Abstract
This work deals with the modelling and simulation of curing phenomena in adhesively bonded piezo metal composites (PMC) which consist of an adhesive layer, an integrated piezoelectric module and two surrounding metal sheet layers. In a first step, a finite strain modelling framework for the representation of polymer curing phenomena is proposed. Based on this formulation, a concretised model is deduced and applied to one specific epoxy based adhesive. Here, appropriate material functions are provided and the thermodynamic consistency is proved. Regarding the finite element implementation, a numerical scheme for time integration and a new approach for maintaining a constant initial volume at arbitrary initial conditions are provided. Finally, finite element simulations of a newly proposed manufacturing process for the production of bonded PMC structures are conducted. Thereby, a representative deep drawing process is analysed with respect to the impact of the adhesive layer on the embedded piezoelectric module.
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Notes
The unimodular part of a tensor is also referred to as isochoric, distortional or volume-preserving part of a tensor.
The isochoric part of the free energy may be extended by additional internal variables. This would be necessary if, for example, models of multiplicative viscoelasticity or viscoplasticity were employed (cf. [16]).
The initial configuration \({\mathcal {K}}_0\) can be interpreted as a new reference [33].
For even more exact prediction of the residual stresses and spring back, material models with nonlinear kinematic hardening are needed [34].
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Acknowledgments
This research is supported by the German Research Foundation (DFG) within the Collaborative Research Centre/Transregio 39 PT-PIESA, and the collaborative research project DFG PAK 273. This support is greatly acknowledged.
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Appendix
Appendix
1.1 Remark on thermodynamic consistency
In Sect. 2.2 the thermodynamic consistency of the general modelling framework has been considered but not finally proved since specific constitutive functions had not been set up to that point. The evaluation of the two remaining conditions (21) and (22) is discussed in this section. To this end, constitutive assumptions presented in Sects. 3.1–3.4 are employed.
Firstly, inequality (22) is considered. To prove this condition, the partial derivative of the isochoric part of the free energy function \(\hat{\psi }_G\) with respect to the intrinsic time scale \(z\) has to be calculated. Since only the viscoelastic parts \(\hat{\psi }_{ve,k}\) include the dependency on \(z\) (cf. Eq. 33), it remains to show that
Furthermore, it is assumed that not only the sum of all Maxwell elements, but also every single Maxwell element meets the consistency condition. Thus, it is sufficient to show that
holds. According to [27] or [42], the thermodynamic consistency of the ansatz (37) is met, if the conditions
hold. Obviously, these conditions are satisfied by the relaxation function (38).
Next, the remaining condition (21) is considered. Since inequality (21) cannot be evaluated in a general form, an estimation under consideration of some physically reasonable assumptions is employed. In the first step, the term \(\partial \hat{\psi }_{\theta C}(\theta ,q) / \partial q \) of Eq. (21) is estimated. Here, an ansatz for the thermochemical part of the specific enthalpy per unit mass is introduced [12, 21]
The functions \(h_{fluid}(\theta )\) and \(h_{solid}(\theta )\) are the specific enthalpy per unit mass of the uncured and fully cured material, respectively. Note that the general ansatz (89) depends on the temperature \(\theta \) and the degree of cure \(q\). Specific models for the consideration of temperature dependent behaviour have been introduced in [12] and [21]. However, for the estimation conducted in this section this is omitted and the values for \(h_{fluid}\) and \(h_{solid}\) are assumed to be constant.
The next step is to calculate the thermochemical free energy \(\psi _{\theta C}\) from \(h_{\theta C}\). This can be accomplished by approaches presented in [21] or [15]. However, here an alternative formulation of this calculation step is used as follows. Firstly, the Legendre transformation
is employed, which relates the free energy and the enthalpy (see, for example, [21, 43]). Therein, \(\eta \) is the specific entropy per unit mass and is the Green-Lagrange strain tensor. Next it is assumed, that DSC experiments take place at zero mechanical stresses [21]. Thus, the last term of (90) is neglected. Furthermore, the constitutive relation \(\eta = - \partial \psi / \partial \theta \) at constant stress state is employed and the resulting equation is formulated with respect to the thermochemical potentials \(\psi _{\theta C}\) and \(h_{\theta C}\). This yields the reduced relation
Eq. (91) is a differential equation that has to be solved for \(\psi _{\theta C}\). Its solution reads as
Here, \(C\) is an integration constant that does not need to be determined in our evaluation. Next, this general solution is applied to the ansatz for the thermochemical enthalpy, which has been introduced in Eq. (89). This yields a specific model for the thermochemical free energy
Based on this solution, the term \(\partial \hat{\psi }_{\theta C}(\theta ,q) / \partial q \) of Eq. (21) is calculated by
To quantify this expression, the maximum specific reaction enthalpy per unit mass \(\Delta h\) of a complete curing experiment has to be taken into account. This quantity has been measured by DSC experiments (see [12, 21] for detailed description) and can be related to the model (89) by the relation
Here, a value of \(\Delta h= h_u \approx - 300 \, \mathrm{J/g}\) has been identified (cf. Sect. 3.1). Furthermore, taking into account the mass density , the first term of Eq. (21) is estimated by
Next, the second term in inequality (21) is examined. Therein, the chemical shrinkage parameter \(\beta _q\) can be identified by the help of Eq. (32). Here, the relation
holds. Furthermore, a relation to the hydrostatic pressure \(p\) is obtained by evaluation of
Finally, inequality (21) can be evaluated. To this end, expressions (97) and (98) are substituted into Eq. (21) which yields
Moreover, the estimation (96) and the chemical shrinkage parameter \(\beta _q=-0.053 \) (cf. Sect. 3.2) are inserted in (99), and the resulting inequality is resolved for the expression \(J\,p\). This finally yields the condition
Since \(J > 0\) holds in general, it can be concluded from (100) that a hydrostatic pressure with \(p > 0\) does not endanger the thermodynamic consistency. However, the condition (100) may be violated if the material is loaded in hydrostatic tension \((p < 0)\). If a constant volume is assumed \((J=1)\), a hydrostatic tension of \(p = -6340 \, \mathrm{MPa}\) would be necessary to violate thermodynamic consistency. Nevertheless, this value seems to be unrealistic to achieve in real experiments. Thus, the thermodynamic consistency can be proved for the case of physically reasonable conditions (see also [14, 15, 21]).
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Landgraf, R., Rudolph, M., Scherzer, R. et al. Modelling and simulation of adhesive curing processes in bonded piezo metal composites. Comput Mech 54, 547–565 (2014). https://doi.org/10.1007/s00466-014-1005-5
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DOI: https://doi.org/10.1007/s00466-014-1005-5