A discrete element model for the investigation of the geometrically nonlinear behaviour of solids

Abstract

A three-dimensional discrete element model for elastic solids with large deformations is presented. Therefore, an discontinuum approach is made for solids. The properties of elastic material are transferred analytically into the parameters of a discrete element model. A new and improved octahedron gap-filled face-centred cubic close packing of spheres is split into unit cells, to determine the parameters of the discrete element model. The symmetrical unit cells allow a model with equal shear components in each contact plane and fully isotropic behaviour for Poisson’s ratio above 0. To validate and show the broad field of applications of the new model, the pin–pin Euler elastica is presented and investigated. The thin and sensitive structure tends to undergo large deformations and rotations with a highly geometrically nonlinear behaviour. This behaviour of the elastica can be modelled and is compared to reference solutions. Afterwards, an improved more realistic simulation of the elastica is presented which softens secondary buckling phenomena. The model is capable of simulating solids with small strains but large deformations and a strongly geometrically nonlinear behaviour, taking the shear stiffness of the material into account correctly.

Appendix

Crystals consisting of a face-centred cubic packing have three independent elastic components in the tensor $\mathbb {C}$. $\hat{\mathrm {C}}_3$ can be expressed as

$$\begin{aligned} \hat{\mathrm {C}}_3=\frac{\hat{\mathrm {C}}_1-\hat{\mathrm {C}}_2}{2} \end{aligned}$$

(28)

if there is fully isotropic behaviour. For the simple face-centred cubic packing (Index fcc) without filled gaps, the tensor of elasticity resulting from the Eq. (19) is

$$\begin{aligned} \mathbb {C}_{fcc}=\left[ \begin{array}{cccccc} \hat{\mathrm {C}}_1&{}\quad \hat{\mathrm {C}}_2&{}\quad \hat{\mathrm {C}}_2&{}\;&{} \;&{}\\ \hat{\mathrm {C}}_2&{}\quad \hat{\mathrm {C}}_1&{}\quad \hat{\mathrm {C}}_2&{}\;&{} \;&{}\\ \hat{\mathrm {C}}_2&{}\quad \hat{\mathrm {C}}_2&{}\quad \hat{\mathrm {C}}_1&{}\;&{} \;&{}\\ \;&{}\;&{} \;&{}\quad \hat{\mathrm {C}}_3 &{}\;&{} \;\\ \;&{}\;&{} \;&{} \;&{}\quad \hat{\mathrm {C}}_3&{} \;\\ \;&{}\;&{} \;&{} \; &{}\;&{}\quad \hat{\mathrm {C}}_3\\ \end{array}\right] , \end{aligned}$$

(29)

where

$$\begin{aligned} \hat{\mathrm {C}}_1&=\frac{\sqrt{2}}{2\mathrm {r}}(k_n+k_s),\quad \hat{\mathrm {C}}_2=\frac{\sqrt{2}}{4\mathrm {r}}(k_n-k_s)\;\;\;\text {and}\nonumber \\ \hat{\mathrm {C}}_3&=\frac{\sqrt{2}}{4\mathrm {r}} (k_n+k_s). \end{aligned}$$

(30)

Evaluating equation (28) yields a condition that

$$\begin{aligned} k_n{\mathop {=}\limits ^{!}}k_s, \end{aligned}$$

(31)

which can only be achieved for $\nu =0$. This states, that the simple face-centred cubic packing has cubic anisotropy for Poisson’s ratio above zero and therefore has three independent elastic constants. Due to the filling of the octahedron gaps, a third parameter is included. Evaluating equation (28) again for the tensor of elasticity from Eq. (20) yields in a new condition for isotropy

$$\begin{aligned} 2\,k_\mathrm{oct}{\mathop {=}\limits ^{!}}k_n+k_s. \end{aligned}$$

(32)

Equation (32) can be fulfilled for every $\nu$. Therefore, the three components can be reduced to two independent elastic constants.