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Archive of Applied Mechanics

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18.06.2022 | Original

Effective behavior of viscoelastic composites: comparison of Laplace–Carson and time-domain mean-field approach

verfasst von: Tarkes Dora Pallicity, O. L. Cruz-González, J. A. Otero, R. Rodríguez-Ramos

Erschienen in: Archive of Applied Mechanics | Ausgabe 8/2022

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Abstract

The paper focuses on deriving the macroscale viscoelastic constitutive laws using asymptotic expansion method. Both the differential and integral form of the linear viscoelastic constitutive relation of the phases is used in deriving the effective incremental potential and effective constitutive relation, respectively. The integral form is handled by considering the correspondence principle and the Laplace–Carson (LC) transform. A closed-form expression for the effective viscoelastic properties in LC domain is obtained by means of the asymptotic homogenization method (AHM). In addition, AHM coupled with finite element simulation of a representative volume element with periodic boundary conditions is used (AHM + FE). The last step in both approaches is the numerical inversion to the time domain. Solution in time domain is obtained with numerical Laplace inversion algorithms. In case of the differential form, using variational approach, the effective incremental potential in time domain is directly obtained using mean-field method. Different homogenization approaches are exemplified for evaluation of the effective relaxation behavior of composite (viscoelastic matrix reinforced by unidirectional elastic fibers), and they are compared. In the approaches based on LC transform, effective modulus and Poisson’s ratio agree well with each other for any property contrast and fiber volume fraction. However, in case of relatively low property contrast, mean field overpredicts as compared to LC approaches in the fiber direction, whereas at relatively higher property contrast, it is vice versa. The difference increases at higher volume fractions due to synergistic effect of the error due to geometrical assumptions involved in the localization tensor and interaction effects of the fiber inclusions. A good agreement in all directions is observed among the three schemes at intermediate volume fractions and property contrast. This study serves as benchmark for further theoretical improvements and experimental investigations.

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Görthofer, J., Meyer, N., Pallicity, T.D., Schöttl, L., Trauth, A., Schemmann, M., Hohberg, M., Pinter, P., Elsner, P., Henning, F., Hrymak, A., Seelig, T., Weidenmann, K., Kärger, L., Böhlke, T.: Virtual process chain of sheet molding compound: development, validation and perspectives. Compos. Part B Eng. 169, 133–147 (2019). https://doi.org/10.1016/j.compositesb.2019.04.001CrossRef

Böhlke, T., Henning, F., Hrymak, A., Kärger, L., Weidenmann, K.A., Wood, J.T.: Continuous-Discontinuous Fiber-Reinforced Polymers. Carl Hanser Verlag GmbH & Co. KG, München (2019)CrossRef

Moulinec, H., Suquet, P.: A numerical method for computing the overall response of nonlinear composites with complex microstructure. Comput. Methods Appl. Mech. Eng. 157, 69–94 (1998). https://doi.org/10.1016/S0045-7825(97)00218-1MathSciNetCrossRefMATH

Bensoussan, A., Papanicolau, G., Lions, J.-L.: Asymptotic Analysis for Periodic Structures. North-Holland (1978)

Sanchez-Palencia, E.: Non-Homogeneous Media and Vibration Theory. Springer-Verlag (1980)

Bakhvalov, N.S., Panasenko, G.: Homogenisation: Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of Composite Materials. Kluwer Academic Publishers (1989)

Eshelby, J.D.: The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. Phys. Soc. Lond. Ser. A. 241, 376–396 (1957). https://doi.org/10.1098/rspa.1983.0054MathSciNetCrossRefMATH

Hashin, Z.: Complex moduli of viscoelastic composites—I. General theory and application to particulate composites. Int. J. Solids Struct. 6, 539–552 (1970). https://doi.org/10.1016/0020-7683(70)90029-6CrossRefMATH

Hashin, Z.: Viscoelastic behavior of heterogeneous media. J. Appl. Mech. 32, 630 (1965). https://doi.org/10.1115/1.3627270CrossRef

10.

Christensen, R.M.: Theory of Viscoelasticity - 2nd Edition An Introduction. Academic Press, Cambridge, MA (1982)

11.

Maghous, S., Creus, G.J.: Periodic homogenization in thermoviscoelasticity: case of multilayered media with ageing. Int. J. Solids Struct. 40, 851–870 (2003). https://doi.org/10.1016/S0020-7683(02)00549-8MathSciNetCrossRefMATH

12.

Lakes, R.: Viscoelastic Materials. Cambridge University Press (2009)

13.

Lahellec, N., Suquet, P.: On the effective behavior of nonlinear inelastic composites: I. Incremental variational principles. J. Mech. Phys. Solids 55, 1932–1963 (2007). https://doi.org/10.1016/J.JMPS.2007.02.003MathSciNetCrossRefMATH

14.

Lahellec, N., Suquet, P.: On the effective behavior of nonlinear inelastic composites: II: A second-order procedure. J. Mech. Phys. Solids. 55, 1964–1992 (2007). https://doi.org/10.1016/J.JMPS.2007.02.004MathSciNetCrossRefMATH

15.

Ricaud, J.-M., Masson, R.: Effective properties of linear viscoelastic heterogeneous media: Internal variables formulation and extension to ageing behaviours. Int. J. Solids Struct. 46, 1599–1606 (2009). https://doi.org/10.1016/J.IJSOLSTR.2008.12.007CrossRefMATH

16.

Vu, Q.H., Brenner, R., Castelnau, O., Moulinec, H., Suquet, P.: A self-consistent estimate for linear viscoelastic polycrystals with internal variables inferred from the collocation method. Model. Simul. Mater. Sci. Eng. 20, 024003 (2012). https://doi.org/10.1088/0965-0393/20/2/024003CrossRef

17.

Lavergne, F., Sab, K., Sanahuja, J., Bornert, M., Toulemonde, C.: Homogenization schemes for aging linear viscoelastic matrix-inclusion composite materials with elongated inclusions. Int. J. Solids Struct. 80, 545–560 (2016). https://doi.org/10.1016/J.IJSOLSTR.2015.10.014CrossRef

18.

Miled, B., Doghri, I., Brassart, L., Delannay, L.: Micromechanical modeling of coupled viscoelastic–viscoplastic composites based on an incrementally affine formulation. Int. J. Solids Struct. 50, 1755–1769 (2013). https://doi.org/10.1016/J.IJSOLSTR.2013.02.004CrossRef

19.

Doghri, I., Adam, L., Bilger, N.: Mean-field homogenization of elasto-viscoplastic composites based on a general incrementally affine linearization method. Int. J. Plast. 26, 219–238 (2010). https://doi.org/10.1016/j.ijplas.2009.06.003CrossRefMATH

20.

Berbenni, S., Dinzart, F., Sabar, H.: A new internal variables homogenization scheme for linear viscoelastic materials based on an exact Eshelby interaction law. Mech. Mater. 81, 110–124 (2015). https://doi.org/10.1016/J.MECHMAT.2014.11.003CrossRef

21.

Kowalczyk-Gajewska, K., Petryk, H.: Sequential linearization method for viscous/elastic heterogeneous materials. Eur. J. Mech. A/Solids. 30, 650–664 (2011). https://doi.org/10.1016/J.EUROMECHSOL.2011.04.002CrossRefMATH

22.

Paquin, A., Sabar, H., Berveiller, M.: Integral formulation and self-consistent modelling of elastoviscoplastic behavior of heterogeneous materials. Arch. Appl. Mech. 69, 14–35 (1999). https://doi.org/10.1007/s004190050201CrossRefMATH

23.

Sanahuja, J.: Efficient homogenization of ageing creep of random media: application to solidifying cementitious materials. In: Mechanics and Physics of Creep, Shrinkage, and Durability of Concrete. pp. 201–210. American Society of Civil Engineers, Reston, VA (2013)

24.

Lahellec, N., Suquet, P.: Effective behavior of linear viscoelastic composites: a time-integration approach. Int. J. Solids Struct. 44, 507–529 (2007). https://doi.org/10.1016/j.ijsolstr.2006.04.038MathSciNetCrossRefMATH

25.

Sabar, H., Berveiller, M., Favier, V., Berbenni, S.: A new class of micro-macro models for elastic-viscoplastic heterogeneous materials. Int. J. Solids Struct. 39, 3257–3276 (2002). https://doi.org/10.1016/S0020-7683(02)00256-1MathSciNetCrossRefMATH

26.

Molinari, A.: Averaging models for heterogeneous viscoplastic and elastic viscoplastic materials. J. Eng. Mater. Technol. Trans. ASME 124, 62–70 (2002). https://doi.org/10.1115/1.1421052CrossRef

27.

Pallicity, T.D., Böhlke, T.: Effective viscoelastic behavior of polymer composites with regular periodic microstructures. Int. J. Solids Struct. 216, 167–181 (2021). https://doi.org/10.1016/j.ijsolstr.2021.01.016CrossRef

28.

Adolfsson, K., Enelund, M., Olsson, P.: On the fractional order model of viscoelasticity. Mech. Time Depend. Mater. 9, 15–34 (2005)CrossRef

29.

Bažant, Z.P., Huet, C.: Thermodynamic functions for ageing viscoelasticity: integral form without internal variables. Int. J. Solids Struct. 36, 3993–4016 (1999). https://doi.org/10.1016/S0020-7683(98)00184-XCrossRefMATH

30.

Chatzigeorgiou, G., Charalambakis, N., Chemisky, Y., Meraghni, F.: Periodic homogenization for fully coupled thermomechanical modeling of dissipative generalized standard materials. Int. J. Plast. 81, 18–39 (2016). https://doi.org/10.1016/j.ijplas.2016.01.013CrossRef

31.

Castañeda, P.P.: New variational principles in plasticity and their application to composite materials. J. Mech. Phys. Solids. 40, 1757–1788 (1992). https://doi.org/10.1016/0022-5096(92)90050-CMathSciNetCrossRefMATH

32.

Castañeda, P.P., Willis, J.R.: Variational second-order estimates for nonlinear composites. Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 455, 1799–1811 (1999). https://doi.org/10.1098/rspa.1999.0380MathSciNetCrossRefMATH

33.

Ponte Castañeda, P.: Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: I—theory. J. Mech. Phys. Solids 50, 737–757 (2002). https://doi.org/10.1016/S0022-5096(01)00099-0MathSciNetCrossRefMATH

34.

Huang, Y., Abou-Chakra Guéry, A., Shao, J.-F.: Incremental variational approach for time dependent deformation in clayey rock. Int. J. Plast. 64, 88–103 (2015). https://doi.org/10.1016/J.IJPLAS.2014.07.003CrossRef

35.

Guinovart-Díaz, R., Bravo-Castillero, J., Rodríguez-Ramos, R., Sabina, F.J.: Closed-form expressions for the effective coefficients of fibre-reinforced composite with transversely isotropic constituents. I: elastic and hexagonal symmetry. J. Mech. Phys. Solids 49, 1445–1462 (2001). https://doi.org/10.1016/S0022-5096(01)00005-9CrossRefMATH

36.

Rodríguez-Ramos, R., Sabina, F.J., Guinovart-Díaz, R., Bravo-Castillero, J.: Closed-form expressions for the effective coefficients of a fiber-reinforced composite with transversely isotropic constituents - I Elastic and square symmetry. Mech. Mater. 33, 223–235 (2001). https://doi.org/10.1016/S0167-6636(00)00059-4CrossRefMATH

37.

Bravo-Castillero, J., Guinovart-Díaz, R., Rodríguez-Ramos, R., Sabina, F.J., Brenner, R.: Unified analytical formulae for the effective properties of periodic fibrous composites. Mater. Lett. 73, 68–71 (2012). https://doi.org/10.1016/j.matlet.2011.12.106CrossRef

38.

Otero, J.A., Rodríguez-Ramos, R., Guinovart-Díaz, R., Cruz-González, O.L., Sabina, F.J., Berger, H., Böhlke, T.: Asymptotic and numerical homogenization methods applied to fibrous viscoelastic composites using Prony’s series. Acta Mech. 231, 2761–2771 (2020). https://doi.org/10.1007/s00707-020-02671-1MathSciNetCrossRefMATH

39.

Rodríguez-Ramos, R., Otero, J.A., Cruz-González, O.L., Guinovart-Díaz, R., Bravo-Castillero, J., Sabina, F.J., Padilla, P., Lebon, F., Sevostianov, I.: Computation of the relaxation effective moduli for fibrous viscoelastic composites using the asymptotic homogenization method. Int. J. Solids Struct. 190, 281–290 (2020). https://doi.org/10.1016/j.ijsolstr.2019.11.014CrossRef

40.

Cruz-González, O.L., Rodríguez-Ramos, R., Otero, J.A., Ramírez-Torres, A., Penta, R., Lebon, F.: On the effective behavior of viscoelastic composites in three dimensions. Int. J. Eng. Sci. (2020). https://doi.org/10.1016/j.ijengsci.2020.103377MathSciNetCrossRefMATH

41.

Cruz-González, O.L., Ramírez-Torres, A., Rodríguez-Ramos, R., Otero, J.A., Penta, R., Lebon, F.: Effective behavior of long and short fiber-reinforced viscoelastic composites. Appl. Eng. Sci. (2021). https://doi.org/10.1016/j.apples.2021.100037CrossRefMATH

42.

Penta, R., Gerisch, A.: Investigation of the potential of asymptotic homogenization for elastic composites via a three-dimensional computational study. Comput. Vis. Sci. 17, 185–201 (2015). https://doi.org/10.1007/s00791-015-0257-8MathSciNetCrossRefMATH

43.

Penta, R., Gerisch, A.: The asymptotic homogenization elasticity tensor properties for composites with material discontinuities. Contin. Mech. Thermodyn. 29, 187–206 (2017). https://doi.org/10.1007/s00161-016-0526-xMathSciNetCrossRefMATH

44.

Valsa, J., Brančik, L.: Approximate formulae for numerical inversion of Laplace transforms. Int. J. Numer. Model Electron. Netw. Devices Fields 11, 153–166 (1998)CrossRef

45.

Juraj: Numerical Inversion of Laplace Transforms in Matlab - File Exchange - MATLAB Central, https://www.mathworks.com/matlabcentral/fileexchange/32824-numerical-inversion-of-laplace-transforms-in-matlab

46.

Penta, R., Gerisch, A.: An introduction to asymptotic homogenization. Lect. Notes Comput. Sci. Eng. 122, 1–26 (2017). https://doi.org/10.1007/978-3-319-73371-5_1MathSciNetCrossRefMATH

47.

Kehrer, L.M.: Thermomechanical mean-field modeling and experimental characterization of long fiber-reinforced sheet molding compound composites. (2019)

48.

Trauth, A., Bondy, M., Weidenmann, K.A., Altenhof, W.: Mechanical properties and damage evolution of a structural sheet molding compound based on a novel two step curing resin system. Mater. Des. 143, 224–237 (2018). https://doi.org/10.1016/J.MATDES.2018.02.002CrossRef

49.

Böhlke, T., Brüggemann, C.: Graphical representation of the generalized Hooke’s Law. Tech. Mech. 21, 145–158 (2001)

50.

Kehrer, L., Wood, J.T., Böhlke, T.: Mean-field homogenization of thermoelastic material properties of a long fiber-reinforced thermoset and experimental investigation. J. Compos. Mater. (2020). https://doi.org/10.1177/0021998320920695CrossRef

51.

Ye, B.S., Svenson, A.L., Bank, L.C.: Mass and volume fraction properties of pultruded glass fibre-reinforced composites. Composites 26, 725–731 (1995). https://doi.org/10.1016/0010-4361(95)91140-ZCrossRef

52.

Ghossein, E., Lévesque, M.: Homogenization models for predicting local field statistics in ellipsoidal particles reinforced composites: comparisons and validations. Int. J. Solids Struct. 58, 91–105 (2015). https://doi.org/10.1016/j.ijsolstr.2014.12.021CrossRef

53.

Böhm, H.J.: A short introduction to basic aspects of continuum micromechanics - ILSB Report / ILSB-Arbeitsbericht 206. (2021)

54.

Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behaviour of polycrystals. J. Mech. Phys. Solids 10, 343–352 (1962). https://doi.org/10.1016/0022-5096(62)90005-4MathSciNetCrossRefMATH

55.

Walpole, L.J.: On the overall elastic moduli of composite materials. J. Mech. Phys. Solids 17, 235–251 (1969). https://doi.org/10.1016/0022-5096(69)90014-3CrossRefMATH

Titel: Effective behavior of viscoelastic composites: comparison of Laplace–Carson and time-domain mean-field approach
verfasst von: Tarkes Dora Pallicity
O. L. Cruz-González
J. A. Otero
R. Rodríguez-Ramos
Publikationsdatum: 18.06.2022
Verlag: Springer Berlin Heidelberg
Erschienen in: Archive of Applied Mechanics / Ausgabe 8/2022
Print ISSN: 0939-1533
Elektronische ISSN: 1432-0681
DOI: https://doi.org/10.1007/s00419-022-02181-7

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