In this manuscript, we propose an analytical equivalent linear viscoelastic constitutive model for fiber-reinforced composites, bypassing general computational homogenization. The method is based on the reduced-order homogenization (ROH) approach. The ROH method typically involves solving multiple finite element problems under periodic conditions to evaluate elastic strain and eigenstrain influence functions in an ‘off-line’ stage, which offers substantial cost savings compared to direct computational homogenization methods. Due to the unique structure of the fibrous unit cell, “off-line” stage calculation can be eliminated by influence functions obtained analytically. Introducing the standard solid model to the ROH method enables the creation of a comprehensive analytical homogeneous viscoelastic constitutive model. This method treats fibrous composite materials as homogeneous, anisotropic viscoelastic materials, significantly reducing computational time due to its analytical nature. This approach also enables precise determination of a homogenized anisotropic relaxation modulus and accurate capture of various viscoelastic responses under different loading conditions. Three sets of numerical examples, including unit cell tests, three-point beam bending tests, and torsion tests, are given to demonstrate the predictive performance of the homogenized viscoelastic model. Furthermore, the model is validated against experimental measurements, confirming its accuracy and reliability.
Homogenized relaxation modulusviscoelasticstandard solid modelreduced order homogenizationfibrous composite materialNational Key R&D Program of China2023YFA1008901National Natural Science Foundation of China1198810212172009Introduction
Fiber-reinforced composites offer advantages such as lightweight properties, high strength, extended maintenance intervals, and corrosion resistance, making them widely used in the aerospace and automotive industries [1–5]. Thermosetting or thermoplastic matrices with cylindrical microstructures exhibit transversely isotropic viscoelastic behavior, as documented in the literature [6,7]. The overall mechanical behavior of fibrous composites is analyzed using multiscale methods. These methods provide detailed insights into the underlying mechanisms, enabling material performance optimization and guiding innovative material design.
Initially, homogenization methods for viscoelastic composites were based on the correspondence principle. Micromechanical theoretical schemes developed for elastic composites were extended to viscoelastic composites to estimate their effective elastic moduli in the Laplace domain. Various approaches, including mean-field homogenization methods [8–10], the asymptotic homogenization method [11–13], the variational principle [14,15], and the sequential linearization approach [16], have been employed to simulate viscoelastic composites. Results obtained in the Laplace domain are then transformed back to the time domain to determine the effective response of the viscoelastic materials. However, performing an analytical inverse transform is usually too complex, so the macroscopic response in the time domain can only be obtained through numerical inversion [12] or approximate inversion methods [17], which can be costly and less accurate.
To overcome the limitations of the Laplace transformation, various homogenization methods have been developed to operate directly in the time domain [18–20]. Mean-field homogenization methods, initially formulated for linear elastic composite materials, have since been extended to address nonlinear behaviors, such as elastoplastic [21,22] and viscoelastic [23] responses, among others. These extensions are typically implemented by linearizing local constitutive laws incrementally, an approach necessitated by the Eshelby problem’s assumption that mechanical interactions within phases are purely elastic. Additionally, stress and strain fields within each phase are approximated as phase averages, treating them as homogeneous. However, in nonlinear problems, where stress and strain distributions are inherently non-uniform across phases, this approximation can lead to an over-stiffness issue. To address this challenge, techniques such as the isotropization method [24] and higher-order theories [25] have been proposed. Bleiler et al. [26] introduced a tangent second-order homogenization method to estimate the effective mechanical response of fibrous composites with hyperelastic, anisotropic, and incompressible phases. Furthermore, Barral et al. [27] advanced the Mori-Tanaka method by incorporating a modified transformation field analysis (TFA) approach first proposed by Dvorak and his coworkers [28] and introducing a specialized coating between fibers and the matrix, effectively mitigating over-stiffness in the analysis of fiber-reinforced composites. In the TFA method, the inelastic strains are considered as given eigenstrains. This method assumes that eigenstrains in each phase of the composite are piecewise constant, this lower-order approximation can lead to inaccurate solutions for overall properties. To overcome the problem, Michel et al. [29,30] have introduced the nonuniform transformation field analysis, where full field calculations take place and the plastic behavior is traced with the help of several plastic modes. Strategies such as subdividing each phase into numerous partitions, using asymptotically consistent eigenstrain fields have also been employed [31].
The asymptotic homogenization method is a mathematically rigorous methodology developed by Sanchez-Palencia [32], Bensoussan and Alain [33]. Compare to the Mean-field homogenization method, it effectively captures the interactions between different phases within composite materials, enabling more precise predictions of their overall performance. It has been proved that the considered fields converge towards the homogeneous macroscopic solution as the micro-structural parameter ξ=x/y tends to 0. The classic asymptotic homogenization method can obtain the linear homogenized modulus by solving a linear unit cell problem with given phase material elastic properties. Thus, one can use the homogenized elastic properties in a macroscopic simulation without downscaling or upscaling between the macro- and meso-scales. In this manner, the linear homogenization process is named as “off-line” process. However, if material nonlinearities at the mesoscale are considered, a unit cell problem must be solved at every integration point of all the macroscopic elements during each macroscopic Newton-Raphson iteration. Additionally, the unit cell problem itself requires a Newton-Raphson iterative process due to material nonlinearities. This repeating process is known as the “on-line” process since it depends on the “on-the-fly” kinematic measurements at the macro-scale, significantly increasing the computational cost. Yu et al. [34] used the asymptotic homogenization method to solve thermo-viscoelastic problems, considering both multiple spatial and temporal scales. Zhai et al. [18] developed a time-domain asymptotic homogenization method based on the integral form of the Kelvin–Voigt viscoelastic model and used it for woven fabric composites.
Multiscale computational techniques based on the finite element method such as FE2, offer the possibility of computing the macro-structural response of heterogeneous materials with an arbitrary microscopic geometry and constitutive behavior. A scheme based on the discrete homogenization method are developed to predict the effective mechanical properties of 3D dry textiles [35]. Recently, the isogeometric analysis method (IGA), proposed by Hughes and his coworkers [36,37], has been incorporated into computational homogenization [38–40]. The method aims to unify the representation of geometric models and mesh models by directly employing the spline functions as the shape functions. The errors produced by geometric approximations inherent in conventional finite-element and finite-volume techniques could be greatly reduced because the same model is used for both modeling and analysis. Cylindrically orthotropic [41] and viscoelasticity [42] are considered for fiber-reinforced composites using IGH methods.
To decrease computational costs at the meso-scale without significantly compromising solution accuracy, the reduced-order-homogenization (ROH) multiscale method was developed, combining the asymptotic homogenization method with transformation field analysis (TFA) [43,44]. The ROH method was first used in composite plasticity and failure analysis [43,45], and higher-order ROH methods have been developed for three-scale nonlinear problems [46–49]. In this paper, a two-phase model with the assumption of one partition per phase is employed for fiber-reinforced composites. Due to the unique structure of the fiber-reinforced unit cell, analytical transformation influence functions can be derived. Based on ROH theory, an analytical homogenization constitutive model for unidirectional fibrous composites is proposed, eliminating the need for computational homogenization. As demonstrated in the following section, the proposed model does not exhibit significant over-stiffness and helps reduce computational time.
This paper is organized as follows. In Section 2, we introduce the analytical viscoelastic fibrous composites constitutive model based on the ROH homogenization method. Special cases where the fiber volume fraction approaches zero are considered to confirm the model’s consistency under extreme conditions. We investigate the impact of material parameters of each component on the relaxation times of the homogenized viscoelastic material. Additionally, numerical implementation specifics for the proposed model are detailed in this section. Section 3 presents numerical verifications of the proposed model through three sets of numerical examples: unit cell tests, three-point beam bending tests, and torsion tests. Experimental results from the literature are used to validate the proposed constitutive model. Conclusions are provided in Section 4.
Methodology
In this section, we introduce an analytical framework to determine homogenized viscoelastic material properties for the fibrous composite material based on the so-called reduced order homogenization (ROH) method. Here we assume the fiber phase follows transversely isotropic symmetry and keeps elastic that
σijf=Lijklfεklf
On the other hand, the matrix phase is viscoelastic and described by the standard solid model [50], and the stress response is written as the following form:
σijm=Lijklm,∞εklm+∫0te−(t−η)/τ1Lijklm,isoε˙klm(η)dηwhere
Lijklm,∞=2μm,∞Iijkldev+3κm,∞Iijklvol,μm,∞=Em,∞2(1+νm),κm,∞=Em,∞3(1−2νm)Lijklm,iso=2μisomIijkldev+3κisomIijklvol,μisom=Eisom2(1+νm),κisom=Eisom3(1−2νm)
Here Em,∞ is the equilibrium Young’s modulus of the matrix phase, and νm is the corresponding Poisson’s ratio. Eisom is the isotropic relaxation modulus. Finally, the fourth-order tensors Iijklvol and Iijkldev are defined as
Iijklvol=13δijδkl,Iijkldev=12(δikδjl+δilδjk)−13δijδklrespectively. Here the quantities associated with the fiber and the matrix phase are marked with superscription Ef and Em, respectively.
Our goal is to obtain an analytical homogeneous relaxation tensor Rijklc(t) that
σijc=Lijklc,∞εklc+∫0tRijklc(t−η)ε˙klc(η)dη=Lijklc,∞εklc+Rijklc(t)∗ε˙klc(t)based on given material properties of the fiber and the matrix phase without computational homogenization process, where σijc, εijc are the macroscopic stress and strain tensor, respectively, Lijklc,∞ is homogenized linear elastic stiffness tensor, and f(t)∗g(t) denotes the convolution of function f(t) and g(t). Here the macroscopic properties are marked with superscription εc.
Reduced Order Homogenization (ROH) for Fibrous Composite Material
The reduced-order homogenization (ROH) method is based on two key assumptions: 1) It is possible to distinguish between two length scales associated with macroscopic and microscopic phenomena; 2) The microstructure is sufficiently regular to be considered periodic; 3) The material within the same partition maintains a consistent status throughout, which reduces computational costs at the meso-scale. A complete derivation of the ROH method would be too extensive for this section, so we will present only the essential formulas. For a detailed explanation of the methodology, please refer to reference [44]. The fibrous composite material consists of the fiber and matrix phases. We introduce eigenstrains μij defined as [44]
μij=εij−Mijklσklwhere Mijkl is the elastic compliance tensor. The ROH aims to solve the following system of nonlinear equations [44]:
εijf−Pijkl,fmμklm−Pijkl,ffμklf=Eijkl,fεklcεijm−Pijkl,mmμklm−Pijkl,mfμklf=Eijkl,mεklcwhere Eijkl,f and Eijkl,m are the elastic strain influence functions, Pijkl,fm, Pijkl,mm, Pijkl,ff and Pijkl,mf are the eigenstrain influence functions, and εijc is the macroscopic strain tensor.
Since the fiber phase keeps elastic, we can obtain μijf=0 directly. The eigenstrains of the matrix phase are given as
μijm=εijm−Mijklσijm=εijm−(Lijklm)−1σklm=εijm−(Lijklm,∞)−1σklm=−ψm∫0te−(t−η)/τ1ε˙ijm(η)dηby the standard solid model defined in Eq. (2), with consideration of material isotropy Eqs. (3a) and (3b), and define ψm≡Eisom/Em,∞. Here we define
Lijklm=Lijklm,∞
By linear elastic stiffness tensor of the fiber phase and the matrix phase given in Eqs. (1) and (9), respectively, we can first obtain the homogenized elastic stiffness tensor Lijklc that
Lijklc=cfLijmnfEmnkl,f⏟≡Lijklcf+cmLijmnmEmnkl,m⏟≡Lijklcmwhere cf and cm=1−cf are the volume fraction for the fiber and the matrix phase, respectively. The macro-scopic averaged stress then is given by volumetric averaging of phase stresses that
σijc=cfLijklfεklf+cmLijklmεklm=Lijklcεklc+Aijklmμklmwhere
Aijklm=cfLijstfPstkl,fm+cmLijstm(Pstkl,mm−Istkl)
Usually, the fibrous composite material is assumed to satisfy the transversely isotropic symmetry so that the linear elastic stiffness tensor can be written with Hill’s constants nc, lc, mc, kc, and pc [51], with the fibers aligned along the longitudinal x1−axis:
{Lijklc}=[nclclclckc+mckc−mclckc−mckc+mcmcpcpc]
The fiber phase also is transversely isotropic, where its Hill’s constants are written as nf, lf, mf, kf, and pf. For fibrous composite material, the following universal connections are satisfied [51]:
kc−kflc−lf=kc−(λm,∞+μm,∞)lc−λm,∞=lc−cflf−cmlmnc−cfnf−cmnm=kf−(λm,∞+μm,∞)lf−λm,∞where λm,∞ and μm,∞ are the Lamé constants of the matrix phase.
Analytical Influence Functions for Fibrous Composite Material
The elastic strain and eigenstrain influence functions are usually evaluated by solving a number of finite element problems over the given fibrous unit cell [44]. In this manuscript, we aim to provide an analytical method to evaluate those functions with given linear homogenized stiffness tensor Lijklc, which can be estimated by Mori-Tanaka or self-consistent method [51].
First of all, the elastic strain influence functions satisfy the following constraint that [44]
cfEijkl,f+cmEijkl,m=Iijkl
One can obtain the elastic strain influence functions by Eqs. (10) and (15) [44]:
Eijkl,f=1cf(Lijmnf−Lijmnm)−1(Lmnklc−Lmnklm)Eijkl,m=1cm(Lijmnm−Lijmnf)−1(Lmnklc−Lmnklf)
In other words, for a two-phase unit cell, as long as Lijklc is given, the elastic strain influence functions can be obtained directly.
Next, we continue to evaluate the eigenstrain influence functions. Define the following supportive tensors:
{E~ijpq,f}≡[100−νatf00−νatf00000]and
{E~ijpq,m}≡[100cfνatf/cmf220cfνatf/cm0f33f23f13f12]with νatf denoting the Poisson’s ratio in longitudinal direction of the fiber phase, and
fij=Lijijc/Lijijcm,ij=22,33,23,13,12,no summation over repeated indiceswhere Lijijcm is defined in Eq. (10). Then we obtain the eigenstrain influence functions:
Pijkl,mm=Iijkl−Eijmn,m(E~mnkl,m)−1,Pijkl,f=Iijkl−Eijkl,m−Pijkl,mmPijkl,fm=(Eijmn,f−E~ijmn,f)(E~mnkl,m)−1,Pijkl,ff=Iijkl−Eijkl,f−Pijkl,fm
However, Pijkl,mf and Pijkl,ff are not used since μijf=0.
Due to the symmetry of fibrous unit cell, one can use only 5 non-zero parameters to define one influence tensor. For the linear strain influence tensors, we have (E=f,m)
[100E11,22E22,22E33,22E11,33E22,33E33,33E23,23E13,13E12,12]⇒[100E1E2E3E1E3E2E4E5E5]and for the eigenstrain influence tensors, we also have (E=f,m)
[100P2211,mmP2222,mmP2233,mmP3311,mmP3322,mmP3333,mmP2323,mmP1313,mmP1212,mm]⇒[100P1mmP2mmP3mmP1mmP3mmP2mmP4mmP5mmP5mm]
From Eqs. (21) and (22), we can quickly obtain the following results:
ε11m=ε11f=ε11cwhich means the axial strains of the matrix and the fiber phases are the same and equals to the macroscopic axial strain ε11c.
Viscoelastic Model for the Fibrous Composite Material
In this section, we propose a homogenized viscoelastic model with the analytical elastic strain and eigenstrain influence functions. Substituting Eqs. (8) into (7b) with μijf=0 yields
εijm+Pijkl,mmψmm∫0te−(t−η)/τ1ε˙klm(η)dη=Eijkl,mεklc,ij=22,33,23,13,12,summation convention over k,lwhere ij=11 is not considered due to Eq. (23). In the rest of this section, we derive system of ordinary differential equations (ODEs) for ε˙ijm with ij=22,33,23,13,12. First, taking the derivative of Eq. (24) with respect to time t yields
ε˙ijm(t)+Pijkl,mmψmm[ε˙klm(t)−1τ1∫0te−(t−η)/τ1ε˙klm(η)dη]=Eijkl,mε˙klc(t)
Thus we obtain
Pijkl,mmψm∫0te−(t−η)/τ1ε˙klm(η)dη=τ1[Iijkl+ψmPijkl,mm]ε˙klm(t)−τ1Eijkl,mε˙klc(t)
Next, substituting Eqs. (26) into (24) yields
εijm+τ1[Iijkl+ψmPijkl,mm]ε˙klm(t)=Eijkl,m[εklc+τ1ε˙klc(t)],ij=22,33,23,13,12
By solving the system of ordinary differential equations in Eq. (27), we obtain the matrix phase strain εijm, where ij=22,33,23,13,12. The detailed derivation is provided in Appendix A. The solutions for the normal strains ε22m(t) and ε33m(t) can be expressed as
ε22m(t)=E1mε11c(t)+E2mε22c(t)+E3mε33c(t)+[−E1mτ2−τ1τ2−τ1τ2ψmP1mm]e−t/τ2∗ε˙11c(t)+[−12(E2m+E3m)τ2−τ1τ2e−t/τ2−12(E2m−E3m)τ3−τ1τ3e−t/τ3]∗ε˙22c(t)+[−12(E2m+E3m)τ2−τ1τ2e−t/τ2+12(E2m−E3m)τ3−τ1τ3e−t/τ3]∗ε˙33c(t)ε33m(t)=E1mε11c(t)+E3mε22c(t)+E2mε33c(t)+[−E1mτ2−τ1τ2−τ1τ2ψmP1mm]e−t/τ2∗ε˙11c(t)+[−12(E2m+E3m)τ2−τ1τ2e−t/τ2+12(E2m−E3m)τ3−τ1τ3e−t/τ3]∗ε˙22c(t)+[−12(E2m+E3m)τ2−τ1τ2e−t/τ2−12(E2m−E3m)τ3−τ1τ3e−t/τ3]∗ε˙33c(t)
The solutions for the shear strains
ε23m(t)=E4mε23c(t)−τ4−τ1τ4E4me−t/τ4∗ε˙23c(t)ε13m(t)=E5mε13c(t)−τ5−τ1τ5E5me−t/τ5∗ε˙13c(t)ε12m(t)=E5mε12c(t)−τ5−τ1τ5E5me−t/τ5∗ε˙12c(t)respectively. With
τ2=τ1(1+ψmP2mm+ψmP3mm),τ3=τ1(1+ψmP2mm−ψmP3mm),τ4=τ1(1+ψmP4mm),τ5=τ1(1+ψmP5mm)here τk,k=2,3,4,5 are characteristic relaxation times for the fibrous composite materials.
So far, we are able to find the matrix phase strain εijm by the macroscopic strain εijc and the corresponding strain rate ε˙ijc. In order to obtain an analytical result for macroscopic stress defined in Eq. (11), we continue to derive the eigenstrain of the matrix phase. For μ11m, we can directly obtain
μ11m=−ψm∫0te−(t−η)/τ1ε˙11m(η)dη=−ψm∫0te−(t−η)/τ1ε˙11c(η)dη=−ψme−t/τ1∗ε˙11c(t)by considering Eq. (23). For the rest components, expand Eq. (7b) into
[P2mmP3mmP3mmP2mm]{μ22m(t)μ33m(t)}=[ε22m(t)−E1mε11c(t)−E2mε22c(t)−E3mε33c(t)−P1mmμ11mε33m(t)−E1mε11c(t)−E3mε22c(t)−E2mε33c(t)−P1mmμ11m]and
P4mmμ23m(t)=ε23m(t)−E4mε23c(t)P5mmμ13m(t)=ε13m(t)−E5mε13c(t)P5mmμ12m(t)=ε12m(t)−E5mε12c(t)
Finally, one can introduce a fourth-order tensor Yijkl,m(t) to represent the eigenstrain μijm(t) as the following:
μijm(t)=Yijkl,m(t)∗ε˙klc(t)
First, from Eq. (31), we can obtain the components corresponding to μ11m that
Y1111,m(t)=−ψme−t/τ1
From Eqs. (28a) and (28b), we obtain
Y2211,m(t)=Y3311,m(t)=P1mmψmP2mm+P3mme−t/τ1+[−τ2−τ1τ2E1mP2mm+P3mzm−τ1τ2P1mmψmP2mm+P3mm]e−t/τ2Y2222,m(t)=−τ2−τ1τ2E2m+E3m2(P2mm+P3mm)e−t/τ2−τ3−τ1τ3E2m−E3m2(P2mm−P3mm)e−t/τ3Y3322,m(t)=−τ2−τ1τ2E2m+E3m2(P2mm+P3mm)e−t/τ2+τ3−τ1τ3E2m−E3m2(P2mm−P3mm)e−t/τ3Y2233,m(t)=Y3322,m(t),Y3333,m(t)=Y2222,m(t)
In addition, from Eq. (29b) and (29c), we obtain
Y2323,m(t)=−τ4−τ1τ4E4mP4mme−t/τ4,Y1313,m(t)=Y1212,m(t)=−τ5−τ1τ5E5mP5mme−t/τ5and all other components of Yijkl are zero. In summary, the components of the eigenstrain μijm(t) are written as the following:
μ11m(t)=Y1111,m(t)∗ε˙11c(t)μ22m(t)=Y2211,m(t)∗ε˙11c(t)+Y2222,m(t)∗ε˙22c(t)+Y2233,m(t)∗ε˙33c(t)μ33m(t)=Y3311,m(t)∗ε˙11c(t)+Y3322,m(t)∗ε˙22c(t)+Y3333,m(t)∗ε˙33c(t)μ23m(t)=Y2323,m(t)∗ε˙23c(t)μ13m(t)=Y1313,m(t)∗ε˙13c(t)μ12m(t)=Y1212,m(t)∗ε˙12c(t)
Substituting Eqs. (33) into (11) yields
σijc=Lijklcεklc+AijmnmYmnkl,m(t)⏟Rijklc(t)∗ε˙klc(t)
Model Consistency in an Extreme Case
In this section, we exam the homogenized viscoelastic properties of the fibrous composite material by letting cf=0 and cm=1. Under this situation, the homogenized viscoelastic properties should approach the viscoelastic response of pure matrix phase as well.
By setting cm=1, one obtains Lijklc=Lijklcm=Lijklm through Eq. (10) and Eijkl,m=Iijkl. In addition, we have Aijklm=−Lijklm by Eq. (12). The value of Eijkl,f does not have any meaning. Similarly, by Eq. (18), we obtain E~ijpq,m=Iijkl and thus leads to Pijkl,mm=0 by Eq. (20a). This result guarantees that τk=τ1, k=2,3,4,5 by Eq. (30).
Considering the aforementioned conditions, we can obtain εijm=εijc from Eqs. (23), (28a) and (29a), by applying ε~22m(t)=ε~33m(t)=0, E2m=E4m=E5m=1, E1m=E3m=0. Finally, one can obtain Yijkl,m=−ψmIijkl and
σijc=σijm=Lijklm,∞εklm+ψmLijklm∗ε˙klm(t)=Lijklm,∞εklm+Lijklm,iso∗ε˙klm(t)which traces back to the pure matrix case.
Influence of Component Parameters on Homogenization Relaxation Times
In this section, we analytically calculate the homogenization relaxation times τi,i=1,2,…,5 using known component parameters, investigating how fiber and matrix properties influence the overall viscoelastic response. It is important to note that relaxation time is closely related to material viscosity. A smaller viscosity results in a shorter relaxation time. When viscosity (η) approaches zero, the material behaves like an inviscid fluid, whereas when it approaches infinity, the material behaves like an elastic solid. We observe that changes in component parameters lead to variations in the homogenization relaxation times, with certain parameters exerting more significant influence compared to others.
We first investigate the influence of fiber parameters on homogenized relaxation times. Parameters are chosen as detailed in Table 1, with one parameter varied within specified ranges. Fig. 1a illustrates the impact of fiber volume fraction on viscoelastic behavior: as the fiber fraction increases, the relaxation times also increase, indicating that, in our research, fibrous composites with higher fiber content tend to exhibit more solid-like characteristics in the homogenized state. It is noteworthy that relaxation time associated with the fiber direction is always equal to the matrix relaxation time, as indicated by Eq. (36), implying that the viscoelastic behavior in this direction is solely influenced by matrix components, while viscoelastic behavior in other directions is influenced by both fiber and matrix components. Fig. 1b demonstrates how homogenized relaxation time varies with changes in νtf=0.20,0.25,0.35,0.40, revealing that νtf has a limited impact on viscosity behavior. A slight decrease in homogenized relaxation time is observed with increasing νtf.
Material parameters of the fibrous composite material
Fiber
Eaf
13,800
(MPa)
Young’s modulus in axial direction
Etf
9500
(MPa)
Young’s modulus in transverse direction
νatf
0.28
[−]
Poisson’s ratio in axial-transverse plane
νtf
0.4
[−]
Poisson’s ratio in transverse plane
μatf
5200
(MPa)
Shear modulus in axial-transverse plane
Matrix
E∞m
4500
(MPa)
Young’s modulus
νm
0.35
[−]
Poisson’s ratio
Eisom
500–3500
(MPa)
Isotropic relaxation modulus
τ1m
10
(s)
Relaxation time
Influence of fiber parameters on homogenized relaxation times: (a) Fiber volume; (b) Poisson ratio in the transverse plane <inline-formula id="ieqn-125"><mml:math id="mml-ieqn-125"><mml:msubsup><mml:mi>ν</mml:mi><mml:mrow><mml:mrow><mml:mtext>t</mml:mtext></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mtext>f</mml:mtext></mml:mrow></mml:mrow></mml:msubsup></mml:math></inline-formula>
Next, we explore the influence of matrix parameters on homogenized relaxation times. As shown in Fig. 2a, an increase in Eisom leads to higher homogenized relaxation times due to the corresponding increase in viscosity. Similarly, Fig. 2b demonstrates that higher E∞m results in increased homogenization relaxation times, paralleling the trend observed in Fig. 1a. Fig. 2c indicates that νm has minimal impact on viscosity behavior, albeit showing a slight increase with higher νm.
Influence of matrix parameters on homogenized relaxation times: (a) Relaxation modulus <inline-formula id="ieqn-126"><mml:math id="mml-ieqn-126"><mml:msubsup><mml:mi>E</mml:mi><mml:mrow><mml:mrow><mml:mtext>iso</mml:mtext></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mtext>m</mml:mtext></mml:mrow></mml:mrow></mml:msubsup></mml:math></inline-formula>; (b) Elastic modulus <inline-formula id="ieqn-127"><mml:math id="mml-ieqn-127"><mml:msubsup><mml:mi>E</mml:mi><mml:mrow><mml:mi mathvariant="bold">∞</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mtext>m</mml:mtext></mml:mrow></mml:mrow></mml:msubsup></mml:math></inline-formula>; (c) Poisson ratio <inline-formula id="ieqn-128"><mml:math id="mml-ieqn-128"><mml:msup><mml:mi>ν</mml:mi><mml:mrow><mml:mrow><mml:mtext>m</mml:mtext></mml:mrow></mml:mrow></mml:msup></mml:math></inline-formula>
Numerical Algorithm
In this section, we briefly introduce the numerical algorithm for the homogenized viscoelastic model of fibrous composite material. In finite element analysis, we update material stress as well as consistent tangent modulus over one increment [tn,tn+1]. The stress at tn+1, namely, σijc,n+1 is given as
σijc,n+1=Lijklcεklc,n+1+Aijklmμklm,n+1=Lijklcεklc,n+1+AijklmYklst,m(t)∗ε˙stc,n+1(t)=Lijklcεklc,n+1+Rijklc(t)∗ε˙klc,n+1(t)again Rijklc(t)≡AijmnmYmnkl,m(t) is the homogeneous relaxation tensor. The second term in the right-hand-side of Eq. (39) is given by the classic integration form:
AijklmYklst,m(t)∗ε˙stc,n+1(t)=Aijklm∫0tn+1Yklst,m(tn+1−η)ε˙stc(η)dη=Aijklm∫0tnYklst,m(tn+1−η)ε˙stc(η)dη+Aijklm∫tntn+1Yklst,m(tn+1−η)ε˙stc(η)dη
The numerical integration scheme is the same as the classic viscoelastic stress update procedure. For example:
∫0tnY1111,m(tn+1−η)ε˙11c,n+1(η)dη=∫0tn−ψme−(tn+1−η)/τ1ε˙11c(η)dη=∫0tn−ψme−(tn+1−tn)/τ1e−(tn−η)/τ1ε˙11c(η)dη=e−Δt/τ1∫0tn−ψme−(tn−η)/τ1ε˙11c(η)dη∫tntn+1Y1111,m(tn+1−η)ε˙11c,n+1(η)dη=∫tntn+1−ψme−(tn+1−η)/τ1ε˙11c(η)dη≈−ψme−Δt/(2τ1)Δε11c
With Δt≡tn+1−tn, and Δε11c the incremental strain at 11 component.
From the convolution form in Eq. (36), define the following 9 state variables:
s1n+1=∫0tn+1e−(tn+1−η)/τ1ε˙11c(η)dη,s2n+1=∫0tn+1e−(tn+1−η)/τ2ε˙11c(η)dηs3n+1=∫0tn+1e−(tn+1−η)/τ2ε˙22c(η)dη,s4n+1=∫0tn+1e−(tn+1−η)/τ3ε˙22c(η)dηs5n+1=∫0tn+1e−(tn+1−η)/τ2ε˙33c(η)dη,s6n+1=∫0tn+1e−(tn+1−η)/τ3ε˙33c(η)dηs7n+1=∫0tn+1e−(tn+1−η)/τ4ε˙23c(η)dη,s8n+1=∫0tn+1e−(tn+1−η)/τ5ε˙13c(η)dηs9n+1=∫0tn+1e−(tn+1−η)/τ5ε˙12c(η)dη,
From (41a) and (41b), we can obtain
s1n+1=e−Δt/τ1s1n+e−Δt/(2τ1)Δε11and
{∂S1n+1∂εijc,n+1}=[e−Δt/(2τ1)00000]T
Similar equations can be derived for sin+1,i=2,3,⋯,9. Thus, Eq. (36) can be written with the aforementioned state variables that
μ11m,n+1=−ψms1n+1μ22m,n+1=P1mmψmP2mm+P3ms1n+1+[−τ2−τ1τ2E1mP2mm+P3mm−τ1τ2P1mmψmP2mm+P3mm]s2n+1−τ2−τ1τ2E2m+E3m2(P2mm+P3mm)(s3n+1+s5n+1)−τ3−τ1τ3E2m−E3m2(P2mm−P3mm)(s4n+1−s6n+1)μ33m,n+1=P1mmψmP2mm+P3mms1n+1+[−τ2−τ1τ2E1mP2mm+P3mm−τ1τ2P1mmψmP2mm+P3mm]s2n+1−τ2−τ1τ2E2m+E3m2(P2mm+P3mm)(s3n+1+s5n+1)+τ3−τ1τ3E2m−E3m2(P2mm−P3mm)(s4n+1−s6n+1)μ23m,n+1=−τ4−τ1τ4E4mP4mms7n+1μ13m,n+1=−τ5−τ1τ5E5mP5mms8n+1μ12m,n+1=−τ5−τ1τ5E5mP5mms9n+1
In addition, one can derive Tijklm≡∂μijm,n+1/∂εklm,n+1 so that the consistent tangent material modulus is given as
∂σijc,n+1∂εklc,n+1=Lijklc+Aijmnm∂μmnm,n+1∂εklc,n+1=Lijklc+AijmnmTmnklm
In this section, we propose several groups of numerical examples to verify and validate the proposed analytical homogenized viscoelastic model for fibrous composite materials. The numerical examples and validation are organized as follows: In Section 3.1, unit cell tests under uniaxial tension and pure shear conditions are performed to demonstrate consistency between FEM and the proposed homogenized model under a uniform strain field. In Sections 3.2 and 3.3, we introduce pure bending beam experiments and torsion experiments to further investigate the ability of the proposed homogenized method. In Section 3.4, theoretical predictions are compared with experimental results from the literature to validate prediction accuracy. All simulations are conducted using the commercial FEM software ABAQUS/Standard 2022 with a user material subroutine (UMAT).
Unit Cell Tests
We first verify the accuracy of the homogenized viscoelastic model using a uniform strain field problem. We explicitly construct a unit cell of the fibrous composite material for the direct numerical simulation (DNS), where the fiber is transversely isotropic elastic and the matrix follows the standard solid model, as depicted in Fig. 3. The size of the unit cell is set as lx=1.732(mm), ly=1(mm), and lz=0.5(mm). By applying periodic boundary conditions and a macroscopic strain history, we obtain the corresponding averaged stress history. For comparison, we construct a one-element model associated with the homogenized viscoelastic constitutive model and apply the same strain history. Table 1 lists the parameters for both the fiber and the matrix phases.
Unit cell model with size of <inline-formula id="ieqn-142"><mml:math id="mml-ieqn-142"><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>1.732</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtext>mm</mml:mtext></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>, <inline-formula id="ieqn-143"><mml:math id="mml-ieqn-143"><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mi>y</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtext>mm</mml:mtext></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>, <inline-formula id="ieqn-144"><mml:math id="mml-ieqn-144"><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtext>mm</mml:mtext></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>
Various numerical tests are proposed with different fiber volume fractions: cf=0.3,0.4,0.5,0.6 and different relaxation moduli: Eisom=500,1500,2500,3500(MPa). For each unit cell problem, we perform three uniaxial tension tests in the x-, y-, and z-direction, and three pure shear tests in the xy-, xz-, and yz- planes, where x refers to the axial direction of the fiber. In order to capture the viscoelastic response, we apply a loading-holding-unloading curve for the prescribed macroscopic strain, as depicted in Fig. 4. In the unit cell problem, we use the loading rate defined in Fig. 4a. For pure bending and torsion tests, the loading rate is defined as in Fig. 4b for a clearer observation of the viscoelastic response.
Loading rates: (a) Unit cell test; (b) Pure bending and torsion tests
A group of DNS simulations is depicted in Fig. 5, while the corresponding group of one-element simulation using the proposed homogenized viscoelastic constitutive model is depicted in Fig. 6. Clearly, the DNS results provide more detail but at a higher computational cost. To avoid redundancy, not all simulation results will be displayed. Figs. 7 and 8 show comparison of strain-stress responses between the DNS and the homogenized model (HM for short) for Eisom=500(MPa) and Eisom=3500(MPa). Note that the x-direction of the material coordinate system coincides with the z-direction of the overall coordinate system. Figs. 7c and 8c show that the composite response is fiber-dominated in this direction, while in other directions, the response is matrix-dominated. A more significant viscoelastic response is observed as Eisom increases. All the results show good agreement between the homogenized model and the DNS simulations.
Demonstrations of DNS model and six loading cases: (a) Honeycomb unit cell; (b) uniaxial tension test in <italic>x</italic>-direction; (c) uniaxial tension test in <italic>y</italic>-direction; (d) uniaxial tension test in <italic>z</italic>-direction; (e) pure shear test in xy plane; (f) pure shear test in <italic>xz</italic> plane; (g) pure shear test in <italic>yz</italic> plane
Demonstrations of homogenized model and six loading cases: (a) One-element model; (b) uniaxial tension test in <italic>x</italic>-direction; (c) uniaxial tension test in <italic>y</italic>-direction; (d) uniaxial tension test in <italic>z</italic>-direction; (e) pure shear test in <italic>xy</italic> plane; (f) pure shear test in <italic>xz</italic> plane; (g) pure shear test in <italic>yz</italic> plane
Relaxation curves under different fiber volume ratios with <inline-formula id="ieqn-157"><mml:math id="mml-ieqn-157"><mml:msubsup><mml:mi>E</mml:mi><mml:mrow><mml:mrow><mml:mtext>iso</mml:mtext></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mtext>m</mml:mtext></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>500</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtext>MPa</mml:mtext></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>: uniaxial tension test in (a) <italic>x</italic>-direction; (b) <italic>y</italic>-direction; (c) <italic>z</italic>-direction; pure shear test in (d) <italic>xy</italic> plane; (e) <italic>xz</italic> plane; (f) <italic>yz</italic> plane
Relaxation curves under different fiber volume ratios with <inline-formula id="ieqn-158"><mml:math id="mml-ieqn-158"><mml:msubsup><mml:mi>E</mml:mi><mml:mrow><mml:mrow><mml:mtext>iso</mml:mtext></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mtext>m</mml:mtext></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>3500</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtext>MPa</mml:mtext></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>: uniaxial tension test in (a) <italic>x</italic>-direction; (b) <italic>y</italic>-direction; (c) <italic>z</italic>-direction; pure shear test in (d) <italic>xy</italic> plane; (e) <italic>xz</italic> plane; (f) <italic>yz</italic> plane
Pure Bending Tests
The unit cell tests verify the homogenized viscoelastic model under uniform strain fields. To extend this validation to non-uniform strain fields, we propose a series of pure bending tests with geometric setups and boundary conditions outlined in Fig. 9. The size of beam is lx=39.836(mm),ly=5(mm),lz=7(mm). The applied moment M(t) follows the loading rate defined in Fig. 4b, with a maximum value of Mmax=50(N⋅mm). Both direct numerical simulation (DNS) and homogenized models, as illustrated in Fig. 10, are employed to compare numerical efficiency and accuracy. The results are presented in Fig. 11. It is observed that the stress concentration effect cannot be effectively captured due to the piecewise constant eigenstrain assumption. This approach sacrifices some detail for the sake of efficiency.
Geometric setup of pure bending beam and boundary conditions
Pure bending beam models: (a) DNS; (b) homogenized model
Pure bending beam simulation result demonstrations: (a) DNS; (b) homogenized model
We tested different fiber volume fractions, cf=0.3,0.4,0.5,0.6, with the matrix relaxation moduli set to Eisom=3000(MPa). The results of the homogenized model and the reference results for comparison are shown in Fig. 12. We used the displacement at t=100 to estimate the relative displacement error between the DNS and homogenized models, as depicted in Fig. 13. The proposed method demonstrates good consistency with the DNS method, with relative errors below 3%. It is also noteworthy that the DNS method takes approximately 7600(s), while the homogenized model requires only 50(s). The simulation is significantly faster than DNS, and the overall response remains highly accurate.
Pure bending beam experiment under different fiber volume fractions: DNS and homogenized model result comparison with <inline-formula id="ieqn-168"><mml:math id="mml-ieqn-168"><mml:msubsup><mml:mi>E</mml:mi><mml:mrow><mml:mrow><mml:mtext>iso</mml:mtext></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mtext>m</mml:mtext></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>3000</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtext>MPa</mml:mtext></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>
Pure bending beam tests under different fiber volume fractions: displacement relative error between DNS and homogenized model result with <inline-formula id="ieqn-169"><mml:math id="mml-ieqn-169"><mml:msubsup><mml:mi>E</mml:mi><mml:mrow><mml:mrow><mml:mtext>iso</mml:mtext></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mtext>m</mml:mtext></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>3000</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtext>MPa</mml:mtext></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>
Torsion Tests
In this section, the beam used for the pure bending tests is employed to conduct torsion tests. We select viscoelastic parameters as Eisom=3000(MPa) and investigate creep behavior under four sets of fiber volume fractions: cf=0.3,0.4,0.5,0.6. The left boundary of the beam is fixed, and a moment of Mmax=500(N⋅mm) is applied to the right boundary with the loading rate shown in Fig. 4b. The simulation results for both the DNS and homogenized models are demonstrated in Fig. 14. The comparison of relaxation curves is depicted in Fig. 15. It is evident from the analysis that the discrepancy between the two curves is minimal, suggesting that the proposed model effectively captures torsional deformation.
Beam torsion simulation result demonstrations: (a) DNS; (b) homogenized model
Beam torsion experiment under different fiber volume fractions, DNS and homogenized model result comparison with <inline-formula id="ieqn-173"><mml:math id="mml-ieqn-173"><mml:msubsup><mml:mi>E</mml:mi><mml:mrow><mml:mrow><mml:mtext>iso</mml:mtext></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mtext>m</mml:mtext></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>3000</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtext>MPa</mml:mtext></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>
Experiment Validation
We obtained experimental data from a paper published by Thiruppukuzhi et al. [52] and referenced another paper by Chen et al. [53]. In the first paper, Thiruppukuzhi et al. conducted experiments using S2-glass/8553-40 epoxy fiber-reinforced composite with a fiber volume fraction of 65%. They performed a series of off-axial tensile tests at three strain rates: 0.0001[1/s], 0.01[1/s], 1[1/s], and four off-axial tensile directions: 0∘, 15∘, 30∘, and 90∘. The combination of loading strain rate and off-axis tensile direction in the experiment is shown in Table 2, where markers represent the experiments conducted under the corresponding combinations. The setup of off-axial tensile tests is illustrated in Fig. 16. The specimen dimensions are lx=100(mm),ly=17.8(mm) as described in the original experiment, and we set lz=1(mm). We simulate the actual experimental boundary conditions by clamping the two ends of the specimen with fixtures, resulting in an over-constrained setup. Symmetric boundary conditions are applied in the z-direction to represent the specimen’s real-life thickness.
Combination of loading strain rate and off-axis tensile direction in the experiment
0∘
15∘
30∘
90∘
0.0001
[1/s]
□
□
□
□
0.01
[1/s]
□
□
1
[1/s]
□
□
Setup of off-axial tensile tests
The elastic material parameters of both the fiber and matrix are identified inversely using experimental data of composite materials under two sets of conditions: 0.0001[1/s],0∘ and 0.0001[1/s],90∘. The viscoelastic material parameters for the matrix are determined using experimental data with the following test conditions:0.0001[1/s],15∘, 0.01[1/s],15∘, and 1[1/s],15∘. Table 3 presents the material parameters of the fiber and matrix obtained by fitting the experimental data. Fig. 17 shows the homogenized model results and experimental data along the 0∘,15∘,30∘,90∘ directions under a loading strain rate of 0.0001[1/s]. The simulation results along the 15∘ and 30∘ directions also show good agreement with the experimental data, indicating that the parameters are a good fit. The Von Mises stress contour plots for the homogenized models along the four directions are shown in Fig. 18. Theoretical predictions begin to diverge from experimental results due to significant nonlinearity induced by plasticity or damage at relatively large strains.
Material parameters for fibrous composite material in experiment
Fiber
Em
71,000
(MPa)
Young’s modulus
νf
0.2
[−]
Poisson’s ratio
Matrix
E∞m
1900
(MPa)
Young’s modulus
νm
0.35
[−]
Poisson’s ratio
E∞iso
304
(MPa)
Isotropic relaxation modulus
τ1m
0.8
(s)
Relaxation time
Strain-stress curves obtained from experiment results and homogenized model predictions for four off-axis tensile directions <inline-formula id="ieqn-226"><mml:math id="mml-ieqn-226"><mml:msup><mml:mn>0</mml:mn><mml:mrow><mml:mo mathvariant="bold">∘</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>, <inline-formula id="ieqn-227"><mml:math id="mml-ieqn-227"><mml:msup><mml:mn>15</mml:mn><mml:mrow><mml:mo mathvariant="bold">∘</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>, <inline-formula id="ieqn-228"><mml:math id="mml-ieqn-228"><mml:msup><mml:mn>30</mml:mn><mml:mrow><mml:mo mathvariant="bold">∘</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>, and <inline-formula id="ieqn-229"><mml:math id="mml-ieqn-229"><mml:msup><mml:mn>90</mml:mn><mml:mrow><mml:mo mathvariant="bold">∘</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>, loading strain rate <inline-formula id="ieqn-230"><mml:math id="mml-ieqn-230"><mml:mrow><mml:mover><mml:mi>ε</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mn>0.0001</mml:mn><mml:mrow><mml:mo>[</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mtext>s</mml:mtext></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula>
Von Mises stress cloud for homogenized models of four off-axis tensile directions: (a) <inline-formula id="ieqn-231"><mml:math id="mml-ieqn-231"><mml:msup><mml:mn>0</mml:mn><mml:mrow><mml:mo mathvariant="bold">∘</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>; (b) <inline-formula id="ieqn-232"><mml:math id="mml-ieqn-232"><mml:msup><mml:mn>15</mml:mn><mml:mrow><mml:mo mathvariant="bold">∘</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>; (c) <inline-formula id="ieqn-233"><mml:math id="mml-ieqn-233"><mml:msup><mml:mn>30</mml:mn><mml:mrow><mml:mo mathvariant="bold">∘</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>; (d) <inline-formula id="ieqn-234"><mml:math id="mml-ieqn-234"><mml:msup><mml:mn>90</mml:mn><mml:mrow><mml:mo mathvariant="bold">∘</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>, loading strain rate <inline-formula id="ieqn-235"><mml:math id="mml-ieqn-235"><mml:mrow><mml:mover><mml:mi>ε</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mn>0.0001</mml:mn><mml:mrow><mml:mo>[</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mtext>s</mml:mtext></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula>
Figs. 19 and 20 present the homogenized model results and experimental data along directions 15∘,30∘ under loading strain rates of 0.01[1/s] and 1[1/s], respectively. The corresponding Von Mises stress contour plots for the homogenized models are shown in Figs. 21 and 22. The experimental data along 15∘ were used to fit the viscoelastic parameters, while the data along 30∘ were used to validate the accuracy of the proposed homogenized model. From Figs. 19 and 20, the simulation results along 30∘ exhibit good consistency with experimental data, indicating that our homogenized model can provide reasonable predictions for fibrous composite materials with viscoelastic effects.
Strain-stress curves obtained from experiment results and homogenized model predictions for two off-axis tensile directions <inline-formula id="ieqn-236"><mml:math id="mml-ieqn-236"><mml:msup><mml:mn>15</mml:mn><mml:mrow><mml:mo mathvariant="bold">∘</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>, <inline-formula id="ieqn-237"><mml:math id="mml-ieqn-237"><mml:msup><mml:mn>30</mml:mn><mml:mrow><mml:mo mathvariant="bold">∘</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>, loading strain rate <inline-formula id="ieqn-238"><mml:math id="mml-ieqn-238"><mml:mrow><mml:mover><mml:mi>ε</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mn>0.01</mml:mn><mml:mrow><mml:mo>[</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mtext>s</mml:mtext></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula>
Strain-stress curves obtained from experiment results and homogenized model predictions for two off-axis tensile directions <inline-formula id="ieqn-239"><mml:math id="mml-ieqn-239"><mml:msup><mml:mn>15</mml:mn><mml:mrow><mml:mo mathvariant="bold">∘</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>, <inline-formula id="ieqn-240"><mml:math id="mml-ieqn-240"><mml:msup><mml:mn>30</mml:mn><mml:mrow><mml:mo mathvariant="bold">∘</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>, loading strain rate <inline-formula id="ieqn-241"><mml:math id="mml-ieqn-241"><mml:mrow><mml:mover><mml:mi>ε</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>[</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mtext>s</mml:mtext></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula>
Von Mises stress contour plot for homogenized models of two off-axis tensile directions: (a) <inline-formula id="ieqn-242"><mml:math id="mml-ieqn-242"><mml:msup><mml:mn>15</mml:mn><mml:mrow><mml:mo mathvariant="bold">∘</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>; (b) <inline-formula id="ieqn-243"><mml:math id="mml-ieqn-243"><mml:msup><mml:mn>30</mml:mn><mml:mrow><mml:mo mathvariant="bold">∘</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>, loading strain rate <inline-formula id="ieqn-244"><mml:math id="mml-ieqn-244"><mml:mrow><mml:mover><mml:mi>ε</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mn>0.01</mml:mn><mml:mrow><mml:mo>[</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mtext>s</mml:mtext></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula>
Von Mises stress contour plot for homogenized models of two off-axis tensile directions: (a) <inline-formula id="ieqn-245"><mml:math id="mml-ieqn-245"><mml:msup><mml:mn>15</mml:mn><mml:mrow><mml:mo mathvariant="bold">∘</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>; (b) <inline-formula id="ieqn-246"><mml:math id="mml-ieqn-246"><mml:msup><mml:mn>30</mml:mn><mml:mrow><mml:mo mathvariant="bold">∘</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>, loading strain rate <inline-formula id="ieqn-247"><mml:math id="mml-ieqn-247"><mml:mrow><mml:mover><mml:mi>ε</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>[</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mtext>s</mml:mtext></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula>
Conclusion
In this manuscript, we propose an analytical homogenized viscoelastic model for fibrous composite materials using the reduced-order homogenization (ROH) method. Compared to numerical homogenization methods, our analytical model significantly reduces simulation costs while providing accurate predictions of the overall mechanical behavior. However, some deviations in local stress are observed, attributable to the one-partition-per-phase assumption. We model the fiber phase as transversely isotropic elastic, while the matrix phase follows a standard solid model. This approach allows for the analytical determination of effective moduli and relaxation times in fibrous composites. As the fiber volume fraction approaches zero, our homogenized viscoelastic model simplifies to a pure viscoelastic model. We verify our model through numerical unit cell tests, pure bending beam tests, and torsion tests, demonstrating its accuracy and computational efficiency compared to direct numerical simulations. Experimental validations further confirm the reliability of our approach.
The authors would like to acknowledge the support of the National Key R&D Program of China and the National Natural Science Foundation of China. The research is also supported by Peking University for this publication.
Funding Statement
Financial support by the National Key R&D Program of China (Grant No. 2023YFA1008901) and the National Natural Science Foundation of China (Grant Nos. 11988102, 12172009) is gratefully acknowledged. The research is also supported by “The Fundamental Research Funds for the Central Universities, Peking University”.
Author Contributions
The authors confirm contribution to the paper as follows: Huilin Jia contributed to the formula derivation, and simulation. Shanqiao Huang assisted in offering methodology and revising the manuscript. Zifeng Yuan contributed to the initial idea, the overall structure and final editing of the manuscript. All authors reviewed the results and approved the final version of the manuscript.
Availability of Data and Materials
Data available on request from the authors.
Ethics Approval
Not applicable.
Conflicts of Interest
The authors declare no conflicts of interest to report regarding the present study.
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In this section, we solve the ODE equation system in Eq. (27) to obtain the matrix phase strains. With the reduced 1-index defined in Eq. (22), we can rewrite Eq. (27) into the following system of ODEs:
ddt{ε22m(t)ε33m(t)}=−12[1/τ2+1/τ31/τ2−1/τ31/τ2−1/τ31/τ2+1/τ3]{ε22m(t)ε33m(t)}+12[1/τ2+1/τ31/τ2−1/τ31/τ2−1/τ31/τ2+1/τ3][E1mE2mE3mE1mE3mE2m]{ε11c(t)+τ1ε˙11c(t)ε22c(t)+τ1ε˙22c(t)ε33c(t)+τ1ε˙33c(t)}−{τ1τ2ψmP1mε˙11c(t)τ1τ2ψmP1mε˙11c(t)}and
dε23m(t)dt=−1τ4ε23m(t)+1τ4E2323,m[ε23c(t)+τ1ε˙23c(t)]dε13m(t)dt=−1τ5ε13m(t)+1τ5E1313,m[ε13c(t)+τ1ε˙13c(t)]dε12m(t)dt=−1τ5ε12m(t)+1τ5E1212,m[ε12c(t)+τ1ε˙12c(t)]with
τ2=τ1(1+ψmP2mm+ψmP3mm),τ3=τ1(1+ψmP2mm−ψmP3mm),τ4=τ1(1+ψmP4mm),τ5=τ1(1+ψmP5mm)
Here τk,k=2,3,4,5 are characteristic relaxation times for the fibrous composite materials.
For the coupled ODEs defined in Eq. (A1a), notice that
12[1/τ2+1/τ31/τ2−1/τ31/τ2−1/τ31/τ2+1/τ3]=[1−111][1/τ2001/τ3][1/21/2−1/21/2]
Define the following two strains:
{ε~22m(t)ε~33m(t)}=[1/21/2−1/21/2]{ε22m(t)−E1mε11c(t)−E2mε22c(t)−E3mε33c(t)ε33m(t)−E1mε11c(t)−E3mε22c(t)−E2mε33c(t)}and the corresponding ODEs become:
ddt{ε~22m(t)ε~33m(t)}=−[1τ2001τ3]{ε~22m(t)ε~33m(t)}−[τ2−τ1τ200τ3−τ1τ3][1/21/2−1/21/2][E1mε˙11c(t)+E2mε˙22c(t)+E3mε˙33c(t)E1mε˙11c(t)+E3mε˙22c(t)+E2mε˙33c(t)]−[τ1τ2ψmP1mmε˙11c(t)0]
The solution of Eqs. (A5) can be written as
ε~22m(t)=−τ2−τ1τ2e−t/τ2∗{(E1m+τ1τ2−τ1ψmP1m)ε˙11c(t)+12(E2m+E3m)(ε˙22c(t)+ε˙33c(t))}ε~33m(t)=−τ3−τ1τ3e−t/τ3∗{12(E2m−E3m)(−ε˙22c(t)+ε˙33c(t))}
Substituting Eqs. (A6a) and (A6b) into (A4) yields the solution that
ε22m(t)=E1mε11c(t)+E2mε22c(t)+E3mε33c(t)+ε~22m(t)−ε~33m(t)=E1mε11c(t)+E2mε22c(t)+E3mε33c(t)+[−E1mτ2−τ1τ2−τ1τ2ψmP1m]e−t/τ2∗ε˙11c(t)+[−12(E2m+E3m)τ2−τ1τ2e−t/τ2−12(E2m−E3m)τ3−τ1τ3e−t/τ3]∗ε˙22c(t)+[−12(E2m+E3m)τ2−τ1τ2e−t/τ2+12(E2m−E3m)τ3−τ1τ3e−t/τ3]∗ε˙33c(t)ε33m(t)=E1mε11c(t)+E3mε22c(t)+E2mε33c(t)+ε~22m(t)+ε~33m(t)=E1mε11c(t)+E3mε22c(t)+E2mε33c(t)+[−E1mτ2−τ1τ2−τ1τ2ψmP1m]e−t/τ2∗ε˙11c(t)+[−12(E2m+E3m)τ2−τ1τ2e−t/τ2+12(E2m−E3m)τ3−τ1τ3e−t/τ3]∗ε˙22c(t)+[−12(E2m+E3m)τ2−τ1τ2e−t/τ2−12(E2m−E3m)τ3−τ1τ3e−t/τ3]∗ε˙33c(t)
The shearing response Eqs. (A1b)–(A1d), can be solved similarly. Taking Eq. (A1b) as an example. Rewrite Eq. (A1b) into
ddt{ε23m(t)−E4mε23c(t)}=−1τ4{ε23m(t)−E4mε23c(t)}−τ4−τ1τ4E4mε˙23c(t)so that the solution can be given by the convolution form:
ε23m(t)=E4mε23c(t)−τ4−τ1τ4E4me−t/τ4∗ε˙23c(t)
Similarly, the solutions of (A1c) and (A1d) are written with convolution form that
ε13m(t)=E5mε13c(t)−τ5−τ1τ5E5me−t/τ5∗ε˙13c(t)