Validation of axial and transverse force–displacement responses and principal strain rate ratios in the critical zone as a precursor to anisotropic damage prediction in metal sheets

Abstract

Most ductile damage models account for the triaxiality η-dependence of fracture strains, provided by the triaxiality failure diagram (TFD). η-values are commonly obtained from FE analysis. By contrast with von Mises materials, for anisotropic materials, η depends on yield criterion and material. The common procedure of matching numerical to experimental axial force–displacement (FD_L) responses may be insufficient because the transverse response (FD_W) is not validated. Further, the use of extensometers blurs local events. In this study, assuming different anisotropic yield criteria, we investigate for highly anisotropic Zirlo sheets (i) on a global level the FD_L and FD_W curves and (ii) on a local level the evolutions of the principal strain rate ratio β’, which is related to η. Tests with uniaxial tension, notched and shear specimens were carried out. Strains were measured by DIC. The pre-critical region reveals that the yield model needs to account for anisotropy regarding r-values and flow. The extended Barlat89 yield criterion is therefore chosen for damage modeling. For selected specimens, representative η-values are determined as input to the anisotropic damage model MAGD. The other specimens serve validation. MAGD properties are obtained by optimization. Despite some deviation for shear, comparison of FD_L and FD_W responses with experiment shows favorable agreement for the combination of anisotropic yield and damage models used here and that more accurate prediction with strain-dependent r-values. β’ agree well with experimental data up to fracture, indicating accurate prediction of η up to fracture. By comparison with “undamaged” FE results, the single contributions of damage and the strain-dependence of r-values to β’ become apparent.

Appendix – Extended Barlat89 yield model

The extended Barlat89 yield model was developed for highly anisotropic sheet materials, especially materials with similar yield stresses but quite dissimilar r-values – a behavior that the original Barlat89 model could not reflect because highly different r-values meant highly different yield curves [54]. Anomalous behavior, including 2nd order anomalous behavior [55], can be simulated, too. The r-values affect both the yield function and the flow rule. The following presentation is based on [51, 56, 57].

Yield function

A generalization of the original Barlat89 model, the yield condition is expressed as

$$f\left( {{{\varvec{\upsigma}}},\overline{\varepsilon }_{p} } \right) = \sigma_{eff} - \overline{\sigma }\left( {{{\varvec{\upsigma}}},\overline{\varepsilon }_{p} } \right) = 0,$$

(10)

where the effective stress

$$\sigma_{eff} = \left\{ {\frac{1}{2}\left( {a\left| {K_{1} + K_{2} } \right|^{m} + a\left| {K_{1} - K_{2} } \right|^{m} + c\left| {2K_{2} } \right|^{m} } \right)} \right\}^{1/m}$$

(11)

with its invariants K₁ and K₂ and material coefficients a, c, p and m are identical to the one in the original Barlat89 criterion [58]

$$K_{1} = \frac{{\sigma_{1} + h\sigma_{2} }}{2}\;\;\;;\;\;\;K_{2} = \sqrt {\left( {\frac{{\sigma_{1} - h\sigma_{2} }}{2}} \right)^{2} + 2p^{2} \sigma_{12}^{2} } .$$

(12)

The coefficients a, c, and h are directly related to r₀ and r₉₀, while p is computed iteratively

$$a = 2 - 2\sqrt {\frac{{r_{0} }}{{1 + r_{0} }}\frac{{r_{90} }}{{1 + r_{90} }}} \;\;\;;\;\;\;c = 2 - a\;\;\;;\;\;\;h = \sqrt {\frac{{r_{0} }}{{1 + r_{0} }}\frac{{1 + r_{90} }}{{r_{90} }}} \;\;\;;\;\;\;\frac{{2m\sigma_{y}^{m} }}{{\left( {\frac{\partial \Phi }{{\partial x}} + \frac{\partial \Phi }{{\partial y}}} \right)\overline{\sigma }_{45} }} - 1 - r_{45} = 0.$$

Here, Φ = (σ_eff)^m; x and y correspond to 11 and 22-directions; and $\overline{{\sigma }_{45}}$ is the uniaxial yield stress in 45° direction.

The yield function and the flow rule are expressed in terms of weighted sums, where the weights depend on the stress state and the accumulated plastic strain. Internal fitting is performed to calculate the material coefficients that enter K₁ and K₂ (a, c, h, and p) from the r_θ and hardening curves.

The yield stress is written as a weighted convex sum of the effective plastic strain-dependent contributions from the flow curves in three main directions

$$\overline{\sigma }\left( {{{\varvec{\upsigma}}},\overline{\varepsilon }_{p} } \right) = \alpha_{0} \overline{\sigma }_{0} \left( {\overline{\varepsilon }_{p} } \right) + \alpha_{45} \overline{\sigma }_{45} \left( {\overline{\varepsilon }_{p} } \right) + \alpha_{90} \overline{\sigma }_{90} \left( {\overline{\varepsilon }_{p} } \right).$$

(13)

${\alpha }_{\theta }$ are so to weight the yield stresses according to the flow curves and depend on the stress state. The 0° direction is the reference direction. Additional input of shear and equibiaxial stress–strain data and r-values is available but since this option is not used here it is excluded from the presentation. If the stress is uniaxial and coincides with one of the three main directions (θ = 0°, 45°, 90°), then the corresponding ${\alpha }_{\theta }$ is 1 and the other ${\alpha }_{\theta }$ are 0. For consistency, ∑${\alpha }_{\theta }$ = 1; ${\alpha }_{\theta }$ = [0, 1].

The weighting is derived from the stress as follows: To begin with, the stress tensor (in component form) is normalized by $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma } = \sqrt {\sigma_{11}^{2} + \sigma_{22}^{2} + 2\sigma_{12}^{2} }$

$$\hat{\sigma }_{ij} = \frac{{\sigma_{ij} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma } }}.$$

(14)

It has the largest eigenvalue $\tilde{\sigma }$ (normalized maximum principal stress) and the associated eigenvectors q_i (normalized principal stress directions). To indicate the hydrostatic and deviatoric contributions, two additional stress measures (normalized stress invariants of the stress tensor) are employed

$$\begin{gathered} \hat{\sigma }_{v} = \frac{{\hat{\sigma }_{11} + \hat{\sigma }_{22} }}{2} \hfill \\ \hat{\sigma }_{d} = \sqrt {\left( {\frac{{\hat{\sigma }_{11} - \hat{\sigma }_{22} }}{2}} \right)^{2} + \hat{\sigma }_{12}^{2} } . \hfill \\ \end{gathered}$$

(15)

From these measures, new measures, C, B and Q, are defined to uniquely characterize the stress state. While C is a measure to distinguish between uniaxial/deviatoric and hydrostatic, Q and B provide the direction with respect to the 0° and 90° directions and whether it is 0° or 90° direction, respectively. They are defined as follows

$$\begin{gathered} C = 2\hat{\sigma }_{v}^{2} \left[ {1 - \left( {2\tilde{\sigma }^{2} - 1} \right)^{2} } \right] \hfill \\ Q = 4Q_{1}^{2} \left( {1 - Q_{1}^{2} } \right) \hfill \\ B = \tilde{\sigma }^{2} Q_{1}^{2} + \left( {1 - \tilde{\sigma }^{2} } \right)\left( {1 - Q_{1}^{2} } \right). \hfill \\ \end{gathered}$$

(16)

For uniaxial and deviatoric states, C = 0, while for hydrostatic states, C = 1. Q becomes 0 for 0° and 90° directions and 1 for 45° direction. (Note that Q = [0, 1] assumes the same value when Q₂ substitutes Q₁.) B becomes 0 and 1 when the stress is directed in 0° and 90° directions, respectively.

For uniaxial tension in 45° direction, for example, we have ${{\hat{\sigma }_{v}^{2} = 1} \mathord{\left/ {\vphantom {{\hat{\sigma }_{v}^{2} = 1} 2}} \right. \kern-\nulldelimiterspace} 2}$, $\tilde{\sigma }^{2} = 1$, ${{Q_{1}^{2} = 1} \mathord{\left/ {\vphantom {{Q_{1}^{2} = 1} 2}} \right. \kern-\nulldelimiterspace} 2}$, which means C = 0, Q = 1 and B = 1/2. The three uniaxial convex parameters are expressed in terms of C, Q and B as

$$\begin{gathered} \alpha_{0} = \left( {1 - C} \right)B\left( {1 - Q} \right) + C/4 \hfill \\ \alpha_{45} = \left( {1 - C} \right)Q + C/2 \hfill \\ \alpha_{90} = \left( {1 - C} \right)\left( {1 - B} \right)\left( {1 - Q} \right) + C/4. \hfill \\ \end{gathered}$$

(17)

Above example then gives [α₀, α₄₅, α₉₀] = [0, 1, 0]. As other examples, uniaxial loading in 30° and hydrostatic pressure give [α₀, α₄₅, α₉₀] = [3/16, 3/4, 1/16] and [1/4, 1/2, 1/4], respectively.

Flow rule and variable r-values

The flow rule is written as

$${\dot{\mathbf{\varepsilon }}}_{p} = \left[ {\begin{array}{*{20}c} {\dot{\varepsilon }_{11}^{p} } & {\dot{\varepsilon }_{12}^{p} } & {\dot{\varepsilon }_{13}^{p} } \\ {\dot{\varepsilon }_{21}^{p} } & {\dot{\varepsilon }_{22}^{p} } & {\dot{\varepsilon }_{23}^{p} } \\ {\dot{\varepsilon }_{31}^{p} } & {\dot{\varepsilon }_{32}^{p} } & {\dot{\varepsilon }_{33}^{p} } \\ \end{array} } \right] = \dot{\lambda }\left( {\frac{\partial \Phi }{{\partial {{\varvec{\upsigma}}}}} + \Delta {\mathbf{n}}} \right),$$

(18)

where λ is the plastic multiplier and $\dot{\varepsilon }_{22}^{p}$ and $\dot{\varepsilon }_{33}^{p}$ are related through r_θ. The first part on the right-hand side is known from the associated flow rule of the original Barlat89 model; the correction ∆n, introduced to match the input r-values (from experiment), makes the criterion a non-associated one. ∆n is obtained as follows.

The strain-dependent r-values are processed in a fashion similar to the yield stress. From the three r-curves, ${r}_{0}(\overline{{\varepsilon }_{p}})$, ${r}_{45}((\overline{{\varepsilon }_{p}})$, and ${r}_{90}((\overline{{\varepsilon }_{p}})$, weighted according to the stress state, an effective r-value is computed as

$$R\left( {{{\varvec{\upsigma}}},\overline{\varepsilon }_{p} } \right) = \alpha_{0} r_{0} \left( {\overline{\varepsilon }_{p} } \right) + \alpha_{45} r_{45} \left( {\overline{\varepsilon }_{p} } \right) + \alpha_{90} r_{90} \left( {\overline{\varepsilon }_{p} } \right).$$

(19)

where the coefficients ${\alpha }_{\theta }$ are given in Eq. (17). The ${\alpha }_{\theta }$ represent the position on the yield surface and n gives the direction of plastic flow. The expression in terms of the stress is generalized to

$$\left[ {\hat{\sigma }_{22} + \left( {\hat{\sigma }_{11} + \hat{\sigma }_{22} } \right)R} \right]n_{1} + \left[ {\hat{\sigma }_{11} + \left( {\hat{\sigma }_{11} + \hat{\sigma }_{22} } \right)R} \right]n_{2} - \hat{\sigma }_{12} n_{4} = 0,$$

(20)

where n_i gives the direction of the (non-associated) plastic flow and where the dependence of stress and plastic strain in R is suppressed. The plastic flow and R “corrections”, ∆n_i and ∆R, can be expressed as

$$\begin{gathered} \Delta n_{1} = n_{1} - \frac{\partial f}{{\partial \sigma_{11} }}\;\;\;;\;\;\;\Delta n_{2} = n_{2} - \frac{\partial f}{{\partial \sigma_{22} }}\;\;\;;\;\;\;\Delta n_{4} = n_{4} - \frac{\partial f}{{\partial \sigma_{12} }} \hfill \\ \Delta R = R\left( {\overline{\varepsilon }_{p} } \right) - R\left( 0 \right) \hfill \\ \end{gathered}$$

(21)

Inserting Eq. (21) into Eq. (20) gives

$$\left[ {\hat{\sigma }_{22} + \left( {\hat{\sigma }_{11} + \hat{\sigma }_{22} } \right)R} \right]\Delta n_{1} + \left[ {\hat{\sigma }_{11} + \left( {\hat{\sigma }_{11} + \hat{\sigma }_{22} } \right)R} \right]\Delta n_{2} - \hat{\sigma }_{12} \Delta n_{4} = - \left( {\frac{\partial f}{{\partial \sigma_{11} }} + \frac{\partial f}{{\partial \sigma_{22} }}} \right)\left( {\hat{\sigma }_{11} + \hat{\sigma }_{22} } \right)\Delta R.$$

(22)

The consistency condition of the plastic flow gives

$$\hat{\sigma }_{11} \Delta n_{1} + \hat{\sigma }_{22} \Delta n_{2} + \hat{\sigma }_{12} \Delta n_{4} = 0$$

(23)

Linear independence requires that

$$\hat{\sigma }_{v} \hat{\sigma }_{d} = \frac{{\hat{\sigma }_{11} + \hat{\sigma }_{22} }}{2}\sqrt {\left( {\frac{{\hat{\sigma }_{11} - \hat{\sigma }_{22} }}{2}} \right)^{2} + \hat{\sigma }_{12}^{2} } \ne 0$$

(24)

so that

$$- \hat{\sigma }_{12} \Delta n_{1} + \hat{\sigma }_{12} \Delta n_{2} + \left( {\hat{\sigma }_{11} - \hat{\sigma }_{22} } \right)\Delta n_{4} = 0$$

(25)

becomes an additional condition. Eqs. (22), (23), and (25) are then employed to compute the deviation of the plastic flow direction from the normal direction (∆n) in a least-square sense. For numerical stability and continuity of n with respect to the stress, Eq. (22) is rewritten as

$$\begin{gathered} \left[ {\hat{\sigma }_{22} + \left( {\hat{\sigma }_{11} + \hat{\sigma }_{22} } \right)R} \right]\Delta n_{1} + \left[ {\hat{\sigma }_{11} + \left( {\hat{\sigma }_{11} + \hat{\sigma }_{22} } \right)R} \right]\Delta n_{2} - \hat{\sigma }_{12} \Delta n_{4} \hfill \\ \;\;\; = - \left( {\frac{\partial f}{{\partial \sigma_{11} }} + \frac{\partial f}{{\partial \sigma_{22} }}} \right)\left( {\hat{\sigma }_{11} + \hat{\sigma }_{22} } \right)\left[ {\left( {\hat{\sigma }_{11} + \hat{\sigma }_{22} } \right)^{2} + 4\hat{\sigma }_{12}^{2} } \right]\Delta R \hfill \\ \end{gathered}$$

(26)