# A Level-Set Based Representative Volume Element Generator and XFEM Simulations for Textile and 3D-Reinforced Composites

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## Abstract

**:**

## 1. Introduction

## 2. A RVE Generator for 3D Reinforced Woven Composites

#### 2.1. Principles of the Weave Generator

- An iterative procedure updates the initial situation as explained below in Section 2.2 to mimic the yarns behavior under tension based on geometrical arguments. At the contacting points, the relative tension of the contacting yarns at equilibrium is implicitly controlled by the relative displacements of the contacting points to be imposed in order to enforce a non-overlapping configuration.
- A post-processing of the obtained configuration is used subsequently in order to avoid potential “residual” overlaps or to produce qualitatively and geometrically (i.e., without account of the yarn’s transversal stiffness) noncircular yarn cross sections at contacting regions.

#### 2.2. Iterative Procedure

**Figure 1.**(

**a**) Input data used for the weave generation process of a simple weft-warp yarn system; (

**b**) 3D weaving similar to that reported in [22]; and (

**c**) Fundamental operation (A) for tensioning of the yarns. Black lines represent the position of the yarn centerline prior to the operation (A), while green lines represent its position after operation (A). Grey lines are segments linking the neighbors of each of the vertices. Red dots represent fixed vertices at the extremities of the yarn.

**Figure 2.**(

**a**) Geometry of two equally tensioned contacting yarns; (

**b**) and (

**c**) schematic representation of fundamental operations (A) and (B) treating geometrical contact between two yarns during one iteration and illustration of the gradual movement of the contact point in the non-equilibrium configuration; (

**b**) represents the operation (A); and (

**c**) the operation (B). Both yarns are represented as circles for simplicity even if the real 3D geometry is more complex. Black circles depict the initial position, while green color denotes the final position associated with both operations. Red arrows highlight the resulting displacement after the iteration.

#### 2.3. Accounting Implicitly for Different Tensions

**Figure 3.**(

**a**,

**b**) Schematic representation of operation (A) and (B) respectively, with one yarn twice as much affected by operation (B) as the other, the situation is stationary since operation (A) also produces displacements two time larger for this yarn; and (

**c**) corresponding configuration obtained for two yarns of the same initial length.

#### 2.4. Optimization, Conditions at the Boundaries and Practical Implementation

**Figure 4.**Alignment of the yarns’ centerline description points (vertices) between contacting points to increase the efficiency of the iterative process (red dots in this figure denote two successive contact points along a yarn).

- Vertices located on the RVE boundary are constrained to remain on that planar boundary (planar constraint);
- Yarns are looped at opposite boundaries of the RVE, i.e., the first and last vertices of a yarn will use the one but last and second vertices respectively to perform operation (A);
- Finally, for the detection of contacts and for the post-processing step (see Section 2.5), distances of points with respect to the yarns’ surfaces have to be computed in a periodic manner.

#### 2.5. Post-Processing for Residual Contacts

_{i}(x) the distance field to a given yarn i, and DS

_{0}(x) the distance field to all the yarns present except yarn i. DS

_{0}(x) can be computed as

_{i}(x) − DS

_{0}(x) takes negative values in any point closer to yarn i than to the other yarns, and positive values elsewhere. If yarn i does not overlap with any other yarn, the zero level surface of this function is a closed surface that encloses the yarn i, and that encloses all the points that are closer to yarn i than to the others, see Figure 5a for a 2D illustration. Conversely, if yarn i does overlap another yarn, this closed surface cuts the interpenetration zone in two parts, being locally the surface of equal distance between the boundaries of both yarns. The intersection between the volume bounded by the zero-level surface of the function DS

_{i}(x) − DS

_{0}(x) and the original yarn i defined by

_{i}(x) at its zero level. If this operation is performed for all the yarns, the overlapping zone can be transformed into an exact contact, modifying locally the shape of the cross section of yarns (Figure 5b). Subtracting a constant term from the function O

_{i}(x) defined in relationship (2) before taking its zero-level surface furthermore allows introduction of a (uniform) clearance as described in Figure 5c.

**Figure 5.**(

**a**) 2D representation of the function DS

_{i}(x) − DS

_{0}(x); (

**b**) use of level set functions to eliminate residual interpenetration zones on arbitrary shapes, dashed lines are original shapes and full lines are the final situation. The background function is min

_{i}$\left({O}_{i}\left(x\right)\right)$; (

**c**) introduction of a clearance between yarns for further processing by eXtended Finite Element Method (XFEM) mechanical simulations; and (

**d**) increase of the yarns’ diameters to mimic contact zone deformations of the yarns’ cross section.

## 3. EXtended Finite Element Method (XFEM) and Computational Homogenization

_{i}being the regular nodal displacements. The first summation represents the usual finite element polynomial interpolation containing the standard shape functions as a partition of unity. The second sum introduces the enrichment with a

_{j}the additional unknowns and Ψ(x) the enrichment functions. For heterogeneous materials, the level set function was shown to be a good basis to introduce the required strain jump at the material boundary. This principle is illustrated in Figure 6 for the simple case of a 1D structure. In the present contribution, the function defined by relationship (2) was used in order to construct the enrichment Ψ(x) based on nodal values of the level set function (index j refers here to nodes) according to

**Figure 6.**Principle of eXtended Finite Element enrichment by a level set function for a bi-material 1D example. The polynomial variation described by nodal displacements (blue curve with related d unknowns) is complemented by a nodal-based enrichment with limited support (red curve with a unknowns) to obtain the real displacement field without placing nodes at the material interface.

**E**is applied to a RVE, the displacement of a point inside the RVE is given by

**ε**resulting from (5) can be shown to be equal to

**E**. Next, the Hill-Mandel condition (energy equivalence between the fine-scale and macroscopic descriptions)

**Σ**is obtained as the volume average of the microstructural stress tensor

**σ**. Using periodicity, the macroscopic stress tensor is obtained based on the RVE tying forces at nodes controlling the macroscopic loading as

**Figure 7.**Representative volume element (RVE) controlling nodes using periodic boundary conditions. (

**a**) RVE within a plate/shell structure; and (

**b**) macroscopic loading control points of the RVE.

## 4. Weave and RVE Generation Illustrations

_{weft}= 0.3. The weft yarns undergoing a higher tension exhibit a straighter shape. This effect is even more pronounced in Figure 8d where an almost completely straight family of yarns is obtained, with p

_{weft}= 0.05. In this case, this tension difference is combined with different diameters for the yarns to show the robustness of the weave geometry generation.

**Figure 8.**Generation of a reinforcement scheme for a one layer thick weaving. (

**a**) Single set of yarns in orthogonal directions with same diameters; (

**b**) same weaving scheme as (

**a**) with different diameters of yarns; (

**c**) same weaving scheme with increase in the tension of one family of yarns; and (

**d**) combination of diameter change with fully tensioned yarns.

**Figure 9.**Generation of a reinforcement scheme with two families of warp yarns alternating on respectively one and two sets of weft yarns; (

**a**) fully tensioned weft yarns; (

**b**) effect of a decrease in weft yarns tension; (

**c**) further decrease of the weft yarns—axonometric view; and (

**d**) further decrease of the weft yarns—side view.

**Figure 10.**Reinforcement scheme with warp yarns and binders alternating on three plies of weft yarns. (

**a**) Normally tensioned weft yarns (p

_{weft}= 0.3)—axonometric view; (

**b**) normally tensioned weft yarns—side view; and (

**c**) effect of excessive tension in binders (p

_{weft}= 0.8).

**Figure 11.**Woven composite with post-processing of the weave configuration to reach contact between yarns. (

**a**) Weaving principle; (

**b**) RVE before level set-based post-processing; and (

**c**) RVE after level set-based post-processing.

**Figure 12.**Woven composite with post-processing of the weave configuration to reach contact between yarns; (

**a**) axonometric view; and (

**b**–

**d**) cuts of the weave perpendicular to the warp yarns.

**Figure 13.**Generation of a 3D reinforced composite similar to that studied in [22]. (

**a**) Axonometric view; (

**b**) side view illustrating binders in thickness direction; and (

**c**) top view.

## 5. Homogenization of Woven and 3D Composites

**Figure 14.**RVEs investigated using the level set-based XFEM simulations: (

**a**) 3D RVE with vertical wefts; (

**b**) 2.5Da RVE with non-alternating looped binders; and (

**c**) 2.5Db RVE with alternating looped binders

**.**

Set | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

E_{m} (GPa) (Matrix) | 3 | ||||

v_{m} (−) (Matrix) | 0.35 | ||||

E_{l} (GPa) (Yarns) | 230 | 230 | 230 | 230 | 230 |

E_{t} (GPa) (Yarns) | 230 | 200 | 150 | 100 | 50 |

v_{tt} (−) (Yarns) | 0.225 | 0.2285 | 0.2343 | 0.2400 | 0.2458 |

v_{lt} (−) (Yarns) | 0.225 | 0.2215 | 0.2157 | 0.2099 | 0.2042 |

G_{lt} (GPa) (Yarns) | 93.878 | 71.745 | 42.176 | 21.870 | 10.960 |

_{t}/E

_{l}) for each RVE. All three RVEs exhibit the same trend but with substantially different values. In particular, the 3D RVE yields a lower stiffness of the composite with respect to the 2.5D RVEs which are believed to be better representations of a real planar (plate) composite structure. The lower stiffness obtained with the 3D RVE is also potentially accompanied by stress underestimation with respect to the 2.5D RVEs. In addition, the difference between the two 2.5D RVEs resulting from the binders position difference is clearly visible, indicating that the alternate RVE (2.5Db) is softer than the other (2.5Da) and potentially that lower peak stresses can be expected.

**σ**being the stress tensor and n the outward unit normal at the yarns surface, this interfacial tangential stress is given by

**Figure 15.**Homogenized Young modulus of RVEs (

**a**) 2.5Da RVE with non-alternating binders; (

**b**) 2.5Db RVE with alternating binders; and (

**c**) fully 3D RVE with purely vertical (weft) binders.

**Figure 16.**Interfacial tangential stress (relationship (8)) for a tensile strain of 1% in the warp average direction for the three considered RVEs. (

**a**–

**c**) Interface between warp yarns and matrix; and (

**d**–

**f**) Interface between binders (2.5D RVEs) or vertical wefts (3D RVEs) and matrix. The maximum stress value in the scale is 0.25 GPa.

**Figure 17.**Maximal interfacial tangential stress present in the RVE as a function of the anisotropy ratio in the transversely isotropic behavior of yarns for the three RVEs: (

**a**) 2.5Da RVE with non-alternating binders; (

**b**) 2.5Db RVE with alternating binders; and (

**c**) fully 3D RVE with purely vertical (weft) binders.

**Figure 18.**Difference in interfacial tangential stresses between the isotropic (set 1) and transversely isotropic (set 5) assumptions for the behavior of yarns. (

**a**) 3D RVE; (

**b**) 2.5Da RVE; and (

**c**) 2.5Db RVE. The maximum value in the scale is 0.05 GPa.

## 6. Discussion and Perspectives

## 7. Conclusions

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Sonon, B.; Massart, T.J.
A Level-Set Based Representative Volume Element Generator and XFEM Simulations for Textile and 3D-Reinforced Composites. *Materials* **2013**, *6*, 5568-5592.
https://doi.org/10.3390/ma6125568

**AMA Style**

Sonon B, Massart TJ.
A Level-Set Based Representative Volume Element Generator and XFEM Simulations for Textile and 3D-Reinforced Composites. *Materials*. 2013; 6(12):5568-5592.
https://doi.org/10.3390/ma6125568

**Chicago/Turabian Style**

Sonon, Bernard, and Thierry J. Massart.
2013. "A Level-Set Based Representative Volume Element Generator and XFEM Simulations for Textile and 3D-Reinforced Composites" *Materials* 6, no. 12: 5568-5592.
https://doi.org/10.3390/ma6125568