Fiber-reinforced polymers contribute significantly to weight-reducing components for various industrial applications. A discontinuous glass fiber-reinforced thermoset resin is considered which is produced by the sheet molding compound (SMC) process. Related to the production process, the samples considered in this work exhibit an anisotropic fiber orientation distribution which highly affects the thermomechanical properties. The thermoviscoelastic material behavior of three selected samples is characterized by means of dynamic mechanical analysis. These tests show the temperature-dependent elastic modulus and the glass transition of the composite. Measurements of the thermal expansion of the SMC composite provide data on the coefficient of thermal expansion (CTE). These experimental investigations provide data for the thermoelastic material modeling. Aiming at the prediction of the effective thermal and mechanical properties, a Hashin–Shtrikman-based homogenization method is presented. Based on an eigenstrain formulation, the effective Young’s modulus and CTE are computed in two steps. Moreover, the mean-field method is given in dependence of a variable reference stiffness allowing to tailor the approach to the material system. The influence of this variable reference stiffness on the effective quantities as well as the predicted behavior is analyzed with respect to the experiments. The presented numerical results are in good agreement with the experimental data.
TrauthAWeidenmannKA. Continuous-discontinuous sheet moulding compounds – effect of hybridisation on mechanical material properties. Compos Str2018; 202: 1087–1098.
2.
FenglerBKärgerLHenningF, et al.Multi-objective patch optimization with integrated kinematic draping simulation for continuous–discontinuous fiber-reinforced composite structures. J Compos Sci2018; 2: 1–19.
3.
GörthoferJMeyerNPallicityTD, et al.Virtual process chain of sheet molding compound: development, validation and perspectives. Compos Part B Eng2019; 169: 133–147.
4.
MaireEWithersPJ. Quantitative X-ray tomography. Int Mater Rev2014; 59: 1–43.
5.
FeldkampLDavisLKressJ. Practical cone-beam algorithm. J Opt Soc Am A Opt Image Sci1984; 1: 612–619.
6.
KrauseMHausherrJMBurgethB, et al.Determination of the fibre orientation in composites using the structure tensor and local X-ray transform. J Mater Sci2010; 45: 888–896.
7.
OhserJSchladitzK. 3D images of materials structures: processing and analysis, Weinheim: John Wiley & Sons, 2009.
8.
BernasconiACosmiFHineP. Analysis of fibre orientation distribution in short fibre reinforced polymers: a comparison between optical and tomographic methods. Compos Sci Technol2012; 72: 2002–2008.
9.
BayRSTuckerCLIII. Fiber orientation in simple injection moldings. Part I: theory and numerical methods. Polym Compos1992; 13: 317–331.
10.
KanataniKI. Distribution of directional data and fabric tensors. Int J Eng Sci1984; 22: 149–164.
11.
AdvaniSGTuckerCLIII. The use of tensors to describe and predict fiber orientation in short fiber composites. J Rheol1987; 31: 751–784.
12.
Müller V. Micromechanical modeling of short-fiber reinforced composites (Schriftenreihe Kontinuumsmechanik im Maschinenbau Nr. 6). Doctoral Dissertation, KIT Scientific Publishing, Karlsruhe, Germany, 2016.
13.
SchemmannMGörthoferJSeeligT, et al.Anisotropic meanfield modeling of debonding and matrix damage in SMC composites. Compos Sci Technol2018; 161: 143–158.
14.
SchneiderMOspaldFKabelM. Computational homogenization of elasticity on a staggered grid. Int J Numer Methods Eng2016; 105: 693–720.
15.
Dvorak G. Micromechanics of composite materials (Solid mechanics and its applications). Dordrecht, Heidelberg, New York, London: Springer, 2012.
16.
VoigtW. Ueber die Beziehung zwischen den beiden Elasticitätsconstanten isotroper Körper. Annalen der Physik1889; 274: 573–587.
17.
ReussA. Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. J Appl Math Mech1929; 9: 49–58.
18.
HillR. The elastic behaviour of a crystalline aggregate. Proc Phys Soc A1952; 65: 349–354.
19.
HashinZShtrikmanS. A variational approach to the theory of the elastic behaviour of polycrystals. J Mech Phys Solids1962; 10: 343–352.
20.
WillisJR. Bounds and self-consistent estimates for the overall properties of anisotropic composites. J Mech Phys Solids1977; 25: 185–202.
21.
WillisJR. Variational and related methods for the overall properties of composites. Adv Appl Mech1981; 21: 1–78.
22.
WalpoleL. On bounds for the overall elastic moduli of inhomogeneous systems–I. J Mech Phys Solids1966; 14: 151–162.
23.
WalpoleL. On the overall elastic moduli of composite materials. J Mech Phys Solids1969; 17: 235–251.
24.
KrönerE. Bounds for effective elastic moduli of disordered materials. J Mech Phys Solids1977; 25: 137–155.
25.
MiltonG. The coherent potential approximation is a realizable effective medium scheme. Commun Math Phys1985; 99: 463–500.
26.
Milton G. The theory of composites (Cambridge Monographs on Applied and Computational Mathematics). Cambridge, UK: Cambridge University Press, 2002.
27.
TorquatoS. Random heterogeneous materials: microstructure and macroscopic properties, New York: Springer, 2002.
28.
WillisJR. The overall elastic response of composite materials. J Appl Mech1983; 50: 1202–1209.
29.
TalbotDWillisJ. Variational principles for inhomogeneous non-linear media. IMA J Appl Math1985; 35: 39–54.
30.
CastañedaPP. The effective mechanical properties of nonlinear isotropic composites. J Mech Phys Solids1991; 39: 45–71.
31.
IdiartMCastañedaPP. Second-order estimates for nonlinear isotropic composites with spherical pores and rigid particles. Comptes Rendus Mécanique2005; 333: 147–154.
32.
LiuYCastañedaPP. Second-order theory for the effective behavior and field fluctuations in viscoplastic polycrystals. J Mech Phys Solids2004; 52: 467–495.
33.
BöhlkeTLobosM. Representation of Hashin–Shtrikman bounds of cubic crystal aggregates in terms of texture coefficients with application in materials design. Acta Materialia2014; 67: 324–334.
34.
EshelbyJD. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc A1957; 241: 376–396.
35.
MoriTTanakaK. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica1973; 21: 571–574.
36.
BenvenisteYDvorakGChenT. Stress fields in composites with coated inclusions. Mech Mater1989; 7: 305–317.
37.
ChenTDvorakGBenvenisteY. Stress fields in composites reinforced by coated cylindrically orthotropic fibers. Mech Mater1990; 9: 17–32.
38.
ZhaoYTandonGWengG. Elastic moduli for a class of porous materials. Acta Mechanica1989; 76: 105–131.
39.
BenvenisteY. A new approach to the application of Mori-Tanaka’s theory in composite materials. Mech Mater1987; 6: 147–157.
40.
Brylka B. Charakterisierung und Modellierung der Steifigkeit von langfaserverstärktem Polypropylen (Schriftenreihe Kontinuumsmechanik im Maschinenbau Nr. 10). Doctoral Dissertation, KIT Scientific Publishing, Karlsruhe, Germany, 2017.
41.
Schemmann M. Biaxial characterization and mean-field based damage modeling of sheet molding compound composites (Schriftenreihe Kontinuumsmechanik im Maschinenbau Nr. 13). Doctoral Dissertation, KIT Scientific Publishing, Karlsruhe, Germany, 2018.
42.
Schemmann M, Gajek S and Böhlke T. Biaxial tensile tests and microstructure-based inverse parameter identification of inhomogeneous SMC composites. In: Altenbach H, Jablonski F, Müller W, Naumenko K and Schneider P (eds) Advances in mechanics of materials and structural analysis. Berlin, Germany: Springer, 2018, pp.329–342.
43.
HersheyAV. The elasticity of an isotropic aggregate of anisotropic cubic crystals. J Appl Mech Trans ASME1954; 21: 236–240.
44.
KrönerE. Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls. Zeitschrift für Physik1958; 151: 504–518.
45.
BudianskyB. On the elastic moduli of some heterogeneous materials. J Mech Phys Solids1965; 13: 223–227.
46.
HillR. A self-consistent mechanics of composite materials. J Mech Phys Solids1965; 13: 213–222.
47.
LawsN. On the thermostatics of composite materials. J Mech Phys Solids1973; 21: 9–17.
48.
LawsNMcLaughlinR. Self-consistent estimates for the viscoelastic creep compliances of composite materials. Proc R Soc London A1978; 359: 251–273.
49.
ChristensenRLoK. Solutions for effective shear properties in three phase sphere and cylinder models. J Mech Phys Solids1979; 27: 315–330.
50.
BenvenisteY. Revisiting the generalized self-consistent scheme in composites: clarification of some aspects and a new formulation. J Mech Phys Solids2008; 56: 2984–3002.
51.
ZhengQSDuDX. An explicit and universally applicable estimate for the effective properties of multiphase composites which accounts for inclusion distribution. J Mech Phys Solids2001; 49: 2765–2788.
52.
DuDXZhengQS. A further exploration of the interaction direct derivative (IDD) estimate for the effective properties of multiphase composites taking into account inclusion distribution. Acta Mechanica2002; 157: 61–80.
53.
KlusemannBBöhmHSvendsenB. Homogenization methods for multi-phase elastic composites with non-elliptical reinforcements: comparisons and benchmarks. Eur J Mech A/Solids2012; 34: 21–37.
54.
MüllerVBrylkaBDillenbergerF, et al.Homogenization of elastic properties of short-fiber reinforced composites based on measured microstructure data. J Compos Mater2016; 50: 297–312.
55.
LielensGPirottePCouniotA, et al.Prediction of thermo-mechanical properties for compression moulded composites. Compos Part A Appl Sci Manuf1998; 29: 63–70.
56.
CamachoCWTuckerCLYalvaçS, et al.Stiffness and thermal expansion predictions for hybrid short fiber composites. Polym Compos1990; 11: 229–239.
57.
DoghriITinelL. Micromechanical modeling and computation of elasto-plastic materials reinforced with distributed-orientation fibers. Int J Plasticity2005; 21: 1919–1940.
58.
DoghriIFriebelC. Effective elasto-plastic properties of inclusion-reinforced composites. Study of shape, orientation and cyclic response. Mech Mater2005; 37: 45–68.
59.
MeraghniFBenzeggaghM. Micromechanical modelling of matrix degradation in randomly oriented discontinuous-fibre composites. Compos Sci Technol1995; 55: 171–186.
60.
JendliZMeraghniFFitoussiJ, et al.Multi-scales modelling of dynamic behaviour for discontinuous fibre SMC composites. Compos Sci Technol2009; 69: 97–103.
61.
AnagnostouDChatzigeorgiouGChemiskyY, et al.Hierarchical micromechanical modeling of the viscoelastic behavior coupled to damage in SMC and SMC-hybrid composites. Compos Part B Eng2018; 151: 8–24.
62.
Truesdell C and Toupin R. The classical field theories. In: Flügge S (ed) Principles of classical mechanics and field theory. Berlin, Heidelberg, Wien: Springer, 1960, pp.226–793.
63.
Kehrer ML. Thermomechanical mean-field modeling and experimental characterization of long fiber-reinforced sheet molding compound composites (Schriftenreihe Kontinuumsmechanik im Maschinenbau Nr. 15). Doctoral Dissertation, KIT Scientific Publishing, Karlsruhe, Germany, 2019.
64.
ColemanBDNollW. The thermodynamics of elastic materials with heat conduction and viscosity. Arch Ration Mech Anal1963; 13: 167–178.
HillR. Elastic properties of reinforced solids: some theoretical principles. J Mech Phys Solids1963; 11: 357–372.
67.
MandelJ. Généralisation de la théorie de plasticité de WT Koiter. Int J Solids Str1965; 1: 273–295.
68.
Castañeda PP and Suquet P. Nonlinear composites. In: Erik van der Giessen and Theodore Y Wu (eds) Advances in applied mechanics. Vol. 34. Amsterdam, Netherlands: Elsevier, 1998, pp.171–302.
BöhlkeTBrüggemannC. Graphical representation of the generalized Hooke’s law. Technische Mechanik2001; 21: 145–158.
72.
BöhlkeTHenningFHrymakAN, et al.Continuous-discontinuous fiber-reinforced polymers. An integrated engineering approach, Munich: Hanser Fachbuchverlag, 2019.
73.
BenvenisteYDvorakGChenT. On diagonal and elastic symmetry of the approximate effective stiffness tensor of heterogeneous media. J Mech Phys Solids1991; 39: 927–946.
74.
QiuYWengG. On the application of Mori-Tanaka’s theory involving transversely isotropic spheroidal inclusions. Int J Eng Sci1990; 28: 1121–1137.
75.
WengG. The theoretical connection between Mori-Tanaka’s theory and the Hashin-Shtrikman-Walpole bounds. Int J Eng Sci1990; 28: 1111–1120.
76.
Lobos Fernández M. Homogenization and materials design of mechanical properties of textured materials based on zeroth-, first-and second-order bounds of linear behavior (Schriftenreihe Kontinuumsmechanik im Maschinenbau Nr. 12). Doctoral Dissertation, KIT Scientific Publishing, Karlsruhe, Germany, 2018.
77.
GörthoferJSchneiderMOspaldF, et al.Computational homogenization of sheet molding compound composites based on high fidelity representative volume elements. Comput Mater Sci2020; 174: 109456–109456.
