Numerical and experimental investigation of the structural behavior of a carbon fiber reinforced ankle-foot orthosis

Medical Engineering and Physics 37 (2015) 505–511

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Medical Engineering and Physics journal homepage: www.elsevier.com/locate/medengphy

Technical note

Numerical and experimental investigation of the structural behavior of a carbon fiber reinforced ankle-foot orthosis Bertram Stier∗, Jaan-Willem Simon, Stefanie Reese Institute of Applied Mechanics, RWTH Aachen University Mies-van-der-Rohe-Str. 1, 52074 Aachen, Germany

a r t i c l e

i n f o

Article history: Received 12 June 2014 Revised 6 February 2015 Accepted 16 February 2015

Keywords: Ankle-foot orthosis Carbon fiber reinforced composite Finite element analysis Multi-scale modeling Digital image correlation

a b s t r a c t Ankle-foot orthoses (AFOs) are designed to enhance the gait function of individuals with motor impairments. Recent AFOs are often made of laminated composites due to their high stiffness and low density. Since the performance of AFO is primarily influenced by their structural stiffness, the investigation of the mechanical response is very important for the design. The aim of this paper is to present a three dimensional multi-scale structural analysis methodology to speed up the design process of AFO. The multi-scale modeling procedure was applied such that the intrinsic micro-structure of the fiber reinforced laminates could be taken into account. In particular, representative volume elements were used on the micro-scale, where fiber and matrix were treated separately, and on the textile scale of the woven structure. For the validation of this methodology, experimental data were generated using digital image correlation (DIC) measurements. Finally, the structural behavior of the whole AFO was predicted numerically for a specific loading scenario and compared with experimental results. It was shown that the proposed numerical multi-scale scheme is well suited for the prediction of the structural behavior of AFOs, validated by the comparison of local strain fields as well as the global force-displacement curves. © 2015 IPEM. Published by Elsevier Ltd. All rights reserved.

1. Introduction Ankle-foot orthoses (AFOs) are designed to enhance the gait function of individuals with motor impairments. In many cases, the materials used in recent AFOs are either low-temperature thermoplastics or fiber reinforced thermoset composites. The latter are particularly suited for these applications due to their high stiffness and low density [1]. On the other hand, the disadvantage of fiber reinforced composites in this context is the complex, time consuming process of design and manufacture [2]. Since the performance of AFOs is dominated by their structural response, information about the mechanical behavior is crucial for the design. There are different experimental setups to measure the mechanical characteristics of AFO or to investigate failure mechanisms, such as the device BRUCE [3] and the ankle-foot simulator presented by Lai et al. [4]. However, these experimental investigations can be very expensive, and the production of prototypes is necessary before the designed structure can be investigated. Hence, numerical investigations are used to support the design process and reduce the time and costs of the prototype phase. In particular, the finite element method (FEM) has proven its potential to

∗

Corresponding author. Tel.: +49 241 80 25006; fax: +49 241 80 22001. E-mail address: [email protected] (B. Stier).

http://dx.doi.org/10.1016/j.medengphy.2015.02.002 1350-4533/© 2015 IPEM. Published by Elsevier Ltd. All rights reserved.

investigate the structural behavior of orthopaedic devices. For example, Bougherara et al. [5] presented a three-dimensional finite element model for the investigation of metallic and carbon fiber composite hip implants, whereas a statistics-based finite element (FE) analysis of pressure-relieving foot orthoses has been presented by Cheung and Zhang [6]. Further, the FEM has been applied by Omasta et al. [7] as a numerical tool to determine the stress-strain behavior of lower limb prosthetics. Further, the FEM has also been used to predict the peak stress of AFO during normal and pathological motions [8] as well as for structural characterization [9]. The material’s intrinsic micro-structure has not been taken into account in the works mentioned above. However, for carbon fiber reinforced plastics (CFRPs), it is crucial to account for micro structural effects at the macro-scale. Various attempts were presented to overcome this problem. Among them, the asymptotic homogenization method for periodic structures [10–12] and the FE2 -method [13–16] are prominent. As an alternative, the special character of the micro-structure can be taken into account by means of an anisotropic material model using the concept of structural tensors. The majority of such models formulated for finite strains were developed in the field of biomechanics, see e.g. [17–19]. These formulations can be efficiently applied to model the anisotropic behavior of fiber-reinforced composites as shown by Reese [20], Stier et al. [21], and Simon et al. [22]. An overview of anisotropic material models developed for reinforced fiber composites can be found e.g. in the work of

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B. Stier et al. / Medical Engineering and Physics 37 (2015) 505–511 Table 1 Material parameters of fibers and matrix. Young’s modulus (N/mm2 )

Shear modulus (N/mm2 )

Poisson’s ratio

Fibers

EF = 230 000 EF⊥ = 15 000

GF⊥ = 15 000 GF⊥⊥ = 7 000

νF⊥ = 0.2

Matrix

EM = 3 600

EM GM = 2 (1+ νM )

νM = 0.38

material parameters for the fiber [26] and matrix given in Table 1. The fiber volume fraction is ϕUD = 0.426. The FE-model contains 20 fibers with a random distribution embedded in the matrix material (Fig. 1(a) left). The mesh consists of 35,000 hexahedral solid C3D8 elements. It is generated such that the meshes on opposite faces are identical. This way, periodic boundary conditions can be applied by linking the nodal degrees of freedom of the corresponding nodes using equation constraints. On this micro-structure, far field strains εˆ¯ j ( j = 1, . . . , 6) are successively applied. The three components εˆ1 , εˆ2 and εˆ3 are the normal strains and εˆ4 , εˆ5 and εˆ6 denote the shear strains in Voigt notation, indicated by ( ˆ· ). For each of these εˆ¯ j , the corresponding six local stress fields σˆ i (i = 1, . . . , 6) are computed. Then, the volume averaged (or far field) stress σˆ¯ i is calculated by Fig. 1. Schematic representation of hierarchical multi-scale modeling strategy.

Ansar et al. [23]. For the experimental validation and implementation into a finite element framework of such models, the reader is referred to Lomov et al. [24]. Modeling techniques for layered composites, including micro-macro scale transitions, are presented by Toledo et al. [25]. However, it is challenging to determine the various material parameters for these models only from constituent data. 2. Methods In this paper, the FEM is applied to predict the structural behavior of an ankle-foot orthosis, which consists of multiple layers of a thermoset matrix reinforced by carbon fibers. In order to adapt the mechanical response of the orthosis to the natural behavior of the human body, plydrops and fiber orientations as well as different architectures of the reinforcements are used. In particular, some layers are made of unidirectional (UD) carbon fiber reinforced composites (CFRPs), in which all fibers are aligned parallel, whereas others contain textile reinforcements which consist of woven tows embedded in the matrix material. In the current work, a hierarchical multi-scale modeling strategy is used. For unidirectional CFRP, a two-scale model is applied which includes the scale transition from the micro- to the macro-scale (Fig. 1(a)) and is discussed in detail in Section 2.1. Further, for textile CFRP, a three-scale method is used to achieve the scale transitions from micro- to meso-scale and from meso- to macro-scale (Fig. 1(b)). The latter model will be explained in Section 2.2. The implicit numerical investigations are carried out in the commercial FE-software Abaqus (Version 6.13, Dassault Systemes). Finally, to validate these models,l experimental results are presented in Section 2.3. 2.1. Modeling of unidirectional CFRP The first step of the two-scale model for UD CFRP (Fig. 1(a)) is the generation of a representative volume element (RVE) of the micro-scale. On this scale, the fibers and the matrix are discretized separately. Both, fibers and matrix, are assumed to be linear elastic. Further, the matrix material is considered to be isotropic, whereas the fibers are modeled as transversely isotropic. Here, Toray T300 carbon fibers are embedded in an epoxy matrix with the corresponding

 σˆ dV ¯ σˆ i = V i V dV

(1)

For linear elastic materials, the relation between the volume averaged stresses and the far field strains can be expressed via the elastic constitutive equation:

σˆ¯ i = Cˆ¯ ij εˆ¯ j .

(2)

ˆ¯ ij are computed, Thereby, the components of the elasticity tensor C which can easily be transferred to the engineering constants given in Table 3. Following the numerical homogenization scheme described previously, the transversely isotropic material tangent is obtained and used on the higher scale to describe the behavior of unidirectional CFRPs on the macro-scale. 2.2. Modeling of textile CFRPs For textile CFRPs an additional intermediate scale is mandatory: the so-called meso-scale (Fig. 1(b)) in which the woven tows and the matrix are modeled separately. The corresponding three-scale strategy works as follows. First, an RVE is set up for the micro-scale, in which the fibers and the matrix are modeled separately (Fig. 1(b) left). Then, the same numerical homogenization scheme as described in Section 2.1 is applied to obtain the homogenized material properties of the tows in the RVE at the meso-scale (Fig. 1(b) middle). A second scale transition from the meso- to the macro-scale is performed following the same procedure. Due to the complex internal architecture of the weave, the meso-scale RVE is generated using the generative shape design tool of Catia V5 (Dassault Systemes). Again, periodicity of the internal structure as well as coinciding meshes are taken care of. Finally, the macro-mechanical properties of the whole weave, which accounts for the composite’s intrinsic micro- and meso-structure, are obtained analogously to the micro- meso-scale transition. The fiber volume fraction ϕtow = 0.73 of the tows is very high. Hence, generating a random distribution for the fiber locations at the micro-scale would be very challenging. Thus, a hexagonal fiber packing is assumed on the micro-scale to obtain the material constants of the tows (Fig. 1(b) left). The material parameters computed for the tows in the meso-scale RVE are:

E = 168 680 N/mm2 E⊥ = 9670 N/mm2

G⊥ = 5670 N/mm2

ν⊥ = 0.22

G⊥⊥ = 3920 N/mm2

The tows are woven into a 4 × 4 twill weave, which is embedded in matrix material. The total fiber volume fraction of the weave is

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Fig. 2. Geometry and FE-model of the meso-scale RVE for 4 × 4 twill weave textile CFRP.

ϕBIAX = 0.407. In Fig. 2(a), the real woven structure is shown as well as the RVE chosen on this scale, which contains 8 × 8 tows in order to be periodic. Describing the geometry of the tows’ crimp regions accurately is crucial for a realistic modeling of the behavior of the whole RVE. The details are provided in Fig. 2(b). The FE-model and the mesh of the meso-scale RVE on the textile scale are shown in Fig. 2(c). The mesh consists of 1,527,206 elements, where tetrahedral elements are used for the matrix and hexahedral elements for the tows. The total number of nodes is 2,603,981.

Table 2 Characteristic dimensions. Parameter

Value (mm)

s  w tUD tBIAX

300.0 200.0 40.0 1.5 0.8

Fiber angles UD 0◦ 15◦ 30◦ 45◦ 60◦ 75◦ 90◦ ◦ ◦ BIAX 0 −90 15 −105◦ 30◦ −120◦ 45◦ −135◦ ◦

2.3. Experimental Investigations To validate the models, experimental investigations of carbon fiber reinforced epoxy resin plates under tensile loading conditions were performed. The setup included fully coupled digital image correlation (DIC) system which provided accurate non-tactile strain measurements at the surface of the specimen during the test procedure. Tabs were applied to the specimen to prevent fiber damage during clamping and testing. Furthermore, a random pattern was sprayed on one of the specimens faces which was mandatory for the DIC measurement. Titanium dioxide powder spray was used for the white base coat while graphite spray speckles induced the contrast. The test setup included a ‘Zwick Z100’ load frame and an ‘Aramis M4’ DIC system, which allowed synchronized recording of force and displacement values. The Zwick’s load cell signal as well as the displacement signals of the DIC system were recorded by the control software ‘TestXpert II’. Analog online signal transfer—e.g. length change measured by the DIC system—from Aramis to the testing machine’s software was realized by the integration of an ‘HBM Spider 8’ measuring amplifier. The ARAMIS system supported the observation of up to ten points at the specimens’ surface in the real-time-sensor (rts) mode. In this mode, ARAMIS replaced an optical extensometer. Assuming a homogeneous deformation state at the center of the specimen, only two points in longitudinal direction needed to be observed to determine the ‘global’ length change and strain values. Recording the displacements of two additional points in transverse direction provided the data for the Poisson’s ratio.

To validate the macro-mechanical model for UD CFRP obtained by the scale transition from the micro- to the macro-scale as described in Section 2.1, a test series of unidirectional CFRP plates with different fiber orientations was performed. Analogously, to validate the macromechanical model for textile CFRP obtained by the two scale transition from the micro- to the meso-scale and from the meso- to the macroscale as described in Section 2.2, a test series of textile CFRP plates with different fiber orientations was performed. In this test series six specimen were tested for every fiber angle. Each UD plate consisted of four layers of carbon fiber prepreg material. The specimen’s geometry and dimensions were given in Fig. 3 and Table 2. The Young’s modulus for every laminate was calculated by evaluating the slope of a linear fit to the initial values of the experimentally obtained stress-strain data. 2.4. Experimental and numerical analysis of the ankle foot orthosis The AFO is an off-the-shelf product which has been provided by the manufacturer. The number of layers in the composite is eleven at the thickest location; varying due to plydrops. Most of the layers consist of the unidirectional CFRP described in the previous Section 2.1. The covering layers are textile CFRP, as discussed in Section 2.2. For the quasi static investigation of the most critical zone, the ankle support, the AFO is clamped at two positions at the sole. Displacement loading with 2 mm/min is applied at the calf support through a punch (Fig. 3(e) and (f)). The actual operating loading rates experienced by

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Fig. 3. Geometry and dimensions of specimen (a)–(d) and comparison of experimental (e) and numerical (f) setup.

the AFO could be higher, but neither rate effects, nor any dynamic effects were considered in this investigation. The whole AFO is analyzed using the static implicit solver (Abaqus/Standard) of the commercial software package Abaqus (Version 6.13) using 36,800 C3D8 elements. The geometry is based on the tool face CAD data set provided by the manufacturer of the AFO. Different layers are discretized separately. A constant offset surface is created from the given tool surface with the corresponding layer thickness describing the top and bottom of the layers. Every layer has its own boundary, which is generated from the cutting path of the cutting tool and wrapped around the 3D geometry defining the sides of the layers. Together with the covering sheets, they provide a hull for every layer that matches perfectly at the interfaces of the layers. The discretization (meshing) is carried out separately in order to provide flexibility for the meshing algorithm. Then, at the interfaces, tie constraints are applied to model perfect bonding between the layers. The punch is modeled as a cylindrical analytical rigid body. Frictionless contact is defined between the punch and calf support. Thereby, loading can be applied by prescribing the displacement of the punch. The run time of the structural simulation is about 15 min using 10 Intel Xeon E5606 cores with 2.13 GHz each.

experimental data from Section 2.3. The mean values and the standard deviations of the Young’s modulus for every fiber angle are plotted in polar diagrams for UD CFRPs (Fig. 4(a)) and for textile CFRPs (Fig. 4(b)). Furthermore, the numerical results for the engineering elastic constants resulting from the numerical homogenization are compared with the experimental results in Table 3 for UD CFRPs and in Table 4 for textile CFRPs. In addition, approximate analytical results based on the classical lamination theory [27] are given to verify the results. 3.2. Comparison of the predicted mechanical behavior of the ankle-foot orthosis to the measurements As expected, the most critical zone of the AFO in this quasi static loading case is the transition region between the sole and the ankle support. The maximum principle strains in this region are therefore investigated in more detail. In particular, the comparison between the local strain field obtained from the full field measurement (Fig. 5(c)) and the FE prediction (Fig. 5(d)) is shown. Finally, the predicted force-displacement curve is compared to the experimental data (Fig. 5). 4. Discussion

3. Results 3.1. Validation of the numerical homogenization procedure For the validation of the multi-scale homogenization procedure, the numerical results from Sections 2.1 and 2.2 are compared to the

The comparison between numerically homogenized values and experimentally measured ones (Fig. 4 and Tables 3 and 4) clearly validates the applied numerical homogenization approach. Both for the UD as well as the textile CFRP, the macro-scale material properties after one or two scale transitions, respectively, are in very good

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Table 3 Comparison of numerical, experimental, and analytical results for engineering elastic constants for unidirectional CFRP. Young’s modulus (N/mm2 )

Shear modulus (N/mm2 )

Poisson’s ratio

Numerical

E = 99 760 E⊥ = 6750

G⊥ = 2850 G⊥⊥ = 2350

ν⊥ = 0.30

Experimental

E = 102 500 E⊥ = 6400

G⊥ = 3050 G⊥⊥ not measured

ν⊥ = 0.31

Analytical

E = 100 050 E⊥ = 7250

G⊥ = 3400 G⊥⊥ = 2630

ν⊥ = 0.30

Table 4 Comparison of numerical, experimental, and analytical results for engineering elastic constants for textile CFRP. The indices 1 and 2 are corresponding to the directions of the warp- and weft yarn (in plane directions), the 3 indicates the thickness direction (out of plane direction).

90° 110

75°

100

Young’s modulus (N/mm2 )

Shear modulus (N/mm2 )

Poisson’s ratio

Numerical

E1 = 46 220 E3 = 7470

G12 = 3080 G13 = 2230

ν12 = 0.074 ν13 = 0.35

Experimental

E1 = 45 910 E3 not measured

G12 = 2800 G13 not measured

ν12 = 0.08

Analytical

E1 = 49 640 E3 = 5340

G12 = 2470 G13 = 2000

ν12 = 0.16 ν13 = 0.37

Experiment Simulation 60°

90 45°

80 70 60

30°

50 40 15°

30 20 10 0

0° Young s modulus [GPa]

(a) Unidirectional CFRP

90° 50 45 40 35

75°

Experiment Simulation 60° 45°

30 30°

25 20 15

15°

10 5 0

0° Young s modulus [GPa]

(b) Textile CFRP Fig. 4. Comparison of numerical and experimental results: polar diagram of Young’s modulus.

agreement with the experimental results. A big advantage of the presented procedure is that no fitting is necessary. Only the material parameters of the constituents and the fiber volume fraction of the composite are required as input variables. Noteworthy, these experiments are necessary only for the validation of the presented approach

and thus have to be performed only once. If a different FRP should be of interest, only the input parameters for fiber and matrix need to be adjusted without the need of any additional experiments. In consequence, the suggested approach is extremely flexible. Further, the results presented in Section 2.4 show that the presented method is suitable for the analysis of AFOs. The computed and experimentally obtained strain fields are, as shown in Fig. 3, in agreement both qualitatively and quantitatively. Thus, stress and strain distributions can be directly evaluated even in critical zones. In addition, the global behavior of the AFO represented by the force-displacement curves (Fig. 5) is predicted very well. The difference between the numerical and experimental results is less than the deviation between the two tested orthoses – even for the nonlinear regime at large displacements. It can be concluded that the presented approach is ideally suited to numerically obtain global and local results for AFOs without the need of additional experiments or parameter fitting. Only the material properties of fiber and matrix, the fiber volume fraction, and the geometry of the textile must be known in advance. Hence, this method can serve as a strong design tool for AFOs. From the local investigations, one can directly detect the most critical zones and the corresponding stresses and strains. If these are not in the expected range, then the layup of the laminates can be easily changed. For example, the fiber direction of each layer can be changed separately, different fiber and matrix materials can be used, or UD layers could be exchanged by textile ones and vice versa. For this design step, there is no need to build a prototype at all since no experiments are necessary. Thus, this approach gives very important information to the designer of AFOs, leading to a better structural reliability and finer adjustment of the gradual structural stiffness needed. To further improve the results in future studies, the boundary conditions in the simulation should be modified to more realistically reflect the loading state during gait. This effect should not be neglected as shown e.g. in a comparable study on a prosthetic foot [7], where a motion analysis and ground reaction force measurement have been used as basis for the definition of the boundary conditions. Additionally, in the work of Takahashi and Stanhope [28], an analysis of AFO stiffness sensitivity to loading conditions implies that the loading angle has a non-negligible effect. Moreover, the consideration of the foot-insole interaction shall be taken into account in future works. As shown e.g. in the publication of Jamshidi et al. [29], the design of the sole can be optimized to the

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Fig. 5. Comparison of numerical and experimental results for the orthosis: Force-displacement curves (a); comparison of experimental (b) and numerical (c) local maximum principal strains (the scales are identical in the two subfigures.

kinetic data of each individual patient in principle. Following Cheung and Zhang [6], a sensitivity analysis of peak plantar pressure relief to several design parameters for the AFO can be performed using statistics-based FEM. Finally, fatigue analysis of the orthopaedic devices is an important aspect. There are experimental investigations on failure mechanisms of AFOs during cyclic walk and cyclic stepping, see e.g. [4]. Nonetheless, to incorporate the fatigue behavior to the numerical procedure is challenging and has not been performed in the present study.

Hence, it has a high potential to enhance the design process of such structures since different layups of composite laminate or different material combinations can be investigated without the need of testing. However, once the presumably optimal design is achieved from the numerical investigations, final experiments at the prototype level are still advisable. Conflict of interest None of the authors has any conflict of interest.

5. Conclusions References The structural behavior of an AFO has been investigated both numerically and experimentally. The orthosis consists of a composite laminate with plydrops, ranging from two to eleven layers of either unidirectional or textile CFRP. In order to take into account the intrinsic micro-structure of the material within the structural computation on the macro-scale, a hierarchical multi-scale modeling strategy has been applied. For this, two different RVEs have been considered. One on the micro-scale for unidirectional CFRP and another on the mesoscale for the textile CFRP. Both steps of numerical homogenization have been validated by comparison with experimental as well as analytical results. In consequence, the structural behavior of the AFO could be predicted numerically. The results of this simulation are in agreement with the data measured in the experiments. This holds for the local strain fields as well as for the global force-displacement curves. The difference between the numerically predicted results and the experimental ones is smaller than the deviation between the two different orthoses tested. It can be concluded, that the proposed modeling strategy is suitable to predict the behavior of carbon fiber reinforced orthoses.

[1] Scholz MS, Blanchfield JP, Bloom LD, Coburn BH, Elkington M, Fuller JD, et al. The use of composite materials in modern orthopaedic medicine and prosthetic devices: a review. Compos Sci Technol 2011;71:1791–803. [2] Faustini MC, Neptune RR, Crawford RH, Stanhope SJ. Manufacture of passive dynamic ankle-foot orthoses using selective laser sintering. IEEE Trans Bio-Med Eng 2008;55(2):784–90. [3] Bregman DJJ, Rozumalski A, Koops D, De Groot V, Schwartz M, Harlaar J. A new method for evaluating ankle foot orthosis characteristics: Bruce. Gait Posture 2009;30:144–9. [4] Lai HJ, Yu CH, Kao HC, Chen WC, Chou CW, Cheng CK. Ankle-foot simulator development for testing ankle-foot orthoses. Med Eng Phys 2010;32:623–9. [5] Bougherara H, Zdero R, Dubov A, Shah S, Khurshid S, Schemitsch CH. A preliminary biomechanical study of a novel carbon-fibre hip implant versus standard metallic hip implants. Med Eng Phys 2011;33:121–8. [6] Cheung JTM, Zhang M. Parametric design of pressure-relieving foot orthosis using statistics-based finite element method. Med Eng Phys 2008;30:269–77. [7] Omasta M, Paloušek D, Navrát T, Rosický J. Finite element analysis for the evaluation of the structural behaviour of a prosthesis for trans-tibial amputees. Med Eng Phys 2012;34:38–45. [8] Chu TM, Reddy NP, Padovan J. Three-dimensional finite element stress analysis of the polypropylene, ankle-foot orthosis: static analysis. Med Eng Phys 1995;17(5):372–9. [9] Syngellakis S, Arnold AM, Rassoulian H. Assessment of the non-linear behaviour of plastic ankle foot orthoses by the finite element method. Proc Inst Mech Eng 2000;214(5):527–39.

B. Stier et al. / Medical Engineering and Physics 37 (2015) 505–511 [10] Ladevèze P, Nouy A. On a multiscale computational strategy with time and space homogenization for structural mechanics. Comput Methods Appl Mech Eng 2003;192:3061–87. [11] Kouznetsova VG, Geers MGD, Brekelmans WAM. Multiscale second order computational homogenization of multi-phase materials: A nested finite element strategy. Comput Methods Appl Mech Eng 2004;193:5525–50. [12] Violeau D, Ladevèze P, Lubineau G. Micromodel-based simulations for laminated composites. Compos Sci Technol 2009;69(9):1364–71. [13] Zienkiewicz OC, Taylor RL. The finite element method for solid and structural mechanics. 6th ed. Amsterdam: Elsevier; 2005. [14] Kanoute P, Boso DP, Chaboche JL, Schrefler BA. Multiscale methods for composites: a review. Arch Comput Methods Eng 2009;16:31–75. [15] Klusemann B, Svendsen B. Homogenization modeling of thin-layer-type microstructure. Int J Solids Struct 2012;49(13):1828–38. [16] Klinge S, Hackl K. Application of the multiscale fem to the modeling of nonlinear composites with a random microstructure. Int J Multiscale Com 2012;10(3): 213–27. [17] Holzapfel G, Eberlein R, Wriggers P, Weizsäcker H. A new axisymmetrical membrane element for anisotropic, finite strain analysis of arteries. Commun Numer Methods Eng 1996;12:507–17. [18] Weiss J, Maker B, Govindjee S. Finite element implementation of incompressible, transversely isotropic hyperelasticity. Comput Methods Appl Mech Eng 1996;135:107–28. [19] Rüter M, Stein E. Analysis, finite element computation and error estimation in transversal isotropic nearly incompressible finite elasticity. Comput Methods Appl Mech Eng 2000;190:519–41.

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[20] Reese S. Meso-macro modelling of fiber-reinforced rubber-like composites exhibiting large elastoplastic deformation. Int J Solids Struct 2003;40:951–80. [21] Stier B, Simon JW, Reese S. Finite element analysis of layered fiber composite structures accounting for the material’s micro-structure and delamination. Appl Compos Mater. 2014. DOI: 10.1007/s10443-013-9378-8. [22] Simon JW, Stier B, Reese S. Numerical analysis of layered fiber composites accounting for the onset of delamination. Adv Eng Soft 2014; DOI: 10.1016/j.advengsoft.2014.09.022. [23] Ansar M, Xinwei W, Chouwei Z. Modeling strategies of 3d woven composites: a review. Compos Struct 2011;93(8):1947–63. [24] Lomov SV, Ivanov DS, Verpoest I, Zako M, Kurashiki T, Nakai H, et al. Full-field strain measurements for validation of meso-fe analysis of textile composites. Compos Part A 2008;39(8):1218–31. [25] Toledo M, Nallim L, Luccioni B. A micro-macromechanical approach for composite laminates. Mech Mater 2008;40(11):885–906. [26] Soden PD, Hinton MJ, Kaddour AS. Lamina properties, lay-up configurations and loading conditions for a range of fibre-reinforced composite laminates. Compos Sci Technol 1998;58:1011–22. [27] Schürmann H. Konstruieren mit Faser-Kunststoff-Verbunden. 2nd ed. Berlin: Springer; 2007. [28] Takahashi KZ, Stanhope SJ. Estimates of stiffness for ankle-foot orthoses are sensitive to loading conditions. J Prosthet Orthot 2010;22(4):211–19. [29] Jamshidi N, Hanife H, Rostami M, Najarian S, Menhaj MB, Saadatnia M, et al. Modelling the interaction of ankle-foot orthosis and foot by finite element methods to design an optimized sole in steppage gait. J Med Eng Technol 2010;34(2): 116–23.