Elasticity

The elastic stress rate is

$$C_{ijkl} \dot{{\epsilon_{kl}}}^{{\rm elas}}$$

where $C_{ijkl}$ is the elastic modulus tensor (which is a doubly symmetric tensor: $C_{ijkl}=C_{jikl}$, $C_{ijkl}=C_{ijlk}$ and $C_{ijkl}=C_{jilk}$), and $\dot{{\epsilon_{kl}}}^{{\rm elas}}$ is the elastic strain rate. See the plasticity section for a definition of the elastic strain rate.

For an isotropic material

$$C_{0000} = C_{1111} = C_{2222} = \frac{E(1-\nu)}{(1+\nu)(1-2\nu)}$$

$$C_{0011} = C_{0022} = C_{1122} = \frac{E\nu}{(1+\nu)(1-2\nu}$$

$$C_{0101} = C_{0202} = C_{1212} = \frac{E}{1+\nu}$$

with $E$ group_materi_elasti_young modulus and $\nu$ group_materi_elasti_poisson ratio (the remaining non-zero moduli follow from the double symmetry conditions).

For a transverse isotropic material the material has one unique direction (think of an material with fibers in one direction). Here we take 'a' as the unique direction; 'b' and 'c' are the transverse directions. The material is fully defined by $C_{aaaa}$, $C_{bbbb}$, $C_{aabb}$, $C_{abab}$ and $C_{bcbc}$ and the unique direction in space (see group_materi_elasti_transverse_isotropy). The other non-zero moduli follow from $C_{cccc}=C_{bbbb}$, $C_{acac}=C_{abab}$, $C_{bbcc}=C_{bbbb}-2C_{bcbc}$ and from the double symmetry conditions.

The nonlinear elasticity polynomials is a strain dependent model. In this model, the 'young's stiffness' modulus is made dependend of the size of the strains via a series of polynomials

$$E = E_0 + E_1 \epsilon^1 + E_2 \epsilon^2 + \ldots$$ (1)

where

$$\epsilon = \sqrt ( \epsilon_{ij} \epsilon_{ij} )$$ (2)

with $\epsilon_{ij}$ the components of the strain matrix. The parameters $E_0$ etc. need to be specified in the group_materi_elasti_young_polynomial record.

The power law nonlinear elasticity is a stress dependent model which typically is used to model the elastic behavior of granular materials. It can be combined with plastic models, by example with the di Prisco plasticity model for soils, and with a poisson ratio.

In this model, the 'young's stiffness' modulus is made a function of the average stress state:

$$E = E_0 (p/p_0)^\alpha$$ (3)

where $p$ is the pressure. Furthermore, $E_0$ is the reference stiffness at reference pressure $p_0$, and $\alpha$ is a soil dependent power coefficient. The parameters $E_0$, $p_0$, and $\alpha$ need to be specified in the group_materi_elasti_young_power record.

The Lade nonlinear elasticity is a stress dependent model which typically is used to model the elastic behavior of granular materials. It can be combined with plastic models, by example with the di Prisco plasticity model for soils.

The stress rates are linked to the strain rates by the equation:

$$\dot{\epsilon_{ij}} = \frac{ \partial W^2 }{ \partial \sigma_{ij} \partial \sigma_{hk} } ~ \dot{\sigma_{hk}}$$ (4)

where the function $W$ is

$$W = \frac{ X^{1-\lambda} }{ 2 B (1-\lambda) }$$

where

$$X = p^2 + R^* \mathrm{abs}( s_{ij} s_{ij} )$$

with pressure $p=(\sigma_{11}+\sigma_{22}+\sigma_{33})/3$ and deviatoric stresses $s_{ij} = \sigma_{ij} - p \delta_{ij}$.

The model contains three user specified constants $B$, $R$, $\lambda$ which need to be specified in the group_materi_elasti_lade record. $B$ and $\lambda$ are defined by means of an isotropic unloading test, and $R$ by means of an unloading-standard-triaxial-compression test. For example for a loose sand $B=1028$, $R=0.25$, $\lambda=0.28$. See [7] for the details.

The model cannot be used in combination with a poisson ratio.