symmetric part of which is called the strain tensor defined by:

$$\begin{aligned} {\varvec{\varepsilon }} =\frac{1}{2}\left( {\varvec{\nabla }}\mathbf {u}+({\varvec{\nabla }}\mathbf {u})^\mathrm {T}\right) ,\quad \varepsilon _{ij} = \varepsilon _{ji}= \frac{1}{2}\left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) . \end{aligned}$$

(7.1)

In linear elasticity, the deformation energy is a quadratic function of the deformation tensor, its volume density being written

$$\begin{aligned} W = \frac{1}{2} \; \mathsf {C}_{ijkl} \; \varepsilon _{ij} \; \varepsilon _{kl}, \end{aligned}$$

(7.2)

where the fourth-rank tensor C is the stiffness tensor. [We use the implicit summation convention of repeated Cartesian indices i, j, k and l over the values 1, 2 and 3.] The antisymmetric part of ${\varvec{\nabla }}\mathbf {u}$, which corresponds to solid-body rotations, does not enter expression (7.2) since the deformation energy is invariant with respect to such rotations. Moreover, since $\varepsilon _{ij}=\varepsilon _{ji}$, one has

$$\begin{aligned} \mathsf {C}_{ijkl} =\frac{\partial ^2 W}{\partial \varepsilon _{ij}\partial \varepsilon _{kl}} =\mathsf {C}_{jikl} =\mathsf {C}_{ijlk} =\mathsf {C}_{klij}. \end{aligned}$$

(7.3)

The stress tensor ${\varvec{\sigma }}$ derives from the energy:

$$\begin{aligned} \sigma _{ij}=\frac{\partial W}{\partial \varepsilon _{ij}}=\mathsf {C}_{ijkl}\varepsilon _{kl}, \end{aligned}$$

(7.4)

the component $\sigma _{ij}$ being the elastic force along the i-direction transmitted across a unit area of a surface normal to the j-direction. This proportionality relation between stress and strain is Hooke’s law.

1.2 Relations for an isotropic material

Among the $3^4=81$ components of the stiffness tensor C, only 21 are independent in the general case due to relations (7.3) [24]. Symmetries of the material further reduce this number; the most symmetric materials being isotropic, in which case $C^{(iso)}$ is independent of the orientation of the reference axes and can be written

$$\begin{aligned} \mathsf {C}^{(iso)}_{ijkl} = \lambda \;\delta _{ij}\delta _{kl} +G\; (\delta _{ik}\delta _{jl}+\delta _{il}\delta _{jk}) \end{aligned}$$

(7.5)

in terms of the two Lamé constants $\lambda$ and $G$, where $G$ is named the shear modulus (the relation between stiffness tensor and material symmetry is derived in [24].) The deformation energy of an isotropic material then reads

$$\begin{aligned} W^{(iso)} =&\frac{\lambda }{2} Tr(\varepsilon )^2 + G\;Tr (\varepsilon )^2= \frac{\lambda }{2} (\varepsilon _{kk})^2 + G\;\varepsilon _{ij}\varepsilon _{ij} \end{aligned}$$

(7.6)

and the stress tensor is

$$\begin{aligned} \sigma _{ij}^{(iso)}=\lambda \; \delta _{ij} \;\varepsilon _{kk}+2\; G\; \varepsilon _{ij} \end{aligned}$$

(7.7)

Besides, an alternative set of elastic constants, widely used to wholly describe an isotropic elastic material, is the Young modulus E and the Poisson ratio $\nu$ which are related to Lamé parameters by:

$$\begin{aligned} E=G\;\frac{3\lambda +2G}{\lambda +G} \quad \text {and}\quad \nu =\frac{\lambda }{2\; (\lambda +G)}. \end{aligned}$$

(7.8)

In particular, they are most useful in uniaxial stress situations: in a stress state such that the only nonzero stress component is $\sigma _{xx}^{(iso)}$, the nonzero strain components are $\varepsilon _{xx}=\sigma ^{(iso)}_{xx}/E$ and $\varepsilon _{zz}=\varepsilon _{yy}=-\nu \,\varepsilon _{xx}$.

A last elastic constant of interest is the bulk modulus

$$\begin{aligned} K=\lambda +\frac{2\; G}{3} \end{aligned}$$

(7.9)

such that, in an uniform dilation $\varepsilon _{ij}=\frac{1}{3}\delta _{ij}\varDelta V/V$ changing volume V by $\varDelta V=V\varepsilon _{kk}$, the stress is $\sigma ^{(iso)}_{ij}=K\varDelta V/V\delta _{ij}=K\varepsilon _{kk}\delta _{ij}$.