Fiber orientation interpolation for the multiscale analysis of short fiber reinforced composite parts
 688 Downloads
Abstract
For short fiber reinforced plastic parts the local fiber orientation has a strong influence on the mechanical properties. To enable multiscale computations using surrogate models we advocate a twostep identification strategy. Firstly, for a number of sample orientations an effective model is derived by numerical methods available in the literature. Secondly, to cover a general orientation state, these effective models are interpolated. In this article we develop a novel and effective strategy to carry out this interpolation. Firstly, taking into account symmetry arguments, we reduce the fiber orientation phase space to a triangle in \({\mathbb {R}}^2\) . For an associated triangulation of this triangle we furnish each node with an surrogate model. Then, we use linear interpolation on the fiber orientation triangle to equip each fiber orientation state with an effective stress. The proposed approach is quite general, and works for any physically nonlinear constitutive law on the microscale, as long as surrogate models for single fiber orientation states can be extracted. To demonstrate the capabilities of our scheme we study the viscoelastic creep behavior of short glass fiber reinforced PA66, and use Schapery’s collocation method together with FFTbased computational homogenization to derive single orientation state effective models. We discuss the efficient implementation of our method, and present results of a component scale computation on a benchmark component by using ABAQUS ®.
Keywords
Multiscale simulation Viscoelasticity Homogenization Fiber orientation tensor FFT based method1 Introduction
1.1 State of the art
To identify these models, experimental methods are supported by mean field and computational upscaling techniques (cf. Zaoui [61] and Matouš et al. [35] for respective surveys), mathematically formalized by the theory of homogenization [6].
For nonlinear material behavior or large differences in the elastic properties of the constituents, numerical methods resolving the microstructure on representative volume elements (RVEs) for the mechanical behavior under considerations [28] are often the only accurate option, and a variety of numerical approaches specialized to homogenization problems have been developed [2, 5, 40].
Solving twoscale problems with nested volume element problems, known as FE\(^2\) analysis or heterogeneous multiscale method [17], is limited by their formidable size— despite impressive computational results like in Mosby–Matouš [38]. The micromechanical problems under consideration, however, share the same geometry, but involve different macroscopic loading conditions. Thus, a huge amount of very similar problems is solved. Based on this observation a variety of acceleration techniques have been developed, among them reduced order models [9, 55, 59], the (nonuniform) transformation field analysis [16, 20, 36], response surface models [4, 12, 46, 50, 60] and machine learning approaches like neural networks [21, 24, 33, 34, 52] giving rise to accurate surrogate models with significantly lower computational demands.
Surrogate models can than be used in a database concept [49, 51].
1.2 Problem setting
Short fiber reinforced composite parts constitute an example for multiscale problems where the parameters determining the microstructure may vary continuously on the macro scale. For illustration, Fig. 1 shows a quick release buckle socket^{1} for which an injection molding simulation was conducted (see Sect. 4 for the exact simulation parameters). The colors indicate the occurring fiber orientation (FO) see Fig. 2: cyan corresponds to isotropic, yellow to planar and magenta to a unidirectional, i.e. an aligned, fiber orientation state. We see that large parts of the socket exhibit a rather aligned orientation. Yet, there are distinguished parts, like the flat stern where the orientation is mostly planar. Furthermore, there are regions where the melt fronts have collided (welding lines). Capturing the mechanical behavior of these regions is particularly important, as they correspond to weak spots of the structure.
The finite element mesh, cf. Fig. 1a, consists of more than 300,000 elements. This large number of elements is necessary for resolving the component. Notice that the fiber orientation is varying continuously, and large parts of the component are characterized by very similar fiber orientation states. Thus, it appears natural to employ an interpolation ansatz to cover the different orientations. Unfortunately, using representative volume elements (RVEs) for computing the effective mechanical response is, at first sight, incompatible with an interpolation scheme: the volume element does not depend continuously on the fiber orientation! The situation is even more delicate: one and the same fiber orientation state might be represented by two completely different volume elements. Thus, the association \(\{\hbox {FO}\}\mapsto \{\hbox {RVE}\}\) is not even a welldefined function. Similar problems arise if surrogate models of different type, in particular with different internal variables, are used for different orientations. This situation is characteristic for models obtained by model order reduction, for instance.
In this work, we propose a workaround for these problems, based on the insight that the effective stress is a welldefined function of the fiber orientation (and the applied load history).
1.3 Our contribution and organization of this article
Within the framework of generalized standard materials [23, 25] we propose a method to interpolate arbitrary effective models. Suppose we have a collection of microstructures, indexed by \(\left\{ 1,\ldots , M \right\} \). For every microstructure whose constituents are generalized standard materials, the effective mechanical behavior can be described by a generalized standard material, see Suquet [48]. Thus, we obtain a free energy \(W_i\) and a dissipation potential \(\varPhi _i\) for each index i. Notice, however, that \(\varPhi _i\) depends also on an internal variable vector \(Q_i\), which lives in some space \(X_i\).
In the context of short fiber reinforced composites we apply the proposed method to the interpolation of fiber reinforced structures described by the fiber orientation tensor of second order, see Sect. 2, corresponding—if objectivity is accounted for—to a twodimensional configuration space.
To study the creeping behavior of short fiber reinforced composites, we equip the matrix with a Burgers model for PA66 [57, 58], use FFTbased computational homogenization [39, 40] to carry out the microstructure computations and identify a surrogate model by Schapery’s collocation method [43], see Sect. 3. Last but not least we check the accuracy of the fiber orientation interpolation and investigate the behavior of the quick release buckle socket of Fig. 1, see Sect. 4. Care has been taken to ensure applicability of the scheme in state of the art engineering computations. For the micromechanical simulation, we use the Fraunhofer ITWM software FeelMath [18], distributed as the ElastoDict module of GeoDict [22]. The database containing the different effective models is filled by a script written in Python [42], and can be accessed by a userdefined material function, which is compatible with the UMAT interface of ABAQUS [1].
2 Fiber orientation interpolation
2.1 The fiber orientation tensor
2.2 The framework of generalized standard materials
For the constitutive modelling we rely upon the twopotential framework of the generalized standard materials [23, 25]. The constitutive relationships are derived from two thermodynamic potentials, the Helmholtz free energy density \(w({{\mathrm{\varepsilon }}},q)\) and the dissipation potential \(\phi ({\dot{q}})\) which are convex functions of the state variables, the infinitesimal strain \({{\mathrm{\varepsilon }}}\) and other internal variables q, and their timederivative. The free energy w is the energy available in the system to trigger its evolution, whereas \(\phi \) describes the evolution of the irreversible mechanisms.
2.3 Effective properties of short fiber reinforced plastics and the invariance principle
Thus, to derive effective models for all fiber orientation states \(A\) it is sufficient to consider the fiber orientations \(\varLambda =\text {diag}(\lambda _1,\lambda _2,1\lambda _1\lambda _2)\) with \(\lambda _1\), \(\lambda _2\) lying in the triangle (3). Effective constitutive laws for general orientation tensors are obtained by subsequent orthogonal transformations.
2.4 Fiber orientation interpolation
Thus, the nodal stresses are interpolated, and the nodal internal variables evolve independently of each other. Of course, if \(s_i=0\) for some \(i=1,2,3\), the evolution for \(Q_{\varLambda _i}\) becomes irrelevant for the evaluation of the stress. In particular, if \(\varLambda =\varLambda _i\) for some \(i=1,2,3\), we consistently recover the constitutive equations for \(\varLambda _i\).
In this formulation, three sets of internal variables are stored. Each node receives the same strain, but uses its own constitutive law to produce a stress, and the evolution of the internal variables depends on the node only. The three stresses are averaged according to the convex representation (12), which is meaningful, see Fig. 5.
We have presented the method for a single triangle of the triangulation only. However, the method can be interpreted for the whole triangulation in a similar fashion, where only the adjacent nodal constitutive laws are active. Notice at this point that for every orientation only three material laws are required, regardless of the number of nodes used for the triangulation of the fiber orientation triangle. This property contrasts starkly with the method of pseudograins [13, 15], which requires the full number of grains to be evaluated for calculating the effective response of the composite.
 1.
Offline phase: triangulate the fiber orientation triangle, cf. Fig. 3, and compute an effective model for each node in the triangulation
 2.Online phase: for each macroscopic Gauss point and given strain E

Spectrally decompose the local fiber orientation tensor \(A=R^T\varLambda R\), cf. (2)

Determine the local triangle T such that \(\varLambda \in T=\text {conv}\{ \varLambda _1, \varLambda _2, \varLambda _3 \}\) with nodes \(\varLambda _1,\varLambda _2,\varLambda _3\) and weights \(s_1,s_2,s_3\), cf. (12)

Transform the strain \(E\mapsto RER^T\), compute the local stress \(\varSigma _i\) and update the internal variable \(Q_{\varLambda _i}\) at node \(\varLambda _i\)

Return the effective stress \(\varSigma =R^T\left[ \sum _{i=1}^3 s_i \varSigma _i\right] R\)

Return the effective consistent tangent matrix \(\mathbb {C}= \mathbb {R}^{1}:\left[ \sum _{i=1}^{3} s_i \mathbb {C}_i\right] : \mathbb {R},\) where \(\mathbb {C}_i\) is the consistent tangent at \(\varSigma _i\) in the transformed orientation, \(\mathbb {R}\) is the action of R on twotensors \(\mathbb {R}_{ijkl}=R_{ik}R_{jl}\), and \(\mathbb {R}^{1}_{ijkl}=R_{ki}R_{lj}\)

For the online ABAQUS computation a dummy UMAT is used, which takes Euler angles, weights and the fiber orientation element as input parameters, and calls the three UMATs associated to the adjacent nodes of the fiber orientation triangle.
3 Identifying a viscoelastic surrogate model using FFTbased computational homogenization and Schapery’s collocation method
To test the orientation interpolation technique we consider a fiber reinforced polymer with linearly viscoelastic matrix and linear elastic reinforcements. Linear viscoelasticity has the advantage that on the one hand the derivation of effective models is comparatively wellunderstood, but on the other hand due to the dependence on the material history complex constitutive behavior can be observed.
3.1 Generating fiberfilled volume elements
To generate periodic volume elements with prescribed fiber orientation and volume fraction we rely upon the Sequential Addition and Migration (SAM) method [44]. As input parameters, the fiber volume fraction \(\phi \), the second order fiber orientation tensor A, the fiber length L, the fiber diameter D and the edge length \(L_x(=L_y=L_z)\) of the cubical volume element needs to be specified. In a preliminary step, the required number N of fibers is calculated, s.t. the fiber volume fraction \(\phi \) is matched with highest precision. Furthermore, the fourth order fiber orientation tensor \(\mathbb {A}\) is computed from the A tensor with the help of the exact closure approximation [37].
In the first step of SAM, N overlapping cylindrical fibers are randomly placed in the volume. Then, in the second step, the fibers are displaced and rotated, s.t. the overlap is removed and the prescribed fiber orientation tensor \(\mathbb {A}\) is matched to prescribed accuracy (typically \(10^{5}\) absolute error in \(L^2\) of the Voigt matrices).
For the study at hand, we force \(\lambda _3\) to be at least 0.01. These orientation tensors are sufficiently close to planar for practical studies, and can easily be generated by the SAM method.
3.2 The Burgers model for the viscoelastic behavior of the matrix
As matrix material we consider a commercial polyamide 6.6 (PA66, DuPont Zytel 101), for which experimental creep curves and fitted viscoelastic model parameters are available in the literature, see Yang et al. [57, 58].
Viscoelastic matrix material parameters [58] for Burgers’ model
\(E_M\) [MPa]  \(E_K\) [MPa]  \(\eta _M\) [GPa h]  \(\tau \) [h] 

3709  16.617  889  14.2 
3.3 Computational homogenization of linear viscoelasticity
 balance of linear momentum$$\begin{aligned} \text {div }\sigma (t)=0, \end{aligned}$$(18)
 compatibility$$\begin{aligned} {{\mathrm{\varepsilon }}}(x,t)=E(t)+\nabla ^s u(x,t) \quad \text {for every }x \in Y, \end{aligned}$$(19)
 linear viscoelastic material law in relaxation function form$$\begin{aligned} {{\mathrm{\varepsilon }}}(x,t)=\int _0^t \mathbb {J}(x,ts):{\dot{\sigma }}(x,s)\, {d}s\quad \text {for every} x\in Y,\nonumber \\ \end{aligned}$$(20)
 fixed macroscopic stress$$\begin{aligned} \langle \sigma (t) \rangle _Y = \varSigma (t). \end{aligned}$$(21)
FFTbased computational homogenization [39, 40] constitutes our method of choice. For this method, a discretization on a regular grid or mesh decomposing Y is chosen. Due to the regularity of the grid and the periodic boundary conditions, problems of the form (23) with homogeneous \(\mathbb {D}^{tan}\) can be solved directly in terms of the discrete Fourier transform. The Operator of this auxiliary homogeneous problem serves to precondition the linear system (23).
For the problem at hand, we choose a discretization on a staggered grid [45], the stressbased formulation of Bhattacharya–Suquet [11] and the conjugate gradient method [62] for the resolution of (23). All mentioned methods are integrated into the Fraunhofer ITWM C++ code FeelMath [18].
3.4 On the resolution and the RVE size necessary for the precomputations
In this section we investigate the necessary resolution and representative volume element (RVE) size for solving (18)–(21), or, equivalently, (23), to engineering accuracy.
 Case 1

The instantaneous response, corresponding to \(t=0\), i.e. (24) becomes simply the linear elastic relationship \({{\mathrm{\varepsilon }}}=\mathbb {D}_0:\sigma _0\)
 Case 2
 The creep rate at infinity. Differentiating (24) in time yieldsi.e., \(\lim _{t\rightarrow \infty } {\dot{{{\mathrm{\varepsilon }}}}}(t)=\mathbb {F}:\sigma _0\), whereas \({\dot{{{\mathrm{\varepsilon }}}}}=0\) in the fibers. To study the creep rate at infinity, we study the linear elastic problem with local compliance$$\begin{aligned} {\dot{{{\mathrm{\varepsilon }}}}}(t)=\mathbb {F}:\sigma _0 + \tau ^{1} e^{t/\tau } \mathbb {D}_1:\sigma _0, \end{aligned}$$(26)As the fibers are rigid, this problem is much more difficult to solve than the instantaneous elastic problem.$$\begin{aligned} \mathbb {D}(x)=\left\{ \begin{array}{ll} \mathbb {F}, &{}\quad x \text { in the matrix,}\\ 0, &{}\quad \text {in the fiber.} \end{array} \right. \end{aligned}$$
We have chosen to check four different resolutions, measured in voxels per fiber diameter. Recall that the fibers have a diameter of 10 µm, so that 5 voxels per diameter correspond to a voxel edge length of 2 µm, i.e. the (600 µm)\(^3\) volume is discretized by \(300^3\) elements. Accordingly, 10, 15 and 20 voxels per diameter correspond to \(600^3\), \(900^3\) and \(1200^3\) elements, respectively.
For each of these resolutions, the homogenized instantaneous elastic tensor \(\mathbb {C}^{hom}\) was computed (Case 1). The relative error in the Frobenius norm of the corresponding Voigt matrices, measured relative to the highest resolution \(1200^3\), is shown in Fig. 7a. Even for the crudest resolution, the relative error for all three considered orientations was below \(1\%\).
The same computation was carried out for the longterm, i.e. viscous, response (Case 2). Upon replacing the strain rate \({\dot{{{\mathrm{\varepsilon }}}}}\) by the strain \({{\mathrm{\varepsilon }}}\) in (26) the problem is formally equivalent to an elasticity problem with rigid fibers and elastic matrix. The computed homogenized elastic tensor for the equivalent elastic problem is in fact a homogenized viscous tensor \(\mathbb {V}^{hom}\), and the relative errors of the results are plotted in Fig. 7b, again relative to the highest resolution.
For the crudest resolution, the relative errors exceed \(10\%\) for all three considered orientations. The highest error of about \(19.1\%\) occurs for the planar fiber orientation state, and the lowest error (\(11.7\%\)) is reached for the isotropic orientation. Doubling the resolution decreases the error for all three orientations below \(2.5\%\). For \(900^3\), the relative error is even smaller than \(1\%\).
Number of fibers for different volume element sizes
Edge length  \(2 \times L\)  \(3 \times L\)  \(4 \times L\)  \(5 \times L\)  \(6 \times L\) 

Number of fibers  686  2298  5447  10,640  18,386 
Average number of iterations per load case to compute effective tensors for different volume element sizes
Volume element size  

Orientation  \(2 \times L\)  \(3 \times L\)  \(4 \times L\)  \(5 \times L\) 
(a) Stiffness  
Isotropic  25.0  22.2  23.5  24.2 
Unidirectional  19.5  19.5  19.5  19.5 
Planar isotropic  19.7  19.8  21.2  20.8 
(b) Viscosity  
Isotropic  498.3  495.2  492.2  490.0 
Unidirectional  461.3  454.8  451.2  450.7 
Planar isotropic  494.8  486.3  486.5  485.0 
3.5 Schapery’s collocation method
3.6 On the number of collocation points for Schapery’s method
In this section we study the applicability of Schapery’s method in our context.
With the previously established parameters, an RVE edge length of three times the fiber length and a resolution of 1 µm, we have computed solutions to the Schapery problems (30) for a different number of collocation points, ranging from 1 to 7, and the basis \(b=10\).
For the isotropic fiber orientation state, see Fig. 10a, a single collocation point starts to strongly deviate from the reference curve at about 40 h, whereas 3 collocation points start to deviate at about 120 h. 5 collocation points slightly overestimate the creep strain, whereas the stress–strain curve for 7 collocation points is hard to distinguish from the reference solution.
The behavior for the unidirectional and the planar isotropic geometries are similar to the isotropic case.
To gain further insight into the quality of the Schapery approximation we consider the maximal relative error in the strain for the different collocation methods, computed relative to the fullfield solution. Here, maximal means the maximum error up to \(10^4\) h of applied stress. Figure 11 shows that there are strong differences in the error, depending on the direction of applied stress and the microstructure. The relative error is smallest for the unidirectional structure and tension in transverse direction, remaining below \(1\%\) even for a single collocation point. This is not unexpected, as this loading scenarios is dominated by the matrix response, and the matrix is described by Burgers’ model, which features only a single collocation point. The situation in transverse direction strongly contrasts to extension in fiber direction for the unidirectional structure. For up to 3 collocation points, the relative error is larger than \(10\%\), but decreases quickly for increased number of collocation points, and remains below \(1\%\) for seven collocation points. The behavior of the planar isotropic structure, loaded transversely, is somewhat similar to the unidirectional structure loaded in fiber direction. Also, there are similarities between the isotropic fiber orientation and the planar isotropic microstructure loaded in fiber direction: for these scenarios, the errors are largest, and decrease rather slowly. A relative error below \(1\%\) is reached for at least 9 collocation points. All in all we see that a relative error below \(1\%\) is guaranteed for more than 9 collocation points.
As a side remark notice that the error in Fig. 11 increases for more than 13 collocation points. This is no mistake, as these frequencies have little impact for creep up to \(10^4\) h, but change the optimum in the minimization (29) which covers also much larger times \(t\in [0,\infty )\). To further increase the accuracy, the base b would have to be changed from its current value 10. However, for the problem at hand, we consider the accuracy to be sufficient.
Last but not least we take a look at the computational costs of the simulations, which were carried out on 64 MPI compute nodes, equipped with a dual Intel Xeon E52670 and 64 GB RAM, on our cluster. The runtimes are plotted in Fig. 12 as a function of the frequency \(\tau b^k\) with \(\tau =14.2\) h and varying integers k, see Table 1.
For \(k=0\) a runtime of 3.08 min is required. Then, as k decreases, the time slightly increases up to the elastic case “at infinity” requiring 3.5 min. For increasing k, the runtimes also increase compared to \(k=0\), at about 4 min for \(k=1\) and about 6 min for \(k=2\). For \(k=4\), already about 43 min are necessary for resolving the unit cell problem. Thus, the viscous computations are the most timedemanding, confirming the findings of Sect. 3.4.
3.7 Discussion of the results and errors
The database was filled with the previously determined parameters, i.e. an RVE length of thrice the fiber length, a resolution of 1 µm and 9 collocation points for Schapery’s method, and a regular triangulation of the fiber orientation triangle with 15 nodes, as shown in Fig. 3. The entire computation took about 15 h on 64 MPI compute nodes, equipped with a dual Intel Xeon E52670 and 64 GB RAM, on our cluster.
As the prescribed fiber orientation tensors of fourth order are orthotropic, the effective tensors \(\mathbb {K}_l\), and thus, the tensors \(\mathbb {D}^{hom}\), \(\mathbb {B}_k^{hom}\) and \(\mathbb {F}^{hom}\) from (28) are orthotropic as well, see Schneider [44]. Hence it is reasonable to investigate the corresponding orthotropic engineering constants.
First, in Fig. 13, we take a look at the orthotropic Young’s moduli of the instantaneous elastic response, i.e. those which can be read from \(\mathbb {D}^{hom}\). Figure 13a shows the three Young’s moduli \(E_1\), \(E_2\) and \(E_3\) as graphs plotted over the fiber orientation triangle, whereas Fig. 13b restricts the view to the edges of the fiber orientation triangle. The distances between ud (unidirectional), isotropic (iso) and planar isotropic (piso) on the xaxis of Fig. 13 correspond to the lengths of the edges connecting the corners of the fiber orientation triangle. We see that the graphs are smooth, but nonlinear, and two of the three Young’s moduli coincide on the edges connecting udiso and isopiso, as they correspond to transversely isotropic fiber orientations. The contrasts between the Young’s moduli is largest at the ud state (by about a factor of 2), much smaller at the piso state (about \(\tfrac{4}{3}\)), and of course, minimal for the isotropic state.
For comparison, the “viscous” Young’s moduli, i.e. those which can be read from \(\mathbb {F}^{hom}\), are plotted in Fig. 14. The qualitative behavior of the graphs is similar to the instantaneous Young’s moduli. However, the contrasts are much larger. Indeed, for the ud case, the Young’s modulus in fiber direction \(E_1^v\) and the transverse Young’s modulus \(E_2^v=E_3^v\) differ by a factor of roughly 18, and the contrast for the piso case computes as about 5. Thus, the creeping behavior of a composite exhibits a much higher anisotropy than the elastic response.
Next we investigate the instantaneous shear moduli \(G_{12}\), \(G_{13}\) and \(G_{23}\) in Fig. 15. In this case, the differences between the different shear moduli is largest for the piso structure, with a contrast of about 1.5. Overall, the graphs appear to be at least quadratic.
Last but not least we take a look at the “viscous” shear moduli, which, as for the Young’s moduli, exhibit qualitative similarities to their instantaneous counterparts, but differ strongly quantitatively. Indeed, the contrast between the shear moduli \(G_{12}^v\) and \(G_{13}^v=G_{23}^v\) at the piso state is about 5 (Fig. 16).
We have shown piecewise linear approximations of the orthotropic constants, considered as functions on the fiber orientation triangle. It remains to assess the quality of the approximation. For this purpose we have repeated the computation of effective elastic and viscous tensors \(\mathbb {D}^{hom}\) and \(\mathbb {F}^{hom}\) for the fiber orientations corresponding to the centers of mass of the triangles of our chosen triangulation and compared those tensors to their counterparts arising by interpolation. We investigate the relative error of the orthotropic elastic constants of the instantaneous elastic response in Fig. 17. The overall level of the relative error is rather small, bounded above by \(0.7\%\).
For the orthotropic constants of the viscous tensor \(\mathbb {F}^{hom}\) the relative errors arising from interpolation are depicted in Fig. 18. The largest error of about \(6\%\) occurs for Poisson’s ratio \(\nu _{31}\). For the Young’s moduli \(E_1^v\), \(E_2^v\) and \(E_3^v\) the relative error is smaller than \(2\%\) except for the tip of the fiber orientation triangle and \(E_2^v\), corresponding to the vicinity of the ud orientation, where the relative error rises to about \(4\%\). This results from the strong contrast between the in fiber Young’s modulus and the transverse Young’s moduli in the viscous case, see Fig. 15b. The relative errors for the shear moduli are bounded by \(4\%\).
All in all, the interpolation scheme is very accurate, except for some extreme cases (the Poisson’s ratio in the viscous case in the vicinity of the ud state). Still, we consider the maximum error of about \(6\%\) acceptable since in most other cases, the relative errors are well below \(5\%\) relative error and thus within engineering accuracy.
4 A computational example
To prove our fiber orientation interpolation concept we investigate the viscoelastic creep behavior of a quick release buckle, whose CAD geometry is publicly available,^{2} see Fig. 1a
Material parameters used for the injection molding simulation
Parameter  Value 

Density  \(1410\,\text {kg}/\text {m}^{3}\) 
Injection temperature  \(275\,^\circ \text {C}\) 
Mold temperature  \(40\,^\circ \text {C}\) 
Specific heat  \(2400\,\text {J}/\text {K}\) 
Thermal conductivity  \(0.25\,W/(\text {m}\,\text {K})\) 
Initial orientation  Isotropic 
Fiber aspect ratio  \(r_a=20\) 
Folgar–Tucker coefficient  \(C_i=0.01\) 
Particle number  \(N_{p}=0\) 
Glass transition temperature  \(T_{ref}=230\,^\circ \text {C}\) 
\(A_0\)  \(0.1\,\text {s}\) 
\(A_1\)  0.65 
\(A_2\)  \(0.021\,1/\text {K}\) 
\(\eta _0\)  \(100\,\text {Pa}\,\text {s}\) 
The calculated fiber orientation tensors serve as input for a component analysis using the commercial finite element software ABAQUS [1]. The mesh consists of approximately 310.000 linear tetrahedral elements (C3D4), see Fig. 20. A viscoelastic creep test up to 200 h was computed with a constant surface force 8N applied as shown in Fig. 20, taking into account the identified viscoelastic database identified in Sect. 3. The computation comprised 30 time increments, spaced equidistantly on a logarithmic scale, taking about 2 h computation time on a desktop PC (single core). To assess the influence of the fiber orientation, for comparison, the computation was repeated for the same component, but where the fiber orientation was assumed isotropic throughout the whole geometry.
In Fig. 21 the strain in xdirection (cf. Fig. 20 for the axes) is shown, comparing the constant isotropic to the distributed fiber orientations. For the initial elastic response (\(t=0\)), the strain distributions are even qualitatively similar. However, after 200 h strong differences in the strain fields can be observed. Indeed, for the isotropic fiber orientation spots with much higher strains are visible (in yellow and orange).
To further investigate this effect we show the corresponding von Mises equivalent stress in Fig. 22, and investigate the local relaxation behavior. For the constant isotropic component, the von Mises stress apparently changes only slightly during the loading history. For the distributed orientation, however, there is a load rebalancing comparing the stress fields at \(t=0\) h and \(t=200\) h.
To gain deeper insight, we investigated the von Mises stress history up to 200 h in the most heavily loaded element for the distributed orientation, cf. Fig. 23. At the beginning of the loading, the stress level lies slightly above 25.5 MPa. After 200 h, the stress level increased to about 28 MPa, i.e. by about \(10\%\).
We see that the constant isotropic computing overestimates the occurring local strains and does not account for the strongly anisotropic relaxation behavior of the component under consideration. In particular, an anisotropic component design necessitates larger safety factors than the computations with anisotropic relaxation behavior.
5 Conclusion
This work concerned the multiscale simulation of short fiber reinforced thermoplastics. Even though the fibers are much smaller than the typical component scale, the mechanical properties on the component scale are not homogeneous, but depend on the local fiber orientation. Thus, working with a single surrogate model for a fixed fiber orientation is insufficient, and the need to interpolate surrogate models corresponding to different fiber orientations arises.
We have proposed a versatile and robust approach to interpolate surrogate models, rigorously derived from the twopotentialframework [23, 25]. The advantage of the approach is the ability to interpolate models of completely different type, in particular in terms of internal variables.
To test the framework on a nontrivial example, we investigated the viscoelastic creep behavior of PA66, reinforced by short glass fibers, and used Moulinec–Suquet’s FFTbased computational homogenization scheme and Schapery’s collocation method to identify anisotropic viscoelastic surrogate models for different fiber orientation states. It is shown that even for a rather rough triangulation of the fiber orientation triangle the interpolation errors are small.
Finally, the full identified viscoelastic model was used as a material model for an ABAQUS simulation of a benchmark component, exhibiting an anisotropyinduced strain rearrangement on the component scale.
This work contributes to bridging the gap between the insights provided by multiscale modelling of fiber reinforced composites achieved in recent years and fast computational analyses carried out by the constructing engineer. Indeed, even though the filling of the database is computationally demanding, the resulting material card enables extremely fast computations and is universal in the sense that it can be (re)used for different components built from the same material.
As already mentioned, the proposed fiber orientation interpolation scheme is not restricted to viscoelastic material behavior. The availability of surrogate models for different fiber orientations is the only prerequisite. In particular, viscoplastic or damage effects could be investigated. Furthermore, the interpolation scheme is not limited to problems at small strains, but extends effortlessly to the finite strain framework. As the interpolation is based on the free energies and the dissipation potentials, the question which strains and stresses should be interpolated does not arise but follows directly from the energetic formulation. Also, the scheme could be extended to incorporate local variations in fiber volume fraction.
Footnotes
 1.
Source for the CAD geometry: https://grabcad.com/library/quickreleasebuckle19mm.
 2.
Notes
Acknowledgements
We are grateful to Matthias Kabel, Sarah Staub, Ralf Müller and Fabian Welschinger for fruitful discussions. Felix Ospald gratefully acknowledges financial support by the German Research Foundation (DFG), Federal Cluster of Excellence EXC 1075 “MERGE Technologies for Multifunctional Lightweight Structures”.
References
 1.Abaqus (2014) Abaqus 6.14: analysis user’s manual. Dassault Systemes Simulia Corp, ProvidenceGoogle Scholar
 2.Aboudi J (2004) The generalized method of cells and highfidelity generalized method of cells micromechanical models—a review. Mech Adv Mater Struct 11(4–5):329–366CrossRefGoogle Scholar
 3.Advani SG, Tucker CL (1987) The use of tensors to describe and predict fiber orientation in short fiber composites. J Rheol 31(8):751–784CrossRefGoogle Scholar
 4.Andrä H, Kabel M, Staub S, Krizikalla F, Schulz V (2012) Numerische Homogenisierung für viskoelastische Faserverbundwerkstoffe. NAFEMS OnlineMag 21:70–83Google Scholar
 5.Bakhvalov NS, Knyazev AV (1991) Efficient computation of averaged characteristics of composites of a periodic structure of essentially different materials. Sov Math Dokl 42(1):57–62zbMATHGoogle Scholar
 6.Bakhvalov NS, Panasenko GP (1989) Homogenization: averaging processes in periodic media. Kluwer, DordrechtCrossRefzbMATHGoogle Scholar
 7.Bella P, Otto F (2014) Corrector estimates for elliptic systems with random periodic coefficients. Multiscale Model Simul 14(4):1434–1462MathSciNetCrossRefzbMATHGoogle Scholar
 8.Bhat HA, Subramaniam S, Pillai AK, L EK Elangovan M (2014) Analysis and design of mold for plastic side release Buckle using Moldflow software. Int J Res Eng Technol 3(5):366–372CrossRefGoogle Scholar
 9.Bhattacharjee S, Matouš K (2016) A nonlinear manifoldbased reduced order model for multiscale analysis of heterogeneous hyperelastic materials. J Comput Phys 313:635–653MathSciNetCrossRefzbMATHGoogle Scholar
 10.Bhattacharya K, Suquet P (2005) A model problem concerning recoverable strains of shapememory polycrystals. Proc R Soc Lond A Math Phys Eng Sci 461(2061):2797–2816MathSciNetCrossRefzbMATHGoogle Scholar
 11.Bhattacharya K, Suquet PM (2005) A model problem concerning recoverable strains of shapememory polycrystals. Proc R Soc Lond A Math Phys Eng Sci 461(2061):2797–2816MathSciNetCrossRefzbMATHGoogle Scholar
 12.Box G, Draper N (2007) Response surfaces, mixtures and ridge analyses, 2nd edn. Wiley, HobokenCrossRefzbMATHGoogle Scholar
 13.Camacho C, Tucker CL, Yalvac S, McGee R (1990) Stiffness and thermal expansion predictions for hybrid short fiber composites. Polym Compos 11:229–239CrossRefGoogle Scholar
 14.Chung DH, Kwon TH (2001) Improved model of orthotropic closure approximation for flow induced fiber orientation. Polym Compos 22(5):636–649CrossRefGoogle Scholar
 15.Doghri I, Tinel L (2005) Micromechanical modeling and computation of elastoplastic materials reinforced with distributedorientation fibers. Int J Plast 21:1919–1940CrossRefzbMATHGoogle Scholar
 16.Dvorak G (1992) Transformation field analysis of inelastic composite materials. Proc R Soc Lond A 4317:311–327MathSciNetCrossRefzbMATHGoogle Scholar
 17.Engqvist EW, Lao B, Ren X, VandenEijnden E W (2007) Heterogeneous multiscale methods: a review. Commun Comput Phys 2:367–450MathSciNetGoogle Scholar
 18.FeelMath (2017) FeelMath. Fraunhofer Institute for Industrial Mathematics. http://www.itwm.fraunhofer.de/en/fraunhoferitwm.html
 19.Findley W, Lai J, Onaran K (1989) Creep and relaxation of nonlinear viscoelastic materials: with an introduction to linear viscoelasticity. Dover, MineolazbMATHGoogle Scholar
 20.Fritzen F, Leuschner M (2013) Reduced basis hybrid computational homogenization based on a mixed incremental formulation. Comput Methods Appl Mech Eng 260:143–154MathSciNetCrossRefzbMATHGoogle Scholar
 21.Furukawa T, Yagawa G (1998) Implicit constitutive modelling for viscoplasticity using neural networks. Int J Numer Methods Eng 43(2):195–219CrossRefzbMATHGoogle Scholar
 22.GeoDict (2017) GeoDict. Math2Market GmbH, Kaiserslautern, Germany. http://www.geodict.de. Accessed on 4 Jan 2017
 23.Germain P, Nguyen QS, Suquet P (1983) Continuum thermodynamics. J Appl Mech 50:1010–1020CrossRefzbMATHGoogle Scholar
 24.HajAli R, Pecknold DA, Ghaboussi J, Voyiadjis GZ (2001) Simulated micromechanical models using artificial neural networks. J Eng Mech 127:730–738CrossRefGoogle Scholar
 25.Halphen B, Nguyen QS (1975) Sur les matériaux standard généralisés. J Méc 14:39–63zbMATHGoogle Scholar
 26.Hashin Z (1965) Viscoelastic behavior of heterogeneous media. J Appl Mech Trans ASME 32:630–636CrossRefGoogle Scholar
 27.Hashin Z (1970) Complex modulis of viscoelastic composites—I. General theory and application to particulate composites. Int J Solids Struct 6:539–552CrossRefzbMATHGoogle Scholar
 28.Hill R (1963) Elastic properties of reinforced solids: some theoretical principles. J Mech Phys Solids 11:357–372CrossRefzbMATHGoogle Scholar
 29.Jack DA, Smith DE (2005) An invariant based fitted closure of the sixthorder orientation tensor for modeling shortfiber suspensions. J Rheol 49(5):1091–1115CrossRefGoogle Scholar
 30.Janovsky V, Shaw MKWS, Whiteman JR (1995) Numerical methods for treating problems of viscoelastic isotropic solid deformation. J Comput Appl Math 63(1–3):91–107MathSciNetCrossRefzbMATHGoogle Scholar
 31.Kennedy PK (2013) Flow analysis of injection molds, 2nd edn. Hanser, MunichCrossRefGoogle Scholar
 32.Lai J, Bakker A (1996) 3D Schapery representation for nonlinear viscoelasticity and finite element implementation. Comput Mech 18(3):182–191CrossRefzbMATHGoogle Scholar
 33.Le BA, Yvonnet J, He QC (2015) Computational homogenization of nonlinear elastic materials using neural networks. Int J Numer Methods Eng 104:1061–1084MathSciNetCrossRefzbMATHGoogle Scholar
 34.Lefik M, Schrefler B (2003) Artificial neural network as an incremental nonlinear constitutive model for a finite element code. Comput Methods Appl Mech Eng 192:3265–3283CrossRefzbMATHGoogle Scholar
 35.Matouš K, Geers MGD, Kouznetsova VG, Gillman A (2017) A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials. J Comput Phys 330:192–220MathSciNetCrossRefGoogle Scholar
 36.Michel J, Suquet P (2003) Nonuniform transformation field analysis. Int J Solids Struct 40(25):6937–6955MathSciNetCrossRefzbMATHGoogle Scholar
 37.MontgomerySmith S, He W, Jack DA, Smith DE (2011) Exact tensor closures for the threedimensional Jeffery’s equation. J Fluid Mech 680:321–335MathSciNetCrossRefzbMATHGoogle Scholar
 38.Mosby M, Matouš K (2016) Computational homogenization at extreme scales. Extreme Mech Lett 6:68–74CrossRefGoogle Scholar
 39.Moulinec H, Suquet P (1994) A fast numerical method for computing the linear and nonlinear mechanical properties of composites. Comptes rendus de l’Académie des sciences Série II, Mécanique, physique, chimie, astronomie 318(11):1417–1423zbMATHGoogle Scholar
 40.Moulinec H, Suquet P (1998) A numerical method for computing the overall response of nonlinear composites with complex microstructure. Comput Methods Appl Mech Eng 157:69–94MathSciNetCrossRefzbMATHGoogle Scholar
 41.Ospald F (2014) Numerical simulation of injection molding using OpenFOAM. Proc Appl Math Mech (PAMM) 14(1):673–674CrossRefGoogle Scholar
 42.Python (2017) Python language reference, version 2.7. Python Software Foundation, WilmingtonGoogle Scholar
 43.Schapery RA (1962) Approximate methods of transform inversion in viscoelastic stress analysis. In: Proceedings of the 4th US national congress on applied mechanics, vol 2, pp 1075–1085Google Scholar
 44.Schneider M (2017) The sequential addition and migration method to generate representative volume elements for the homogenization of short fiber reinforced plastics. Comput Mech 59(2):247–263MathSciNetCrossRefzbMATHGoogle Scholar
 45.Schneider M, Ospald F, Kabel M (2016) Computational homogenization on a staggered grid. Int J Numer Methods Eng 105(9):697–720MathSciNetCrossRefGoogle Scholar
 46.Steiner K, für Technound Wirtschaftsmathematik FI (2012) MISESFOK: Multiskalenintegrierende Struktureigenschaftssimulation der Faserorientierung für faserverstärkte Kunststoffe im Automobil und Flugzeugbau: Abschlussbericht zum Forschungsvorhaben 03x0513F im BMBFRahmenprogramm WING : Bearbeitungszeitraum vom 01.03.2007 bis 31.12.2011Google Scholar
 47.Suquet P (1985) Local and global aspects in the mathematical theory of Plasticity. In: Sawchuk A, Bianchi G (eds) Plasticity today: modelling, methods and applications. Elsevier, London, pp 279–310Google Scholar
 48.Suquet P (1987) Elements of homogenization for inelastic solid mechanics. In: SanchezPalencia E, Zaoui A (eds) Homogenization techniques for composite media, vol 272. Lecture notes in physics. Springer, New York, pp 193–278CrossRefGoogle Scholar
 49.Takano N, Zako M, Ohnishi Y (1996) Macromicro uncoupled homogenization procedure for microscopic nonlinear behavior analysis of composites. Mater Sci Res Int 2(2):81–86Google Scholar
 50.Temizer I, Wriggers P (2007) An adaptive method for homogenization in orthotropic nonlinear elasticity. Comput Methods Appl Mech Eng 35–36:3409–3423MathSciNetCrossRefzbMATHGoogle Scholar
 51.Terada K, Kikuchi N (1995) Nonlinear homogenization method for practical applications. ASME Appl Mech Div Publ AMD 212:1–16Google Scholar
 52.Unger JF, Könke C (2008) Coupling of scales in multiscale simulation using neural network. Comput Struct 86:1994–2003CrossRefGoogle Scholar
 53.Ward IM (2013) Mechanical properties of solid polymers, 3rd edn. Wiley, HobokenGoogle Scholar
 54.Weller HG, Tabor HJG, Fureby C (1998) A tensorial approach to computational continuum mechanics using objectoriented techniques. Comput Phys 12(6):620–631CrossRefGoogle Scholar
 55.Wirtz D, Karajan N, Haasdonk B (2015) Surrogate modeling of multiscale models using kernel methods. Int J Numer Methods Eng 101:1–28MathSciNetCrossRefzbMATHGoogle Scholar
 56.Woldekidan M (2011) Response modelling of bitumen, bituminous mastic and mortar. Ph.D. thesis, Delft University of TechnologyGoogle Scholar
 57.Yang JL, Zhang Z, Schlarb AK, Friedrich K (2006) On the characterization of tensile creep resistance of polyamide 66 nanocomposites. Part I. Experimental results and general discussions. Polymer 47:2791–2801CrossRefGoogle Scholar
 58.Yang JL, Zhang Z, Schlarb AK, Friedrich K (2006) On the characterization of tensile creep resistance of polyamide 66 nanocomposites. Part II. Modeling and prediction of longterm performance. Polymer 47:6745–6758CrossRefGoogle Scholar
 59.Yvonnet J, He QC (2007) The reduced model multiscale method (R3M) for the nonlinear homogenization of hyperelastic media at finite strains. J Comput Phys 223:341–368MathSciNetCrossRefzbMATHGoogle Scholar
 60.Yvonnet J, Monteiro E, He Q (2013) Computational homogenization method and reduced database model for hyperelastic heterogeneous structures. Int J Multiscale Comput Eng 11:201–225CrossRefGoogle Scholar
 61.Zaoui A (2002) Continuum micromechanics: survey. J Eng Mech 128:808–816CrossRefGoogle Scholar
 62.Zeman J, Vondřejc J, Novak J, Marek I (2010) Accelerating a FFTbased solver for numerical homogenization of periodic media by conjugate gradients. J Comput Phys 229(21):8065–8071MathSciNetCrossRefzbMATHGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.