# RVE modelling of short fiber reinforced thermoplastics with discrete fiber orientation and fiber length distribution

**Part of the following topical collections:**

## Abstract

This study presents an analysis of modelling aspects on the effective composite properties of short glass fiber reinforced thermoplastics using representative volume elements (RVE). Although, many investigations were published showing effects of different modelling parameters of RVEs, we further elaborate in this contribution the parameters: influence of fiber packing, fiber shape, bonding of the fibers to the matrix, fiber length distribution and fiber orientation. The knowledge of these influences is used to determine the extent to which the increased modelling accuracy and thus the computational effort leads to an improved RVE’s forecast quality. This objective is achieved by creating and comparing different RVE models of a PBT-GF20 composite. The information required for the RVE models is obtained by experimental characterization of the PBT-GF20 and the PBT matrix material. It can be concluded based on the results of the numerical investigations in conjunction with the experimental tests of the composite that fiber packing, fiber length distribution, fiber orientation and fiber geometry are essential for a precise determination of the effective composite properties.

## Keywords

RVE Short fiber reinforced Fiber geometry Fiber orientation Homogenization## 1 Introduction

The mechanical properties of the short fiber-reinforced plastics depend on a number of factors i.e. on the length and on the orientation of the fibers. Due to the injection molding process, these two factors vary spatially in an injection-molded part, resulting in locally varying composite properties. For an exact dimensioning and design of a technical lightweight application, the spatial distribution of composite properties must be known. These can be calculated by different theories. From the first approaches of Reuss [1] and Voigt [2] up to the commonly two-stage homogenization method [3], the information content about the microstructure, which is included in the calculation of the effective composite properties, increases. This increasing information content is the basis for the increasing forecasting quality of the effective composite properties. Accordingly, efforts are obvious to further increase the forecasting quality by further increasing the information content that is used to calculate the effective composite properties. In this context, the term representative volume elements (RVE) is well known. Hill [4] uses the term RVE for sections of a composite that are statistically representative for the entire composite. According to Hill, the number of inclusions must be sufficiently high so that the calculated composite properties are independent of the selected boundary conditions. Drugan and Willis [5] give a further definition of the term RVE: They define an RVE as the smallest volume with which an effective mean value of a composite property can be calculated. Gitman et al. [6] work out the existence respectively the non-existence of RVEs depending on the used constitutive equations as well as the influence of size and periodicity on the effective composite properties on the basis of 2 dimensional RVEs with circular inclusions. It has also been shown in the literature that an effective mean value can be calculated over several small RVE’s rather than over a large one [7, 8, 9]. Especially Monte-Carlo simulations can be used, in which the realization of a single RVE is randomly generated [10].

The advantage of using RVE’s instead of micromechanical models is that less assumptions and simplifications of the underlying microstructure have to be made. This has the advantage that arbitrary shapes of inclusions or an influence of inclusions on each other can be considered in RVEs in contrast to the micromechanical models using homogenization techniques. But, solving RVEs is generally very time-consuming and numerical expensive.

The central question is therefore which information must be at least contained in a RVE to achieve a reasonable representation of the composite properties. Therefore, it is the motivation for this paper to investigate this question by an investigation of the influence of the modelling aspects of short fiber reinforced plastics on the effective composite properties. For this special case in the field of applied material- and engineering science, the influence of modeling aspects has not yet been fully investigated in the known literature. So far, the influence of modeling aspects has been investigated in recently published publications: Wang and Smith, for example, use a parameter study of a unit cell to determine the influence of fiber volume fraction, aspect ratio and distance between the fibers on the effective composite properties [11]. The influence of fiber orientation and fiber volume fraction is investigated in [12, 13]. Furthermore, Chen et al. investigates the composite properties with randomly oriented fibers in all spatial directions in [14, 15]. The distance between fibers as well as the influence of perpendicular fiber patterns are investigated by Pan et al. [16]. The influence of bent or more specifically kinked fibers is also investigated by Pan et al. in another study [17]. The dependence of linear-viscoelastic properties with respect to the fiber volume fraction is considered in [18]. Wang and his co-authors investigate the effect of an considered interphase between fiber and matrix material in [19]. However, some modelling aspects remain unconsidered despite the known works.

The investigation is carried out using PBT-GF20, exemplarily. Several RVE’s considering different modelling aspects are created and evaluated regarding their effective composite stiffness. The aim of this study is to identify the influence of these modelling aspects and to conclude which of them should be modelled with particular precision in order to achieve the best possible RVE’s forecast quality results.

## 2 Modelling of RVE

Since the aspect ratio of the RVE influences the fiber packing, this is the first modeling aspect to be investigated in Sect. 4.1.

According to Hill’s definition, RVE’s must be independent of boundary effects. This can be achieved by creating the RVE large enough so that the boundary effects are negligible. A more efficient approach is to use periodic boundary conditions [20, 21]. For this purpose, the RVE has to be created in such a way that a periodic fiber constellation is generated.

*n*fibers are derived from the fiber length distribution and the fiber probability orientation. The

*n*fibers are then placed one by one in the volume. The position of the fibers are randomly selected and periodically continued. This means that fibers that extend the boundary of the RVE are cut off and continued on the opposite side. Figure 3 illustrates this by a sketch of a two-dimensional RVE. The fiber marked by “1” protrudes beyond one edge of the RVE and is continued periodically on the opposite side. Furthermore, in the creation of the RVE, each fiber and each periodically shifted part of a fiber is checked for collision with other fibers. If a collision is detected, a new random position of the fiber is determined. The described procedure is carried out until all considered fibers are placed.

Following the geometry creation, the RVE is discretized by finite elements. The finite element mesh has a significant influence on the result of the finite element analysis. In this paper tetrahedral elements with a quadratic shape functions are used only.

*u*denotes the displacement of opposite nodes on the surfaces

*j*+ and

*j*− in the direction of \(\varepsilon_{i}\) denotes the displacement of opposite points

*x*on the surface

*j*+ and \(j\)−. The implementation of the periodic boundary conditions into a finite element analysis is carried out with linear constraint equations. Here the displacement \(\varepsilon_{i} \left( {x^{j + } - x^{j - } } \right)\) is provided by a reference point:

This reference point is not connected to the mesh of the RVE. Although the structure of the RVE is periodic, the finite element mesh may not be periodic. Opposite surfaces of the RVE are identical, but below the surface the geometries are different. Therefore, enforcing a periodic mesh simply by shifting of the nodes is not advantageous, since strongly distorted elements can result. Therefore, an interpolation approach is used instead that projects the periodic boundary conditions to a non-periodic mesh. This publication uses an interpolation approach based on quadratic interpolation functions that is presented in the following.

## 3 Characterization

The definition of the microstructure of a short fiber reinforced plastic requires knowledge about the fiber orientation and fiber length distribution as well as of the mechanical properties of the individual phases of the composite. A PBT-GF20 (Type 2300 GV1/20, 20 wt% glass fiber) from Celanex^{®} is used for all following investigations [23]. The mechanical material behavior of the glass fibers is considered to be isotropic, linear elastic with a Young’s modulus \(E_{f} = 73.000 \,{\text{MPa}}\) and a Poisson’s ratio \(\nu_{F} = 0.22\) [24, 25].

The mechanical behaviour of the matrix material PBT is tested by temperature-dependent mechanical tensile tests on bar-shaped specimens cut from 2 mm thick injection moulded plates in the dimensions w: 2 mm, d: 3 mm, l: 20 mm. The mechanical behaviour of the composite material PBT-GF20 is also measured using the same specimen geometry. The fibers of the specimens are oriented in the longitudinal direction of the specimens. The tensile tests are carried out at 0 °C, 50 °C and 80 °C, since these temperatures classify roughly the rubbery-, transition- and glassy regime of the PBT. The measured Young’s moduli at these three temperatures of the PBT are \(E_{M} = 500;\; 1000 \;{\text{and}}\; 2000\, {\text{MPa}}\) and \(E_{C} = 3050; \;4500 \;{\text{and}}\; 6800 \,{\text{MPa}}\) for the PBT-GF20.

The fiber orientation is determined by a micro computed tomography (*μ*CT) measurement of a cylindrical specimen with a 1 mm radius, extracted out of the injection moulded plates. The voxel edge length measures 1.80 μm. The two-stage fiber orientation tensor \(\varvec{a}\) is determined on a cylindrical sample of an injection-molded plate [26]. The averaged fiber orientation tensor is represented by its eigenvalues \({\text{eig}}(\varvec{a}) = (0.244, 0.711, 0.045)\).

*μ*CT can not be used to measure the fiber length distribution, as only fibers with an length exceeding 30 μm can be detected due to the quality of detection of these fibers with the

*μ*CT. The fibers shown in Fig. 5 are generated by incineration of a sample cut out of the injection-molded plates. The fibers are mixed with ethanol and distributed on a Petri dish so that they are arranged as randomly as possible. Randomly selected sections of the Petri dish are then viewed under a light microscope and the length of the fibers is measured. The results in Fig. 5 proves that not negligibly few fibers are shorter than the µCT resolution. The length distribution is determined by measuring 420 fibers from 3 different samples, which is shown in Fig. 6. A particularly high number of very short fibers can be seen here. The arithmetic mean is \(172.7 \,\upmu{\text{m}}\), which corresponds to an aspect ratio of 15.4 since the mean diameter of the fibers is 11.4 [26].

## 4 Influence of modeling aspects

This section examines the influence of modelling aspects. First, the influence of the fiber packing is investigated, followed by the fiber geometry. The influence of the fiber length distribution and the fiber orientation is then examined individually. Finally, the mentioned modeling aspects are combined. Each of the following finite element analysis is repeated with a refined mesh using 50% smaller element edges until a convergence of 1% deviation of the effective composite stiffness is reached.

### 4.1 Influence of the fiber packing

*low RVE aspect ratio*, the distance between the fibers is small in the direction of the fibers and large at right angles to them. In the

*high RVE aspect ratio variant*, the distance in the fiber direction is larger than across it.

In the case of multiple fibers in an RVE, the package is no longer determined exclusively by the RVE dimension. In addition, the packing is determined by the arrangement within the RVE. The package, which is determined by the RVE dimension, is less important the more fibers are used in the RVE. This is illustrated by the mean distance between the fibers compared to the RVE dimension. In the case of several fibers, the mean value of the distance is \(1/n\) of the RVE dimension, where *n* is the number of fibres. However, this means that the influence decreases with an increasing number of fibers, but remains significant for appropriate numbers of fibers (< 30). Therefore, the RVE dimension for n fibers of different lengths, diameters and orientation is determined in such a way that the mean distance between the fibers is identical in all spatial directions: First, the volume of the RVE \(V_{RVE}\) is determined according to Eq. 1.

*s*, the condition that the product of the dimension of the RVE is equal to the volume of the RVE applies:

This cubic equation can be solved either analytically or numerically, for example using the Newton method. Among the three possible mathematical solutions only one is a physically feasible solution. This one solution is not a complex solution.

### 4.2 Fiber shape

Next, the influence of the geometric shape of the fibers on the effective composite behavior is investigated. For this purpose, the stiffness of the composite in relation to the aspect ratio of the fiber is examined using a unit cell with one fiber and homogeneous fiber packing. The fiber volume fraction is constant with \(v_{F} = 0.116\). The fibers are modeled as ellipsoids and as cylinders. In the case of fibers represented by cylinders, a distinction is made between a matrix fully tied to the fiber and a matrix tied to the lateral surface of the fiber only. This strategy is motivated by the fiber sizing. The sizing serves as an adhesion agent between matrix and fiber and is applied during the manufacturing process of endless fibers before they are cut to short fibers. In the injection molding process, the fibers break further into smaller pieces. Accordingly, there is no sizing on the fiber’s cross section surfaces, leading to a poor adhesion between fiber and matrix at these positions.

### 4.3 Length distribution

*n*RVE’s are generated and the effective stiffness of each is determined by finite element analyses and the homogenization scheme described in Sect. 2. A probability distribution is then determined from the calculated values. In contrast to the unit cell with a single fiber, the result depends on the spatial position of the fibers in the RVE due to fiber interacting. Additionally, the Monte-Carlo simulation is necessary for the fiber length probability, since reconstructions with few fibers do not provide a good approximation to the measured fiber length distribution. This means that the reconstruction is carried out several times (min. 100 times) for both concepts. Both concepts are repeated for different numbers of fibers. The result of the study is shown in Fig. 11.

The normal distributions of the calculated effective stiffness of the composite in fiber direction are shown for both concepts and 10, 30 and 50 fibers respectively. The mean values of both concepts converge to different values. For a uniform length distribution, the mean value for 50 fibers is approximately 6500 MPa, using a fiber length distribution it is approximately 7600 MPa. The mean value of the effective stiffness is thus significantly higher with fiber length distribution than with uniform fiber length. In both concepts, the mean values with 10, 30 and 50 fibers are approximately identical, with the exception of 10 fibers in the case of the fiber length distribution. In this case the dimensions of the RVE are smaller than some fibers. Therefore, the RVE may not be modelled as short fibers, but as endless fibers. This leads to a higher mean value of the effective composite stiffness. In addition, it becomes apparent that the standard deviation of the effective composite stiffness is greater with the concept of fiber length distribution than with a uniform fiber length, because it contains another random variable. Accordingly, the convergence rate is lower when using the fiber length distribution, so that more RVE’s have to be considered in order to determine the stiffness with the same confidence level.

Another important aspect becomes also evident with the distribution of the homogenized composite stiffness. A higher homogenized stiffness means that the local stresses within such an RVE are higher with the same deformation than the local stresses within RVE’s which lead to a lower homogenized stiffness. The local stresses are therefore dependent on the fiber arrangement and are statistically depended variable.

### 4.4 Fiber orientation

### 4.5 Combination of modelling aspects

## 5 Discussion

In this section, a discussion of the investigated modelling aspects follows. The fiber package in short fiber reinforced plastics has a large influence on the resulting effective composite properties. For a unit cell with one fiber, the fiber package is only determined by the dimension of the RVE and its influence is principally studied frequently in the literature [11, 29, 30, 31, 32]. With many fibers, in the sense of the definition of an RVE according to Hill the fiber packing is not effected by the dimensions of the RVE. With only a few fibers, however, as for the Drugan and Willis approach to determine an RVE as the smallest possible volume, the fiber package depends both on the realized arrangement and on the dimension. As shown in this paper by the investigation of the influence of the fiber package, it is very significant for the effective composite stiffness of short fiber reinforced plastics. Therefore, it is important to model the fiber package accurately for a precise prediction of real composite materials. For the modelling of a RVE in the sense of the Drugan and Willis approach, it seems to be reasonable to determine the dimension of the RVE in such a way that a homogeneous fiber package results on average with the Monte-Carlo approach, which can be achieved with the presented approach in Sect. 4.1.

Using the Monte-Carlo approach with many fibers (unidirectional and equal length), the mean value of the effective stiffness corresponds to that of a homogeneous fiber package (see Figs. 9, 11). It is important to note, however, that this only applies to the average stiffness. The effective stiffness of a single RVE always depends on the fiber constellation realized. As the size of the RVE increases, it is only more likely to produce a fiber constellation that corresponds to a homogeneous fiber packing.

The dependence of the aspect ratio on the effective composite stiffness with ellipsoidal fibers is well known in the literature from the model of Mori–Tanaka [3]. Qualitatively, this dependence is the same for the fully and partially bonded cylindrical fibers, as shown in this study. Thereby the fully bonded cylinders results in higher effective composite stiffness compared to ellipsoids, while the partially bonded are in-between. The difference between fully and partially bonded decreases with increasing aspect ratio. The overall influence of the fiber shape and bondage on the effective composite stiffness can therefore be classified as medium, especially for aspect ratios which typically occur in technical applications. However, it remains to be investigated to what extent this is also the case in the calculation of the strength of the composite.

The results of the investigation with the combination of the different modelling aspects are also compared with experimental results of a characterization of PBT-GF20. For this purpose, the converged mean value of the calculated stiffness by the RVE is used. In addition, the calculated stiffness using the Mori–Tanaka [3] approach, the approach of Duschelbauer [33] and the self-consistent [34] method are used as an additional comparison. To consider the effects of fiber orientation and fiber length distribution, a two-step homogenization is used.

## 6 Conclusion

The fiber package has a very significant influence on the calculated homogeneous composite stiffness. This can be conducted by an RVE with an elliptical fiber and varied RVE dimensions. A homogeneous fiber package, i.e. the distance in all directions is the same, reproduces the results of the Mori–Tanaka model.

The fiber shape has a medium influence on the homogeneous composite stiffness. Cylindrical fibers are stiffer than ellipsoid fibers with the same fiber volume content and aspect ratio. Fibers with a matrix tied on their lateral surface only owns a homogenized stiffness between the stiffness of cylindrical and ellipsoid modelled fibers.

A fiber length distribution influences the effective stiffness compared to a constant average fiber length. In addition to the mean value of the stiffness, the scattering width of the effective stiffness is higher.

The combined consideration of modeling aspects in the RVE provides a very accurate prediction of a composite’s effective stiffness, in the case of the used PBT-GF20 for the validation more accurate than the Mori-Tanka-, Duschelbauer- or Self-Consistent Method [3, 33, 34].

From the results of the numerical investigations it can be concluded that a precise knowledge of the fiber packing, fiber length distribution, fiber orientation and fiber geometry is essential for a precise determination of the effective composite properties.

In further studies it has to be clarified to what extent the influence of the modelling parameters remains the same for other load directions as well as for a plastic or visco-plastic modelling of the matrix material.

## Notes

### Compliance with ethical standards

### Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

## References

- 1.Reuss A (1929) Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. Z Angew Math Mech. https://doi.org/10.1002/zamm.19290090104 CrossRefzbMATHGoogle Scholar
- 2.Voigt W (1889) Ueber die Beziehung zwischen den beiden Elasticitätsconstanten isotroper Körper. Ann Phys. https://doi.org/10.1002/andp.18892741206 CrossRefzbMATHGoogle Scholar
- 3.Mori T, Tanaka K (1973) Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. https://doi.org/10.1016/0001-6160(73)90064-3 CrossRefGoogle Scholar
- 4.Hill R (1963) Elastic properties of reinforced solids. Some theoretical principles. J Mech Phys Solids. https://doi.org/10.1016/0022-5096(63)90036-X CrossRefzbMATHGoogle Scholar
- 5.Drugan WJ, Willis JR (1996) A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites. J Mech Phys Solids. https://doi.org/10.1016/0022-5096(96)00007-5 MathSciNetCrossRefzbMATHGoogle Scholar
- 6.Gitman IM, Askes H, Sluys LJ (2007) Representative volume. Existence and size determination. Eng Fract Mech. https://doi.org/10.1016/j.engfracmech.2006.12.021 CrossRefGoogle Scholar
- 7.Huet C (1990) Application of variational concepts to size effects in elastic heterogeneous bodies. J Mech Phys Solids. https://doi.org/10.1016/0022-5096(90)90041-2 MathSciNetCrossRefGoogle Scholar
- 8.Hazanov S, Huet C (1994) Order relationships for boundary conditions effect in heterogeneous bodies smaller than the representative volume. J Mech Phys Solids. https://doi.org/10.1016/0022-5096(94)90022-1 MathSciNetCrossRefzbMATHGoogle Scholar
- 9.Kanit T, Forest S, Galliet I, Mounoury V, Jeulin D (2003) Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int J Solids Struct. https://doi.org/10.1016/S0020-7683(03)00143-4 CrossRefzbMATHGoogle Scholar
- 10.Gusev AA (1997) Representative volume element size for elastic composites: a numerical study. J Mech Phys Solids. https://doi.org/10.1016/S0022-5096(97)00016-1 CrossRefzbMATHGoogle Scholar
- 11.Wang Z, Smith DE (2019) Numerical analysis on viscoelastic creep responses of aligned short fiber reinforced composites. Compos Struct. https://doi.org/10.1016/j.compstruct.2019.111394 CrossRefGoogle Scholar
- 12.Babu KP, Mohite PM, Upadhyay CS (2018) Development of an RVE and its stiffness predictions based on mathematical homogenization theory for short fibre composites. Int J Solids Struct. https://doi.org/10.1016/j.ijsolstr.2017.10.011 CrossRefGoogle Scholar
- 13.Berger H, Kari S, Gabbert U, Rodríguez-Ramos R, Bravo-Castillero J, Guinovart-Díaz R, Volume D (2007) Nº: evaluation of effective material properties of randomly distributed short cylindrical fiber composites using a numerical homogenization technique. J Mech Mater Struct. https://doi.org/10.2140/jomms.2007.2.1561 CrossRefGoogle Scholar
- 14.Chen L, Gu B, Tao J, Zhou J (2019) The average response and isotropy of 3D representative volume elements for random distributed short fibers reinforced elastomer. Compos Struct. https://doi.org/10.1016/j.compstruct.2019.02.049 CrossRefGoogle Scholar
- 15.Chen L, Gu B, Zhou J, Tao J (2019) Study of the effectiveness of the RVEs for random short fiber reinforced elastomer composites. Fibers Polym. https://doi.org/10.1007/s12221-019-1178-9 CrossRefGoogle Scholar
- 16.Pan Y, Iorga L, Pelegri AA (2008) Analysis of 3D random chopped fiber reinforced composites using FEM and random sequential adsorption. Comput Mater Sci. https://doi.org/10.1016/j.commatsci.2007.12.016 CrossRefGoogle Scholar
- 17.Pan Y, Iorga L, Pelegri AA (2008) Numerical generation of a random chopped fiber composite RVE and its elastic properties. Compos Sci Technol. https://doi.org/10.1016/j.compscitech.2008.06.007 CrossRefGoogle Scholar
- 18.Burgarella B, Maurel-Pantel A, Lahellec N, Bouvard J-L, Billon N, Moulinec H, Lebon F (2019) Effective viscoelastic behavior of short fibers composites using virtual DMA experiments. Mech Time Depend Mater. https://doi.org/10.1007/s11043-018-9386-z CrossRefGoogle Scholar
- 19.Wang L, Nygren G, Karkkainen RL, Yang Q (2019) A multiscale approach for virtual testing of highly aligned short carbon fiber composites. Compos Struct. https://doi.org/10.1016/j.compstruct.2019.111462 CrossRefGoogle Scholar
- 20.Okereke MI, Akpoyomare AI (2013) A virtual framework for prediction of full-field elastic response of unidirectional composites. Comput Mater Sci. https://doi.org/10.1016/j.commatsci.2012.12.036 CrossRefGoogle Scholar
- 21.Nguyen V-D, Béchet E, Geuzaine C, Noels L (2012) Imposing periodic boundary condition on arbitrary meshes by polynomial interpolation. Comput Mater Sci. https://doi.org/10.1016/j.commatsci.2011.10.017 CrossRefGoogle Scholar
- 22.Advanced Finite Element Methods (ASEN 6367) Course Material. https://www.colorado.edu/engineering/CAS/courses.d/AFEM.d/. Accessed 16 Feb 2018
- 23.Celanese (2017) Celanex 2300 GV1/20 Datasheet. http://catalog.ides.com/Datasheet.aspx?I=26793&E=73487. Accessed 12 July 2017
- 24.Kaiser J-M, Stommel M (2014) Modified mean-field formulations for the improved simulation of short fiber reinforced thermoplastics. Compos Sci Technol. https://doi.org/10.1016/j.compscitech.2014.05.010 CrossRefGoogle Scholar
- 25.Parra-Venegas EJ, Campos-Venegas K, Martinez-Sanchez R, Herrera-Ramirez JM, Rodriguez-Miranda A (2012) The tensile behavior of E-glass fibers. Microsc Microanal. https://doi.org/10.1017/S1431927612005776 CrossRefGoogle Scholar
- 26.Kaiser J-M (2013) Beitrag zur mikromechanischen Berechnung kurzfaserverstärkter Kunststoffe—Deformation und Versagen. Dissertation, Universität des SaarlandesGoogle Scholar
- 27.Leckie FA, Onat ET (1981) Tensorial nature of damage measuring internal variables. In: Hult J, Lemaitre J (eds) Physical non-linearities in structural analysis. Springer, Berlin, pp 140–155CrossRefGoogle Scholar
- 28.Advani SG, Tucker CL (1987) The use of tensors to describe and predict fiber orientation in short fiber composites. J Rheol. https://doi.org/10.1122/1.549945 CrossRefGoogle Scholar
- 29.Swolfs Y, Verpoest I, Gorbatikh L (2016) A review of input data and modelling assumptions in longitudinal strength models for unidirectional fibre-reinforced composites. Compos Struct. https://doi.org/10.1016/j.compstruct.2016.05.002 CrossRefGoogle Scholar
- 30.Landis CM, McMeeking RM (1999) Stress concentrations in composites with interface sliding, matrix stiffness and uneven fiber spacing using shear lag theory. Int J Solids Struct. https://doi.org/10.1016/S0020-7683(98)00193-0 CrossRefzbMATHGoogle Scholar
- 31.Sun CT, Vaidya RS (1996) Prediction of composite properties from a representative volume element. Compos Sci Technol. https://doi.org/10.1016/0266-3538(95)00141-7 CrossRefGoogle Scholar
- 32.Bargmann S, Klusemann B, Markmann J, Schnabel JE, Schneider K, Soyarslan C, Wilmers J (2018) Generation of 3D representative volume elements for heterogeneous materials: a review. Prog Mater Sci. https://doi.org/10.1016/j.pmatsci.2018.02.003 CrossRefGoogle Scholar
- 33.Duschlbauer D, Pettermann HE, Böhm HJ (2003) Mori–Tanaka based evaluation of inclusion stresses in composites with nonaligned reinforcements. Scripta Mater. https://doi.org/10.1016/S1359-6462(02)00390-1 CrossRefGoogle Scholar
- 34.Hill R (1965) A self-consistent mechanics of composite materials. J Mech Phys Solids. https://doi.org/10.1016/0022-5096(65)90010-4 CrossRefGoogle Scholar

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