Damage effects of adhesives in modern glass façades: a micro-mechanically motivated volumetric damage model for poro-hyperelastic materials

Abstract

This paper presents a micro-mechanically motivated volumetric damage model accounting for cavitation effects in modern glass connections, e.g. laminated glass connections. The volumetric part of an arbitrary Helmholtz free energy function is equipped with an isotropic damage formulation. To develop a micro-mechanical damage model, the porous micro-structure of a transparent structural silicone adhesive is analyzed numerically applying hydrostatic loading conditions. Based on the structural responses of different types of cubic representative volume elements incorporating an initial void fraction, three damage parameters are fitted utilizing the Levenberg–Marquard algorithm. The present volumetric damage model is implemented into ANSYS FE Code using a UserMat subroutine, where the algorithmic setting is described in detail in the present paper. To compare the structural responses of cubic equivalent homogeneous materials with representative volume elements, benchmark tests under hydrostatic loading are performed. The results indicate that the novel damage model accounts adequately for volumetric damage due to the cavitation effect. A special form of the pancake test is described briefly. The test allows for visualizing the cavitation effect during experimental testing. The experimental results of the pancake test are compared with numerical results, where the pancake test is simulated incorporating the micro-mechanical damage model. The micro-mechanically motivated scalar, internal damage variable is equipped with the obtained damage parameters from the structural response of the representative volume elements. The results show an adequate approximation of the experiment through the simulation. However, to optimize the results of the simulation, an optimization study on the damage parameters is conducted utilizing the Downhill-Simplex algorithm. Using the optimized damage parameters, the simulation of the pancake tests is further improved. Hence, it is shown that the novel micro-mechanically motivated volumetric damage model is excellently suited to represent the cavitation effect in poro-hyperelastic materials.

Appendix A: scalar damage variables

The explicit representation of the scalar damage variables for the Helmholtz free energy is given by

$$\begin{aligned} \begin{array}{c} {\xi _\varPsi } = \left( \begin{array}{c} {2\, \arctan \left( {{{2\, {\kappa _4}\left( {J - 1} \right) + {\kappa _3}} \over {\sqrt{4\, {\kappa _2}{\kappa _4} - {\kappa _3}^2} }}} \right) {\kappa _2}{\kappa _3}} \\ \vdots \\ { - 2\, \arctan \left( {{{{\kappa _3}} \over {\sqrt{4\, {\kappa _2}{\kappa _4} - {\kappa _3}^2} }}} \right) {\kappa _2}{\kappa _3}} \\ \vdots \\ \left( \begin{array}{c} { - \ln \left( {{\kappa _4}{{\left( {J - 1} \right) }^2} + {\kappa _3}\left( {J - 1} \right) + {\kappa _2}} \right) {\kappa _2}} \\ \vdots \\ { + \ln \left( {{\kappa _2}} \right) {\kappa _2} + {\kappa _4}{{\left( {J - 1} \right) }^2}} \end{array} \right) \\ \vdots \\ {\sqrt{4\, {\kappa _2}{\kappa _4} - {\kappa _3}^2} } \end{array} \right) \\ \vdots \\ {\kappa _4}^{ - 1}{1 \over {\sqrt{4\, {\kappa _2}{\kappa _4} - {\kappa _3}^2} }}{\left( {J - 1} \right) ^{ - 2}}, \end{array} \end{aligned}$$

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whereas the scalar damage variables for the volumetric pressure is represented by

$$\begin{aligned} {\xi _P } = {{\left( {J - 1} \right) \left( {{\kappa _4}\left( {J - 1} \right) + {\kappa _3}} \right) } \over {{\kappa _4}{{\left( {J - 1} \right) }^2} + {\kappa _3}\left( {J - 1} \right) + {\kappa _2}}}. \end{aligned}$$

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Finally, the scalar damage variable for degradation of the bulk modulus is given by the following equation

$$\begin{aligned} {\xi _K } = \begin{array}{c} \left( \begin{array}{c} {{\kappa _4}^2{{\left( {J - 1} \right) }^3} + 2\, {\kappa _3}{\kappa _4}{{\left( {J - 1} \right) }^2}} \\ \vdots \\ { + 3\, {\kappa _2}{\kappa _4}\left( {J - 1} \right) + {\kappa _3}^2\left( {J - 1} \right) + 2\, {\kappa _2}{\kappa _3}} \end{array} \right) \\ \vdots \\ {1 \over {{{\left( {{\kappa _4}{{\left( {J - 1} \right) }^2} + {\kappa _3}\left( {J - 1} \right) + {\kappa _2}} \right) }^2}{{\left( {J - 1} \right) }^{ - 1}}}}. \end{array} \end{aligned}$$

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