1-s2.0-s0263822313006326-main

ffeo

Keywords:Stiffened composite laminated plateLow velocity impactProgressive failure modelLayerwise/solid-elements method

elements method (LW/SE). The LW/SE method, which was developed in our previous works (Li, Qing,

lored for specic applications. Particularly, in the eld of aviation

or debris from runways could reduce the strength of the structuresignicantly and furthermore the internal damages are not detect-able through visible observation. If the impact-induced internaldamages are not detected and repaired in time, the damage areawill continuously grow and nally lead to complete structural col-lapse. Some comprehensive reviews can be found in Ref. [3,4] for

statical. When the structure reaches its maximum deection, thethe initial kineticn most completeructure, the non-are repre

structure aimpactor are represented by the mass models. Therefore, tpact dynamic system can be simplied to a two-degree-of-frmodel or one-degree-of-freedom model. The stiffness usedspring-mass models can be determined from theoretical formulasavailable in many handbooks, or numerically using the nite ele-ment method. With a complete model, the dynamic behavior ofthe structure is described accurately. Usually, in many cases theclassical plate theory can be used but, in some cases, transverseshear deformations become signicant and higher-order theoriesmust be used. For the plate behaving in a quasi-static manner,

Corresponding author. Tel.: +86 1062782078.E-mail addresses: [email protected] (D.H. Li), [email protected] (Y. Liu),

Composite Structures 110 (2014) 249275

Contents lists availab

Composite S

[email protected] (X. Zhang).industry, its proportion in the structure has been considered asthe one of most important indicators for the overall improvementof the aircraft. The behavior of composite laminated materials un-der low velocity impact is of concern in recent decades, since thedamages induced by low velocity impact such as a dropped tool

velocity of the impactor becomes zero and allenergy has been used to deform the structure. Ispring-mass models, the linear stiffness of the stlinear membrane stiffness and the contact forceby the spring models, the effective mass of the0263-8223/$ - see front matter 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compstruct.2013.12.011sentednd thehe im-eedomin theComposite materials are used in almost all aspects of the indus-trial and commercial manufacturing elds of aircrafts, ships, vehi-cles and other high performance structures due to their highspecic stiffness and strength, excellent fatigue resistance, longerdurability as compared to metallic structures, and ability to be tai-

According to Abrate [5], the analysis models used to obtain theimpact response can be classied into three categories: energy-balance models, spring-mass models and complete models. In theenergy-balance models, the impact dynamic is to consider the bal-ance of energy in the system and the structure behaviors are quasi-1. Introductionet al., 2013; Li, Liu, et al., 2013) [1,2] for the composite laminated stiffened shells and sandwich plates,not only can obtain accurate displacements and stresses for composite laminates but also can considercomplex stiffeners without any assumptions. In the present analysis method, impact responses are deter-mined by the nite element code of LW/SE and the nonlinear Hertzs contact law which enables consid-eration of local indentation produced by the indentor on the impacted surface. The 3D Hashin criteria inquadratic strain is employed to predict the initiation of the impact-induced damages. The effect of theaccumulation of the damages on the stiffness of laminated plate and stiffeners is computed by using con-tinuum damage mechanics. Several numerical examples are carried out to demonstrate the excellent pre-dictive capability of current method and to study the inuence of parameters on the impact responsesand impact-induced damages. In addition, the present analysis method is used to study the multi-impacts problem of the stiffened composite laminated plate.

2013 Elsevier Ltd. All rights reserved.

the literatures focused on low velocity impact on compositematerials.Article history:Available online 18 December 2013

Low-velocity impact responses and impact-induced damages evaluation problems are investigated forthe stiffened composite laminated plates based on the progressive failure model and layerwise/solid-Low-velocity impact responses of the stiplates based on the progressive failure mlayerwise/solid-elements method

D.H. Li, Y. Liu, X. Zhang School of Aerospace, Tsinghua University, Beijing 100084, China

a r t i c l e i n f o a b s t r a c t

journal homepage: www.elned composite laminateddel and the

le at ScienceDirect

tructures

ier .com/locate /compstruct

wer

tiffe

al s

ructY

Impactor

Nonlinear spring(Hertzian)

Lo

Compatibility relations between the plate and

stiffeners

S

Fig. 1. A composite laminated plate or cylindric

250 D.H. Li et al. / Composite Stthe bending waves induced by impact travel from the contact pointto the boundary of the plate and back many times during thepredicted contact duration, the boundary controlled impact andthe spring mass model or an energy balance approach might beadequate. If the deformation never reaches the edges of the plateduring the predicted contact duration, very good results can beprovided by the wave-controlled impact and the approximate solu-tion. For the intermediate cases, reected waves will affect thecontact force history signicantly, so a complete model taking intoaccount the full dynamic behavior of the plate and the boundaryconditions is necessary.

Without considering the impact-induced damage, there are anumber of research works focusing on the responses analysis prob-lem of low velocity impact for the composite laminated shells/plates. Sun [6] and Dobyns [7] used the analytical method of platedeveloped by Whitney and Pagano [8] to analyze a simply sup-ported orthotropic plate subjected to a central impact. Sun andChen [9] investigated the impact response behaviors of the initiallystressed composite laminates by using nite element method andNewmark integration algorithm. Cairns and Lagace [10] studiedthe inuence of different parameters on the impact behaviors oflaminated composite plates by using RayleighRitz method to spa-tially discretize the time-varying boundary value problem and

L1 L2

L1 L2 =

XZ

(a)

(c)Fig. 2. Finite element discretization. (a) Uniform nite elements; (b) nonuniform nY

surface of plate

ners

Upper surface of stiffeners

hell with stiffeners impacted by a steel sphere.

ures 110 (2014) 249275using Newmark integration algorithmmethod to integrate the mo-tion equations. The effects of shearing deformation, bending-twist-ing coupling, and nonlinear contact behavior were considered. Wuand Chang [11] carried out a transient dynamic nite elementanalysis for the responses of the composite laminated plates sub-jected to transverse foreign object impact. An 8-point brick ele-ment with incompatible modes was developed in this analysis,and the direct Gauss quadrature integration scheme was usedthrough the element thickness to account for the change in mate-rial properties from layer to layer within the element. The New-mark scheme was adopted to perform time integration from stepto step, and a contact law incorporated with the NewtonRaphsonmethod was applied to calculate the contact force during impact.Gong et al. [12] developed a spring-mass model to determine thecontact force between the shell and impactor during impact. In thismodel, the contact force was described by an analytic function interms of the material properties and mass of the shell and impac-tor, as well as the impact velocity. And then, based on a higher-order shear deformation theory (HSDT), Gong et al. [13] presenteda set of analytical solutions to predict the dynamic response of thesimply-supported, doubly curved, cross-ply laminated shells im-pacted by a solid striker. Nosier et al. [14] used a layerwise theoryand model superposition technique to investigate the low velocity

L1

1.5

XZ

(b)

ite elements; and (c) type Two way bias used in nonuniform nite elements.

truct5x 105

D.H. Li et al. / Composite Simpact response of composite laminated plates, and the contactarea is time dependent. Chandrashekhara and Schroeder [15]developed a nite element model for the nonlinear impactresponse of composite laminated cylindrical and doubly curved

0 50 100 150 200 250 300 350 400 450 5000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time (s)

Cont

act f

orce

(10

3 N)

Elem. 100 (10 10)Elem. 400 (20 20)Elem. 900 (30 30)Elem. 1600 (40 40)Elem. 2025 (45 45)Elem. 2916 (54 54)

0 50 100 150 200 250 300 350 400 450 5001000

500

0

500

1000

1500

2000

2500

3000

Time (s)

Vel

ocity

of i

mpa

ctor

(mm/

s)


(a)

(c)

Fig. 3. Convergence test. (a) Contact force; (b) displacement w at the contact poin

0 50 100 150 200 250 300 350 400 450 5000

0.5

1

1.5

2

2.5

3

3.5 x 105

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act f

orce

(10

3 N

)

Elem. 900 (30 30) uniform Elem. 900 (30 30) nonuniformElem. 1600 (40 40) uniformElem. 1600 (40 40) nonuniformElem. 2916 (54 54) uniform

Fig. 4. Effect of the nite element scheme on the contact force.0 50 100 150 200 250 300 350 400 450 5000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time (s)

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ral d

efle

ctio

n of

pla

te (m

m)


0

0.05

0.1

(b)

ures 110 (2014) 249275 251shells based on Sanders shell theory and the modied Hertziancontact law. Using a linearized contact law, Choi and Lim [16] stud-ied low-velocity impact analysis of composite laminates, it couldbe shown that the linearized contact law could be well appliedto the low-velocity impact analysis of composite laminates.

In all above works the impact-induced damages predicitionwas not considered. Choi and Chang [17,18] performed an inves-tigation consisting of both analysis and experiments to fundamen-tally understand the failure mechanisms and mechanics ofber-reinforced composites resulting from impact and to identifythe essential parameters governing the impact damage. And then,Choi and Chang [19] presented a model for predicting the initia-tion of the damages and the extent of the nal damages as a func-tion of material properties, laminate conguration and impactorsmass. The impact damages in terms of matrix cracking and delam-inations resulting from a point-nose impactor were the primaryconcern. Kim et al. [20] investigated the dynamic behavior andimpact-induced damage of the composite laminated curved struc-tures by using the incompatible eight-noded brick elements withTaylors modication and a modied Hertzian contact law. Krish-namurthy et al. [21,22] determined the impact responses of acomposite laminated cylindrical shell by a nonlinear Hertzs con-tact law. The impact-induced damage was evaluated by thesemi-empirical damage prediction model of Choi-Chang [17,18].The effects of important parameters, such as impactor mass and

0 50 100 150 200 250 300 350 400 450 5000.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

Time (s)

Dep

th o

f ind

enta

tion

(mm)


(d)

t (x = y = a/2, z = h/2), (c) velocity of the impactor and (d) indentation depth.

auth

ruct0 50 100 150 200 250 300 350 400 450 5000

0.5

1

1.5

2

2.5

3

3.5

4x 105

Time (ms)

Cont

act f

orce

(10

3 N

)

PresentCairns & LagaceWuSun & Chen

Plate: 200mm 200mm 2.69mm, Simply supported [0/90/0/90/0]s T300/934 graphite/epoxyImpactor: 12.7mm diameter steel rigid ball 3.0 m/s initial velocity

(a)Fig. 5. Comparison of the impact responses with the results obtained by other

Table 1Material properties of the composite laminates.

Elastic properties GPa E11 E22 E33 G12 G13 G23156.5 13.0 13.0 6.96 6.96 3.45m12 m13 m230.23 0.23 0.4

Strength properties MPa rf ;t11 rf ;c11 rf ;t22 rf ;c221516.8 1592.7 44.5 253.0

252 D.H. Li et al. / Composite Stvelocity, shell curvature and stacking sequence of plies, are inves-tigated as well.

In contrast to the number of studies dealing with impact on thecomposite plates and shells, few papers focus on the topic of im-pact on stiffened composite plates and shells. Since stiffened com-posite plates are commonly utilized in a variety of engineeringstructures, an understanding of the various effects of stiffenerson the impact response is important to assess the strength and reli-ability of the structures. Gong and Lam [23] presented an approx-imate solution to predict the transient response of stiffenedcomposite plates impacted by a solid striker. In this approximatesolution, the stiffened composite plate is characterized by a cti-tious orthotropic layer whose bending and extensional propertiesare those of the individual stiffeners averaged out over representa-tive areas. Seydel and Chang [24,25] developed a real-time identi-cation technique for the contact force history of low-velocityimpacts on the composite panels with beam stiffeners by bothanalysis and experiments. Faggiani and Falzon [26] presented anintralaminar damage model for the damage mechanisms occurringin carbon ber composite structures incorporating ber tensile andcompressive breakage, matrix tensile and compressive fracture,and shear failure on the basis of a continuum damage mechanicsapproach. The aim of the present work is to estiblish a methodfor low velocity impact responses and impact-induced damagesprediction of the stiffened composite laminated plates. Further-more, the developed method can be used to obtain the low-impactresponses and impact-induced damages of the composite lami-nated plate stiffened with complex stiffeners without any assump-tion and simplication.

The widespread application of the stiffened plates/shells hasresulted in different methods of performing appropriate structural

rf12 rf13 r

f23

106.9 106.9 106.9

Fracture energy N=mm Gt;1 Gc;1 Gt;2 Gc;2 Gt;3 Gc;320 20 0.23 0.76 0.23 0.76analysis. A large number of studies on bending, buckling, andvibration are available in the literature. During last 20 years, a lotof analysis schemes have been developed such as the methodologybased on energy principles [27,28], the semi-analytical methods[29,30], the differential quadrature methods [31], the nite ele-ment methods (FEM) [3234,1,3542], the meshfree methods[4347], and the boundary element methods (BEM) [48,49] or acombination of nite element methods and boundary elementmethods [50]. Among these approaches, the nite element methodis the most widely used numerical method for the stiffened com-posite plates/shells. There are two main schemes to deal with therelationship between the plate and stiffeners in aforementionedmethods: the one is that the closely spaced stiffeners are averagedout or smeared over the shell surface, and anther is that thestiffeners are equivalent as beams. These approximation may beapplicable only if the ratios of spacing between two consecutivestiffeners to plate dimensions are small enough to ensure approx-imate homogeneity of stiffness, and the ratio of stiffener rigidity tothe plate rigidity must not become so large that the beam action ispredominant. So several assumptions must be made in order tofacilitate a solution if the stiffeners are not identical or unequallyspaced.

Moreover, stiffened composite structures have been widespread

(b)

0 200 400 600 800 1000 1200 1400 16000.2

0.1

0

0.1

0.2

0.3

0.4

0.5

Time (s)

Cent

ral d

eflec

tion

of p

late (

mm)

PresentCairns & LagaceWuSun & Chen

Plate: 200mm 200mm 2.69mm, Simplysupported [0/90/0/90/0]

s T300/934 graphite/epoxy

Impactor: 12.7mm diameter steel rigid ball 3.0 m/s initial velocity

ors. (a) Contact force history; and (b) central deection at the neutral surface.

ures 110 (2014) 249275in recent years due to the economic and structural advantages ofsuch systems. The complication would further increase when ana-lyzing stiffened composite laminated plate/shell structures. There-fore, it is very necessary to develop an analysis method which notonly can obtain the accurate displacements and stresses of com-posite laminated plates/shells but also can consider the complexstiffeners without any assumption and simplication. In the earlierstudies of the present authors [1,2], a layerwise/solid-element(LW/SE) method was established based on the layerwise theoryand the solid elements for the composite stiffened laminated cylin-drical shells. In LW/SE method, the layerwise theory was used tomodel the behavior of the composite laminated cylindrical shells,and the eight-noded solid element is employed to discrete the stiff-eners without any assumption and simplication. The displace-ments freedoms of layerwise theory in the surface of plates/shells appear in the governing equations, and the freedom of thein-plane nodes of the layerwise theory is equal to that of the brickelement. It means that the governing equations of the plates andshells established based on the layerwise theory can be simulta-neously solved with the governing equations of stiffeners discretedby the solid elements and therefore enables to ensure the compat-ibility of displacements at the interface between shells and stiffen-ers through the compatibility conditions.

truct0 50 100 150 200 250 300 3500

0.5

1

1.5

2

2.5

3

3.5x 106

Time (s)

Cont

act f

orce

(10

3 N

)

Lower surface with progressive failure modelMiddle surface with progressive failure modelUpper surface with progressive failure modelLower surface without progressive failure modelMiddle surface without progressive failure modelUpper surface without progressive failure model

0.5

1

1.5

2

2.5

3x 104

impa

ctor

(mm/

s)

Lower surface with progressive failure modelMiddle surface withprogressive failure modelUpper surface with progressive failure modelLower surface without progressive failure modelMiddle surface without progressive failure modelUpper surface without progressive failure model

(a)

D.H. Li et al. / Composite SIt is well known that damage in composite material is generallycomplicated consisting of multiple failure modes such as berbreakage, ber pullout, matrix cracking, bermatrix debonding,delamination between plies, etc. Methods for modeling materialdamage of composite materials can be divided into micromechan-ics of damage and continuum damage mechanics approaches(CDM) [51]. Micromechanics has been developed and applied incomposite failure mechanism related to the damage of either thebers or the matrix separately. It is valuable tools to gain insightsinto mechanisms and failure processes at the micro-scale. It alsocan be used in the global response of the composite structures.However, a signicant limitations with the micromechanics mod-els is the large number of material parameters need to identifythe constitutive model and the high computational cost. Sincecomposite structures can accumulate damage before structuralcollapse, the use of failure criteria is not sufcient to predict ulti-mate structural failure. To bridge this gap, the progressive failurehas been developed and enhanced for composite materials. Simpli-ed models, such as the ply discount method, can be used to pre-dict ultimate failure, but cannot represent with satisfactoryaccuracy the quasi-brittle failure characteristic of a laminate thatresults from the accumulation of different damage mechanisms.In order to conjugate simplicity of application and accuracy ofresults, the concept of distributed damage and the use of formula-tions based on the thermodynamics of continuum media play afundamental role. In this framework, CDM considers damagedmaterials as a continuum, in spite of heterogeneity, micro-cavities,and micro-defects.

0 50 100 150 200 250 300 3501.5

1

0.5

0

Time (s)

Vel

ocity

of

(c)

Fig. 6. The impact responses of composite laminated plate with/without the progressivesurface, (c) velocity of impactor; and (d) displacement of the impactor.0 50 100 150 200 250 300 3500

0.5

1

1.5

2

2.5

3

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Cent

ral d

efle

ctio

n of

mid

dle

surfa

ce (m

m)


1.5

2

2.5

3

t of i

mpa

ctor

(mm)

(b)

ures 110 (2014) 249275 253In the present work, the LW/SE method, modied nonlinearHertzs law and progressive failure model are employed to developa method for low impact responses and impact-induced damageprediction of the stiffened composite laminated plates. Firstly,the transient response analysis model of the stiffened compositelaminated plate is estiblished by using the LW/SE method andNewmark method, the contact force between the stiffened plateand impactor is obtained by the modied nonlinear Hertzs lawwith Newmark method and NewtonRaphson iterative method,and the prediction of impact-induced damages is carried out bythe 3D Hashin criteria and progressive failure model. And then,numerical examples are carried out to demonstrate the excellentpredictive capability of current method and to study the effect ofthe load application point of equivalent contact force in thicknessdirection on the impact responses. For two kinds of stiffened com-posite laminated plates, the inuence of important parameters onthe impact responses and impact-induced damages is investigatedas well, such as the impact velocity, radius of impactor and thick-ness of stiffeners. At last, the present analysis method is used tostudy the multi-impacts problem of stiffened composite laminatedplate.

2. Contact force of low-speed impact based on Hertzs law

Krishnamurthy and Mahajan [22,21] developed a nonlinearHertzs law for the low-impact problem of the composite lami-nated cylindrical shell. A brief theoretical deviration of themodied nonlinear Hertzs contact low for the plates and the

0 50 100 150 200 250 300 3500

0.5

1

Time (s)

Disp

lace

men


(d)damage analysis model. (a) Contact force; (b) central deection of plate at middle

5 s

70 s

160 s

15 s

70 s

160 s

5 s

70 s

160 s

(a) (c)(b)Fig. 8. The damage variations of the composite laminated plate subjected to impact load at the middle surface. (a) Damage variation d1 (ber breakage); (b) damage variationd2 (matrix cracking); and (c) damage variation d3 (delamination).

30 s

100 s

240 s

5 s

100 s

240 s

12 s

100 s

240 s

(a) (c)(b)Fig. 7. The damage variations of the composite laminated plate subjected to impact load at the lower surface. (a) Damage variation d1 (ber breakage); (b) damage variationd2 (matrix cracking); and (c) damage variation d3 (delamination).

254 D.H. Li et al. / Composite Structures 110 (2014) 249275

s

0 s

truct10 s

150 s

5

15

D.H. Li et al. / Composite Scylindrical shells is presented in this section together with thereloading phase. According to Newtons second law the equationof motion for a rigid impactor is given by

mi wi fc 1where mi and wi are the mass and the acceleration of the impactor,respectively. fc is the contact force between impactor and targetbody.

Three kinds of phases (loading, unloading and reloading phase)probably occur during the impact. By using the Hertzian contacttheory, the contact force fc can be related to the indentation depthas follow [52]

fc

ka1:5 loading

fmaa0ama0

2:5unloading

fmaa0ama0

1:5reloading

8>>>>>>>:

2

where at wit wtt is the depth of indentation, wtt is thedisplacement of composite laminated plate at the contact point,fm and am are the maximum contact force and indentation at thebeginning of unloading, k is the modied constant of the Hertzscontact theory which is given by

320 s 320

(a) (b)Fig. 9. The damage variations of the composite laminated plate subjected to impact loadd2 (matrix cracking); and (c) damage variation d3 (delamination).

Table 2Initial time points of the ber breakage, the matrix cracking and the delaminations.

Load position in thicknessdirection

Damage form (ls)

Fiberbreakage

Matrixcracking

Delaminations

Lower surface 21 3 9Middle surface 13 4 5Upper surface 7 2 3k 43

1ri 12Rc

0:5= 1mi

2

Ei 1E2

h icylindrical shell

8>:

where ri and Rc are the radius of the impactor and the cylindricalshell, respectively; Ei and E2 are the Youngs modulus of the impac-tor and the elastic modulus transverse to the ber direction, respec-tively; mi is the Poissons ratio of the impactor.

a0 in Eq. (2) denotes the permanent indentation in a loding-unloading cycle which is given by

a0 0 am < acr

am 1 acram 2=5

am P acr

8>>>>>>>>:

6where wi wni _wni Dt wni Dt

2

4

.

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4.5x 106

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20 m/s (with progressive failure model)30 m/s (with progressive failure model)35 m/s (with progressive failure model)20 m/s (without progressive failure model)30 m/s (without progressive failure model)35 m/s (without progressive failure model)

0 50 100 150 200 250

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te (m

m)


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ocity

of i

mpa

ctor

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Time (s)

Disp

lace

men

t of i

mpa

ctor

(mm)


(a) (b)

(c) (d)Fig. 11. Inuences of the impact velocity on the impact responses of concentrically stiffened composite laminated plates with/without the progressive damage analysismodel. (a) Contact force; (b) displacement w of plates in contact point, (c) velocity of the impactor and (d) displacement of the impactor.

(b) (a)

a

Hh

1b 2b

2a 1a

a

Hh

2a 1a

a = 200 mm; H - h= 2.69 mm; h = 3.0 mm, 5.0 mm, 7.0 mm

= 55 mm; = 90 mm;1b = 80 mm; 2b = 50 mm;

y

x

1a 1a

Fig. 10. Two kinds of stiffened composite laminated plates. (a) Concentrically stiffened laminated plates with four stiffeners; and (b) parallelly stiffened laminated plates withtwo stiffeners.

256 D.H. Li et al. / Composite Structures 110 (2014) 249275

s

truct30

X Y

Z

1.110.90.80.70.60.50.40.30.20.10

X Y

Z

1.8

X Y

Z

0.45

X Y

Z

0.450.40.350.30.250.20.150.10.05

D.H. Li et al. / Composite S3. Mathematic model of the composite laminated stiffenedplates/shells subjected to impact

3.1. Layerwise theory of composite laminated plates/shells

In contrast to the equivalent single layer theory [5356], thelayerwise theories [5762] are developed by assuming that the dis-placement eld exhibits only C0-continuity through the laminatethickness. Thus the displacement components are continuousthrough the laminate thickness but the derivatives of the displace-ments with respect to the thickness coordinate may be discontin-uous at various points through the thickness, thereby allowing forthe possibility of continuous transverse stresses at interfaces sep-arating dissimilar materials.

The Layerwise Shell Theory (LWST) of Reddy [59] gives anaccurate description for the displacement eld in the thicknessdirection. The three-dimensional displacement eld is expanded

75s

195 s

255 s(b)(a)

1.61.41.210.80.60.40.20

X Y

Z

1.31.21.110.90.80.70.60.50.40.30.20.10

X Y

Z

0.850.80.750.70.650.60.550.50.450.40.350.30.250.20.150.10.05

0.40.350.30.250.20.150.10.05

X Y

Z

0.450.40.350.30.250.20.150.10.05

X Y

Z0.450.40.350.30.250.20.150.10.05

Fig. 12. The deformation and the damage histories of concentrically stiffened componephogram of displacement w; (b) damage variation d1 (ber breakage); (c) damage vaX Y

Z

0.90.8

X Y

Z

0.90.8

X Y

Z

0.950.90.850.80.750.70.650.60.550.50.450.40.350.30.250.20.150.10.05

X Y

Z

0.90.850.80.750.70.650.60.550.50.450.40.350.30.250.20.150.10.05

ures 110 (2014) 249275 257as a function of a surface-wise two dimensional displacement eldand a one-dimensional interpolation function through the thick-ness. The use of higher order polynomial interpolation functionsor more sub-divisions through the thickness improves the accuracyof the displacement eld. In the layerwise laminate theory, the dis-placements at point (x,y,z) in the composite laminated plates areassumed to be as

ux; y; z XN1i1

uix; y/iz;

vx; y; z XN1i1

v ix; y/iz;

wx; y; z XN1i1

wix; y/iz;

7

(c) (d)

0.70.60.50.40.30.20.1

0.70.60.50.40.30.20.1

X Y

Z

0.950.90.850.80.750.70.650.60.550.50.450.40.350.30.250.20.150.10.05

X Y

Z

0.950.90.850.80.750.70.650.60.550.50.450.40.350.30.250.20.150.10.05

X Y

Z

0.950.90.850.80.750.70.650.60.550.50.450.40.350.30.250.20.150.10.05

X Y

Z

0.950.90.850.80.750.70.650.60.550.50.450.40.350.30.250.20.150.10.05

site laminated plates subjected to impact velocity 30 m/s. (a) Deformation andriation d2 (matrix cracking); and (d) damage variation d3 (delamination).

(d1

ructDamage variation

258 D.H. Li et al. / Composite Stwhere u, v and w represent the displacement components of pointPx; y; z in the x, y and z directions, respectively. /i is a 1D linearLagrangian interpolation function through the thickness of the lam-inated plate. The laminate thickness dimension is subdivided into aseries of N one-dimensional nite elements (Ne = N + 1 nodes)whose nodes are located in planes parallel to xy plane in the unde-formed laminated facesheets. ui, v i and wi are the nodal values ofthe 1D linear Lagrangian interpolation function through the thick-ness direction. N is also the number of mathematical layers of thelaminated plates, which may be equal or less than the number ofphysical layers.

In order to develop the nite element formulation, the nodaldisplacement functions ui;v i and wi are approximated on the ithplane of the plate by

uix; y Xnelmn1

uni wnx; y;

v ix; y Xnelmn1

vni wnx; y;

wix; y Xnelmn1

wni wnx; y;

8

where nelm is the number of nodes in each 2D element, ux; y isstandard 2D nite element shape function, uni , vni and wni are the

Damage variation (

Damage variation

d2

d3(a)

Fig. 13. Inuence of impact velocity on damage variations distributions of concentricaimpactor 6.0 mm and the thickness of stiffeners is 7.0 mm. (a) Impact velocity 30 m/s;fibre breakage)ures 110 (2014) 249275displacement components of nth node of the 2D nite element rep-resenting the ith plane of the physical laminates element.

The nite element formulation of the present layerwise theorycan be derived using the principle of virtual displacements in ma-trix form as

M U KU F 9Since the contact area is so small in comparison with the

dimensions of the plate, the contact pressure between the impac-tor and plates can be considered as a concentrated (point) load atthe central of plates/shells for the low-impact response analysisbased on Hertzs law. Therefore, except the nodel component ofthe contact point in the direction of impact, all the componentsof load vector F in Eq. (9) are zero. The load vector F can be rewrit-ten as

F f0;0; . . . ; fc; . . . ;0;0gT 10

3.2. Layerwise/solid-elements method of the composite laminatedstiffened plates/shells

A layerwise/solid-element method has been developed in ourprevious work [1] for the linear static and free vibration analysisof composite stiffened laminated cylindrical shell. Only the essen-tial details are provided in this section, see Ref. [1] for more details.

(b)

matrix cracking)

(delamination)

lly stiffened composite laminated plates with four stiffeners, where the radius ofand (b) impact velocity 35 m/s.

truct1.5

2

2.5

3 x 104

r (mm

/s)

Radius of impactor 5.0 mmRadius of impactor 6.0 mmRadius of impactor 6.5 mm

0 50 100 150 200 250 3000

0.5

1

1.5

2

2.5

3x 106

Time (s)

Cont

act f

orce

10

3 N


(a)

D.H. Li et al. / Composite SThe schematic diagram of the LW/SE method for the compositelaminated plate with stiffeners impacted by a steel sphere is shownin Fig.1, where the layerwise laminate theory is used to model thebehavior of the composite laminated plates, and the traditional so-lid element is employed to discrete the stiffeners. And then, basedon the governing equations of the lamianted plate and stiffeners,the nal governing equations of the composite stiffened laminatedplate can be assembled by using the interface conditions to ensurethe compatibility of displacements at the interface between plateand stiffeners as follow

MP11MS11 MP12 MS12MP21 M

P22 0

MS21 0 MS22

264

375

UP1UP2US2

8>:

9>=>;

KP11KS11 KP12 KS12KP21 K

P22 0

KS21 0 KS22

264

375

UP1UP2US2

8>:

9>=>;

0FP2FS2

8>:

9>=>;11

where the superscript P and S denote plates and stiffeners, respec-tively. The subscripts 1 and 2 denote the interface displacementsvector and internal displacements vector, respectively. U; U and Fare the displacements, accelerations and external loads vector,respectively. Maiji; j 1;2;a S;B is the mass matrixes,Kaiji; j 1;2;a P; S is the stiffness matrixes, and their detailedexpressions can be found in [1]. In the present work, the nodes ofthe stiffeners at the interface are coincide with the nodes of theplate to ensure the fully coordination of displacements and forcesat the interface between plate and stiffeners. However, in order to

0 50 100 150 200 250 3001.5

1

0.5

0

0.5

1

Time (s)

Velo

city

of i

mpa

cto

(c)Fig. 14. Inuences of the radius of impactor on the impact responses of concentrically stContact force; (b) displacement w of plates in contact point, (c) velocity of impactor and0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (s)

Cent

ral d

efle

ctio

n of

pla

te (m

m)


1

1.5

2

acto

r (mm

)

(b)

ures 110 (2014) 249275 259reduce the complexity which results from the consistency require-ment of the nodes at the interface, some of the other methodswould be employed to deal with this coupling problem in our fur-ther works, such as the meshfree method and the tied interfacescheme.

Eq. (11) represents a set of ordinary differential equations intime. In order to solve those equations, we must fully discretizethem and reduce them to algebraic equations by using a numericalintegration method. In present work the Newmark time integra-tion scheme is employed to approximate the time derivatives [59].

The computational algorithm of the contact force betweenimpactor and composite laminated plate can be found in Ref.[21,22]. For the low-impact problem studied in this paper, theinitial conditions are _w0i v0 (initial velocity of impactor),w0i w0t 0. Firstly, an approximate value of fc is obtained fromimplicit expressions of the loading law (the rst equation of Eq.(6)) by using the initial conditions and a root nding aglorithm,where the NewtonRaphson method is employed. Then, theapproximate f 1c is now applied as external load at the contact nodeof laminated plate and the displacement of laminated plate at thecontact node w1t is next found from Eq. (11). Using the displace-ment w1t ; f

1c is recomputed by Eq. (6), as is done previously. This

iterative process is repeated till the required accuracy is achieved.The nal contact force is now used to calculate acceleration, veloc-ity and displacement of the impactor for the iterative process of thenext time step.

0 50 100 150 200 250 3001.5

1

0.5

0

0.5

Time (s)

Disp

lace

men

t of i

mp


(d)iffened composite laminated plates with the progressive damage analysis model. (a)(d) displacement of impactor.

(d1


260 D.H. Li et al. / Composite St4. Progressive failure analysis for the impact-induced damage

4.1. Three-dimensional continuum damage model

Distributed microscopic damage is quantied by the use of anappropriate tensor eld that describes the orientation and densityof microcracks in the material. The microcracks and microvoidscause the cross-sectional area A reducing to A. The damage variableand effective Cauchy stress can be dened as follows

d A A=A 12r r=1 d 13where r is the Cauchy stress tensor. However, For the convenienceof the nite element equations, Voigt form of the Cauchy stress isused in the other sections. The damage variables d1; d2 and d3,known as three scalar damage parameters, represent modulusreductions under different loading conditions due to microdamagein the material. For fabric plies d1 and d2 are associated with dam-age or failure in the principle ber directions, and controls in-planeshear damage or failure. The scalar damage parameters di repre-sents the effective fractional reduction in load carrying area onplanes that are perpendicular to theith principal material direction.In the present work, the rst principal material direction is chosento coincide with the ber direction. The second principal material

Damage variation (m

Damage variation

d2

d3(a)

Fig. 15. Inuences of the radius of impactor on damage variations distributions of concenvelocity is 30 m/s and the thickness of stiffeners is 7.0 mm. (a) Radius of impactor 5.0 mfibre breakage)

ures 110 (2014) 249275direction is chosen to coincide with the transverse direction in lam-ina plane. The third principal material direction is chosen to coin-cide with the thickness direction. The scalar damage parametershave values 0 6 di 1, where di 0 corresponds to a complete lackof microcracks at the ith principal material direction, while di 1corresponds to a complete separation of the material across planesat the ith principal material direction.

Based on the damage tensor, the effective stress of the orthotro-pic material with mircodamages can be dened as follows

r Md r 14

where Md diag 1=x11 1=x22 1=x33 1=x23 1=x13 1=x12 ;x11 1 d1; x22 1 d2; x33 1 d3, x12

1 d1 1 d2

p;

x13 1 d1 1 d3

p, x12

1 d1 1 d2

p.

According the strain energy for the material without damages,the equivalence strain energy for the damaged orthotropic mate-rial is following

Wd 12 r C1 r 1

2r MTd C1 Md r

12r Cd1 r 15

where Cd M1d C1 MTd is the constitutive of the damagedorthotropic composite laminates.

(b)

atrix cracking)

(delamination)

trically stiffened composite laminated plates with four stiffeners, where the impactm; and (b) radius of impactor 7.0 mm.

truct0 50 100 150 200 2500

0.5

1

1.5

2

2.5

3x 106

Time (s)

Cont

act f

orce

10

3 N

Thickness of stiffeners 3.0 mmThickness of stiffeners 5.0 mmThickness of stiffeners 7.0 mm

1.5

2

2.5

3 x 104

or (m

m/s)


(a)

D.H. Li et al. / Composite S4.2. Damage initiation criteria

The failure initiation criteria employed in the present work is3D Hashin criteria in quadratic strain [63,64], in which failuremode indexes in three principle directions, ber tension or com-pression failure Ff , matrix tension or compression failure Fm anddelaminations Fz, are dened by

F2f e11ef ;t11

2 e12

ef ;t12

2 e13

ef ;t13

2P 1 e11 > 0

e11ef ;c11

2P 1 e11 < 0

8>>>>>:

16

F2m

e22e33ef ;t22ef ;t22

2 e22 e33

ef23 2

e12ef12

2 e13ef

13

2 e23ef

23

2P 1 e22 e33 > 0

e22e33ef ;c22 e

f ;c22

2 e22e33

ef ;c22

ef ;c222ef12

1

e22 e33ef23 2

e12ef12

2 e13ef

13

2 e23ef

23

2P 1 e22 e33 < 0

8>>>>>>>>>>>>>>>>>:

17

F2z e33ef ;t33

2 e13ef13 2

e23ef23 2

P 1 e33 > 0

e33ef ;c33

2 e13ef13 2

e23ef23 2

P 1 e33 < 0

8>>>>>:

18

0 50 100 150 200 2501.5

1

0.5

0

0.5

1

Time (s)

Velo

city

of i

mpa

ct

(c)Fig. 16. Inuence of the thickness of stiffeners on the impact responses of concentrically(a) Contact force; (b) displacement w of plates in contact point, (c) velocity of impactor0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (s)

Cent

ral d

efle

ctio

n of

pla

te (m

m)


1.2

1.4

1.6

1.8

pact

or (m

m)

(b)

ures 110 (2014) 249275 261where ef ;tii rf ;tii =Cii; i 1;2;3; ef ;cii rf;cii =Cii; i 1;2;3;ef12 rf12=C66; ef23 rf23=C44; ef13 rf13=C55. rf;tii and rf;cii are the ten-sile and compression strength of three principle directions of mate-rial, respectively; rf12; rf13 and rf23 are shear strength in three planesof material.

4.3. Damage evolution law

In a nite-element-based failure analysis procedure, an elementfailure is rst identied through the use of one of the above failurecriteria and later through a stiffness reduction scheme. The failedelement is replaced with an equivalent element according a mate-rial property degradation model. A number of post-failure materialproperty degradation models have been proposed for the progres-sive failure analysis. Most of these material degradation modelsbelong to one of three general categories [65,66]: instantaneousunloading, gradual unloading and constant stress at failure mate-rial point. In instantaneous unloading scheme the selected materialproperties of the failure element are reduced to zero when the fail-ure is detected. One of the most common instantaneous unloadingcategories used for degradation of material properties is the ply-discount theory for composite structures. This kind stiffnessdegradation scheme is independent of the mesh form used in theanalysis. However, it may be noted that the size of the actual dam-age in the form of micro or meso cracks is very small compared tothe size of the elements. Hence it appears unjustied in reducing

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

Time (s)

Disp

lace

men

t of i

m


(d)stiffened composite laminated plates with the progressive damage analysis model.and (d) displacement of impactor.

(d1


262 D.H. Li et al. / Composite Stthe stiffness properties of the whole element to zero. For the con-stant stress case, the material properties associated with that modeof failure are degraded such that the material cannot sustain addi-tional load. For the gradual unloading case, the material propertyassociated with that mode of failure is degraded gradually (expo-nentially for example) until it reaches zero, and the degradationcan be either independent or interactive corresponding to themode of failure.

In the present work, the reduction of the stiffness coefcients iscontrolled by damage variables d1; d2 and d3 which take values be-tween zero (undamaged state) and one (fully damaged state for themode corresponding to this damage variable). The gradual unload-ing scheme of damage variables d1; d2 and d3 can be divided intotwo types: linear and nonlinear. For the nonlinear type, the dam-age variable is given as follows

d1 1exp rf11dfeq;11 Ff 1 =Gc;1

h iFf

d2 1exp rf22dfeq;22 Fm 1 =Gc;2

h iFm

d3 1exp rf33dfeq;33 Fz 1 =Gc;3

h iFz

19

(a)

Damage variation (m

Damage variation

d2

d3

Fig. 17. Inuences of thickness of stiffeners on damage variations distributions of concenvelocity is 30 m/s and the radius of impactor is 6.0 mm. (a) Thickness of stiffeners 3.0 mfibre breakage)

ures 110 (2014) 249275where Gc;1; Gc;2 and Gc;3 are the fracture energy of three principledirections of material, respectively. dfeq;11, d

feq;22 and d

feq;33 are the

equivalent displacements at which the material is fully damaged.

dfeq;ii efii Lc 20

where Lc is a characteristic length of the element. Different methodshave been suggested for computing the characteristic length.Bazant and Oh [67] proposed the following relation for squareelement:

Lc AIP

p

cos h21

where AIP is the area associated with an integration point and h isthe angle between the mesh line along which the crack bandadvances and the crack direction. According to Maimi [68], for anunknown direction of crack propagation the average of above

expression can be used, Lc p4R p=40 Lcdc 1:12

AIP

p. Lapczyk and

Hurtado [69] assumed that the characteristic length at a materialpoint is equal to the square root of the area associated with it,although any other of the methods mentioned previously could beincorporated easily in the model.

(b)

atrix cracking)

(delamination)

trically stiffened composite laminated plates with four stiffeners, where the impactm; and (b) thickness of stiffeners 5.0 mm.

truct0 50 100 150 200 2500

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 x 106

Time (s)

Cont

act f

orce

10

3 N


2

3

4 x 104

acto

r (mm

/s)


(a)

D.H. Li et al. / Composite S5. Numerical examples

5.1. Validation of the present method

Example 1. Validation of the current nite element code has beencarried out by the same example as that used in many otherliteratures [9,70,10]. A simply supported graphite/epoxy compositeplate [0/90/0/90/0]s of the dimensions 200 mm 200 mm2:69 mm is impacted by a steel sphere of 12.7 mm diameter at avelocity of 3 m/s. All the layers of the laminated plate have thesame thickness and material properties.

E11 141:2GPa; E22 9:72GPa; G12 5:53GPa; G23 3:74GPa;v12 v23 0:30; q 1536Kg=m3

The uniform and nonuniform nite element scheme arepresented in Fig. 2, where the nonuniform elements are to renethe local grid in the area of impact point. Convergence of thecontact force, the displacement w of the plate at the contactpoint, the velocity of the impactor and the indentation depth ofthe plate are shown in Fig. 3, where the nite elements areuniformly distributed. It can be seen from Fig. 3 that theresponses of impact converge to the value of mathematicalmodel monotonically as the number of the nite elementsincreased. The reasonable convergence values are achievedwhen the nite elements scheme is 1600(40 40). The contactforce, the displacement w of plate at the contact point, the

0 50 100 150 200 2502

1

0

1

Time (s)

Vel

ocity

of i

mp

(c)Fig. 18. Inuence of the impact velocity on the impact responses of parallelly stiffened coContact force; (b) displacement w of plates in contact point, (c) velocity of impactor and0 50 100 150 200 250

0

0.5

1

1.5

2

Time (s)

Cent

ral d

efle

ctio

n of

pla

te (m

m)


1.5

2

2.5

f im

pact

or (m

m)

(b)

ures 110 (2014) 249275 263residual velocity of impactor and the indentation increase withthe increasing of the number of the nite elements. It suggeststhat the stiffness of composite laminated plate decreases andthe secondary impact delays with the increasing of the numberof the nite elements. For the nonuniform discretization (seeFig. 2), convergence of the contact force, the displacement w ofplate at the contact point, velocity of impactor and indentationof plate are shown in Fig. 4. It can be observed from Fig. 4 thatthe convergence rate of the maximum contact force of the rstloadingunloading cycle is improved signicantly by the non-uniform nite element scheme. However, it almost has no effecton the convergence rate of the contact force peak value andstarting time of the secondary impact. Therefore, the nonuni-form nite element scheme could reduce the computational costfor the evaluation of impact-induced damage, since the damageof the low-impact is caused in the rst impact periods. But ifthe complete impact responses are needed, there is no differ-ence between the uniform and nonuniform nite elementschemes.

Comparison of the contact force and the deection historiesobtained by the present method with the results obtained by otherliteratures [9,70,10] are shown in Fig. 5. It may be seen in Fig. 5 thatthe contact force history and the time history of the centraldeection obtained by the current nite element code are inreasonable agreement with the results reported in previousinvestigations.

0 50 100 150 200 2500

0.5

1

Time (s)

Disp

lace

men

t o


(d)mposite laminated plates with/without the progressive damage analysis model. (a)(d) displacement of impactor.

s

ruct30

X Y

Z

0.90.850.80.750.70.650.60.550.50.450.40.350.30.250.20.150.10.050

X Y

Z

1.8

X Y

Z

0.7

X Y

Z

0.450.40.350.30.250.20.150.10.05

264 D.H. Li et al. / Composite StExample 2. The composite laminated plate studied in thisnumerical example have the same geometry as in Example 1. Butthe stacking sequence is [0/90/0/90/0], and the densityq 1389:2 Kg=m3. Material properties of the composite laminatesare listed in Table 1 [71].

A number of investigations have demonstrated that uponimpact by a low velocity, the main part of damage in the compositelaminates is caused by matrix cracking and delaminations, becausethe tensile failure strength of the ber is high, and the damageinduced by ber breakage is generally very limited and conned tothe region under and near the contact area between the impactorand the laminates. Choi et al. [17] reported that intraply matrixcracking is the initial damage mode, and the delamination damageinitiates once the matrix crack reaches the interface between theply groups having different ber orientaions after propagatingthroughout the thickness of the ply group consisting of the crackedply.

75s

195 s

255 s(b)(a)

1.61.41.210.80.60.40.20

X Y

Z

1.81.61.41.210.80.60.40.20

X Y

Z

1.61.51.41.31.21.110.90.80.70.60.50.40.30.20.1

0.650.60.550.50.450.40.350.30.250.20.150.10.05

X Y

Z

0.70.650.60.550.50.450.40.350.30.250.20.150.10.05

X Y

Z

0.70.650.60.550.50.450.40.350.30.250.20.150.10.05

Fig. 19. The deection and damage variations histories of parallelly stiffened compositeof displacement w; (b) damage variation d1 (ber breakage); (c) damage variation d2 (mX Y

Z

0.950.90.850.80.750.70.650.60.550.50.450.40.350.30.250.20.150.10.05

X Y

Z

0.95

X Y

Z

0.90.850.750.70.650.60.550.50.450.40.350.30.250.150.10.05

X Y

Z

0.95

ures 110 (2014) 249275If the three-dimensional theory of plate and Herlzs theory areemployed to analyze the low velocity impact problem, rst weneed to determine the load application point of the equivalentcontact force in thickness direction. The major aim of thisnumerical example is to investigate the effect of the load applica-tion point of the equivalent contact force in thickness direction onthe impact responses. The impact responses of composite lami-nated plate with/without the progressive damage analysis modelare shown in Fig. 6, where the initial velocity of the impactor is30 m/s and the radius of the impactor is 6.0 mm. Because thematerial property associated with the failure is degraded graduallywhen the failure is detected, the maximum of the contact force, thecentral deection of plate at middle surface and the displacementof the impactor without tht progressive failure model are greaterthan those with the progressive failure model. It can be seen fromFig. 6 that the load point in thickness direction has no effect on theimpact responses except the maximum contact force of the

(c) (d)

0.90.850.80.750.70.650.60.550.50.450.40.350.30.250.20.150.10.05

X Y

Z

0.950.90.850.80.750.70.650.60.550.50.450.40.350.30.250.20.150.10.05

X Y

Z

0.950.90.850.80.750.70.650.60.550.50.450.40.350.30.250.20.150.10.05

0.850.80.750.70.650.60.550.450.40.350.30.250.20.150.05

X Y

Z

0.950.850.80.750.70.650.60.550.450.40.350.30.250.20.150.05

X Y

Z

0.950.90.850.80.750.70.650.60.550.50.450.40.350.30.250.20.150.10.05

laminated plates subjected to impact velocity 40 m/s. (a) Deection and nephogramatrix cracking); and (d) damage variation d3 (delamination).

(d1

tructDamage variation

D.H. Li et al. / Composite Ssecondary impact if the impact-induced damage is not taken intoaccount, while with the progressive damage model the load pointin thickness direction has signicant inuence on the impactresponses. One important reason is that the impact-induceddamage always initiates from area nearby the load point sincethe contact force is equivalent to a point load in the low velocityimpact analysis method estiblished by Hertzs low.

The deection and damage variations of the composite lami-nated plate subjected to impact load on the lower surface, themiddle surface and the upper surface are shown in Figs. 79,respectively. Initial times of the ber breakage, matrix crackingand delaminations are listed in Table 2. It can be observed fromFigs. 79 and Table 2 that the matrix cracking and delaminationsare the main part of damage in the composite laminates, and thedamage variation of ber breakage is much smaller than those ofmatrix cracking and delaminations and conned to the region nearthe contact area. The matrix cracking is the initial damage mode,and then the delamination damage initiates, at last the berbreakage propagates. When the equivalent point load resultedfrom impact is located on the lower surface, the impact-induceddamage initiates and develops from the lower surface of laminatedplate. Athougth the ber damage conned to the region under andnear the contact area between the impactor and the laminates islogical, it is not resonable for the matrix cracking and delamination

(a)

Damage variation (m

Damage variation

d2

d3

Fig. 20. Inuence of impact velocity on damage variations distributions of parallelly stiffthe thickness of stiffeners is 7.0 mm. (a) Impact velocity 30 m/s; and (b) impact velocityfibre breakage)ures 110 (2014) 249275 265which are the main part of damage. When the equivalent pointload resulted from impact is located on the upper surface, theimpact-induced damages are initiated and developed on the uppersurface of laminated plate. It is not reasonable for three kinds ofdamages. In contrast, when the equivalent point load resulted fromimpact is located on the middle surface, the initiation anddevelopment of the matrix cracking and delamination are alongthe thickness direction of laminated plate. Although the berdamage initiated and developed on the middle region of thicknessdirection is not resonable, it is not the main part of damage anddoes not inuence the impact responses signicantly. Therefore, inthe numerical examples of the next sections the equivalent pointload resulted from impact is acted on the middle surface oflaminated plates.

5.2. Composite laminated plates with stiffeners

The response of stiffened structures is affected signicantly bythe form and distribution of stiffeners. Therefore, two kinds of stiff-ened composite laminated plates (see Fig. 10) are investigated inthis section, concentrically stiffened laminated plates with fourstiffeners and parallelly stiffened laminated plates with two stiff-eners. The inuence of different parameters of the stiffeners and

(b)

atrix cracking)

(delamination)

ened laminated plate with two stiffeners, where the radius of impactor 6.0 mm and40 m/s.

ruct0 100 200 300 400 500 6000

0.5

1

1.5

2

2.5

3x 106

Time (s)

Cont

act f

orce

10

3 N


1.5

2

2.5

3x 104

ctor

(mm/

s)


(a)

266 D.H. Li et al. / Composite Stimpactor, such as the size and form of stiffeners, impact velocity,and the mass of impactor, on the responses and damages is studiedas well.

All the layers of laminated plate and stiffeners have same mate-rial properties as those listed in Table 1. The stacking sequence oflaminated plate is [0/90/0/90/0] and the stacking sequence of stiff-eners is [0]. All the single layers of laminated plate have the samethickness. The stiffened composite laminated plates are subjectedto low velocity impact at the lower surface, but the load applica-tion point of the equivalent contact force is subjected at the middlesurface. The density of the impactor is 7960 Kg=m3. When the ef-fect of the impact velocity on the responses and damages is inves-tigated, the radius of impactor is 6.0 mm and the thickness ofstiffeners is 7.0 mm. When the effect of the impact mass (radiusof impactor) on the responses and damages is investigated, the im-pact velocity is 30 m/s and the thickness of stiffeners is 7.0 mm.When the effect of the thickness of stiffeners on the responsesand damages is investigated, the radius of impactor is 6.0 mmand the impact velocity is 30 m/s.

5.2.1. Concentrically stiffened laminated plate with four stiffenersThe impact responses of concentrically stiffened composite

laminated plates subjected to different impact velocities with/without the progressive damage analysis model are shown inFig. 11, where the radius of impactor is 6.0 mm and the thicknessof stiffeners is 7.0 mm. It is observed that with the increase of

0 100 200 300 400 500 6001.5

1

0.5

0

0.5

1

Time (s)

Velo

city

of i

mpa

(c)Fig. 21. Inuence of the radius of impactor on the impact responses of parallelly stiffeContact force; (b) displacement w of plates in contact point, (c) velocity of impactor an0 100 200 300 400 500 6001

0.5

0

0.5

1

1.5

2

2.5

Time (s)

Cent

ral d

efle

ctio

n of

pla

te (m

m)


1

1.5

2

2.5

pact

or (m

m)


(b)

ures 110 (2014) 249275impact velocity the effect of the progressive damage analysis mod-el on the responses of plate and impactor increased. Therefore,damage analysis model is necessary for the impact responses anal-ysis problem when the impact velocity is larger. With the increaseof the impact velocity, the maximum contact force, the impactcompletion time, the central deection of plate and the displace-ment of impactor also increased. In the rst loadingunloadingcycle, the contact force uctuated greatly, especially in the loadingphase, because the impact-induced damages are mainly producedin this period.

The deformation and damage histories of the concentricallystiffened composite laminated plates subjected to impact velocity35 m/s are shown in Fig. 12, where the radius of impactor is6.0 mm and the thickness of stiffeners is 7.0 mm. The inuencesof the impact velocity on damage variations distributions is shownin Fig. 13, where the damage variations distributions are the nalresults after the impact. It is observed that in the early stages ofimpact the deformation of laminated plate is limited in the rectan-gular region surrounded by four stiffeners, and then with the elas-tic resilience the deformation of laminated plate began to expandout the rectangular region surrounded by stiffeners. In the processof the impact, the ber breakage is generally very limited(d1 < 0:45) and conned to the region near the contact area be-tween the impactor and the laminates. Serious matrix crackingand delamination arised near the contact region are the main partof damage in the composite laminates. The ber breakage and

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D.H. Li et al. / Composite Sdelaminations are mainly induced in the rst loadingunloadingcycle, while the matrix failure is extending constantly to far awayfrom the contact area especially the middle of the x-direction (berdirection) stiffeners and the interface region between the y-direc-tion stiffeners and laminated plate (see the second subgraph ofFig. 13(b)). Because the stacking sequence of laminated plate is[0/90/0/90/0] and stacking sequence of stiffeners is [0], the stiff-ness of the stiffened composite laminated plate in the ber direc-tion (x-direction) is greater than stiffness in the transversedirection (y-direction). It results in that the deection of laminatedplate and stiffeners in the y-direction is larger than that in the x-direction, which is the reason why the impact-induced damagestend to spread to the middle of the x-direction stiffeners and theinterface region between the y-direction stiffeners and laminatedplate. Consequently, there are three danger zones for the concen-trically stiffened composite laminated plates subjected to lowvelocity impact: (a) the contact region between impactor and lam-inated plate; (b) the middle region of the stiffeners along the direc-tion with the weaker stiffness; and (c) the interface region betweenthe laminated plate and the stiffeners along the direction with thestronger stiffness.

With the increase of impact velocity, the values and distri-bution range of the ber breakage and the delamination in-creased along the lower surface of the lamianted plate near

(a)

Damage variation (m

Damage variation

d2

d3

Fig. 22. Inuence of radius of impactor on damage variations distributions of parallelly stthe thickness of stiffeners is 7.0 mm. (a) radius of impactor 5.0 mm; and (b) radius of imfibre breakage)

ures 110 (2014) 249275 267the contact area. While the values and distribution range ofthe matrix cracking not only increased along the lower surfacewith the increase of impact velocity, but also appeares at theupper surface near the interface region between the y-directionstiffeners and laminated plate (see the second subgraph ofFig. 13(b)). In addition, the values and distribution range ofthe matrix cracking in thex-direction stiffeners increased andextanded rapidly.

The inuences of the radius of impactor on the impact re-sponses of the concentrically stiffened composite laminated plateswith the progressive damage analysis model are shown in Fig. 14,where the impact velocity is 30 m/s and the thickness of stiffenersis 7.0 mm. And the inuences of the radius of impactor on damagevariations distributions is shown in Fig. 15. It is observed that theradius of impactor has signicant inuence on the impactresponses of the concentrically stiffened laminated plate andimpactor since both of the mass and constant of the Hertzs contacttheory (see Fig. 3) increased with the increase of the radius ofimpactor. With the increase of the radius of impactor, the contactforce, the impact completion time, the central deection of plateand the displacement of impactor increased rapidly. The valuesand distribution range of the ber breakage, the matrix crackingand the delaminations rapidly increased and extanded with the in-crease of the radius of impactor.

(b)

atrix cracking)

(delamination)

iffened laminated plate with two stiffeners, where the impact velocity is 30 m/s andpactor 7.0 mm.

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268 D.H. Li et al. / Composite StInuences of the thickness of stiffeners on the impact responsesof the concentrically stiffened composite laminated plates with theprogressive damage analysis model are shown in Fig. 16, where theradius of impactor is 6.0 mm and the impact velocity is 30 m/s. It isobserved that the thickness of stiffeners has no effect on the im-pact responses during the rst loadingunloading cycle. The reasonmay be that in the rst loadingunloading cycle (the early stages ofimpact) the deformation of plate is limited in the rectangular re-gion surrounded by four stiffeners. With the increase of thicknessof stiffeners, the maximum contact force of the second loadingunloading cycle, the impact completion time and the residualvelocity of impactor increased.

Inuences of thickness of stiffeners on damage variations distri-butions of the concentrically stiffened composite laminated plateswith four stiffeners are shown in Fig. 17. With the increase of thethickness of stiffeners, the range of the ber breakage and delam-inations increased. The reason is that the increase of the thicknessof stiffeners results in the increase of the overall stiffness of thestiffened composite laminated plate. However, the range of thematrix cracking in the y-direction stiffeners rst increased andthen decreased with the increase of the thickness of stiffeners(see Figs. 13(a) and 17).

5.2.2. Parallelly stiffened laminated plate with two stiffenersFor the parallelly stiffened composite laminated plates, inu-

ences of the impact velocity on the impact responses with/withoutthe progressive damage analysis model are shown in Fig. 18, wherethe radius of impactor is 6.0 mm and the thickness of stiffeners is

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ures 110 (2014) 2492757.0 mm. It can be seen from Fig. 18 that with the increase of theimpact velocity the effect of the progressive damage analysis mod-el on the responses of laminated plate and impactor increasesed.The contact force increased with the increase of the impact veloc-ity, and the inuence of the impact velocity on contact force duringthe rst loadingunloading cycle is more signicant than that dur-ing the second impact. The deection of laminated plate and thedisplacement of impactor increased proportionately as the impactvelocity increased.

The deformation and damage variations histories of the parall-elly stiffened composite laminated plates with two stiffeners sub-jected to impact velocity 40 m/s are shown in Fig. 19, where theradius of impactor is 6.0 mm and the thickness of stiffeners is7.0 mm. Inuences of impact velocity on damage variations distri-butions of the parallelly stiffened laminated plate are shown inFig. 20. In the early stages of impact the deformation of plate is lim-it in the rectangular region between two parallelly stiffeners in thex direction, and then with the elastic resilience the deformation ofplate began to expand out this region. The extent of the damage(the value of damage variations) and range increased as the impactvelocity increased.

The ber breakage is very limited and conned to the regionnear the contact area between the impactor and the laminates.The matrix cracking initiates at the contact point, and rapidly ex-pand during the rst loadingunloading cycle in the area nearthe contact point. With the elastic resilience the matrix crackingcontinued to extend along the lower surface during the second im-pact. The matrix cracking rstly appears at the upper surface near

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tructD.H. Li et al. / Composite Sthe interface between laminated plate and stiffeners and then theit initiates at the lower surface (see Fig. 20(b)). The delaminationalso initiates at the contact point, and rapidly expand during therst loadingunloading cycle in the area near the contact point.There are two danger zones for the parallelly stiffened compositelaminated plates subjected to low velocity impact: (a) the contact

Damage variation (

(a)

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d1

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Fig. 24. Inuence of thickness of stiffeners on damage variations distributions of paralleland the radius of impactor is 6.0 mm. (a) Thickness of stiffeners 3.0 mm; and (b) thickn

Table 3Effects of the parameters on the residual velocity for the concentrically stiffened plate an

Concentrically stiffened plate

vr vr=v0 10Initial velocity (m/s)20 8.3303 41.651530 12.8645 42.881740 (35) 15.2766 43.6474

Radius of impactor (mm)5.0 7.6256 25.41876.0 12.8645 42.88177.0 (6.5) 13.9746 46.5820

Thickness of stiffeners (mm)3.0 7.1551 23.85035.0 10.1293 33.76437.0 12.8645 42.8817ures 110 (2014) 249275 269region between impactor and laminated plate; and (b) the inter-face region between the laminated plate and the stiffeners alongthe direction with the stronger stiffness.

The inuences of the radius of impactor on the impactresponses of the parallelly stiffened composite laminated platesare shown in Fig. 21, where the impact velocity is 30 m/s and the

fibre breakage)

(b)

atrix cracking)

(delamination)

ly stiffened laminated plate with two stiffeners, where the impact velocity is 30 m/sess of stiffeners 5.0 mm.

d the parallelly stiffened plate.

Parallelly stiffened plate

0 vr vr=v0 100

4.7011 23.51557.1945 23.9817

10.4774 26.1935

6.2506 20.83537.1945 23.9817

14.3573 47.8577

5.4546 18.18206.3986 21.32877.1945 23.9817

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270 D.H. Li et al. / Composite Stthickness of stiffeners is 7.0 mm. And the inuences of radius ofimpactor on damage variations distributions are shown inFig. 22. It is observed that the radius of impactor has also signi-cant inuence on the impact responses of the parallelly stiffenedlaminated plate and impactor. With the increase of the radius ofimpactor, the contact force, the impact completion time, the cen-tral deection of plate and the displacement of impactor increasedrapidly. The values and distribution range of the ber breakage, thematrix cracking and the delamination rapidly increased and extan-ded with the increase of the radius of impactor.

Inuences of the thickness of stiffeners on the impact re-sponses of the parallelly stiffened composite laminated plateswith the progressive damage analysis model are shown inFig. 23, where the radius of impactor is 6.0 mm and the impactvelocity is 30 m/s. Similar to the concentrically stiffened compos-ite laminated plates, the thickness of stiffeners of the parallellystiffened composite laminated plates has not effect on the impactresponses during the rst loadingunloading cycle, and the peakvalue of the second impact increased with the increase of thethickness of stiffeners.

Inuences of thickness of stiffeners on damage variations distri-butions of the parallelly stiffened laminated plate are shown inFig. 24. It can be seen from Fig. 24 that the thickness of stiffenersof the parallelly stiffened composite laminated plates has a greatinuence on the ber breakage and matrix cracking, while thedelamination increased slightly with the increase of the thicknessof stiffeners.

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ures 110 (2014) 249275Effects of the parameters on the residual velocity of the impac-tor v r for the concentrically stiffened plate and the parallelly stiff-ened plate are listed in Table 3, where the values in the bracketsare adopted for the concentrically stiffened composite laminatedplate. It can be seen from Table 3 that the percentages of the resid-ual velocity in the initial velocity v0 are increased with the increas-ing of the initial velocity, the radius of the impactor and thethickness of stiffeners. The percentages of the residual velocity inthe initial velocity of the concentrically stiffened plate are largerthan that of the parallelly stiffened plate, so the absorption capac-ity of the impact energy of the parallelly stiffened plate is betterthan that of the concentrically stiffened plate. Comparing twokinds of stiffened composite laminated plates studied in this sec-tion, the deection of the concentrically stiffened composite lamin-ted plate at the contact point is smaller than that of the parallellystiffened compoiste laminated plate, while the impact-induceddamages just the opposite. Therefore, the impact resistant abilityand the stiffness of the stiffened composite laminated plate arecontradictory.

5.3. Multi-impacts on stiffened composite laminated plate

Internal damages arised due to low-speed impact event can re-duce the residual strength of composite structures signicantly.Furthermore, such kind internal damage as delamination and berbreakage are imperfections. This results in a reduction in stiffnessand strength and consequent progagation under further loading.

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D.H. Li et al. / Composite SThe accumulation of these aws over time may result in failure ofthe apparently undamage structure. Hence the safe use of compos-ite structure requires an understanding of the progression ofdamage because of multi-impacts. Although the impact strengthunder single-impact loading is widely studied, the problem ofcomposites under repeated impacts has attracted less attention[72]. In this section, the present analysis model was used to studythe multi-impacts problem of stiffened composite laminated plate,and the damage accumulation problem of the multi-impacts wasalso investigated. In the analysis process of the multi-impacts, allof the parameters for each impact kept unchanged except thematerial properties which would be affected by the damageaccumulation.

Impact responses of concentrically stiffened composite lami-nated plates subjected to multi-impacts are shown in Fig. 25. Inthis numerical example, the thickness of stiffeners is 7.0 mm andthe radius of impactor is 6.0 mm. The impact velocity of the rstand second impacts is 24.7487 m/s, and the total impact energyof two impacts equals to that results from one impact with initialvelocity 35 m/s. In the rst loadingunloading cycle, the loadingspeed of contact force of the 1st impact is larger than that of the2nd impact, while the maximum contact force of the 2nd impactis larger than that of the 1st impact. The reason is that the damagesinduced by the 1st impact signicantly reduce the stiffness nearthe contact region. In the second loadingunloading cycle, the peak

(a)

Damage variation (m

Damage variation

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Fig. 26. Damage variations distributions of concentrically stiffened laminatedfibre breakage)

ures 110 (2014) 249275 271values of contact force of the 1st and 2nd impact are roughly equal.Although the start time of loading and unloading of the 1st impactis earlier that of the 2nd impact, the end time of the whole impactprocess of 1st and 2nd impact is the same. For the single impact atthe initial velocity 35 m/s, the percentages of the residual velocityin the initial velocity are 43.6474% (residual velocity 15.2766 m/s).For the multi-impact at the initial velocity 24.7487 m/s (rst im-pact) and 24.7487 m/s (second impact), the percentages of theresidual velocity in the initial velocity are 41.9060% (residualvelocity 10.3712 m/s) and 43.1178% (residual velocity 10.6711 m/s), respectively. The central deection of laminated plate and dis-placement of impactor of the 2nd impact are larger than those ofthe 1st impact because of the stiffness reduction near the contactregion resulted from damage accumulation of the 1st impact.

Damage variations distributions of concentrically stiffened lam-inated plate subjected to multi-impacts (1st and 2nd impact) areshown in Fig. 26. It can be seen from Fig. 26 that compared withthe damages induced by 1st impact the range of damages havebeen broadened by the 2nd impact even though the damagesarised due to 1st or 2nd impact are limited. Composite structuresare prone to sudden failure under the action of repeated small en-ergy impacts, and whats even worse, the extension and cumula-tive process is not detectable from visible observation. Therefore,the dangers of repeated small energy impact for composite struc-tures are greater than the dangers of single larger energy impact.

(b)

atrix cracking)

(delamination)

plate subjected to multi-impacts. (a) First impact; and (b) second impact.

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272 D.H. Li et al. / Composite StCompared with the damage induced by the single impact withinitial velocity 35 m/s (see Fig. 13), the damage accumulations in-duced by the multi-impacts (1st and 2nd impact) are signicantlysmaller than those resulted from the single impact, though the sumof impact energy of the multi-impacts is equal to the impact en-ergy of the single impact (see Fig. 27).

Impact responses of parallelly stiffened composite laminatedplates subjected to multi-impacts are shown in Fig. 25. And thedamage variations distributions of parallelly stiffened laminatedplate subjected to multi-impacts (1st and 2nd impact) are shownin Fig. 28. In this numerical example, the thickness of stiffeners is7.0 mm and the radius of impactor is 6.0 mm. The impact velocityof the rst and second impacts is 28.2843 m/s, and the total impactenergy of two impacts equal to that results from one impact withinitial velocity 40 m/s. For the single impact at the initial velocity40 m/s, the percentages of the residual velocity in the initialvelocity are 26.19% (residual velocity 10.4770 m/s). For themulti-impact at the initial velocity 28.2843 m/s (rst impact) and28.2843 m/s (second impact), the percentages of the residualvelocity in the initial velocity are 23.0969% (residual velocity6.5328 m/s) and 23.4218% (residual velocity 6.6247 m/s),respectively.

It can be seen from Figs. 25 and 28 that the multi-impactsbehavior of parallelly stiffened composite laminated plates issimilar to that of concentrically stiffened laminated plates. Theloading speed of contact force of the 1st impact is larger than that

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ures 110 (2014) 249275of the 2nd impact in the rst loadingunloading cycle, and thepeak values of contact force of the 1st and 2nd impact are roughlyequal in the second loadingunloading cycle. The difference is thatthe maximum contact force of the 2nd impact is slightly largerthan that of the 1st impact.

6. Concluding remarks

In the present work, a LW/SE method, modied nonlinearHertzs law and the progressive failure model are combined todevelop a method for low impact responses and impact-induceddamages analysis of the composite stiffened plates. In this analysismodel, the transient response analysis model of the stiffened com-posite laminated plate is estiblished by using the LW/SE methodand Newmark method, the contact force between the stiffenedplate and impactor is obtained by the modied nonlinear Hertzslaw with Newmark method, and the predicition of impact-induceddamages is carried out with the progressive failure model. Withoutany assumption about the stiffeners, the accurate stress and strainresponses can be obtained by using the present low-impact analy-sis method for the composite laminated plates with arbitrarilycomplex stiffeners. As a result of the adoption of the progressivefailure model in this method, the accumulation and extension ofthe impact-induced damages can be predicted accurately as well.

Several numerical examples are carried out to study the inu-ence of parameters on the impact response and impact-induced

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