A finite strain constitutive model for non-quadratic yield criteria and nonlinear kinematic/isotropic hardening: application to sheet metal forming

Abstract

In this paper, a finite strain material model for complex plastic anisotropy with nonlinear isotropic and kinematic hardening is consistently derived. The model is based on the classical multiplicative decomposition of the deformation gradient and derived in a thermodynamically consistent way. An important new aspect of the work is the straightforward implementation of general and anisotropic yield criteria into a constitutive model, which is entirely formulated in the reference configuration. Nevertheless, and for the sake of illustrating the potential of the model, in this work a Barlat-type (Yld2004-18p) yield criterion is employed. The kinematic hardening formulation follows the continuum mechanical extension of the classical rheological model of Armstrong–Frederick hardening. The numerical integration of the evolution equations is carried out using the exponential map approach, which is able to preserve plastic incompressibility. For numerical efficiency purposes, the exponential tensor functions are evaluated in a closed form using the spectral decomposition, and special attention is given to the preservation of the internal variables’ symmetry. The model is assessed by means of several numerical simulations for anisotropic materials at finite strains, including sheet metal forming processes with comparison to experimental data.

Appendix: Representation of Hill’s 1948 [20] yield function in the reference configuration

Regarding the yield function proposed by Hill [20], the effective stress is given by $\bar{\sigma }=\sqrt{{\varvec{\eta }}^{*}:{{\mathbb {A}}}:{\varvec{\eta }}^{*}}$. Introducing the relation (20) in this yield function, one obtains

$$\begin{aligned} {\varvec{\eta }}^{*}:{{\mathbb {A}}}:{\varvec{\eta }}^{*}= & {} {{\eta }}_{ij}^{*} {A}_{ijkl} {{\eta }}_{kl}^{*}= \left( {\mathrm{det {\mathbf{F}} }}\right) ^{-2} ({{\mathbf{R}}^{*{\mathrm{T}}}} {\mathbf{F}}^{-\mathrm{T}} {\mathbf{C}}_{p})_{im} {Y}_{mn} ({\mathbf{F}}^{\mathrm{T}} {\mathbf{R}}^{*})_{nj} {A}_{ijkl} ({{\mathbf{R}}^{*{\mathrm{T}}}} {\mathbf{F}}^{-\mathrm{T}} {\mathbf{C}}_{p})_{ko} {Y}_{op} ({\mathbf{F}}^{\mathrm{T}} {\mathbf{R}}^{*})_{pl}\nonumber \\= & {} {Y}_{mn} \bar{A}_{mnop} {Y}_{op} = {\mathbf{Y}}: {\bar{\mathbb {A}}}: {\mathbf{Y}}, \end{aligned}$$

(36)

The fourth-order tensor ${\bar{\mathbb {A}}}$ acts on stress quantities defined in the reference configuration and is related to the anisotropic tensor ${\mathbb {A}}$ following the expression

$$\begin{aligned} {\bar{A}}_{mnop} = \left( {\mathrm{det {\mathbf{F}} }}\right) ^{-2} ({\mathbf{C}}_{p} {\mathbf{F}}^{-1} {{\mathbf{R}}^{*}})_{mi} ({\mathbf{F}}^{\mathrm{T}} {\mathbf{R}}^{*})_{nj}({\mathbf{C}}_{p} {\mathbf{F}}^{-1} {{\mathbf{R}}^{*}})_{ok} ({\mathbf{F}}^{\mathrm{T}} {\mathbf{R}}^{*})_{pl} {A}_{ijkl} \end{aligned}$$

(37)

The fourth-order anisotropic tensor ${\mathbb {A}}$ can be represented by means of structural tensors (see, e.g. [47]), following

$$\begin{aligned} {\mathbb {A}}= a_1\mathbb {I} + a_2{\mathbf{M}}_{1}\otimes {\mathbf{M}}_{1} + a_3{\mathbf{M}}_{2}\otimes {\mathbf{M}}_{2} + a_4({\mathbf{M}}_{1}\otimes {\mathbf{M}}_{2} + {\mathbf{M}}_{2}\otimes {\mathbf{M}}_{1}) + a_5 \mathbb {G}_1 + a_6 \mathbb {G}_2 \end{aligned}$$

(38)

where the auxiliary fourth-order tensors $\mathbb {G}_1$ and $\mathbb {G}_2$ are given by

$$\begin{aligned} (G_{\alpha })_{ijkl} = \frac{1}{2} \left( (M_{\alpha })_{ik} \delta _{jl} + (M_{\alpha })_{il} \delta _{jk} + (M_{\alpha })_{jk} \delta _{il} + (M_{\alpha })_{jl} \delta _{ik} \right) , \ \forall \alpha =1,2. \end{aligned}$$

(39)

The coefficients $a_i$, with $i=1 \ldots 6$ are related to the classical Hill coefficients (F, G, H, L, M and N). The structural tensors represent the symmetry of the material and are expressed as

$$\begin{aligned} {\mathbf{M}}_{i} = {\mathbf{N}}_{i} \otimes {\mathbf{N}}_{i}, \ \forall i=1,2,3 \end{aligned}$$

(40)

where ${\mathbf{N}}_{i}$ are the privileged directions of the material.

Applying Eq. 37 to the anisotropic tensor defined by Eq. 38, and considering, for the sake of simplicity, only the second term on the right-hand side, it yields

$$\begin{aligned} {\mathbf{M}}_{1}\otimes {\mathbf{M}}_{1} \rightarrow \left( {\mathrm{det {\mathbf{F}} }}\right) ^{-2} ({\mathbf{C}}_{p} {\mathbf{F}}^{-1} {{\mathbf{R}}^{*}})_{mi} ({\mathbf{F}}^{\mathrm{T}} {\mathbf{R}}^{*})_{nj} ({\mathbf{C}}_{p} {\mathbf{F}}^{-1} {{\mathbf{R}}^{*}})_{ok} ({\mathbf{F}}^{\mathrm{T}} {\mathbf{R}}^{*})_{pl} ({{\mathbf{M}}_1})_{ij} ({{\mathbf{M}}_1})_{kl}= ({\bar{\mathbf{M}}_1})_{mn} ({\bar{\mathbf{M}}_1})_{op},\nonumber \\ \end{aligned}$$

(41)

where the structural tensor in the reference configuration is expressed as

$$\begin{aligned} {\bar{\mathbf{M}}}_{1}&= \left( {\mathrm{det {\mathbf{F}} }}\right) ^{-1} {\mathbf{C}}_{p} {\mathbf{F}}^{-1} {{\mathbf{R}}^{*}} {\mathbf{M}}_{1} {{\mathbf{R}}^{*{\mathrm{T}}}} {\mathbf{F}}\nonumber \\&=\left( {\mathrm{det {\mathbf{F}} }}\right) ^{-1} ({\mathbf{C}}_{p} {\mathbf{F}}^{-1} {{\mathbf{R}}^{*}} {\mathbf{N}}_{1}) \otimes ({\mathbf{N}}_{1} {{\mathbf{R}}^{*{\mathrm{T}}}} {\mathbf{F}}) = {\bar{\mathbf{N}}}_{1}^{\prime } \otimes {\bar{\mathbf{N}}}_{1}^{\prime \prime }, \end{aligned}$$

(42)

which is dependent on two privileged directions that evolve distinctly according to

$$\begin{aligned} {\bar{\mathbf{N}}}_{1}^{\prime } = \left( {\mathrm{det {\mathbf{F}} }}\right) ^{-\frac{1}{2}} {\mathbf{C}}_{p} {\mathbf{F}}^{-1} {{\mathbf{R}}^{*}} {\mathbf{N}}_{1} \quad {\mathrm{and}} \quad {\bar{\mathbf{N}}}_{1}^{\prime \prime } = \left( {\mathrm{det {\mathbf{F}} }}\right) ^{-\frac{1}{2}} {\mathbf{N}}_{1} {{\mathbf{R}}^{*{\mathrm{T}}}} {\mathbf{F}}. \end{aligned}$$

(43)

The representation of the remaining terms of Eq. 38 in the reference configuration may be done following the same procedure.