Experimental and numerical investigation of impact resistance of aluminum–copper cladded sheets using an energy-based damage model

Abstract

Cladded multilayer multi-material sheets are now extensively being used in different engineering fields. These sheets are well known for their excellent mechanical and functional properties, corrosion resistance and electrical conductivity. In the present study, low velocity impact of cladded two-layer sheets has been studied experimentally and numerically. The desired sheets were fabricated from aluminum AA1200 and copper (OFHC), and then, they were impacted at the impact energy of 45 J using two different impactor nose shapes. The effect of stacking sequence of the constituent layers on impact behavior of the cladded sheets was also investigated. Numerical modeling of the problem was carried out using finite element method where an energy-based rate-dependent damage model was developed and employed to model the behavior of the impacted sheets including damage initiation and evolution. The required rate constants were achieved using a direct experimental procedure as well as a calibration process via genetic algorithm. Also, for the numerical part of the research, one more impactor nose shape and two more impact energies were considered to be studied. Accuracy of both the finite element modeling and developed damage model was admitted by comparing the numerical results with experimental ones. Moreover, the achieved results prove the advantage of the Al/Cu sheet over the Cu/Al one in terms of both the energy absorption and impact strength.

Graphic abstract

Appendices

Appendix A

1.1 Plastic corrector scheme

Step 1. Calculation of the trial values for the strain increment of $\Delta \varepsilon$:

$$ \varvec{\varepsilon}_{n + 1}^{{e {\text{trial}}}} =\varvec{\varepsilon}_{n}^{e} + \Delta\varvec{\varepsilon},\,\varepsilon_{{{\text{eq}}, n + 1}}^{\text{p trial}} = \varepsilon_{{{\text{eq}},n}}^{\text{p}} ,\varvec{s}^{\text{trial}} = \varvec{s}_{n} + 2G\Delta \varvec{e}, p^{\text{trial}} = p_{n} + K\Delta v,\xi_{n + 1} = \xi_{n} $$

Step 2. Check the yield condition:

$$ f^{\text{trial}} = \sqrt {\frac{3}{2}} \|\varvec{s}^{\text{trial}}\| - \left[ {\left( {\sigma_{Y}^{0} + R\left( {\varepsilon_{{{\text{eq}},n}}^{\text{p}} } \right)} \right).\left( {1 + C . \log \left( {\dot{\varepsilon }^{*} } \right)} \right)} \right] $$

And then check:

IF$f^{\text{trial}} \le 0$THEN (Elastic region)

Update $\left( \cdot \right)_{n + 1} = \left( \cdot \right)^{\text{trial}}$RETURN

ELSE (Plastic region)

Step 3. Obtain $\Delta \gamma$:

$$ \sqrt {\frac{3}{2}} \|\varvec{s}^{\text{trial}}\| - 3G\Delta \gamma - \left[ {(\sigma_{Y}^{0} + R\left( {\varepsilon_{{{\text{eq}},n}}^{\text{p}} + \Delta \gamma } \right)).\left( {1 + C . \log \left( {\dot{\varepsilon }^{*} } \right)} \right)} \right] = 0 $$

Step 4. Update the required parameters:

$$ \begin{aligned} \varvec{s}_{n + 1} & = \left( {1 - \sqrt {\frac{3}{2}} \frac{2G\Delta \gamma }{{\|\varvec{s}^{\text{trial}}\|}}} \right)\varvec{s}^{\text{trial}} ,p_{n + 1} = p^{\text{trial}} \\\varvec{\sigma}_{n + 1} & = \varvec{s}_{n + 1 } + p_{n + 1} \varvec{I,}\varepsilon_{{{\text{eq}},n + 1}}^{\text{p}} = \varepsilon_{{{\text{eq}},n}}^{\text{p}} + \Delta \gamma \\\varvec{\varepsilon}_{n + 1}^{e} & = \frac{1}{2G}\varvec{s}_{n + 1} + \frac{1}{3K}p_{n + 1} \varvec{I},\xi_{n + 1} = \xi_{n} + \Delta \gamma \left( {\frac{{ - {\text{SE}}_{n + 1} }}{r}} \right) \\ \end{aligned} $$

ENDIF

RETURN

Appendix B

2.1 Algorithm of damage evolution

Step 1. Calculation of the trial values for the strain increment of $\Delta \varepsilon$:

$$ \varvec{\varepsilon}_{n + 1}^{{e {\text{trial}}}} =\varvec{\varepsilon}_{n}^{e} + \Delta\varvec{\varepsilon},\varepsilon_{{{\text{eq}}, n + 1}}^{\text{p trial}} = \varepsilon_{{{\text{eq}},n}}^{\text{p}} ,\varvec{s}^{\text{trial}} = \left( {1 - D} \right)\varvec{s}_{n} + 2G\Delta \varvec{e}, p^{\text{trial}} = \left( {1 - D} \right)p_{n} + K\Delta v $$

Step 2. Check the yield condition:

$$ f^{\text{trial}} = \sqrt {\frac{3}{2}} \frac{{\|\varvec{s}^{\text{trial}}\|}}{{\left( {1 - D} \right)}} - \left[ {(\sigma_{Y}^{0} + R\left( {\varepsilon_{{{\text{eq}},n}}^{\text{p}} } \right)} \right).\left( {1 + C . \log \left( {\dot{\varepsilon }^{*} } \right)} \right)] $$

And then check:

IF$f^{\text{trial}} \le 0$THEN (no evolution of damage parameter)

Update $\left( \cdot \right)_{n + 1} = \left( \cdot \right)^{\text{trial}}$RETURN

ELSE (evolution of damage parameter)

Step 3. Obtain $\Delta \lambda$:

$$ \sqrt {\frac{3}{2}} \|\varvec{s}^{\text{trial}}\| - 3G\Delta \lambda - \left( {1 - D} \right)\left[ {\left( {\sigma_{Y}^{0} + R\left( {\varepsilon_{{{\text{eq}},n}}^{\text{p}} + \Delta \lambda } \right)} \right).\left( {1 + C . \log \left( {\dot{\varepsilon }^{*} } \right)} \right)} \right] = 0 $$

Step 4. Calculate the equivalent plastic strain

$$ \varepsilon_{{{\text{eq}},n + 1}}^{\text{p}} = \varepsilon_{{{\text{eq}},n}}^{\text{p}} + \Delta \lambda $$

Step 5. Update the damage variable using the new equivalent plastic strain:

$$ u_{\text{eq}}^{\text{p}} = L_{\text{c}} \varepsilon_{{{\text{eq}},n + 1}}^{\text{p}} ,\,D = 1 - \exp \left( { - \mathop \int \limits_{0}^{{u_{\text{eq}}^{\text{p}} }} \frac{{\sigma_{y} {\text{d}}u_{\text{eq}}^{\text{p}} }}{{G_{\text{f}} }}} \right) $$

Step 6. Update the required parameters:

$$ \begin{aligned} \varvec{s}^{\text{trial}} & = \left( {1 - D} \right)\varvec{s}_{n} + 2G\Delta \varvec{e},\, p^{\text{trial}} = \left( {1 - D} \right)p_{n} + K\Delta v \\ \varvec{s}_{n + 1} & = \left( {1 - \sqrt {\frac{3}{2}} \frac{2G\Delta \lambda }{{\|\varvec{s}^{\text{trial}}\|}}} \right)\varvec{s}^{\text{trial}} ,p_{n + 1} = p^{\text{trial}} \\\varvec{\sigma}_{n + 1} & = \varvec{s}_{n + 1 } + p_{n + 1} \varvec{I},\varvec{\varepsilon}_{n + 1}^{e} = \frac{1}{2G}\varvec{s}_{n + 1} + \frac{1}{3K}p_{n + 1} \varvec{I} \\ \end{aligned} $$

ENDIF

RETURN