Predicting anisotropic crushable polymer foam behavior in sandwich structures

Abstract

An anisotropic elastic-plastic, viscoelastic-damage model was developed to predict the multiaxial crushing behavior of polymer foams used in the core of sandwich structures. This model was based on pressure vessel experiments on Divinycell H100, whereby the post-yield response of the foam was characterized by anisotropic hardening during plastic flow, as well as damage and viscoelastic hysteresis. Post-yield properties from uniaxial compression/tension and simple shear material hysteresis experiments were used to develop three-dimensional material constitutive equations for the foam. Plastic flow was governed by the Tsai–Wu failure criterion, which was verified to accurately predict yielding of the foam in the triaxial pressure experiments. The solution methodology proved to be very effective in predicting the hysteresis response of the foam under triaxial compression, triaxial compression–tension, and triaxial compression–shear. Good agreement was found between theoretical predictions and experimental results.

Appendices

Appendix A

1.1 Damage stiffness of equilibrium spring

For the equilibrium spring, the actual damage cases, ${\varvec{\upsigma}}_{eq}$ and ${\varvec{\upvarepsilon}}_{e}$, are the stress and elastic strain acting at a material point:

$${\varvec{\upsigma}}_{eq} = {\bar{\mathbf{C}}}_{0} {\varvec{\upvarepsilon}}_{e} .$$

(54)

Assuming there is no damage, the corresponding stress ${\tilde{\mathbf{\sigma }}}_{eq}$ and strain ${\tilde{\mathbf{\varepsilon }}}_{e}$ are called effective stress and effective elastic strain, respectively:

$${\tilde{\mathbf{\sigma }}}_{eq} = {\mathbf{C}}_{0} {\tilde{\mathbf{\varepsilon }}}_{e} .$$

(55)

They are defined based on equality of the elastic strain energy in the equivalent spring, given as

$$\varPi = {\varvec{\upsigma}}_{eq}^{T} {\varvec{\upvarepsilon}}_{e} = {\tilde{\mathbf{\sigma }}}_{eq}^{T} {\tilde{\mathbf{\varepsilon }}}_{e} .$$

(56)

Substituting Eqs. (54) and (55) into Eq. (56) gives

$${\varvec{\upsigma}}_{eq}^{T} {\bar{\mathbf{C}}}_{0}^{ - 1} {\varvec{\upsigma}}_{eq} = {\tilde{\mathbf{\sigma }}}_{eq}^{T} {\mathbf{C}}_{0}^{ - 1} {\tilde{\mathbf{\sigma }}}_{eq} .$$

(57)

The damage tensor ${\mathbf{D}}_{eq}$ is defined to build the connection between ${\varvec{\upsigma}}_{eq}$ and ${\tilde{\mathbf{\sigma }}}_{eq}$ as follows:

$${\varvec{\upsigma}}_{eq} = {\mathbf{D}}_{eq} {\tilde{\mathbf{\sigma }}}_{eq} .$$

(58)

Since ${\varvec{\upsigma}}_{eq}$ and ${\tilde{\mathbf{\sigma }}}_{eq}$ should be the same stress states, ${\mathbf{D}}_{eq}$ has to be a diagonal matrix given as

$${\mathbf{D}}_{{eq}} = \left[ {\begin{array}{*{20}l} {D_{{eq11}} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {D_{{eq22}} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {D_{{eq33}} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & {D_{{eq23}} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {D_{{eq13}} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {D_{{eq12}} } \hfill \\ \end{array} } \right]$$

(59)

Substituting Eq. (58) into Eq. (57), we have

$${\tilde{\mathbf{\sigma }}}_{eq}^{T} {\mathbf{D}}_{eq}^{T} {\bar{\mathbf{C}}}_{0}^{ - 1} {\mathbf{D}}_{eq} {\tilde{\mathbf{\sigma }}}_{eq} = {\tilde{\mathbf{\sigma }}}_{eq}^{T} {\mathbf{C}}_{0}^{ - 1} {\tilde{\mathbf{\sigma }}}_{eq} .$$

(60)

Equation (57) can be simplified as

$${\bar{\mathbf{C}}}_{0} = {\mathbf{D}}_{eq}^{T} {\mathbf{C}}_{0} {\mathbf{D}}_{eq} .$$

(61)

Considering uniaxial loading case in 11-direction, we have

$$\tilde{\sigma }_{eq1} = E_{11} \cdot \tilde{\varepsilon }_{e11}$$

(62)

and

$$\sigma_{eq1} = \bar{E}_{11} \cdot \varepsilon_{e11} ,$$

(63)

where $\bar{E}_{11}$ is the actual damaged modulus measured from the test.

Substituting Eqs. (62) and (63) into Eq. (56) and considering all the other stresses as zero, we get

$$\sigma_{eq11} = \sqrt {\frac{{\bar{E}_{11} }}{{E_{11} }}} \tilde{\sigma }_{eq11} .$$

(64)

From Eqs. (58) and (59), the damage coefficient $D_{eq11}$ is obtained as

$$D_{eq11} = \sqrt {\frac{{\bar{E}_{11} }}{{E_{11} }}} .$$

(65)

Similarly, all the other damage coefficients inside the ${\mathbf{D}}_{eq}$ matrix can be obtained from the uniaxial and simple shear tests and the damage tensor finally takes the form:

$${\mathbf{D}}_{\text{eq}} = \left[ {\begin{array}{*{20}l} {\sqrt {\frac{{\bar{E}_{11} }}{{E_{11} }}} } & 0& 0& 0& 0 & 0 \\ 0& {\sqrt {\frac{{\bar{E}_{22} }}{{E_{22} }}} } & 0& 0 & 0 & 0 \\ 0& 0& {\sqrt {\frac{{\bar{E}_{33} }}{{E_{33} }}} } & 0 & 0 & 0 \\ 0 & 0 & 0 & {\sqrt {\frac{{\bar{G}_{23} }}{{G_{23} }}} } & 0 & 0 \\ 0 & 0 & 0 & 0 & {\sqrt {\frac{{\bar{G}_{13} }}{{G_{13} }}} } & 0 \\ 0 & 0 & 0 & 0 & 0 & {\sqrt {\frac{{\bar{G}_{12} }}{{G_{12} }}} } \\ \end{array} } \right].$$

(66)

Appendix B

2.1 Post-yield plasticity and viscoelastic-damage functions

The normalized flow stress and viscoelastic-damage parameters are plotted in Figs. 14, 15, 16 and 17.

Fig. 14

Normalized flow stress (hardening functions): a compression, b tension, and c shear

Fig. 15

Viscoelastic spring stiffness and damping associated with in-plane compression

Fig. 16

Viscoelastic spring stiffness and damping associated with out-of-plane compression

Fig. 17

Viscoelastic spring stiffness and damping associated with shear