Zhenyu Jiang
email: zhenyu.jiang@ivw.uni-kl.de

Lada Antonova Gyurova

Alois K Schlarb

Klaus Friedrich

Zhong Zhang

Study on friction and wear behavior of polyphenylene sulfide composites reinforced by short carbon fibers and sub-micro TiO 2 particles

Keywords: A. Short-fiber composites, A. Particle-reinforced composites, B. Friction/wear, D. Scanning electron microscopy, Artificial neural network

Polyphenylene sulfide (PPS) composites filled with short carbon fibers (SCFs) (up to 15 vol.%) and sub-micro-scale TiO 2 particles (up to 7 vol.%) were prepared by extrusion and subsequently injection-molding. Based on the results of sliding wear tests, the tribological behavior of these materials was investigated using an artificial neural network (ANN) approach. A synergistic effect of the incorporated short carbon fibers and sub-micro TiO 2 particles is reported. The lowest specific wear rate was obtained for the composition of PPS with 15 vol.% SCF and 5 vol.% TiO 2 . A more optimal composition of PPS with 15 vol.% SCF and 6 vol.% TiO 2 was estimated according to ANN prediction. The scanning electron microscopy (SEM) observation revealed that this hybrid reinforcement could be interpreted in terms of a positive rolling effect of the particles between the two sliding surfaces, which protected the short carbon fibers from being pulled out of the PPS matrix.

Introduction

Polymer composites occupy a considerable market share nowadays as one of the most common engineering materials. They provide a combination of various advantages, such as ease in manufacturing, cost effectiveness and excellent performance, which cannot be attained by metals, ceramics, or polymers alone [START_REF] Palabiyik | Tribological srtudies of polyamide 6 and high-density polyethylene blends filled with PTFE and copper oxide and reinforced with short glass fibers[END_REF]. In recent years a tremendous interest was raised in scientific and industrial communities to apply polymer composites in sliding components, where their self lubricating properties can be exploited to avoid the need for oil or grease lubrication accompanied with the problems of contamination [START_REF] Sviridyonok | Self lubricantion mechanism in polymer composites[END_REF].

The choice of an appropriate matrix is of great importance in the design of wear resistant polymer composites. The required properties of such a tribo-matrix includes high service temperature, good chemical resistance and outstanding cohesive strength [START_REF] Tewari | Recent development in tribology of fibre reinforced composites with thermoplastic and thermosetting matrices[END_REF]. Within this frame, PPS is regarded as a proper tribo-matrix material [START_REF] Bucher | Polyphenylene Sulfide[END_REF]. Neat PPS, however, is a brittle material with relatively low impact strength [START_REF] Fred | Polymer science and technology[END_REF]. Therefore, various fillers have been incorporated in order to enhance the property profile of PPS. Short carbon fibers (SCFs), which are widely advocated as a decisive reinforcement component, show a remarkable capability to increase the wear resistance of PPS [START_REF] Lhymn | Friction and wear of fiber reinforced PPS composites[END_REF]. As for the incorporation of particulate fillers, the problem becomes more sophisticated. Some micro-particles such as Ag 2 S, CuS, NiS, SiC and Cr 3 C 2 have been reported to improve the wear resistance of PPS, but others like PbTe, PbSe, ZnF 2 , SnS and Al 2 O 3 have been found to exercise an adverse influence [START_REF] Yu | An investigation of the friction and wear behaviors of ceramic particle filled polyphenylene sulfide composites[END_REF][START_REF] Yu | An investigation of the friction and wear behaviors of polyphenylene sulfide filled with solid lubricants[END_REF][START_REF] Zhao | A study of the modification of the friction and wear behavior of polyphenylene sulfide by particulate Ag 2 S and PbTe fillers[END_REF][START_REF] Zhao | The mechanism of filler action and the criterion of filler selection for reducing wear[END_REF][START_REF] Schwartz | The role of filler deformability, filler-polymer bonding, and counterface material on the tribological behavior of polyphenylene sulfide (PPS)[END_REF][START_REF] Cho | Friction and wear studies using Taguchi method on polyphenylene sulfide filled with a complex mixture of MoS 2 , Al 2 O 3 , and other compounds[END_REF]. The concept of adding particles of sub-micro-or nano-scale into polymers is one of the most intriguing subjects in the recent decades. On the one hand, the action of the particle angularity, which is detrimental to the wear resistance, is greatly diminished on those scales. On the other hand, proper particulate nano-fillers contribute positively towards the development of a thin and uniform transfer film and the better adhesion of transfer film to the counterpart during sliding [START_REF] Schwartz | Studies on the tribological behavior and transfer film-ounterface bond strength for polyphenylene sulfide filled with nanscale alumina particles[END_REF][START_REF] Bahadur | Effect of transfer film structure, composition and bonding on the tribological behavior of polyphenylene sulfide filled with nano particles of TiO 2 , ZnO, CuO and SiC[END_REF], which play a crucial role in the enhancement of wear resistance [START_REF] Bahadur | The development of transfer layers and their role in polymer tribology[END_REF]. However, there are few papers that investigated the influence of the hybrid reinforcement of sub-micro/nano-particles and short carbon fibers on the tribological behavior of polymer composites [START_REF] Zhang | Tribological characteristics of micro-and nanoparticle filled polymer composites[END_REF][START_REF] Chang | Effect of nanoparticles on the tribological behaviour of short carbon fibre reinforced poly(etherimide) composites[END_REF][START_REF] Chang | Tribological properties of epoxy nanocomposites Part II. A combinative effect of short carbon fibre with nano-TiO 2[END_REF]. Very recently, Cho and Bahadur [START_REF] Cho | Study of the tribological synergistic effects in nano CuO-filled and fiber-reinforced polyphenylene sulfide composites[END_REF] reported that a lower wear rate was attained in the case that nano CuO particles and short carbon/aramid fibers were added into PPS, as compared to that including one of the two fillers alone.

The objective of the present study is to investigate on the tribological behavior of PPS composites filled with short carbon fibers and sub-micro TiO 2 particles. An artificial neural network (ANN) approach, a powerful analytical tool, is introduced to model the functional relationship between the wear properties of PPS composites and the chosen parameters, including material compositions and testing conditions.

According to the visualized profiles of the wear properties predicted by the ANNs, an optimal combination of short carbon fibers and sub-micro TiO 2 particles could be obtained to fulfill the hybrid reinforcement, which significantly improves the wear resistance of PPS composites. Moreover, the wear mechanism for the synergistic effect of the two kinds of fillers was also studied based on the SEM analysis of the surfaces of sliding bodies.

Experimental

Polyphenylene-sulfide (Fortron 214 C, Ticona GmbH) was used as a matrix material.

Pitch-based short carbon fibers (Kureha M-2007S, Kureha Chemicals GmbH) and TiO 2 (Kronos 2310, Kronos Titan GmbH) were selected as fillers. The total content of the fillers in the matrix was up to 22 vol.%. The average diameter and length of the short carbon fiber were approximately 14.5 m and 90 m, respectively. The average size of TiO 2 sub-micro-particle is about 300 nm.

The compounding of the fillers with the matrix was achieved via twin screw extruder (ZE25Ax44D, Berstorff). The compounds were further molded into plates (80×80×4 mm 3 ) using an injection molding machine (Allrounder, Arburg GmbH). For the purpose of testing, samples were cut into pins with a contact surface of 4×4 mm 2 . In order to ensure identical flow conditions, only the middle section of the plates were chosen for machining the samples.

Sliding wear tests were completed on a pin-on-disc set-up under different pv-conditions at room temperature. The counterpart (steel disc 100Cr6, LS 2542, INA Scheffler KG) was cleaned with acetone prior to test. The surface roughness of the counterpart was measured as average roughness R a = 0.19 m and peak-to-valley roughness R t = 2.20 m respectively using a Mahr Perthometer (Perthen, Mahr-Perthen).

The applied pressure was varied between 1 and 3 MPa, the sliding speed was set as 1 or 3 m/s, and the testing time was fixed at 20 hours. For some of the materials, e.g. neat PPS, the test was stopped after 1 hour due to the severe wear. In the course of the experiments both the normal and frictional forces were recorded to determine the friction coefficient. The specific wear rate, w s , was calculated by the following equation:

[ / ] s N m w mm Nm v t F ρ ∆ = ⋅ ⋅ ⋅ ( 1
)
where m is the mass loss, is the density of the material, v is the sliding speed, and t is the duration of test. F N represents the normal force imposed on the specimen during sliding.

Since the two testing parameters, i.e. the sliding speed v and the applied pressure p

( N F p Contact area =
), are changed independently, a time related depth wear rate, w t , is introduced to evaluate the wear behavior under various pv-conditions.

[ / ]

t s h w w p v mm s t

∆ = ⋅ ⋅ = (2)
where h represents the height reduction of the specimen after the test.

Finally the morphologies of selected worn pins and wear tracks were examined by scanning electron microscopy (SEM) (Jeol 6300, Jeol).

3 Artificial neural network analysis

Principle

The ANN technology is inspired by the biological neural system, and has been used

to solve a wide variety of problems in diverse fields (see Reference [START_REF] Zeng | Neural computing in mechanics[END_REF][START_REF] Zhang | Artificial neural networks applied to polymer composites: a review[END_REF][START_REF] Kadi | Modeling the mechanical behavior of fiber-reinforced polymeric composite materials using artificial neural networks-A review[END_REF] for the reviews about its application in material science). ANN is ideally suitable for some complex, non-linear and multi-dimensional problems because it is able to imitate the learning capability of human. This means the network can learn directly from the examples without any prior formulae about the nature of the problem, and generalize by itself some knowledge, which could be applied for new cases. A neural network is a system composed of many cross linked simple processing units, also called neurons. As illustrated in Fig. 1, a neural network is generally defined as three parts connected in series: input layer, hidden layer and output layer [START_REF] Haykin | Neural networks: a comprehensive foundation[END_REF]. The coarse information is accepted by the input layer, and processed in the hidden layer. Finally the results are exported via the output layer. For convenience, the structure of the neural network is described as the following notation:

[ ]

1 2 ... in h out h N N N N N - --- - (3)
where N in and N out represent the number of neurons in the input and output layer, respectively. N h is the number of neurons in each hidden layer, and h denotes the [ ]

2 1 1 ( ) ( ) 2 L t E d t p t L = = - (4)
where L refers to the number of training patterns, d(t) is the desired output value, and p(t) is the target output value predicted by the ANN for the tth pattern. During the training process, the network is presented with the data for hundreds of cycles, the weights and biases are adjusted until the expected error level is achieved or the maximum iteration is reached. This iterative adjustment of the weights and biases follows the equations

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1)
( ) ( 1)

n n ji ji n ji n n j j n j E W k W k W E b k b k b α α ∂ = --∂ ∂ = --∂ (5)
where α represents the learning rate, and k refers to the iteration.

Procedure

Table 1 lists the chosen input variables (material compositions and testing conditions) it is clear that they can work simultaneously and make the respective effects superposed.

Influence of testing conditions on tribological behavior

Fig. 3 displays the ANN predicted 3D profiles of the specific wear rate and the friction coefficient as function of the sliding speed and the applied pressure for PPS with 15 vol.% SCF and 5 vol.% TiO 2 . The measured data are plotted as the black points with error bars. In Fig. 3a, the specific wear rate increases obviously with the increase in applied pressure at low sliding speed (1 m/s), whereas it becomes less sensitive to the applied pressure at high sliding speed (3 m/s). Fig. 3b shows the variation of friction coefficient versus the testing conditions. The friction coefficient exhibits a monotonic decrease trend when ether sliding speed or applied pressure was increased. It can be estimated that a relatively lower value will be achieved under the pv-conditions of 3×3 MPa•m/s (3 MPa × 3 m/s). However, the stick-slip phenomenon occurred during the experiments under this condition and resulted in that the friction coefficient can not be accurately determined.

When plotting the results in terms of the depth wear rate w t (Fig. 4), one can see according to the fitting curve that w t increases with the product of applied pressure and sliding speed in a quasi-linear fashion, at a slope which corresponds to an average specific wear rate of 0.7×10 -6 mm 3 /Nm over a pv-product of 9 MPa•m/s. Only under the pv-conditions of 3×1 MPa•m/s, there is a remarkable deviation from the fitting curve.

Discussions

Hybrid reinforcement: short carbon fibers and sub-micro TiO 2 particles

The mechanical properties of multi-phase materials can be generally considered as a function of the respective contribution of each phase [START_REF] Friedrich | Effects of various fillers on the sliding wear of polymer composites[END_REF]. The situation is more complicated with regard to the wear behavior, since some chemical reactions and physical interactions among these phases may occur during the wear process and dominate the wear performance of polymer composites. It is well known that short carbon fibers serve as a fundamental load bearing element in the wear process of polymer composites [START_REF] Lhymn | Friction and wear of fiber reinforced PPS composites[END_REF][START_REF] Jain | Investigation of the wear mechanism of carbon-fiber-reinforced acetal[END_REF]. Moreover, they have a lubricating function and enhance the thermal conductivity of matrix [START_REF] Xian | Sliding wear of polyetherimide matrix composites I. Influence of short carbon fiber reinforcement[END_REF]. Generally, the wear process of short carbon fiber reinforced composites can be summarized in the following stages: (i) after wear of the matrix material on and around the fibers, (ii) the fibers get exposed to the counterpart asperities, and are thinned down. (iii) Subsequently they get cracked, eventually pulverized and removed, or pulled-out from matrix [START_REF] Friedrich | Wear of reinforced polymers by different abrasive counterparts[END_REF]. These mechanisms can be recognized in the SEM microphotograph of the worn surfaces, as displayed in Fig. 5a.

In Fig. 5b, the wear grooves and the material fragments indicate that the solely incorporated sub-micro TiO 2 particles, in particular at high content, result in a severer wear, though the hard particles also help to develop the toughness and stiffness of the matrix to some extent. This effect is ascribed to that the particles act as an abrasive third body and meanwhile increase the discontinuities in matrix. The combination of the two kinds of fillers, however, promotes a pronounced change in wear process. It can be seen in Fig. 5c that a considerable amount of compacted wear debris are collected around the fibers, acting temporarily as a distance holder between the composite pin and the steel counterpart, and thus protect the fibers. Furthermore, the matrix is much smoother with fewer small scratches in comparison to the PPS filled with short carbon fibers alone. An additional positive effect of the sub-micro-particles is considered to be occurring. The sub-micro TiO 2 particles, when being used in combination with the much larger short carbon fibers, are not easily ploughed out of the matrix by counterpart asperities.

Instead, they are gradually released and likely start to act as a rolling body between the two mating surfaces. As illustrated in Fig. 6a, the effect of sub-micro-particles may involve two aspects:

1.

The rolling action of sub-micro-particles protects the short carbon fibers, especially at their edges, from being easily pulled out of the matrix [START_REF] Chang | Tribological properties of epoxy nanocomposites Part II. A combinative effect of short carbon fibre with nano-TiO 2[END_REF]. Moreover, the particles can polish the exposed fiber surfaces during the sliding, since they are slightly harder than short carbon fibers. In this way, they reduce the hard abrasion between the two contact surfaces to mild one.

2.

The sub-micro-particles, either operating alone or being wrapped by the matrix material, act as a ball-bearing component, which means that the particles roll rather than slide between the two mating surfaces, and reduce the shear stress, the friction coefficient and the contact temperature. This concept was also reported with the fullerene-like WS 2 nano-particles as additives for lubrication fluids [START_REF] Rapoport | Slow release of fullerene-like WS 2 nanoparticles as a superior solid lubrication mechanism in composite matrices[END_REF]. When counterpart tip comes to the front edge of fiber, the complexes prevent the tip from directly scraping the fiber (Fig. 6c). Meanwhile, they are hindered by the exposed fiber and accumulate there. (iii) The particles become an abrasive component and polish the fiber surface (Fig. 6d). (iv) Finally, the counterpart tip moves through the whole surface of fiber, sometimes damaging the back edge of the fiber (Fig. 5c.), and then starts to plough the soft matrix again (Fig. 6e).

Wear under high pv-conditions

The wear of materials usually becomes severer under high pv-conditions. However, sliding speed and applied pressure has different influence on the wear behavior. In the present work, the rise in sliding speed leads to a relatively gentler increase of wear in comparison with that in applied pressure. As already shown in Fig. 4, pv-condition of × MPa•m/s leads to a higher depth wear rate than that at 1×3 MPa•m/s or even 2×3 MPa•m/s. It is believed that this phenomenon is related to the contact temperature in the process of wear, which is mainly controlled by the sliding speed. Table 2 This indicates that under these pv-conditions the PPS matrix on the surface of the composite pin can easily deform plastically when transfer film is initialized on the counterpart surface, and contribute to develop a compact and smooth transfer film with better adhesion to the disk. In consequence, the wear was reduced from hard abrasion to milder one. 7c) is much smoother than that at pv-product of 3×1 MPa•m/s (Fig. 7a). Despite much higher sliding speed and similar applied pressure, the fibers remain well embedded in the matrix, whilst less pulverization and peeling-off of the fibers takes place on the worn surface. Likewise, the wear track on the steel disk at pv-product of 2×3 MPa•m/s is smoother with fewer apparent scratches (Fig. 7d), in contrast with that at pv-product of 3×1 MPa•m/s (Fig. 7b). It is noteworthy that there are substantially more sphere-shape debris evenly distributing on the wear track at pv-product of 2×3 MPa•m/s, as shown in Fig. 7d. The debris (presumably TiO 2 wrapped by PPS), which act in a rolling fashion, is considered to be one of main factors that reduce the friction coefficient as well as severity of wear.

Therefore, it is considered that the TiO 2 particles can be released faster at an appropriately high sliding speed. This may enhance the positive rolling effect, which provides a lasting protection to the worn surface from the aggressive damage by the hard asperity tips of counterpart.

Captions of figures

3

 3

  amount of hidden layers. The neurons in the contiguous layers are cross-linked by the weighted inter-connections, which resemble the bioelectricity passing through the real axons. Besides the weight, the neurons also have a bias value, which refers to their initial states. During the training process, the generalized knowledge is memorized in terms of the combination of weights and biases. In the hidden layers and the output layer, the individual neuron gets the information from the neurons in the previous layer, and modulates the information with a transfer function, weight as well as bias, and then outputs the result.The neural network in this study is operated by a backpropogation algorithm, which is one of the most widely used algorithms for multi-layer perceptron. The weights and biases are initialized as small random values. Then the network is fed with a series of training data. During an iterative training process, the weights and biases are adjusted by the network itself to optimize its performance, namely to minimize the difference between output values and real values. A quadratic form is used as the performance function:

4 Results 4 . 1

 441 Fig.2ashows both the measured and the predicted results of the specific wear rate versus the volume content of short carbon fibers and TiO 2 particles. It can be seen that the sole incorporation of TiO 2 particles leads to an increase in specific wear rate w s . The significant reductions (by a factor of 10 2 ) in w s show that short carbon fibers plays a key role in improving the wear resistance of PPS, both as a single or a second reinforcement phase. This improvement, however, tends to a saturation state when the content of short carbon fiber exceeds 10 vol.%. When both of short carbon fibers and TiO 2 particles are incorporated to PPS matrix, a hybrid reinforcement effect can be found. For both 10 vol.% and 15 vol.% short carbon fibers, the specific wear rate exhibited a drop-off tendency with the increase of TiO 2 content in the range 3-7 vol.%. According to the experimental results the composition of PPS with 15 vol.% SCF and 5 vol.% TiO 2 gives the highest wear resistance. Furthermore, it is estimated from the predicted trends that the lowest specific wear rate can be obtained for PPS with 15 vol.% SCF and 6 vol.% TiO 2 . Fig.2bshows the trend of measured and predicted friction coefficient. In comparison to the specific wear rate, both two fillers result in a monotonic decrease in friction coefficient. Although the mechanisms of the reduction in friction coefficient contributed by each of the two fillers are different, which will be discussed in section 5,

Fig. 6 (

 6 Fig. 6(b-e) elucidate the positive rolling effect of sub-micro-particles in detail, which can be divided into four stages. (i) During the wear the particles are ploughed out by the counterpart tip together with matrix material. These complexes play a role as a rolling third body between the hard metal disk and the soft polymer composite pin (Fig. 6b). (ii)

  lists the maximum value of contact temperature under different pv-conditions for PPS with 15 vol.% SCF and 5 vol.% TiO 2 . This peak temperature generally occurs at the early stage of wear, accompanied with the peak of friction coefficient. It is found that the maximum contact temperatures at pv-product of 2×3 MPa•m/s and 3×3 MPa•m/s reach 113.9 o C and 120.1 o C respectively, which are higher than the glass transition temperature of the PPS composie (110.3 o C, measured by dynamic mechanical thermal analysis (DMTA)).

Fig. 7

 7 Fig. 7 gives the SEM micrographs of the worn surface of PPS with 15 vol.% SCF and 5 vol.% TiO 2 and the corresponding wear track on the steel counterpart at different pv-products ((a-b) 3×1 MPa•m/s; (c-d) 2×3 MPa•m/s). It can be seen that the worn surface of the sample at pv-product of 2×3 MPa•m/s (Fig. 7c) is much smoother than

6 Conclusions 1 .

 61 The application of artificial neural network technology permitted to analyze the wear and friction behavior of the PPS composites filled with short carbon fibers and sub-micro TiO 2 particles. The ANN prediction and the experimental observation show a good agreement. 2. A synergistic effect of the two fillers on improving the wear resistance was reported.The lowest specific wear rate of approximately 4.0×10 -7 mm 3 /Nm was found for PPS with 15 vol.% SCF and 5 vol.% TiO 2 . A more optimal composition of PPS with 15 vol.% SCF and 6 vol.% TiO 2 was estimated by ANN prediction.3. The possible interaction of short carbon fibers and TiO 2 could be interpreted in terms of a positive rolling action of sub-micro TiO 2 particles, which protected the short carbon fibers from being pulled-out from the matrix by the counterpart asperities and finally resulted in an enhanced wear resistance of the composites.4. Sliding speed and applied pressure have their respective influence on the wear of PPS with 15 vol.% SCF and 5 vol.% TiO 2 . Under higher pv-conditions, the increase in sliding speed results in a relative milder increase of depth wear rate in comparison to that in applied pressure. The phenomenon indicates that a combination of appropriately high sliding speed and applied pressure can augment the hybrid reinforcement effect of the two fillers and develop a compact and smooth transfer film, and consequently reduce the severity of wear.

Fig. 1 .

 1 Fig. 1. A schematic illustration of the artificial neural network.

Fig. 2 .

 2 Fig. 2. Measured and predicted (a) specific wear rate and (b) friction coefficient vs. the

Fig. 3 .

 3 Fig. 3. ANN predicted 3D profiles of (a) the specific wear rate, (b) the friction coefficient as function of the sliding speed and applied pressure for PPS with15 vol.% SCF and 5 vol.% TiO 2 . The measured data are plotted as black points with error bars.

Fig. 4 .

 4 Fig. 4. Measured and predicted depth wear rate as a function of pv-product. A quasi-linear increase trend can be observed.

Fig. 5 .

 5 Fig. 5. SEM micrographs of the worn surfaces under standard test conditions (sliding speed = 1 m/s, applied pressure = 1 MPa) for (a) PPS with 15 vol.% SCF, (b) PPS with 5 vol.% TiO 2 , (c) PPS with 15 vol.% SCF and 5 vol.% TiO 2 . The arrows in the top-left corner indicate the sliding direction of the pin.

Fig. 6 .

 6 Fig. 6. Illustration of the rolling effect of sub-micro-particles protecting the short carbon fibers. (a) A whole view. (b) Matrix-wrapped particles roll on the matrix, and reduce the abrasion of matrix. (c) Particles prevent the counterpart tip from directly scraping the fiber. However, the debris ploughed by the tip are blocked by the exposed fiber and accumulate nearby. (d) Particles act as an abrasive component to polish the fiber surface. (e) The counterpart tip moves through the fiber surface, sometimes result in the damage of fiber edge.

Fig. 7 .

 7 Fig. 7. SEM micrographs of PPS with 15 vol.% SCF and 5 vol.% TiO 2 : (a) worn surface and (b) wear track at 3×1 MPa•m/s; (c) worn surface and (d) wear track at 2×3 MPa•m/s. The arrows on the top-left corner indicate the sliding direction of the pin.

Fig. 1 .Fig. 2 .Fig. 3 .

 123 Fig. 1. A schematic illustration of the artificial neural network.

Fig. 4 .

 4 Fig. 4.Measured and predicted depth wear rate as a function of pv-product. A quasi-linear increase trend can be observed.

Fig. 5 .

 5 SEM micrographs of the worn surfaces under standard test conditions (sliding speed = 1 m/s, applied pressure = 1 MPa) for (a) PPS with 15 vol.% SCF, (b) PPS with 5 vol.% TiO 2 , (c) PPS with 15 vol.% SCF and 5 vol.% TiO 2 . The arrows in the top-left corner indicate the sliding direction of the pin.

Fig. 6 .Fig. 7 .

 67 Illustration of the rolling effect of sub-micro-particles protecting the short carbon fibers. (a) A whole view. (b) Matrix-wrapped particles roll on the matrix, and reduce the abrasion of matrix. (c) Particles prevent the counterpart tip from directly scraping the fiber. However, the debris ploughed by the tip are blocked by the exposed fiber and accumulate nearby. (d) Particles act as an abrasive component to polish the fiber surface. (e) The counterpart tip moves through the fiber surface, sometimes result in the damage of fiber edge. SEM micrographs of PPS with 15 vol.% SCF and 5 vol.% TiO 2 : (a) worn surface and (b) wear track at 3×1 MPa•m/s; (c) worn surface and (d) wear track at 2×3 MPa•m/s. The arrows on the top-left corner indicate the sliding direction of the pin.s

Table 1

 1 Measured parameters for input and output of ANN.

	Input
	Material	PPS matrix	Short carbon fiber	Nano-TiO 2	Lubricant
	compositions	(65-100 vol.%)	(0, 10, 15 vol.%)	particle	PTFE
				(0, 1, 3, 5, 7	(0, 5, 10 vol.%)
				vol.%)	Graphite
					(0, 5, 10 vol.%)
	Testing	Sliding speed	Applied pressure
	conditions	(1, 3 m/s)	(1, 2, 3 MPa)
	Output

Table 2

 2 Maximum contact temperature of PPS with 15 vol.% SCF and 5 vol.% TiO 2 at different pv-products.

	pv-product [MPa•m/s]	1	2	3	3	6	9
	Sliding speed [m/s]	1	1	1	3	3	3
	Applied pressure [MPa]	1	2	3	1	2	3
	Maximum contact temperature [ o C] 45.0	47.3	54.6	57.5 113.9 120.1

(a) (b)

Acknowledgement

The authors gratefully acknowledge the financial support of the German Research Foundation (DFG FR 675/45-1: Wear prediction of polymers and composites using an artificial neural network approach). We would like to thank F. Richter (Ensinger GmbH, Germany) and Dipl.-Ing. B. Lehmann (IVW, Germany) for their assistance in experiments. Dr. G. Zhang (IVW, Germany) is appreciated for his valuable discussions. K. Friedrich is grateful to the Australian Research Council for his International Professorial Fellowship at the University of Sydney in 2006/2007.


G Wimmer
email: wimmer@ilsb.tuwien.ac.at

H E Pettermann

A semi-analytical model for the simulation of delamination in laminated composites

Keywords: Layered structures A, Fracture B, Non linear behavior B, Delamination C

A semi-analytical model for the simulation of delamination in laminated composite components is presented. Two well established approaches are combined in a way to use their specific advantages. The Griffith crack growth criterion is used in order to predict the load required to propagate the delamination, the delamination growth direction, and the growth stability. The compliance functions used in the criterion are evaluated numerically, the Finite Element Method is applied for this purpose.

The proposed model allows to treat non-linear structural behavior, including snapthrough and snap-back behavior, in a computational efficient way. Furthermore,

Introduction

Fiber reinforced laminated composites have become more and more important over the past years due to their great potential in weight saving. They are successfully used in applications where high strength, high stiffness, and low weight are required, such as aircraft structures, sports equipment, ship hulls, etc. Even if widely used their potential is not yet fully exploited due to a lack of predictive simulation tools for the non-linear behavior. More precise and more reliable tools are required to handle progressive damage and to predict the ultimate load of laminated composites.

One of the critical failure modes in laminated composites is delamination, as it can change the structural stiffness significantly and is difficult to detect during inspection. Delamination causes a non-linear structural response and can lead to snap-through or snap-back behavior. It is caused by overloading of the interface due to out-of-plane shear and normal stresses attributed to, e.g. transverse loading, free edge effect, ply drop offs, or local load introduction. An overview about delamination and its effects is given in [1,[START_REF] Bolotin | Delamination in composite structures: its origin, buckling, growth and stability[END_REF]. In the following, delamination is considered as the growth of an existing flaw in the interface between two plies. The formation of a first flaw in an initially flawless interface is not considered to be part of the delamination process.

In order to treat the discontinuity at the delamination front fracture mechanics can be used. It is widely accepted that conventional Carbon Fiber Reinforced Polymers (CFRP) show a brittle fracture behavior. Consequently, local material non-linearities in the vicinity of the delamination front are neglected and Linear Elastic Fracture Mechanics (LEFM) is used [START_REF] Hild | Discrete versus continuum damage mechanics: a probabilistic perspective[END_REF]. For the prediction of delamination growth the Griffith crack growth criterion is widely utilized [START_REF] Krueger | Virtual crack closure technique: History, approach, and applications[END_REF].

It states that the released energy at delamination growth has to be equal or larger than the energy required to create new delaminated area. The energy released can be computed from the load applied and the change of the structural compliance caused by the delamination progression [START_REF] Anderson | Fracture Mechanics, Fundamentals and Applications[END_REF].

Within the framework of the Finite Element Method (FEM) methods such as the J-integral [START_REF] Rice | A path independent integral and the approximate analysis of strain concentration by notches and cracks[END_REF], the Virtual Crack Extension Technique [START_REF] Hellen | On the method of the virtual crack closure extension[END_REF], and the Virtual Crack Closure Technique (VCCT) [START_REF] Krueger | Virtual crack closure technique: History, approach, and applications[END_REF][START_REF] Raju | Calculation of strain-energy release rates with higher order and singular finite elements[END_REF][START_REF] Rybicki | A finite element calculation of stress intensity factors by a modified crack closure integral[END_REF] are used successfully for the simulation of delamination. Another approach to treat delamination progression was developed in [START_REF] Dugdale | Yielding of steel sheets containing slits[END_REF][START_REF] Barenblatt | The mathematical theory of equilibrium cracks in brittle fracture[END_REF] where the concept of a cohesive zone was introduced.

Based on this idea interface elements have been developed within the FEM that are used successfully for the simulation of delamination [START_REF] Alfano | Finite elemente interface models for the delamination analysis of laminated composites: mechanical and computational issues[END_REF][START_REF] Camanho | Numerical simulation of delamination growth in composite materials[END_REF]. These approaches, however, are numerically expensive and the achievement of convergence can be tricky.

In the present paper two well established approaches, the Griffith criterion and the FEM, are combined to use their specific advantages and to circumvent their shortcomings. Both approaches are widely used for the simulation of delamination. Their experimental verification is treated in depth in the literature, e.g. [START_REF] Krueger | Virtual crack closure technique: History, approach, and applications[END_REF][START_REF] Camanho | Numerical simulation of delamination growth in composite materials[END_REF]. To apply the Griffith criterion the structural compliance is required as function of the parameters describing the delamination. Since this function cannot be given analytically for general cases, it is computed by a numerical parameter study employing the FEM. A particular consideration of the actual mode mix at the delamination front is also required, the VCCT is used for this purpose. The numerical parameter study may be costly in terms of CPU-time but can be done in an fully automated procedure. Once the compliance and the mode mix are evaluated the delamination behavior of the considered structure is at hand and further steps are very inexpensive in terms of computational effort. All further conclusions concerning delamination growth and its stability are directly deduced from existing data. Any delamination which is described by the parameters considered in the numerical study can be assessed in terms of the direction of the delamination growth, the load required to propagate the delamination, and the growth stability. Stable and unstable delamination growth and the corresponding nonlinear structural response, including snap-through and snap-back behavior, can be handled in a computational efficient way. If conventional FEM tools are used, every particular configuration of the delamination has to be investigated separately. For each delamination a non-linear analysis is required for the simulation of delamination growth. This way, only information about particular chosen delaminations is obtained and the achievement of convergence might be tricky. The proposed semi-analytical approach, however, is set up in a way to gain a complete picture of the delamination problem and the corresponding non-linear structural response. It offers a systematic understanding of the influence of the delamination size and position on the growth, the growth stability, and the structural response. Critical configurations where delamination growth changes from stable to unstable, or vice versa, are determined.

In order to show the application and the capabilities of the proposed semi-

The semi-analytical model

The Griffith criterion is adapted in order to treat delamination in laminated composite components. The compliance functions used in the criterion are evaluated numerically; in the present study the FEM is used for this purpose.

Only plane problems with through-width delaminations are considered here, allowing a two dimensional description of the structures analyzed.

The Griffith energy criterion

Crack growth is a irreversible process that is accompanied by the formation of new crack surfaces. According to Griffith, the total energy, Π * , of an elastic body containing a crack does not change during equilibrium crack growth (see e.g. [START_REF] Anderson | Fracture Mechanics, Fundamentals and Applications[END_REF]). The energy balance for an increase in crack area, ∂A, reads,

∂Π * ∂A = ∂Π ∂A + ∂Γ ∂A = 0 . ( 1
)
The potential energy, Π, is the sum of the the strain energy and the potential of the external forces.

-
G G c = 1 . ( 2
)
Growth is stable if the total energy is a minimum. The stability condition reads,

∂ 2 Π * ∂A 2 > 0 . ( 3
)
From Eqs. ( 1) and (3) growth and stability conditions for a structure loaded by a single concentrated force, P , or a prescribed displacement, u, are derived.

The structure is assumed to have unit thickness, thus, an increase in crack length, ∂a, is equal to an increase in crack area, ∂A. In the linear elastic range the relation between the force and the displacement of the load introduction point in load direction is described by the compliance, C, as,

C = u P . (4)
In the case of displacement controlled loading, the variation of the potential of the external forces is zero and the potential energy of the structure is equal to the strain energy,

Π u = u 2 2C . ( 5
)
An increase in crack length leads to an increase of the compliance and to a decrease of the potential energy of the structure. From Eqs. ( 1) and ( 5) the displacement required to propagate the crack, u 0 , is computed as,

u 0 = 2G c C 2 ∂C ∂a . ( 6
)
Using Eq. ( 3) the stability condition for stable crack growth for displacement controlled loading reads,

- G c ∂C ∂a ⎛ ⎝ ∂ 2 C ∂a 2 - 2 C ∂C ∂a 2 ⎞ ⎠ + ∂G c ∂a > 0 . ( 7
)
In the case of force controlled loading the external and the internal forces contribute to the potential energy of the structure, which reads [START_REF] Anderson | Fracture Mechanics, Fundamentals and Applications[END_REF],

Π P = - P 2 C 2 . ( 8
)
The force required to propagate the crack, P 0 , is computed from Eqs. ( 2) and ( 8) as,

P 0 = 2G c ∂C ∂a , ( 9
)
and the stability condition for force controlled loading reads,

- G c ∂C ∂a ∂ 2 C ∂a 2 + ∂G c ∂a > 0 . ( 10
)
Above equations are based on the assumption of linear elasticity, this means that the structural compliance is load independent. In the case of closed cracks this assumption is violated as friction and contact take place. However, many delamination problems of practical relevance have open crack fronts and can be treated with the proposed semi-analytical model.

Mode mix

In general plane problems crack growth is caused by a combination of mode I and mode II loading, see e.g. [START_REF] Krueger | Virtual crack closure technique: History, approach, and applications[END_REF][START_REF] Turon | An interface damage model for the simulation of delamination under variable-mode ratio in composite materials[END_REF]. From experimental material tests the critical energy release rate for pure mode I and pure mode II loading, G Ic and G IIc , and a criterion for mixed mode cases are known. Here, a quadratic crack growth criterion [16] is used and Eq. ( 2) extends to,

G I G Ic 2 + G II G IIc 2 = 1 , ( 11
)
with individual contributions of fracture mode I and fracture mode II. From the Griffith energy balance, Eq. ( 1), only the total energy release rate, G I +G II ,

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can be assessed and, thus, Eq. ( 11) cannot be applied directly. A mode mix variable is introduced as,

m = G II G I + G II . ( 12
)
In the linear elastic range the mode mix depends on the crack geometry and the loading scenario but is independent of the actual load magnitude. Equation ( 11) together with Eq. ( 12) allows to define a critical energy release rate for mixed mode loading,

G c = 1 1-m G Ic 2 + m G IIc 2 . ( 13
)
Substituting this equation into Eqs. ( 6) to [START_REF] Dugdale | Yielding of steel sheets containing slits[END_REF] yields crack growth and stability conditions for mixed mode loading. Note that the same results concerning delamination growth are obtained if the mode mix is used to derive the individual mode I and mode II energy release rates and Eq. ( 11) is applied.

However, for the computation of the stability the critical energy release rate for mixed mode cases needs to be defined.

Delamination growth

In a laminate build up by plies of a unidirectional CFRP, the fracture toughness of the plies is much greater than the fracture toughness of the interfaces between the plies. Thus, it can be assumed that delaminations only grow along the interface between the plies. In Fig. 1 a laminate containing a throughwidth delamination in one of the interfaces is sketched. The delamination can grow in both directions individually. For plane problems a delamination is entirely described by two parameters. In the present paper the distances between the edges of the structure and the delamination fronts, a 1 and a 2 , are used as parameters, see Fig. 1. Note that a decrease in the parameter a 1 or a 2 is equal to an increase of the delaminated area. For general three dimensional problems some assumptions concerning the shape of the delamination have to be made in order to keep the number of parameters required to describe the delamination low. Such problems are beyond the scope of the present paper, but the semi-analytical approach can handle such problems too. As the mode mix at each delamination front, m a 1 and m a 2 , does not need to be equal, the individual values of the critical energy release rate at each front, G c,a 1 and G c,a 2 , have to be considered. From Eq. ( 6) and ( 9) the load (force or displacement) required to propagate the delamination can be computed for each delamination front. The delamination will grow at the higher loaded delamination front, printed symbolically as,

∂C -∂a i G c,a i > ∂C -∂a j G c,a j ⇒ Delamination growth in direction a i ; i, j = 1, 2 . ( 14
)

Numerical evaluation of the compliance and the mode mix

The analytical equations presented above allow to determine the delamination growth direction, to compute the load required to propagate the delamination, and to predict the growth stability. In order to perform these computations the compliance and the mode mix at each delamination front, m a 1 and m a 2 , need to be provided for the current loading scenario as functions of a 1 and a 2 .

For general cases an analytical solution for these functions cannot be given, thus, an approximation is used. Discrete values of a 1 and a 2 are chosen in a range that covers all delamination configurations of interest. For each combination of a 1 and a 2 the compliance and the mode mix at each delamination front are evaluated numerically. The results of the numerical evaluation allow

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for a pointwise description of the compliance and the mode mix as functions of a 1 and a 2 . From these results the difference quotients are computed instead of the derivatives (e.g. ∂C ∂a ⇒ ΔC Δa ). Due to this approximation only incremental delamination growth can be simulated, the size of the increment being equal to the incrementation in a 1 or a 2 .

The compliance is a monotonic and continuous function of the parameters a 1 and a 2 . Their discretization has to be chosen fine enough to allow for a proper approximation of the compliance as well as its first and second derivatives.

Visual inspection of the functions shows whether or not the approximation is proper in the entire range of a 1 and a 2 . If not, a finer discretization has to be used around the corresponding values.

In the present study the FEM package ABAQUS/Standard/V6.6 (ABAQUS Inc., RI, USA) is used for the numerical evaluation. An FEM analysis procedure is set up, parameterised in a 1 and a 2 . All combination of a 1 and a 2 are analysed in a fully automated procedure. The structure is loaded with a unit load and the compliance as well as the mode mix at each delamination front are computed within a linear elastic analysis. For the computation of the mode mix the VCCT is employed. An implementation is provided by ABAQUS as an add on tool [START_REF]ABAQUS/VCCT for ABAQUS, User's Manual[END_REF]. The VCCT is based on the assumption that the energy released during the delamination growth of Δa is equal to the energy required to close the delamination over the length of -Δa. According to [START_REF] Raju | Calculation of strain-energy release rates with higher order and singular finite elements[END_REF] the energy released at delamination growth can be computed within the FEM from the nodal forces at the delamination front and the relative nodal displacement behind the delamination front. This assumption hold true if the stress and displacements fields with respect to the delamination front change only slightly during delamination growth, i.e. self similar delamination growth takes place.

This means that the increase in the delaminated area needs to be small com-pared to the total delaminated area and that the shape of the delamination does not change during delamination growth (i.e. no crack kinking) [START_REF] Krueger | Virtual crack closure technique: History, approach, and applications[END_REF]. From the consideration of the relative normal and shear forces and displacements, contributions of mode I and mode II energy release rates as well as the mode mix are computed for both delamination fronts.

From the results of the FEM procedure the compliance and its difference quotients up to the second order are computed. Furthermore, the critical energy release rate and its difference quotient are evaluated at each delamination front. Based on these data growth of any delamination can be treated which is described by the considered range of a 1 and a 2 .

Capabilities of the proposed approach

Once the functions of the compliance and the mode mix are obtained, the further computational effort for the prediction of delamination growth is extremely low. The load required to propagate the delamination and the corresponding delamination growth (Δa 1 or Δa 2 ) are accessible. The latter leads to an increase in the delaminated area and a change in the compliance. For the new delamination size, again, the load required to propagate it and the change in a 1 or a 2 are computed. Repeating these considerations allows a pointwise description of the delamination growth process and the corresponding forcedisplacement curve.

In Fig. 2 generic force-displacement curves are shown that can be handled with the proposed approach. Curves a and b show stable delamination growth for force and displacement controlled loading, respectively. The corresponding load has to be increased monotonously in order to propagate the delamina-

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tion. Delamination growth is, therefore, entirely controlled by the applied load.

Curve c shows unstable equilibrium delamination growth. Once the growth load is reached the load required to propagate the delamination decreases as the delamination size increases, which characterizes a snap-back response.

Note that such non-monotonous loading conditions cannot be realized in structural tests, consequently curve c is only a "fictitious" force-displacement curve. Curve d shows the structural response for the same delamination but monotonously increasing force. Curve e shows the results for monotonously increasing displacement. Such loading conditions lead to a energy release rate that is greater than the critical energy release rate and dynamic effects take place during delamination growth. Dynamic effects are not considered in the present paper, indicated by the dashed lines.

As additional capability the critical load, force or displacement, of a structure containing a delamination of a certain size but unknown location can be estimated, allowing for a fast interpretation of the results from non-destructive testing. The load required to propagate a delamination of a certain size is evaluated for all possible locations of the delamination. The smallest load obtained is a conservative estimate of the critical load of the structure, the corresponding location is the most detrimental.

Example: L-shaped component

The application of the semi-analytical model is demonstrated in an example.

Delamination in an L-shaped laminated structure is analyzed. Input data for the fracture mechanics based part of the model is evaluated numerically, employing the FEM. Based on this data delamination growth and its stability are treated.

Details concerning the geometry of the structure are given in Fig. 3. The structure is built up by 15 plies of a unidirectional fiber reinforced carbon/epoxy material. The plies are oriented in alternating 0 • and 90 • orientation, where the angle is measured from the xy-plane. The inner and the outer ply are oriented in 0 • direction. The load (displacement or force) is prescribed in horizontal direction at the vertical leg of the structure. The displacements in xand y-direction are locked along the bottom of the horizontal leg. Ply material data and interface properties are taken from the literature [16], see Table 1.

The structure has a considerable length in z-direction, thus, generalized plane strain conditions are used. An FEM model of the structure, parameterised in a 1 and a 2 , is set up, using two dimensional solid continuum elements with linear shape functions and full integration. Each ply is represented by three elements over the ply thickness, all element aspects ratios are close to one.

Delamination of the interface between ply five and six is analyzed. This interface is chosen because it faces the highest normal traction in the flawless case and, therefore, is the one most critically loaded. The delaminations investigated are described by two parameters a 1 and a 2 . Ninety discrete values of each parameter in the range from 1.0 mm to 9.0 mm are considered. The incrementation of the parameters is equal to the length of 3 elements along the interface (about 0.1 mm). The length of delaminations described by the chosen parameters ranges from 0.5 mm to 12 mm. For each combination of a 1 and a 2 the parameterised FEM model is used to compute the compliance and the mode mix at each delamination front. In total 4700 combinations of a 1 and a 2 are analyzed within a fully automated procedure.

The compliance is shown in Fig. 4 as function of size and location of the delamination expressed in terms of a 1 and a 2 . Note that delaminations of the

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same length but different position lie on straight lines with a slope of minus one in the a 1 -a 2 plane. Similar plots can be given for the mode mix at each delamination front.

Based on the pointwise approximations of the compliance and the mode mix growth of any delamination which is described by a 1 and a 2 can be predicted.

Selected results for delaminations with an initial size of 0.6 mm and different initial positions are given in Fig. 5. It shows the incremental change in a 1 and a 2 due to delamination growth caused by a force load. Depending on the initial position of the delamination it starts growing either in direction a 1 or a 2 . As soon as a certain size is reached, simultanous growth in both directions takes place. Independent of the initial position all delaminations converge to the same final size and position.

The delamination growth stability for displacement controlled loading is shown in Fig. 6 as function of a 1 and a 2 . The figure shows that small delaminations will grow in an unstable manner while larger delaminations grow in a stable manner. Note that here "small" and "large" are used in a qualitative sense and refer to the size of the delamination compared to the entire range of delamination sizes considered. From Fig. 6 two delaminations are picked, one (a 1 = 5.3 mm, a 2 = 4.6 mm) that will grow in a stable manner and one (a 1 = 3.9 mm, a 2 = 3.8 mm) that will grow in an unstable manner. For these delaminations growth under displacement controlled loading is predicted and the corresponding force-displacement curves are shown in Fig. 7. The curves show linear elastic response until the load required to propagate the delamination is reached. Then the small delamination grows in an unstable manner while the large one grows stable. The difference of the initial slope of the force-displacement curves corresponds to the difference in the size of the delaminations.

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The delamination growth stability under force controlled loading is shown in Fig. 8. Comparison to displacement controlled loading (Fig. 6) shows that the range of delaminations that grow in a stable manner is smaller under force controlled loading. From Fig. 8, again, two delaminations are picked, one (a 1 = 5.5 mm, a 2 = 4.4 mm) that will grow in an unstable manner and one (a 1 = 3.6 mm, a 2 = 2.4 mm) that will grow in a stable manner. Force-displacement curves for force controlled loading are shown for both delaminations in Fig. 9.

Again the small delamination grows in an unstable manner, indicated by the dashed line. The large delamination grows in a stable manner and the entire growth process is controlled by the force applied.

In Fig. 10 a force-displacement behavior for non-monotonous loading is shown.

A structure containing a small initial delamination (a 1 = 2.4 mm, a 2 = 7.6 mm) is analyzed and a pronounced snap-back behavior is found. This structural response confirms the finding that small delaminations grow in an unstable manner, while large delaminations grow in a stable manner.

Finally, the most detrimental position of a delamination with respect to the load required to propagate it is analysed. In Fig. 11 the minimum load (force or displacement) required to propagate delaminations of different sizes is shown.

The figure points out that the larger the delamination is, the lower is the load required to propagate it. As example a delamination length of 2 mm is assessed and the load required to propagate it, if it lies in its most detrimental position, is predicted to be 0.215 mm or 14.2 N, respectively.

To highlight the advantage of the proposed semi-analytical model a comparison to FEM simulations, using the delamination propagation capability of the ABAQUS/VCCT tool is done. The same FEM model as set up before for the computation of the compliance and the mode mix is used. Growth of the large delamination discussed in Fig. 7 is simulated and, of course, the results are identical with those obtained from the semi-analytical model. However, unstable delamination growth and the stability of the delamination growth process can only be analysed with the proposed semi-analytical model.

Summary

A semi-analytical model for the simulation of delamination in laminated composite components is presented. The model is based on the Griffith crack growth criterion and is able to predict equilibrium delamination growth and its stability. Compliance and mode mix functions used in the analytical part of the model are evaluated numerically by means of a fully automated FEM procedure. Plane problems with concentrated loads (force or displacement) are considered. The proposed semi-analytical approach allows to predict delamination growth and the corresponding non-linear structural response, including snap-through and snap-back behavior. On the one hand, a systematic and general understanding of the influence of size and position of a delamination on the growth stability and the structural response is gained. On the other hand, growth of delaminations of specific size and location is fully contained in the results. The load required to propagate and the growth stability are predicted. Critical configurations where growth changes from stable to unstable, or vice versa, are determined. Furthermore, for each delamination size its most detrimental position, concerning the load required to propagate it, is determined. This allows for a conservative estimation of the critical load for a structure with a delamination of a certain size but unknown location. To conclude, the proposed semi-analytical approach provides deeper understanding on delamination growth and its stability. It can be applied to laminated

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structures as well as to other problems where the crack path is known.

As an example delamination in an L-shaped structure is analyzed. The results

show that small delaminations grow in an unstable manner while large delaminations grow in a stable manner. Equilibrium delamination growth is predicted for different delamination sizes. For small delaminations a pronounced snap-back behavior is found. Table 1 Material and interface data of carbon/epoxy UD-layer, T300/976, taken from [16] Elastic constants

E 1 E 2 = E 3 G 12 = G 13 ν 12 = ν 13 ν 23 [GP a] [GP a] [GP a]

2 Fig. 1 .

 21 Fig. 1. Delamination in a laminated composite component under generalized plane strain conditions, the delamination is entirely described by two parameters a 1 and a 2 .

Fig. 2 .Fig. 3 .

 23 Fig. 2. Generic examples for the force-displacement behavior caused by delamination growth; a) stable growth under force controlled loading; b) stable growth under displacement controlled loading; c) unstable growth under non-monotonous loading; d) and e) unstable growth under monotonous loading;

Fig. 4 .

 4 Fig. 4. Compliance of the laminated component as function of size and location of a delamination expressed in terms of a 1 and a 2 .

Fig. 5 .Fig. 6 .

 56 Fig. 5. Equilibrium delamination growth in the laminated component for different initial configurations; all converging to the same final configuration.

Fig. 7 .

 7 Fig. 7. Structural response of the laminated component under displacement controlled loading for two different sizes of initial delaminations as marked in Fig 6.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Delamination growth stability in the laminated component under force controlled loading as function of size and location of the delamination; large delaminations grow in a stable manner while small delaminations grow in an unstable manner.

Fig. 10 .

 10 Fig. 10. Structural response of the laminated component under non-monotonous loading conditions for a delamination that grow in an unstable manner.

Fig. 11 .

 11 Fig. 11. Minimum load required to propagate a delamination of a certain size in its most detrimental position.

Acknowledgement

The funding of the Austrian Aeronautics Research (AAR)/Network for Materials and Engineering by the Austrian Federal Ministry of Economics and Labor is gratefully acknowledged.


Rezak Ayad
email: rezak.ayad@univ-reims.fr

Nabil Talbi

Tewfik Ghomari
email: tewfikghomari@yahoo.com

Modified discrete mindlin hypothesises for laminated composite structures

Keywords: Plate, laminated composites, finite element, bending-shear, discrete Mindlin

The present work deals with a procedure of defining discrete kinematic and mechanic hypothesises for Reissner-Mindlin plate finite elements. They are associated with the transverse shear strains for which the corresponding classical approximations caused the shear locking problem. Used in the past for isotropic homogeneous bending-shear plates, these hypothesises have been locally modified in order to be used for laminated composite plates. The derived plate finite element model, labelled DDM "Displacement Discrete Mindlin" has been applied to a 3-node triangular (DMTP: Discrete Mindlin Triangle for Plates) and validated on some standard problem tests, including one or several orthotropic layers.

ANALYTICAL ASPECTS OF THE DDM MODEL

The formulation of simple and efficient finite elements has been always the main concern of the researchers in the field of bending-shear plates. The most interesting approach was certainly the use of discrete Kirchhoff hypothesis for formulating the well known finite element for thin plates DKT (Bathe et al,80). The authors used first a quadratic interpolation of the normal rotations β difficulties. They we will be detailed in the full paper. For isotropic plates, Katili (Katili,93) and Ayad (Ayad et al,01) proposed to use two discrete Mindlin hypothesises on the three sides of the element, defined initially with 6 nodes. These hypothesises are written as All these previous equations will contribute to final and modified expressions of curvatures, shear strains and stiffness matrices in consequence. These matrices are exactly integrated using only three hammer points. More details will be presented in the full paper.

( ) 0
) ( 0 , = + - ds w k L s s s β γ (kinematic hypothesis) (2) 0 0 = - ds D T k L s s s γ ; kGh D s = ; ) 1 ( 2 ν + = E G (mechanical hypothesis) (3)

NUMERICAL RESULTS

Simply supported under uniform loading of a sandwich plate

  x and β y . Their element had initially 6 nodes with the previous two rotations per node: the bending behaviour was thus well represented. After introducing the following hypothesis (called Discrete Kirchhoff hypothesis) strains: thin plate) (1) the rotational mid-side degrees of freedom (dofs) will be written in terms of the transverse displacement w (deflection) on the three corner nodes. The final element, labelled DKT (Discrete Kirchhoff Element), is very simple (3 nodes and 3 dofs per node: w,β x ,β y ) and performs well all standard tests for thin isotropic homogeneous plates. It has been implemented in the majority of finite element programs. This popular element remained unfortunately limited for thin structures without shear effects. It became then necessary to think about new procedures leading to simple and efficient finite elements which would take into account the effects of transverse shear. The first contributions in this field (Aalto, 88) (Lardeur et al, 89) (Onate et al, 92) proposed interesting triangular elements but with some

φ

  γ s and T s are the shear strain and force on the side s, E the elasticity modulus, ν the Poisson coefficient, k the shear correction factor. The derived 3-node finite elements DKMT (displacement model proposed by Katili (Katili, 93)) and MiSP3+ (hybrid-mixed model proposed by Ayad (Ayad et al, 01)) are very simple and robust. They performed well all tests for isotropic bending-shear isotropic plates, from thin to thick situations. The second hypothesis (3) shows clearly the limitation to isotropic homogeneous plates for these elements. In order to compute laminated composite structures, we have introduced in our recent works (Ayad, 02) (Ayad et al, 03) a new modified mechanical Mindlin hypothesis, generalising then the equation (3). This new hypothesis takes the following form: inv ck H 12 are two behaviour terms of the following shear inverse matrix 1 ] is the transverse shear behaviour matrix, defined for one or several layers (laminated composites). It will be detailed with other matrices in the full paper. Introducing the bending behaviour equation {M}=[H f ]{χ}, quadratic and linear interpolations respectively for the edge rotations β s and β n , and the following equilibrium equations: being the new shear influence factor for a laminated composite plate):

Figure 1 .

 1 Figure 1. Modified Mindlin hypothesis on a side ij of a multilayered finite element

A simply supported square sandwich plate (3-layers) is submitted to a uniform loading f z .

Problem data are given by figure 2. The skin material (Layers 1 and 3) is orthotropic. The material properties of layer 2 (heart) are proportional to those of layers 1 and 3 (shear modulus of the heart = shear modulus of skins divided by 10). The results of central deflection w c are reported on the table 1. They are compared to a 3D solution given by Srinivas (Srinivas, 73) (

).

A good correlation between our model DMTP (Discrete Mindlin Triangular for Plates) and the 3D solution is then obtained. DMTP also performs well other multilayered plate examples. We'll present some of them in the full paper.

A Simply supported 9-layer square plate under uniform pressure load

A simply supported 9-layer unidirectional square laminate with length L=1000 and thickness t=250, 100

and 20, is analysed with the proposed element using a number of mesh size (Figure 3). The plate is subjected to a distributed transverse load of intensity q = q 0 sin( x/L) sin( y/L) where q 0 is its amplitude at the plate centre. the PHSDT and PFSDT (Table 1) as expected.

It is obvious that the results from element DMTS are in good agreement with the results from 3D solution compared to the element DST.

CONCLUSION


Y Aminanda

B Castanié
email: bruno.castanie@supaero.fr

J-J Barrau

P Thevenet

EXPERIMENTAL AND NUMERICAL STUDY OF COMPRESSION AFTER IMPACT OF SANDWICHS STRUCTURES WITH METALLIC SKINS

Keywords:

A finite element model is proposed to determine the residual print of sandwiches structures with Nomex honeycomb core and metallic skins indented by a spherical indenter and to simulate its behavior when this indented structure is subjected to lateral compressive loading (known as CAI/ Compression after impact). The particularities of this model rely on representing the honeycomb with a grid of non-linear springs which its behavior law is calibrated from uniform compression test. This simple model, after ACCEPTED MANUSCRIPT 2 integrating the cycle behavior law of honeycomb, allow predicting the geometry of residual print with a good precision. This model is then developed to propose a complete computation from indentation, residual print geometry to lateral compressive loading after indentation (CAI). This model allows also predicting numerically the residual strength of structure in CAI and the elliptical evolution of residual print geometry during CAI loading. A good correlation with test results is obtained except for the very small residual print depth.

INTRODUCTION

The study presented in this paper is a development of ones that have been done in ( [START_REF] Aminanda | Modelisation de l'indentation des structures sandwichs a peaux métalliques[END_REF], [START_REF] Aminanda | Experimental analysis and Modelling of Honeycomb Cores Crush[END_REF]), in which a phenomenological analysis of damage modes of cellular structure have been presented. This phenomenological study has given a possibility to propose a simple model based on grid of non-linear spring to represent Nomex honeycomb subjected to uniform compressive load. Thereafter this model has been generalized to the study of indentation on sandwiches structures with metallic skin and a good correlation with test results has been obtained. Even if the final goal of the study was to model the indentation (quasi-static or low energy and low speed impact) on aeronautic type of sandwiches structures, so with composite skins, the use of metallic skins has been taken first to avoid the complexity and difficulty of modeling the composite skins damage modes [START_REF] Abrate | Localized impact on sandwich structures with laminated facings[END_REF].

Experiments have shown that the depth of residual print after indentation would not be corresponded to the maximal depth one [START_REF] Guedra-Degeorges | Damage Tolerance of sandwich structures[END_REF][START_REF] Horrigan | Modelling of crushing due to impact in honeycomb sandwich[END_REF]. However, in aviation regulation point of view, limit of detectable damage called BVID (barely visible Impact damage) after impact is defined and beyond this limit the structure should be designed for damage tolerance [START_REF] Guedra-Degeorges | Damage Tolerance of sandwich structures[END_REF]. Thus, it is important to be able to determine first this residual print geometry which, in our knowledge, only a small number of studies in term of numerical and analytical model have been done on this subject [START_REF] Herup | Low-velocity impact damage initiation in graphite/epoxy/nomex honeycomb-sandwich plates[END_REF][START_REF] Hwang | Numerical modeling of impact-damaged sandwich composites subjected to compression-after-impact loading[END_REF][START_REF] Horrigan | Modelling of crushing due to impact in honeycomb sandwich[END_REF]. Palazotto [START_REF] Herup | Low-velocity impact damage initiation in graphite/epoxy/nomex honeycomb-sandwich plates[END_REF] has proposed a finite elements model of impact using 3 loops of iterative computation to determine successively damage on honeycomb, damage on composite skin and geometry of print.

Horrigan [START_REF] Horrigan | Modelling of crushing due to impact in honeycomb sandwich[END_REF] has proposed a continuum damage model to model the behavior of honeycomb. But, due to the continuum approach, the core damaged area could not be predicted correctly. In [START_REF] Aminanda | Modelisation de l'indentation des structures sandwichs a peaux métalliques[END_REF], honeycomb is represented by a grid of vertical springs which behavior law in compression is calibrated from uniform compressive loading experiments. In this paper, the law is developed further by integrating the cycled behavior (compression loading and unloading) of honeycomb that allows simulating the defect recovery of honeycomb after compression and the residual print geometry of the sandwiches structures after indentation. This preliminary study is important because the residual print geometry after indentation has a direct relation with residual strength of indented sandwiches structures subjected to lateral compressive loading (CAI) [START_REF] Abrate | Localized impact on sandwich structures with laminated facings[END_REF][START_REF] Guedra-Degeorges | Damage Tolerance of sandwich structures[END_REF]. Several analytical models have been proposed to simulate the CAI phenomena [START_REF] Minguet | A model for predicting behavior of impact-damaged minimum gage sandwich panels under compression[END_REF] and, recently, Lacy [START_REF] Hwang | Numerical modeling of impact-damaged sandwich composites subjected to compression-after-impact loading[END_REF] has proposed a numerical model where the initial damage geometry after impact has been measured directly from the specimen by destructive and non-destructive inspection. He used also a fixed law behavior for composite skins on the impacted area (50 % of module degradation). In general, all references concerning CAI study shows that the phenomena have not been known totally and some developments were necessary to be developed before its application in industry. The proposed model in this paper will integrate the complete computation from indentation, residual print geometry after indentation to compression after indentation (CAI). With this complete computation, the initial conditions before CAI computation in term of geometry, core damage and state of stress on skin will be identified and will be used to determine the residual strength of sandwiches structures subjected to CAI loading.

Description of CAI Tests

Specimens (figure 1 and table 1) are prepared by taking care a high surface smoothness and high dimensional precision (+-0.01mm) which are the necessary conditions to obtain a correct uniform compressive test. The standard dimension of specimen is 150x100mm.

Resins are mould into both extremities of each specimen. To transfer properly the compressive load, 2 plates of brass with 1mm of thickness are added as reinforcement at resin-nomex junction area. These specimens are previously indented on flat support (figure 2) and the indentation is carried out by imposing the displacement with a constant speed of 0.5mm/mn. A spherical indenter made from steel with radii of 57,25mm is used and several depths of maximum indentation d=0; 0.5; 1; 1.5 and 2 mm are realized to obtain a different damage area.

For compression (CAI) tests after indentation, test supports and specimens correspond to the AIRBUS standards (figure 3) that allow us to compare test results with the industrial ones. These supports and specimens are realized within the laboratory. In order to observe the evolution of damage geometry in term of its depth and its form during compressive loading, a method of 3D Image correlation using 2 cameras is employed (figure 3).

Strains gages are also used to observe the distribution of compression flux on both skins.

Compressive load acting on the specimens are measured directly from INSTRON machine and the displacement of compression surface is measured using LVDT displacement measurement.

Analysis of CAI test results

Analysis of indentation test results was described in detail in [START_REF] Aminanda | Modelisation de l'indentation des structures sandwichs a peaux métalliques[END_REF] and will not be presented again in this paper. The curves of compressive load in function of its displacement for 5 indented specimens with different depth of indentation d are plotted on figure 4. For all of the specimens, the initial stiffness is identical and corresponds to the elastic behavior of skins. Analytical calculation shows that the point where the slope is changed just after the elastic behavior of skins coincides with the yield limit of skin.

Thus, it seems that at the beginning of compression, the dimension of damage area caused by indentation do not play a significant influence. The second slope is also practically identical for all specimens. This slope seems to be controlled by plastic behavior of skins. As the force of compression increases, an inflexion is observed just after the passage of maximum force which can be qualified as the residual strength of indented structure. This residual strength depends on the depth of the maximum indentation d. It decreases significantly with the increasing of the maximum indentation depth d so when the damage in the core due to indentation is more important (figure 5).

The evolution of the damage geometry (more precisely: residual print geometry) after indentation in term of several parameters such as its depth and its radii about lateral axis a and longitudinal axis b during the compressive loading is studied. The elliptical evolution of residual print geometry during loading is described on figure 6 and the evolution of damage geometry in term of the parameters mentioned before in function of compressive load is drawn also in figure 6. By observation from 3D Image correlation, the evolution of damage area can be described as follows:

-In the region of elastic behavior of skins, the circle form of residual print geometry after indentation remains circle and its depth almost does not vary.

-At the beginning of plastic behavior of skins, the form of the print begin to be elliptical progressively in the direction of lateral axis a. In the same time, in the direction of longitudinal axis b, no evolution is observed. The print depth is also observed to be increased progressively.

-Approaching maximum compressive load (residual strength of the structure), the print depth increases brutally, the same observation is also obtained for the evolution of radii about the lateral axis a which finally it reaches the edges of specimen. It's interesting to note also that the deflection of non-indented skin below the indented area increases also rather quickly.

For specimen that was indented only with maximum indentation depth d= 0,5 mm and which has a corresponding residual print depth of 0.1 mm, the evolution of its damage geometry is quite different. This specimen, with only 2 % of decreasing in term of residual strength, behaves almost as a non-indented specimen. However, the elliptical evolution of the damage geometry is also observed with a brutal progression when approaching the maximum compressive loading. This behaviour is similar from all qualitative points of view to that observed for a thin composite skin [9].

Modeling of Compression after indentation (CAI) tests

Globally, the same finite element model using implicit SAMCEF software that was used for the study of indentation [START_REF] Aminanda | Modelisation de l'indentation des structures sandwichs a peaux métalliques[END_REF] is employed (see Figure 7). The principals of the model are as follows:

-Indented skin is modeled by Mindlin type of element. This skin has free boundary conditions except at the position of the CAI test supports (vertical blockages). These blockages did not exist during indentation (figure 3). However, literatures showed that indentation caused a local damage area and for the size of specimen used for this study, the boundary conditions are not sensitive. This insensitivity is confirmed by a posteriori numerical computation. The compression law for the brass skin was obtained from tests on a virgin (non-indented) sandwich specimen and it appeared that car the yied stress on compression was 110 MPa instead of the 100 MPa on traction one and also the hardening law was slightly different.

-Hexagonal Nomex honeycomb is modeled by a grid of non-linear vertical springs placed geometrically on the same position of honeycomb verticals edges. Its law behavior is obtained experimentally from cycled compression uniform loading test on a small block of Nomex honeycomb (figure 8). Until 0.4 mm of displacement of uniform compression surface, compression unloading give a linear return of displacement with the same slope as compression one. Beyond that displacement, the return is no more linear (ellipse with the power of 2.8). By assumption, the hysteresis behavior found from the test is judged negligible and is not taken into account in FEM computation.

-Non-indented skin had to be modeled with the same model as indented skin. But almost all the degree of freedom are blocked except its in-plan displacement (in this case translation in x and y) to allow non-indented skin deforming like a membrane during compression.

The lower skin was added because the distribution of the forces between the two skins varies as the defect progresses. It is obvious that the discrete model used before cannot directly represent the bending of the sandwich and thus the out-of-plane displacement of the non-indented skin. In fact, all the degrees of freedom of this skin are locked except for the displacements in the plane of the sandwich (x and y axis) so that the skin can deform during the compression. However, this assumption is justified because the deflection of non-indented skin below the indented area measured during tests is very small and only become significant when the compressive load approaches the maximum one. Also globally, the deflection of non-indented skin still remains negligible compared to the depth of indented area. Moreover, the computation is more robust numerically when the boundaries conditions are conserved for complete computation involving different phases.

The load acting on structure is simulated by imposing first a vertical displacement of the indenter toward Z negative (compression loading) and Z positive (compression unloading) then followed by imposing the displacement on the edges of structure to simulate the lateral compression load(CAI) on indented structure (figure 9).

Test/Computation comparison

Test/computation comparison during indentation phase are described on figure 10 In compressive loading after indentation phase, the evolution of residual print is studied by analyzing the 2 parameters: depth of residual print and profile about major axis a. The minor axis does not vary significantly. Test/Computation comparisons for these 2 parameters are represented in figure 11 for the specimen indented with maximum indentation d=1.5mm. Globally, numerical computations give the same evolution as test results. The small difference at the beginning of compression is due to the difference of residual print depth between computation and test from the previous phase. However, there is also a significant difference in term of final depth of damage area (in the maximum compressive load region) because the out-of-plan deflection of the structure is not taken into account in computation. In the other hand, the maximum compressive load, so the residual strength of indented sandwich structure is perfectly simulated. Also, the evolution of damage area in term of its profile about major axis is perfectly simulated by computation (figure 11).

A good correlation between test results and computations in term of print geometry evolution and maximum compressive load is obtained for almost all indented specimens except in the case of small indentation d=0.5mm (figure 12). For this test, the depth of residual print is so small with only 0.15mm. Investigations are carrying out to know whether the initial geometry defect such as skin curving would stabilize the indented structure whereas this defect is not taken into account in computation. It is also possible that, regarding the smallness of residual print depth, the difference comes from the insufficiency of refined mesh that creates early numerical instability.

We can note that the reactions of these springs (number 1 and 2) are very weak and increase a little bit during the elliptical progression of print geometry. We can see it for the spring placed just in the circumference of initial damage area before compression loading (spring number 1). It is interesting to note that the collapse of this spring after reaching its maximum force, more or less precedes the brutal evolution of elliptical damage geometry that is produced when the neighbor spring (number 2) placed about lateral axis a just after spring 1 collapses in its turn.

To describe mechanical behavior of print geometry evolution during compressive load, it is interesting to analyse the reaction of springs placed about major axis and in the circumference of residual print. This can be done by extracting the force of the corresponding spring (n°1,2,3, figure 13) for the computation case d max = 1.5 mm. The reaction of these springs (1 à 3) is seen to be very weak and do not increase much during denting and the gradual extension of the ellipse. After a drop of spring force which is probably due to appearance of little bump that stretch the springs, a brutal increase of force until reaching its maximum (peak force of spring) is observed and quickly reaches the critical force (the peak) for the first spring at the periphery (n°1). It is very interesting to note that the collapse of this first edge occurs only shortly before the abrupt progression of the ellipse takes place when the second edge (spring n°2) situated on the major axis of the ellipse, collapes in its turn. Numerically, it is shown here that the advance of the defect coincides with the physical phenomenon of edge buckling. So the collapse of the edge situated on the major axis of the ellipse can be proposed as the criterion of structural failure. This criterion is interesting as it could be used as an inexpensive calculation in industry to predict a breaking force that is not too conservative. These criteria should also avoid an expensive and complex numerical computation of damage propagation in term of CPU time in the case of composite skins.

CONCLUSIONS

A finite elements model is proposed to simulate the behavior of indented sandwiches structures with Nomex honeycomb core and metallic skins subjected to compressive loading ( CAI ). The model has globally a good correlation with test results for the specimens studied in term of residual strength of structure and evolution of damage area (residual print geometry) except for the very smalls indented area below BVID. The original modeling of honeycomb using a grid of non-linear springs allows describing and explaining the mechanical behavior of the damage area evolution during compressive loading on indented structure (CAI) and proposing failures criteria of the structure. This study will be continued by integrating the degradations of carbon laminate skins.

[9] Castanié B., Barrau J.J., Jaouen J.P., Rivallant S., 2004. Combined Shear/Compression Structural Testing of Asymmetric Sandwich Structures. Exp mech 44(5), 461-472.

Displacement (mm) Compression Force (N)

Yielding of the skin d=0 d=2 d=1,5 d=1 d=0,5

Maximum indentation depth (mm) Relative residual strength (%)

Figure 6 : Evolution of the dent under compression-after-impact loading for maximum dent d max of 0.5, 1 and 1.5 mm

Residual dent depth

Dent depth Compression force (N) C o l l a p s e P r o g r e s s i v e d e e p e n i n g o f d e n t

Non-evolution

Remain circular

Progressive ovalization

  for the case of maximum indentation d=2 mm. The curve of indentation force in function of indenter displacement is plotted on figure 10(a) and the profile of residual print about longitudinal axis b is plotted on figure 10 (b). The depth of residual print obtained from computation is 1.42 mm whereas 1.32 mm is obtained from test measurement that give the difference of 7%. For all specimens, the difference of residual print between computation and test varies from 25% for the smallest indentation depth (d=0.5 mm) to 7 % for the deepest one (d=2 mm). The curves obtained from the indentation test are correctly simulated by computation [1] also globally for the geometry of residual print after indentation.

Figure 1 :Figure 2 :

 12 Figure 1 : Specimen for CAI tests

Figure 3 Figure 4 :

 34 Figure 3 : Test device

Figure 5 :

 5 Figure 5 : Relative residual strength function of indentation depth.

Figure 7 :

 7 Figure 7 : Finite element model for compression-after-impact.

Figure 8 :Figure 9 :Figure 10 :Figure 11 :Figure 12 : 25 Figure 13 :

 891011122513 Figure 8 : Compression behaviour of Nomex honeycomb with cycling.

Skins

Honeycomb core