C Mangavel

N Rossignol

A Perronnet

J Barbot

Y Popineau

J Guéguen

Properties and Microstructure of Thermo-Pressed Wheat Gluten Films: a Comparison with Cast Films

Wheat gluten films were prepared by thermo-pressing, and their mechanical properties were compared to those of cast films. The stress-strain relationship was established for films with various amounts of glycerol. Both relationships were quite different, revealing a different network organization. Thermo-pressed films presented higher stress values than cast films, but the effect of the glycerol amount was similar in both cases, an increase of the glycerol amount leading to a decrease of both films stress. The glycerol influence on the strain at break of thermo-pressed films was very limited, with strain values reaching a maximum around 200%. The role of disulfide bridges on themomoulded films mechanical properties was investigated, and it was shown that some rearrangements and a significative protein insolubilization occurred during the process. The effective flow porosity of the protein network for thermo-pressed films was estimated by water capillary rise measurements to about 7%. Scanning electron microscopy was used to obtain some information about the microstructure of both cast and thermo-pressed films.

Introduction

Because of the growth of environmental concerns, the interest for biodegradable materials in packaging applications has increased over the past few years. Proteins, either from plant or from animal origins, seem to be valuable candidates as raw substrates for manufacturing, and their potentials in film forming have been largely described. 1 Various processes can be employed to prepare films from plant proteins. Whereas the casting process involves a solubilization step of the proteins, [2][3][4] it is also possible to prepare films by a low-moisture process, such as thermo-pressing. 1 The main advantage of this low moisture process is that it is not far from processes used presently in the plastics industry. Moreover, as the proteins do not need to be in solution, it appears to be very suitable in the case of wheat gluten, because these proteins are highly insoluble in water. 5 A lot of studies have been done on wheat gluten films obtained by the casting process. [6][7][8] In many cases, it was necessary to add some reducing agents (cysteine, sodium sulfite) or to work in denaturating conditions (highly alkaline pH) in order to solubilize these proteins. In the case of the thermo-pressing process, the combination between temperature and pressure gives rise to the protein network. Some studies have been aimed at the characterization of films from various proteins (soy, pea, sunflower) obtained by thermopressing. [9][10][11] In the case of gluten proteins, some rheological data concerning the behavior of plasticized wheat gluten upon mixing are available. 12,13 Extrusion studies were also undertaken and gave interesting results: 14 it was shown that extrusion of wheat proteins plasticized with glycerol was feasible under steady-state conditions, but the authors emphasized the influence of operating conditions on the aspect of extrudates. The use of a thermo-pressing process to prepare biomaterials from gluten was recently reported, and the gluten network formation upon thermosetting was investigated. 15 Another study was focused on gluten plasticization with fatty acids; 16 in this case, the films water vapor permeability was investigated. In the present study, we tried to establish some relationships betweeen the macroscopic properties of thermo-pressed gluten films and the microstructure of the protein network. The features of thermopressed gluten films were compared to those of cast gluten films, to get information about the influence of the filmforming process on these parameters.

Materials and Methods

Materials.

A commercial gluten with a protein content of 70.6% and a starch content of 11.8% was supplied by Amylum (Aalst, Belgium). Glycerol, NEMI, and other chemicals were supplied by Sigma-Aldrich (St Quentin, France).

Modification of Gluten by NEMI. A gluten dispersion (3% in water) was stirred during 1.5 h in the presence of 0.1 M N-ethylmaleimide (NEMI). After the reaction time, the modified gluten was freeze-dried. The control gluten followed the same procedure (dispersion and freeze-drying) without addition of NEMI.

Preparation of Films by Thermo-Pressing. Gluten was mixed with the glycerol in a mortar with a pestle and was allowed to stay overnight in saturating relative humidity. The blend was then placed between two Teflon sheets and pressed at 120 °C and 200 bar for 15 min in a heated press (hydraulic press "Pinette" Shell Tellus T27). A metal spacer was used to produce films of various thickness: for mechanical measurements and biochemical characterization, films thickness was about 180 µm. For water uptake measurements, thicker films of about 1.8 mm were prepared.

After cooling, the film was peeled and stored at 20 °C and 60% RH (produced by a saturated solution of sodium bromide) during 72 h. Each film was made in duplicate.

Preparation of the Protein Films by Casting. Protein films were prepared by casting as described previously. 4 Briefly, gluten (12.5% w/w) was dispersed into 0.1 N NaOH by using a Polytron (Kinematica, Luzern, Switzerland) homogenizer (25 000 rpm). After addition of the glycerol (glycerol/protein ratios from 0.20 to 0.61 w/w), the mixture was stirred again and then centrifuged (115 × g, 30 min) to remove entrapped air bubbles. Two types of films were prepared, thin films (about 80 µm thickness) for mechanical properties and biochemical composition studies and thick films (about 1.8 mm thickness) for water uptake measurements and scanning electron microscopy.

For the thinner films, the procedure was as follows: 30 mL of the solution was spread on a glass plate (15 × 25 cm) covered with a polyester sheet and then dried during 1 h at 70 °C in a ventilated oven. The film was peeled off from the plate and equilibrated during 72 h in a cabinet under standard conditions of temperature (20 °C) and relative humidity (60% RH). For thicker films, the film-forming solution was poured into a Petri dish and allowed to dry 30 h at 70 °C. In both cases, the drying time was adapted in order to have a constant weight for the material. The thick films were then taken off the Petri dish and equilibrated in the same conditions than the thin films (20 °C, 60% RH).

Determination of the Glycerol/Protein Ratio. Total protein content (N × 5.7) was determined after digestion of the gluten films in concentrated sulfuric acid by an automated ammonia/salycilate reaction. 17 The method was slightly modified as follows: selenium was used as a catalyst and the digestion was realized at 300 °C during 2 h.

The residual amount of glycerol in the films was determined by high performance liquid chromatography as previously described. 18 Film samples (500 mg) were extracted twice with water (25 mL) for 60 and 30 min, and the total volume of the extract was adjusted to 250 mL. The solution was applied on an ion-exchange column (Sucrex-CIL Ca 2+ ) heated at 60 °C. The glycerol was eluted by water and detected by a differential refractometer (Jobin-Yvon). The concentration was calculated by reference to a calibration curve. The measurements were done in duplicate on each film.

Taking into account the protein content in the films and assuming an average molecular weight of 121 g mol -1 for all amino acids, the glycerol content was expressed in moles of glycerol/moles of amino acid.

Residual Water. The water content in films was determined from weight difference of the samples (500 mg) after 24 h drying in a desiccator containing P 2 O 5 under reduced pressure. Two measurements were made on each film.

Mechanical Properties. Mechanical properties were evaluated on 5A-type dumbbell-shape specimens according to the ISO-527-2 standard (gauge length l 0 ) 20 mm). Film thickness was evaluated at five points with a micrometer. Elongation at break and tensile strength were measured in a cabinet (ERATIS, Bouloc, France) at 20 °C and 60% RH with a Synergie 100 MTS Systems device (Cre ´teil, France), equipped with a 10 N load cell. The initial grip separation was set at 50 mm, and the cross-head speed was 20 mm min -1 . Each measurement was replicated five times.

Prolamin Composition by SE-HPLC. Size distribution and aggregation of prolamins in gluten and in gluten films were analyzed by SE-HPLC on a Superose 6 column (1 × 30 cm; flow rate 0.3 mL/min; detection at 220 nm) equilibrated in 12.5 mM sodium borate buffer 0.1% SDS (Sodium Dodecyl Sulfate). Samples were prepared by a three-step extraction. 19,20 The first step consisted of stirring in 12.5 mM sodium borate buffer pH 8.5, 0.5% SDS. Residual proteins were sonicated under controlled conditions (6W, 30 s) in 12.5 mM sodium borate buffer with 2% SDS after being stirred overnight. In addition a third extraction step was achieved on the pellet with 12.5 mM sodium borate buffer, 2% SDS containing 1% DTT (Dithiothreitol). All three protein extracts were analyzed by SE-HPLC. Chromatographic patterns of the first solubilisation step were divided into three peakssP1, P2, and P3scorresponding respectively to large-size glutenin polymers (M w > 500 kDa), medium-size glutenin polymers (500 kDa < M w < 70 kDa) and gliadin monomers. Each type of protein was quantified by measuring the surface areas of peaks and was normalized with respect to the total area. Scanning Electron Microscopy. Films sections were observed by scanning electron microscopy (SM 840A, JEOL, Tokyo, Japan) with an accelerating voltage of 5 kV; samples were previously cut with a scalpel and gilded. The thickness of the gold layer was about 15 nm.

Water Uptake Measurements. The experimental device for capillary rise, also called a Baumann apparatus, is shown in Figure 1. Samples (about 1.8 mm thickness) were cut in order to expose their inner structure to water rising. The size of the samples was adapted in order to have a section of 0.2 cm 2 (in contact with water) and a total volume comprised between 0.2 cm 3 and 0.5 cm 3 .

They were then coated on each of their faces (except the contact surface) with epoxide resin (Buehler, Limonest, France) to prevent their swelling and put on fritted glass in contact with distilled water. The absorbed water volume versus time data was then recorded as water ascended into the sample.

Results and Discussion

Mechanical Properties of Cast and Thermo-Pressed Gluten Films. Thermo-pressed gluten films were prepared with various amounts of glycerol as a plasticizer. Assuming a mean molecular weight of 121 g mol -1 per amino acid, the mechanical properties of thermo-pressed films versus the molar ratio of the glycerol per amino acid are plotted in Figure 2. A set of data previously obtained for cast films 4 is also reported for comparison. For the same molar glycerol ratios, thermo-pressed films presented always a higher stress and a lower strain than cast films (for example 4.15 ( 0.78 MPa and 179 ( 46% for thermo-pressed films versus 0.60 ( 0.15 MPa and 732 ( 90% for cast films at a glycerol/ amino acid ratio of 0.62); thus, we can assume that casting and thermo-pressing induced different protein network organizations. The evolution of the maximal stress of the films versus the amount of glycerol were similar in both cases (Figure 2a), in the range of 0.20-0.70 mol of glycerol/mol of amino acid; that is, the decrease of the stress followed parallel curves for cast and thermo-pressed films. However, the behavior of strain at break for both films was very different, thermo-pressed films being far less affected by the variation of glycerol amounts than cast films (Figure 2b).

Concerning cast films, it was previously noticed that the plasticizing effect seemed to reach a plateau beyond a threshold value of glycerol concentration: 8 values of stress and strain remained constant. This was attributed to a saturation of the protein network by the glycerol molecules. It is interesting to notice that the behavior of thermo-pressed films was different. Whereas strain values remained almost constant over a glycerol amount of 0.60 mol/mol of amino acid, stress values kept on decreasing. Thus, for larger concentrations, glycerol seemed to still have an effect on stress but not on strain: this behavior must be related to the way glycerol interacts with the protein network and to the structure of the network. The plateau value of stress, if existing in this case, might start at higher glycerol concentration; in this study, films with higher ratios than 1.35 mol of glycerol/mol of amino acid (protein: glycerol weight ratio 1:1) were not prepared.

Higher stress and lower strain values of thermo-pressed films compared to cast films suggested the dry process resulted in a more densely cross-linked network. Moreover, the difference in network organization was emphasized by a different stress-strain relationship for both types of films (Figure 3). Covalent bonds such as disulfide bridges could be involved, as their rearrangements could have been favored by the high-temperature conditions of the thermo-pressing process. Such an effect of temperature was shown previously: increasing the drying temperature in the casting process favored protein cross-linking, increased maximal stress, and decreased strain at break of gliadin films. 21 Polymer Size and Role of the Disulfide Bridges. The aggregation state of the prolamin in the vital gluten as well as in a thermo-pressed film was analyzed by a three-step extraction procedure followed by a SE-HPLC analysis (Table 1). The transition from gluten powder to the plastic film was characterized by an important decrease of proteins extractability: whereas 70.8% of the prolamins of gluten were soluble in 0.5% SDS, this amount dropped to 35.8% for the film. In the same time, there was a clear increase of 2% SDS soluble proteins (second extract, from 14.2% to 33.9%) and of insoluble proteins (from 15.1% to 30.3%). The study of the distribution of soluble proteins (i.e., the relative amounts of polymers P1, P2, and P3) in the gluten powder and in the film revealed that this insolubilization concerned all three types of soluble prolamins, gliadins (P3 peak) being affected as well as glutenins soluble polymers (P1 and P2 peaks). A previous work concerning the extrusion of wheat gluten plasticized with glycerol lead to the same conclusion that gliadins could be involved in polymerization and could act as cross-linking agents. 14 Note that the so-called "insoluble proteins" were finally extracted from gluten powder as well as from the thermo-pressed film for SE-HPLC analysis by a buffer containing DTT, a disulfide reducing agent, pointing out a possible involvment of disulfide bonds.

Thus, the film formation may involve the formation of covalent bonds between prolamin chains. These crosslinkages should be due to disulfide bridges rearrangements resulting from the temperature treatment applied during thermo-pressing, since sulfhydryl/disulfide exchanges occur at high temperature in gluten. [22][23][24][25] It is the common suggestion that disulfide bridges may participate in the network connectivity and thus are responsible of at least a certain part of films mechanical strength. 4,26,27 To further investigate the role of these bridges and their rearrangements, gluten was treated with Nemi, to block free sulfhydryl groups and thus avoid or limit disulfide bridges rearrangements. A film was prepared from this Nemi-treated gluten by thermopressing, and both the gluten and the film were analyzed by sequential extraction and SE-HPLC (Table 1). Compared to the untreated gluten, the Nemi-treated gluten presented a slightly higher content of soluble proteins, but the difference (73.4% versus 70.8%) was not significant. As for untreated gluten, the transition from the gluten powder to the film induced a decrease of solubility of Nemi-treated proteins. However, this loss of extractability was lower than for native gluten proteins: there were still 58.1% of soluble proteins in the Nemi-treated film, and the amount of insoluble proteins (third extract) remained unaffected between the Nemi-treated gluten and the Nemi-treated film (16.1% versus 14.6%). A striking feature is that the amount of gliadins (37.7% of the total proteins in the film versus 44.7% in the Nemi-treated gluten) was much less affected than in the absence of Nemi, revealing that in this case gliadins were not, or less, involved in cross-linking reactions because gliadins contain only intramolecular disulfide bonds. 28,29 This can be related to the fact that, when disulfide bridges rearrange, intermolecular bridges are often affected before intramolecular bridges. 27 The loss of soluble proteins (first extract) in the Nemi-treated film corresponded to an increase of the fraction extracted in 2% SDS and sonicated (from 10.5% to 27.3%). Because no reducing agent was used to obtain the second extract, this increase was due to noncovalent aggregation of proteins and to a very limited covalent cross-linking associating only a few gliadins with existing glutenin polymers.

To evaluate the role of rearrangement of disulfide bridges on mechanical properties, these films (from native gluten and Nemi-treated gluten) were submitted to tensile tests (Table 2). The film made from Nemi-treated gluten presented both lower strength and strain; blocking free sulfhydryl groups in the gluten by Nemi thus limited strongly the protein aggregation by covalent bonds and affected the whole network resistance. The effect on stress is supposed to be related to a lower density of intermolecular cross-linkages, as the presence of Nemi will unfavor rearrangements. The effect on strain, though less important, is significant and might be related to the shorter mean length of the gluten polymers. However, the whole network rearrangements induced by the Nemi treatment cannot be completely explained on the basis of what was previously observed after increased cross-linking treatment (thermal treatment, for example), since there was a simultaneous decrease of both stress and strain. Most of the time, the strain decrease was coupled with a stress increase. 21,30,31 We can suggest that hindering disulfide bridges rearrangements prevented network development, thus leading to an overall decrease of films mechanical properties.

Microstructure of the Network. Parallel and crosssections of a thermo-pressed and a cast wheat gluten film were examined by SEM in order to have information about their inner microstructures (Figure 4). The main feature of the parallel section of the thermo-pressed film (Figure 4a) was the presence of some residual starch granules that were not destroyed during the thermo-pressing process. The micrograph of the cross section showed a rough microstructure, with pores of about 200-500 nm in diameter (Figure 4b). The microstructures observed here for a thermo-pressed gluten film were quite similar to those reported for a soy protein isolate film produced by thermal compaction, thus suggesting that the process may play a major role in these features. 32 This was confirmed by the comparison with the microstructure of a cast gluten film, which was very different (Figure 4, parts c and d). First of all, the parallel section did not show large size starch granules as in thermo-pressed films but only small ones, probably damaged. The large starch granules were probably lost by pelleting during the centrifugation of the film-forming solution before casting. Some starch granules might also have been damaged by the presence of NaOH in the film-forming solution. In addition, the cross-section of the cast film is much smoother than that of the thermo-pressed one. The kinetics of water uptake of the cross section of a thermo-pressed film by capillarity is shown in Figure 5. Water absorption increased until it reached a plateau. Though the reproducibility of the experiment was not excellent, we noticed that the plateau value was between 5% and 7% (water volume/sample volume) for all of the thermo-pressed samples. Since no swelling of the sample was detected, and because previous studies (not shown) had revealed that gluten proteins in thermo-pressed films were completely insoluble in water, we supposed that there was a direct relationship between the plateau value of the capillary rise and the sample porosity. Therefore, the effective flow porosity of thermo-pressed films, i.e., the volume fraction of pore space open to flow, should be in the order of magnitude of 7%. These results, revealing a quite compact structure, are in accordance with the microscopic observations of thermo-pressed film sections. The behavior of cast film was very different since a very large swelling of the sample out of the resin occurred (Figure 5). This macroscopic swelling of the casting samples revealed a high degree of interaction between water and the protein network and probably a strong rearrangement of this network due to protein hydration. These results are in accordance with a previous work on β-casein cast films 33 showing differences in microstructures when the films were equilibrated under various conditions of water activities. The inner microstructure of the casein film conditioned at high water activity (76% RH) presented some porosity, whereas films conditioned at low water activity (11% RH) were completely smooth. Thus, in the case of cast films, it was not possible to determine an effective flow porosity.

Conclusion

This work enligthed the drastic influence of the process (casting or thermo-pressing) on film properties and overall protein network organization. In general, cast films presented higher elongation properties, whereas thermo-pressed films were more resistant to rupture. The properties of these films could be modulated by adjusting the amount of glycerol, but the stress-strain relationship remained very different in both cases, suggesting that the whole organization of the protein network was affected by the process. The role of disulfide bridges rearrangements due to the process was clearly demonstrated. The microstructures revealed by scanning electron microscopy, as well as the water exposure behavior of both types of films, were completely different.

Acknowledgment. This research was supported by the European Community in the frame of "Gluten Biopolymer" (FAIR CT 96 1979). The authors thank Amylum for providing gluten.

Figure 1 .

 1 Figure 1. Principle of a Baumann device.

Figure 2 .

 2 Figure 2. Evolution of maximal stress (a) and of strain at break (b) of thermo-pressed ([) and cast (0) wheat gluten films versus the molar amount of glycerol.

Figure 3 .

 3 Figure 3. Stress-strain relationship for thermo-pressed ([) and cast (0) wheat gluten films.

Figure 4 .

 4 Figure 4. SEM micrographs of gluten films: (a) parallel (×1000) and (b) cross-section (×5000) of a thermo-pressed film; (c) parallel (×1000) and (d) cross-section (×5000) of a cast film.

Figure 5 .

 5 Figure 5. Water capillary rise of wheat gluten films with various sample volumes: thermo-pressed films, (×) V ) 0.20 cm 3 ; ([) V ) 0.25 cm 3 ; (0) V ) 0.40 cm 3 ; and cast film, (b), V ) 0.20 cm 3 .

Table 1 .

 1 Prolamin Distribution of Gluten (Control and Nemi-Treated) and Gluten Films (Control and Nemi-Treated), Determined by a Three Step Extraction Procedure and SE-HPLC This fraction was extracted in the presence of 0.5% SDS. b This fraction was extracted in the presence of 2% SDS. c This fraction was extracted in the presence of 2% SDS +DTT.

	first extract a
	P1	P2	P3	second extract b third extract c
	9.3 16.9 44.6
	gluten	70.8		14.2	15.1
	11.3 17.5 44.7
	gluten Nemi	73.4		10.5	16.1
	1.4	6.8 27.6
	film	35.8		33.9	30.3
	5.9 14.5 37.7
	film Nemi	58.1		27.3	14.6

a

Table 2 .

 2 Mechanical Properties of Thermo-Pressed Gluten Films from a Control Sample and from a Nemi-Treated Gluten

		maximal stress (MPa)	strain at break (%)
	gluten film	2.76 ( 0.33	190 ( 35
	Nemi-gluten film	1.65 ( 0.18	144 ( 28


P.-Y Hicher

C S Chang

Elastic Model for Partially Saturated Granular Materials

Keywords: Granular materials, Stress strain relations, Elastic deformation, Unsaturated soils, Soil suction, Micromechanics

This paper presents the development of an elastic model for partially saturated granular materials based on micromechanical factor consideration. A granular material is considered as an assembly of particles. The stress-strain relationship for an assembly can be determined by integrating the behavior at all interparticle contacts and by using a static hypothesis, which relates the average stress of the granular assembly to a mean field of particle contact forces. As for the nonsaturated state, capillary forces at grain contacts are added to the contact forces created by an external load. These are then calculated as a function of the degree of saturation, depending on the grain size distribution and on the void ratio of the granular assembly. Hypothesizing a Hertz-Mindlin law for the grain contacts leads to an elastic nonlinear behavior of the particulate material. The prediction of the stress-strain model is compared to experimental results obtained from several different granular materials in dry, partially saturated and fully saturated states. The numerical predictions demonstrate that the model is capable of taking into account the influence of key parameters, such as degree of saturation, void ratio, and mean stress.

Introduction

A granular material can be considered as a collection of particles of different sizes and shapes. This discontinuous medium can be represented by a fictitious continuous medium whose mechanical properties should depend on the geometric arrangement and contact interactions between interacting particles. For dry granular materials, the interparticle forces are related to the applied external stresses. When a certain amount of water is added to the grain assembly, significant changes, such as an increase in stiffness and strength, can be observed. These changes can be explained by the formation of water menisci between neighboring particles, which creates capillary forces. The amplitude of these capillary forces depends upon the degree of saturation and the morphology at particle level ͑grain and pore sizes͒ ͑Bishop and Blight 1963; Hicher 1998͒. A number of studies have addressed the determination of the capillary forces for a two-particle system ͑Fredlund and Rahardjo 1993; [START_REF] Lian | A theoretical study of the liquid bridge forces between two rigid spherical bodies[END_REF][START_REF] Taibi | Comportement mécanique et hydraulique des sols partiellement saturés[END_REF][START_REF] Cho | Unsaturated particulate materials-Particle-level studies[END_REF]Molenkamp and Nazemi 2003a͒. However, fewer results are available for more complex particle packing ͑Molenkamp and Nazemi 2003b; [START_REF] Dangla | Nonlinear poroelasticity for unsatur-ated porous materials: An energy approach[END_REF] Macroscopic analyses of the mechanical behavior of unsaturated granular materials in the elastic domain can be classified into two categories. Most of the data are analyzed on the basis of the double-variable theory by considering that the strain tensor is governed by the total net stress tensor ij -p g ␦ ij , p g being the gas pressure, and the negative pore water pressure or suction s ͑Fredlund and Rahardjo 1993; [START_REF] Alonso | A constitutive model for partially saturated soils[END_REF]. However, several authors have shown that the experimental results could be interpreted using the concept of a generalized effective stress Ј by introducing a capillary stress c , function of the suction s ͑Biarez et al. 1993͒ Ј = -p g Ic ͑1͒

The above equation can be considered as a general formulation of the equation proposed by Bishop and Blight ͑1963͒ for unsaturated soils Ј = -p g I + X͑S l ͒͑p g -p l ͒I ͑2͒

where p l = liquid pressure; and X͑S l ͒ϭfunction of the degree of saturation S l of the liquid phase ͑X = 0 for a dry material, X = 1 for a fully saturated material͒. Dangla and Coussy ͑1998͒ have confirmed theoretically the validity of the effective stress approach in the elastic domain.

An interesting method for granular material modeling is to develop homogenization techniques in order to derive a stressstrain relationship from forces and displacements at the particle level. Two different approaches can be followed based on the so-called kinematic hypothesis, which states that every particle displaces in accordance with a uniform deformation field ͑Chang 1988͒, or on the static hypothesis, which relates the average stress of the granular assembly to a mean field of particle contact forces ͑Cambou et [START_REF] Cambou | Homogenization for granular materials[END_REF][START_REF] Chang | Kinematic and static hypotheses for constitutive modeling of granulates considering particle rotation[END_REF][START_REF] Liao | Stressstrain relationships for granular materials based on the hypothesis of best fit[END_REF]. Hicher and Chang ͑2005͒ demonstrated the capability of these two approaches to provide a good approximation of the nonlinear elastic behavior of dry granular materials along different loading paths. In this study, the static hypothesis was considered and the capillary forces were introduced at each grain contact in order to extend the model to unsaturated granular media. The determina-tion of the elastic parameters of the granular assembly based on the characteristics of the particle contacts will be discussed and the predictions of the model along different stress loading will be compared to experimental results obtained from different granular materials, in dry, partially, or fully saturated states.

Micromechanical Approach

In this model, we envision a granular material as a collection of particles. The deformation of a representative volume of this material is generated by the mobilization of contact particles inside the whole volume. Thus, the stress-strain relationship can be derived as an average of the mobilization behavior at all interparticle contact planes in the representative volume. At the ␣th contact plane, the local forces f i ␣ and the local movements ␦ i ␣ can be denoted as follows:

f i ␣ = ͕f n ␣ , f s ␣ , f t ␣ ͖ and ␦ i ␣ = ͕␦ n ␣ , ␦ s ␣ , ␦ t ␣ ͖
, where the subscripts n, s, and t represent the components in the three directions of the local coordinate system. The direction normal to the plane is denoted as n; the other two orthogonal directions, s and t, are tangential to the plane ͑Fig. 1͒.

The forces and movements at all contact planes are suitably superimposed to obtain the macroscopic stress-strain tensors. The macroscopic stiffness tensor is obtained by suitably superimposing the stiffness properties at all interparticle contacts. In such a formulation, it has usually been assumed that the microstructure is statically constrained, which means that the forces acting on each contact plane are assumed equal to the resolve components of the macroscopic stress tensor.

Contact Forces and Capillary Forces

For dry samples, the contact forces are directly determined from the external stresses applied on the granular assembly. In the case of wetted samples, different stages of saturation can be identified. The fully saturated regime corresponds to a two-phase material with water filling completely the voids between grains. The water pressure p w can be positive or negative ͑suction͒, but in both cases the effective stress concept ͑Terzaghi 1925͒ can be applied and the contact forces determined by considering the effective stresses Ј as the external stresses ͑De Buhan and Dormieux 1996; Hicher 1998͒ Ј = -p w I ͑3͒

In the case of partially saturated samples, the liquid phase is distributed in menisci located between close grains and as a consequence, capillary forces are applied on the grains and are added to the contact forces defined above. When the water content decreases inside a saturated granular sample, the air breaks through at a given point. The negative water pressure or suction corresponding to that point is called the air-entry pressure. Its value depends on the pore sizes. Afterwards, the sample becomes unsaturated with the water phase being connected inside the pores. This state is called the funicular regime, in which a constant decrease of the degree of saturation corresponds to a gentle decrease in pore water pressure, which is homogenized along the continuous water phase. The menisci start to form between two grains, not necessarily in contact, and are connected to each other. The pendular regime starts when the water menisci become disconnected. The water is no longer continuous and equilibrium in the water pressure is obtained by the vapor pressure. At this stage, the capillary forces start to increase drastically according to the pore size. If the drying process continues, the water bridges begin to fail, at first with the noncontacting grains, until a completely dry state is achieved and no capillary forces are left inside the granular assembly.

Several studies have shown that the attractive capillary forces between two grains connected by a water bridge are a decreasing function of the distance between the grains until the bridge fails ͓see, for example Biarez et al. ͑1993͒, Taibi ͑1994͒, and Chateau et al. ͑2002͔͒. This function depends on the volume of liquid found between the grains. Different mathematical expressions have been proposed for these capillary forces, which are the sum of the pressure forces exerted by the liquid on the wetted area and the surface tension forces acting along the wetted perimeter. In this study, we retained the following expression:

f n cap = f max e -C͑d/R͒ ͑4͒
where f cap = capillary force between two neighboring grains, not necessarily in contact; f max = value of f cap for two grains in contact; and R = mean grain radius; d = distance between two grains and is equal to l -2R, l being the branch length given as a distribution function of the grain size and the void ratio; c = material parameter, dependent on the grain morphology and on the water content; f max depends on the capillary pressure defined as the pressure jump across the liquid-air interface, on the liquid-air interface surface tension, as well as on the geometry of the menisci governed by the solid-liquid contact angle and the filling angle. In this study, a simplified approach was to consider an empirical relation between f max and the degree of saturation S r , without taking into account the hysteresis along drying and wetting paths ͑Hicher 1998͒

f max = f 0 S r S 0 for 0 Ͻ S r Ͻ S 0 f max = f 0 S 0 ͑1 -S r ͒ S r ͑1 -S 0 ͒ for S 0 Ͻ S r Ͻ 1 ͑5͒
where f 0 and S 0 = material parameters; f 0 depends on the grain size distribution; S 0 = degree of saturation at which any further drying of the specimen will cause substantial breaking of the menisci in the pendular domain; S 0 depends on the nature of the granular material. Since the menisci are not necessarily all formed in the funicular regime, Eq. ͑5͒ may not be applicable for high degrees of saturation. However, in this first approach, we decided to extend it to the whole range of saturation, considering that the amplitudes of capillary forces were small for degrees of saturation higher than 80% and could, therefore, be approached with sufficient accuracy by using the same equation.

Interparticle Behavior

The contact stiffness of an orientation includes normal stiffness k n ␣ and shear stiffness k t ␣ of the contact plane. The elastic stiffness tensor is defined in

f i ␣ = k ij ␣ ␦ j ␣ ͑6͒
where k ij ␣ = interparticle contact stiffness tensor; ␦ j ␣ = relative displacement at the interparticle contact.

Let k n be the compressive contact stiffness in normal direction and k s be the shear contact stiffness. Assuming the shear contact stiffness to be the same in s and t directions and that there is no coupling effect between normal and shear directions, the contact stiffness tensor k qk c can then be expressed in terms of the unit vectors n, s, and t, as

k ij ␣ = k n ␣ n i ␣ n j ␣ + k t ␣ ͑s i ␣ s j ␣ + t i ␣ t j ␣ ͒ ͑ 7͒
For each particle contact, the corresponding auxiliary local coordinate system is related to the global coordinate system according to ͑see Fig. 1͒ n = ͑cos ␥,sin ␥ cos ␤,sin ␥ sin ␤͒

s = ͑-sin ␥,cos ␥ cos ␤,cos ␥ sin ␤͒ t = ͑0,-sin ␤,cos ␤͒ ͑ 8͒
The vector s is on the plane defined by x and n. The vector t is perpendicular to this plane and can be obtained by the crossproduct of n ϫ s. The rolling resistances between two particles are not discussed in this paper.

The value of the stiffness for two elastic spheres can be estimated from Hertz-Mindlin's formulation ͑Mindlin 1969͒. For sand grains, a revised form can be adopted ͑Chang et al. 1989͒, given by

k n = k n0 ͩ f n f ref ͪ n k t = k t0 ͩ f n f ref ͪ n ͑9͒
where f n = contact force in normal direction; f ref = reference force to make the units consistent. In this paper, f ref is selected to be 1 N. k n0 , k t0 , and n = material constants. For two spherical particles, if the parameters are chosen to be

k n0 = G g rͩ ͱ 3 1 -g ͪ 2/3 ͩ f ref G g R 2 ͪ 1/3 k t0 = k n0 2͑1 -g ͒ 2 -g ͑10͒
where G g and g = respectively, the shear modulus and the Poisson's ratio for the grains; and r = mean radius of the grains. Eq. ͑9͒ is equivalent to Hertz-Mindlin's contact formulation.

Macro-Micro Relationship

The stress-strain relationship for an assembly can be determined from integrating the behavior of interparticles at all contacts. In the integration process, a micro-macro relationship is required.

For simplicity, the effects of particle spin and the change of interparticle orientations are not considered during a small strain increment ͑i.e., finite-strain condition is not considered͒. Using the static hypotheses proposed by Liao et al. ͑1997͒, we obtain the relation between the strain and interparticle displacement

u ˙j,i = A ik -1 ͚ ␣=1 N ␦ ˙j ␣ l k ␣ ͑11͒
where the branch vector l k ␣ is defined as the vector joining the centers of two particles; ␦ j ␣ = relative displacement at the interparticle contact ͓see Eq. ͑6͔͒; and the fabric tensor is defined as

A ik = ͚ ␣=1 N l i ␣ l k ␣ ͑12͒
The length of branch vectors for the unsaturated case may be greater than the summation of the two particle radius due to the film of water between particles. Thus, the branch vectors do not have the same value for all particle pairs. It is noted that in Eqs. ͑11͒ and ͑12͒, the summation is not limited to direct contacts. The summation should also include indirect contacts of neighboring particles within a Voronoi polygon, as discussed by Cambou et al. ͑2000͒.

Using Eq. ͑11͒ and invoking the condition that the rate of energy dissipation expressed in terms of the assembly stress and strain must be equivalent to that expressed in terms of interparticle forces and movements, we are seeking to establish the expression of interparticle forces in terms of assembly stress. It is obvious that there exist an infinite number of solutions that can satisfy the conditions stated above. However, without invoking further constraints, it can be proved that the following relationship is always one of the possible solutions ͑Liao et al. 1997͒:

f ˙j ␣ = ˙ij A ik -1 l k ␣ V ͑13͒
Eq. ͑13͒ also satisfies the stress definition in terms of interparticle contact forces ͑Cauchy theory of stress mean͒

˙ij = 1 V ͚ ␣=1 N f ˙j ␣ l i ␣ ͑14͒
Because of its approximation nature, Eq. ͑13͒ can be viewed as an averaged solution. The interparticle force can be regarded as the mean value for forces on all contact planes of the same orientation. For convenience, we also regard the branch vector as the mean value for all contact planes of the same orientation.

Using the definition of Eq. ͑14͒, the stress induced by capillary forces can be computed and is termed as capillary stress, given by

͑ ˙ij ͒ cap = 1 V ͚ ␣=1 N f ˙j ␣͑cap͒ l i ␣ ͑15͒
It is noted that this term is not analogous to the usual concept of capillary pressure or suction, which represents the negative pore water pressure inside the unsaturated material ͑Bishop and Blight 1963͒. In this model, the capillary stress depends on the geometry of the pores and is a tensor rather than a scalar. Only for an isotropic distribution of the branch lengths l ␣ , the capillary stress can be reduced to an isotropic tensor. This can be the case for an initially isotropic structure during isotropic loading, but during deviatoric loading, an induced anisotropy is created and the capillary tensor is no longer isotropic.

By adding the contact forces due to the external stresses ͓Eq. ͑14͔͒ and the capillary forces ͓Eq. ͑15͔͒, we can define a generalized effective stress tensor * as

* = + cap

͑16͒

This equation represents a generalization of Eq. ͑2͒ proposed by Bishop, in which the capillary stress is reduced to an isotropic tensor. Dangla and Coussy ͑1998͒ demonstrated the validity of the effective stress approach in elasticity by means of an energy approach. They obtained an expression similar to Eq. ͑2͒, but with an additional term corresponding to the work of the interfaces. As pointed out before, the capillary forces in our model, and as a consequence, the capillary stresses depend on the negative pore water pressure, or suction, and on the water-air interface surface tension. A similar approach can be found in the work of Fleureau and et al. ͑2003͒ who obtained an explicit expression of the capillary stress as a function of the suction for regular arrangements of spherical grains by neglecting the surface tension. The definition of the capillary stress tensor in Eq. ͑15͒ can, therefore, be considered as an extension of the results obtained from these previous studies to cases of nonisotropic granular assemblies.

Stress-Strain Relationship

Using Eqs. ͑6͒, ͑11͒, and ͑13͒, the following relationship between stress and strain can be obtained:

u ˙i,j = C ijmp ˙mp where C ijmp = A ik -1 A mn -1 V ͚ ␣ ͑k jp ␣ ͒ -1 l k ␣ l n ␣ ͑17͒
For elastic behavior, Eq. ͑17͒ can lead to a closed-form solution in the case of a random packing of equal-size particles ͑Liao et al. 1997͒. It is noted that for contacts in the same orientation, the branch vector and contact stiffness are their mean values.

The orientation of each interparticle contact can be expressed by two angles ␤ and ␥ in a spherical coordinate system as shown in Fig. 1. The orientational distribution of interparticle contacts can be described by a probability density function E͑␤ , ␥͒. In the case of random packing with sufficient number of particles, the discrete probability density function can be approximated by a continuous probability density function. Integration of E͑␤ , ␥͒ over the surface of unit sphere should equal to one ͑i.e., selfconsistency equation͒

1 = ͵ 0 /2 ͵ 0 2 E͑␤,␥͒sin ␥d␤d␥ ͑18͒
For isotropic packing structure, E͑␤ , ␥͒ =1/ 2. For any arbitrary parameter h c , which represents interparticle contact property, the summation of h c over all interparticle contacts can be approximated by an expression of an integral, given by 1

N ͚ c=1 N h c = ͵ 0 /2 ͵ 0 2 E͑␤,␥͒h͑␤,␥͒sin ␥d␤d␥ ͑19͒
where N = total number of interparticle contacts; and h͑␤ , ␥͒ = continuous expression of h c . Using the definition of Eq. ͑19͒ and assuming an isotropic packing structure, the following summation can be replaced by an integral form:

A ik = ͚ c=1 N l i c l k c = Nl 2 2 A ¯ik where A ¯ik = ͵ 0 /2 ͵ 0 2 n i n k sin ␥d␤d␥ ͑20͒
For convenience, we define a packing density index ͑normalized number of contacts per unit volume͒, = Nl 3 / V. Note that the total number of contacts N per volume V is normalized by the cube of the mean branch length l. The packing density index is unitless. The branch length is defined as the length between centroids of two contact particles. For round particles, the mean branch length is approximately equal to the mean particle size D. The packing density index for an assembly of equal-sized spheres can be related to the mean coordination number n ¯and the void ratio e of the assembly ͑Chang et al.

1989͒ = Nl 3 V = 3n ͑1 + e͒ ͑21͒
A relationship between n ¯and e obtained for a regular packing of spheres is plotted in Fig. 2. For an isotropic packing structure, using the packing density index, we can express the flexibility tensor

C ijmp -1 = 2l A ¯ik -1 A ¯mn -1 ͵ 0 /2 ͵ 0 2 k jp -1 ͑␤,␥͒n k n n sin ␥d␤d␥ ͑22͒
Although the packing structure is isotropic, the contact stiffness can be orientational dependent because the contact forces are orientational dependent ͓see Eq. ͑9͔͒. Thus, the assembly can exhibit anisotropic behavior due to the anisotropy of applied stresses.

A numerical integration method can be used for the integrals in Eqs. ͑17͒ and ͑22͒. The integrals are computed based on a set of integration points on the surface of a sphere. Each integration point represents a contact orientation and has a weighing factor. We found that the results were more accurate by using a set of fully symmetrical integration points. A study of the performance of using different numbers of orientations showed 37 points to be adequate ͑Chang and Hicher 2005͒.

Elastic Properties of Dry Granular Materials

As a first step, we will describe the procedure used to model the elastic behavior of dry granular materials. If one assumes an isotropic fabric for the material, the parameters of the proposed model are the following: • packing density index: Nl 3 / V. • mean particle size, D.

• interparticle elastic constants: k n0 , k t0 , and n. The unit volume V and the mean particle size D are in fact related, so that only one of these two values has to be considered.

The number of contacts per unit volume governs the density of the packing. If we assume that the same relationship as the one presented in Fig. 2 for a regular packing of spheres can be roughly applied to irregular packing ͑Liao et al. 2000͒, a relationship can be obtained between N / V and void ratio e, for R =1.

This relationship, applicable to spherical grains, has to be adapted for grains with irregular shapes, which can only be done by an empirical method. Based on the study of different granular materials behavior ͑Hicher 2001͒, a chart, giving an empirical relationship between N / V and void ratio e for various coefficients of uniformity Uc = d 60 / d 10 , was proposed ͑Hicher and Chang 2005͒ and is presented in Fig. 3.

One can, therefore, determine theoretically the shear modulus of an assembly of particles, knowing the elastic properties ͑G g , g ͒ of the particles, the grain size distribution, and the void ratio ͓Eq. ͑22͔͒. In fact, Chang and Gao ͑1996͒ have demonstrated that static approaches underestimate the true stiffness of the packing. As a consequence, the approach hereby adopted was to determine the parameters of the model through the following procedure. From the test results, we could directly determine the coefficient of nonlinearity n as well as the ratio ␣ = k t0 / k n0 , which is directly related to the Poisson's ratio of the assembly. The ratio N / V could be determined from Fig. 3. The value of k n0 could, therefore, be determined by fitting predicted and experimental values obtained for the material stiffness ͑E or G͒. A typical value of k n0 = 2,420 was derived by fitting predicted and experimental values in the case of a glass ballotini assembly as well as for quartzic sands. Using this procedure, Hicher and Chang ͑2006͒ showed the capacity of the present model to reproduce with very good accuracy the elastic behavior of different granular materials.

The same procedure was used in this work to determine the parameters of the model for the studied granular materials in dry state. The grain size distribution curves of the studied materials are presented in Fig. 4.

Elastic Properties of Unsaturated Granular Materials

Resonant column tests made on fine-grained cohesionless soils by Wu et al. ͑1984͒ showed the influence of the capillary stresses on the values of the shear modulus G. The unsaturated samples were compacted in a mold that was placed on the pedestal of the resonant column device, at a desired density and water content. Dry samples were prepared with a vibrator to obtain the required density. Fully saturated samples were prepared by introducing distilled water into prepared dry samples. For all the tested soils, the authors found a significant increase in the shear modulus with the decrease of the degree of saturation. A maximum influence was obtained for degrees of saturation between 5 and 20%, according to the nature of the soil. For lower degrees of saturation, a decrease of the shear modulus was observed, and completely dry samples had the same modulus as the fully saturated samples. The authors proposed a correlation of the optimum degree of saturation, giving the maximum shear modulus, with the grain size, represented by the d 10 value in mm ͑S r ͒ opt = ͑-6.5 log͑d 10 ͒ + 1.5͒/100 ͑23͒ in which ͑S r ͒ opt is equivalent to S 0 in Eq. ͑5͒, d 10 being the effective grain size in mm. Fig. 5 presents the results obtained for the glacier way silt for two different confining pressures ͑see grain size distribution curve in Fig. 4͒. The value of G was the same for S r = 0% and S r = 100%. A well-marked peak for the two confining pressures could be observed, which occurred for a value of S r around 17%. The maximum value of G is 1.8 times its value for dry and saturated states at a confining pressure of 50 kPa and 1.6 times at a confining pressure of 100 kPa. The capillary effect appears more pronounced at lower confining pressures. The general trend showing the evolution of the shear modulus with the degree of saturation is in agreement with the model concepts. That implies an identical value of G at dry and fully saturated states in the absence of capillary forces and a maximum influence of these forces at an optimum degree of saturation, as given by Eq. ͑5͒ in which S 0 corresponds to ͑S r ͒ opt in Eq. ͑23͒. To determine the capillary forces, Eq. ͑4͒ includes two material parameters c and d. Very little information could be gathered on these two parameters by simply using the test results. Therefore, we decided to take a constant value c = 8 and a value d function of the mean grain size d 50 : d = 1.1d 50 . These values give a standard evolution for the capillary forces, function of the distance between particles, as well as an initial isotropic distribution of these forces. For the numerical simulations, the following parameters had to be determined: • The parameters of the elastic nonlinear law: k n0 , k t0 , and n.

• The parameters of the capillary forces equation: f 0 and S 0 .

k n0 and n were determined from the values of G obtained at dry ͑or saturated states͒. A ratio k t0 / k n0 = 0.35 was assumed, leading to a Poisson's ratio n = 0.2, which corresponds to a standard value for this type of material ͑Hicher 1996͒.

The value of N / V was determined by using the chart in Fig. 3. For a material with U c Ͼ 5, a void ratio e = 0.58 corresponds to N / V = 0.31.

S 0 was considered equal to ͑S r ͒ opt in Eq. ͑23͒ for d 10 = 0.0024 mm. f 0 was computed in order to fit the maximum value of G for the confining pressure of 100 kPa. The material parameters for Glacier Way Silt are summarized in Table 1.

The numerical results are presented in Fig. 5 together with the experimental ones. A good overall agreement was obtained for the two confining pressures. Starting from the fully saturated state, the shear modulus values increase first slowly for degrees of saturation varying between 100 and 70% and then faster up to the maximum values obtained for S r = S 0 . The computed peak value of G was close to the experimental one for the confining pressure of 50 kPa, showing a more significant effect of the capillary forces at low confining pressure. For degrees of saturation lower than S 0 , the shear modulus values decrease toward the values obtained for dry states.

Cho and Santamaria ͑2001͒ also measured the elastic properties of granular materials, using, however, a different preparation technique. The samples were prepared in a oedometric cell. They were mixed with distilled water at saturated conditions and gradually poured into the cell. They were then subjected to gradual drying in an incubator at 50°C. Shear wave velocity was measured continuously during drying by means of bender elements placed on top and bottom platens. In this study, we concentrated on two of the tested granular materials: monosize glass beads and sandboil sand ͑see grain size distribution curves in Fig. 4͒. The results obtained on glass beads are presented in Fig. 6. They show a continuous increase of the shear modulus with the decrease of the degree of saturation up to a peak value obtained for S r = 0.007 and then a sharp decrease as drying continues. The G value at the dry state remains slightly higher than the one obtained at a fully saturated state. The authors explain the difference by an increase in internal coordination during drying and residual compressive stress. The value of S r at peak shear modu-lus was found to be equal to 0.007, which differs from the value computed from Eq. ͑23͒: ͑S r ͒ opt = 0.053. This could be due to the mode of preparation, which is different in the two cases and could lead to different specimen states at small degrees of saturation. For a specimen prepared at a given low water content, menisci might not form between two wet particles because the water film is not thick enough, whereas during drying, the menisci may subsist longer before starting to fail when the water continues to evaporate.

For the numerical simulations, k n0 and n were determined from the values of G obtained at saturated state, using previous studies on elastic properties on glass beads ͑Hicher and Chang 2006͒. S 0 was chosen equal to 0.007 and f 0 was determined by fitting the G peak value ͑Table 2͒.

The experimental testing was performed under a very small applied vertical stress: 1.5 kPa. Numerical simulations were performed at a higher stress ͑10 kPa͒ to prevent numerical errors. As a result, we decided to present the evolution of the ratio G / G 0 , rather than the absolute values of G, as function of S r . A comparison of experimental and numerical results in Fig. 6 shows an overall agreement in the pendular domain ͑S r Ͻ 5%͒ except for the very dry state where the computed ratio G / G 0 converges toward 1, while it remains around 1.4 in the test results. However, the numerical values are smaller than the experimental ones in the funicular domain. We can see in Fig. 6 that the difference is mainly due to the fact that the experimental values increase significantly between S r = 100% and S r = 80%: for S r = 80%, G / G 0 = 1.31, while this increase remains more limited between 80 and 10%: for S r = 10%, G / G 0 = 1.51. A similar trend was obtained in the numerical simulations in this intermediate range of degrees of saturation, while the initial strong increase failed to appear. Its physical meaning lies in the drying process, which follows a mechanism often termed "ink bottle." When water is progressively removed from the sample, the pore space remains saturated as long as the negative water pressure is not big enough to allow the formation of a meniscus. The air entry point corresponds to the suction value for which an air bubble will form inside the pore. When the material is highly heterogeneous, this phenom- enon is progressive, as water is removed first from the biggest pores and then from smaller and smaller pores during the drying process. On the contrary, for monosized grain assemblies, the air entry point is well marked and corresponds to a given value of the negative water pressure and function of the grain size. In the studied case, the mean grain size was equal to 0.32 mm with a maximum value of 0.5 mm and a minimum value of 0.2 mm. A theoretical study by Taibi ͑1994͒ gave the following relation between the pressure of desaturation and the grain size for monosized assemblies:

-u wd ͑kPa͒ = 300 bd͑m͒ ͑24͒

where u wd = pressure of desaturation; d = diameter of the grains; and b = parameter that depends on the density of the assembly. For a void ratio e = 0.601 and 200 m Ͻ d Ͻ 500 m, we obtain: 1.2 kPaϽ -u wd Ͻ 3.1 kPa. This small value of the suction is large enough to affect the material stiffness, due to the very small value of the confining stress applied to the specimen. Since we performed our numerical simulations at larger confining stress, the effect cannot be as marked, even if we replace the net applied stress by the effective stress, adding to the effect of the suction. Therefore, we decided to perform another series of numerical simulations, starting from the value of the shear modulus corresponding to S r = 80%, as a reference point. The f 0 value was readjusted ͑Table 3͒ and the numerical results are presented in Fig. 6. They are, this time, in much better agreement with the experimental values in the range 0 Ͻ S r Ͻ 80%.

The same kind of test was conducted on a natural sand: The sandboil sand. For this material, the shear modulus increased continuously and showed no drop even at perfectly dry conditions ͑Fig. 7͒. The authors explained the absence of drop at dry states as a consequence of chemical reactions, such as salt precipitation and bonding, at particles contacts. Our model is not elaborate enough at this stage to be able to deal with this kind of phenomenon. Therefore, we decided to choose a small value for S 0 , typically S 0 =10 -3 and to investigate only the influence of degrees of saturation higher than S 0 . The same procedure, as explained before, was used to determine the material parameters, which are presented in Table 4. Fig. 7 shows the comparison between experimental and numerical results for sandboil sand. A good agreement was obtained, in particular a gentle increase of G / G 0 for degrees of saturation higher than 0.2 in the funicular regime and then a strong increase for lower values of S r in the pendular regime. The value of the negative pore water pressure at the air entry point was probably small enough in this case, due to the existence of bigger pores created by a more heterogeneous size distribution, to prevent any noticeable influence on the stiffness values at high degrees of saturation.

Hadiwardoyo ͑2002͒ reported results on an Indonesian silty sand. The material is classified as a silty sand with spread distribution ͑see grain size distribution curve in Fig. 4͒. The samples were all prepared according to the same procedure: The soil was mixed with a given quantity of water and then compacted to the chosen water content and density. The elastic properties were measured in a precision triaxial cell, equipped with local sensors for measuring axial and radial strains. The enhanced precision of the transducers allows the measurement of strains as small as 10 -6 . Tests were performed at different confining pressures on samples having different void ratios and water contents. Fig. 8 presents the evolution of the Young's modulus E with the confining pressure p 0 for dry specimen with void ratio equal to 0.74 and unsaturated specimen with void ratio varying from 0.48 to 0.7 and water content from 8.3 to 16%, corresponding to various degrees of saturation from 33 to 90%.

The elastic parameters were determined by using the experimental relation between E and p for dry samples ͑Table 5͒. Comparison between experimental and computed values is shown in Fig. 8. The test results did not allow us to determine S 0 . Using the results by Wu et al. ͑1984͒ and Eq. ͑23͒, a typical value of S 0 = 0.26 was selected for d 10 = 0.2 m. f 0 was then determined from the test results on unsaturated specimen at a given confining pressure ͑p 0 = 52 kPa͒. The set of parameters for the Indonesian silty sand is presented in Table 5. Numerical results, as well as experimental ones, are presented in Fig. 8. A good overall agreement could be obtained. For unsaturated samples at a given void ratio and water content, the relation between Young's modulus and confining pressure in a bilogarithmic coordinate plane is linear. The slope value decreases with the degree of saturation down to 0.3 for S r = 33% and remains smaller than the one obtained on dry samples, which was equal to 0.67. This decrease is due to the increased influence of the capillary forces compared to the applied forces when S r decreases. Using Eq. ͑15͒, we can calculate the capillary stress tensor for each initial state. The material is assumed to have an isotropic structure and, therefore, the capillary stress tensor is also isotropic. The capillary stress is a decreasing function of the degree of saturation, which depends on the grain size distribution curve.

Using the definition of the generalized effective stress tensor in Eq. ͑16͒, we can now plot the relation between the Young's modulus E and the generalized mean effective stress for all the tests. The results are presented in Fig. 9. One can see that, in this plane, the relationship between log͑E͒ and log͑p*͒ is linear and depends only on the void ratio e. This relationship can be written

E = a͑e͒͑p * ͒ n ͑25͒
n being equal here to 0.67. The Young's modulus appears, therefore, to be a function of the generalized mean effective stress and of the void ratio.

In conclusion, the model is able to take into account the evolution of the elastic properties of a granular material with the degree of saturation varying from a perfectly dry state to a fully saturated state. The determination of the material parameters requires to know the elastic coefficients at dry or saturated state, assuming that they are equal in both cases. If we examine the model parameters obtained for each material in Tables 12345, it appears that the parameters of the contact law k n0 , k t0 , and n have very similar values, except for n in the case of the Indonesian silty sand, which has a significantly higher value than the standard value n = 0.5 obtained for the other materials. This difference can be explained by the higher content of clay material ͑20% in weight Ͻ2 m͒.

The model also requires the determination of the value of the parameters controlling the capillary force amplitude S 0 and f 0 . These can be obtained by knowing the values of the elastic coef-ficients of the granular material at a given degree of saturation different from 0 and 1, and the degree of saturation corresponding to the peak stiffness value. If this last parameter is not determined experimentally, an approximate value can be derived from the grain size distribution using Eq. ͑23͒. We have seen, however, that this parameter can be sensitive to the mode of preparation. For the two materials prepared by compaction at a given water content, i.e., Glacier Way silt and Indonesian silty sand, Eq. ͑23͒ can be applied and leads to values of S 0 function of the grain size of the smallest particles d 10 . In these conditions, f 0 is also a function of the grain size distribution and increases when d 10 decreases: f 0 = 0.02 N for d 10 = 0.0024 ͑glacier way silt͒ and f 0 = 0.1 N for d 10 = 0.2 mm ͑Indonesian silty sand͒. This shows the influence of size of the smallest pores on the amplitude of the capillary forces. The same conclusions can be drawn by comparing the S 0 and f 0 values determined for the two materials prepared by progressive drying. Eq. ͑23͒ cannot be applied to determine S 0 , since the degree of saturation corresponding to the maximum stiffness appears to be smaller in this case than the computed one. However, one can see that S 0 is here also an increasing function of d 10 : S 0 = 0.007 for d 10 = 0.26 mm ͑glass beads͒ and S 0 = 0.001 for d 10 = 0.17 mm ͑sandboil sand͒. In the same way, f 0 is a decreasing function of d 10 : f 0 = 0.025 N to 0.045 N for d 10 = 0.26 mm and f 0 = 0.051 N for d 10 = 0.17 mm.

In the case of uniform size grain material, such as the assembly of glass beads, the negative pore pressure corresponding to the air entry point needs to be considered and the calculation of the elastic coefficients has to be performed using the effective stress as long as the sample remains saturated.

Summary and Conclusion

An elastic model for granular materials has been developed based on micromechanics considerations. The material is considered as an assembly of particles. The stress-strain relationship for the assembly can be determined from integrating the behavior of interparticle contacts in all orientations, given a static hypothesis, which relates the average stress of the granular assembly to a mean field of particle contact forces. For each contact plane, an elastic behavior based on the Hertz-Mindlin contact formulation was assumed. Under these conditions, the stress-strain relationship leads to an elastic nonlinear behavior of the material.

The elastic coefficients at the level of the assembly depend on the parameters used to define the contact law at the level of the particles. These parameters comprise the contact law itself, i.e., the normal stiffness k n0 and the shear stiffness as a function of the normal stiffness k t0 = ␣k n0 , as well as the number of contacts per unit volume N / V along a given plane orientation. The dependency of the normal stiffness on the normal force gives a pressure-dependent behavior for the assembly, while the ratio ␣ determines the value of Poisson's ratio. The influence of the density or void ratio of the assembly on the elastic properties is taken into account during the integration procedure by the ratio N / V. An empirical relationship has been proposed in order to link N / V values to the void ratio. This relationship was based on results obtained for different granular materials with different grain size distributions.

For dry samples, the contact forces are directly determined from the external stresses applied on the granular assembly. In the case of partially saturated samples, the liquid phase is distributed in menisci located between close grains and, as a consequence, capillary forces are applied on the grains and added to the contact forces. The capillary forces were computed as a function of the pore morphology and of the water content, using a simple empirical relationship. Further improvement of the model will require a better understanding of the capillary force amplitude and evolution, based more closely on the physics of the phenomenon.

Comparison between numerical and experimental results on several granular materials demonstrated the ability of the model to reproduce the main characteristics of the elastic behavior of an unsaturated granular material. In particular, the model was able to reproduce the evolution of the material stiffness with the degree of saturation. The integration of the capillary forces over all spatial orientations permitted a definition of a capillary stress tensor. A generalized effective stress tensor could, therefore, be determined by adding the total stress and the capillary stress tensors. It was demonstrated that the evolution of the Young's modulus obtained on a given granular material at different water contents was controlled by the generalized effective stress.

Fig. 1 .

 1 Fig. 1. Local coordinate at interparticle contact

Fig. 2 .

 2 Fig. 2. Relationship between void ratio and coordination number

Fig. 3 .Fig. 4 .Fig. 5 .

 345 Fig. 3. Empirical relationship between contact density N / V and void ratio as a function of grain size distribution

Fig. 6 .

 6 Fig. 6. Evolution of shear modulus with degree of saturation for glass beads

Fig. 8 .

 8 Fig. 8. Young's modulus versus net confining pressure for Indonesian silty sand

Table 1 .

 1 Model Parameters for Glacier Way Silt

						f 0
	k n0	␣	n	N / V	S 0	͑N͒
	2,200	0.35	0.5	0.31	0.175	0.02

Table 2 .

 2 Model Parameters for Glass Beads ͑Simulation 1͒

						f 0
	k n0	␣	n	N / V	S 0	͑N͒
	2,420	0.35	0.5	0.61	0.007	0.025

Table 3 .

 3 Model Parameters for Glass Beads ͑Simulation 2͒

						f 0
	k n0	␣	n	N / V	S 0	͑N͒
	2,420	0.35	0.5	0.61	0.007	0.045

Table 4 .

 4 Model Parameters for Sandboil Sand

						f 0
	k n0	␣	n	N / V	S 0	͑N͒
	2,420	0.35	0.5	0.52	0.001	0.051

Fig. 7. Shear modulus versus degree of saturation for sandboil sand

Table 5 .

 5 Model Parameters for Indonesian Silty Sand

						f 0
	k n0	␣	n	N / V	S 0	͑N͒
	2,420	0.35	0.67	0.33	0.26	0.10

Fig. 9. Young's modulus versus generalized effective stress for Indonesian silty sand

Professor, Research Institute in Civil and Mechanical Engineering, UMR CNRS 6183, Ecole Centrale Nantes, BP 92101, 44321 Nantes Cedex 3, France. E-mail: Pierre-Yves.Hicher@ec-nantes.fr

Professor, Dept. of Civil and Environmental Engineering, Univ. of Massachusetts, Amherst, MA 01002.


Pascal Casari

Frédéric Jacquemin

Peter Davies

Gem

Characterization of residual stresses in wound composite tubes

Keywords: Residual stress, Composite, Tube, Moisture

de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The increasing use of composite laminates in engineering applications, due to their high strength-to-weight ratio as well as excellent corrosion and fatigue resistance, has resulted in greater interest in their long term behaviour. Composites tubes are widely used for a range of industrial applications and have been the subject of many previous studies. The manufacture of composite tubes by filament winding generates process-induced stresses. When curing is completed, the tubes are then subjected to ambient conditions which may generate hygrothermal internal stresses which add to the residual manufacturing stresses. When polymer matrix composites are exposed to elevated temperatures and wet environments they undergo dimensional and stress state changes due to moisture induced swelling and thermal expansion. Such stresses can have a significant effect on the mechanical performance of composite structures by inducing warpage and initiating matrix cracks and delaminations. However it is not a trivial task to measure internal stresses accurately during the service life of composite structures. The aim of the present study is to measure internal stresses and to compare them to numerical predictions. An experimental investigation into internal stress measurement is described here, based on measuring the strains revealed by cutting composite tubes. Axial and transverse cuts release the strains caused by internal stresses. The corresponding residual stress model, based on continuum mechanics within the framework of hygrothermoelasticity, is presented. Then the comparison is given through the calculation of axial and circumferential bending moments.

Experimental study

Principle of residual stresses characterisation

Many techniques have been developed to measure internal stresses. Embedded sensors such as optical fibres [START_REF] Vacher | Embedded fibre optic sensors for real time RFI process monitoring Proceedings of structural health monitoring[END_REF] and strain gages have been used to measure the local internal stress state. At a ply or macroscopic scale the hole drilling method [START_REF] Sicot | Determination of residual stresses in composite laminates using the incremental hole-drilling method[END_REF], the layer removal method [START_REF] Manson | Process simulated laminate (PSL): a methodology to internal stress characterization in advanced composite materials[END_REF] and the compliance method [START_REF] Ersoy | Measurement of residual stresses in layered composites by compliance method[END_REF] have been used in the past. The experimental technique used in the current work is similar to the compliance method and consists of sectioning the tubes (Fig. 1) and measuring the change in strains on the external and internal surfaces with two biaxial strain gages, one bonded to each side of the tube wall (Fig. 2). This leads to the estimation of axial and circumferential internal bending moments in order to give an order of magnitude of quantities commonly used in industrial design for the calculation of the properties of filament wound tubes.

Testing operation

The test chosen to reveal residual stresses is a destructive one. It involves removing all the tube surrounding the area on which strain gages are bonded. A series of cuts is performed, and strains follow a non-monotonic evolution which depends on the order of cutting operations. The testing procedure is detailed below in order to analyse accurately the effect of each step by taking the typical response of a R glass tube, with a stacking sequence of [G85], equipped with strain gages during cutting: † 1st step: Samples. The length of the tube is chosen to ensure that the cut of a first ring at the edge has no effect on the internal strains measured in the middle by strain gages. The tubes are 55 mm diameter and 6.5 mm thickness, The minimum length of the tubes was 76 mm. † 2nd step: Instrumentation. Biaxial strain gages are bonded and sealed in the middle of the piece of tube on the inside and outside surfaces. Strain gages have a grid length of 5 mm and a supporting area of 10 mm diameter. An HBM Spider8e data acquisition system is used for strain gage quarter bridge conditioning. The electric supply is based upon a carrier frequency principle which ensures that even on glass-epoxy material, which has a lower thermal conductivity than carbon-epoxy, there is no drift of the signal due to strain gage heating. Before handling and clamping the tube, the gage readings are set at zero. † 3rd step: Machining. A series of cuts are made in a convenient order for machining (Fig. 3). A lubricated diamond saw is mounted on the machine instead of the regular tool (Fig. 4). The cut is obtained by the rotation of the saw and only one turn of the chuck, so that the wires of the strain gages remain long enough to follow the motion of the tube.

The following numbers refer to Figs. 3 and5. (0)-The tube is clamped in the chuck of the lathe.

(1-4)-Four rings are cut from the end of the tube. The rings are 6 mm wide and the saw is 2.5 mm thick, so the total thickness of the removed material is 8.5 mm. The last ring cut is stopped as close as possible to the support of the strain gage. Several rings were cut in order to ensure that the tube was sufficiently long to avoid end effects. (5)-A fifth cut is done as close as possible to the strain gages. (6)-The remaining part of the tube is unclamped from the lathe. ( 7)-A cut is made parallel to the x axis in order to open the tube. (8-9)-The last two cuts are made around the strain gages, and there remains only a square piece of material on which a zero stress state is assumed.

The strains reported in Table 1 correspond to those measured after the end of all cutting operations.

The first observations are: † A significant strain level with hundreds of microstrains is measured. † Tubes tend to close after axial cuts. † The transverse cuts generate mostly a relaxation of the longitudinal strains and the longitudinal cuts generate a relaxation of the transverse strains. Fig. 2. Piece of tube and biaxial strain gages bonded on each side of the tubesurface. † Strains from inner and outer surfaces are roughly opposite to each other, except for the carbon tubes, for which a thermal drift seems to exist. Different heat transfer conditions may explain the difference compared with glass tubes.

Results from experiments

Table 1 shows the inner and outer circumferential and axial final strains for nine epoxy composite tubes, reinforced with E glass, R glass and T700 carbon fibres. Three winding angles are considered, [G558], [G358] and [G858] with respect to the tube axis. The tubes were manufactured in 1998 applying a cure temperature of 125 8C for 7 h, then stored at ambient temperature for five years.

The strains are generally negative on the inner surfaces of the tubes while the strains on the outer surfaces of the tubes are generally positive. Global equilibrium considerations ensure that only bending internal loads are released from cut-outs. Indeed axial and circumferential loads are released because of free boundary conditions. However, strains measured inside and outside which should be opposite to each other don't respect this. Table 1 gives also the calculation results of membrane and bending strains (defined on Fig. 2). This may be due to a thermal drift of the strain gages due to the machining warm up, even if lubricant is removing heat from the tube. Then, only the bending part of strains is considered for the comparison with simulations in the next sections. Moreover, only one test could be performed on each type of tube, which may lead to some potential errors in strain results.

Numerical study

Process-induced stresses

A model has been developed in parallel with the experimental study, which predicts the stress level in the constitutive plies of the tube. The residual stresses, induced by the temperature differential (between cure and room temperatures), for each ply at any time, are calculated by using the classical equations of solid mechanics (see Ref. [START_REF] Hyer | Thermally induced stresses and deformations in angle-ply composite tubes[END_REF] for details): constitutive laws of thermoelastic orthotropic materials (1), strain-displacement relationship, compatibility and equilibrium equations and boundary conditions.

The constitutive laws of a thermoelastic orthotropic material are given as follows:

s Z L : ð3 K aðT K T 0 ÞÞ (1)
where a and L are respectively the tensors of thermal expansion coefficients and stiffness. a and L are assumed to be constants. At the beginning of the cool-down, a stress-free state is assumed. Previous studies have considered the material behavior of a thermosetting resin during cure to be highly viscoelastic, the residual stresses to be immediately relaxed and stress-free conditions to exist at the cure temperature so that an elastic model may be sufficient to predict residual stresses during cool-down [START_REF] White | Process-induced residual stress analysis of AS4/3501-6 composite material[END_REF]. This is a strong assumption but little experimental data is available to allow alternative initial conditions to be defined, though results from a recent study suggest that it may be possible to design experiments to verify this [START_REF] Gigliotti | Development of curvature during the cure of AS4/8552 [0/90] unsymmetric composite plates[END_REF].

The axial and circumferential components of the displacement field of the ith ply, respectively u and v, are then expressed:

u Z R 1 x v Z R 2 xr R 1 ; R 2 are constants 8 > < > : (2)
The radial component of the displacement field w satisfies the following equation: It is shown that the general solution of Eq. ( 3) can be written as the sum of a solution of the homogeneous equation and of a particular solution.

r 2 v 2 w vr 2 C r vw vr K L qq L rr w Z ðL xq K L xr ÞR 1 r C ðL sq K 2L rs ÞR 2 r 2 L rr C KrðT K T 0 Þ L rr ð3Þ
wðrÞ Z R 3 r ffiffiffiffiffiffiffiffiffi ffi L qq =L rr p C R 4 r K ffiffiffiffiffiffiffiffiffi ffi L qq =L rr p C ðL xq K L xr ÞR 1 r ðL rr K L qq Þ C ðL sq K 2L rs ÞR 2 r 2 ð4L rr K L qq Þ C Kr ðL rr K L qq Þ ðT K T 0 Þ (4)
Finally for an n-layer tube the displacement field depends on 4n constants (R 1 , R 2 , R 3 and R 4 ) to be determined. The constants are calculated from the following conditions [START_REF] Hyer | Thermally induced stresses and deformations in angle-ply composite tubes[END_REF]: † traction-free conditions at the inner and outer radii and continuity of the radial stress at each interply; † continuity of the displacement components at each interface between adjacent plies; † global force balance of the cylinder.

Internal stresses generated by the ambient conditions

When curing is completed, the tube is exposed to the environmental conditions. Ambient conditions generate internal stresses which add to the residual manufacturing stresses. The internal stresses in the tube wall are computed within the framework of hygroelasticity. We consider an orthotropic hygroelastic constitutive law ( 5), strain-displacement relationship and compatibility and equilibrium equations [START_REF] Jacquemin | A closed-form solution for the internal stresses in thick composite cylinders induced by cyclical environmental conditions[END_REF].

The constitutive laws of a hygroelastic orthotropic material are given as follows:

s Z L : ð3 K bðm K m 0 ÞÞ (5)
where b and L are respectively the tensors of hygroscopic expansion coefficients and stiffness and m the moisture content. b and L are assumed to be constants.

Example

The properties of the three materials used for the simulations are presented in Table 2. An example of a predicted stress distribution in a [G55] E glass/epoxy tube due to cooling from 125 to 20 8C is shown in Fig. 6. This shows that shear stresses are not negligible and must be considered at the design stage.

The traction-free conditions imply that the radial stress and the shear stresses in the planes (r,q) and (x,r) are null at the inner and outer radii. Moreover, these shear stresses are null through the thickness of the pipe. The inplane shear stress (x,q), corresponding to the Figs. 6 and7, is not affected by these traction-free conditions.

Modelling the change of the moisture concentration from the dry conditions to saturation at an ambient humidity of 50% [9] indicates compressive stresses which counteract the residual manufacturing stresses (Fig. 7). This shows that the residual stress state can vary significantly during the service life of tubes such as these.

G858 3 i xx Z 160; 3 i qq ZK640 3 o xx Z 18; 3 o qq Z 550 3 M xx Z 89; 3 M qq ZK45 3 xx Z71; 3 qq ZK595 R glass G358 3 i xx ZK72; 3 i qq ZK560 3 o xx Z 34; 3 o qq Z 550 3 M xx ZK19; 3 M qq ZK5 3 xx ZK53; 3 qq ZK555 G558 3 i xx ZK300; 3 i qq ZK390 3 o xx Z 270; 3 o qq Z 470 3 M xx ZK15; 3 M qq Z 40 3 xx ZK285; 3 qq ZK430 G858 3 i xx Z 170; 3 i qq ZK590 3 o xx Z 6; 3 o qq Z 390 3 M xx Z 88; 3 M qq ZK100 3 xx Z82; 3 qq ZK490 Carbon T700 G358 3 i xx ZK74; 3 i qq Z 150 3 o xx Z 18; 3 o qq Z 550 3 M xx ZK28; 3 M qq Z 350 3 xx ZK46; 3 qq ZK200 G558 3 i xx ZK360; 3 i qq ZK100 3 o xx Z 350; 3 o qq Z 610 3 M xx Z 5; 3 M qq Z 255 3 xx ZK355; 3 qq ZK355 G858 3 i xx ZK120; 3 i qq Z 930 3 o xx Z 170; 3 o qq Z 680 3 M xx Z 25; 3 M qq Z 805 3 xx ZK145; 3 qq Z125

Comparison between experimental and numerical approaches

To make a comparison between experimental and predicted behaviour, the corresponding bending moments are calculated by the following relations (thin tubes are considered) [START_REF] Hyer | Laminated Plate and Shell Theory[END_REF]:

M x Z ðH 2 KH 2 s xx z dz and M q Z ðH 2 KH 2 s qq z dz ( (6)
where H is the thickness of the tube.

Fig. 8 shows a comparison for the bending moments between the experimental and numerical results. The experimental bending moments are deduced from the measured strains and the corresponding stresses are calculated by using the classical laminate theory. The order of magnitude of these moments is significant and needs to be taken into account in design. The results show reasonable agreement for all cases. Experimental errors due to the high sensitivity of strain gages to temperature transitions may affect some of the reported data (Table 1), especially the so-called 'membrane strains' which may be produced by local temperature changes during cutting. The remaining mismatch may be reduced by improved characterisation of the hygrothermal properties of the composite tubes. For example, for the simulations, a transverse coefficient of moisture expansion (b) of 0.4 is used. The influence of this coefficient on the calculated internal stresses and thus on the corresponding bending moments is illustrated in Fig. 9.

All the properties presented in Table 2 are retained except the transverse coefficient of moisture expansion which is allowed to take values of 0.2, 0.4 or 0.6. The stacking sequence considered is [G55] 9 with respect to the longitudinal axis. A coefficient of moisture expansion of 0.6 can induce a change of the stress sign (Fig. 9(d)). Since the internal stress depends so strongly on this coefficient, it must be accurately known in order to achieve good agreement with experimental results. The intralaminar shear stress, non-zero on the inner and outer surfaces whatever the transverse coefficient of moisture expansion, is characteristic of the ply anisotropy and stacking sequence [START_REF] Jacquemin | Analytical calculation of the transient thermoelastic stresses in thick walled composite pipes[END_REF].

Conclusion

This paper presents a simple method to measure the internal stress state in composite tubes. The internal stresses due to the manufacturing process and the environmental conditions are revealed by cutting the composite tubes and measuring corresponding strains by means of strain gages. They appear to be signficant and should be taken into account in design of structures made of such tubes. A comparison with an hygrothermoelastic stress model shows reasonable agreement for glass reinforced composite tubes, particularly in the hoop direction. Some higher discrepancies are observed for carbon/epoxy tubes but these may be reduced by improved characterization of the hygrothermal properties, in particular the coefficient of moisture expansion, and by improved representation of material behaviour in the numerical model (viscoelasticity, ageing, damage.). Temperature and moisture may affect mechanical behaviour and cause stress relaxation, especially in the long term. Improved identification of the point during the cure cycle at which stresses start to build up would also improve the model. Further studies are examining these parameters, but the proposed approach makes it possible to check the influence of time-variable process temperature and pressure, and hygrothermal inservice conditions on the behaviour of polymer matrix composites.

Fig. 1 .

 1 Fig. 1. Release of bending moments due to internal stresses.

Fig. 3 .

 3 Fig. 3. Cutting operations on the tube.

Fig. 4 .

 4 Fig. 4. Saw mounted on the lathe.

Fig. 5 .

 5 Fig. 5. Typical response of a tube equipped with strain gages during cutting.

Fig. 8 .

 8 Fig. 8. Comparison between tests and calculations for bending moments.

Fig. 7 .

 7 Fig. 7. Stress distribution in a [G55] E glass/epoxy tube at saturation.

Fig. 6 .

 6 Fig. 6. Stress distribution in a [G55] E glass/epoxy tube-Cooling from 125 to 20 8C.

Table 1

 1 Inner and outer strains for three composite materials and three winding angles

	Material	Winding angle	Inner Strains!10 6	Outer Strains!10 6	Membrane!10 6	Bending!10 6
	E glass	G358	3 i xx ZK62; 3 i qq ZK296	3 o xx Z 9; 3 o qq Z 270	3 M xx ZK26:5; 3 M qq ZK13	3 xx ZK35.5; 3 qq ZK283
		G558	3 i xx ZK79; 3 i qq ZK600	3 o xx Z 60; 3 o qq Z 500	3 M xx ZK9:5; 3 M qq ZK50	3 xx ZK69.5; 3 qq ZK550

Table 2

 2

	Properties used for the simulations
	Material	T700/epoxy	E glass/epoxy	R glass/epoxy
	E 1 [GPa]	155	49.5	64
	E 2 , E 3 [GPa]	9	16	15.9
	n 12 , n 13	0.27	0.26	0.26
	n 23	0.4	0.4	0.4
	G 12 [GPa]	5.6	5.6	6
	a 1 [K K1 ]	0.09!10 K6	8.6!10 K6	8.6!10 K6
	a 2 , a 3 [K K1 ]	28.8!10 K6	26.4!10 K6	26.4!10 K6
	b	0	0	0
	b 2 , b 3	0.4	0.2	0.2