An integrated method for predicting damage and residual tensile strength of composite laminates under low velocity impact

An integrated method for predicting damage and residual tensile strength of composite laminates under low velocity impact

Computers and Structures 87 (2009) 456–466 Contents lists available at ScienceDirect Computers and Structures journal homepage: www.elsevier.com/loc...

2MB Sizes 5 Downloads 104 Views

Computers and Structures 87 (2009) 456–466

Contents lists available at ScienceDirect

Computers and Structures journal homepage: www.elsevier.com/locate/compstruc

An integrated method for predicting damage and residual tensile strength of composite laminates under low velocity impact Hai-Po Cui a,*, Wei-Dong Wen b, Hai-Tao Cui b a b

College of Medical Device and Food, University of Shanghai for Science and Technology, Shanghai 200093, People’s Republic of China College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 19 July 2008 Accepted 7 January 2009 Available online 14 February 2009 Keywords: Impact damage Composite laminates Progressive damage Residual tensile strength

a b s t r a c t An integrated approach is presented to analyze the whole process of damage initiation and development for composite laminates under impact loading as well as tensile loading after impact using the 3D progressive damage theory. The real impact damage status of composite laminates is employed to analyze the residual tensile strength instead of the artificial premises adopted by traditional methods. This integrated approach can not only improve the prediction accuracy of the ultimate strength but also avoid large numbers of experiments for obtaining the impact damage parameters. A parametric modeling program package based on the analytical method has been developed. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Fiber reinforced composite laminates are widely used in many engineering fields, especially in aeronautic and aerospace structures owing to their high strength-to-weight and stiffness-toweight ratios. However, they are susceptible to damage due to low velocity impact loading during manufacture and in service. Low velocity impact loading can cause extensive sub-surface damage that may not be visible on the laminate surface but can lead to a significant reduction in the residual strength of composite laminates. Therefore, considerable research has been done to better understand the impact properties and residual strength after impact. Kevin and Reza [1] studied the impact damage process of laminates but they did not consider the residual strength after impact (RSAI). Johnson et al. [2] and Allix [3] performed the impact damage analysis in a way similar to that presented in Ref. [1]. Husman et al. [4] proposed a model relating the RSAI of composite laminates to the impact energy, in which the theoretical prediction exhibited good agreement with the experimental results. Caprino and Lopresto [5] presented a model for predicting the RSAI of laminates with an indentation law, where the RSAI was a function of the indentation depth. Although these models exhibit good correlations with the corresponding experimental data, many of the parameters have to be obtained by a number of experiments and the internal damage of laminates cannot be appropriately predicted. To overcome these disadvantages, more detailed stress * Corresponding author. Tel.: +86 21 55271115; fax: +86 21 55270695. E-mail address: [email protected] (H.-P. Cui). 0045-7949/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2009.01.006

analyses of the laminates should be performed. Davies et al. [6,7] predicted the threshold impact energy for onset of delamination and the relationship between the impact force and the delamination using the force-driven model and energy-balance model. Good agreement was obtained by comparison the theoretical prediction with the experimental data. More importantly, the CPU time spent on the analysis by the models was greatly shortened. However, Davies et al. did not predict the other failure modes such as transverse matrix cracking, matrix crushing, and fiber failure, which have an insignificant influence on the residual strength of laminates. Furthermore, the prediction error of the threshold impact energy by the models would increase if the fiber breakage takes place. El-Zein and Reifsnider [8] adopted the average stress criterion to predict the RSAI, and the damaged zone due to impact was simulated as an elliptical inclusion. Soutis and Curtis [9] and Gottesman et al. [10] proposed two different models for predicting the RSAI, where the damaged zone generated by impact was based on the assumption about a regular geometry existing in these models. These premises, however, still have two evident drawbacks in that the prediction accuracy of the RSAI cannot be guaranteed and that many parameters have to be obtained by performing numerous experiments. For the research mentioned above, the impact process is analyzed individually or the RSAI is predicted based on the assumption about impact damage. The objective of this paper is to propose a whole-process analysis approach to exploring the impact process and the RSAI, where the entire damage initiation and development process of composite laminates under the impact loading, the tensile loading after impact and prediction of RSAI are all involved. The 3D progressive damage theory is used in the model. The real

457

H.-P. Cui et al. / Computers and Structures 87 (2009) 456–466

impact damage status of composite laminates is applied to analyzing the residual tensile strength instead of the artificial premises adopted by many traditional methods for the damage status of composite laminates after impact. This integrated approach can not only improve the prediction accuracy of the ultimate strength but also bypass large numbers of experiments for obtaining the impact damage parameters. Furthermore, the RSAI can be predicted directly from the impact energy by this approach. Based on the analytical method, a parametric modeling program has been developed to predict the impact damage process and RSAI of composite laminates with any ply orientation.

The 3D progressive damage theory is applied in the analysis. The theoretical model consists of a stress analysis for determining the stress distributions inside the composite laminates and a failure analysis for predicting the initiation and the extent of the damage. The solving procedure is briefly summarized as follows. 2.1. Stress analysis 2.1.1. Stress analysis for the impact process For the small strain theory, the equilibrium equation for composite laminates can be expressed as [11]

rij;j ¼ qu€i

ð1Þ

where rij, ui and q are stresses, displacements and mass density, respectively. The stress–strain relation is

rij ¼ C ijkl ekl

€i C ijkl uk;lj ¼ qu

ð3Þ

In terms of finite element terminology, the equations of equilibrium for a finite element system in motion can be expressed as

€ þ KU ¼ F MU

ð4Þ

Where M and K are the mass and stiffness matrices, F is the external € are the displacement and acceleration vectors force vector, U and U of the finite element assemblage. The displacements of each element can be calculated by solving Eq. (4) utilizing the initial and boundary conditions. The stresses of each element may then be obtained from Eq. (1). 2.1.2. Stress analysis for the tensile process For the orthotropic composite laminates, the strain distributions throughout the laminated plate are given by

@u @v @w ex ¼ ; ey ¼ ; ez ¼ ; @x @z @y @w @u @u @ v czx ¼ þ ; cxy ¼ þ @x @z @y @x

@ v @w cyz ¼ þ ; @z @y

ð7Þ

2 2 2 2 2 2  66 @ u þ Q  55 @ u þ 2Q  16 @ u þ Q  26 @ v  16 @ v þ Q  11 @ u þ Q Q 2 2 2 2 @x @y @z @x@y @x @y2 2 2 @2v  12 þ Q  66 Þ @ v þ ðQ  55 Þ @ w  13 þ Q þ ðQ 2 @z@x @z @x@y

 45 Þ  36 þ Q þ ðQ

@2w ¼0 @y@z

ð8Þ

2 2 2 2 2  16 @ u þ Q  26 @ u þ Q  45 @ u þ ðQ  12 þ Q  66 Þ @ u þ Q  66 @ v Q 2 2 2 @x @y @z @x@y @x2 2 2 2 2  22 @ v þ Q  44 @ v þ 2Q  26 @ v þ ðQ  45 Þ @ w  36 þ Q þQ @z@x @y2 @z2 @x@y 2  44 Þ @ w ¼ 0  23 þ Q þ ðQ @y@z

 45 Þ  36 þ Q ðQ

 45 þ 2Q

ð9Þ

2 2 @2u  55 Þ @ u þ ðQ  23 Þ @ v  13 þ Q  44 þ Q þ ðQ @y@z @z@x @y@z

 45 Þ  36 þ Q þ ðQ

2 2 2 @2v  44 @ w þ Q  33 @ w  55 @ w þ Q þQ 2 2 @x @y @z2 @z@x

@2w ¼0 @y@z

ð10Þ

 ij is related to the material constants and the ply orienwhere the Q tation of the layer in a laminate. From Eqs. (8)–(10), the displacements of each element can be calculated by the finite element method utilizing the initial and boundary conditions. The strain and stress of each element can then be obtained from Eqs. (5) and (6), respectively. 2.2. Failure analysis 2.2.1. Failure criteria for the impact process Typical low velocity impact damage appears in the form of transverse matrix cracking, matrix crushing, delamination and fiber fracture. Hou et al. [12] summarized the stress-based failure criteria for the four types of damage. These criteria have been adopted in many previous researches [13,14]. Here, we adopt the criteria presented in Ref. [12], which are formulated below: Transverse matrix cracking

e2m ¼



r22

2

Yt

 þ

r12 S12

2

 þ

r23 Sm23

2

P 1 ðr22 P 0Þ

ð11Þ

ð5Þ

The stress–strain relation is expressed as

2 Q 11 6Q 6  12 6 6 Q 13 rz ¼6 6 0 > s yz > > > 6 > > > > 6 > > > 4 0 > s > zx > > > ; :  16 sxy Q

@ sxy @ ry @ syz þ þ ¼ 0; @x @y @z

By substituting Eqs. (5) and (6) into Eq. (7), the equilibrium equations can be written as

ð2Þ

where Cijkl is the elastic modulus of material, ekl is strain. By substituting Eq. (1) into Eq. (2), the governing equation of the composite laminates can be written as

9 > > > > > > > > =

@ rx @ sxy @ szx þ þ ¼ 0; @x @y @z @ szx @ syz @ rz þ þ ¼0 @x @y @z

 45 þQ

2. Analytical model

8 rx > > > > > r y > > > <

The equilibrium equations can be expressed as

 12 Q  22 Q

 13 Q  23 Q

 23 Q 0

 33 Q 0

0  44 Q

0  55 Q

0  26 Q

0  36 Q

 44 Q

 55 Q

0

0

0

0

0

0

9 8  16 3> ex > Q > > > > > > > ey >  26 7 > Q > > 7> > > > > 7 = <  e Q 36 7 z 7 7 0 7> > > cyz > > > > 7> > > > c 0 5> > zx > > > > > :  66 cxy ; Q

Table 1 Material property degradation rules.

ð6Þ

Damage mode

Property degradation rule

Matrix cracking Matrix crushing Fiber–matrix shear-out Delamination Fiber failure

Eyy ¼ 0:2Eyy Gxy ¼ 0:2Gxy Gyz ¼ 0:2Gyz Eyy ¼ 0:4Eyy Gxy ¼ 0:4Gxy Gyz ¼ 0:4Gyz Gxy ¼ txy ¼ 0 Ezz ¼ Gyz ¼ Gxz ¼ tyz ¼ txz ¼ 0 Exx ¼ 0:07Exx

Note: Superscript ‘‘*” indicates the material property after degradation.

458

H.-P. Cui et al. / Computers and Structures 87 (2009) 456–466

Fig. 1. Flowchart of the whole-process analysis for laminates under impact loading and tensile loading after impact.

Fig. 2. Typical surface impact damage of the laminate.

459

H.-P. Cui et al. / Computers and Structures 87 (2009) 456–466

Matrix crushing

 2  2 1 r22 Y 2 r22 r22 r12 e2d ¼ þ c2  þ P 1 ðr22 < 0Þ 4 S12 Y S12 c 4S12 Y c

ð12Þ

Delamination

e2l ¼



r33 Zt

2

 þ

Fiber failure

r23 Sl23

2

 þ

r31 S31

2

P 1 ðr33 P 0ÞÞ

ð13Þ

e2f

 ¼

r11 Xt

2 þ

r212 þ r213 S2f

! P1

ð14Þ

where rij and Sij are the layer stress components and shear strength referring to a local coordinate system. In this system, the x- and yaxes are parallel and transverse to the fiber’s direction, respectively, while the z-axis coincides with the normal direction. In the denominators of the criteria, the corresponding strengths in which the subscripts are t and c refer to the tensile and compressive value, respectively. If the stress components of an element satisfy the failure criterion, the corresponding failure will occur.

Fig. 3. Typical X-radiographs for impact damage.

Fig. 4. Progressive damage X-radiographs under tensile loading.

460

H.-P. Cui et al. / Computers and Structures 87 (2009) 456–466

Fig. 5. Typical tensile failure photographs of laminates.

2.2.2. Failure criteria for the tensile process The damage of composite laminates induced by tensile loading is often a complex mixture consisting of: matrix cracking, fiber– matrix shear-out, delamination and fiber failure. Tserpes et al. [15] developed a 3D progressive damage model with these four types of damage to predict the residual strength of bolted composite joints. We used the failure criteria in the model to predict the RSAI of composite laminates with impact damage. Each of the damage modes is predicted by the following expressions: Matrix cracking



r22

2

 þ

Yt

r12

2

 þ

S12

r23

2

S23

P1

ð15Þ

P1

ð16Þ

P1

ð17Þ



r11 Xt

 P1

ð18Þ

As long as the stress components within a specific layer of an element satisfy the failure criterion, the corresponding damage mode will occur. 2.2.3. Property degradation models Once damage occurs, the material constants in this layer of the element should be modified. The degradation technique proposed by Tan and Perez [16] assumed that the effect of damage on the material constants can be represented using internal state variables. This approach is the most reliable one because it can adjust the value of the internal state variables according to the predicted

Fiber–matrix shear-out



r11

2

 þ

Xt

r12

2

 þ

S12

r13

2

S13

Delamination



r33

2

Zt

 þ

r13

2

S13

 þ

r23 S23

2

Fibre failure

Table 2 Material properties of the T300/BMP-316 laminates. Exx/GPa Eyy/GPa Ezz/GPa Gxy/GPa Gxz/GPa Gyz/GPa

txy txz tyz Xt/MPa Xc/MPa Yt/MPa Yc/MPa Sxy/MPa Sxz/MPa Syz/MPa Zt/MPa Zc/MPa

128.8 8.3 8.3 4.1 4.1 4.1 0.355 0.355 0.355 1298.2 1269.4 53.6 185.0 102.0 102.0 102.0 53.6 185.0

Fig. 6. Sensitivity evaluation of the effect of number of elements on the calculated residual strength.

Fig. 7. FE model of the laminates.

H.-P. Cui et al. / Computers and Structures 87 (2009) 456–466

461

type of failure mode, which leads to a more realistic simulation of damage accumulation. Camanho and Matthews [17] extended the degradation technique of Tan to account for the 3D cases. We adopted the property degradation models of Camanho and Matthews [17] to analyze the impact process and the RSAI. The property degradation rules are shown in Table 1.

can no longer sustain the accumulated damage is required. It is assumed that the total failure of the composite laminates occurs when the damage in the fibers propagates to the laminate freeedge. Such an approach has been successfully used by other researchers to predict the ultimate strength of composite laminates [18,19].

2.3. Final failure criterion

2.4. Flowchart of the model

Besides the failure criteria used to predict the damage, a criterion for predicting the load under which the composite laminates

Based on the proposed numerical model, a parametric modeling program has been developed to predict the impact damage process

Fig. 8. Progressive damage modeling of laminates under impact loading.

462

H.-P. Cui et al. / Computers and Structures 87 (2009) 456–466

and RSAI of composite laminates with any ply orientation. The flowchart is shown in Fig. 1. The module A represents the flowchart of impact damage analysis, while the module B describes the flowchart of the damage progression and residual strength prediction of composite laminates with impact damage under tensile loading. 3. Experimental work The material system used to study the impact damage and RSAI was T300/BMP-316. The dimensions of the laminates were 200  45  2.5 mm3. The following configurations were tested: [45/45/90/0/45/0/45/0/90/0]S and [45/45/90/90/45/0/45/ 90/90/0]S. The specimens were indexed according to the following

method: PNN. When P was A, it indicated the stacking sequence of [45/45/90/0/45/0/45/0/90/0]S. When P was B, it indicated the stacking sequence of [45/45/90/90/45/0/45/90/90/0]S. NN indicated the impact energy. An instrumented dropweight machine was used to impact the specimens which were clamped between two plates each having a square (40  40 mm2) opening and struck by a 16 mm in diameter hemi-spherical tup weighing 2.28 kg. The impact energy was varied by changing the velocity of impact rather than the weight. The repeated impacts were avoided when the impactor bounced against the laminates. The specimens were examined by penetrant enhanced X-ray radiographs to ensure that they were damage-free before impact.

Fig. 9. Progressive damage modeling of laminates with impact damage under tensile loading.

463

H.-P. Cui et al. / Computers and Structures 87 (2009) 456–466

Fig. 2 shows typical surface impact damage photographs while Fig. 3 shows the typical X-radiographs of impact damage. The light color corresponds to damage. The damaged specimens were then loaded to failure, in tension, using an MTS machine. The progressive damage X-radiographs under tensile loading are shown in Fig. 4. The 0 MPa represents the impact damage X-radiographs of laminates before applying tensile loading. Fig. 5 shows typical tensile failure photographs of laminates with impact damage.

4. Model validation In order to verify the proposed analytical method, numerical results were generated from the method to compare with the experimental data. The material system was T300/BMP-316. The stacking sequence for the laminates was [45/45/90/0/45/0/45/ 0/90/0]S, and the impact energy was 5.9 J, which corresponded to the specimen of A5.9. The material properties of the laminates are listed in Table 2.

Fig. 10. Damage development process of A5.9.

300

900 A8.6

A8.6

200

800

B5.9

150 100 50 0

A5.9

A5.9

Damage area (mm 2)

Damage area (mm 2)

250

B5.9

700 600 500 400 300

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

Time (ms)

(a) damage area v.s. impact time

200

0

50 100 150 200 250 300 350 400 450 500

Load (MPa)

(b) damage area v.s. tensile loading

Fig. 11. Predicted damage area versus time and loading curves.

464

H.-P. Cui et al. / Computers and Structures 87 (2009) 456–466

4.1. Mesh sensitivity study In order to evaluate the effect of finite element mesh size on the convergence of the numerical prediction based on the model, a series of numerical calculations were obtained based on different meshes, ranging from a rather coarse to a very fine one. Fig. 6 shows the comparisons of the calculations in which the meshes were generated by decreasing or increasing the number of elements along length (L) and width (W) direction of the laminate while keeping the number of total elements through the thickness (T) fixed. It can be seen that the predicted residual strength quickly converged as the number of the elements was more than 1736 [L(=62)  W(=28)]. Accordingly, it seems that the proposed model does not require the use of an extensive fine mesh. In the model finally chosen, the mesh is graded so that the location of the most severely loaded zone is made up of the smallest elements. A total of 34720[L(=62)  W(=28)  T(=20)] elements are used in the numerical calculations for generating the results in comparison with the test data. The typical finite element model of the laminates is shown in Fig. 7, which is a top view figure. For each layer, one element is used along the through-thickness direction. 4.2. Progressive damage analysis for each layer of laminates 4.2.1. Progressive damage analysis under impact loading Fig. 8 presents the progressive damage process of laminates. The first layer was in contact with the impactor. When the time reached 0.59 ms, very limited matrix crushing appeared in the first layer because of the local high compression stress, and a little delamination started around the contact zone between the impactor and the laminates. Limited delamination also appeared in the second layer and the third layer. When the time reached 0.74 ms, very limited fiber failure initiated in the first layer. Delamination appeared in the fourth layer. A little matrix cracking initiated in the 19th layer and the 20th layer (those farthest away from the im-

pact face) owing to the high tensile stress under the impact loading. When the time reached 1.33 ms, the stresses in the laminates were maximum. Damage appeared on each layer of the laminates. From the damage figures of each layer, it can be seen that the damage distribution had the following rules: the matrix cracking centered on the layers near the un-impacted surface, matrix crushing centralized on the layers near the impacted surface, delamination centralized mainly on the middle layers, and fiber failure centered on the layers near the impacted and un-impacted surface. 4.2.2. Progressive damage analysis under tensile loading after impact After impact analysis, the material parameters representing the damage status of each element can be obtained. According to these parameters, we may initialize the damage of each element of tensile analysis, which was the initial condition for tensile analysis. The initial damage of tensile analysis was the same as the damage shown in Fig. 8, which avoided the artificial premises on damage status after impact, and realized the whole-process analysis for the impact process and the RSAI of composite laminates. As the 0o plies sustain the main tensile loading for the stacking sequence of [45/45/90/0/45/0/45/0/90/0]S, it is assumed that the total failure of the composite laminates occurs when the fibre failure of 0o plies extends to the laminate free-edge. The progressive damage process of the 0o plies is graphically presented at different load levels in Fig. 9. Because of symmetry, only a quarter of the laminates is shown. Fig. 9a represents the impact damage of composite laminates before applying tensile loading. It can be seen that the fibre failure initiated from the center of 0o plies under the load of about 470 MPa. The other three types of damage, i.e. the matrix cracking, fibre-matrix shear-out and delamination, also appeared. The fiber failure mode quickly propagated to the free edges along the direction normal to that of loading, resulting in a total catastrophic failure. The collapse load was about 475 MPa.

Table 3 Impact damage comparison of the laminates. Specimen

A5.9

A8.6

B5.9

Experimental result

Predicted result

Table 4 Comparisons of the predicted impact damage area and RSAI with the experimental results. Specimen

A 5.9 A 8.6 B 5.9

Damage area(mm2)

Residual strength (MPa)

Experimental result

Numerical result

Error (%)

Experimental result

Numerical result

Error (%)

246 310 256

231 280 242

6.1 9.7 5.5

502 354 341

475 385 310

5.4 8.8 9.1

465

H.-P. Cui et al. / Computers and Structures 87 (2009) 456–466

ported, and the impact was performed using a blunt tup made of steel with a 12.7 mm diameter, the impact energy being 3.24 J, 4.59 J and 5.4 J. The predicted damage area and RSAI are compared with the experimental results of El-Zein and Reifsnider [8] in Table 5. From Table 5, it can be seen that the predicted accuracy of RSAI is improved by using the current analysis. The main reason is that the whole-process analysis method for the impact process and the RSAI does not employ the premises on the damage status of composite laminates after impact. Instead the real impact damage status of composite laminates has been used to analyze the RSAI.

4.3. Progressive damage analysis for the whole laminates The damage initiation and development process of the whole laminates under impact loading and the tensile loading after impact are shown in Fig. 10. For the damage development process under impact loading, from Fig. 10a it can be seen that the damage initiated in the center of the laminates in contact with the impactor and spread toward the exterior with the passage of time. When the impact process ended, the main failure mode was matrix cracking. From Fig. 10b, it can be seen that the impact damage expanded along the width direction of the laminates under tensile loading. The matrix cracking and fiber failure propagated faster. When the fiber failure spread to the free edges, total failure of the structure occurred. The predicted damage propagation rules of the composite laminates with impact damage under tensile loading are consistent with the experimental results. The predicted damage area versus time and loading curves of three kinds of laminates are given in Fig. 11a and b.

5. Conclusions In this contribution, an integrated method for the prediction of the low velocity impact damage and residual tensile strength of composite laminates is presented. The following conclusions can be drawn: A whole-process analysis method for the impact process and the RSAI is proposed to study the whole damage initiation and development process of composite laminates under impact as well as tensile loading after impact. The real impact damage status of composite laminates has been applied to analyze the residual tensile strength instead of the artificial premises adopted by traditional methods for damage status of composite laminates after impact, which can not only improve the prediction accuracy of the ultimate strength but also avoid large numbers of experiments for obtaining the impact damage parameters. Furthermore, the residual strength after impact can be predicted directly from the impact energy by this method. The 3D progressive damage theory and analysis technique are used to simulate the impact process and tensile process after impact of composite laminates. The correlation between the experimental and theoretical results for the impact damage area and residual tensile strength within 10% were rational.

4.4. Comparison with the experimental results Table 3 shows that the experimental and predicted impact damage figures of the laminates are in good agreement with each other. The comparison data of the damage area and RSAI are listed in Table 4 that exhibit a good correlation. The comparison between the predicted load–displacement curves and the experimental results are given in Fig. 12. Good consistency is obtained. To further validate the capability and applicability of the proposed method, the results obtained using the theoretical model has been compared with experimental data in Ref. [8]. The material system was Fiberite T300/934 Graphite/epoxy. The stacking sequence was [(0/90)4S. The dimensions of the laminates were 152 mm  89 mm. The edges of the laminates were simply sup-

50

60

50 Experiment

Experiment

30 20

0 0.0

30 20

0.2

0.4

0.6

0.8

1.0

1.2

1.4

30 20 10

10

10

Prediction

40

Load (KN)

40

Experiment

Prediction

40

Prediction

Load (KN)

Load (KN)

50

0 0.0

0.2

Displacement (mm)

(a) A 5.9

0.4

0.6

0.8

0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

1.0

Displacement (mm)

Displacement (mm)

(b) A 8.6

(c) B5.9

Fig. 12. Comparisons of the predicted load–displacement curves and the experimental ones.

Table 5 Comparison of predicted results with the test data of El-Zein and Reifsnider [8]. Impact energy (J)

3.24 4.59 5.4

Damage area (mm2)

Residual strength (MPa)

AEX

APR

Error (%)

PEX

PEL

PPR

Error (%) EEL

EPR

225.8 309.6 374.2

212.4 328.7 401.5

5.9 6.2 7.3

524 448 365

462 421 400

560 465 381

11.8 6.0 9.5

6.9 3.8 4.4

Note: Subscript EX = experimental result of El-zein. EL = predicted result by El-zein’s method. PR = predicted result by current method.

466

H.-P. Cui et al. / Computers and Structures 87 (2009) 456–466

A parametric modeling program has been developed to reliably assess not only the residual tensile strength of composite laminates with any ply orientation after impact but also the accumulation of damage under both impact and tensile loading, which provides a better analysis tool for the structure design and damage analysis of composite laminates. References [1] Kevin VW, Reza V. Application of a damage mechanics model for predicting the impact response of composite materials. Comput Struct 2001;79(10):997–1011. [2] Johnson AF, Pickett AK, Rozycki P. Computational methods for predicting impact damage in composite structures. Compos Sci Technol 2001;61(15):2183–92. [3] Allix O. A composite damage meso-model for impact problems. Compos Sci Technol 2001;61(15):2193–205. [4] Husman GE, Whitney JM, Halpin JC. Residual strength characterization of laminated composites subjected to impact loading. ASTM STP 568. Philadelphia: American Society for Testing and Materials; 1975. p. 92–113. [5] Caprino G, Lopresto V. The significance of indentation in the inspection of carbon fibre-reinforced plastic panels damaged by low-velocity impact. Compos Sci Technol 2000;60(7):1003–12. [6] Davies GAO, Zhang X. Impact damage prediction in carbon composite structures. Int J Impact Eng 1995;16(1):149–70. [7] Davies GAO, Hitchings D, Wang J. Prediction of threshold impact energy for onset of delamination in quasi-isotropic carbon/epoxy composite laminates under low-velocity impact. Compos Sci Technol 2000;60(1):1–7.

[8] El-Zein MS, Reifsnider KL. On the prediction of tensile strength after impact of composite laminates. J Compos Technol Res 1990;12(3):147–54. [9] Soutis C, Curtis PT. Prediction of the post-Impact compressive strength of CFRP laminated composites. Compos Sci Technol 1996;56(6):677–84. [10] Gottesman T, Girshovich S, Drukker E, et al. Residual strength of impacted composites: analysis and tests. J Compos Technol Res 1994;16(3):244–55. [11] Lee JD, Du S, Liebowitz H. Three-dimensional finite element and dynamic analysis of composite laminate subjected to impact. Comput Struct 1984;19(4):807–13. [12] Hou JP, Petrinic N, Ruiz C, Hallett SR. Prediction of impact damage in composite plates. Compos Sci Technol 2000;60(2):273–81. [13] Li CF, Hu N. Low-velocity impact-induced damage of continuous fiberreinforced composite laminates: Part I. An FEM numerical model. Composites Part A 2002;33(8):1055–62. [14] Qi B, Herszberg I. An engineering approach for predicting residual strength of carbonepoxy laminates after impact and hydrothermal cycling. Compos Struct 1999;47(1):483–90. [15] Tserpes KI, Labeas G, Papanikos P, Kermanidis Th. Strength prediction of bolted joints in graphite/epoxy composite laminates. Composites Part B 2002;33(7):521–9. [16] Tan SC, Perez J. Progressive failure of laminated composites with a hole under compressive loading. J Reinf Plast Compos 1993;12(10):1043–57. [17] Camanho PP, Matthews FL. A progressive damage model for mechanically fastened joints in composite laminates. J Compos Mater 1999;33(24):2248–80. [18] Chang FK, Lessard LB. Damage tolerance of laminated composites containing an open hole and subjected to compressive loadings: Part I. Analysis. J Compos Mater 1991;25(1):2–43. [19] Tan SC. A progressive failure model for composite laminates containing openings. J Compos Mater 1991;25(5):556–77.