Plastic anisotropy of AA7075-T6 alloy under quasi-static compression: experiments, classical plasticity and artificial neural networks modeling

Abstract

This paper presents the experimental observations, theoretical analysis and machine learning model of plastic anisotropy of rolling AA7075-T6. Compression responses have been discussed by obtaining the instantaneous stress–strain relationship. The analytical solution of anisotropic initial yielding and hardening has been derived through generalizing the J2 flow theory and applying the evolutive constitutive parameters. In addition, a machine learning model consisting of artificial neural network optimized by genetic algorithm (GA-ANN) is utilized to simulate the plastic anisotropy of AA7075-T6. According to the comparisons among experimental, theoretical and predicted (GA-ANN) results, the machine learning model provides flexible application and is found easy to be generalized for solving such mechanical problems, but with difficultly in assessment of the model’s reliability. Multi-index estimation is a feasible approach to ensure the objective of evaluation in machine learning model.

Appendices

Appendix 1: Shear test of AA7075-T6 in in-plane direction and validation of constitutive model based on stress integral algorithm

Figure 19 shows the geometric schematic of in-plane shear specimen. The shear stress state can be obtained through stretching short edge of specimen along RD direction. Then, the shear stress can be calculated by dividing the cross-sectional area into force and the shear strain also obtained through DIC method. The complete responses of AA7075-T6 under shear are shown in Fig. 20 and the combined Voce–Hockett–Sherby hardening model was utilized to fit such responses:

$$\sigma_{{{\text{eq}}}} = (1 - \omega )(\sigma_{0} + A_{0} (1 - e^{{ - n_{1} \varepsilon_{{{\text{eq}}}}^{p} }} )) + \omega (\sigma_{d} - (\sigma_{d} - \sigma_{e} )e^{{ - n_{2} \varepsilon_{{{\text{eq}}}}^{p} }} )).$$

(31)

In addition, the hardening parameters are listed in Table 8. Based on Eq. 31, the equivalent stress at four different levels of equivalent strain can be obtained as displayed in Table 9. It should be point out that the fracture failure has occurred when $\varepsilon_{{{\text{eq}}}}^{p}$ reaches 0.2 and the corresponding $\sigma_{{{\text{eq}}}}$ is extrapolated through Eq. 31. Then, the plastic work of in-plane shear can be integrated and cubic polynomial was adopted to guarantee the continuous description:

$$W_{{{\text{eq}}}}^{p} = - 1.297 + 272.075\varepsilon_{{{\text{eq}}}}^{p} + 69.939\varepsilon_{{{\text{eq}}}}^{{p^{2} }} - 64.197\varepsilon_{{{\text{eq}}}}^{{p^{3} }}$$

(32)

where $W_{{{\text{eq}}}}^{p}$ is equivalent plastic work.

Table 8 Parameters of combined Voce–Hockett–Sherby hardening rule

Table 9 Equivalent stress at different levels of equivalent plastic strain

Fig. 19

Size of specific shear specimen for describing hardening behaviors

Fig. 20

Stress–strain curve of AA7075-T6 under in-plane shear and the fitted combined voce–Hockett–Sherby hardening model

Fig. 21

Plastic work–equivalent plastic strain curve and its cubic fit

In order to validate the accuracy of the constitutive model used in this paper, the return mapping algorithm and implicit integration algorithm are applied to carry out stress update at the level of integration point with the assumption of associated flow rule (AFR). Based on above considerations, the integration algorithm can be written as (Fig. 21)

$$\begin{gathered} {{\varvec{\upvarepsilon}}}_{n + 1} = {{\varvec{\upvarepsilon}}}_{n} + \Delta {{\varvec{\upvarepsilon}}} \hfill \\ {{\varvec{\upvarepsilon}}}^{p}_{n + 1} = {{\varvec{\upvarepsilon}}}_{n}^{p} + \Delta \lambda_{n + 1} {\mathbf{g}}_{n + 1} \hfill \\ {\mathbf{q}}_{n + 1} = {\mathbf{q}}_{n} + \Delta \lambda_{n + 1} {\mathbf{h}}_{n + 1} \hfill \\ {{\varvec{\upsigma}}}_{n + 1} = {\mathbf{C}}:({{\varvec{\upvarepsilon}}}_{n + 1} - {{\varvec{\upvarepsilon}}}_{n + 1}^{p} ) \hfill \\ f_{n + 1} = f({{\varvec{\upsigma}}}_{n + 1} ,{\mathbf{q}}_{n + 1} ) = 0, \hfill \\ \end{gathered}$$

(33)

where $\lambda$, ${\mathbf{g}}$, ${\mathbf{q}}$, ${\mathbf{h}}$, ${\mathbf{C}}$ and n denote the plastic multiplier, flow direction, internal variables, hardening parameters, elastic modulus tensor and time step. Ignoring the subscript n + 1, the following equations can be obtained [40]:

$$\begin{gathered} {\mathbf{a}} = - {{\varvec{\upvarepsilon}}}^{p} + {{\varvec{\upvarepsilon}}}_{n}^{p} + \Delta \lambda {\mathbf{g}} = 0 \hfill \\ {\mathbf{b}} = - {\mathbf{q}} + {\mathbf{q}}_{n} + \Delta \lambda {\mathbf{h}} = 0 \hfill \\ f = f_{Y} ({{\varvec{\upsigma}}},{\mathbf{q}}) = 0. \hfill \\ \end{gathered}$$

(34)

Based on Taylor expansion, the linearized equation system can be obtained (k denotes the iterative step):

$$\begin{gathered} {\mathbf{a}}^{(k)} + {\mathbf{C}}^{{{ - }{1}}} :\Delta {{\varvec{\upsigma}}}^{(k)} + \Delta \lambda^{(k)} ({\mathbf{g}}_{{\upsigma }}^{(k)} :\Delta {{\varvec{\upsigma}}}^{(k)} + {\mathbf{g}}_{{\text{q}}}^{(k)} \cdot \Delta {\mathbf{q}}^{(k)} ) + \delta \lambda^{(k)} {\mathbf{g}}^{(k)} = 0 \hfill \\ {\mathbf{b}}^{(k)} - \Delta {\mathbf{q}}^{(k)} + \Delta \lambda^{(k)} ({\mathbf{h}}_{{\upsigma }}^{(k)} :\Delta {{\varvec{\upsigma}}}^{(k)} + {\mathbf{h}}_{{\text{q}}}^{(k)} \cdot \Delta {\mathbf{q}}^{(k)} ) + \delta \lambda^{(k)} {\mathbf{h}}^{(k)} = 0 \hfill \\ f^{(k)} + f_{{\upsigma }}^{(k)} :\Delta {{\varvec{\upsigma}}}^{(k)} + f_{{\text{q}}}^{(k)} \cdot \Delta {\mathbf{q}}^{(k)} = 0. \hfill \\ \end{gathered}$$

(35)

Therefore, the following equation system can be built:

$$\left[ {\begin{array}{*{20}c} {{\mathbf{C}}^{{ { - }1}} + \Delta \lambda^{(k)} {\mathbf{g}}_{{\upsigma }}^{(k)} } & {\Delta \lambda^{(k)} {\mathbf{g}}_{{\text{q}}}^{(k)} } & {{\mathbf{g}}^{(k)} } \\ {\Delta \lambda^{(k)} {\mathbf{h}}_{{\upsigma }}^{(k)} } & {\Delta \lambda^{(k)} {\mathbf{h}}_{{\text{q}}}^{(k)} - {\mathbf{I}}} & {{\mathbf{h}}^{(k)} } \\ {f_{{\upsigma }}^{(k)} } & {f_{{\text{q}}}^{(k)} } & {\mathbf{o}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\Delta {{\varvec{\upsigma}}}^{(k)} } \\ {\Delta {\mathbf{q}}^{(k)} } \\ {\delta \lambda^{(k)} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - {\mathbf{a}}^{(k)} } \\ { - {\mathbf{b}}^{(k)} } \\ { - f^{(k)} } \\ \end{array} } \right].$$

(36)

Considering the combination of anisotropic hardening and isotropic hardening, the internal variables ${\mathbf{q}}$ only involve hardening parameters:

$${\mathbf{q}} = \left[ {\begin{array}{*{20}c} {a_{1} (\varepsilon_{{{\text{eq}}}}^{p} )} & {a_{2} (\varepsilon_{{{\text{eq}}}}^{p} )} & {a_{3} (\varepsilon_{{{\text{eq}}}}^{p} )} & {a_{4} (\varepsilon_{{{\text{eq}}}}^{p} )} & {a_{5} (\varepsilon_{{{\text{eq}}}}^{p} )} & {a_{6} (\varepsilon_{{{\text{eq}}}}^{p} )} & {\varepsilon_{{{\text{eq}}}}^{p} } \\ \end{array} } \right]^{{\text{T}}} .$$

(37)

Based on AFR, the increase of plastic multiplier equals to the increase of equivalent plastic strain:

$${\text{d}}\lambda = {\text{d}}\varepsilon_{{{\text{eq}}}}^{p} .$$

(38)

Then, the equation system B4 can be rewritten as

$$\left[ {\begin{array}{*{20}c} {{\mathbf{C}}^{{{ - }{1}}} + \Delta \lambda^{(k)} {\mathbf{g}}_{{\upsigma }}^{(k)} } & {\Delta \lambda^{(k)} {\mathbf{g}}_{{\text{q}}}^{(k)} } & {{\mathbf{g}}^{(k)} } \\ {\mathbf{o}} & { - {\mathbf{I}}} & {{\mathbf{h}}^{(k)} } \\ {{\mathbf{f}}_{{\upsigma }}^{(k)} } & {{\mathbf{f}}_{{\text{q}}}^{(k)} } & {f_{{\varepsilon_{eq}^{p} }}^{(k)} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\Delta {{\varvec{\upsigma}}}^{(k)} } \\ {\Delta {\mathbf{q}}^{(k)} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - {\mathbf{a}}^{(k)} } \\ { - {\mathbf{b}}^{(k)} } \\ { - f^{(k)} } \\ \end{array} } \right].$$

(39)

Through solving above equations, the updated stresses satisfy the consistency condition which confirms that the stress loci are located on the yield surface.

It should be point out that the two-dimension stress state is considered, and the isotropic elastic relation can be written as

$$\sigma_{ij} = {\text{C}}_{ijkl} \varepsilon_{kl} ,$$

(40)

where

$$\left\{ \begin{gathered} \sigma_{ij} = \left[ {\begin{array}{*{20}c} {\sigma_{xx} } & {\sigma_{yy} } & {\sigma_{xy} } \\ \end{array} } \right]^{{\text{T}}} \hfill \\ \left[ {\text{C}} \right] = \left[ {\begin{array}{*{20}c} {\frac{E}{{(1 - \upsilon^{2} )}}} & {\frac{E\upsilon }{{(1 - \upsilon^{2} )}}} & 0 \\ {\frac{E\upsilon }{{(1 - \upsilon^{2} )}}} & {\frac{E}{{(1 - \upsilon^{2} )}}} & 0 \\ 0 & 0 & {\frac{E}{1 + \upsilon }} \\ \end{array} } \right] \hfill \\ \varepsilon_{ij} = \left[ {\begin{array}{*{20}c} {\varepsilon_{xx} } & {\varepsilon_{yy} } & {\varepsilon_{xy} } \\ \end{array} } \right]^{{\text{T}}} . \hfill \\ \end{gathered} \right.$$

(40-1)

The flow directions can be expressed as

$$g = \left[ {\frac{1}{{J_{2}^{0} (\sigma_{ij} )}}\left[ {\frac{{a_{1} }}{6}(\sigma_{xx} - \sigma_{yy} ) - \frac{{a_{2} }}{6}( - \sigma_{xx} )} \right]\quad \frac{1}{{J_{2}^{0} (\sigma_{ij} )}}\left[ {\frac{{a_{3} }}{6}\sigma_{yy} - \frac{{a_{1} }}{6}(\sigma_{xx} - \sigma_{yy} )} \right]\quad \frac{{a_{4} }}{{J_{2}^{0} (\sigma_{ij} )}}\sigma_{xy} } \right].$$

(41)

Based on above descriptions, the comparisons between experimental results and constitutive model of compression responses at loading angle of 0° and 90° are presented in Fig. 22. In 0° direction, the predicted values of model agree well with experimental results while that predicted values are slightly higher than experimental results as the loading angle equals to 90°.

Fig. 22

Comparisons between experimental results and constitutive model of compression responses at loading angle of 0° and 90°

Appendix 2: Determination of number of neurons and fitness histories with generations of GA

Figures 23, 24, 25, 26, 27 and 28 show the determination of number of neurons of ANN and fitness histories versus generations of GA with different datasets of case 1, case 2 and case 3. The number of neurons can be determined through finding the neurons whose MSE is minimum. Fitness curves that also can be found are convergent to small values with the increasing generation which indicates the reliable optimization of the initial $\omega_{ij}$, $\omega_{ki}$, $b_{i}$,$b_{k}$ in ANN.

Fig. 23

Determination of number of neurons with dataset of case 1

Fig. 24

Fitness generations of GA with dataset of case 1

Fig. 25

Determination of number of neurons with dataset of case 2

Fig. 26

Fitness generations of GA with dataset of case 2

Fig. 27

Determination of number of neurons with dataset of case 3

Fig. 28

Fitness generations of GA with dataset of case 3

Appendix 3: Performance and training histories of ANN

Figures 29, 30, 31, 32, 33, and 34 show the performance and training histories of ANN with datasets of case 1, case 2 and case 3. For case 1, the R of training set, validation set, test set and total dataset are 0.99991, 0.99985, 0.9998 and 0.99989, respectively. The best validation performance is 4.94e−5 at epoch 277. For case 2, the R of training set, validation set, test set and total dataset are 0.99619, 0.99739, 0.99529 and 0.99621, respectively and the best validation performance 8.396e−4 is found at epoch 39. In addition, the R of training set, validation set, test set and total dataset are 0.99929, 0.99917, 0.99876 and 0.9992, respectively, as the dataset turned into case 3. The corresponding best validation performance is 2.509e−4 at epoch 152.

Fig. 29

Performance of trained ANN with dataset of case 1

Fig. 30

Training histories of ANN with dataset of case 1

Fig. 31

Performance of trained ANN with dataset of case 2

Fig. 32

Training histories of ANN with dataset of case 2

Fig. 33

Performance of trained ANN with dataset of case 3

Fig. 34

Training histories of ANN with dataset of case 3

Appendix 4: Weights and bias of ANN with different datasets of case 1, case 2 and case 3 optimized by GA

Tables 10, 11 and 12 list the optimized initial values of weight and bias of ANN with different dataset of case 1, case 2 and case 3 by GA.

Table 10 Weights and bias of ANN with dataset of case 1

Table 11 Weights and bias of ANN with dataset of case 2

Table 12 Weights and bias of ANN with dataset of case 3