Constitutive Models in Finite Element Codes

Abstract

In finite element simulations the constitutive information is usually handled by a user-supplied subroutine. For a prescribed strain increment, this subroutine provides the finite element code with the corresponding stress increment and the Jacobian, which is required to build the consistent tangent operator. We propose an approach that relieves the user from computing and coding the Jacobian information. Instead, this information is computed automatically together with the stress increment. This approach requires reliable and efficient numerical integration. In particular, adaptivity and automatic error control are highly desirable features. Such integrators are presented in this article. The underlying ideas of the approach are first elucidated at simple one-dimensional problems from geotechnics. However, it is also discussed how this concept can be used in a fully three-dimensional framework. We expect that this new approach will strongly enhance the development of constitutive models and help to identify the most appropriate ones.

Appendix: Hypoplastic Models

For the sake of completeness, we outline the used hypoplastic model and the parameters used for our calculations. Tensors of second order are denoted with bold letters (e.g., $\mathbf{D},\mathbf{T},\boldsymbol{\delta },\mathbf{N}$) and tensors of fourth order with calligraphic letters (e.g., $\mathcal{L},\mathcal{M}$). Different kinds of tensorial multiplication are used: $\mathbf{TD} = T_{\mathit{ij}}D_{kl}$, $\mathbf{T}: \mathbf{D} = T_{\mathit{ij}}D_{\mathit{ij}}$, $\mathcal{L}: \mathbf{D} = L_{ijkl}D_{kl}$, $\mathbf{T} \cdot \mathbf{D} = T_{\mathit{ij}}D_{jk}$. The Euclidian norm of a tensor is $\|\mathbf{D}\| = \sqrt{D_{\mathit{ij } } D_{\mathit{ij }}}$. Unit tensors of second and fourth orders are denoted by I and $\mathcal{I}$, respectively.

1.1.1 A.1 Basic Model

The basic hypoplastic model was proposed in [20]:

$$\displaystyle{ \mathring{\mathbf{T}} = \mathcal{L}(\mathbf{T},e): \mathbf{D} + \mathbf{N}(\mathbf{T},e)\|\mathbf{D}\| }$$

(1.125)

with the linear term

$$\displaystyle{ \mathcal{L} = f_{s} \frac{1} {\hat{\mathbf{T}}:\hat{ \mathbf{T}}}\Big({F}^{2}\mathcal{I} + {a}^{2}\hat{\mathbf{T}}\hat{\mathbf{T}}\Big) }$$

(1.126)

and the nonlinear term

$$\displaystyle{ \mathbf{N} = f_{s}f_{d} \frac{aF} {\hat{\mathbf{T}}:\hat{ \mathbf{T}}}\Big(\hat{\mathbf{T}} +\hat{{ \mathbf{T}}}^{{}^{{\ast}} }\Big)\,. }$$

(1.127)

The employed stress variables are defined as follows

$$\displaystyle{ \hat{\mathbf{T}} = \frac{\mathbf{T}} {\mbox{ tr}\,\mathbf{T}}\,,\qquad \hat{{\mathbf{T}}}^{{}^{{\ast}} } =\hat{ \mathbf{T}} -\frac{1} {3}\mathbf{I}\,. }$$

The factors for pressure and density dependency (barotropy and pyknotropy) are given by

$$\displaystyle{ \begin{array}{c} a = \frac{\sqrt{3}(3-\sin \varphi _{c})} {2\sqrt{2}\sin \varphi _{c}} \,,\qquad f_{d} ={ \left ( \frac{e-e_{d}} {e_{c}-e_{d}}\right )}^{\alpha }, \\ f_{s} = \frac{h_{s}} {n}{ \left (\frac{e_{i}} {e} \right )}^{\beta }\frac{1+e_{i}} {e_{i}}{ \left (\frac{-\mbox{ tr}\,\mathbf{T}} {h_{s}} \right )}^{1-n}{\Bigg[3 + {a}^{2} - a\sqrt{3}{\left (\frac{e_{i0}-e_{d0}} {e_{c0}-e_{d0}} \right )}^{\alpha }\Bigg]}^{-1}. \end{array} }$$

The factor F for adapting the deviatoric yield surface to that of Matsuoka–Nakai is

$$\displaystyle{ F = \sqrt{\frac{1} {8}{\tan }^{2}\psi + \frac{2 {-\tan }^{2}\psi } {2 + \sqrt{2}\tan \psi \cos 3\theta }} - \frac{1} {2\sqrt{2}}\tan \psi }$$

with

$$\displaystyle{ \tan \psi = \sqrt{3}\,\big\|\hat{{\mathbf{T}}}^{{}^{{\ast}} }\big\|\quad \mbox{ and}\quad \cos 3\theta = -\sqrt{6}\frac{\mbox{ tr}\,(\hat{{\mathbf{T}}}^{{}^{{\ast}} }\cdot \hat{{\mathbf{T}}}^{{}^{{\ast}} }\cdot \hat{{\mathbf{T}}}^{{}^{{\ast}} })} {{\big[\hat{{\mathbf{T}}}^{{}^{{\ast}}}:\hat{{ \mathbf{T}}}^{{}^{{\ast}}}\big]}^{3/2}} \,. }$$

The void ratios are assumed to fulfill the compression model

$$\displaystyle{ \frac{e_{i}} {e_{i0}} = \frac{e_{c}} {e_{c0}} = \frac{e_{d}} {e_{d0}} =\exp \left [-{\left (\frac{-\mbox{ tr}\,\mathbf{T}} {h_{s}} \right )}^{n}\right ]\,. }$$

(1.128)

This hypoplastic relation has eight parameters: the critical friction angle $\varphi _{c}$, the granular hardness h _s , the void ratios e _i0, e _c0, and e _d0, and the exponents n, α, and β. They can be determined easily from simple index and element tests [13].

Since the mass is assumed to remain constant, the evolution of the void ratio e is described by

$$\displaystyle{ \dot{e} = (1 + e)\,\mbox{ tr}\,\mathbf{D}\,. }$$

(1.129)

1.1.2 A.2 Extended Hypoplastic Model

The here used extended version of hypoplasticity with intergranular strain was proposed in [16]. The general stress–strain relation is written as

$$\displaystyle{ \mathring{\mathbf{T}} = \mathcal{M}: \mathbf{D}\,, }$$

(1.130)

where $\mathcal{M}$ is a fourth-order tensor that represents the stiffness. It depends on the hypoplastic tensors $\mathcal{L}(\mathbf{T},e)$ and N(T, e) and is defined as follows:

$$\displaystyle{ \begin{array}{rcl} \mathcal{M}& =&\big[{\rho }^{\chi }m_{T} + (1 {-\rho }^{\chi })m_{R}\big]\mathcal{L} + \\ & &\quad \left \{\begin{array}{ll} {\rho }^{\chi }(1 - m_{ T})\mathcal{L}:\hat{ \boldsymbol{\delta }}\hat{\boldsymbol{\delta }} {+\rho }^{\chi }\mathbf{N}\hat{\boldsymbol{\delta }}&\quad \mbox{ for }\ \hat{\boldsymbol{\delta }}: \mathbf{D} > 0\,, \\ {\rho }^{\chi }(m_{R} - m_{T})\mathcal{L}:\hat{ \boldsymbol{\delta }}\hat{\boldsymbol{\delta }} &\quad \mbox{ for }\ \hat{\boldsymbol{\delta }}: \mathbf{D} \leq 0\,, \end{array} \right. \end{array} }$$

(1.131)

where $\boldsymbol{\delta }$ is the intergranular strain, and m _R , m _T , χ, and R denote material parameters. The normalized magnitude of $\boldsymbol{\delta }$ is defined as

$$\displaystyle{ \rho = \frac{\|\boldsymbol{\delta }\|} {R}\,. }$$

(1.132)

Further, the direction of the intergranular strain $\boldsymbol{\delta }$ is set

$$\displaystyle{ \hat{\boldsymbol{\delta }} = \left \{\begin{array}{ll} \boldsymbol{\delta }/\|\boldsymbol{\delta }\| &\quad \mbox{ for }\ \boldsymbol{\delta }\neq \mathbf{0}\,,\\ \mathbf{0} &\quad \mbox{ for } \ \boldsymbol{\delta } = \mathbf{0}\,. \end{array} \right. }$$

(1.133)

The evolution equation of the intergranular strain tensor $\boldsymbol{\delta }$ is postulated as

$$\displaystyle{ \mathring{\boldsymbol{\delta }} = \left \{\begin{array}{ll} (\mathcal{I}-\hat{\boldsymbol{\delta }}\hat{{\boldsymbol{\delta }}\rho }^{\beta _{r}}): \mathbf{D}&\quad \mbox{ for }\ \hat{\boldsymbol{\delta }}: \mathbf{D} > 0\,, \\ \mathbf{D} &\quad \mbox{ for }\ \hat{\boldsymbol{\delta }}: \mathbf{D} \leq 0\,, \end{array} \right. }$$

(1.134)

where $\mathring{\boldsymbol{\delta}}$ is the objective rate of intergranular strain and the exponent β _r is a material parameter.

For a monotonic continuation of straining with $\mathbf{D} \sim \hat{\boldsymbol{\delta }}$, the stiffness is

$$\displaystyle{ \mathcal{M} = \mathcal{L} + \mathbf{N}\hat{\boldsymbol{\delta }}\,. }$$

(1.135)

Note that $\mathbf{D} =\hat{ \boldsymbol{\delta }}\|\mathbf{D}\|$ and $\mathbf{N}\hat{\boldsymbol{\delta }}: \mathbf{D} = \mathbf{N}\|\mathbf{D}\|$ in this case. Thus we obtain the basic hypoplastic equation (1.125).

1.1.3 A.3 Material Parameters

The parameters used in all calculations are listed in Tables 1.4 and 1.5.

Table 1.4 Parameters for the basic model

Table 1.5 Parameters for the extended model