Journal of Composite Materials

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Low-Velocity Impact and Damage Process of Composite Laminates Zhidong Guan and Chihdar Yang Journal of Composite Materials 2002; 36; 851 DOI: 10.1177/0021998302036007512 The online version of this article can be found at: http://jcm.sagepub.com/cgi/content/abstract/36/7/851

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Low-Velocity Impact and Damage Process of Composite Laminates ZHIDONG GUAN AND CHIHDAR YANG* Department of Mechanical Engineering Wichita State University Wichita, KS 67260-0133, USA (Received March 19, 2000) (Revised February 9, 2001)

ABSTRACT: Several important issues regarding damage simulation of composite laminates due to low velocity impact were investigated including contact law, damage initiation and the corresponding change of stiffness, and damping. Continuum damage mechanics was applied to account for the change of mechanical properties of damaged materials. The Hertzian contact law was modified in order to accommodate the serious damage in the plate. A semi-empirical delamination damage criterion was introduced. A finite element program was written in FORTRAN using twenty-noded solid elements with layered structure to analyze the transient dynamic response of composite laminates. In the simulation computations, variable material damping coefficients were applied to the elements according to damage in order to stabilize the computation. Damage in the forms of matrix cracking, delamination, and fiber breakage were included and analyzed. Results including the force history and delamination areas were found to correlate well with the experiments. KEY WORDS: low-velocity impact, damage mechanics, residual strength, composite laminate, impact simulation.

INTRODUCTION OMPOSITE MATERIALS HAVE been demonstrated to be very sensitive to impact damage. For example, barely visible impact damage can contribute up to 60% loss in an aircraft composite structure’s compressive strength. Because composite materials are relatively new and complex in nature when compared with traditional metal materials, many of their characteristics still remain mysterious, especially the theories regarding impact tolerance and after-impact residual properties. There have been overwhelming studies in this area and many publications can be found in the literature. The contact law proposed by Yang and Sun [1] has successfully coupled the equations of motions of the

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*Author to whom correspondence should be addressed.

Journal of COMPOSITE MATERIALS, Vol. 36, No. 07/2002 0021-9983/02/07 0851–22 $10.00/0 DOI: 10.1106/002199802023512 ß 2002 Sage Publications


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impactor and composites. It has also simplified the computation of the damage simulation. However, the effect of serious impact damage was not taken into account. Until now, little effort has been made to consider this issue. On the other hand, the change of material moduli due to damage is another changing issue because of the fact that no data can be retrieved directly from impact experiments. Belingardi et al. [2] and Luo et al. [3] adopted the simple assumption that when damage occurs during impact, the mechanical properties of the damaged materials would reduce to certain levels regardless of the extent of damage. However, based on experimental results, impact damage would continue to develop after the initial damage appears. A model is needed to simulate the gradual change of material properties according to the extent of damage. The impact induced delamination is the most important damage mode because the level of impact energy to initiate delamination is low and the post-impact compressive strength is dramatically reduced due to delamination. Because of the complex nature of laminated composites, a simple and effective criterion for the delamination initiation is still lacking. The stability of simulation computation is another important issue to be considered. This is especially true when the tremendous changes of material properties due to the damage in composites are taken into account. The instability or divergence of the computation might be caused by the specific algorithms in some cases. However, in most cases, they are the results of inappropriate physical modes. Increasing the damping coefficient at damage areas is an effective way to stabilize the computation and this can be explained based on impact rests. However, not many papers on the use of damping coefficients can be found in the literature. The above complications and other issues such as the accurate calculation of stresses in the laminated composites prevent the precise simulation of impact process and damage process. They also limit the thorough understanding of impact damage characteristics as well as the accurate computational study of the mechanical properties of composites with impact damage. The objective of this work is two-fold: (1) to improve the impact response methodology by addressing each of the above concerns and to incorporate them through the enhancement in the methodologies and validating through correlation with the experimental test results, (2) to use this simulation model to analyze typical impact events to gain further understanding of the characteristics of impact damage processes. A finite element program was written in FORTRAN using 20-noded solid hexahedral elements with layered structure [4]. The Wilson- scheme was adopted to perform time integration for the motion equations for the impactor and composites plate system. Simulation results were satisfactory when compared with experimental results from the literature.

CONTACT LAW

Contact law plays a very important role in impact analyses because it facilitates the solution process by coupling the motion of the impactor and the target plates and it greatly simplifies the damage simulation. Based on experiment results, impact damage can be regarded as local damage and global damage. The local damage is located directly under the impactor and is essentially associated with the geometry and material properties of the impactor. On the other hand, the global damage is generally referred to the damage initiated from the impact point and is mainly in the form of delamination. Even though the local damage has relatively small effects on the post-impact mechanical properties


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of composite laminates, it is one of the essential parameters when formulating the impact simulation. Yang and Sun [1] modified the Hertzian contact law through a systematic experimental study. The modified contact law can be applied to determine the relation between the contact force and indentation during both loading and unloading. f ¼ k1:5 o 2:5 Unloading f ¼ fmax max o Loading

ð1Þ ð2Þ

where k is the contact stiffness, and 0 are indentation and permanent indentation, respectively, fmax is the maximum contact force, and max is its corresponding indentation. k and 0 are material dependent and can be determined by experiments [1]. Even though these equations do not account for the change of stiffness due to impact damage, they have been widely used in the impact simulation studies. Lagace et al. [5] addressed the effect of damage on the indentation and impact behavior in their experimental investigation. Based on their observation, they have concluded that the contact stiffness of damaged laminates is larger than the undamaged laminates and that there are no detectable effects on the global structural response due to impact damage. On the contrary, Shahid and Chang [6] assumed that impact damage affects the contact behavior. They modified the contact stiffness of Hertzian law using the weighted average stiffness of the plies, including undamaged and damaged, through the laminate thickness so that the contact stiffness decreases as the impact damage develops. A computer program developed by the authors in FORTRAN using the finite element method was executed in order to evaluate the effect of damage induced change of contact stiffness in the contact law. In the simulation, a simply supported 126 mm 76 mm 3.76 mm [(45/0/ 45/90)s]4 graphite/epoxy laminate was impacted using an impactor of 5.17 kg at 2.03 m/s. The following two assumptions for contact stiffness were adopted for comparison. 1. k remains as constant k1 throughout the impact duration. 2. k equals to k1 before the impact force reaches 4,000 N and equals to bk1 when the impact force is between 4,000 and 7,000 N. The impact forces of 4,000 N and 7,000 N are the forces assumed to cause initial and serious damage of the plates, respectively [7], and b is a linear factor which varies from 1.0 to 0.6 according to the impact force from 4,000 N to 7,000 N [6]. k remains as 0.6 k1 for the rest of the impact duration after the contact force reaches 7,000 N. For the cases which the maximum force do not reach 7,000 N, k remains as the smallest value according to the maximum impact force for the rest of the impact period [6]. The reduction of contact stiffness due to damage would certainly result in a reduction in contact force during the impact process. However, depending on the magnitude of the forces causing initial and serious damage, the level of impact energy, and other parameters, the effect of the change of contact stiffness on impact process might not be appreciable. As shown in Figure 1, the calculated maximum impact force and length of the impact duration based on the two assumptions deviate only 4.2% and 2.0%, respectively. Therefore, the contact law in Equation (1) with a constant contact stiffness was adopted in this investigation.


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Figure 1. Contact force history with different contact stiffness.

Experimental results from low velocity impact of composite laminates show that there is a distinct dent left on the impacted surface after an impact of a certain energy level. As the impact energy increases, the permanent dent becomes deeper until the plate is perforated. In computer simulation, the impact usually begins when the impactor comes into contact with the plate and ends when the impactor rebounds and loses contact with the plate. The criterion most researchers use to define the end of impact is that the impactor rebounds to the location which is 0 under the location of initial contact. However, the location where the impactor loses contact with the plate is d0, depth of permanent dent, under the location of initial contact and 0 is the permanent indentation, which is the permanent change of plate thickness. This commonly used criterion is actually using Equation (2) and setting the depth of permanent dent d0 equal to the permanent indentation 0 because the deformation of the backside of the plate is not taken into account. This is considered appropriate if the impact energy is at the moderate level when the back surface of the plate is not protruded. However, if the impact energy is so high that d0 is no longer equal to 0 due to the protrusion on the back side of the plate, this approach will result in a major error in modeling the impact process. On the other hand, once serious damage has occurred and permanent deformation is present at the backside of the plate, the accurate measurement of permanent indentation becomes very difficult. Due to the limitation of computation power and the huge number of iterations needed for impact simulation with serious damages, linear finite element models with allowable mesh size cannot accurately determine the displacement of the back surface. This has prohibited the use of indentation as the criterion to determine the end of impact process. Furthermore, when serious damages have occurred, the permanent indentation is beyond the range considered by the existing empirical function which correlates the maximum impact force and permanent indentation. Therefore, the following equation describing the contact force during unloading as a function of the impact force at the beginning of rebound fm, impactor displacement u and impactor displacement at the beginning of rebound um, and depth of permanent dent d0 is proposed in this present study. Unloading f ¼ fm

u d0 um d0

n ð3Þ

In Equation (3), n is used to adjust the nonlinear relation of the contact force and the displacement caused by the damage of the plate. The depth of permanent dent at impact



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side d0 can be obtained from experiments or can be estimated, after further studies, from other impact damage state parameters. The laminate responds to the impactor in two modes: vibratory mode and the major loading and unloading mode. The former is caused by the dynamic response of the impacted plate and the latter is based on the contact law as well as the flexural displacement of the plate. As discussed previously, Equation (1) with constant contact stiffness was adopted for the loading portion of both modes of response. Due to the fact that the unloading of the vibratory mode is of much smaller scale compared with the major loading and unloading mode, Equation (2) is used for the unloading of the vibratory mode. Equation (3) is used to govern unloading contact law only in the major unloading mode.

FAILURE CRITERIA

The damage of composite laminates due to impact usually consists of several modes such as matrix cracking, delamination, fiber and matrix debonding, and fiber breakage, etc. Among these damage modes, delamination is the most important mode because it contributes the most to the reduction in post-impact residual compressive strength. In the present study, the impact damage model developed by Choi and Chang [8] is adopted. It is summarized as follows: 1. Intraply matrix cracking caused by transverse impact is the initial damage mode. 2. Delaminations occur immediately after the presence of matrix cracking in the intimate plies. For the interface between two given plies, the interlaminar shear stress 13 and the transverse in-plane stress 22 of the lower ply and the interlaminar transverse shear stress 23 in the upper ply contribute significantly to the growth of delamination. 3. Additional matrix cracking can occur subsequently in other plies and can result in additional delaminations along other interfaces. According to this model, the failure criteria must have the capability of predicting the initiation of the impact damage, i.e., the critical matrix cracking and the propagation of delamination in the laminates. In the present study, three modes of damage, fiber breakage, matrix cracking, and delamination are considered. The maximum stress criterion along the fiber direction of each ply is used for fiber breakage. Tsai–Wu quadratic failure criterion is chosen for predicting the initiation of matrix cracking. The following semi-empirical equation is proposed, based on the damage model mentioned previously, to predict the initiation and propagation of delamination of the nth interface 33, n 2 ð 23, n Þ2 þ ð 13, nþ1 Þ2 þ Ds sin n Zc S2 2 eD 1 damaged 22, nþ1 þ Dt sin n ¼ eD Y eD < 1 undamaged

ð4Þ

where, Ds and Dt are material constants, represents the average stress in each ply, the subscripts 1, 2, and 3 denote the fiber, in-plane transverse, and out-of-plane directions, respectively, of each ply. The subscripts n and n þ 1 correspond to the upper and lower


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plies of the nth interface, respectively. Zc is out-of-plane compression strength, and S is in-plane shear strength. Y is the transverse tensile strength Yz when 22, nþ1 > 0 and the transverse compressive strength Yc when 22;nþ1 < 0 of each ply. n is the difference in fiber angles between the two plies adjacent to the nth interface. As can be seen from Equation (4), all the important stresses described in the damage model are included. The coefficient sinn in the second and third terms represents the effects of the angle difference between two adjacent plies, which is a unique characteristic of laminate delamination. In fact, Ds and Dt are two weighting factors which determine the contribution of interlaminar stress and bending stress to delamination damage, respectively. They can be estimated from experiments and trial calculations [7], as described in the ‘‘Results and Discussion’’ section, Ds and Dt are mainly material dependent, but they might be case-dependent to certain extent because the ply strength is slightly different if the stacking sequence is different [8]. Equation (4) is proposed directly on the bases of experiment observation. It is simple, effective, and easy to use in predicting the impact delamination damage of laminated composites, especially in the cases shown later where the target laminates contain dozens of plies.

CONSTITUTIVE RELATIONS FOR DAMAGED COMPOSITES

After the occurrence of damage, the behavior of material will change, leading to stiffness reduction and stress redistributions. As mentioned previously, the impact damage can be regarded separately as local damage and global damage. The effect of local damage on the impact process is mainly through indentation and depth of the dent created during impact and is reflected in the contact law aforementioned. On the other hand, global damage affects not only the impact process, as it changes the mechanical response of the laminates during impact, but also the post-impact behavior. However, global damage and its effect on the impact process are very complicated and difficult to analyze either analytically or experimentally. A few articles addressing this issue can be found in the literature. The strategy of reducing the stiffness of damaged plies to a certain fraction without considering the degree of damage was used by some investigators [2,3]. Other researchers applied damage theory of continuum mechanics to address this problem. Shen et al. [9] proposed a generalized damage theory for anisotropic elastic material including composites. They initiated the stress–strain relation for the actual damaged material by introducing a fourth-order damage operator to transform the compliance matrix according to the damage states. Based on this work, several investigations were conducted in the subjects such as tension of composite laminates with holes and high velocity impact of composite laminates [9,10]. Shen’s concept is adopted in the present investigation to account for the reduction in stiffness due to impact damage. The stress–strain relation of anisotropic elastic materials with small deformation is given by ekl ¼ Cklop rop

ð5Þ

where ekl and rop are the symmetric second-order elastic strain and stress tensor, respectively, Cklop is the initial compliance tensor of the material. With the introduction of fourth-order operators Nijkl and Mmnop to reflect material damage, the actual elastic



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stress and strain tensors corresponding to the damaged material, edtj and rdmn , are obtained from the undamaged stress and strain tensors as edij ¼ Nijkl ekl

rdmn ¼ Mmnop rop

ði, j, k, l, m, n, o, p ¼ 1, 2, 3Þ

ð6Þ

where Nijkl and Mmnop are symmetric with respect to ij and mn. Based on Equations (5) and (6), the strain tensor of damaged material can be obtained as 1 d Cklop rdmn ¼ Cijmn rdmn edij ¼ Nijkl Mmnop

ð7Þ

Considering the symmetry of Cklop and Cdijmn and letting t be a symmetric second-order tensor depending on D which is a fourth-rank damage state tensor for general anisotropic materials, Equation (7) can be written as d rdmn edij ¼ tik ðDÞtjl ðDÞtmo ðDÞtnp ðDÞCklop rdmn ¼ Cijmn

ð8Þ

d ¼ tik ðDÞtjl ðDÞtmo ðDÞtnp ðDÞCklop Cijmn

ð9Þ

where

Equation (9) can be regarded as the transformation of compliance tensor from the initial undamaged stage to damaged stage and this relation is used to account for the effects due to damage in impact simulation. With the assumption that laminate orthotropy and principal material axes remain unchanged after damage occurs, the components tij(D) of t(D) are zero [when i 6¼ j according to Equation (9)]. The relation between the initial undamaged and damaged compliance components in Equation (9) can be rewritten as d ¼ t411 ðDÞC1111 C1111

ð10Þ

d C2222 ¼ t422 ðDÞC2222

ð11Þ

d C3333 ¼ t433 ðDÞC3333

ð12Þ

d C3311 ¼ t233 ðDÞt211 C3311

ð13Þ

d C2233 ¼ t222 ðDÞt233 ðDÞC2233

ð14Þ

d C1122 ¼ t211 ðDÞt222 ðDÞC1122

ð15Þ

d ¼ t233 ðDÞt211 ðDÞC3131 C3131

ð16Þ

d C2323 ¼ t222 ðDÞt233 ðDÞC2323

ð17Þ

d C1212 ¼ t211 ðDÞt222 ðDÞC1212

ð18Þ

Because the subscript ‘‘1’’ denotes the fiber direction, it is assumed that t11(D) regards only the state of fiber damage mode and can be expressed as t11(Df) with Df as a scalar which


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reflects the extent of fiber breakage. As can be seen from Equations (10–18), t11(D) affects only the compliance matrix in its in-plane normal component C1111, interlaminar shear component C3131, in-plane shear component C1212, and the coupling component between the fiber direction and both in-plane and out-of-plane transverse normal directions C1122 and C3311. This is generally coincident with the effect of fiber breakage. Similarly, t22(D) and t33(D) are assumed to be associated with matrix cracking and delamination damage, respectively. With two indices Dm, Dd denoting the extent of matrix cracking and delamination damage, respectively, t22(D) and t33(D) become t22(Dm) and t33(Dd). In the present study, Df, Dm, and Dd are defined as the numbers of occurrences that the three damage criteria are met, respectively, during the time step iterations. Taking Df for example, when no fiber breakage has occurred, Df is initiated as 0. When the fiber breakage criterion is met for the first time, Df is set to be 1. As Df changes its value from 0 to 1, the compliance components of the specific element at the specific ply will change to a higher value for the proceeding calculations. As the impact process continues, the impactor continues to move into the laminate. The stresses at the same location might increase to certain values that the criterion for fiber breakage is satisfied again, even though the stiffness has dropped due to the damage in the previous iteration. The damage criterion is repeatedly used to detect successive damage and the value of Df increases by 1 each time the damage criterion is satisfied. As Df increases, fiber breakage becomes more severe so the corresponding compliance components become larger. Through the following relation, the compliance transformation factor from undamaged plies to damaged plies, t11, is represented as a function of Df t11 ðDf Þ ¼ kDf

ð19Þ

where k is a material-dependent parameter determined by comparing the impactor and laminate displacements between trial computation and impact experiments. For T300/ QY8911 (graphite/epoxy) [11], which is used in the present study, k is chosen as 1.3. The same approach is used to evaluate the degree of damage due to matrix cracking and delamination with Dm and Dd as the indices. The similar function in Equation (19) is used to determine t22 and t33 with Dm and Dd, respectively. t22 ðDm Þ ¼ kDm

ð20Þ

t33 ðDd Þ ¼ kDd

ð21Þ

In reality, there should be different expressions to correlate the degree of damage to the extent of reduction in stiffness under different damage modes. However, by adjusting the k value, Equations (19)–(21) can reflect the extent of damage and provide computational results that compare well with experimental results.

DAMPING AND COMPUTATION STABILITY

The dynamic response of the impactor can be ignored during the simulation because the impactor is generally made of steel with high density and mass [7]. However, the dynamic response of the laminate can be remarkable. The dynamic response of the laminate is shown as the oscillation of both the impact force and the flexural displacement of the



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laminate. When the stiffness of the laminate is reduced due to impact damage according to the present damage model, the amplitude of the laminate vibration will be enlarged at the damaged areas. This change in laminate displacement enlarges the difference between the laminate displacement and the impactor displacement, leading to severe oscillation in impact force according to the contact law. The abrupt change in impact force in turn enhances the dynamic response of the laminate causing very complicated interactions. In most cases, a computational error due to divergence will occur. Material damping is considered in order to eliminate the computational instability. Usually, material damping is ignored in low velocity impact analysis because it has little effects on the impact process if damage is not considered. However, when impact damage is present, the properties of damaged material will change and the material damping will increase dramatically. The material damping has important effects on the dynamic behavior of the plate at the damaged locations. Energy dissipation due to damping can actually stabilize computation. In the present study, only material damping is taken into account: C ¼ 0 K

ð22Þ

where C and K are the damping matrix and structure stiffness matrix, respectively, 0 is material damping coefficient to be determined from trial computations. A damping ratio of 0.015 was obtained experimentally by Luo et al. [3] on free vibration of a circular [04/904]S composite laminate with a diameter of 100 mm. Even though the geometry of Luo’s sample is not the same as the sample geometry used in the present study, their damping ratios should be approximately of the same order. Therefore, a damping ratio of 0.015 was used to determine the material damping coefficient 0. Figure 2 shows the results from the finite element model with 0 ¼ 5.0E-5 where a damping ratio of 0.015 was achieved. To include the effect on damping due to damage, the following relation is used to define the material damping coefficient as a function of damage condition. bi, j ¼ b0 þ r t11 ðDf Þ þ t22 ðDm Þ þ t33 ðDd Þ

ð23Þ

where bij is material damping coefficient of the jth ply in the ith element, 0 is material damping coefficient for undamaged laminate, and r is a constant to be determined from

Figure 2. Displacement history at center of plate with b0 ¼ 5.0E-5.


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Figure 3. Displacement history at center of plate with 0 ¼ 1.87E-3.

Figure 4. Finite element mesh.

trial calculations. Based on trial calculations, r ¼ 0.0007 is the lowest value which can result in stable computations for the impact simulation in the following sections. Figure 3 shows the plots of the free vibration of a composite laminate using the damping matrix C where r ¼ 0.0007, Dm ¼ 1 and Dd ¼ 1 that are used for the eight elements at the center of the finite element model shown in Figure 4. Based on Equation (23), bi,j changes when damage develops. However, the variation of bi,j in the range involved in this study has not much effect on the calculation stability and the simulation results. In order to simplify the calculations, b i,j remains the value corresponding to the initial damage of the material at each location after impact damage occurs.

RESULTS AND DISCUSSION

Computer simulation of low-velocity impact were performed to demonstrate the impact process and damage progress in detail. T300/QY8911 (graphite/epoxy) composite


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laminates, with material properties listed in Tables 1 and 2, were selected for this study. The specimens were [(45/0/ 45/90)S]4, 126 mm 76 mm 3.76 mm laminates. The four corners of the plates were clamped in order to simulate the support condition in the impact experiment. Figure 4 shows the finite element mesh consisted of 336 20-noded solid hexahedral elements with layered structure [4]. A steel impactor of mass 5.17 kg with a hemispherical tip of 12.7-mm in diameter was used. During the impact simulation, the time steps for integration using Wilson- method and the time steps for reforming the stiffness matrix to account for damage were 0.75 and 75 ms, respectively. Ds and Dt in Equation (5) were assumed to be 6.5 and 1.5, respectively, and a value of 1.3 was selected for k in Equation (13). The selection procedures of Ds, Dt, and k values are described later in this section. An impact of 25.18 J in energy was simulated as an example to demonstrate the impact and damage process. The results are given in Figures 5–8. Figure 5 shows the simulated force history using models with and without laminate damage during the impact duration. It can be seen that the results based on the damaged model is consistent with the experimental force history shown in Figure 9 [7]. Figure 6 shows the impact force versus the flexural displacement of the impacted laminate at contact point and Figure 7 shows the time history of the impactor displacement and the laminate’s flexural displacement. It should be noted that the flexural displacement (global displacement) of the plate is obtained from the impact force and the global stiffness matrix with laminate damage taken into account without the inclusion of local indentation. In the loading portion of

Table 1. Material properties of T300/QY8911. E11 (GPa)

E22 (GPa)

G12

v12

(kg/m3)

8.8

4.5

0.33

1.614

135

Table 2. Strength characteristics of T300/QY8911. Xt (MPa) 1.548

Xc (MPa)

Yt (MPa)

Yc (MPa)

S12 (MPa)

1.226

55.5

218

89.9

Figure 5. Impact force history.


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Figure 6. Impact force vs. flexural displacement of plate.

Figure 7. Displacement of impactor and flexural displacement of plate.

the curve (up to P4 and shown as solid curve) in Figure 7, Equation (1) was used as the loading contact law and the difference between the impactor displacement and the plate displacement represents the indentation. However, in the unloading portion of the impact duration, Equation (3) was used to calculate the impact force and the laminate displacement is presented in dashed curve. Due to the difference in nature between Equations (1) and (3), i.e., the first half of the laminate displacement emphasizes the relations among impactor, laminate global displacement, and laminate indentation, while the second half of the curve emphasizes only the impactor displacement and laminate global displacement. The comparison of the depth of dent between the loading and unloading portions are not readily available. In reality, this is the result caused by the use of linear model to simulate a nonlinear behavior (permanent dent on the impacted laminate). The simulated velocity and change of kinetic energy of the impactor as functions of time are shown in Figure 8. The impact and damage processes of the composite plate can be observed as five stages as shown in Figures 5 and 6 where P1, P2, P3, and P4 are used to specify the beginning or the end of each stage. The calculated percentage delamination as a function of time is shown in Figure 10 and the delamination area of each ply-interface at P1, P2, P3, and P4 are shown in Figure 11. The simulated final delamination area in each ply-interface after impact is compared with experimental results obtained by Guan [7] using thermaldeply damage examination technique and is shown in Figure 12. It should be noted from



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Figure 8. Velocity and variation of kinetic energy of impactor.

Figure 9. Experimental impact force and kinetic energy change of impactor.

Figure 10. Development of total delamination in all interfaces during impact.

Figures 11 and 12 and the 4th, 8th, 12th, 16th, 20th, 24th, and 28th interfaces showed no delamination and this is because they are interfaces between plies of same fiber direction. The impact process and damage progression mainly in the form of delamination within each stage are discussed below.


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Figure 11. Delamination area of each interface at different stages.

Figure 12. Comparison of delamination area between simulation and experiment.

Stage 1 – From the beginning of impact to P1: The impact force and the flexural displacement of the plate at the contact point increases rapidly but relatively smooth (Figures 5–7). The regular undulations of the impact force are caused by the vibration effects of the impacted plate. As shown in Figure 5, the impact force of both the ‘‘damaged’’ and ‘‘undamaged’’ laminates in this stage are coincident because no damage has occurred at this time. At P1, slight damage in the forms of matrix cracking and delamination has initiated (Figure 11). Therefore, P1 can be regarded as the initiation of impact damage. Stage 2 – P1 to P2: In this stage, the impact force reaches its maximum at P2 with intensive random fluctuation caused by rapid development of impact damage. The rate of increase of impact force, however, is a little lower than the first stage. Due to the change of stiffness caused by damage, the impact force starts its deviation from the impact force history of an ‘‘undamaged’’ laminate (Figure 5). The impact damage developed quickly in this stage which can be seen in Figure 10. It should be noted that the delamination areas near the front and back of the laminate have almost reached their peak at P2 (Figure 11). Stage 3 – P2 to P3: The impact force drops from its maximum to a comparatively steady value while its fluctuation remains very intensive (Figure 5). The flexural displacement of the plate at impact continues to increase, demonstrating that the impact damage is still developing (Figures 6 and 10). Figure 11 shows that the delamination expanded in this



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stage is mainly near the middle of the laminate. This indicates that the stiffness deduction due to damage near the front and back sides of the plate from the previous stages would lead to an increase of stresses at the middle of the laminate. Figure 11 also indicates that the delamination area at every interface in the plate has nearly researched its maximum at P3. Stage 4 – P3 to P4: The impact force becomes steady compared with the two previous stages. The amplitude of force fluctuation decreases but the impactor is still pressing into the plate until P4, where the velocity of impactor becomes zero (Figures 5 and 6). It should be noted that, as described earlier, due to the different contact laws used in the loading and unloading portion, the indentation at P4 cannot be used to compared with the indentation at the end of impact. The velocity of impactor becomes zero at P4 and the energy absorbed by the laminate reaches its maximum at P4 (Figure 8). The absorbed energy has either converted to the elastic energy or dissipated through the damage development inside the plate. Due to the continued motion of the impactor into the laminate, the damage continues to develop, mainly reflected as the continuously increasing value of Dd in Equations (19–21). The delamination area reaches its maximum at the end of this stage (Figures 10 and 11). Stage 5 – P4 to the end of the impact: This is the last stage of impact process. The impact force begins to decrease, the amplitude of the random fluctuation becomes small, and the motion of the impacted plate and impactor are all in their rebounding state (Figures 5–8). No further damage development is accessed in this stage. Because of serious impact damage and plastic deformation in the laminate, there would be a dent left on the plate. Therefore, the impact process is ended when the impactor rebounds to the position which is do, the depth of permanent dent, below the position where the initial contact took place. This is shown in Figure 7 where the end of impact is determined when the displacement of the impactor equals the depth of permanent dent which is 1.8 mm from experimental observation for this case [7]. At this time, the impactor leaves the plate at a rather low velocity compared with the initial impact velocity (Figure 8). The cost kinetic energy of the impactor is absorbed by the damage of composite plate. Even though the impact response varies with different impact energy, the above distinction of the five stages for impact process is still effective to some extent. Figures 13 and 14 are the impact force history with impact energy of 16.78 J and 50.35 J, respectively. Figures 15 and 16 also show the impact force versus the global flexural displacement of the

Figure 13. Impact force history with impact energy of 16.78 J.


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Figure 14. Impact force history with impact energy of 50.35 J.

Figure 15. Impact force vs. flexural displacement of plate (16.78 J).

Figure 16. Impact force vs. flexural displacement of plate (50.35 J).

impacted laminate at contact point. It is obvious that the impact energy has significant effects on Stages 3 and 4. As described previously regarding impact damage, the largest delamination area near the backside of impacted plate has already occurred at the end of the Stage 2 and the delamination area around the middle of the plate is developed mainly during Stage 3. The delamination does not expand much during Stage 4 but the continuous motion of the impact into the plate causes more serious damage.



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As the impact energy increases, for example 50.35 J in this study, Stages 4 and 5 become more prominent (Figures 14 and 16). The periods of these two stages are prolonged and the deformation at impact site is very large. The flexural displacement is about 11 mm at P4 and the impact force is reduced dramatically after it reached its peak value at P2. The phenomenon of abrupt drop in impact force after it reaches the maximum at P2 is consistent with the experimental observation by Liu et al. [12]. This high value of flexural displacement suggests plate perforation. Another phenomenon worth mentioning is regarding the delamination area in different interfaces through the thickness of the plate. Although the higher the impact energy the more severe the impact damage, the delamination areas in the interfaces close to the backside of the laminate are similar for different impact energy levels. However, larger delamination areas occur at the middle of the laminate for higher impact energy as shown in Figure 17. This is because at higher energy impact the impact force increases so quickly in Stages 1 and 2 that the delamination cannot fully develop. While Stages 3 and 4 are prominent, the impactor presses deeper into the plate and the damage formulated in Stages 3 and 4 can develop to the full extent resulting in large areas of delamination around the middle of the plate and serious damage just beneath the impactor. This phenomenon reveals the reason that after the impact energy increases to a certain level, the reduction of compression strength of the impacted plate levels off [12], under the assumption that the post-impact compression strength of the laminate depends on delamination area. This high value of flexural displacement suggests plate perforation. The impact model proposed in the present study provides accurate details on all aspects of low velocity impact on laminated composite including impact force, displacements, velocity, energy response, and damage development during impact. There are four parameters in the model to be determined by experiments: the depth of permanent dent d0 in Equation (3), Dt and Ds in Equation (4), and k in Equations (19)–(21). Among the four parameters, d0 is used in unloading law for the rebound portion of impact and is not related to damage formulation because the impact damage is established before the impactor starts to rebound. Ds and Dt are the primary factors which dominate the delamination areas in the interfaces close to the middle and backside of laminate, respectively. Their effects can be seen from the delamination areas in a simulated static indentation test shown in Figures 18 and 19. In the simulated indentation test, a simply supported 126 mm 76 mm 3.76 mm[(45/0/ 45/90)S]4 T300/QY8911 graphite/epoxy laminate is applied with an indentation force of 7,000 N which is approximately the

Figure 17. Comparison of delamination areas at different impact energy levels.


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Figure 18. Delamination under indentation force of 7,000 N (Dt ¼ 0).

Figure 19. Delamination under indentation force of 7,000 N (Ds ¼ 0).

peak contact force in the impact test shown in Figure 9. It can also be seen that no delamination occurs at the backside of the laminate if Dt is set to be 0 and no delamination occurs at the middle of the laminate if Ds is set to be 0. Because the damage due to low velocity impact is similar to the damage due to indentation on composite laminates [13], the values of Ds and Dt can be estimated using results from simulated static indentation tests. Let Sbottom represent the summation of delamination area in the 29th, 30th, and 31st interfaces and Smiddle represent the summation of delamination area of the 13th, 14th, and 15th interfaces from the impact side. Figure 20 shows a near-linear relationship between the Sbottom-to-Smiddle ratio and Ds-to-Dt ratio regardless of the individual values of Ds and Dt in the simulated indentation tests. A Ds-to-Dt ratio of 4.33 is chosen based on the value of 1.1 for the Sbottom-to-Smiddle ratio which is determined from the experimental results of 25.18 J impact. In order to further validate the selection of Ds and Dt values, an error function on the total delamination area is defined as Err ¼

N 1X jSe, i Sc, i j N i¼1 Se, i

ð24Þ

where Se,i and Sc,i are the experiment and calculated delamination areas at the ith interface of the laminate, respectively, and N is the total number of interfaces. A parametric study reveals that with Ds-to-Dt ratio remaining 4.33, the error function reaches a minimum value when Dt is approximately 1.5, as shown in Figure 21. Therefore, Dt is chosen as 1.5 and Ds is chosen as 6.5 in this study.



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k determines the reduction in material stiffness due to damage and has great influence on the variation of impact force during impact. k can be obtained by the impact force response in experiments and trial calculations. Figure 22 shows the stiffness reduction factors versus damage degrees for different k values. As the value of k determines the extent of stiffness reduction, it affects the simulated impact and damage processes. Figure 23 is the impact simulation results of the example shown in Figure 5 with different k values.

Figure 20. Effects of Ds/Dt on delamination of middle and backside of laminate.

Figure 21. Error function vs. Dt (with Ds/Dt ¼ 4.33).

Figure 22. Stiffness reduction for different k values.


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Figure 23. Impact force history with different k values.

In general, when k increases, the impact duration becomes longer and the peak impact force becomes smaller.

SUMMARY

A computational model has been developed in this investigation to simulate the process and damage of low-velocity impact of composite laminates. The existing loading contact law for low velocity impact was adopted and the unloading contact law was modified. The contact law greatly simplified the impact and damage analysis, precluding the complicated contact and damage analysis around the impact point. A delamination criterion is proposed to be used in impact simulation. The coefficients Ds and Dt in the delamination criterion have defined physical meanings and can be determined by experiments and trial calculations. The continuum damage mechanics analysis is introduced into impact damage with assumed compliance variation functions. There is only one coefficient in the functions which needs to be adjusted and this can be done based on the impact force response from experiments. The damping effects of damaged material are considered in the simulation to increase the calculation stability and to improve the accuracy of the simulation. The developed model is comparable with impact test data including the dent left on the impact point, the damage inside the plate, and the impact force history.

ACKNOWLEDGEMENTS

This material is based upon work supported by NASA under Grant No. NCC5-168 and the Faculty Fellowship from the National Institute for Aviation Research, Wichita State University. The support of Kansas NSF Cooperative Agreement EPS-9874732 and the Wichita State University High Performance Computing Center is also acknowledged.

REFERENCES 1. Yang, S.H. and Sun, C.T. (1982). Indentation Law for Composite Laminates. Composite Materials: Testing and Design (6th conference), ASTM STP-787, pp. 425–449.



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2. Belingardi, G., Gugliotta, A. and Vadori, R. (1998). Numerical simulation of fragmentation of composite material plates due to impact. International Journal of Impact Engineering, 21: 335–347. 3. Luo, R.K., Green, E.R. and Morrison, C.J. (1999). Impact damage analysis of composite plates. International Journal of Engineering, 22: 435–447. 4. Jones, R.R., Callinan, R., Hen, K.K. and Brown, K.C. (1984). Analysis of multi-layer laminates using three-dimensional super-elements. International Journal for Numerical Methods in Engineering, 20: 583–587. 5. Lagace, P.A., Ryan, K.F. and Graves, M.J. (1994). Effect of damage on the impact response of composite laminates. AIAA Journal, 32: 1328–1330. 6. Shahid, I., Chang, F.K. and Shah, B.M. (1996). Impact damage resistance and damage tolerance of composite with progressive damage. AIAA-96-1403-CP. 7. Guan, Z.D. (1994). Transient dynamic analysis of impact and damage processes of laminated composite panels and stiffened panels due to low velocity impact. PhD Dissertation, Department of Flying Vehicle Design and Applied Mechanics, Beijing University of Aeronautics and Astronautics. 8. Choi, H.Y. and Chang, F.K. (1992). A model for prediction damage in graphite/epoxy laminated composites resulting from low-velocity point impact. Journal of Composite Materials, 26: 2134–2169. 9. Shen, W., Peng, L.H., Yang, F. and Shen, Z. (1987). Generalized elastic damage theory and its application to composite plate. Engineering Fracture Mechanics, 28: 403–412. 10. Shen,W. and Wu, C.S. (1992). Computer simulation for damage-failure process of composite plate under high-speed impact. Engineering Fracture Mechanics, 42: 159–168. 11. The science and technology of aeronautics and astronautics institute (PRC) (1990). Manual of Composite Structure Design, p. 114. 12. Liu, Dahsin, Raju, B.B. and Dang, X.L. (1998). Size effects on impact response of composite laminates. International Journal of Impact Engineering, 21: 837–854. 13. Lee, Shaw Ming and Zahuta, P. (1991). Instrumented impact and static indentation of composites. Journal of Composite Materials, 25: 204–222.


Journal of Composite Materials - CiteSeerX

Journal of Composite Materials - CiteSeerX

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