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Modelling and simulation of adhesive curing processes in bonded piezo metal composites

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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

  1. The unimodular part of a tensor is also referred to as isochoric, distortional or volume-preserving part of a tensor.

  2. 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]).

  3. The initial configuration \({\mathcal {K}}_0\) can be interpreted as a new reference [33].

  4. For even more exact prediction of the residual stresses and spring back, material models with nonlinear kinematic hardening are needed [34].

References

  1. Wiedemann M, Sinapius M (2013) Adaptive, tolerant and efficient composite structures. Springer, Berlin

    Book  Google Scholar 

  2. Messler RW (2004) Joining of materials and structures: from pragmatic process to enabling technology. Butterworth-Heinemann, London

    Google Scholar 

  3. Prasad SE, Waechter DF, Blacow RG, King HW, Yaman Y (2005) Application of piezoelectrics to smart structures. In: II ECCOMAS Thematic Conference on Smart Structures and Materials, Lisbon, Portugal

  4. Lloyd JM (2004) Electrical properties of macro-fiber composite actuators and sensors. Ph.D. Thesis, Virginia Polytechnic Institute and State University

  5. Neugebauer R, Lachmann L, Drossel WG, Nestler M, Hensel S (2010) Multi-layer compounds with integrated actor-sensor-functionality. Prod Eng 4(4):379–384

    Article  Google Scholar 

  6. Neugebauer R, Lachmann L, Drossel WG, Nestler M, Hensel S (2013) Smart semi-finished parts for the application in sheet-metal structures. Future trends in production engineering. Springer, Berlin, pp 121–130

    Google Scholar 

  7. Drossel WG, Hensel S, Kranz B, Nestler M, Goeschel A (2009) Sheet metal forming of piezoceramic-metal-laminar structures: simulation and experimental analysis. Cirp Ann Manuf Tech 58(1):279–282

    Article  Google Scholar 

  8. Seemann W, Sattel T (1999) Influence of bonding on the efficiency of piezoceramic patches as actuators in smart structures. In: 1999 Symposium on Smart Structures and Materials, pp. 677–687. International Society for Optics and Photonics

  9. Hossain M, Possart G, Steinmann P (2009) A finite strain framework for the simulation of polymer curing. Part I: elasticity. Comput Mech 44(5):621–630

    Article  MathSciNet  MATH  Google Scholar 

  10. Hossain M, Possart G, Steinmann P (2010) A finite strain framework for the simulation of polymer curing. Part II. Viscoelasticity and shrinkage. Comput Mech 46(3):363–375

    Article  MATH  Google Scholar 

  11. Klinge S, Bartels A, Steinmann P (2012) The multiscale approach to the curing of polymers incorporating viscous and shrinkage effects. Int J Solids Struct 49(26):3883–3900

    Article  Google Scholar 

  12. Kolmeder S, Lion A, Landgraf R, Ihlemann J (2011) Thermophysical properties and material modelling of acrylic bone cements used in vertebroplasty. J Therm Anal Calorim 105(2):705–718

    Article  Google Scholar 

  13. Liebl C, Johlitz M, Yagimli B, Lion A (2012) Three-dimensional chemo-thermomechanically coupled simulation of curing adhesives including viscoplasticity and chemical shrinkage. Comput Mech 49(5):603–615

    Article  MathSciNet  MATH  Google Scholar 

  14. Lion A, Höfer P (2007) On the phenomenological representation of curing phenomena in continuum mechanics. Arch Mech 59(1):59–89

    MATH  Google Scholar 

  15. Mahnken R (2013) Thermodynamic consistent modeling of polymer curing coupled to viscoelasticity at large strains. Int J Solids Struct 50(13):2003–2021

    Article  Google Scholar 

  16. Landgraf R, Ihlemann J (2011) Zur Modellierung von Aushärtevorgängen in Polymeren unter Verwendung von Stoffgesetzen der Viskoelastizität und Viskoplastizität. PAMM 11(1):399–400

    Article  Google Scholar 

  17. Ihlemann J (2006) Beobachterkonzepte und Darstellungsformen der nichtlinearen Kontinuumsmechanik. Habilitation Thesis, Leibniz Universität Hannover

  18. Haupt P (2002) Continuum mechanics and theory of materials. Springer, Berlin

    Book  MATH  Google Scholar 

  19. 3M Deutschland: 3M\(^{{\rm TM}}\) Scotch-Weld\(^{{\rm TM}}\) EPX Epoxy Adhesive DP410—Technical datasheet. 3M Deutschland GmbH (2003).

  20. Kolmeder S, Lion A (2010) On the thermomechanical-chemically coupled behavior of acrylic bone cements: experimental characterization of material behavior and modeling approach. Technische Mechanik 30(1–3):195–202

    Google Scholar 

  21. Lion A, Yagimli B (2008) Differential scanning calorimetry: continuum mechanical considerations with focus to the polymerisation of adhesives. Z Angew Math Mech 88(5):388–402

    Article  MATH  Google Scholar 

  22. Halley PJ, Mackay ME (1996) Chemorheology of thermosets: an overview. Polym Eng Sci 36(5):593–609

    Article  Google Scholar 

  23. Fournier J, Williams G, Duch C, Aldridge GA (1996) Changes in molecular dynamics during bulk polymerization of an epoxide-asmine system as studied by dielectric relaxation spectroscopy. Macromolecules 29(22):7097–7107

    Article  Google Scholar 

  24. DiBenedetto AT (1987) Prediction of the glass transition temperature of polymers: a model based on the principle of corresponding states. J Polym Sci Polym Phys 25(9):1949–1969

    Article  Google Scholar 

  25. Pascault JP, Williams RJJ (1990) Glass transition temperature versus conversion relationships for thermosetting polymers. J Polym Sci Polym Phys 28(1):85–95

    Article  Google Scholar 

  26. Lion A, Johlitz M (2012) On the representation of chemical ageing of rubber in continuum mechanics. Int J Solids Struct 49(10):1227–1240

    Article  MathSciNet  Google Scholar 

  27. Haupt P, Lion A (2002) On finite linear viscoelasticity of incompressible isotropic materials. Acta Mech 159(1):87–124

    Google Scholar 

  28. Häßler R, Kunze W (2010) Thermische Eigenschaften von Klebstoffen und Harzen: DMA–DSC–TGA–TMA. Stoffsammlung thermoanalytischer Messkurven, TA-Instruments

  29. Dušek K (1984) Cross-linking of epoxy resins. In: Keith RC, GJ K (eds) Rubber-Modified Thermoset Resins, chap. 1, pp. 3–14. American Chemical Society, Washington DC. doi:10.1021/ba-1984-0208.ch001.

  30. Meuwissen MH, de Boer HA, Steijvers HL, Schreurs PJ, Geers MG (2004) Residual stresses in microelectronics induced by thermoset packaging materials during cure. Microelectron Reliab 44(12):1985–1994

    Article  Google Scholar 

  31. ANSYS (2011) ANSYS User Manual, Release 14. ANSYS Inc, USA

  32. Rendek M, Lion A (2010) Amplitude dependence of filler-reinforced rubber: experiments, constitutive modelling and FEM-Implementation. Int J Solids Struct 47(21):2918–2936

    Article  MATH  Google Scholar 

  33. Shutov AV, Pfeiffer S, Ihlemann J (2012) On the simulation of multi-stage forming processes: invariance under change of the reference configuration. Materialwissenschaft und Werkstofftechnik 43(7):617–625

    Article  Google Scholar 

  34. Shutov AV, Kreißig R (2008) Finite strain viscoplasticity with nonlinear kinematic hardening: phenomenological modeling and time integration. Comput Methods Appl Mech Eng 197(21–24):2015–2029

    Article  MATH  Google Scholar 

  35. Häuselmann Metall (2002) Data sheet EN AW-5083 (AlMg4, 5Mn). Häuselmann Metall GmbH

  36. Giddings PF, Bowen CR, Kim HA (2009) A coupled field finite element model to predict actuation properties of piezoelectrically actuated bistable composites. ICCM-17: 17th International Conference on Composite Materials

  37. Smart Material (2012) Macro Fiber Composite—MFC. Smart Material GmbH

  38. Winter HH, Chambon F (1986) Analysis of linear viscoelasticity of a crosslinking polymer at the gel point. J Rheol 30:367–382

    Article  Google Scholar 

  39. Daue TP, Kunzmann J, Schönecker A (2010) Energy harvesting systems using piezo-electric macro fiber composites. 6th Annual Symposium—Nanotechnology: state of the art and applications

  40. Landgraf R, Ihlemann J, Kolmeder S, Lion A (2012) Thermo-mechanically coupled curing processes of acrylic bone cements used in vertebroplasty: Material modelling and Finite-Element-Simulation. In: Proceedings of 10th International Symposium Computer Methods in Biomechanics and Biomedical Engineering, Berlin, Germany, April 7–11, 2012, pp. 19–24. Published by ARUP, ISBN: 978-0-9562121-5-3

  41. Landgraf R, Ihlemann J, Kolmeder S, Lion A, Lebsack H, Kober C (2013) Modelling and simulation of acrylic bone cement injection and curing within the framework of vertebroplasty. arXiv preprint:1307.0880

  42. Lion A, Kardelky C (2004) The Payne effect in finite viscoelasticity: constitutive modelling based on fractional derivatives and intrinsic time scales. Int J Plast 20(7):1313–1345

    Article  MATH  Google Scholar 

  43. Lubarda V (2004) On thermodynamic potentials in linear thermoelasticity. Int J Solids Struct 41(26):7377–7398

    Article  MATH  Google Scholar 

Download references

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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Authors and Affiliations

  1. Department of Solid Mechanics, Institute of Mechanics and Thermodynamics, Faculty of Mechanical Engineering, Chemnitz University of Technology, Chemnitz, Germany

    Ralf Landgraf, Martin Rudolph, Robert Scherzer & Jörn Ihlemann

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  1. Ralf Landgraf
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Correspondence to Ralf Landgraf.

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

$$\begin{aligned} - \sum _{k=1}^{N_k} \dfrac{\partial \hat{\psi }_{ve,k}}{\partial z} \ge 0 . \end{aligned}$$
(86)

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

$$\begin{aligned} - \dfrac{\partial \hat{\psi }_{ve,k}}{\partial z} \ge 0 \end{aligned}$$
(87)

holds. According to [27] or [42], the thermodynamic consistency of the ansatz (37) is met, if the conditions

$$\begin{aligned} G_k(t) \ge 0 \ , \qquad \dfrac{\mathrm{d}}{\mathrm{d}t} G_k(t) \le 0 \ , \qquad \dfrac{\mathrm{d}^2}{\mathrm{d}t^2} G_k(t) \ge 0 \end{aligned}$$
(88)

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]

$$\begin{aligned} h_{\theta C}(\theta ,q) = h_{fluid}(\theta ) \, (1-q) + h_{solid}(\theta ) \, q . \end{aligned}$$
(89)

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

(90)

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

$$\begin{aligned} \psi _{\theta C}(\theta ,q) - \theta \, \dfrac{\partial \psi _{\theta C}(\theta ,q)}{\partial \theta } = h_{\theta C}(\theta ,q) . \end{aligned}$$
(91)

Eq. (91) is a differential equation that has to be solved for \(\psi _{\theta C}\). Its solution reads as

$$\begin{aligned} \psi _{\theta C}(\theta ,q) = C \, \dfrac{\theta }{\theta _0} - \theta \, \int \dfrac{1}{\theta ^2}\,h_{\theta C}(\theta ,q) \, \mathrm{d}\theta . \end{aligned}$$
(92)

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

$$\begin{aligned} \psi _{\theta C}(\theta ,q) = C \, \dfrac{\theta }{\theta _0} + h_{fluid}\, (1-q) + h_{solid} \, q . \end{aligned}$$
(93)

Based on this solution, the term \(\partial \hat{\psi }_{\theta C}(\theta ,q) / \partial q \) of Eq. (21) is calculated by

(94)

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

$$\begin{aligned} \Delta h = h(\theta ,q=1) - h(\theta ,q=0) = h_{solid} - h_{fluid} . \end{aligned}$$
(95)

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

$$\begin{aligned} \dfrac{\partial \hat{\psi }_{\theta C}(\theta ,q)}{\partial q} = -336 \ \mathrm{MPa} . \end{aligned}$$
(96)

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

$$\begin{aligned} \dfrac{1}{J_{\theta C}}\, \dfrac{\partial J_{\theta C}}{\partial q} \, = \beta _q \end{aligned}$$
(97)

holds. Furthermore, a relation to the hydrostatic pressure \(p\) is obtained by evaluation of

(98)

Finally, inequality (21) can be evaluated. To this end, expressions (97) and (98) are substituted into Eq. (21) which yields

$$\begin{aligned} - \dfrac{\partial \hat{\psi }_{\theta C}(\theta ,q)}{\partial q} - \beta _q\,J\,p \ \ge 0 . \end{aligned}$$
(99)

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

$$\begin{aligned} J\,p \ge - 6340 \, \mathrm{MPa} . \end{aligned}$$
(100)

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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