**2. Kinematic approach**

mechanism during forming of woven or nonwoven-reinforced composites is shear, which causes a change in fiber orientations. Fiber reorientation is one of the major factors causing fabric distortions, shrinking, and warpage defect. The fiber reorientation is an important factor that should be taken into account when designing composite products, since it will influence

In this context, numerical simulation methods are needed to anticipate the performance of the final product, but also to predict the reinforcement preforming and the resin injection. Several modeling approaches have been developed in the literature to account for the evolution of the fiber orientation [8-11]. The earliest technique is based on discrete mapping approaches. In contrast to these mapping schemes, the constitutive behavior is required for continuum

The mapping approach, the so-called kinematic method, is used to determine the de‐ formed shape of draped fabrics. The main assumptions are that the warp and weft fibers are inextensible, intersection points between warp and weft yarns are fixed during preforming, and the angle between warp and weft yarns are free. This method, where the fabric is placed progressively from an initial line, provides a close enough resemblance to

The alternative to the kinematic approach consists of the use of Finite Element (FE) methods to simulate the fabric deformation under the boundary conditions prescribed by the forming process by considering the fabric as a homogeneous material using computationally efficient constitutive laws and continuum FEs. The limitation of the FE method is that the fabric is not really a continuum but can be more closely likened to a structure comprising discrete rods, possibly intertwined (for woven fabric), or loosely held together with stitching (for nonwoven fabric). The draping of composite fabric using a mechanical approach requires the resolution of equilibrium PDE's problems by the FE method. In general, in the case of complex surfaces, the boundary conditions are not well-defined and the contact between the surface and the fabric is difficult to manage [18, 19, 20]. Furthermore, the resolution of such a problem can be too long in CPU time and is detrimental to the optimization stage of draping regarding the initial fiber directions. All of these facts lead us to consider rather a kinematic approach, which is very fast and more robust, allowing simultaneously to define the stratification sequences and the flat pattern for different plies and to predict difficult impregnated areas that involve manual operation like dart insertion or, on the contrary,

This paper presents an optimization-based method for simulation of forming processes of woven and nonwoven fabric reinforced composites using geometrical approach. Two draping simulation examples are given. These simulations are performed using the geometrical analysis computer code. For each example, we assume that a mesh of the mold to drape is given. The first example is the draping of woven fabric on double dome mold geometry. The second example shows the influence of the woven fabric and nonwoven on the draping process. The effects of the initial conditions (fiber orientation and start point) on the draped

the overall thermomechanical properties and performance.

mechanical approaches [9, 12, 13].

144 Non-woven Fabrics

hand-made draping [14-17].

the shortage of fabric [21-38].

preform are discussed.

Several methods are used for predicting the fiber reorientation of the fabric. The geometrical model, also referred to as the kinematics or fishnet model is a widely used model to predict the resulting fiber reorientation for doubly curved fabric reinforced products [39, 40]. Based on a pinned-joint description of the weave, the model assumes inextensible fibers pinned together at their crossings, allowing free rotation at these joints. They analytically solved the fiber redistribution of a fabric orientated in the bias direction on the circumference of simple surfaces of revolution, such as cones, spheres, and spheroids. The resulting fiber orientations were solved as a function of the constant height coordinate of the circumference.

In the last 20 years, many authors have presented numerically based drape solutions, based on the same assumptions. The author refers to [6, 15, 41]. Typically, geometrical or kinematic fabric draping starts from a start point and two initial warp–weft fiber directions. Further points are then generated on the mold surface at a fixed equal distance from the previous points, creating a fabric mesh of quadrilateral element.

As the surface drape is generally complex and that the layup depends on the starting point and two ply directions, so there is no unique solution for the iterative geometrical simulation. This problem is generally solved by defining two fiber paths on the drape surface. Based on draping criteria such as maximum mold surface covering, minimum fabric drape covering, and minimum shear angle between warp and weft fibers, the geometrical approach can constitute the predimensioning or the preoptimization stage of the manufacturing in the product development process. The local change in composite properties must be taken into account to predict the properties of a composite product.

In this study, we propose a new discrete geometrical algorithm that takes into account the true geometry of the nonwoven fabric mesh element plotted onto the surface. The proposed approach is based on the fishnet method for which a fabric mesh element is subjected only to shear deformation. The difficulty of such a method is the mapping of the nonwoven fabric mesh element onto any surface [42].

Such a fabric mesh element is then defined by a curved quadrilateral, whose edges are geodesic lines with the same length plotted onto the surface to drape. Given three vertices of the fabric mesh element on the surface, we propose an optimization algorithm to define the fourth vertex of the fabric mesh element. The nonwoven fabric is a fabric-like material made from long fibers and expandable filaments bonded together by mechanical weaving, in order to obtain suitable shear deformation. If the shear angles of fibres are significant, then is allowed step-by-step elongations of filaments and this through an optimized iterative procedure.

Let Σ denote the surface of the part to drape and we assume that a geometrical mesh **T<sup>Σ</sup>** of surface is known. Let ℱ be the woven composite fabric modeled by two families (warp and weft) of mutually orthogonal and inextensible fiber described by the local coordinates x=(*ξ*, *η*). These families constitute regular quadrilateral fabric mesh **TF** of the fabric ℱ (Fig. 1 gives example of draping steps of complex surface). The problem of geometrical draping of ℱ onto the surface Σ consists of calculating each node displacement of fabric mesh **TF** with a point of the surface mesh **TΣ** such that the lengths of the edge of the corresponding mesh **TF <sup>Σ</sup>** on the surface are preserved (no extensible). This problem presents infinity of solutions depending on:


Thus, to ensure a unique solution, we suppose that the points of impact on the part surface as well as the fabric orientation are given. The draping scheme is given by the following step [31]:


The nodes of **TF <sup>S</sup>** associated with nodes (*ξ*, *<sup>η</sup>*0) and (*ξ*0, *<sup>η</sup>*) of **TF** and the *<sup>α</sup>* <sup>−</sup>**nodes** are located on the surface along the geodesic lines emanating from the point of impact. Regarding the *β* −**nodes**, various algorithms are proposed. Most of them use an analytical expression of the surface and formulate the draping problem in terms of nonlinear partial differential equations. Other algorithms are also proposed to simplify these equations by using a finite element discretization of the surface by flat triangular face (i.e., a mesh of the surface). Based on this latter approach, we propose a new algorithm. *β* −**nodes** are computed by solving an optimi‐ zation problem corresponding to determine a vertex of an equilateral quadrilateral plotted on the surface from the data of the three other vertices. This optimization problem concerns the following:


The problem is to find point Q such that curve P1Q and curve P2Q are the geodesic lines with a given length. Thus, we have to determine the directions *u* → <sup>1</sup> and *u* <sup>→</sup> 2 of these geodesic lines from

5

P1 and P2. Initially, these directions are set to *u* → <sup>1</sup> =*PP*<sup>2</sup> → and *<sup>u</sup>* → <sup>2</sup> =*PP*<sup>1</sup> <sup>→</sup> and geodesic curves P1Q1 and P2Q2 are obtained. Then iteratively these directions are set to *u* → <sup>1</sup> =*P*1*Q*<sup>12</sup> → and *<sup>u</sup>* → <sup>2</sup> =*P*2*Q*<sup>12</sup> →, where Q12=(P1+P2)/2; while Q1 is different than Q2.

surface are preserved (no extensible). This problem presents infinity of solutions depending

Thus, to ensure a unique solution, we suppose that the points of impact on the part surface as well as the fabric orientation are given. The draping scheme is given by the following step [31]: **1.** Choose a starting point A (corresponding to the point of impact of the machine: to drape)

<sup>Σ</sup> =(*ξ*0, *<sup>η</sup>*0).

**<sup>S</sup>** associated with nodes (*ξ*, *<sup>η</sup>*0) and (*ξ*0, *<sup>η</sup>*) of **TF** and the *<sup>α</sup>* <sup>−</sup>**nodes** are located on

the surface along the geodesic lines emanating from the point of impact. Regarding the *β* −**nodes**, various algorithms are proposed. Most of them use an analytical expression of the surface and formulate the draping problem in terms of nonlinear partial differential equations. Other algorithms are also proposed to simplify these equations by using a finite element discretization of the surface by flat triangular face (i.e., a mesh of the surface). Based on this latter approach, we propose a new algorithm. *β* −**nodes** are computed by solving an optimi‐ zation problem corresponding to determine a vertex of an equilateral quadrilateral plotted on the surface from the data of the three other vertices. This optimization problem concerns the

**1. Problem 1**: Determine the geodesic curve with a given length from a point of surface according to a given woven or nonwoven fabric direction. This problem is solved by isometrically unfolding the mesh elements along a given direction. The geodesic curve is then a strait segment. The latter is then mapped back to the surface using the unfolded

**2. Problem 2**: Determine the directions of two geodesic curves with a given length from two different points of surface reaching to the same point. To solve the second problem, an iterative approach is applied to find the searched directions. Let us consider a starting point P; its successor P1 along a warp direction and its successor P2 along a weft direction

The problem is to find point Q such that curve P1Q and curve P2Q are the geodesic lines with

→ <sup>1</sup> and *u*

**<sup>Σ</sup>**, classified as *α* −**nodes**, from the starting point,

**<sup>Σ</sup>**, classified also as *α* −**nodes**, from

**<sup>Σ</sup>**, classified as *β* −**nodes**, from *x*<sup>0</sup>

<sup>→</sup> 2 of these geodesic lines from

**Σ**

**1.** Starting point P associated with a node of fabric **TF**

on the surface on geometrical part mesh **x**<sup>0</sup>

**3.** Compute iteratively step-by-step the weft nodes of **TF**

the starting point, associated with nodes (*ξ*0, *η*) of **TF**.

**4.** Compute iteratively cell-by-cell all the other nodes of **TF**

elements to obtain the desired geodesic line.

a given length. Thus, we have to determine the directions *u*

and the nodes associated with nodes (*ξ*, *η*0) and (*ξ*0, *η*) of **TF**.

**2.** Compute iteratively the warp nodes of **TF**

associated with nodes (*ξ*, *η*0) of **TF**.

**2.** Initial warp-and-weft orientation *θ*

on:

146 Non-woven Fabrics

The nodes of **TF**

following:

(see Fig. 1).

The proposed algorithm can easily extended to the case of nonwoven intended, if the shear angle at the point Q is greater than a threshold value depending on nonwoven properties. In order to obtain suitable deformation of nonwoven during the draping operation, the algorithm to find the optimized position of the point Q is based on the shear angle criterion. If the shear angles are lower than the locking angle, the algorithm is identical to the case of woven fabric, and if the shear angles are significant, then the algorithm allows step-by-step elongations of filaments and this through an optimized iterative procedure.

The kinematic approach is well-adapted to preliminary design level. It is based on a modified MOSAIC algorithm, which is suitable to generate a regular quad mesh representing the layup of the curved surfaces. The method is implemented in the GeomDrap software, which is now integrated in the ESI-Pam QUICK software. Pam QUICK software allows to estimate a fiber quality charter (showing distortion of fiber, drop rate, and drape surface ratio) to predict local bending due to overlapping of the fibers in the shear exceeds the limit value depending on the properties of the fabric. It can be used to optimize the draping process by improving the layup directions or the marker data location [43].

Figure 1. Geometrical draping steps of woven composite fabric

The simulated results for the complex geometry are presented here. Draping is simulated with a geometrical draping method described in section 2. For each example, we assume that a mesh of the mold to drape is given. The material properties of the composite product can be predicted from the local fiber volume fraction and the shear angle, which represents the angle between the local warps/weft fiber directions. In the first example, we present the draping of woven or nonwoven composite mold on nondevelopable geometry in order to validate the proposed model. In this case, we compare the fiber distribution and distribution of the shear angle between the fibers in the deformed fabric. In this example, we study the effect of the excessive fiber distortion and the cutting chisel on the draping process of half hemisphere connected by a half cylinder. Comparison with experimental results is illustrated in order to validate the proposed approach. The second example compares the draping of woven and nonwoven carbon fabric on complex mold. The third and the last example show the

influence of the initial start point and fiber orientation on the flat pattern of the nonwoven fabric.

The highest procedure presented above is used to simulate the draping of composite woven dry fabric on complex mold. In this case, we consider the draping of woven carbon fabric on two half hemispheres with a radius of R = 38.8 mm connected by a half cylinder with a length of L = 85 mm (see Fig. 2). Drape experiments results have been performed by [43]. Boundary conditions consist of an initial start point and warp/weft directions in this point. The mold surface is modeled using triangular and

**Figure 1.** Geometrical draping steps of woven composite fabric

3.1. Validation of the geometrical algorithm

3. Applications

Figure 2. Schematic view of the half hemisphere connected by a half cylinder

shows the resulting 3D surface draping for [0°/90°] and [-45°/+45°] fiber orientations. One can notice

quadrilateral shells elements. In order to assess the influence of the initial constraints, two fabric orientations are considered. The initial contact point P of the drape is the center point of the hemisphere

part of the mold and two fiber directions, 0° and 45° fabric orientations are presented.

orientation) where most shearing occurs are examined (see Fig. 2).

**Figure 2.** Schematic view of the half hemisphere connected by a half cylinder

#### **3. Applications** In order to evaluate qualitatively the drape simulation, the enclosed fiber angle was measured in the area between the half hemisphere and cylinder at the longitudinal and diagonal axis of the mold. Figure 3

The simulated results for the complex geometry are presented here. Draping is simulated with a geometrical draping method described in section 2. For each example, we assume that a mesh of the mold to drape is given. The material properties of the composite product can be predicted from the local fiber volume fraction and the shear angle, which represents the angle between the local warps/weft fiber directions. In the first example, we present the draping of woven or nonwoven composite mold on non-developable geometry in order to validate the proposed model. In this case, we compare the fiber distribution and distribution of the shear angle between the fibers in the deformed fabric. In this example, we study the effect of the excessive fiber distortion and the cutting chisel on the draping process of half hemisphere connected by a half cylinder. Comparison with experimental results is illustrated in order to validate the proposed approach. The second example compares the draping of woven and nonwoven carbon fabric on complex mold. The third and the last example show the influence of the initial start point and fiber orientation on the flat pattern of the nonwoven fabric. that in both cases, the mold is completely draped but the shear angles between warp and weft fibers are very excessive (greater than 80°) which is impossible and can induce defects in the composite properties after resin injection or polymerization. Figure 4 presents shaded contours interpolated from the map of the shear angles for [0°/90°] ply orientation and Fig. 5 presents shear angles for [-45°/+45°] ply orientation. The shear angle for both [0°/90°] and [-45°/+45°] draping is large (70°) but the maximum shear angle localization is different. The evolution of a draping simulation shows that the red areas indicate that with this fabric, a single sheet will not be able to cover the hemisphere without creasing. Splits can be added to the fabric, or fiber stretching can be allowed in order to minimize the shear angle. In order to optimize the draping operation and drape completely the mold without excessive fiber distortion (fiber locking <50°) depending on fabric properties, it is necessary to either: • Make cuts chisel along the line C2 for [0°/90°] fiber orientation (Fig. 4a) and along the line C1 for [-45°/+45°] fiber orientation (Fig. 4b). With the cutting chisel operation, the shear limit reached 38° in the case of [0°/90°] and 68° in the case of [-45°/+45°]. • Allow warp and weft fiber stretching of 20%. Without the cutting chisel operation, the mold is completely draped and the shear limit reached 52° in the case of [0°/90°] fiber orientation (Fig.

#### **3.1. Validation of the geometrical algorithm** To compare the predicted shear angles with the experimental result given by [44], measurements of

5a) and 89° in the case of [-45°/+45°] (Fig. 5b).

The highest procedure presented above is used to simulate the draping of composite woven dry fabric on complex mold. In this case, we consider the draping of woven carbon fabric on two half hemispheres with a radius of R = 38.8 mm connected by a half cylinder with a length of L = 80 mm (see Fig. 2). Drape experiments results have been performed by [43]. Boundary conditions consist of an initial start point and warp/weft directions in this point. The mold surface is modeled using triangular and quadrilateral shells elements. In order to assess the influence of the initial constraints, two fabric orientations are considered. The initial contact point P of the drape is the center point of the hemisphere part of the mold and two fiber directions, 0° and 45° fabric orientations are presented. 6 angles along the lines where the highest shear angles occur were performed. From Fig. 6 it can be

In order to compare the experimental warp/weft angles with the predicted results, two cross sections along the symmetrical line noted L1 (for 45° ply orientation) and the diagonal line noted L2 (for 0° ply orientation) where most shearing occurs are examined (see Fig. 2).

7

In order to evaluate qualitatively the drape simulation, the enclosed fiber angle was measured in the area between the half hemisphere and cylinder at the longitudinal and diagonal axis of the mold. Figure 3 shows the resulting 3D surface draping for [0°/90°] and [-45°/+45°] fiber orientations. One can notice that in both cases, the mold is completely draped but the shear angles between warp and weft fibers are very excessive (greater than 80°) which is impossible and can induce defects in the composite properties after resin injection or polymerization. Figure 4 presents shaded contours interpolated from the map of the shear angles for [0°/90°] ply orientation and Fig. 5 presents shear angles for [-45°/+45°] ply orientation. The shear angle for both [0°/90°] and [-45°/+45°] draping is large (70°) but the maximum shear angle localization is different. The evolution of a draping simulation shows that the red areas indicate that with this fabric, a single sheet will not be able to cover the hemisphere without creasing. Splits can be added to the fabric, or fiber stretching can be allowed in order to minimize the shear angle.

In order to optimize the draping operation and drape completely the mold without excessive fiber distortion (fiber locking <50°) depending on fabric properties, it is necessary to either: concluded that for the [0°/90°] ply orientation along the diagonal line L2, the agreement between the


Figure 3. Drape results on shear angles of 0° and 45° ply orientations **Figure 3.** Drape results on shear angles of 0° and 45° ply orientations

**3. Applications**

148 Non-woven Fabrics

The simulated results for the complex geometry are presented here. Draping is simulated with a geometrical draping method described in section 2. For each example, we assume that a mesh of the mold to drape is given. The material properties of the composite product can be predicted from the local fiber volume fraction and the shear angle, which represents the angle between the local warps/weft fiber directions. In the first example, we present the draping of woven or nonwoven composite mold on non-developable geometry in order to validate the proposed model. In this case, we compare the fiber distribution and distribution of the shear angle between the fibers in the deformed fabric. In this example, we study the effect of the excessive fiber distortion and the cutting chisel on the draping process of half hemisphere connected by a half cylinder. Comparison with experimental results is illustrated in order to validate the proposed approach. The second example compares the draping of woven and nonwoven carbon fabric on complex mold. The third and the last example show the influence of the initial

Figure 2. Schematic view of the half hemisphere connected by a half cylinder

L<sup>2</sup> L<sup>1</sup>

R=38.8mm

Start point P

In order to evaluate qualitatively the drape simulation, the enclosed fiber angle was measured in the area between the half hemisphere and cylinder at the longitudinal and diagonal axis of the mold. Figure 3 shows the resulting 3D surface draping for [0°/90°] and [-45°/+45°] fiber orientations. One can notice that in both cases, the mold is completely draped but the shear angles between warp and weft fibers are very excessive (greater than 80°) which is impossible and can induce defects in the composite properties after resin injection or polymerization. Figure 4 presents shaded contours interpolated from the map of the shear angles for [0°/90°] ply orientation and Fig. 5 presents shear angles for [-45°/+45°] ply orientation. The shear angle for both [0°/90°] and [-45°/+45°] draping is large (70°) but the maximum shear angle localization is different. The evolution of a draping simulation shows that the red areas indicate that with this fabric, a single sheet will not be able to cover the hemisphere without creasing. Splits can be added to the fabric, or fiber stretching can be allowed in order to minimize the shear angle. In order to optimize the draping operation and drape completely the mold without excessive fiber

quadrilateral shells elements. In order to assess the influence of the initial constraints, two fabric orientations are considered. The initial contact point P of the drape is the center point of the hemisphere

In order to compare the experimental warp/weft angles with the predicted results, two cross sections along the symmetrical line noted L1 (for 45° ply orientation) and the diagonal line noted L2 (for 0° ply

part of the mold and two fiber directions, 0° and 45° fabric orientations are presented.

orientation) where most shearing occurs are examined (see Fig. 2).

L=80mm

**Figure 2.** Schematic view of the half hemisphere connected by a half cylinder

The highest procedure presented above is used to simulate the draping of composite woven dry fabric on complex mold. In this case, we consider the draping of woven carbon fabric on two half hemispheres with a radius of R = 38.8 mm connected by a half cylinder with a length of L = 80 mm (see Fig. 2). Drape experiments results have been performed by [43]. Boundary conditions consist of an initial start point and warp/weft directions in this point. The mold surface is modeled using triangular and quadrilateral shells elements. In order to assess the influence of the initial constraints, two fabric orientations are considered. The initial contact point P of the drape is the center point of the hemisphere part of the mold and two fiber

To compare the predicted shear angles with the experimental result given by [44], measurements of angles along the lines where the highest shear angles occur were performed. From Fig. 6 it can be

• Make cuts chisel along the line C2 for [0°/90°] fiber orientation (Fig. 4a) and along the line C1 for [-45°/+45°] fiber orientation (Fig. 4b). With the cutting chisel operation, the shear limit reached

• Allow warp and weft fiber stretching of 20%. Without the cutting chisel operation, the mold is completely draped and the shear limit reached 52° in the case of [0°/90°] fiber orientation (Fig.

6

In order to compare the experimental warp/weft angles with the predicted results, two cross sections along the symmetrical line noted L1 (for 45° ply orientation) and the diagonal line noted L2 (for 0° ply orientation) where most shearing occurs are examined (see Fig. 2).

start point and fiber orientation on the flat pattern of the nonwoven fabric.

5a) and 89° in the case of [-45°/+45°] (Fig. 5b).

38° in the case of [0°/90°] and 68° in the case of [-45°/+45°].

distortion (fiber locking <50°) depending on fabric properties, it is necessary to either:

**3.1. Validation of the geometrical algorithm**

directions, 0° and 45° fabric orientations are presented.

Cutting fabric direction To compare the predicted shear angles with the experimental result given by [44], measure‐ ments of angles along the lines where the highest shear angles occur were performed. From Fig. 6 it can be concluded that for the [0°/90°] ply orientation along the diagonal line L2, the agreement between the experimental shear angles and the predicted draping results is good. On the other side, for [-45°/+45°] ply orientation along the symmetrical line L1, the predicted results do not agree at all with the experimental values. The oversimplification of the fabric deformation in the geometrical model gives shear angles up to 89° in case of the 45° ply orientation, which is impossible for woven fabric. The geometrical model is used with a cutoff

(a) [0°/90°] ply orientation (b) [-45°/+45°] ply orientation

Figure 4. 2D flat pattern results of 0° and 45° ply orientations

shear angle based either on an experimentally determined locking angle, or the maximum orientation that the designer is prepared to tolerate. When defining cutoff angle of 38°, which equals the experimentally determined locking angle, one can, in this case, drape the mould, which has high shear angles, with nonwoven fabric. *(a) [0°/90°] ply orientation (b) [-45°/+45°] ply orientation*  **Figure 3.** *Drape results on shear angles of 0° and 45° ply orientations* 

Excess fabric

concluded that for the [0°/90°] ply orientation along the diagonal line L2, the agreement between the experimental shear angles and the predicted draping results is good. On the other side, for [-45°/+45°] ply orientation along the symmetrical line L1, the predicted results do not agree at all with the experimental values. The oversimplification of the fabric deformation in the geometrical model gives shear angles up to 89° in case of the 45° ply orientation, which is impossible for woven fabric. The geometrical model is used with a cutoff shear angle based either on an experimentally determined locking angle, or the maximum orientation that the designer is prepared to tolerate. When defining cutoff angle of 38°, which equals the experimentally determined locking angle, one can, in this case,

drape the mould, which has high shear angles, with nonwoven fabric.

**Figure 4.** *2D flat pattern results of 0° and 45° ply orientations* **Figure 4.** flat pattern results of 0° and 45° ply orientations

Figure 5. Optimized drape results on shear angles of 0° and 45° ply orientations

7

8

8

The second example concerns the draping of woven and nonwoven carbon fabric on complex plastic mold. The initial rectangular taffeta fabric dimensions are: length = 700 mm and width = 350 mm (Fig. 7). The start point P in the simulation was the center of mass and the initial fiber direction is 45° with the geodesic mold directions (see Fig. 8). Figure 9 shows the resulting 3D draping for the woven and nonwoven fabric. We can note that all part surfaces are completely draped with woven and nonwoven fabric but with defect, the outline shapes of flat pattern are different, and the location of the maximum shear angles is the same as in the draped surface. Figure 10 presents shaded contours interpolated from the map of the fiber shear angles. For the draping of the surface with nonwoven fabric, we note that also all part of surface is completely draped with large shear angle (θ> ° 85 ) with fiber disentanglement and junction unraveling. For drape orientation, the results from geometrical model agree with the experimental results. One can conclude, in the considered cases, the draped surface of the product with 45° fiber orientation with woven or nonwoven fabric is impossible without cutting chisel operation. The

The second example concerns the draping of woven and nonwoven carbon fabric on complex plastic mold. The initial rectangular taffeta fabric dimensions are: length = 700 mm and width = 350 mm (Fig. 7). The start point P in the simulation was the center of mass and the initial fiber direction is 45° with the geodesic mold directions (see Fig. 8). Figure 9 shows the resulting 3D draping for the woven and nonwoven fabric. We can note that all part surfaces are completely draped with woven and nonwoven fabric but with defect, the outline shapes of flat pattern are different, and the location of the maximum shear angles is the same as in the draped surface. Figure 10 presents shaded contours interpolated from the map of the fiber shear angles. For the draping of the surface with nonwoven fabric, we note that also all part of surface is completely draped with large shear angle (θ> ° 85 ) with fiber disentanglement and junction unraveling. For drape orientation, the results from geometrical model agree with the experimental results. One can conclude, in the considered cases, the draped surface of the product with 45° fiber orientation with woven or nonwoven fabric is impossible without cutting chisel operation. The

Figure 6. Shear angles along (a) the line L1 of 45° orientation and (b) along L2 of 0° orientation **Figure 6.** Shear angles along (a) the line L1 of 45° orientation and (b) along L2 of 0° orientation

3.2. Comparison of woven and nonwoven fabric draping

3.2. Comparison of woven and nonwoven fabric draping

9

9
