Effect of a curved fiber on the overall material stiffness

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Effect of a curved fiber on the overall material stiffness Borys Drach∗, Dmytro Kuksenko, Igor Sevostianov Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM, USA

a r t i c l e

i n f o

Article history: Received 2 March 2016 Revised 15 August 2016 Available online xxx Keywords: Micromechanics Stiffness contribution tensor N-tensor Curvilinear fiber Curved tow Replacement relations

a b s t r a c t The effect of fiber curvature on the overall material stiffness of fiber-reinforced composites is discussed using the formalism of stiffness contribution tensor for the special case of a continuous sinusoidal fiber. Stiffness contribution tensors of individual sinusoidal fibers with different crimp ratios are presented for the first time. The tensors are calculated numerically using Finite Element Analysis and analytically by representing fibers as equivalent sets of ellipsoids following the procedure available in the literature for approximation of effective composite stiffness. It is demonstrated that the existing procedure results in large approximation errors in several components of the stiffness contribution tensors of the considered fibers. A modification to the procedure is proposed to improve the accuracy of the predictions. Replacement relations that interrelate stiffness contribution tensors of inhomogeneities having the same shape but different material properties are tested for the studied fiber geometries. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction This paper focuses on the quantitative evaluation of the effect produced by an isolated curved fiber of circular cross-section on the overall elastic properties. We use the concept of stiffness contribution tensor of an individual inhomogeneity (see, for example, Sevostianov and Kachanov (2002a); Kachanov et al. (2003); Drach et al. (2011)). Components of this tensor are calculated numerically and approximated by simple analytical formulas that allow identification of the microstructural parameters governing the aforementioned effect. The problem is motivated mostly by the needs of composite materials industry where the use of curved fibers has increased during the last two decades – in woven composites, composites reinforced with nanofibers, etc. Another important application is related to biomechanics of soft and hard tissues where collagen fibers usually have curved shape. Effect of curvature on the fiber’s mechanical response has attracted attention of researchers starting from 1950s. To the best of our knowledge, the first paper where geometrical parameters of bent yarns were analyzed in the context of stress-strain analysis was the one of Backer (1952). The author derived expressions for local fiber tensile strain, average strain in the helix half loop, etc. in terms of the geometric quantities. The main application of the earlier research in this area (see also works of Platt (1950a);

∗

Corresponding author. Fax: +1 (575) 646 6111; Tel.: +1 (575) 646-8041. E-mail addresses: [email protected] (B. Drach), [email protected] (D. Kuksenko), [email protected] (I. Sevostianov).

Platt (1950b); Schwarz (1951)) was textile materials and their elastic performance. With appearance of fiber reinforced composites, the problem was transferred to the area of optimization of mechanical performance of composites. Bažant (1968) analyzed the influence of the curvature of reinforcing fibers on the mean longitudinal modulus and strength of composite materials. Brandmaier (1970) showed that the maximum composite strength can be obtained when most of the stress is carried in the fiber direction. It was first pointed out that the optimal fiber orientation is different from the principal stress direction depending on the strength properties of the lamina. Later research can be categorized into four main directions: a) optimization of the properties of materials containing curvilinear fibers using numerical methods; b) calculation of the effective stiffness of a single curvilinear fiber considered separately from the matrix material; c) solution of boundary value problems for materials with curved fibers; d) approximation of the effective properties of woven composites. First works on numerical optimization appeared in 1980s. Hyer and Charette (1987) used Finite Element Analysis (FEA) and an iterative scheme to find optimal orientation of curved fibers in laminaes to increase the material strength. The authors also cite PhD dissertation of Cooper (1972) where the effect of fiber curvature was first analyzed in the context of mechanical properties. Gurdal and Olmedo (1993) obtained a solution to the plane elasticity problem for a symmetrically laminated composite panel with spatially varying fiber orientations and discussed the effects of the variable fiber orientation on the displacement fields, stress

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Please cite this article as: B. Drach et al., Effect of a curved fiber on the overall material stiffness, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.08.018

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resultants, and global stiffness. Reuschel and Mattheck (1999) and Mattheck and Bethge (1998) discussed optimization of threedimensional (3-D) fibrous structures mimicking natural structures such as bones and trees to design laminated composites which follow the principal stress direction selection scheme. Setoodeh and Gürdal (2003) used the method of Cellular Automata for curvilinear fiber design of composite laminae for in-plane loading. Duvaut et al. (20 0 0) used variational formulation to develop a numerical method for determining the optimal direction and volume fraction of fibers at each point of a structure. Effective stiffness of a curvilinear fiber without matrix was first addressed by Potierferry and Siad (1992) who applied the doublescale asymptotic expansion method to a corrugated beam when the ratio of the length of the basic cell to the length of the beam tends to zero. Ding and Selig (2004) computed the effective compliance of a helical spring by summing the compliances of infinitesimal elements along the path of the spring. Haussy et al. (2004) modelled a single yarn as an undulated beam and used perturbation method to calculate its effective stiffness. Messager and Cartraud (2008) computed axial stiffness of a helical beamlike structure using homogenization theory of periodic slender domains. Marino and Vairo (2012) developed a general model accounting for fiber 3D geometry, as well as for shear and torsional effects together with the extensional ones. They highlighted the influence of fiber geometric parameters and shear deformability, thus enabling to test the limits of applicability of the commonly used assumptions. Zhao et al. (2014) studied hierarchical helical structures by analyzing the internal forces and deformations of a single helical ply; the effect of hierarchical helical structures was revealed by comparing the properties of a carbon nanotube rope having two-level helical structure with its counterpart bundle consisting of straight carbon nanotubes. The authors examined dependence of mechanical properties of the materials on the initial helical angles, number of fibers, and handedness at different structural levels. Exact solutions for elastic fields in materials containing thin curvilinear fibers loaded along the fiber axis have been obtained in the works of Akbarov and co-authors. In the book of Akbarov and Guz (20 0 0) (Chapter 7) the basic principles of the solution are formulated. Kosker and Akbarov (2003) obtained the stress distribution in the body containing two neighboring periodically co-phase curved fibers. Akbarov et al. (2004); Akbarov et al. (2006) considered infinite elastic matrix containing a row of periodically cophase and antiphase curved fibers, correspondingly. The effective properties of composites with curvilinear fibers and tows can be estimated by considering the microstructure as an aggregate of infinitesimal “subcells” each having the properties of a composite reinforced by straight fibers aligned along the tow path. In the original approach, the subcells were assumed to be reinforced with infinitely long fibers and the overall elastic properties were calculated by averaging stiffness (upper bound) or compliance (lower bound) tensors of the subcells, see for example Kregers and Melbardis (1978); Gowayed and Pastore (1993); Gommers et al. (1996); Matveeva et al. (2014). Gommers et al. (1998) presented a Mori-Tanaka based method for homogenization of the subcells. It was shown in the aforementioned publications that use of straight infinitely long fibers for reinforcement in the subcells resulted in overestimation of the effective elastic properties of the composites. Huysmans et al. (1998) refined the approach by employing subcells reinforced by ellipsoids with finite aspect ratios determined by local tow curvature (also known as “short fiber analogy” ) rather than infinite fibers. The latter approach was shown to produce relatively good predictions for textile composites especially for in-plane elastic properties (Birkefeld et al. (2012); Prodromou et al. (2011)) and it was later implemented in commercial software WiseTex (Verpoest and Lomov (2005); Lomov et al. (2007);

Lomov et al. (2014)). A number of research groups have used the Mori–Tanaka based subcell approach since. For example, Mourid et al. (2013) applied it to homogenization of woven composites with viscoelastic properties, Skocˇ ek et al. (2008) used it to evaluate effective elastic properties of carbon/carbon composites, and Olave et al. (2012); Vanaerschot et al. (2013) utilized it to study the sensitivity of effective elastic properties of textile composites to variations in tow paths and cross-section geometries. We note, however, that for anisotropic multiphase composites, Mori–Tanaka scheme may violate Hashin–Shtrikman bounds (Norris (1989) and Benveniste and Milton (2011)). Yet another inconsistency in the case of anisotropic multiphase composites is often claimed in the literature (Benveniste (1990); Qiu and Weng (1990); Qiu and Weng (1990)): the scheme predictions may violate the symmetry of the effective stiffness tensor. Artificial symmetrization does not resolve the problem (see Sevostianov and Kachanov (2014)). To the best of our knowledge, the results currently available in literature do not provide a tool that allows one to choose a homogenization scheme to evaluate the full set of anisotropic elastic constants of a material containing a given orientation distribution of curvilinear fibers. This work addresses the lack of the tools mentioned above. Two approaches to calculation of the stiffness contribution tensors of individual continuous curvilinear fibers are presented for the first time: direct FEA and analytical approximation. Given elastic (compliance or stiffness) contribution tensor of a single inhomogeneity, effective elastic properties of a composite containing multiple inhomogeneities can be readily obtained using a number of homogenization schemes (e.g. Eroshkin and Tsukrov (2005); Chen et al. (2015); Drach et al. (2016)). Thus calculation of effective properties is outside of the scope of this paper. The presented analytical approximation for the stiffness contribution tensor is based on the approach proposed by Huysmans et al. (1998) for effective stiffness tensor. We demonstrate the applicability of the original approach to approximation of the stiffness contribution tensor of a curvilinear fiber and propose a modification to reduce the approximation error. In the present study we focus on the stiffness contribution tensor of a single continuous sinusoidal fiber of circular cross-section embedded in a large matrix volume, see Fig. 1. The geometry of the fiber is characterized by amplitude a, wavelength λ and radius of the cross-section r. Amplitude and wavelength can be combined into a single parameter called “crimp ratio”, which is calculated as

CR =

a

λ

(1.1)

Both matrix and fiber are assumed to have isotropic properties. We present the effects of the geometric parameters on the components of the stiffness contribution tensor. We note that few results are available for description of the effect of non-ellipsoidal 3D inhomogeneities on the overall elastic properties. Compliance contribution tensors for several examples of pores of irregular shape typical for carbon/carbon composites were calculated by Drach et al. (2011) using FEA. In the narrower context of irregularly shaped cracks, certain results were obtained for compliance contribution tensors by Fabrikant (1989); Sevostianov and Kachanov (2002b) (planar cracks), Grechka et al. (2006) (intersecting planar cracks), Mear et al. (2007) (non-planar cracks), and Kachanov and Sevostianov (2012) (cracks growing from pores). Effects of various concave pores on overall elastic and conductive properties have been discussed in works of Sevostianov et al. (2008); Sevostianov and Giraud (2012); Chen et al. (2015) and Sevostianov et al. (2016). The results of the present research will contribute to the existing library of solutions for property contribution tensors of non-ellipsoidal inhomogeneities.

Please cite this article as: B. Drach et al., Effect of a curved fiber on the overall material stiffness, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.08.018

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Fig. 1. Geometry of the problem considered in this paper: single continuous sinusoidal fiber with wavelength λ, amplitude a and circular cross-section of radius r embedded in a large matrix volume.

This paper is organized as follows. Section 2 introduces the concept of the stiffness contribution tensor and approach to analytical approximation of the tensor of a curvilinear fiber using the subcell method. FEA approach to calculation of the stiffness contribution tensor of an infinite fiber is described in Section 3. Section 4 presents the stiffness contribution tensor results obtained using FEA, the analytical approximation based on Huysmans et al. (1998) and the modified approximation approach. Replacement relations that interrelate stiffness contribution tensors of inhomogeneities having the same shape but different material properties are tested for the studied fiber geometries and the results are given in Section 5. Section 6 presents conclusions of the research. Finally, all non-zero components of the stiffness contribution tensors of sinusoidal fibers for several values of crimp ratio and three sets of fiber/matrix material properties are provided in Appendix A. 2. Analytical approach to estimation of the stiffness contribution tensor of a curved fiber 2.1. Stiffness contribution tensor Elastic property contribution tensor was first introduced in the context of compliance contribution of ellipsoidal pores and cracks by Horii and Nemat-Nasser (1983). For the case of ellipsoidal elastic inhomogeneities, compliance contribution tensor was calculated by Sevostianov and Kachanov (1999). The concept of the stiffness contribution tensor used in the present work was first introduced in Sevostianov and Kachanov (2002a). Kushch and Sevostianov (2015) established the link between these tensors and dipole moments. We assume that the solid is subjected to “remotely applied” strain ɛ that, in the absence of inhomogeneities, would have been uniform within representative volume V (“homogeneous boundary conditions”, see Hashin (1983)). The problem of finding the effective elastic properties is best formulated in terms of the elastic potential g(ɛ), since the structure of the potential aids in identification of the proper microstructural parameters (Kachanov and Sevostianov (2005)). The effective stiffnesses Cijkl are then obtained by differentiation: σi j = Ci jkl εkl = ∂ g/∂ εi j . We represent the elastic potential g(ɛ) as

g = g 0 + g

(2.1)

where g0 = [E0 /2(1 + ν0 )]εi j ε ji + [ν0 E0 /2(1 + ν0 )(1 − 2ν0 )](εkk )2 is the potential in the absence of inhomogeneities. This implies a

0) similar sum for the effective stiffnesses: Ci jkl = Ci(jkl + Ci jkl . We

represent g as a sum of terms corresponding to inhomogeneities contained in volume V

g =



g( p) =

p

 1 ε: V ( p) N ( p) : ε 2V p

(2.2)

where V(p) is the volume of the pth inhomogeneity and N( p ) is the stiffness contribution tensor of the pth inhomogeneity representing additional stress in V due to its presence:

σ ( p ) =

V ( p) ( p) N : ε. V

(2.3)

Thus, the structure of g points to the general microstructural parameter – namely, the sum

1  ( p) ( p) V N V p

(2.4)

Formula (2.2) highlights the fundamental importance of stiffness contribution tensors: it is they that have to be summed up (averaged) in the context of the effective elastic properties. The sum (2.4) reflects stiffness contributions of individual inhomogeneities and constitutes the general microstructural parameter in the context of the effective elastic properties – effective elastic stiffness is the sum of the stiffness tensor of the matrix and sum (2.4):

C = C (0 ) +

1  ( p) ( p) V N V p

(2.5)

Summation over inhomogeneities can be replaced by integration over their orientations when appropriate. Since the N-tensors represent individual inhomogeneities in accordance with their actual contributions to the effective elastic properties, using parameter (2.4) ensures that the effective elastic constants are unique functions of this parameter, at least in the non-interaction approximation. In general case, the parameter (2.4) covers mixtures of inhomogeneities of diverse shapes and orientations. The stiffness contribution tensor of an ellipsoidal inhomogeneity can be expressed in terms of Hill tensor P (see Hill (1963), Walpole (1966)) as

N ( p) =



C (1 ) − C (0 )

−1

−1

+P

(2.6)

where C(1) and C(0) are the stiffness tensors of inhomogeneity and matrix, respectively.

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Fig. 3. FEA model setup: curved fiber with CR = 0.1 in a large reference volume (matrix elements are not shown). Fig. 2. Illustration of the approximation of a continuous curvilinear fiber by a set of equivalent ellipsoids.

2.2. Approximation of a continuous fiber by equivalent set of ellipsoids To approximate the stiffness contribution tensor of a continuous curvilinear fiber, we apply the procedure similar to the one presented for effective stiffness tensors in Gommers et al. (1998) and Huysmans et al. (1998). The fiber in matrix material is subdivided into a sufficiently large number of subcells so that each subcell can be considered to contain a straight segment of the fiber. Then each straight segment is replaced with an equivalent ellipsoid, see Fig. 2. The overall N-tensor of the continuous fiber can then be estimated from the sum (non-interaction approximation):

Ni jkl =

M 1  (z ) N pqrs l (z ) L

(2.7)

z=1

where L and M are the total length and the number of all straight segments approximating the continuous fiber; l(z) is the length (z ) of the zth segment of the fiber path; N pqrs is the 4th rank stiffness contribution tensor of the ellipsoid corresponding to the zth segment of the fiber path. Since we focus on a fiber of circular cross-section only, every replacement ellipsoid is reduced to a prolate spheroid with the long semi-axis aligned with the direction of the corresponding straight segment. Expressions for N-tensor components of prolate spheroids can be found in Sevostianov and Kachanov (2002a). The accuracy of the approximation described above depends on the proper choice of aspect ratios of the approximating spheroids. Huysmans et al. (1998) suggested that the aspect ratios should be a function of local fiber curvature ρ , fiber radius r and geometrical factor γ :

AR = γ

ρ 2r

(2.8)

By fitting the approximation results to FEA data, authors estimated γ to be equal to π (3.14 . . .). In Section 4 we show that the aspect ratios calculated from expression (2.8) lead to significant approximation errors in several components of the stiffness contribution tensor in the case of continuous sinusoidal fiber. In the same section we demonstrate that constant aspect ratio (chosen based on the crimp ratio of the fiber) results in good approximations for all components of the N-tensor.

3. FEA approach to calculation of the stiffness contribution tensor of a curved fiber Stiffness contribution tensors of individual fibers (each embedded in a large matrix volume) were obtained numerically using FEA for different matrix/fiber property combinations as well as fiber crimp ratios. The numerical procedure used to calculate Ntensors is similar to the one used to obtain H-tensors for irregular pore shapes in Drach et al. (2011) and for two interacting spherical pores in Sevostianov et al. (2014). The main steps of the numerical procedure are: a) generation of the fiber geometry and finite element mesh; b) FEA model preparation and simulation including: - meshing of the matrix domain with 3D elements; - prescription of fiber and matrix material properties; - application of periodic boundary conditions; - FEA computation of the elastic load cases; c) processing of the FEA results to calculate the N-tensor components. Detailed description of the steps listed above is provided in Sections 3.2 and 3.3. 3.1. Fiber geometry generation and meshing Geometry generation and meshing of the sinudoidal fiber with volume elements is performed in a custom Matlab script. There are only two parameters that need to be specified to generate the geometry of a sinusoidal fiber given its wavelength – crimp ratio and diameter. All values of the considered geometric parameters are given in Section 4. We start by generating the fiber path with user-specified number of straight-line segments. The number of segments controls the final mesh density of the fiber. Fiber path is modeled as one cycle of a sinusoid with the wavelength equal to the length of the reference volume, see Fig. 3. After that, the circular cross-section is generated and meshed with two-dimensional (2D) triangular elements using the procedure based on the one published in Sherburn (2007) for meshing of elliptical shapes. To generate volume mesh, the 2D cross-section mesh is first duplicated to every segment endpoint of the fiber path and oriented so that the mesh’s normal coincides with the segment’s direction vector, see Fig. 4a. Since we limit our considerations to fibers of constant crosssection, the same 2D mesh is used for all points along the fiber path. This makes it easy to connect two triangular elements in the adjacent cross-sections to create volume elements. The minimum of three tetrahedrons is required to fill the volume between two triangles as shown in Fig. 4b. The resulting volume mesh of the

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Fig. 4. Illustration of fiber meshing procedure: (a) duplication of 2D cross-section mesh to the points along the fiber path; (b) generation of 3D tetrahedral elements based on 2D triangular elements in adjacent cross-sections.

entire fiber including coordinate table of nodes and connectivity table of tetrahedral elements is exported from Matlab in DAT format for further FEA model preparation. 3.2. FEA model preparation and simulation The output of the Matlab script described above is imported into commercial FEA software package MSC Marc Mentat for further model preparation and FEA analysis. All preprocessing in the software package is performed automatically using a custom Mentat script that results in a ready-to-run model upon completion. The major steps of the script are as follows. A box representing the reference volume is created around the fiber. Note that the box must be sufficiently large to simulate a single fiber in an infinite volume in order to eliminate any boundary effects from calculation of stiffness contribution tensors. A sensitivity study was performed to determine the maximum fiber volume fraction for which no such effects are observed. It was found, that simulations with volume fractions of 0.25% result in good estimates for contribution tensors. Fig. 3 shows an example of the domain used for FEA N-tensor calculations – large reference volume with a fiber of crimp ratio CR = 0.1 (matrix elements are not shown). The faces of the reference volume are automeshed with triangular elements. The matrix volume surrounding the fiber is automeshed with tetrahedral elements based on surface meshes of the reference volume and the fiber. A typical model contained about 750,0 0 0 quadratic 10-node tetrahedral elements (#133 in MSC Marc classification). Once the volume meshes are generated the corresponding isotropic material properties are assigned to matrix and fiber elements. Six load cases are simulated to determine the components of the stiffness contribution tensor: three uniaxial extensions corre(0 ) (0 ) sponding to non-zero applied strain components ε11 , ε22 and (0 ) ε33 , and three shear cases corresponding to non-zero applied (0 ) (0 ) (0 ) strain components ε12 , ε23 and ε31 . The boundary conditions of

Fig. 5. Distribution of elastic strains ɛ11 in a large volume with a sinusoidal fiber (0 ) = 10−5 ) obtained from FEA of uniaxial tension in x1 direction (applied strain ε11 for high elastic contrast material combination (the fiber is one hundred times stiffer than the matrix).

the average displacement in xj direction. The PBCs are implemented in MSC Marc Mentat using the “servo-link” feature (Drach et al. (2014)). Servo-links allow prescription of multi-point boundary conditions in terms of a linear displacement function with constant coefficients. Note, that use of servo-links in MSC Marc Mentat requires congruent meshes on the opposite faces. Built-in CASI iterative solver was used for the FEA simulations. Fig. 5 shows the distribution of elastic strains ɛ11 in a volume with a sinusoidal fiber (CR = 0.1) obtained from FEA of uniaxial tension (0 ) in x1 direction (applied strain ε11 = 10−5 ) for high elastic contrast material combination (the fiber is one hundred times stiffer than the matrix). 3.3. Processing of the FEA results

each load case are applied in terms of displacements on the reference volume faces, see, for example, Drach et al. (2011); Sevostianov et al. (2014) for details. To simulate infinitely long fiber, periodic boundary conditions (PBCs) in the longitudinal fiber direction x1 are used. For two opposite faces, PBCs are introduced as follows (see for example, Segurado and Llorca (2002)):

The result files of the FEA simulations are processed via a custom Python script to extract volume averaged stress components from each load case:

i+ i− u(j ) = u(j ) + δ j , ( j = 1, 2, 3)

where σ ij k is the volume average of the stress component ij calculated from the kth load case, V is the total volume of the reference volume, (σi(jz ) )k is the stress component ij at the centroid of

(3.1)

where u(ji+) and u(ji−) are displacements in xj direction of the ith node on the positive and negative faces respectively, and δ j is

σij k =

1   (z )  σij k · v(z) , (i, j = 1, 2, 3; k = 1, 2, . . . , 6 ) V z

(3.2)

the finite element z calculated from kth load case, and v(z) is the

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B. Drach et al. / International Journal of Solids and Structures 000 (2016) 1–12 Table 1 Elastic properties of the considered material systems. Matrix

High contrast Epoxy/E-glass Hydroxyapatite/ collagen

Fiber

Young’s modulus Em , GPa

Poisson’s ratio ν m

Young’s modulus Ef , GPa

Poisson’s ratio ν f

1 3

0.4 0.3

100 73

0.2 0.217

114

0.3

1.5

0.3

volume of the finite element z. Given volume averages from the six load cases, all components of the N-tensor of a fiber can be calculated. For example, from the first load case, component Nij11 is found as (see expression (2.3))

 V  σi j − σ 0  1 0  i j 1 Ni j11 = V (1 ) ε11 1

(3.3)

where (σi0j )1 is component ij of the stress tensor in matrix material (in the absence of the fiber) subjected to the first load case bound0 ) is the applied strain component 11 correary conditions; (ε11 1 (1 )

sponding to the first load case, and ( V V ) is the volume fraction of the fiber. 4. Results and discussion To illustrate the dependence of the stiffness contribution tensor components on the geometry of the sinusoidal fiber and demonstrate the approximation procedure described in Section 2.2, we used the following three material systems: (1) high contrast material combination with the fiber one hundred times stiffer than the matrix, (2) composite material with epoxy matrix and stiff E-glass fiber, and (3) composite material with stiff hydroxyapatite matrix and soft collagen fibers (dense skeletal tissue). The elastic properties of the materials are given in Table 1. Note that crystals of hydroxyapatite have hexagonal structure and thus possess transversely isotropic symmetry (Martin et al. (2015)); however they are randomly oriented in space and thus form an isotropic aggregate. We validated our numerical procedure by comparing the Ntensors of straight fibers obtained numerically with the exact solution obtained analytically (Sevostianov and Kachanov (2002a)). The maximum errors of 0.4%, 0.3% and 0.8% were observed in the cases of high contrast, epoxy/E-glass and hydroxyapatite/collagen material combinations, respectively. 4.1. FEA results All components of N-tensors were calculated numerically for the material combinations given in Table 1 and the following crimp ratios: 0.0 02, 0.0 04, 0.0 06, 0.0 08, 0.01, 0.02, 0.03, 0.04, 0.05, 0.10, 0.15, 0.2. The value of 0.03 was used for the fiber radius divided by the wavelength. The results are shown as data points in Fig. 6. Note that every component is normalized (divided) by the corresponding component of the N-tensor calculated for the straight fiber. N2222 is not plotted since it does not appear to change with crimp ratio. It can be seen from the plots that crimp ratio has little to no effect on N-tensor components when CR ≤ 0.025, i.e. such sinusoidal fibers can be considered straight. Greater values of crimp ratio result in gradual decrease of overall stiffness in direction 1 (component N1111 ) and increase in direction 3 (component N3333 ). Note that even though the magnitude of N1111 increases with crimp ratio in the case of hydroxyapatite/collagen material combination, the overall stiffness decreases because of the negative sign of the non-normalized N-tensor components in the case of this material combination. The components N1122 , N1133 and N3131 increase with

crimp ratio in the case of stiff fiber (high contrast and epoxy/Eglass material combinations), while in the case of soft fiber (hydroxyapatite/collagen) they appear to have minimal dependence on crimp ratio. Components N2233 , N1212 and N2323 are not significantly affected by crimp ratio in all material combinations. 4.2. Approximation using equivalent sets of ellipsoids Following the procedure presented in Huysmans et al. (1998) for effective stiffness tensor and in Section 2.2 of this paper for stiffness contribution tensor we approximated the continuous fibers with equivalent sets of spheroids. The aspect ratios of the spheroids were chosen according to Eq. (2.8). The results are presented as dashed lines in Fig. 6. From the analysis of the plots, it appears that the approximation works well for five out of eight components and fails in the cases of high contrast and epoxy/Eglass material combinations for three components: N1133 , N3333 and N1313 . The errors between the FEA results and the approximations appear to be proportional to the fiber/matrix elastic contrast: the largest errors are observed in the case of high contrast material (38.6%, 88.9% and 163.5% in N1133 , N3333 and N1313 , respectively) and the lowest – in the case of hydroxyapatite/collagen combination (0.21%, 0.01%, 0.03% in N1133 , N3333 and N1313 , respectively). The approximation error may originate from the fact that a sinusoid has large values of radius of curvature ρ in the vicinity of locations with maximum slope (where ρ → ∞) and as a result these path segments are approximated with spheroids of extremely high aspect ratios. At the same time, the contributions of such path segments to the mechanical response of the fiber are not too different from other straight segments. Capping the maximum aspect ratio of the approximating spheroids or using spheroids with the same aspect ratio for the entire fiber path may reduce the error. Good correlation in the case of hydroxyapatite/collagen material system can be explained by the fact that N-tensor components of a moderately prolate spheroid are indistinguishable from those of infinitely long fiber when the spheroid is much softer than the matrix. This can be illustrated by comparing two plots of component N1111 vs spheroid aspect ratio obtained for high contrast and hydroxyapatite/collagen material systems, see Fig. 7. The plot shows that the difference between the spheroid with aspect ratio equal to 3.5 and infinitely long straight fiber is below 5% in the case of hydroxyapatite/collagen material system. Same error in the case of high contrast material combination is observed for spheroid with aspect ratio equal to 150. This means that the approximate N-tensor of a curved fiber is not nearly as sensitive to the aspect ratios of the approximating spheroids in the case of soft fiber as it is when the fiber is stiffer than the matrix. In the next section, we attempt to improve the approximations for high elastic contrast material combinations. 4.3. Modification of the approximation procedure We observed that for the considered fiber geometries the approximation works better when aspect ratios of the approximating spheroids are the same over the fiber path. To determine the

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Fig. 6. N-tensor components of sinusoidal fibers calculated using FEA (data points) and via approximations based on equivalent sets of ellipsoids with aspect ratios given by Eq. (2.8) (dashed lines).

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approximating ellipsoids are not connected to each other and thus do not possess the bending stiffness of a continuous fiber. The error is expected to be more pronounced in material combinations with stiff fiber where fiber bending dominates over stretching due to low load transfer provided by the soft matrix. 4.4. Validation

Fig. 7. N-tensor component N1111 of a spheroidal inhomogeneity vs aspect ratio for two material systems.

expression relating the best choice for aspect ratio of the approximating spheroids with the geometry of curved fiber, we developed a procedure minimizing the root-mean-square error between FEA and approximation results for the components N1111 , N2222 and N3333 . The following expression was used for the error:

To validate the approximation approach proposed in Section 4.3, we estimated N-tensor components for a set of material properties and fiber geometry parameters not used earlier to find the aspect ratio expressions (4.3), and compared the predictions with FEA results. The following material properties were used: E f = 120 GPa, ν f = 0.15, Em = 2 GPa and νm = 0.35. The fiber’s crimp ratio CR and radius r were 0.15 and 0.02, respectively. Table 2 presents the Ntensor components calculated via approximation using equivalent set of ellipsoids with aspect ratios given by Eq. (2.8) and formulas presented in Eq. (4.3), as well as FEA results and corresponding relative errors. It can be seen that compared to the approximation described in Section (2.2), the one based on aspect ratios given by expressions (4.3) results in lower prediction errors for

   approx 2  approx 2  approx 2  F EA F EA F EA 1 N − N N − N N − N 3333 3333 1111 1111 2222 2222 Err = + + F EA N1111

3

F EA N2222

The approximation using aspect ratios obtained from the minimization procedure appeared to correct the error in N3333 component and somewhat improve predictions for N1133 component. However, predictions for N1313 were still considerably off. To determine the aspect ratios that result in good predictions for N1313 , we used the same minimization procedure with the error

 approx  N − N F EA  Err =  1313 F EA 1313 . N

(4.2)

1313

Since the aspect ratios may depend on the fiber radius, a new set of FEA simulations was performed to determine N-tensor components of the fibers with half of the initial radius, r = 0.015. The material combinations and crimp ratio values were the same as before. Fig. 8a and b present the plots of aspect ratios as functions of fiber path geometry for two material combinations (high contrast and epoxy/E-glass) and two fiber radii obtained from the first and second error minimization procedures correspondingly. It is clear from the plots that aspect ratios are functions of fiber radius, but they appear to be independent of material contrast. The analysis of the plots also reveals that the aspect ratios that best approximate N1111 , N2222 and N3333 increase linearly with the reciprocal value of crimp ratio (1/CR and as a result decrease with increase of CR), while aspect ratio function for N1313 increases linearly with CR. The following expressions were obtained from curve fitting of the FEA results (lower aspect ratio values were given more weight due to the sensitivity of N-tensor components to AR < 150 discussed in Section (4.2)):



−1

AR = 5.5 CR r 0.82 (for all components except N1313 ) AR = 30 CR + 0.31.11 (for component N1313 )

(4.1)

F EA N3333

(4.3)

( 2r )

Comparison of the FEA results presented earlier with approximation using equivalent set of ellipsoids based on aspect ratios in Eq. (4.3) is illustrated in Fig. 9. The approximation with the proposed aspect ratios appears to perform well for all components and studied material systems except N1133 in the case of high contrast material for CR > 0.1 – the relative error of approximation for CR = 0.2 is 24.9%. The error may be explained by the fact that the

N1111 (0.61% vs 7.79%), N3333 (8.88% vs 31.32%) and N1313 (3.41% vs 96.54%). At the same time, prediction for N1133 appears to be a little worse (11.96% vs 5.08%). 5. Replacement relations Replacement relations that interrelate stiffness contribution tensors of ellipsoidal inhomogeneities having the same shape but different material properties have been proposed by Sevostianov and Kachanov (2007). Expressing N-tensors of inhomogeneities of ellipsoidal shape in terms of Hill tensor P by equation (2.6), we observe that P depends on the inhomogeneity shape, but not on its elastic constants. Thus, Eq. (2.6) written for two inhomogeneities “A” and “B” having the same shape but different elastic constants (and placed in the same matrix) will contain the same P. Excluding P from the expressions yields the sought replacement relations:



N (A )

−1



− N (B )

−1



= C (A ) − C (0 )

−1



− C (B ) − C (0 )

−1

(5.1)

In particular, if material “B” is rigid, the above relations take the form:



N (A )

−1



− N (B )

−1



= C (A ) − C (0 )

−1

(5.2)

If the inhomogeneity “A” is a pore and “B” is perfectly rigid these relations simplify further:



N (A )

−1



− N (B )

−1

= −S(0 ) ,

(5.3)

where S(0) is the compliance tensor of the matrix material. In the case of a non-ellipsoidal shape, relation (5.1) can sometimes be used as an approximation. The accuracy of this approximation has been checked by Sevostianov and Kachanov (2007) for several 2D pore shapes considered by Tsukrov and Novak (2002) and Tsukrov and Novak (2004), as well as for the 3D cuboidal inhomogeneity discussed by Chen and Young (1977). The reported accuracy is better than 8%. Chen et al. (2016) showed that these relations can be used as accurate approximations in the case of convex shapes only. In the case of concave inhomogeneities the error becomes very large.

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9

Fig. 8. Aspect ratio functions obtained from minimization of errors between FEA and approximation using equivalent sets of ellipsoids for: (a) all components except N1313 ; (b) component N1313 . Table 2 Comparison of N-tensor components (GPa) obtained from FEA with approximations using equivalent ellipsoids for the validation case.

FEA Present approach Error, % Huysmans et al. (1998) Error, %

N1111

N1122

N1133

N2222

N2233

N3333

N1212

N2323

N1313

44.11 44.37 0.61 40.67 7.79

1.44 1.49 3.93 1.50 4.34

17.01 14.97 11.96 17.87 5.08

4.24 4.25 0.15 4.25 0.19

1.70 1.73 1.68 1.66 2.17

11.54 12.57 8.88 15.16 31.32

1.38 1.37 0.85 1.37 0.70

1.28 1.26 1.59 1.26 1.61

9.19 9.50 3.41 18.05 96.54

Table 3 N-tensor components (GPa) calculated via direct FEA and replacement relations (5.1). N1111 Infinitely long straight fiber with circular cross-section (CR = 0.00) FEA Replacement Relative error, % Sinusoidal fiber with circular cross-section and CR = 0.05 FEA Replacement Relative error, % Sinusoidal fiber with circular cross-section and CR = 0.10 FEA Replacement Relative error, % Sinusoidal fiber with circular cross-section and CR = 0.20 FEA Replacement Relative error, %

N1122

N1133

N2222

N2233

N3333

N1212

N2323

N1313

8.719 8.719 0.0 0 0

0.396 0.396 0.0 0 0

0.396 0.396 0.0 0 0

2.041 2.041 0.0 0 0

0.987 0.987 0.0 0 0

2.041 2.041 0.0 0 0

1.203 1.203 0.0 0 0

1.055 1.055 0.0 0 0

1.203 1.203 0.0 0 0

7.939 8.507 7.154

0.413 0.389 5.792

0.710 0.708 0.282

2.006 2.049 2.139

0.936 0.958 2.330

2.018 2.053 1.729

1.196 1.206 0.843

1.063 1.077 1.355

1.606 1.933 20.34

6.366 7.800 22.52

0.498 0.387 22.30

1.307 1.483 13.40

1.999 2.056 2.85

0.872 0.874 0.25

2.121 2.242 5.74

1.179 1.203 2.07

1.082 1.116 3.15

2.408 3.152 30.86

3.979 5.415 36.09

0.657 0.499 24.15

1.873 2.658 41.95

1.992 2.067 3.77

0.743 0.635 14.50

3.004 3.842 27.89

1.141 1.186 3.91

1.121 1.169 4.25

3.380 4.504 33.25

To estimate the applicability of the replacement relations to sinusoidal fibers, we calculated N-tensor components for the material combination E f = 10 GPa, ν f = 0.2, Em = 1 GPa and νm = 0.4 via direct FEA and using relations (5.1) based on the results obtained earlier for the high contrast material combination. The results for fiber radius r = 0.03 and crimp ratios CR = 0.00, 0.05, 0.10, 0.20 are summarized in Table 3. It can be seen that in the case of an infinitely long straight fiber (special case of an ellipsoid), the replacement relations are exact. In the case of a sinusoidal fiber with CR ≥ 0.05, similarly to the case of concave inhomogeneities, the accuracy of the relations is insufficient. This indicates, in particular, that Hill tensor (or Eshelby tensor that is proportional to it, Mura (1987)) is generally irrelevant for the problem of effective properties of heterogeneous materials with non-ellipsoidal inhomogeneities. It may only be used as an approximation in some particular cases of inhomogeneities of convex shape.

6. Concluding remarks In this paper stiffness contribution tensors (N-tensors) of continuous sinusoidal fibers are calculated using numerical and analytical approaches and presented for the first time. The approximation procedure for the N-tensors is based on the method proposed by Huysmans et al. (1998) for effective stiffness tensors, in which elastic response of a continuous fiber is approximated by an equivalent set of ellipsoids. Comparison of the approximation results with FEA calculations revealed large approximation errors in components N1133 , N3333 and N1313 that increase in magnitude with increase of fiber/matrix elastic contrast as well as fiber crimp ratio. We believe that the errors originate from the choice of the aspect ratios of the approximating ellipsoids, which according the procedure, are calculated from the local curvature of the fiber path. In the case of a sinusoidal path, this results in spheroids with extreme aspect ratios due to high values of the radius of curvature

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Fig. 9. N-tensor components of sinusoidal fibers calculated using FEA (data points) and via approximations based on equivalent sets of ellipsoids with aspect ratios given by Eq. (4.3) (lines).

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near the locations corresponding to the maximum slope of the path. We propose an alternative approach in which aspect ratios of approximating spheroids are the same throughout the fiber path and depend on the path crimp ratio and fiber radius only. It is shown that good approximations for all components but N1313 are achieved when aspect ratios are linearly proportional to the reciprocal values of crimp ratio. On the other hand, the “best” aspect ratios in the case of component N1313 are linearly proportional to the crimp ratio. The proposed approach results in a good agreement between FEA and approximations for all components of the N-tensor. Replacement relations that interrelate stiffness contribution tensors of inhomogeneities with the same shape but different material properties were tested for the studied fiber geometries. It was found that while the relations are exact for straight fibers (as expected), the error is significant in the case of sinusoidal fibers even at low crimp ratios. Appendix A: Numerical results

Table A.1 Components of the stiffness contribution tensor (GPa) calculated for the high contrast material combination using FEA (r = 0.03). CR

N1111

N1122

N1133

N2222

N2233

N3333

N1212

N2323

N1313

0.00 0.01 0.02 0.03 0.04 0.05 0.10 0.15 0.20

99.023 97.618 93.632 87.666 80.489 72.828 40.769 23.902 15.579

0.811 0.821 0.847 0.886 0.934 0.985 1.199 1.311 1.364

0.811 0.993 1.507 2.272 3.185 4.146 7.889 9.341 9.583

2.691 2.691 2.690 2.688 2.687 2.687 2.685 2.684 2.684

1.475 1.474 1.470 1.464 1.457 1.450 1.423 1.408 1.397

2.691 2.692 2.696 2.716 2.763 2.851 4.056 6.365 9.346

0.704 0.704 0.704 0.704 0.704 0.704 0.702 0.698 0.691

0.605 0.606 0.607 0.610 0.614 0.618 0.644 0.665 0.679

0.704 0.728 0.798 0.912 1.066 1.256 2.523 3.842 4.859

Table A.2 Components of the stiffness contribution tensor (GPa) calculated for the epoxy/Eglass material combination using FEA (r = 0.03). CR

N1111

N1122

N1133

N2222

N2233

N3333

N1212

N2323

N1313

0.00 0.01 0.02 0.03 0.04 0.05 0.10 0.15 0.20

70.591 70.177 68.962 67.031 64.506 61.531 44.627 31.061 22.367

1.591 1.593 1.598 1.607 1.621 1.639 1.743 1.828 1.883

1.591 1.711 2.061 2.614 3.330 4.158 8.469 11.080 11.905

5.448 5.448 5.440 5.432 5.429 5.427 5.422 5.420 5.419

2.048 2.047 2.042 2.035 2.029 2.022 1.987 1.957 1.932

5.448 5.451 5.463 5.491 5.549 5.650 7.117 10.295 14.523

2.143 2.142 2.139 2.134 2.129 2.122 2.078 2.026 1.975

1.694 1.695 1.698 1.702 1.710 1.718 1.779 1.846 1.902

2.143 2.191 2.332 2.561 2.867 3.237 5.547 7.602 8.850

Table A.3 Components of the stiffness contribution tensor (GPa) calculated for the hydroxyapatite/collagen material combination using FEA (r = 0.03). CR

N1111

N1122

N1133

N2222

N2233

N3333

N1212

N2323

N1313

0.00 0.01 0.02 0.03 0.04 0.05 0.10 0.15 0.20

–243.523 –243.772 –245.667 –248.271 –251.717 –255.938 –284.529 –316.174 –343.846

–218.373 –218.334 –218.772 –219.245 –219.711 –220.244 –223.727 –227.672 –231.246

–218.373 –218.232 –218.065 –217.690 –217.108 –216.391 –212.077 –208.668 –206.882

–481.002 –480.835 –481.377 –481.800 –481.987 –482.119 –482.566 –482.905 –483.154

–246.908 –246.787 –246.875 –246.817 –246.541 –246.150 –243.165 –239.617 –236.315

–481.001 –480.596 –479.635 –477.942 –475.514 –472.450 –449.925 –421.680 –393.977

–85.154 –85.168 –85.440 –85.787 –86.245 –86.808 –90.728 –95.297 –99.502

–117.834 –117.751 –117.610 –117.328 –116.919 –116.408 –112.814 –108.643 –104.842

–85.176 –85.085 –84.801 –84.351 –83.756 –83.048 –78.921 –75.812 –74.390

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Please cite this article as: B. Drach et al., Effect of a curved fiber on the overall material stiffness, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.08.018