Damage in woven-fabric composites subjected to low-velocity impact

Damage in woven-fabric composites subjected to low-velocity impact N.K. Naik*, Y. Chandra Sekher, Sailendra Meduri Aerospace Engineering Department, Indian Institute of Technology, Powai, Mumbai-400 076, India Abstract The behaviour of woven-fabric laminated composite plates has been studied under transverse central low-velocity point impact by using a modi ed Hertz law and a 3D transient nite-element analysis code. The in-plane failure behaviour of the composites has been evaluated by means of a failure function based on the Tsai-Hill quadratic failure criterion. The e ect of fabric geometry on the impact behaviour of woven-fabric composites has been studied. For comparison, the impact behaviour of balanced, symmetric, crossply laminates made of unidirectional layers and unidirectional composites has been included. The studies have been carried out with plate dimensions of 150 mm150 mm6 mm for a supported boundary condition. For these studies, incident impact velocities of 3 and 1 m/s and an impactor mass of 50 gm have been used. It is observed that the in-plane failure function is lower for wovenfabric laminates than for crossply laminates, indicating that woven-fabric laminates are more resistant to impact damage. Keywords: Impact behaviour; Polymer matrix composites; Textile composites; Failure criterion; Finite-element analysis; Fabrics 1. Introduction One of the major concerns in designing composite structures is their susceptibility to impact loading. Fibre-reinforced polymer-matrix composites are known to be highly susceptible to internal damage caused by transverse loads, even under low-velocity impacts. The composites can be damaged on the surface as well as beneath the surface with relatively light impacts causing barely-visible impact damage, while the surface may appear to be undamaged to visual inspection. For the e ective use of bre-reinforced polymer-matrix composites for high-performance applications, understanding the causes for the formation of such damage under lowvelocity impact and improving the damage-resistance characteristics of the composites are important considerations which have been the topic of extensive research for the last few years. Review articles on the impact behaviour of polymer-matrix composites covering contact laws, impact dynamics, stress analysis, damage mechanics, post-impact residual property characterisation and damage-resistance improvements are available in the literature [1±5]. Many research publications are available on the impact behaviour of polymer-matrix composites covering speci c aspects [6±33]. Matrix deformation and micro-cracking, interfacial debonding, lamina splitting, delamination, bre breakage and bre pull-out are the possible modes of failure in composites subjected to impact loading. Even though bre breakage is the ultimate failure mode, the damage would initiate in the form of matrix cracking/lamina splitting and would lead to delamination. Damage-free composites are necessary for their e ective use. Traditionally laminated composites made of unidirectional (UD) layers were used for structural applications. These composites are characterised by high speci c sti ness and high speci c strength. But these composites are highly susceptible to impact damage because of their lower transverse tensile strength. One of the ways of improving impact behaviour of polymer-matrix composites is to use woven-fabric (WF) layers instead of unidirectional layers. A woven-fabric is a fabric produced by the process of weaving in which the fabric is formed by interlacing the warp and ll strands. The integrated nature of the fabric provides the balanced in-plane properties. Woven-fabric composites are characterised by high fracture toughness and ease of handling. The transverse tensile strength of the WF

732 Nomenclature a strand width BŠ strain shape function E 11, E 22, E 33 Young's moduli w r t material coordinate system E S modulus of impactor E yy transverse modulus normal to the bre orientation of the uppermost layer F I impact force vector f, F scalar contact force ffg external force vector ff m g force vector caused by the inertia terms f m peak contact force G 12, G 13, G 23 shear moduli w r t material coordinate system g inter-strand gap h strand thickness I in-plane failure function k Hertz constant KŠ sti ness matrix of the plate L x, L y, L z Lengths along X, Y and Z directions/plate dimensions m, M impactor mass MŠ global mass matrix of the plate r radius of impactor S 12, S 13, S 23 shear strengths w r t material coordinate system t indicates any time `t' t f duration of impact u, v, w displacements along X, Y and Z directions f Ug displacement vector of the plate U I displacement caused by impact force fu m g displacement caused by inertia force fu : g velocity vector of the plate fu g acceleration vector of the plate V 0 incident impact velocity V 0 f, V f overall bre-volume fraction X; Y; Z global coordinate axes X C, Y C, Z C compressive strengths w r t material coordinate system X T, Y T, Z T tensile strengths w r t material coordinate system di erence between the displacement of the centre of the nose of the impactor and that of the centre of the mid-surface of the plate, constants in Newmark time integration " strain c displacement of the centre of the mid-surface of the plate I t m s 12, 13, 23 1, 2, 3 X, Y, Z XY, XZ, YZ 12, 13, 23 Abbreviations 2D 3D FEA UD CP WF WG av displacement of the tip of the impactor small increment in time maximum displacement Poisson's ratio Poisson's ratio of the impactor material Poisson's ratios w r t material coordinate system density normal stresses w r t material coordinate system normal stresses w r t global coordinate system shear stresses w r t global coordinate system shear stresses w r t material coordinate system two-dimensional three-dimensional nite-element analysis unidirectional balanced symmetric crossply laminate (made of UD layers) woven-fabric weave geometry average. composites is much higher than the UD composites. This is one of the possible reasons for the superior impact resistance characteristics of WF composites. A number of studies are available on the impact behaviour of UD laminated composites [1±33]. Even though there are some studies on the impact behaviour of WF composites [34±51], further studies are necessary for their e ective use in structural applications. In this paper, the impact behaviour of plain-weave fabric-laminated composite plates supported on all the four sides and subjected to a transverse central point load is studied. The studies have been carried out for low-velocity impact loading. Stress state in the composite plate is evaluated using 3D transient nite-element analysis. Initiation of damage and the location of damage are predicted using an in-plane quadratic failure criterion. The prediction model was validated with experimental results in our earlier work [42,43]. Behaviour of laminated composite plates made of di erent plainweave fabrics under low-velocity impact is compared with

733 those of balanced symmetric crossply (CP) laminates made of UD layers and also UD composites. 2. Governing equations In the analysis of the composite plate, it is assumed that the material of each layer is linearly elastic and obeys the generalised Hooke's law. In the present study, de ections of the plate are small in comparison to the dimensions of the plate, and hence the small-de ection theory is found to be valid for the impact analysis. For small-strain theory the equilibrium equation for a body in motion by neglecting the damping coe cient is written as MŠ U KŠfUg ˆ ffg where MŠ is global mass matrix, KŠ is sti ness matrix, ffg is external force vector and fug, U are the displacement and acceleration vectors. 1 or bulk deformations of the objects as a whole. The rst attempt to incorporate a theory of local indentation was based on a scheme suggested by Hertz [52], who viewed the contact of two bodies as an equivalent problem in elastostatics. A solution was obtained in the form of a potential which described the stresses and deformations near the contact point as a function of the geometrical and elastic properties of the bodies. This result, although both static and elastic in nature, has been widely applied to impact situations where permanent deformations are produced [53]. The contact force in case of impact between a hemisphere and a plane isotropic surface is obtained from Hertz theory. In low-velocity impact, where the duration of impact is long in comparison to the period of the lowest mode of vibration of the plate, the Hertz contact law can be applied. In this case, Hertz theory which has been modi ed to apply for the case of impact on an anisotropic surface like composites is used to calculate the force caused by the impact on the plate. The Hertz contact law can be expressed as 2.1. Contact force f ˆ k 3=2 2 The knowledge of force vector is important for the solution of Eq. (1). For calculation of the force caused as a result of impact on composite plate, the impactor is modelled as an isotropic elastic body of spherical shape and the target as a plane anisotropic surface. The impactor is assumed to be rigid and of higher sti ness compared to the target in the direction of impact. According to Davies et al. [20] the structure is expected to respond dynamically away from the impactor, where conventional theories of plates and shells can be applied to predict the behaviour of the structure. It has been found that near the point of impact the inertia forces are small compared with those of impactor, so that although the dynamic response of the structure may be needed to nd the impactor-force history, the nature of the stress eld can be analysed as if subjected to a quasi-static force. 2.2. Contact law The classical treatment of impact phenomenon is based primarily on the impulse-momentum law for rigid bodies. The colliding objects are regarded essentially as single mass points. Also, it is assumed that the contact is instantaneous. This requirement can be met when the contacting surfaces are ideal smooth planes located normally to the relative velocity. In practice, however, it is di cult to meet this requirement. Also, one of the colliding surfaces is usually curved or non-planar. In this case, the two bodies su er a relative indentation in the vicinity of the impact point in addition to the gross where f is the scalar contact force, `' is the di erence between the displacement of the centre of the nose of the impactor and that of the centre of the mid-surface of the plate and `k' is modi ed Hertz constant whose value can be calculated by [21,54,55]: k ˆ 4 3 p 1 r 1 2 s E s 1 2 yx E yy where yx ˆ " x " y Here, r, s and E S are radius, Poisson's ratio and modulus of impactor, respectively. E yy is the transverse modulus normal to the bre orientation in the uppermost composite layer. 3. Finite-element formulation for stress analysis 3.1. Boundary conditions The 3D nite-element analysis (FEA) code was used to carry out the stress analysis of laminated composite plates for the following boundary conditions. Supported on all sides: i. At X ˆ 0 and X ˆ Lx u ˆ 0, v 6ˆ 0, w ˆ 0. ii. At Y ˆ 0 and Y ˆ Ly u 6ˆ 0, v ˆ 0, w ˆ 0. 3

734 The dynamic equilibrium equation (1) is expressed in terms of nite-element formulation. For the modelling of composite laminates, eight-noded linear isoparametric brick elements with incompatible modes [56,57] were used. Each node has 3 degrees of freedom. The solution of the dynamic equation was obtained using an implicit directintegration technique like Newmark's method [57]. 3.2. Stress calculation For calculating the stresses, the BŠ matrix is initially evaluated at the Gauss points and using this matrix, strains are calculated. These strains are then extrapolated to the nodes. To overcome the discontinuity of the strain values between two elements because of the use of C o type of nite-element, the strain at any node is calculated as the sum of the strains obtained at that node from all the surrounding elements, divided by the number of surrounding elements. These nodal strains are used to calculate the stresses at the nodes. In case of node lying on the interface the in-plane normal stresses and shear stress are calculated by simply multiplying with the elasticity matrix of the particular layer. But the transverse normal stress and transverse shear stresses are calculated by using the average elasticity matrix of the two layers. The values of stresses YZ and XZ are not exactly zero at the top and bottom surfaces and that of Z is not zero at the bottom, since the equilibrium equations are not explicitly enforced. These can be made exactly zero at the top and the bottom surfaces by back-substituting the values of in-plane normal stresses in the equilibrium equations and then solving the resulting partial di erential equations for Z, YZ and XZ by applying the appropriate boundary conditions. Numerical techniques like nite di erence can be used to obtain their solution. However, the discrepancy seen at the top and the bottom was found to be very small to make any di erence in the failure calculations and hence was neglected in the present study. 3.3. Newmark time integration For the solution of the dynamic equation, Newmark time integration is used. The dynamic equation can be written at time t t as MŠ U t t KŠ U t t ˆ Ft t 4 The acceleration and velocity vectors at time t t are obtained by the following expressions: n o U t t ˆ 1 n t 2 U t t f Ut g t U : o t 1 1 n o 5 2 U t n U : o n t t ˆ U : o t t n 1 n o U t n U t t oo where and are the constants, which are in this case chosen to be 0.25 and 0.5, respectively [57]. Then this method is also called the constant average acceleration method or the trapezoidal method. Substituting Eq. (5) in the dynamic equation (4). KŠ 1 t 2 MŠ U t t ˆ F I t t F m t 7 F m 1 t ˆ MŠ t 2 f U tg 1 n U : o t t 1 n o 8 2 1 U t In Eq. (4) the only unknowns are the de ection fug and force ffg at time `t t'. Rest of the terms, i.e. velocity, displacement and acceleration at time `t' are all known. As there are two unknowns and only one equation, another equation is developed using Hertz contact law. 4. Solution procedure The force caused by the impact of a sphere on a plate cannot be expressed by a simple analytical function of time. Hence, the force Vs time curve is idealised by a suitable step curve having constant forces F1, F2, F3,... over equal time intervals. That means, it is assumed that the continuous action of the force is replaced by a series of constant forces F1, F2, F3,... each acting for a time interval `t'. Here, F1, F2, F3... are the forces at time, (nt), where n=1, 2, 3.... At any time `t' the force at that instant is obtained by the superposition of the e ects of the previous impulses at that time `t' with the impulse at the same time `t'. In the present impact problem, the algorithm given in Ref. [58] was followed to save the computation time. First, the response of the plate to a unit impulse of time duration `t' is calculated for the entire time history and stored in the memory. After that the non-linear Hertz law is applied at every time step to calculate the impact force. The impact force at any time `t' is calculated from the impactor and plate displacements of the previous time step. 4.1. Response of plate to unit impulse The exact procedure for getting the response of the plate as a result of a unit impulse of time duration `t' is given as follows. It is seen in Eq. (7) that the force vector consists of the force caused by an external force 6

735 applied as well as the inertia force caused by the mass of the plate. Hence the displacement vector is written as sum of the displacements caused by the external applied load as well as by the inertia force [55]. U t t ˆ U I t t U m t t 9 Substituting Eq. (9) in Eq. (7), KŠ 1 t 2 MŠ U I t t U m t t ˆ F I t t F m t 10 To proceed with the solution, it is necessary to prescribe the right hand side force vectors. It must be noted that ff m g at time `t' is known since the displacement, acceleration and velocity vectors at time `t' are known. Also the force caused by the impulse F I during time t ˆ 0totˆt is of unit magnitude. After the rst time step `t' the external applied force is made zero. The left-hand-side matrix contains the e ective sti ness matrix, which is decomposed only once before the iterations are begun. This decomposed matrix is used to calculate the displacements of the plate with di erent right-hand-side force vectors at each time step. Newmark time integration scheme is used to get the displacement history of the plate over the required time period. 4.2. Calculation of impact force The impact force is calculated using the Hertz contact law which during the loading phase is given by f t t ˆ k t t 3=2 11 where t t is the indentation depth at any instant of time. At any time `t t' t t ˆ I t t c t t 12 The above equation is approximated by its value calculated from the plate and impactor displacements at time `t' given as below: t t ˆ I t c t 13 Here, c t is the displacement of the centre of the midsurface of the plate in the direction of the impact. I t is the displacement of the tip of the impactor at any time `t'. The position of the tip of the impactor at any time `t' after the contact between the plate and impactor has taken place is given in Eq. (14), where the double integral signi es the resistance of the plate to the motion of the impactor and `f' is the contact force caused by impact. I t ˆ V 0t t t 0 0 f m dtdt Substituting Eq. (14) and Eq. (13) in Eq. (11), f t t ˆ k V 0 t t t 0 0 jˆ1 jˆt 14! f Xt X 1 3=2 dtdt F j c j 15 m Here, F j is a dimensionless quantity de ned as (f j /unit force). As a result of the non-linear nature of the above equation, analytical solution is not possible and the solution is obtained numerically by small time increment method, then Eq. (15) becomes! f t t ˆ k V 0 t t2 X n D n j 1 F j Xt X 1 3=2 F j c j m jˆ1 jˆ1 jˆt 16 The summation in the second term of the Eq. (16) arises from the double integration term and, for a linear continuous approximation of force/time curve can be expressed as X n jˆ1 D n j 1 F j ˆ 2 n 1 F 1 n 2 F 2 F 1 ::: Š 1 3 F n F n 1 F n 2 ::: 17 The solution of Eq. (16) is obtained by the following approximation scheme [53,59]. At time t ˆ t, the force is assumed to be a function of only the local compression and not the de ection of the plate. Hence the force is calculated as f t ˆ kv 0 t 3=2 18 Then the force at time t ˆ 2t is calculated from the known displacements of plate and impactor at time t ˆ t. Similarly, force at any time `t t' is calculated by substituting the impactor and plate displacements of the previous time step using Eq. (17). The value of c t, already available from the response history of the plate, calculated for a unit impulse of duration `t', is multiplied by the impactor force to get the displacement of the plate under the impact load. At any time `t' the displacement of the plate is obtained by taking a summation of the products of the impact force and the displacement response of the plate due to the unit load up to that time `t' as shown by the double summation sign in the third term of the Eq. (16).

736 5. Results and discussion The results have been obtained using the inhouse FEA code developed for impact analysis. This program was used to analyse the woven-fabric composite plates under impact loading. For comparison, the results for a [0/90] S crossply laminates made of UD layers and [0] n UD composites are also presented. For all the cases, the loading was through a point load acting at the centre of the plate. Joshi and Sun [7] have analysed the impact behaviour of composite beams. They used nite-element analysis based on 2D plain-strain formulation. In the present analysis, impact behaviour of composite plates has been analysed using eight-noded linear isoparametric brick elements with incompatible modes. The present analysis is based on solution of equilibrium equations tending towards exact solution. The schematic arrangement of WF laminated composite plate geometry is shown in Fig. 1. For the present study, a square plate of 150 mm150 mm6 mm thickness was considered. Supported boundary condition was considered for all the cases. A steel spherical impactor with radius 6.5 mm, modulus of elasticity of 200 GPa and Poisson's ratio of 0.3 was considered. Di erent plain-weave fabric laminated composite plates were considered for the impact studies. The weave geometrical parameters considered are presented in Table 1. Laminate con guration C2 was considered for all the laminates [60±65]. E-glass/epoxy and T300/ 5208 carbon/epoxy materials were used for the studies. Five weave geometries were considered for the studies with di erent strand width (a), strand thickness (h) and inter-strand gap (g). All the laminates were made with balanced fabrics with the same geometrical and material parameters along both warp and ll directions. The ratio (g=a) was 0.1 for all the cases. The ratio (h=a) was varied in the practical range from 0.01 to 0.25. The Table 1 Plain-weave fabric structure a Weave geometry Warp strand Fill strand h=a g=a h=a g=a WG06 0.01 0.1 0.01 0.1 WG07 0.05 0.1 0.05 0.1 WG02 0.10 0.1 0.10 0.1 WG08 0.15 0.1 0.15 0.1 WG10 0.25 0.1 0.25 0.1 a a=2.0 mm. fabric thickness was minimum for WG06 and maximum for WG10. For the other fabrics it was in between. As given in Tables 1±8, materials WG06, WG07, WG02, WG08 and WG10 refer to plain-weave fabric laminates. Materials CP0.65, CP0.40 and CP0.70 refer to balanced symmetric crossply laminates made of UD layers. Materials UD0.65 and UD0.70 refer to UD composites. The last letter `G' refers to E-glass as reinforcing material and `C' refers to T300 carbon as reinforcing material. The elastic and strength properties are given in the Tables 2±5 [60±66]. The in-house FEA code developed was rst run with di erent meshes for convergence study. Based on the study, a mesh of 20208 was adopted for all the cases. The code was run for an impact problem solved by Karas [55,67] for an isotropic material. The results obtained by the present in-house FEA code and that obtained by Karas were compared, and a good correlation was obtained [42,43]. The code was also used to obtain the failure for a material given by Lammerant and Verpoest [26] and tested under quasi-static loading for a crossply laminate [0/90] S, made of HTA/6376 toughened carbon/epoxy. They observed damage initiation in the form of a matrix cracking/lamina splitting in the bottom layer right under the point of the loading. This rst matrix cracking/lamina splitting appeared at a displacement of 1 mm. Using the present code, it was observed that the failure had just initiated in the form of a matrix cracking/lamina splitting in the bottom most layer for a displacement of 1 mm [42,43]. 5.1. Impact loading Fig. 1. Woven-fabric laminated composite plate geometry. Response plots of E-glass/epoxy laminates are presented for supported boundary conditions in Fig. 2. Contact force and displacement plots are given in Figs. 3 and 4, respectively. It can be seen from Fig. 3 that multiple contacts occur for both WF and CP laminates. The peak contact force was more for WF laminates. Corresponding plate and impactor displacements are nearly the same for the WF and CP laminates considered. It was observed that the loss of contact takes place when the plate starts moving with a higher velocity

737 Table 2 Elastic properties of composites: E-glass/epoxy a Material E 11 (GPa) E 22 (GPa) E 33 (GPa) G 12 (GPa) G 13 (GPa) G 23 (GPa) 12 13 23 V f (kg/m 3 ) UD 48.0 15.3 15.3 5.1 5.1 5.8 0.315 0.315 0.332 0.65 2112 UD 44.6 13.4 13.4 4.4 4.4 5.0 0.317 0.317 0.342 0.6 2040 UD 30.9 8.3 8.3 2.8 2.8 3.0 0.327 0.327 0.395 0.4 1750 WG06G 21.7 21.7 8.5 3.91 4.1 4.1 0.169 0.280 0.280 0.4 1750 WG07G 21.4 21.4 8.6 3.91 4.1 4.1 0.171 0.279 0.279 0.4 1750 WG02G 20.8 20.8 8.7 3.92 4.2 4.2 0.173 0.279 0.279 0.4 1750 WG08G 19.9 19.9 8.8 3.92 4.2 4.2 0.178 0.278 0.278 0.4 1750 WG10G 18.1 18.1 9.0 3.94 4.2 4.2 0.189 0.276 0.276 0.4 1750 a Densities of E-glass bre=2620 kg/m 3 and epoxy resin=1170 kg/m 3. Table 3 Elastic properties of composites: T300/5208 carbon/epoxy a Material E 11 (GPa) E 22 (GPa) E 33 (GPa) G 12 (GPa) G 13 (GPa) G 23 (GPa) 12 13 23 V f (kg/m 3 ) UD 162.0 14.9 14.9 5.7 5.7 5.4 0.283 0.283 0.386 0.7 1583 UD 150.7 13.3 13.3 4.9 4.9 4.8 0.287 0.287 0.390 0.65 1554 UD 94.1 7.9 7.9 2.8 2.8 2.7 0.309 0.309 0.436 0.4 1406 WG06C 53.1 53.1 8.0 3.79 3.7 3.7 0.055 0.074 0.074 0.4 1406 WG07C 50.3 50.3 8.1 3.79 3.7 3.7 0.058 0.073 0.073 0.4 1406 WG02C 44.2 44.2 8.2 3.79 3.7 3.7 0.064 0.072 0.072 0.4 1406 WG08C 38.0 38.0 8.4 3.79 3.7 3.7 0.073 0.071 0.071 0.4 1406 WG10C 28.8 28.8 8.6 3.79 3.7 3.7 0.093 0.070 0.070 0.4 1406 a Densities of T300 carbon bre=1760 kg/m 3 and epoxy resin=1170 kg/m 3. Table 4 Strength properties of composites: E-glass/epoxy a Material X T (MPa) Y T (MPa) Z T (MPa) S 12 (MPa) S 13 (MPa) S 23 (MPa) X C (MPa) Y C (MPa) Z C (MPa) V f UD 1297 27.8 27.8 39.2 39.2 38 820 150 150 0.65 UD 1197 27.7 27.7 38.7 38.7 37 740 145 145 0.6 UD 798 27.1 27.1 36.8 36.8 36 480 140 140 0.4 WG06G 315 315 27.1 28 28 28 210 210 140 0.4 WG07G 290 290 27.1 28 28 28 196 196 140 0.4 WG02G 250 250 27.1 28 28 28 183 183 140 0.4 WG08G 230 230 27.2 28 28 28 170 170 140 0.4 WG10G 210 210 27.3 28 28 28 120 120 140 0.4 a X T and Y T values for WF composites are at pure matrix block failure. Table 5 Strength properties of composites: T300/5208 carbon/epoxy Material X T (MPa) Y T (MPa) Z T (MPa) S 12 (MPa) S 13 (MPa) S 23 (MPa) X C (MPa) Y C (MPa) Z C (MPa) V f UD 1744 52.6 52.6 107.8 107.8 100 1650 260 260 0.7 UD 1622 52.0 52.0 106.4 106.4 99 1525 250 250 0.65 UD 1013 49.5 49.5 100.5 100.5 96 940 220 220 0.4 WG06C 565 565 49.5 66 66 66 420 420 220 0.4 WG07C 500 500 49.5 66 66 66 370 370 220 0.4 WG02C 405 405 49.5 66 66 66 300 300 220 0.4 WG08C 365 365 49.5 66 66 66 255 255 220 0.4 WG10C 345 345 49.5 66 66 66 240 240 220 0.4 compared to the velocity of the impactor, i.e. the displacement of the plate exceeds that of the impactor causing a separation between the two. They once again come into contact when the plate reverses its direction of motion and starts coming back towards its original position. However, it may be noted that the loss of contact does not signify the end of impact event. During the impact event there can be a number of contacts

738 Table 6 Impact behaviour of composites: E-glass/epoxy a Material Plate thickness, L Z (mm) Peak contact force, f m (N) Max. displacement, m (mm) Duration of impact, b t f (ms) reach f m (ms) reach m (ms) Maximum failure in-plane function reach (ms) I1 (bottom) I2 (top) Peak I1 Peak I2 WG10G 6.0 1220 0.628 871 (938) 798 430 0.033 0.207 789 801 WG08G 6.0 1215 0.610 844 (907) 776 418 0.029 0.107 766 769 WG02G 6.0 1213 0.602 831 (893) 763 412 0.025 0.094 754 764 WG07G 6.0 1193 0.598 825 (885) 758 409 0.018 0.083 751 756 WG06G 6.0 1200 0.595 822 (888) 754 408 0.016 0.072 745 755 CP0.65G 5.0 927 c 0.113 c 66 c (66 c ) 66 c 66 c 1.000 0.08 c 66 c 66 c CP0.40G 6.0 880 0.600 1030 (962) 78 514 0.545 0.066 110 104 UD0.65G 5.0 854 c 0.107 c 61 c (61 c ) 61 c 61 c 1.000 0.073 c 61 c 61 c a M=50 gm, V 0 =3 m/s; plate dimensions: L X =150 mm, L Y =150 mm, simply supported. b Duration of impact is given at contact force, F ˆ 0. The quantity in parentheses indicates duration of impact at plate displacement, w ˆ 0. c Indicates the values at in-plane failure. Table 7 Impact behaviour of composites: E-glass/epoxy a Material Plate thickness, L Z (mm) Peak contact force, f m (N) Max. displacement, m (mm) Duration of impact, b t f (ms) reach f m (ms) reach m (ms) Maximum failure in-plane function reach (ms) I1 (bottom) I2 (top) Peak I1 Peak I2 WG10G 6.0 393 0.210 902 (940) 823 442 0.003 0.019 811 813 WG08G 6.0 391 0.204 876 (910) 798 429 0.002 0.009 104 790 WG02G 6.0 392 0.201 863 (895) 784 422 0.002 0.009 103 776 WG07G 6.0 388 0.199 858 (888) 780 419 0.002 0.007 103 772 WG06G 6.0 388 0.199 854 (884) 774 418 0.001 0.007 103 766 CP0.65G 5.0 298 0.201 1026 (962) 72 492 0.131 0.009 118 96 CP0.40G 6.0 275 0.205 1068 (966) 102 526 0.058 0.007 136 122 UD0.65G 5.0 364 0.201 1000 (940) 648 500 0.143 0.011 116 658 a M=50 gm, V 0 =1 m/s; plate dimensions: L X =150 mm, L Y =150 mm, simply supported. b Duration of impact is given at contact force, F ˆ 0. The quantity in parentheses indicates duration of impact at plate displacement, w ˆ 0. Table 8 Impact behaviour of composites: T300/5208 carbon/epoxy a Material Plate thickness, L Z (mm) Peak contact force, f m (N) Max. displacement, m (mm) Duration of impact, b t f (ms) reach f m (ms) reach m (ms) Maximum failure in-plane function reach (ms) I1 (bottom) I2 (top) Peak I1 Peak I2 WG10C 6.00 371 0.187 757 (778) 645 357 0.002 0.006 631 631 WG08C 6.00 379 0.169 693 (703) 551 325 0.002 0.007 570 568 WG02C 6.00 405 0.159 656 (662) 525 308 0.002 0.006 546 545 WG07C 6.00 441 0.151 622 (629) 512 296 0.002 0.004 519 518 WG06C 6.00 437 0.148 618 (612) 495 485 0.001 0.003 516 515 CP0.70C 5.32 369 0.134 664 (614) 472 340 0.019 0.003 124 96 CP0.40C 6.00 305 0.149 762 (702) 114 394 0.010 0.003 150 120 UD0.70C 5.32 399 0.119 628 (708) 540 398 0.032 0.004 530 534 a M=50 gm, V 0 =1 m/s; plate dimensions: L X =150 mm, L Y =150 mm, simply supported. b Duration of impact is given at contact force, F ˆ 0. The quantity in parentheses indicates duration of impact at plate displacement, w ˆ 0. between the impactor and the plate, but the impact event can be considered to be over only when the impactor displacement reverses its sign and the contact between the impactor and the plate is lost during the upward motion of the plate and the impactor, or the plate returns to its original position, whichever takes place later. In the present case, it can be seen that, for WF laminates, the contact between the plate and the impactor was lost even before the plate returned to its original position. On the other hand, for CP laminates,

739 Fig. 2. Response of E-glass/epoxy laminates, CP0.40G and WG02G to unit impulse Ð simply supported. Fig. 4. Variation of impactor and plate displacements with time at X ˆ L X /2 and Y ˆ L Y /2 for E-glass/epoxy laminates, CP0.40G and WG02G Ð simply supported. Fig. 3. Variation of contact force with time at X ˆ L X /2 and Y ˆ L Y /2 for E-glass/epoxy laminates, CP0.40G and WG02G Ð simply supported. the contact between the plate and the impactor was lost beyond the original position. In the present study, only the rst impact was considered and hence the calculations were stopped after the impactor displacement reversed its sign and the contact between the impactor and the plate was lost, or the plate returned to its original position, whichever took place later. 5.2. In-plane stress plots The plots for the variation of the stresses ( X and Y ) through the thickness of the plate are given in Figs. 5±7 for the WF and CP laminates. For WF laminate, time interval 763 ms corresponds to time to reach peak contact Fig. 5. Variation of X and Y through the plate thickness at X ˆ L X / 2 and Y ˆ L Y /2 for E-glass/epoxy laminate, WF, (0) S, WG02G Ð simply supported. force. Time interval 500 ms is during the separation period. For CP laminate, time interval 78 ms corresponds to time to reach peak contact force. In this case also time interval 500 ms is during the separation period. For the incident impact velocity and mass of the impactor considered for these two cases, stresses induced were not su cient to cause damage in the

740 plates. From Figs. 5±7, it can be seen that the magnitudes of compressive normal stresses in the upper layers of the laminate are more than the magnitudes of the tensile normal stresses in the lower layers. This is because of the compression of the upper layers during the impact event when the contact exists between the impactor and the plate. But during the separation period the magnitudes of the stresses in the upper and the lower layers are identical. From Fig. 5, it can be seen that the variation of X and Y through the plate thickness is identical for WF composites. This is because the WF composites have balanced properties in both warp and ll directions. Variation of stresses ( X and Y ) along L Y at peak contact force is presented in Figs. 8±11 for WF and CP laminates. From X and Y plots it can be seen that near the centre the stresses are compressive in the upper layers and tensile in the lower layers. But the sign changes towards the end of the plate. The magnitude of XY is seen to be a very small quantity. 5.3. In-plane failure function plots Fig. 6. Variation of X through the plate thickness at X ˆ L X /2 and Y ˆ L Y /2 for E-glass/epoxy laminate, CP, [0/90] S, CP0.40G Ð simply supported. The three possible macro modes of damage initiation that can occur in the composite laminate under impact load are: i. Initiation of matrix cracking/lamina failure caused by in-plane stresses. In-plane tensile normal stresses and shear stress can lead to matrix cracking/ lamina failure in the lower layer whereas in-plane compressive normal stresses and the shear stress can lead to matrix cracking/lamina failure in the upper layer. These matrix cracks initiated within the layer can lead to delamination when these cracks reach the neighbouring interface. ii. Initiation of delamination caused by the tensile nature of Z and other interlaminar shear stresses. iii. Crushing of upper layers caused by impact loading. Out of these three possible modes of damage initiation, only the rst mode, i.e. the in-plane failure mode, Fig. 7. Variation of Y through the plate thickness at X ˆ L X =2 and Y ˆ L Y /2 for E-glass/epoxy laminate, CP, [0/90] S, CP0.40G Ð simply supported. Fig. 8. Variation of X along L Y at peak contact force for E-glass/ epoxy laminate, WF, (0) S, WG02G Ð simply supported.

has been considered in the present study. Damage initiation caused by Z, XZ and YZ is being studied separately. The damage initiation would occur in the form of matrix cracking/lamina splitting either in the bottom layer or in the top layer. Hence, an in-plane failure criterion by Tsai±Hill [68] was used for damage initiation studies. An in-plane failure function, I, based on Tsai± Hill criterion is de ned as follows: 2 2 2 2 12 1 2 ˆ I 1 X T Y T S 12 Here 1, 2 and 12 are the induced in-plane stress components and X T, Y T and S 12 are the normal and inplane shear-strength values. The damage initiation takes place when the value of in-plane failure function, I just exceeds unity. In-plane failure function plots are presented in Figs. 12±15 for WF and CP laminates. Failure has not taken place for the plate geometry and impact parameters considered for both WF and CP laminates. In-plane failure function value is very high for CP laminates as compared to X T Y T 741 Fig. 9. Variation of Y along L Y at peak contact force for E-glass/ epoxy laminate, WF, (0) S, WG02G Ð simply supported. Fig. 11. Variation of Y along L Y at peak contact force for E-glass/ epoxy laminate, CP, [0/90] S, CP0.40G Ð simply supported. Fig. 10. Variation of X along L Y at peak contact force for E-glass/ epoxy laminate, CP, [0/90] S, CP0.40G Ð simply supported. Fig. 12. Variation of in-plane failure function, I with time at the centre for E-glass/epoxy laminate, WF, (0) S, WG02G Ð simply supported.

742 Fig. 13. Variation of in-plane failure function, I along L Y at peak contact force for E-glass/epoxy laminate, WF, (0) S, WG02G Ð simply supported. Fig. 15. Variation of in-plane failure function, I along LY at peak contact force for E-glass/epoxy laminate, CP, [0/90] S, CP0.40G Ð simply supported. of the lower transverse tensile strength. Woven-fabric laminates are characterised by balanced and equal properties along both warp and ll directions. The tensile failure strength along the transverse direction is signi cantly higher for WF laminates leading to a signi cant reduction in in-plane failure function. It is interesting to note that the in-plane failure initiates from the bottom surface for CP laminated composites whereas it initiates from the top surface for the WF laminates. Woven-fabric laminates have balanced in-plane properties, but the compressive strength properties for the WF laminates are much lower than the tensile strength properties. The upper layer is under compressive stresses for the WF laminates, and hence the failure initiates on the upper layer. Fig. 14. Variation of in-plane failure function, I with time at the centre for E-glass/epoxy laminate, CP, [0/90] S, CP0.40G Ð simply supported. WF laminates. This indicates that WF laminates are more impact-damage resistant compared to CP laminates. From Figs. 12 and 13, it is seen that the in-plane failure function value is higher for the top layer (I2) than for bottom layer (I1) for the WF laminate. On the other hand, it is seen from the Figs. 14 and 15, that the in-plane failure-function value is higher for the bottom layer (I1) than for the top layer (I2) for the CP laminate. For the CP laminate the in-plane failure can initiate in the form of matrix cracking/lamina splitting in the lower layer during impact because of in-plane tensile stresses. The damage initiates in the lower layer because 5.4. E ect of fabric geometry on impact behaviour of composites Impact behaviour of di erent plain-weave fabric composites made of E-glass/epoxy and T300/5208 carbon/epoxy is presented in Tables 6±8. Weave geometries WG06, WG07, WG02, WG08 and WG10 indicate plain-weave fabric composites with decreasing in-plane modulus of elasticity values. The maximum displacement and maximum in-plane failure function decrease with the increase in in-plane modulus of elasticity for both E-glass/epoxy and T300/5208 carbon/epoxy laminates. Nominal plate thickness considered was 6 mm for all these cases. [0/90] S and [0] n laminate results are also presented in Tables 6±8 for comparison. CP0.65G, UD0.65G, CP0.70C and UD0.70C laminate thickness values were changed as shown in Tables 6±8. This was done to have the same mass for all the plates.

743 For CP0.65G and UD0.65G the laminates were failing for the plate dimensions and the impact parameters considered for E-glass/epoxy at V 0 ˆ 3 m/s (Table 6). The in-plane failure was initiating in the bottom layer in the form of matrix cracking/lamina splitting. For E- glass/epoxy, the peak contact force was more for all the plain-weave fabric laminates considered compared to UD and CP laminates (Tables 6 and 7). For T300/5208 carbon/epoxy, UD and CP laminates were not failing for the plate dimensions and the impact parameters considered (Table 8). But, the in-plane failure function is higher for UD and CP laminates compared to WF laminates. From the Tables 6±8, it is clear that the in-plane failure would initiate from the top surface for all the plain-weave fabric laminates, whereas it would initiate from the bottom surface for the UD and CP laminates. 6. Conclusions The behaviour of WF laminated composite plates has been studied under transverse central low-velocity-point impact. For this, a nite-element formulation is presented. For the plate geometry and impact conditions considered, it is observed that: i. Multiple contacts between the impactor and the composite plate occur during the impact event. ii. Maximum displacement and maximum in-plane failure function decrease with the increase in inplane modulus of elasticity for both E-glass/epoxy and T300/5208 carbon/epoxy laminates. iii.the magnitudes of compressive normal stresses X and Y in the upper layers of the laminate are higher than the magnitudes of tensile normal stresses X and Y in the lower layers. iv. The in-plane failure function is lower for the WF laminates than for UD and CP laminates indicating that the WF laminates are more impact resistant. v. In-plane failure function is higher on the top layer than on the bottom layer for WF laminates. It indicates that in-plane failure would initiate on the top layer for WF laminates. vi. 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