For the three-point bending test, several images were obtained for each test. Each image corresponds to a time t during loading/unloading. For the cantilever bending test, only one image was obtained for each test. The software “Fiji” (Schindelin et al., 2012) was used to extract from each image experimental points belonging to the mid-line of the specimen. Once the experimental mid-line was obtained, a polynomial f ( x ) was then sought to approximate this line as closely as possible by the method of least squares (Figures 4A, 5A). Piecewise approximations can be performed if a single polynomial is not enough to properly approximate the experimental mid-line.

The curvature was calculated from f ' ( x ) and f ' ' ( x ) , the first and second derivatives of f ( x ) with respect to x : χ ( x ) = f " ( x ) ( 1 + f ′ 2 ( x ) ) 3 2 (4)

For the three-point bending test, the calculation of the moment was carried out on the mid-line of the deformed sample. The bending moment was determined at the point of application of the load on the updated configuration provided by the images of the sample. At any time t, the point of contact between the supports and the fabric was also updated (Figure 6).

In the initial state, the mid-line of the woven reinforcement is assumed to coincide with the horizontal axis ( x -axis). The origin of the coordinate system is on the mid-line of the sample before deformation, and at the level of the left support (Figure 6).

The Geometric Relationships Give: β = − tan − 1 ( f ′ ( x 1 ) )   ; z 1 = − 0.5 ( d a + e ) ( 1 − cos ⁡ β ) ; f ( x 1 ) = z 1 (5) with d a the diameter of the supports, e the thickness of the fabric, ( x 1 , z 1 ) are the coordinates of the projection of the contact point on the mid-line (point P 1 in Figure 6) and β the angle between the tangent to the mid-line at x 1 abscissa point and the x -axis. Solving the three equations system (Eq. 5) gives us the values of x 1 , z 1 and β.

The studied woven reinforcement remains horizontal on the two supports when it is not subjected to any effort: its own weight is not important enough to make it bend. Thus, the own weight is neglected in the calculation of the moment.

The bending moment at the point of load application is given by: M b = F m 2 ⁡ cos ⁡ β ( d m + z 1 ) sin ⁡ β + F m 2 ( L 0 − x 1 ) (6) with d m and F m , are, respectively, the displacement and load applied by the machine.

For the cantilever bending test, the bending moment at any point A along the deformed sample was calculated as follows: M b ( A ) = ∫ s L w ( u − s ) cos ( ∅ ( u ) ) d u (7) where s is the curvilinear abscissa of point A, L is the length of the sample, w is the weight per unit length, and u and ∅ are the Frenet coordinates of a point B moving along the profile from A to the free end of the profile.

Figure 7A shows the load-displacement curve corresponding to a loading/unloading in bending. The loading phase was controlled in displacement while the unloading phase was controlled in load (unloading until zero force). This curve clearly shows that the behavior of the studied reinforcement is inelastic. In addition, after a loading/unloading, the reinforcement did not return to its initial configuration: a residual deformation was obtained. Figure 7B corresponds to alternating loading/unloading. This figure shows that the residual displacement at zero force of each load/unload depends on the imposed displacement during the loading phase: the greater the imposed displacement, the greater the residual displacement.

In order to get an idea of the interaction between the different fiber layers, the variation of the direction of the cross sections during the loading phase of a three-point bending test was studied. This study is interesting because it allowed, with a macroscopic scale test, to find a possible source of inelasticity which is the friction between the different layers of fibers.

Dot marks have been drawn on the yarns whose direction is in the plane of the photo. These points were used to define the direction of a cross section of the reinforcement at time t. Figure 8 shows the deformed specimen as well as the cross sections studied at the initial time (Figure 8A) and at the time corresponding to a 30 mm displacement of the loading pin (Figure 8B). These different cross sections were chosen in a way to cover the different zones of the specimen: some sections are close to the loading point, others close to the cylindrical supports and others between these two zones.

The cross sections are initially vertical and perpendicular to the mid-line of the reinforcement. Figure 8C shows the evolution of the angle between the different cross sections and the mid-line. The measurement uncertainty is of the order of 3°. This figure shows that the transverse directions do not remain perpendicular to the mid-line of the reinforcement during deformation. This mobility confirms that there is slippage between the layers of fibers, which can cause friction. This explains, in part, the inelastic behavior observed in Figure 7.

Figure 4 shows the results of calculations of the moment and curvature of a loading/unloading in bending, the corresponding load-displacement curve is shown in Figure 7A. After fitting the experimental points belonging to the mid-line by a continuous function (Figure 4A), the curvature was calculated at the point of application of the load using Eq. 4 for each image corresponding to a displacement during the loading/unloading (Figure 4B). This figure shows that the relationship between curvature and displacement is quasi-linear and that this relationship is the same for loading and unloading. The evolution of the moment as a function of the curvature is shown in Figure 4C where a distinction is made between the loading phase and the unloading phase. These two phases take different paths due to the dissipative (inelastic) behavior of the reinforcement.

The bending model of Dahl (Eq. 3) gives the material an inelastic behavior comparable to a perfect elasto-plastic behavior; the bending stiffness is the same for the loading phase and for the unloading phase when the bending moment is zero (Figure 2A). This stiffness is equal to parameter B of the model (Eq. 3).

However, Figure 4C shows that the bending stiffness of the studied reinforcement is not the same for loading and for unloading under zero bending moment: the stiffness at low bending moment is much lower for the unloading phase than for the loading phase. This calls into question the relevance of an analogy with the behavior of a classical elasto-plastic material for which the bending stiffness would be the same for the loading and unloading phases under zero bending moment. This deviation from a classical elasto-plastic material may be due to a phenomenon of reorganization of the fiber networks during the deformation, which modifies the number of contacts between them.

It has been noticed that woven reinforcements have a different mechanical behavior from that of more classical materials: for example, it is possible to fold and unfold a piece of woven reinforcement without damaging the material. Moreover, woven reinforcements are stored in rolls and often flattened to be used. Thus, it is proposed to adapt the inelastic model of Dahl to the woven reinforcements by giving model parameters for the unloading phase that are different from those of the loading phase. The model becomes: d M b ( χ ) d χ = B l ( 1 − M b ( χ ) M 0 l ) n l if   χ ˙ ≥ 0 d M b ( χ ) d χ = B u ( 1 + M b ( χ ) M 0 u ) n u if   χ ˙ < 0 (8) This model was used to describe the bending behavior experimentally found in this work (Figure 4C). For the loading phase the model parameters are: B l = 745  N . mm , M 0 l = 2.6  N and n l = 1 . For the unloading phase: B u = 18.9  N . mm , M 0 u = 0.81  N and n u = 2.5 . These parameters were chosen by the method of least squares to fit the experimental points as well as possible.

The parameters of the unloading phase depend on the loading history and more specifically on the curvature χ m a x and the moment M b   m a x at the end of the loading phase (Figure 4C). Thus, it is necessary to find the parameters of the unloading phase for any previous loading phase (and thus for any value of χ m a x ). For this purpose, several unloadings were carried out corresponding to several values of χ m a x . When the curvature χ m a x changes, the residual curvature χ r (under zero moment) shown in Figure 4C also changes. Figure 9A shows the relationship between the residual curvature χ r and the curvature χ m a x for several unloading phases. This relationship is linear and takes the form: χ r = 0.52   χ m a x (9)

This relationship can be used when determining the parameters of the unloading phase.

For the unloading phase, the representative curve M b ( χ ) of the model of Dahl (Eq. 8) passes through the two points ( χ m a x , M b   m a x ) and ( χ r ,   0 ). The first point corresponds to the end of the loading phase and the second corresponds to the point where the moment is zeroed during the unloading phase (with residual curvature χ r ). This leads to a relationship between the different parameters which is as follows: B u = − ( M b   m a x + M 0 u ) − n u + 1 ( M 0 u ) n u + M 0 u ( χ m a x − χ r ) ( n u − 1 ) (10) The parameter n u is a shape parameter and does not change from one unloading phase to another ( n u = 2.5 ). χ r is estimated using Eq. 9. This leaves only the parameter M 0 u to be determined and the parameter B u will be calculated using Eq. 10. Figure 9B shows the relationship between the parameter M 0 u and the moment M b   m a x for several unloadings. For each unloading phase, the value of M 0 u is chosen by the least squares method to best fit the model of Dahl to the experimental values of the bending moments. The relationship between M 0 u and M b   m a x is almost linear and can be written as: M 0 u = 0.31   M b m a x (11) Thus, from the curvature χ m a x and the moment M b   m a x corresponding to the end of the loading phase, the parameters of the unloading phase can be determined using Eqs 9–11.

Figure 9C shows the experimental relationship between the bending moment and the curvature corresponding to the test shown in Figure 7B. Figure 9C also shows the modified Dahl model whose unloading phase parameters were determined using Eqs 9–11.

Figure 5 shows the results of the cantilever bending test. The curvature was calculated along the specimen (Figure 5B). The curvature is maximum at the level of the fixed side and decreases rapidly away from this zone to become almost zero for x = 100  mm . This rapid decrease is due to a high bending stiffness for low curvatures.

This test allowed to obtain a non-linear relationship between the bending moment and the curvature (Figure 5C). However, this relationship corresponds to the loading phase since this test does not allow the reinforcement to be subjected to unloading in bending. The relations M b ( χ ) obtained by this test and by the three-point bending test for the loading phase are in agreement (Figure 5C). A slight difference is still observed for large curvatures. This difference may be due to the hypothesis made in this work which is the following: the bending moment is a fair function of the curvature (transverse shear is neglected). However, both relationships remain correct to describe the bending behavior of this reinforcement and can be used for finite element simulations.

The simulation of a loading/unloading in a three-point bending test was carried out to validate the bending model of Dahl.

The orientations of the specimen fibers and boundary conditions used for numerical simulation correspond to those of the experimental test: the reinforcement is placed freely on two cylindrical supports of 16 mm diameter and a loading pin moves down/up to impose loading/unloading. The loading phase corresponds to an imposed displacement of 30 mm. The unloading phase corresponds to a return to zero force.

Figure 10A shows that the load-displacement relationship at the point of application of the force obtained by simulation is consistent with that obtained by experimentation. The mid-lines of the deformed sample obtained by simulation and experimentation are shown in Figure 10B for the end of the loading phase and in Figure 10C at the end of the unloading phase. The geometry of the deformed sample obtained by simulation is consistent with that obtained by experimentation. If during the loading phase the loading pin imposes the displacement of the reinforcement at the point of application of the force, this is not the case after its release where the reinforcement is free to have a residual deformation. At the end of the unloading phase, the residual displacement obtained by simulation is consistent with that obtained by experimentation. The reinforcement would return to its initial configuration (flat and horizontal) if the simulation was done with an elastic behavior model. The inelastic model allowed here to predict the non-elastic return of the reinforcement.

The boundary conditions for this test are as follows: the left end of the specimen is embedded while the rest is free and bends under its own weight. The dimensions of the specimen are: 445 × 70 × 3 mm.

The finite element numerical simulation of the cantilever bending test gave a geometry of the deformed sample that is consistent with that obtained experimentally (Figure 11).

Thus, these characterization tests validate the use of the bending model of Dahl and the parameters of this model described in Bending Test Results and Behavior Modelling.