The longitudinal response along fiber direction E 1 ¯ (Figure 3A) is dominated by fibers quasi-inextensibility. Tension behavior can hence be approximated by a simple rule of mixture E L ≈ V f E f where E L is the reinforcement longitudinal modulus, V f the fiber volume fraction and E f the fiber longitudinal modulus in tension known to be constant for carbon fibers and available in the fiber data sheet. As for compression in this same longitudinal direction, it is assumed that the external compaction that occurs in double-membrane forming constrains the transverse displacement on either side of the plies, and therefore largely limits the occurrence of potential out-of-plane fiber micro-buckling that may appear if layers are free (Drapier et al., 1996; Drapier and Wisnom, 1999). Moreover, due to the high volume fraction ( V f ≈ 60 % under 1 bar compaction as measured by Blais (2016)) and in-plane confinement of plies, in-plane fiber micro-buckling could only appear at extremely high loads (Rosen, 1964; Drapier et al., 1996) which cannot be reached in standard forming conditions. In conclusion, the prescribed transverse compaction during forming limits the potential difference in tension-compression response, and the longitudinal compressive modulus can hence be taken equal to the constant elongation modulus E L .

In transverse directions E 2 ¯ and E 3 ¯ , the characterization of tension behavior of UDs is quite a challenge due to the lack of cohesion. Pre-impregnated UDs were tested in transverse direction by Leutz (2015) and Margossian et al. (2016), but results were scattered with low repeatability. In the case of HiTape®, the internal structure with no resin support inside plies exhibits little cohesion only brought by a slight entanglement of the UD dry fibers network, whereas TP veil on either side of the tape brings a superficial cohesion just sufficient for handling and processing. This structure makes both elongational responses in transverse directions almost impossible to characterize. Therefore, for modeling purpose, it can be assumed that transverse stiffnesses correspond to a little fraction of the longitudinal modulus, i.e. E T = α   E L with α ≪ 1 .

In transverse compression, the response of the fibrous material is driven by the fiber network rearrangement as well as fiber-to-fiber contacts (Xiong et al., 2019). Recalling that we consider double-membrane forming processes where the individual layers of the stack are compacted along thickness direction ( E 3 ¯ ) before forming, compression behavior understanding appears to be of prime importance. Such non-linear behavior is already known as it was determined using a standard testing machine to compact HiTape® stacks between circular platens in previous works (Celle et al., 2008; Blais, 2016; Blais et al., 2017).

Considering the small thickness of HiTape® (approximately 0.2 mm), we can assume that transverse shear–i.e. in planes ( O , E 2 ¯ , E 3 ¯ ) and ( O , E 1 ¯ , E 3 ¯ ) as depicted in Figure 3B–is negligible. Conversely, in-plane shear (Figure 3A) may develop during forming. This deformation mechanism indeed dominates the formability of composites semi-products such as woven ones (Cao et al., 2008) due to the relative mobility of the fiber tows (Boisse, 2004). Two types of specific characterization methods were hence developed to evaluate the formability of both dry and pre-impregnated semi-products: the bias-extension (Orawattanasrikul, 2006; Charmetant, 2011) and the picture-frame (Kawabata, 1980) tests.

To the knowledge of the authors, literature on the characterization of the in-plane shear behavior of dry UD reinforcements under forming conditions is quite sparse. Related studies may be found for UD-NCF (Trejo et al., 2020) as well as for pre-impregnated UD tapes with various strategies proposed to maintain the specimen integrity during testing: Leutz (2015) tested 4-ply stacks (Figure 4), McGuinness and Bradaigh (1998) used membranes to maintain a single ply but struggled to isolate the ply response from the membrane one, and Larberg et al. (2012) considered cross-plied. However, all these tests demonstrated a rapid decohesion following the occurrence of out-of-plane instabilities, either fiber micro-buckling or ply buckling. Due to the only superficial cohesion of HiTape®, in-plane shear characterization seems out of reach from a technological point of view. More importantly, due to the compaction which limits the fiber transverse displacements during forming, the formability of a stack of HiTape® plies is expected to be driven more by the relative motion of thin HiTape® individual plies than by intra-ply in-plane shear as it is highlighted in the following.

Many recent studies showed that accounting for ply scale out-of-plane bending behavior of either dry or pre-impregnated semi-products (Cao et al., 2008; Haanappel et al., 2014) in forming simulations allows the prediction of wrinkles occurrence in thick architectures (Boisse et al., 2011), and more generally is a step forward to reliable simulations (Madeo et al., 2015). Moreover, even for thin plies, the bending behavior controls the overall formability to a great extent (Liang et al., 2014). Although the hypothesis of continuity is often made for fibrous media–especially those with high fiber volume fraction–, bending behavior cannot be directly deduced from the elongation stiffness for these media, mainly because of internal relative motion between fibers during deformation (Kang and Yu, 1995). For this reason, reinforcements bending behavior should be characterized.

Due to the low bending stiffness of composite semi-products, classical characterization methods such as three- or four-point bending tests have classically been discarded by the textile industry. Peirce (Peirce, 1930) early proposed a ply scale linear elastic bending model for cloth, extending Euler-Bernoulli’s beam theory to large deflections but small deformations using a correction term. This model led to a simple, cost-effective and rapid bending characterization method called fixed angle flexometer, and involving a strip of fabric bending under its own weight. Using this method Leutz (2015) measured the bending stiffnesses B i about direction E i ¯ ( i = 1,2 ) of pre-impregnated UDs in thermally controled environments and showed that the ratio of bending stiffnesses B 2 / B 1 is about 10 3 . Indeed, bending about E 2 ¯ is dominated by the fiber longitudinal stiffness whereas bending about E 1 ¯ is almost solely controled by the molten resin as illustrated in Figure 3B. In the case of HiTape® this bending ratio is expected to be even higher due to the absence of molten resin. We can therefore infer that B 2 (B thereafter) dominates in forming modeling, i.e. the influence of B 1 on forming simulation may be negligible. Still, the question of how to measure bending stiffness B 1 remains open for future works.

Going a step forward, fibrous semi-products were shown to exhibit a non-linear bending response (Liang et al., 2014). Their tangent behavior can be more generally defined by the differential form B ( κ ) = d ℳ ( κ ) / d κ in the corresponding plane, where ℳ is the bending moment per unit width, B the bending stiffness per unit width, and κ the curvature (Clapp et al., 1990). Following Peirce (1930), various models were introduced to account for the non-linearity of the moment-curvature relationship: bi-linear (Huang, 1979), non-linear for small curvatures and linear for larger curvatures (Abbott et al., 1971; Clapp et al., 1990), with a threshold (Grosberg, 1966), etc.

In order to characterize this particular behavior, a specific apparatus called Kawabata Evaluation System KES-FB2 (Kawabata, 1980) was designed and successfully used on fabrics (Ngo Ngoc et al., 2002) and non-crimp fabric (NCF) multi-plies (Lomov et al., 2003) for example; however this type of apparatus is rather cumbersome and limited to room temperature. Other characterization methods were further derived from the fixed angle flexometer test to characterize non-linear behaviors: de Bilbao et al. (2008) increased step by step the free length of the strip for NCFs and interlocks, while other approaches were based on recording the deformed shape of the sample and using image processing to relate the bending moment to the sample curvature. In order to characterize a wide range of curvatures, Liang et al. (2014) also added a lumped mass at the end of the strip.

Besides, some methods were developed to take into account the temperature-dependency of the bending behavior of pre-impregnated reinforcements, for UD carbon/PEEK ¹ and UD carbon/PPS ² for example (Liang, 2016). In those studies, managing the increase of temperature in the sample before testing was reported to be challenging. A specific apparatus called hot vertical cantilever (Soteropoulos et al., 2011; Alshahrani and Hojjati, 2017) was hence proposed in order to load the sample only when the desired temperature was reached. However, a deformation of the sample was already observed before applying the load due to the temperature rise (Angel and Graef, 2016). In addition, the homogeneity and control of the environment are critical and can significantly influence the results (Angel and Graef, 2016). To overcome this, other tests were proposed on smaller volumes: for instance in a 3-point bending DMA machine (Margossian et al., 2015) or with a rheometer (Sachs et al., 2014).

The two inter-ply deformation modes are depicted on Figure 3C, namely inter-ply opening (decohesion) and sliding. In the case of double-membrane forming processes, it seems wise to assume that decohesion along direction E 3 ¯ is not likely to occur due to external loading. Conversely, sliding between plies is expected to be a key factor controlling the formability of a stack (Guzman-Maldonado et al., 2019); inter-ply sliding needs therefore to be characterized, and is dominated by the presence of the TP veil.

The bending test apparatus, represented in Figure 7B, was composed of a frame with a specimen holder developed to limit edge effect due to the clamp. Thermocouples were used for temperature monitoring. This apparatus was placed in a thermoregulated oven equipped with a glass window. A CCD camera was positioned outside the oven and a LabVIEW program ⁵ recorded in real time both photographs of the sample and temperature of the thermocouples, with a chosen acquisition frequency and duration. A set of masses to be positioned at the free end of the sample was available to increase sample curvature. The grip system to attach the selected mass to the sample tip justified to model it as a point force oriented with gravity.

The sample (either unit plies or stacks obtained with DFP) was first pre-consolidated under vacuum at T proc in order to be as close as possible to double-membrane vacuum hot forming process conditions. The sample was then placed in the oven at T proc . When it no longer deformed, a photograph of the sample was recorded and post-processed.

Using the output photograph of the deformed sample, the cartesian coordinates ( x , z ) of the mean-line of the deformed sample in the orthonormal basis ( E 1 ¯ , E 3 ¯ ) were obtained by digital image processing. Then, a program ran the following steps (corresponding notations are illustrated in Figure 7A): 1.

Derive the relationship z ^ = f ( x ) using a Levenberg-Marquardt algorithm (Marquardt, 1963) to find the optimal couple ( k 1 , k 2 ) minimizing the distance between the experimental data ( x , z ) and the following function:

Voce’s model (Pannier, 2006) is illustrated in Figure 8B and its justification is given in Section 4.1.2.1 .

The outputs of the corresponding program were the following: the range of curvatures of the deformed sample, the optimized couple ( k 1 , k 2 ) in Eq. 3 describing the shape of the deformed sample, the optimized triplet ( R 0 , R inf , κ lim ) in Eq. 8 giving the bending moment vs. curvature relationship, and finally, in any point of the deformed sample: the curvilinear abscissa, the curvature and the bending moment.

The two successive regressions–firstly on coordinates ( x , z ) in step 3.2.3 and secondly on points ( κ , ℳ ) in step 3.2.3 – induced an uncertainty which could be quantified by the two following errors that describe the mean scatter between experimental data ( z i , ℳ i ) and their homologous from the corresponding model ( z ^ i ( x i ) , ℳ ^ i ( κ i ) ) at the ith point of the deformed sample: err % ( z ) = ∑ i = 1 N ( z ^ i ( x i ) − z i ) 2 N | z | max − | z | min ; err % ( ℳ ) = ∑ i = 1 N ( ℳ ^ i ( κ i ) − ℳ i ) 2 N | M | max − | M | min . (9)

For commodity reasons, these errors were normalized by the range of values obtained experimentally. We could then define the two following validity criteria for any tested sample: err % ( z ) < 5 % and err % ( ℳ ) < 5 % . In some cases an angle locally formed on the samples (due to the clamping condition for example); such phenomenon was related to the tendency of the quasi-inextensible fibers to out-of-plane buckle in the inner radius of the curvature, especially for thick stacks. In this case, the samples exhibited an angular mean-line with a poor quality of regression that the validity critera allowed to reject. However it was not always sufficient and a complementary visual check of the quality of the regression was realized. Eventually, the coefficient of variation σ % ( a ) was of interest to measure the mean scatter of a set of N data a i around its mean a ¯ : σ % ( a ) = ∑ i = 1 N ( a i − a ¯ ) 2 a ¯ . (10)

A fixture was designed both to hold the lateral specimens in place and to apply the desired normal pressure and temperature conditions. Figure 5B provides a schematic of the final bench. The closing pressure was prescribed by an actuator. The lateral samples were placed in contact with heating pads and were fixed by means of a specific system. This apparatus was designed to be mounted on a standard traction machine, and the central sample attachement was fixed to the crosshead of the machine. The test was controled using the traction machine software, and an acquisition station enabled the force, displacement, and time data of the traction machine to be recorded in real time, as well as the temperature of the heating pads.

Temperature homogeneity inside the samples was checked by placing thermocouples in the central sample just upstream of the test area and carrying out the test: it was then verified that when the thermocouple came into the test area, i.e. in contact with the heating pads, the setpoint temperature T proc was reached almost immediately. Indeed, the isothermal temperature regime was quasi-instantaneously reached since carbon is a good heat conductor, but also because the pressure applied compacted the sample hence increased fiber-to-fiber contacts, the reinforcement was thin, and the low speed resulted in little contact changes over time.

The test procedure consisted first in fixing both lateral specimens to the heating pads and clamping the central specimen in the upper attachement of the traction machine crosshead. The actuator was then activated to apply the closing pressure, and the temperature of the pads increased up to the setpoint temperature T proc . When T proc was reached homogeneously both on the pads and in the central sample, the test started. It began with a preload of 10 N at a crosshead displacement rate of 10 mm min⁻¹, after which the crosshead kept pulling the central specimen at the preselected displacement rate. The input parameters of the test were: the displacement rate of the crosshead representing the relative speed between tow adjacent plies, and the normal stress applied. The outputs were: displacement, tensile force, normal force, and temperature as a function of time.

The friction coefficient CoF and modified Hersey number H * were calculated as follows: CoF = F T 2   F N     ;     H * = H η = ν F N (11) where F T and F N are, respectively, the tangential force and the normal force, η is the TP veil viscosity and ν the sliding velocity. Let us notice that the factor 2 in the expression of the friction coefficient appears (compared to Eq. 1) since it was assumed that the problem is perfectly symmetrical and consequently the contact area is twice the tested surface. We decided to consider H * , i.e., the Hersey number without the influence of viscosity, since in this study isothermal conditions were fulfilled and hence viscosity was assumed to be constant in the considered normal pressure range.

Let us recall the pressure and speed ranges representative of hot vacuum forming conditions that we introduced in Section 3.3 : pressures around 1 bar and relative speeds of two adjacent plies of the order of magnitude of a few millimeters in some seconds, i.e. 1–10 mm min⁻¹. In order to assess the relative influence of velocity and normal pressure in thoses ranges, five configurations were tested with four to five samples each. Keeping speed at 2.0 mm min⁻¹, tests were carried out at 1.35, 1.00 and 0.50 bar, respectively. Tests at bounding pressures of 1.35 and 0.50 bar were also performed at 0.5 mm min⁻¹. As this work was the first inter-ply sliding characterization of HiTape®, we decided to focus on 0°/0° sliding, i.e. both lateral and central specimen were aligned with loading direction.

The resulting tangent force vs. displacement curves (not presented for confidentiality reasons) exhibited the same shape than wet friction results of friction coefficient vs. displacement from literature, for example from Sachs et al. (2012) (Figure 6B bottom). Such comparison makes sense as friction coefficient is related to tangent force only by normal force which is kept constant during the test. After tensionning of the sample, displacement initiated (at a tangent force value F stat ). Then, as the prescribed displacement increased, the tangent force also increased to a maximum value F max and then decreased. F stat was used to compute the static friction coefficient Co F stat using Eq. 11 with F T = F stat . Similarly F max was also used to compute a so-called maximum friction coefficient Co F max using Eq. 11 with F T = F max . Let us highlight that as the normal force F N was kept constant during the test, the friction coefficient vs. displacement curve also exhibited the same shape than Figure 6B (bottom).

The results of static and maximum coefficients of friction are presented with their standard deviations as a function of pressure in Figure 12 and as a function of the relative speed in Figure 13. A strong influence of pressure and velocity was observed in the ranges of pressure and velocity studied. Note that the standard deviations on every configuration were small (less than 10%) compared to the differences observed from one configuration to another.

The shape of the tangent force-displacement response obtained was consistent with literature results presented for composites pre-impregnated with a TP resin tested in a molten state, as illustrated in Figure 6B taken from Sachs et al. (2012). However, unlike in Ten Thije et al. (2011) and Fetfatsidis et al. (2013), the tangent force measured in our experiment did not clearly stabilize. Consequently, we could not extract from the presented tests a dynamic fiction coefficient. This phenomenon is attributed to a fibrous rearrangement during the test and to possible penetration of the veil inside the UD fiber bed.

When forming a stack, the expected relative displacement between two adjacent plies is of the range of a few millimeters. Moreover, purely geometrical considerations show that forming two 0.2 mm thick inextensible layers on a quadrant implies a relative displacement between of 0.3 mm the two layers. This highlights that only the very first millimeters of sliding of the test are of interest for us. Using bounding values of F stat and F max for the tangent force is therefore justified, and we can state that friction coefficient during forming is bounded by Co F stat and Co F max .

In order to conclude on the type of lubricated regime that characterizes the inter-ply response during hot forming, both coefficients of friction obtained Co F stat and Co F max are plotted as functions of the modified Hersey number (Eq. 11) in Figure 14. In both cases, an increasing linear relationship between the friction coefficients and the modified Hersey number H * may be observed. Recalling the Stribeck curve (Figure 6), we can therefore conclude that a hydrodynamic lubricated regime prevailed, which corresponds to the presence of a lubricating film of thickness larger than the roughness of the reinforcement. In terms of industrial forming, this means that inter-ply mobility is promoted at low Hersey numbers, i.e. at high forming pressures and low velocities, and may not depend of the relative orientations of the plies.