Viscoelastic Interphases in Polymer–Matrix Composites: Theoretical Models and ﬁnite-Element Analysis

Composites Science and Technology 61 (2001) 731–748 www.elsevier.com/locate/compscitech

Viscoelastic interphases in polymer–matrix composites: theoretical models and ﬁnite-element analysis

F.T. Fisher, L.C. Brinson * Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA

Received 21 January 2000; received in revised form 7 December 2000; accepted 18 January 2001

Abstract We investigate the mechanical property predictions for a three-phase viscoelastic (VE) composite by the use of two micromechanical models: the original Mori–Tanaka (MT) method and an extension of the Mori–Tanaka solution developed by Benve- niste to treat fibers with interphase regions. These micro-mechanical solutions were compared to a suitable finite-element analysis, which provided the benchmark numerical results for a periodic array of inclusions. Several case studies compare the composite moduli predicted by each of these methods, highlighting the role of the interphase. We show that the MT method, in general, provides the better micromechanical approximation of the viscoelastic behavior of the composite; however, the micromechanical methods only provide an order-of-magnitude approximation for the effective moduli. Finally, these methods were used to study the physical aging of a viscoelastic composite. The results imply that the existence of an interphase region, with viscoelastic moduli different from those of the bulk matrix, is not responsible for the difference in the shift rates, 22 and 66, describing the transverse Young’s axial shear moduli, found experimentally. # 2001 Published by Elsevier Science Ltd. All rights reserved. Keywords: Interphase; Micromechanics; Viscoelasticity; Finite element analysis; Physical aging

1. Introduction PMC with a viscoelastic interphase may be undertaken. Such analytical models may be used to study the inter- Polymer–matrix composites (PMCs), particularly those phase region as related to: containing fiber reinforcement, have become increasingly popular for use in structural applications. The primary 1. modeling the effective behavior of composite benefit of such systems is the potential for impressive materials, strength-to-weight ratios. Other additional benefits, such 2. optimizing composite properties through engi- as improved corrosion resistance and material property neered interphases, tailorability, can be of significance for specialized appli- 3. measuring the interphase mechanical properties cations. One area of complication in the analysis of through experimental tests, PMCs is the existence of an interphase region of finite 4. more complex composite behavior (aging, damage, dimensions. For PMCs, interphase weight fractions on moisture absorption, etc.). the order of 5% of the total weight of the composite have been estimated experimentally [1]. In this case, one A review of the literature produces a large number of may expect the interphase region to influence the overall models developed for the prediction of the elastic moduli effective behavior of the composite. The goal of this of composite materials. Excellent review articles are work is to extend micromechanical solutions, developed provided by Hashin [2] and Christensen [3]. Several for multiphase elastic materials, so that an analysis of a researchers have used the Mori–Tanaka method to study multiphase composite behavior [4–6]. Other micromechanical analyses of effective composite behavior can also be found [7–9]. Weng [10] noted the relationship * Corresponding author. Tel.: +1-847-467-2347; fax: +1-847-491- between the MT method and the Hashin–Shtrikman– 3915. Walpole bounds for multiphase elastic composites when E-mail address: [email protected] (L.C. Brinson). the matrix is the stiffest or the softest elastic material.

0266-3538/01/$ - see front matter # 2001 Published by Elsevier Science Ltd. All rights reserved. PII: S0266-3538(01)00002-1 732 F.T. Fisher, L.C. Brinson / Composites Science and Technology 61 (2001) 731–748

Analysis of viscoelastic composites using micro- that these tensors remain valid for the related, multiple- mechanical techniques is less developed. Early work in inclusion composite (where the stress which the inclu- this area was pioneered by Hashin [11–14] and Schapery sions now ‘‘feel’’ is the unknown average matrix stress), [15,16]. Some researchers have looked at the effect of the overall composite moduli can be determined. inclusion shape on the mechanical properties of two- For comparison, a finite element analysis with the phase VE composites [17,18]. Brinson and Lin [19,20] included phases aligned in an hexagonal array was also have compared the MT method with a finite-element completed. We use the dynamic correspondence principle analysis of two-phase VE composites. In the present to solve the unit cell boundary value problem necessary to work, two micromechanical methods and a finite-ele- obtain the effective complex moduli of the composite [25]. ment solution will be used to analyze composites with A hexagonal array of inclusions yields transversely iso- VE interphases. As described in the next section, the tropic composite behavior, allowing direct comparison correspondence principle is used to extend these solu- between the FEA and micromechanical solutions. Other tions for viscoelastic materials. researchers have used alternative interface/interphase The first micromechanical method used in this work is models to study the interaction between the fiber and the Mori–Tanaka method [21–23]. The MT method has matrix regions [26–28], as well as FEA to study the been used for a wide range of problems and thus become effective moduli of composites with elastic interphases a popular tool for analysis of multiphase materials. Here [29–31]. Yi et al. [32] found that for composites with we extend the model to study a three-phase fibrous com- viscoelastic interphases, the effective transverse modulus posite consisting of isotropic, viscoelastic phase materi- was largely dependent on the volume fraction of the als. As used here, the MT method does not model the fiber and the viscoelastic behavior of the matrix. geometry of an annular interphase region surrounding The following section provides background informa- the fiber inclusion, but rather treats the fiber and inter- tion concerning viscoelasticity, the correspondence phase regions as separate, physically distinct regions as principle, and the role of the interphase for PMCs. The shown in Fig. 1. Although this approach at first glance extension of the micromechanical methods and the FEA may appear inappropriate for representing the real for viscoelastic materials will then be briefly outlined, fiber-interphase-matrix geometry (see Fig. 2), the MT after which the results of the models will be presented technique is so widely used and simple to employ that it and discussed. Finally, these models will be used to is useful to know how accurate the prediction may be in investigate the impact of a viscoelastic interphase on the this context. physical aging of polymer matrix composites. The second micromechanical solution follows the model proposed by Benveniste et al. [24] to study materials with coated inclusions. This model maintains the 2. Background proper fiber-interphase-matrix geometry as shown in Fig. 2. It utilizes a related auxiliary problem (i.e. a single In this paper, we analyze the mechanical response of a fiber-interphase inclusion within an infinite matrix fiber-reinforced polymer matrix composite with a vis- material) to find the stress-concentration tensors relating coelastic interphase region. Recent experimental evidence an applied far-field stress to the phase-averaged stresses suggests that the interphase region could be larger than of the included (fiber and interphase) phases. Assuming previously believed, making it necessary to explicitly

Fig. 1. Interphase modeled as a distinct inclusion region for the Mori– Fig. 2. Three phase composite model for the Benveniste solution. Tanaka method. a=Radius of fiber; b=outer radius of interphase. F.T. Fisher, L.C. Brinson / Composites Science and Technology 61 (2001) 731–748 733 account for such a region from a design perspective. diffuse into the fiber sizing such that no discernible Properties of the interphase region can be expected to interphase could be detected. Obviously, a better differ from the other bulk materials [33]. Because ana- understanding of the interphase region will be necessary lysis of the time-dependent response with VE phase as models to predict the behavior of composites with materials can become computationally prohibitive, the finite interphase regions are developed. In this paper, a dynamic correspondence principle is used to study the nominal interphase area fraction of 10% was used associated problem in the frequency domain. Two (unless otherwise noted) to maximize the influence of hypothetical VE materials, and a hypothetical elastic the interphase on the effective moduli of the composite. fiber, are used for the numerical simulations presented in this paper. 2.2. Viscoelasticity and the correspondence principle

2.1. The interphase region The constitutive equation for a viscoelastic material can be written in terms of a Stieltjes integral equation of The interphase (or mesophase) refers to the inhomo- the form geneous region surrounding the inclusions in a compo- ð t site material. Interphase regions may be due to voids, d"klðÞ ijðÞ¼t CijklðÞt À d; ð1Þ mechanical imperfections, unreacted polymer compo- À1 d nents, fiber treatments, restricted macromolecular mobility due to the fiber surface, and other inconsistencies where ij and "kl are standard stress and strain tensors [34–36]. In addition, there are ‘‘engineered interphases’’: and Cijkl is the time-dependent modulus. For isotropic the deliberate introduction of a novel third phase (for VE materials, the mechanical response can be char- example, fiber sizings), with mechanical properties on the acterized by the time-dependent bulk and shear moduli, order of magnitude of the matrix, to optimize composite K(t) and G(t), respectively. However, it is possible to performance. In each case, a relatively small volume completely avoid time domain analysis by utilizing the fraction of interphase may play a crucial role in deter- dynamic correspondence principle, where the mechanical mining the overall mechanical response of the composite. characterization of viscoelastic materials is completed in One difficulty that is encountered when modeling the the transformed (frequency) domain [11]. This is interphase is the characterization of the size and prop- accomplished by considering a separable form for the erties of the region. Experimental work to better deter- displacements, mine these parameters are ongoing. In the past, the i!t thickness of a PMC interphase has been estimated to be uiðÞ¼x; t uiðÞx;! e ; ð2Þ between 30 and 240 nm [37], and similar results were reported in an experimental study using scanning force where ! is frequency, x is position, and transformed microscopy [38]. However, other recent experimental find- quantities are denoted by an overbar. Given (2), the ings suggest that the interphase region within PMC mate- constitutive equations for an isotropic viscoelastic rials may be larger. These techniques, including secondary material can be written in terms of transformed devia- ion mass spectroscopy [1] and atomic force microscopy toric and dilitational stress components, [39], suggest interphase thicknesses on the order of 1 mm Ã Ã for glass-fiber epoxy composites. An example of PMC sij ¼ 2 G ðÞ! "ij;kk ¼ 3 K ðÞ! "kk; ð3Þ interphase thicknesses measured experimentally are sum- marized in Table 1. where the complex bulk and shear moduli, K ÃðÞ! and Another atomic force microscopy study [40] showed G ÃðÞ! , are defined as ð that variations in the fiber sizing can impact the creation 1 of the interphase region. In this study, the matrix mate- K ÃðÞ! i!K ðÞ¼! i! KtðÞ eÀi!tdt; ð0 rial was unable to completely diffuse into a thick fiber 1 ð4Þ sizing, and hence a distinct interphase region was rea- G ÃðÞ! i!G ðÞ¼! i! GtðÞ eÀi!tdt: lized. For a thinner fiber sizing, the matrix was able to 0

Table 1 Experimental interphase thicknesses for PMC materials

Researcher Year Method Thickness Material

Theocaris [36] 1985 Differential scanning calorimetry 30–240 nm Glass-fiber/epoxy vf=10–70% Thomason [1] 1995 Secondary ion mass spectroscopy 1 mm Glass-fiber/epoxy vf=65%; df=25 mm Munz et al. [38] 1998 Scanning force microscopy 75–150 nm Carbon-fiber/poly-phenylenesulfide Mai et al. [39] 1998 Atomic force microscopy 1–3 mm Carbon-fiber/epoxy single fiber 734 F.T. Fisher, L.C. Brinson / Composites Science and Technology 61 (2001) 731–748

These complex moduli are typically written in terms where Aj and j represent the relaxation spectra and of the storage and loss moduli, relaxation times, respectively, and A1 is the rubbery asymptotic value of the appropriate modulus. The K ÃðÞ¼! K0ðÞþ! iK00ðÞ! ; parameters used to obtain the viscoelastic behavior of ð5Þ the ‘‘stiff’’ and ‘‘soft’’ materials are given in Table 2. The G ÃðÞ¼! G0ðÞþ! iG00ðÞ! ; fiber, always the stiffest of the three phases, had elastic 1 moduli characterized by Gf=40,000 and Kf=100,000. where single and double primes denote the storage and loss moduli, respectively. The storage modulus is a measure of the elastic response of the material, whereas 3. Micro-mechanical methods the loss moduli represents viscous (damping) behavior [41]. Two micro-mechanical analyzes, the original Mori– Note that the constitutive equations in (3) are analo- Tanaka method and an extension of this solution pro- gous to the standard elasticity formulation, where now posed by Benveniste, are used to predict the mechanical all field quantities are complex. Thus, there is a direct behavior of a three-phase composite. The former is a relationship between elasticity problems in the time standard micromechanical tool which only approx- domain and viscoelasticity problems in the frequency imates the composite geometry, whereas the latter keeps domain. We take advantage of this relationship by the physicality of an annular interphase region intact. transforming appropriate elasticity solutions into the Because the micromechanical solutions are easier to frequency domain and replacing the elastic moduli with implement than a full FEA, it will be useful to ascertain the related complex moduli defined in (4). The response the ability of these methods to provide approximations of the composite can be found by completing the desired of the composite response. analysis for an appropriate range of frequencies. 3.1. Mori–Tanaka method 2.3. Constituent materials The formulation of the Mori–Tanaka method for the Our goal is to determine the effective moduli of a determination of the effective moduli of a viscoelastic three phase viscoelastic composite consisting of cylind- composite is briefly presented. More detailed deriva- rical fibers, surrounded by an annular interphase, and tions can be found in the literature [21–23]. As used embedded in a binding, continuous matrix. Perfect here, the MT method approximates the fiber-interphase bonding between the phases is assumed. Both the matrix problem using a composite with distinct cylindrical and interphase are isotropic, linear viscoelastic materi- inclusions representing the fiber and interphase regions als, whereas the fiber is isotropic elastic. Temperature (see Fig. 1). Since the Mori–Tanaka method is a stan- effects were not studied in the first part of this work, dard micromechanical tool and easy to apply, it will be although the physical aging study presented at the end valuable to determine the approximate solution the of this paper assumes an isothermal temperature below Mori–Tanaka method provides in this context. the glass transition temperatures (Tg) of each viscoelas- The individual phases which comprise the composite tic material. It should be noted that the Tg of the inter- are denoted f, g, and m for the fiber, interphase, and phase may also vary significantly from the other matrix, respectively. Area fractions for each phase are viscoelastic phases of the composite. Processing tempera- denoted ck, where k={f,g,m}, a is the fiber radius, and b tures greater than the interphase Tg will increase mole- is the outer radius of the interphase; thus, cular mobility in the region, enhancing the ability of the interphase to properly bond with the other constituent ÀÁ c ¼ a2; c ¼ b2 À a2 ; c ¼ 1 À c À c : ð7Þ materials [1]. f g m f g For this investigation two hypothetical viscoelastic materials, designated ‘‘stiff’’ and ‘‘soft’’, were used as the matrix and interphase materials in a three-phase Following the standard elastic derivation, and imple- composite. Various area fractions and permutations of menting the correspondence principle to account for the constitutive materials were analyzed. The viscoelas- viscoelasticity, one can show tic bulk and shear moduli of these two materials were represented using a standard Prony series representation of the form 1 While unitless parameters are used here, the ratios of the con- XN stituent moduli (e.g. fiber to matrix) are consistent with those used by Àt=j other researchers for finite element analyzes of polymer matrix mate- AtðÞ¼A1 þ Aj e ; ð6Þ rials [29] and representative of common composite constituent mate- j¼1 rials. F.T. Fisher, L.C. Brinson / Composites Science and Technology 61 (2001) 731–748 735

Table 2 lar interphase, and as such models the actual geometry Prony series terms for the viscoelastic moduli of the ‘‘stiff’’ and ‘‘soft’’ of the interphase region. The local stress fields in the hypothetical materials fiber and interphase are approximated by those of Stiff material Soft material an auxiliary problem,asingle fiber-interphase inclusion embedded in an infinite medium of matrix material and G =100 G =3.162 1 1 subjected to an appropriate far-field stress field. The

K1=8000 K1=200 superposition of auxiliary problems necessary to determine the complex transverse Young’s modulus, E Ã ,is G G G G 22 j j j j shown in Fig. 3. Particle interaction is described 3.0 3.1620.0322.512 through the unknown average matrix stress, in a man- 10.0 17.783 0.100 10.0 ner identical to that used in the MT method. The effec- 32.0 100.0 0.316 56.234 100.0 316.228 1.0 316.228 tive moduli are found by considering different far-field 316.0 1000.0 3.1621000.0 loadings; the reader is referred to the original work of 1000.0 5623.413 10.0 199.526 Benveniste for a complete derivation. 3162.0 10000.0 31.623 50.119 Using the correspondence principle to account for vis- 10000.0 562.341 100.0 19.953 coelastic behavior, the displacements for each auxiliary 31623.0 141.254 316.228 12.589 100000.0 56.234 1000.0 2.512 problem can be written in terms of unknown complex 316228.0 17.783 3162.278 1.698 constants. The analysis then proceeds along the lines of a 1000000.0 5.623 10000.0 1.202 standard elasticity solution in complex space by simply 3162278.0 3.162 31622.777 1.148 carrying through the complex variables. Thus, Hooke’s 10000000.0 1.778 100000.0 1.096 K K Law and the strain-displacement equations provide the j Ki j Ki 10000 40000 100 3000 analytic expressions for the stress and strain fields in each 316.228 100 phase of the auxiliary problem. Enforcing suitable boundary conditions provides the complex displacement constants, allowing the phase-average stresses in each "# included phase to be determined. Appropriate stress- Ã X ÀÁÈÉ Ã Lm Ã concentration tensors, W k for k={f,g}, are then defined S þ A k À cnS A n ¼ÀI; k; n ¼ f; g ; k Ã Ã n Lk À Lm n to relate the far-field applied stress of the auxiliary pro- ð8Þ blem, o, to the primary phase-average stress compo- k nent of the included phases, ij,

Ã where n is a dummy variable, Lm is the complex stiffness Ã Ã auxiliary problem of the matrix, Lk and Sk are the complex stiffness and ÀÁÈÉ k k the Eshelby tensor of the kth inclusion (evaluated using ij ¼ W o: k ¼ f; g : ð10Þ the complex matrix properties), and I is the fourth-order identity tensor. The strain-concentration tensors Af and Ag found via (8) relate the uniform applied strain " to Once these tensors have been determined for the aux- Ã the transformation strain "~k of the included phases via iliary problem, it is assumed that they are also valid for Ã the relationship "~k ¼ Ak ". The effective modulus of the actual (multiple inclusion) composite geometry. In the composite, L Ã, is found to be this case, the applied stress to which the included phases "# are subjected is the average matrix stress, such that X ÀÁÈÉ Ã Ã L ¼ Lm I À ckAk ; k ¼ f; g : ð9Þ k multiple inclusion geometry ÀÁÈÉ k k m ij ¼ W ij : k ¼ f; g ð11Þ Once the effective complex stiffness of the composite has been determined, it is straightforward to determine One can now enforce consistency for the actual com- five transversely isotropic material moduli that describe posite geometry, given the absence of body forces, to the mechanical response. find the average matrix stress within the composite. Using (11), the primary phase-averaged stresses for 3.2. Benveniste solution all phases within the actual composite can be determined. Using these average stresses, the average strain Benveniste et al. [24] have proposed a model to eval- within each phase can be determined; at this point it is uate the effective moduli of a three phase elastic com- then straightforward to determine the complex compo- posite based on the local fields of composites with coated site moduli. This procedure is further outlined in the inclusions. This derivation explicitly considers an annu- appendices. 736 F.T. Fisher, L.C. Brinson / Composites Science and Technology 61 (2001) 731–748

Fig. 3. Superposition of the transverse hydrostatic and transverse shear auxiliary problems for the determination of the transverse Young’s modulus.

4. Finite element formulation independent of inclusion packing [31]. Therefore, the results of this section will provide a baseline by which A two-dimensional finite element analysis (FEA) was the results of the two micromechanical methods may be undertaken on the unit cell models of Figs. 4 and 5. Plane judged, although neither analytical method explicitly strain conditions were assumed. The mesh was refined represents a periodic array of inclusions. Because it is until a convergent solution for a hexagonal array of assumed that the reader is familiar with the finite ele- inclusions was found. These results were then used as the ment method for elastic materials, we will mention only benchmark numerical result for transversely isotropic briefly the modifications necessary for a viscoelastic composite behavior. Although stress fields and related boundary value problem. This finite element work fol- localized phenomena (such as plasticity and failure initia- lows that of Brinson and co-workers [19,25,42] for two tion) are very dependent on the inclusion arrangement, phase viscoelastic materials. Suitable boundary condi- the effective moduli of composite materials are largely tions for the transverse Young’s and transverse shear Ã Ã complex moduli, E22 and G44, are given in Figs. 4 and 5. For the transverse Young’s modulus unit cell shown in Fig. 4, the top face is given a uniform displacement of uo. The prescribed displacement À b at x=W ensures that the right-hand side of the unit cell remains straight and parallel upon deformation. This displacement is chosen by simple interpolation such that the traction boundary condition at x=W is approximated as ð L T x dy ¼ 0; at x ¼ W ð12Þ 0

where T x are the resultant nodal tractions on the right Fig. 4. Unit cell analysis to determine the complex transverse Young’s hand side of the unit cell. Alternative methods to satisfy modulus. (12) are discussed in the literature [25]. The average resulting normal stress y;aveðÞ! on the top face of the unit cell was obtained by summing the resultant nodal forces at y=L and dividing by the area. Given the prescribed normal strain across the top face of the unit cell, uo "y ¼ L , the complex transverse Young’s modulus of the composite can be determined via

ðÞ Ã y;ave ! E22ðÞ¼! i ! E22ðÞ! : ð13Þ "y

A similar procedure is required to determine the Fig. 5. Unit cell analysis to determine the complex transverse shear complex transverse shear modulus of the composite (see modulus. Fig. 5). Solution of this problem does not require inter- F.T. Fisher, L.C. Brinson / Composites Science and Technology 61 (2001) 731–748 737 polation to provide suitable boundary conditions. Also included in these figures are the transformed uo Applying a uniform shear strain xy L across the top Reuss and Voigt bounds (i.e. the Rule of Mixtures). face of the unit cell, one can show Note that in Fig. 7 loss moduli predictions appear to violate these bounds at intermediate frequencies. This ðÞ! apparent discrepancy is admissible because the trans- G Ã ðÞ¼! i!G ðÞ! xy;ave ; ð14Þ 44 44 formed bounds are not in fact valid in the frequency xy domain. Schapery [16] notes that, for composites with isotropic phase materials, transformed bounds hold for where xy;aveðÞ! is the sum of the resultant nodal forces the ‘‘operational moduli’’ (i.e. the transformed moduli in along the top face of the unit cell in the x-direction Laplace space) of the composite. Indeed, it was verified divided by the width of the unit cell. numerically that the micromechanical solutions and FEA results do fall within the bounds when the Laplace transform is used in the above formulations. Gibiansky 5. Results and co-workers [44,45] have developed bounding methods in the Fourier domain for a limited number of The models discussed in the previous sections were moduli for two-phase viscoelastic composites. used to predict composite moduli as a function of the Figs. 6 and 7 indicate that, for stiff-interphase com- Ã properties and volume fractions of the constituent posites, the micromechanical predictions for E22 are in materials. These results will provide insight as to the close agreement with both each other and the FEA ability of the analytical models to capture the mechan- results; this was found to be true for included phase area ical behavior predicted by the more thorough FEA. The fractions (fiber plus interphase) approaching 70%. Since Ã storage and loss components of the transverse Young’s the results for G44 are similar, a detailed discussion of modulus for a composite consisting of 30% elastic fiber, these results will not be included. For comparison, the 10% stiff interphase, and 60% soft matrix are shown in transverse moduli for the complementary composite Figures 6 and 7. Note that 60% fiber cases have also configuration (30% elastic fiber, 10% soft interphase, been examined [43] and show similar trends to those 60% stiff matrix) are shown in Figs. 8 and 9. For soft- presented in this paper. Cases with 30% fiber and 10% interphase composites, a large discrepancy was found to interphase were chosen for illustration in order to exist between the two micromechanical models, with the emphasize the influence of the interphase on the viscoe- FEA results falling between these two methods. Com- lastic moduli of the composite. Here again ‘‘stiff’’ or paring this behavior with that of the constituent phases ‘‘soft’’ refer to the viscoelastic material moduli descri- (also shown in Figs. 8 and 9), we see that the frequency- bed earlier. dependence of the Benveniste solution is in general

Fig. 6. Transverse Young’s storage moduli solutions for a composite with 30% elastic ﬁber, 10% stiﬀ interphase, 60% soft matrix. 738 F.T. Fisher, L.C. Brinson / Composites Science and Technology 61 (2001) 731–748

Fig. 7. Transverse Young’s loss moduli solutions for a composite with 30% elastic ﬁber, 10% stiﬀ interphase, 60% soft matrix.

dominated by the interphase material (most explicitly results for a particular frequency within the transition seen in the loss modulus), even though it is a non-con- region of the constituent materials (!=1E-4), this was tinuous phase making up only a small portion of the found to be true for all configurations and frequencies composite. This result is viewed as a major drawback of tested. It appears that micromechanical models which the Benveniste solution, particularly for the analysis of assume a uniform stress distribution within the con- the soft-interphase composites. stituent materials may be severely oversimplifying the We believe that this flaw is largely due to the inability analysis of the problem. of the Benveniste solution to accurately model the The Mori–Tanaka solution, even though it does not influence of the interphase on the stress transfer between model the actual geometry of the problem, seems to the fibers and matrix. This is illustrated in Fig. 10, better approximate the viscoelastic behavior of the where the normalized (with respect to the average moduli in that the effective moduli are dominated by the matrix stress), phase-averaged real component of yy as matrix material. Because the MT method treats the fiber calculated via the FEA and the Benveniste solution are and interphase as separate inclusions, the solution for compared for the soft-interphase composite configura- soft interphase composites does not rely on the transfer tion discussed above. For this composite configuration, of stress through the interphase. These conclusions and the Benveniste solution severely overestimates the aver- results are consistent with those obtained for other age stress in the included phases at all frequencies. We moduli and phase area fractions [43]. Ã believe this is caused by the smearing of the ‘‘phase-aver- Figs. 12and 13 show the components of E22 as a aged’’ stresses required to determine the stress-con- function of interphase area fraction for a constant cf of centration tensors necessary for this procedure. However, 30% and at fixed frequency of 1E-4; this is within the in the case of a stiff-interphase composite (not shown), transition region of each viscoelastic material. The the Benveniste solution is able to more closely model the results show that both micromechanical solutions pro- stress transfer between the soft matrix and the fibers, vide very good approximations when the interphase is which suggests use of the Benveniste model for stiff- the stiff VE material, whereas the predictions are again interphase composites only [43]. poor when the interphase is the soft material. Again, Fig. 11 shows that the FEA predictions of the stress these results were representative of others obtained for distribution for the two complementary composite con- different composite configurations and at different fre- figurations (soft interphase versus stiff interphase) are quencies. In the limit case of zero interphase area fraction, very different, and in both cases, far from uniform. This the micromechanical solutions are identical since both agrees with published experimental results for a three- utilize the concept of an ‘‘unknown average matrix stress’’ phase epoxy matrix composite [31]. While Fig. 11 shows to account for inclusion interaction. A comparison of the F.T. Fisher, L.C. Brinson / Composites Science and Technology 61 (2001) 731–748 739

Fig. 8. Transverse Young’s storage moduli solutions for a composite with 30% elastic ﬁber, 10% soft interphase, 60% stiﬀ matrix.

Fig. 9. Transverse Young’s loss moduli solutions for a composite with 30% elastic ﬁber, 10% soft interphase, 60% stiﬀ matrix.

Mori–Tanaka method and a comparable ﬁnite element a polymeric material below its glass transition tempera- analysis for a two phase composite was presented in a ture (Tg) is in a non-equilibrated thermodynamic state. previous study [19]. As the polymer slowly approaches equilibrium, changes in mechanical behavior, and other phenomena, are rea- lized. Similar to time-temperature superposition, the 6. Physical aging case study mechanical response at diﬀerent aging times (te) can be superposed via a horizontal shifting of a master refer- Physical aging has been the subject of numerous ence curve by an amount equal to the aging time shift investigations; for an in-depth review of the subject see factor (ate ). These shift factors are related to aging time Ferry [41] and Struik [46]. Physical aging occurs because via the shift rate , 740 F.T. Fisher, L.C. Brinson / Composites Science and Technology 61 (2001) 731–748

ave Fig. 10. Comparison of the phase-averaged real components of yy from the FEA and Benveniste solutions for a 30% elastic fiber, 10% soft interphase, 60% stiff matrix composite at three different frequencies (+xx% refers to the difference between the Benveniste solution and the FEA prediction).

Re Fig. 11. Comparison of the yy stress distribution for transverse normal loading at a frequency of 1E-4 for 60% elastic ﬁber composites with (a) stiﬀ interphase, (b) soft interphase.

d log a mentally. It should be stressed that the investigation ¼À te : ð15Þ d log te presented here is not rigorous; our 2D finite element code cannot determine the axial shear modulus of the composite. Instead we will analyze the aging behavior of Struik [46] has demonstrated that the shift rates the transverse shear modulus, another matrix-dominated describing all mechanical properties for a homogeneous property. Thus, the shift rates 22 and 44=55 will be polymer are identical. However, experimental work on analyzed via FEA to determine whether differences in polymer matrix composites suggests that different shift matrix and interphase shift rates could be responsible for rates, 22 and 66, are required to characterize the phy- the distinct effective shift rates seen experimentally for sical aging of the transverse normal and axial shear PMCs. responses, respectively; this despite the fact that both Shift rates of 0.85 for the interphase and 0.70 for the are considered matrix-dominated properties [47,48]. matrix were assigned for all trials. FEA was then used Ã Ã Here we investigate whether an interphase with aging to calculate E22 and G44 of the composite at different properties distinct from those of the matrix could be aging times. From the complex moduli, the time-depen- responsible for the different shift rates measured experi- dent moduli, E(t) and G(t), could be determined using a F.T. Fisher, L.C. Brinson / Composites Science and Technology 61 (2001) 731–748 741

Fig. 12. Complex transverse Young’s moduli versus interphase volume fraction for composites with 30% ﬁber, stiﬀ interphase, and soft matrix material at a frequency of 1E-4.

Fig. 13. Complex transverse Young’s moduli versus interphase volume fraction for composites with 30% fiber, soft interphase, and stiff matrix material at a frequency of 1E-4. linear least-squares solver [49]. Time-aging time super- Fig. 14 shows the frequency-dependent aging beha- position was performed on the different aged moduli to vior of the transverse shear loss modulus for a soft- determine the shift rates of the composite for each of the interphase composite as predicted by the finite element two moduli. Note that shift rates can only be determined analysis. The reference curve in Fig. 14 (assigned as the if the moduli functions at different aging times are initial aging time tage=1) has been shifted so as to best superposable via a simple shift along the ordinate axis. coincide with the moduli curves at other aging times. This represents physical aging having an equal impact on Note that while the reference curve has been shifted to all relaxation times describing the mechanical behavior. superpose low frequency data, the high frequency ends 742 F.T. Fisher, L.C. Brinson / Composites Science and Technology 61 (2001) 731–748

Fig. 14. Aged transverse shear loss moduli in the frequency domain for composite with 30% ﬁber, 10% soft interphase, 60% stiﬀ matrix via FEA.

Reference curve (tage=1) is shifted (shown as light gray lines) to the aged moduli such that low frequency data is matched.

do not overlap and thus prevent a unique shift factor viscoelastic material, the shift rates 22 and 44=55 from being determined. This type of behavior was found calculated using FEA are strongly dependent on the for all soft-interphase composites studied and is indica- shift rate of the isotropic matrix. This is not the case tive of thermorheologically complex (TRC) behavior. when the interphase is the softest phase; here the overall To circumvent this difficulty in shifting frequency shift rates of the composite are intermediate those of the domain moduli, the data were transformed into the time viscoelastic phases (more evident for the 60% fiber case, domain using a linear least squares solver [49]. Although where there is more interphase material per unit of the behavior of the material in the time domain is still matrix material). This suggests the influence of two TRC (see Fig. 15), the degree of mismatch between the separate factors which determine the effective physical moduli functions at different aging times is relatively aging of the composite: the shift rate of the softest VE small and, in practice, could easily be attributed to material and the shift rate of the matrix phase. For stiff- experimental scatter. If the slight non-overlap of the interphase composites, both conditions suggest aging superposed curves in Fig. 15 is ignored, the results at behavior dominated by the matrix, and in these instances different aging times can be shifted and a characteristic the TRC behavior of the composite response is negli- shift factor for the composite property determined.2 gible. Similar results for other soft-interphase composite These numerical tests in the time domain are then a rea- configurations support this hypothesis [43]. sonable model to ascertain if the interphase is respon- On the other hand, for soft-interphase composites, the sible for the differences in shift rates seen experimentally. shift rate of the matrix and the shift rate of the softest Thus, Eq. (15) was used to determine the effective shift VE material (in this case the interphase) oppose each rates as a function of composite configuration. other, which may be responsible for the TRC behavior Table 3 shows shift rates calculated for different effec- apparent in Fig. 14. This effect is magnified for larger tive moduli via the FEA and the micromechanical solu- fiber area fractions, as the ratio of the viscoelastic pha- tions. The results suggest that the shift rates determined ses approaches unity. While TRC behavior has been via FEA are largely dependent on the composite config- seen in previous numerical studies of two-phase viscoe- uration. For composites where the interphase is the stiff lastic composites [20,25], the present work suggests that a relatively small area fraction of interphase can have a large role in determining the overall TRC behavior of the composite. 2 An important point illustrated by this analysis is that time domain However, as the FEA results presented in Table 3 data is much more forgiving than frequency domain data. Thermo- rheological complexity, which can invalidate time-temperature and indicate, the interphase does not seem to cause a differ- time-aging time superposition, which may be masked in the time ence in the shift rates 22 and 44=55. Although this is domain, may be more easily detected in the frequency domain. circumstantial evidence (since the shift rate 66 was not F.T. Fisher, L.C. Brinson / Composites Science and Technology 61 (2001) 731–748 743

Fig. 15. Aged transverse Young’s moduli in the time domain for composite with 30% ﬁber, 10% soft interphase, 60% stiﬀ matrix composite found via FEA. Reference curve (tage=1) is shifted (shown as light gray lines) to the aged moduli.

Table 3

Eﬀective shift rates found in the time domain for various composite conﬁgurations. Shift rates of the phase materials were mat=0.70 for the matrix and int=0.85 for the interphase

Method Property Geometry

30% fiber 30% fiber 60% fiber 60% fiber 10% stiff interphase 10% soft interphase 10% stiff interphase 10% soft interphase 60% soft matrix 60% stiff matrix 30% soft matrix 30% stiff matrix

FEA 22 0.7003 0.7168 0.7003 0.7904 FEA 44 0.6989 0.7144 0.6988 0.7847

BENV 22 0.7007 0.8395 0.7006 0.8447 BENV 44 0.7016 0.8421 0.7007 0.8461 BENV 66 0.7011 0.7080 0.7008 0.7761

MT 22 0.6989 0.70120.7011 0.7014 MT 44 0.6997 0.7010 0.7000 0.7103 MT 66 0.6987 0.7009 0.6988 0.6933 determined numerically in this work), these results sug- solution predicts a large difference in the transverse and gest that the interphase is not responsible for the differ- axial shear shift rates, 44=55 and 66, respectively, ence in composite aging shift rates that is measured for the soft-interphase composite. While the Benveniste experimentally. Although this should be verified by a 3D method was earlier demonstrated to have difficulties in finite element analysis, these results imply that perhaps predicting the effective moduli of composites with soft other mechanisms (e.g. residual stresses) cause poly- interphases, these results suggest that a full 3D FEA to meric composites to respond to physical aging differ- analyze the interphase-dependence of the axial shear ently than homogeneous polymer materials. shift rate 66 may be warranted. The shift rates determined via the micromechanical methods are also given in Table 3. In general, the solutions found using the Benveniste method are dominated 7. Conclusions by the softest VE phase in the composite, whereas the MT results are dictated by the shift rate of the matrix This paper investigated the results of two micro- material. It is also interesting to note that the Benveniste mechanical methods, the original Mori–Tanaka method 744 F.T. Fisher, L.C. Brinson / Composites Science and Technology 61 (2001) 731–748 and an extension of the Mori–Tanaka model developed B B uf ¼ A r; ug ¼ A r þ g ; um ¼ A r þ m ; ð16Þ by Benveniste, for predicting the effective moduli of a r f r g r r m r three phase viscoelastic composite. The main conclusions of this work are outlined below: where A i (i=f,g,m) and B j (j=g,m) are the unknown displacement constants, which are complex in nature. . The micromechanical methods provide satisfac- These constants are determined by requiring that the tory approximations for the effective moduli in displacements satisfy suitable boundary conditions, those cases where the matrix is the softest of the including the far-field boundary conditions three phases. The difference between the MT and

Benveniste solutions is minimal in these cases. xx r !1 ¼ o ; yy r !1¼ o: ð17Þ . When the interphase is the softest material, the integrity of the micromechanical methods is greatly diminished. Here the MT method is more The analysis then proceeds along the lines of a stan- effective in modeling the viscoelastic behavior of dard elasticity solution. Assuming a state of plane the composite because it predicts matrix-domi- strain, the non-zero stresses in each isotropic phase can nated effective moduli for the composite. be written in terms of the complex Lame and shear Ã Ã . For further work, it may be interesting to compare moduli, li and Gi , respectively, the methods used in this paper to a micromechanical analytic method with periodic unit cell f Ã Ã Ã Ã arrangement, such as Aboudi’s method of cells [50], rr ¼ 2 Gf Af þ lf 2 Af ¼ 2 Af Gf þ lf ! although in general effective moduli have been B shown to be independent of inclusion packing [31]. g ¼ Ã À g þ Ã rr 2 Gg Ag 2 lg 2 Ag . The physical aging study indicated that thermo- r rheologically complex behavior can be masked in Ã Ã Ã Bg time domain data, suggesting that frequency ¼ 2 Ag Gg þ lg À 2 Gg ð18Þ !r2 domain data may be preferred in order to identify B TRC viscoelastic response. The loss modulus exhi- m ¼ Ã À m þ Ã rr 2 Gm Am 2 lm 2 Am bits the greatest sensitivity to TRC behavior. r . The FEA results suggest that the interphase plays Ã Ã Ã Bm a large role in determining the overall shift rates of ¼ 2 A m G þ l À 2 G ; m m m r2 the composite. However, the interphase does not cause a difference in shift rates describing the physical aging of the transverse shear and axial shear moduli, and m =m , respectively. With f Ã Ã Ã Ã 22 44 55 ¼ 2 Gf Af þ lf 2 Af ¼ 2 Af Gf þ lf this evidence, it seems unlikely that the interphase ! is responsible for the difference in and shift B 22 66 g ¼ 2 G Ã A þ g þ lÃ 2 A rates measured experimentally, although further g g r2 g g verification with a 3D finite element analysis may be warranted. Ã Ã Ã Bg ¼ 2 Ag Gg þ lg þ 2 Gg ð19Þ !r2 B m ¼ 2 G Ã A þ m þ lÃ 2 A Acknowledgements m m r2 m m The authors appreciate sponsorship from NSF for Ã Ã Ã Bm ¼ 2 A m G þ l þ 2 G : this work. m m m r2

Appendix A. Transverse hydrostatic loading auxiliary Enforcing the boundary conditions provides the fol- problem for the Benveniste solution lowing five simultaneous equations, from which the unknown displacement constants can be determined, Consider the single-inclusion auxiliary problem for the transverse hydrostatic loading shown as part of B g Fig. 3. Using standard elasticity theory [51] and the A f a ¼ A g a þ a Correspondence Principle, the radial displacements in the transformed domain must take the following Bg Bm A g b þ ¼ A m b þ form, b b F.T. Fisher, L.C. Brinson / Composites Science and Technology 61 (2001) 731–748 745 !( b r 3 r b Ã Ã Ã Ã Ã Bg g o Ã Ã 2 A G þ l ¼ 2 A G þ l À 2 G ur ¼ a2 g À 3 þd2 þ c2 g þ 1 f f f g g g g a2 4 G Ã b b r )g B B 2 A G Ã þ lÃ À 2 G Ã g ¼ 2 A G Ã þ lÃ À 2 G Ã m b 3 g g g g b2 m m m m b2 þ b cos2 2 r Ã Ã 2 Am Gm þ lm ¼ o; a4r4b; !( ð20Þ 3 g b o Ã r r Ã b u ¼ a2 þ 3 Àd2 À c2 À 1 where a and b are the outer radius of the fiber and the Ã g b b g r 4 Gg interphase, respectively. ) 3 Given these displacement constants, one can now b þ b sin2; ð26Þ solve for the phase average stresses in the fiber and 2 r interphase (k=f,g) using the relationship !() Ð Ð ÀÁ 3 2 ro k m b o r Ã b b r dr d ur ¼ 2 þ m þ 1 a3 þ c3 cos2 ~ k ¼ 0 rÀÁi ij : ð21Þ 4 G Ã b r r ij 2 2 m ro À ri r5b !() ÀÁ 3 m b o r Ã b b u ¼ À2 À À 1 a3 þ c3 sin2; Note that phase-averaged quantities are denoted by Ã b m r r 4 Gm ðÞ~ . Following the work of Benveniste [24], we now ð27Þ k define scalar stress concentration factors Wij which Ã Ã relate the applied far-field stress of the auxiliary pro- where k ¼ 3 À 4 k and all other displacements are blem to the corresponding phase-averaged stresses in equal to zero. the included phases (k=f,g), These displacement equations contain 8 unknown constants, which again can be complex in nature. These ~ k ¼ W k ;~ k ¼ W k constants are determined using the set of simultaneous xx XX o yy YY o ; ð22Þ ~ k ¼ W k ¼ 2 Ã W k equations given below, where =b/a: zz ZZ o k XX o hi h Ã Ã 2 Ã Ã Gg a1 f À 3 þ d1 ¼ Gf a2 g À 3 Ã i where k is the complex Poisson ratio of the kth phase. k k 2 Ã 4 6 Note that due to symmetry, W and W are iden- þ d2 þ c2 g þ 1 þ b2 XX YY hi h tical. These stress concentration factors can be written Ã Ã 2 Ã Ã Gg a1 f þ 3 À d1 ¼ Gf a2 g þ 3 explicitly as i À 2 À Ã À 4 þ 6 Ã d2 c2 g 1 b2 f 2 Af f 2 Af lf hi ð28Þ W ¼ G Ã þ lÃ ; W ¼ ; ð23Þ XX f f ZZ Ã Ã Ã o o Gm a2 g À 3 þ d2 þ c2 g þ 1 þ b2 ÂÃÀÁ Ã Ã ¼ G 2 þ a3 þ 1 þ c3 2 A lÃ hig m g 2 Ag Ã Ã g g g WXX ¼ Gg þ lg ; WZZ ¼ : ð24Þ Ã Ã Ã o o Gm a2 g þ 3 À d2 À c2 g À 1 þ b2 ÂÃÀÁ Ã Ã ¼ G À2 À a3 À 1 þ c3 g nom no Ã Ã Ã 2 Ã Ã Ã Ã Ã Appendix B. Transverse shear loading auxiliary pro- 2Gg Gf 3a1 f À3 þd1 þGglf 6a1 f À1 ¼2 Gg Gf blem for the Benveniste solution no Ã 2 Ã 4 6 Ã 3a2 g À 3 þ d2 À c2 g þ 1 À 3b2 Consider the single-inclusion auxiliary problem for no Ã Ã Ã Ã 4 the transverse shear loading, the second of the two þ Gf lg 6a2 g À 1 À 2c2 g À 1 auxiliary problems shown in Fig. 3. The displacements 6a À d 2 ¼ 6a À d 2 À 2c 4 À 3b 6 for this problem can be written [52] as follows, 1 1no2 2 22 ! Ã Ã Ã Ã 2 G G 3a2 À 3 þ d2 À c2 þ 1 À 3b2 3 m g g g f b o Ã r r no u ¼ a1 À 3 þd1 cos2 r Ã f b b þ G Ã lÃ 6a Ã À 1 À 2c Ã À 1 4 Gf ! m g 2 g 2 g 04r4a; b r 3 r ÈÉÀÁ ÈÉÀÁ f o Ã ¼ G Ã G Ã À a Ã þ À c þ G Ã Ã a À Ã u ¼ a1 þ 3 Àd1 sin2; 2 m g 2 3 m 1 3 3 glm 2 3 1 m 4 G Ã f b b f 6a2 À d2 À 2c2 À 3b2 ¼À2 À 2a3 À 3c3: ð25Þ 746 F.T. Fisher, L.C. Brinson / Composites Science and Technology 61 (2001) 731–748

Following the procedure outlined in the previous sec- where cf and cg are the normalized areas of the included k tion, stress concentration factors WTS are defined relat- phases, and cm=1–cf Àcg. Thus the average y-compo- ing the phase-averaged stress in the included phases to nent of the matrix stress can be found, the applied far-field stress in the auxiliary problem, such 2 that ~ m ¼ o : ð33Þ yy f f g g ÈÉ cm þ cf WXX þ WTS þ cg WXX þ WTS ~ k k ~ k k xx ¼ÀWTS o; yy ¼ WTS o: k ¼ f; g ð29Þ Likewise, other average stress components in the After algebraic manipulation, these factors can be matrix can be determined. Once this is completed, we simplified as are in a position to evaluate the transverse Young’s modulus of the composite, as shown in Fig. 2. Consider 3 a a2 d the expression f ¼À 1 þ 1 WTS 2 2 bÀÁ2 ð30Þ X 2 2 yy g 3 a2 b þ a d2 ~k ¼À þ "yy ¼ ¼ ck "yy; ð34Þ WTS 2 : Ã 2 b 2 E22 k

where "yy is the overall composite strain in the y-direction, yy is the applied far-field stress in the same direc- Ã Appendix C. Effective transverse Young’s modulus of tion, and E22 is the effective transverse Young’s the composite via Benveniste solution modulus of the composite. For the superposition problem considered here, we have Having analyzed the previous two auxiliary problems, 2 we are now in a position to superpose the solutions to o ¼ c "~f þ c "~g þ c "~m : ð35Þ Ã f yy g yy m yy determine the effective transverse Young’s modulus of E22 the composite. The procedure to determine the transverse shear modulus is similar and will not be covered Since all phase average stresses can now be deter- here. We first assume that the stress concentration fac- mined, the average strain in the phases can be found k tors Wij, which were calculated using the single-inclu- using Hooke’s Law, sion auxiliary problem, are sufficient to use for the no 1 ÈÉ multiple-inclusion composite. In addition, we assume "~k ¼ ~ k À Ã ~ k þ ~ k þ ~ k : k ¼ f; g; m yy Ã yy k xx yy zz that for the actual composite, the far-field stress to 2 Gk which each inclusion is subjected is the unknown aver- ð36Þ age stress in the matrix, which for the current loading ~ m mode is yy. Substitution into (35) allows the numerical calculation Using the stress-concentration factors derived pre- of the transverse Young’s modulus of the composite. viously, the average stress components in the fiber and This expression is too unwieldy to present in its alge- interphase regions of the composite can be written in braic form. terms of this unknown stress, As a final note, the stress-concentration scalars defined in (30) can be used to calculate the effective transverse shear modulus, G Ã , of the composite. Again following ~ k k k ~ m 44 yy ¼ WXX þ WTS yy; the procedure of [24], the transverse shear modulus of ~ k k k ~ m the composite is found to be, xx ¼ WXX À WTS yy; ~ k k ~ m 1 1 1 zz ¼ WLT yy: ð31Þ f g cm þ cf WTS þ cg WTS 1 2 G Ã 2 G Ã 2 G Ã ¼ m f g : ð37Þ Ã f g Enforcing that the sum of the average phase stresses 2 G44 cm þ cfWTS þ cgWTS in the y-direction must equal the applied far-field stress, in this case 2 o, yields X ~ k ~ m ~ f ~ g Appendix D. Effective axial shear modulus of the 2 o ¼ ck yy ¼ cm yy þ cf yy þ cg yy k composite via Benveniste solution ~ m f f ~ m ¼ cm yy þ cf WXX þ WTS yy The solution for the axial shear auxiliary problem g g ~ m follows an identical procedure. Suitable displacement þ cg W þ W ; ð32Þ XX TS yy equations for this problem are as follows, F.T. Fisher, L.C. Brinson / Composites Science and Technology 61 (2001) 731–748 747 ! [9] Qui YP, Weng GJ. 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