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Nonsingular crack modelling in orthotropic plates by four equivalentsingle layersQ6

Q5 Andr�as Szekr�enyesBudapest University of Technology and Economics Department of Applied Mechanics, Müegyetem rkp. 5, 1111 Budapest, Building MM, HungaryQ2

a r t i c l e i n f o

Article history:Received 4 March 2015Accepted 8 August 2015Available online xxx

Keywords:DelaminationMixed mode II/III fractureEnergy release rateSecond-order plate theoryTheoreom of autocontinuityQ3

a b s t r a c t

In this work the second- and third-order laminated plate theories are applied to model delaminatedcomposite plates with material orthotropy. The method of four equivalent single layers is proposed and ageneral third-order displacement field is utilized in each layer. The kinematic continuity between thelayers is established by the system of exact kinematic conditions. Apart from the continuity of the in-plane displacements between the interfaces of the layers even the continuity of shear strains, theirderivatives and curvatures is imposed. As a novelty a so-called shear strain control condition is intro-duced, which means that the shear strains at two or more points located along the thickness are imposedto be the same. Using the proposed conditions the equilibrium equations are derived for the delaminatedand undelaminated regions of the plate. Plates with different boundary conditions are solved as ex-amples and the theorem of autocontinuity is introduced, which is essentially related to the continuityconditions between the delaminated and undelaminated parts. The stress and displacement fields aswell as the J-integral are determined in the examples and compared to finite element calculations. Theresults indicate that the control condition works very well in the case of the second-order plate theory, incontrast it is rather a disadvantage in the case of the third-order approximation.

© 2015 Published by Elsevier Masson SAS.

1. Introduction

Anisotropic composites (Chaudhuri and Balaraman, 2007;Czig�any and De�ak, 2012; M�esz�aros et al., 2013) are often utilizedin air-, spacecraft (Smojver and Ivan�cevi�c, 2010; Ivan�cevi�c andSmojver, 2011; Smojver and Ivan�cevi�c, 2011, 2012; Langdon et al.,2014) and sport industry (Jiang, 2014; Li and Jing, 2014; Su, 2014;Zhang, 2014; Tang, 2014), composite panels are also applied incars and vehicles (Norhidayah et al., 2014; Wennberg and Stichel,2014; Khan et al., 2014), ships (Chirica et al., 2011; Chirica, 2013),pressure vessels (Gheshlaghi et al., 2006) and many other engi-neering applications (Goch et al., 2012). Themechanical behavior oflaminated composite plates and shells can be described by differenttheories. The classical laminated plate theory (CLPT) is based on theKirchhoff assumption and so, it does not take the shear deformationinto account (Radosavljevi�c and Dra�zi�c, 2010; Eftekhari and Jafari,2012; Hajheidari and Mirdamadi, 2013). The first-order sheardeformation (FSDT or Mindlin) theory assumes constant shearstrain distribution over the thickness of the plate (Kreja and

Schmidt, 2006; Endo and Kimura, 2007; Assie et al., 2012; Batista,2012; Nanda and Sahu, 2012; Sabik and Kreja, 2013; Endo, 2015).As a next step, the second-order plate theory (SSDT) proposes thatthe in-plane displacements are captured by quadratic functions interms of the through-thickness coordinate (Baddour, 2011; Izadiand Tahani, 2010; Shahrjerdi et al., 2011, 2010). The third-ordershear deformation plate theory (TSDT) (Talha and Singh, 2010;Aragh et al., 2013) as well as the refined Reddy third-order theory(Thai et al., 2012; Oktem et al., 2013; Taj et al., 2013; Batista, 2012;Bodaghi and Saidi, 2010; Thai et al., 2012) are also very common todescribe the mechanical behavior of anisotropic plates and shells.

The 3D and elasticity based solutions (Alibeigloo and Zanoosi,2013; Chang and Tarn, 2012; Yang et al., 2012) are also availablein the literature, as well as the different layerwise approximations(Arya et al., 2002; Ferreira et al., 2011; Ovesy et al., 2015; Saeediet al., 2012a,b, 2013a,b; Lerpiniere et al., 2014; Sahoo and Singh,2014b). Other higher-order theories (Steigmann, 2012; Xianget al., 2012; Shi, 2007; Zhao et al., 2013; Lu et al., 2013; Shimpiand Patel, 2006) and those based on trigonometric or otherappropriate functions (Thai and Vo, 2013; Thai and Choi, 2013;Sahoo and Singh, 2014a, 2013a,b) are useful in the case of func-tionally graded (FGM) plates.E-mail address: [email protected].

URL: http://www.mm.bme.hu/~szeki

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Please cite this article in press as: Szekr�enyes, A., Nonsingular crack modelling in orthotropic plates by four equivalent single layers, EuropeanJournal of Mechanics A/Solids (2015), http://dx.doi.org/10.1016/j.euromechsol.2015.08.005

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It is clear that many plate and shell theories are proposed in theliterature, however, the application of these theories to model de-laminations is started only few years ago. The delamination is atypical and common damage mode in composite laminates. Thepresence of delaminations (Rizov, 2012a,b; Ahn et al., 2012) andcracks (Hajikazemi and Sadr, 2014b,a) in the material reduces thestiffness, strength and lifetime, it also changes the dynamic prop-erties and the behavior of the structure against the loss of stability.Therefore it is essential to develop new models for the delamina-tion analysis of composite materials. In this respect the work byDavidson et al. (2000) is noteworthy. It has also to be mentionedthat coupled multilayer/interface models have already been intro-duced in the literature, which are able to provide accurate energyrelease rate (ERR) calculations compared to FE models using solidtype elements (Bruno and Greco, 2001; Bruno et al., 2003, 2005).An important aspect of these formulations is that it avoids thecomplications arising from the oscillatory singularities for adelamination between different materials.

The delaminations may take place because of low-velocityimpact (Ganapathy and Rao, 1998; Rizov et al., 2005; Christoforouet al., 2008; Burlayenko and Sadowski, 2012; Zammit et al., 2011;Wang et al., 2012; Goodmiller and TerMaath, 2014),manufacturing defects (Zhang and Fox, 2007; Zhou et al., 2013) andfree edge effects (Sarvestani and Sarvestani, 2012; Ahn et al., 2013).The resistance against the delamination is characterized by mode-I(Jumel et al., 2011a,b; Salem et al., 2013; Kim et al., 2011; Peng et al.,2011; Gracia et al., 2015) mode-II (Arrese et al., 2010; Argüelleset al., 2011; Petrova et al., 2012; Petrova and Sadowski, 2012;Yoshihara and Satoh, 2009; Rizov and Mladensky, 2012) mode-III(Rizov et al., 2006; Szekr�enyes, 2009a, 2011a,b; Johnston et al.,2014; Johnston and Davidson, 2014; Rodríguez-Gonz�alez et al.,2014; Khoshravan and Moslemi, 2014; L�opez-Men�endez et al.,2014; Marat-Mendes and Freitas, 2009; Mehrabadi and Khosra-van, 2013; Yoshihara, 2006) mixed-mode I/II (de Baere et al., 2012;Davidson et al., 2009; Fern�andez et al., 2013; Islam and Kapania,2014; Peng et al., 2012; Kenane et al., 2010; Marat-Mendes andFreitas, 2010; Nikbakht et al., 2010; Nikbakht and Choupani, 2008;Plain and Tong, 2011; Liu et al., 2015; Pereira et al., 2014) mixed-mode II/III (Szekr�enyes, 2007; Kondo et al., 2011, 2010;Mladensky and Rizov, 2013; Kotousov et al., 2013, 2012;Szekr�enyes, 2012; Mehrabadi, 2013, 2014; Miura et al., 2012;Takeda et al., 2013; Suemasu et al., 2010) mixed-mode I/III (Miuraet al., 2014; Szekr�enyes, 2009b) and mixed-mode I/II/III (Davidsonand Sediles, 2011; Szekr�enyes, 2011a,b) fracture test.

The previous developments e that this paper is based on e aresummarized as follows: In some recent papers the so-calledinterface constraints were applied to model delaminated compos-ite plates by CLPT, FSDT, SSDT and TSDT with midplane cracks andsymmetric lay-up (Szekr�enyes, 2014a,c,e, 2013a). Later, the modelswere generalized to asymmetrically delaminated plates and the so-called system of exact kinematic conditions (SEKC) was proposedfor the FSDT (Szekr�enyes, 2013b), SSDT (Szekr�enyes, 2014b) andReddy third-order theory (Szekr�enyes, 2014d). In these papers themethod of two equivalent single layers (2ESLs) was applied, i.e. thewhole plate was divided into two parts in the plane of the delam-ination: a top and a bottom plate was analyzed. Simply supportedplates were investigated with straight widthwise delamination.Among the mentioned models Reddy third-order theory was foundto be the most promising, however, it will be shown in this paperthat a better performance can be achieved by the present method.

The novelty of this paper is the proposed method of fourequivalent single layers (4ESLs). The top and bottom plates arecaptured by two ESLs including the proper kinematic continuity.Furthermore, the updated form of the SEKC is presented includingthe conditions of continuous shear strain, its derivative and

curvature at the interface planes and a so-called shear strain con-trol condition. From the point of view of the continuity conditionsbetween the delaminated and undelaminated regions the theoremof autocontinuity is formulated. The novel technique is applied todelaminated L�evy plates with different geometry and boundaryconditions.

2. Semi-layerwise laminated plate theory e the method offour ESLs

Fig. 1 shows the section of the layered plate element in the X�Zplane, while in Fig. 2 the Y�Z plane is shown. The elements containan interfacial delamination parallel to the Y axis, i.e. it goes acrossthe entire platewidth. The delamination divides the plate into a topand a bottom subplate. Each subplate is modelled by further twoESLs. The interface planes between the ESLs are called the pertur-bation planes. In accordance with the literature review, it is clearthat the ESLs can be captured by different plate theories. In thisworkwe apply the SSDTand TSDT theories. The general third-orderTaylor series expansion of the displacement functions results in thefollowing displacement field (Talha and Singh, 2010; Panda andSingh, 2011; Singh and Panda, 2014; Panda and Singh, 2009):

ui�x;y;zðiÞ

�¼ u0ðx;yÞþu0iðx;yÞþ qðxÞiðx;yÞzðiÞ þfðxÞiðx;yÞ

�zðiÞ�2

þlðxÞiðx;yÞ�zðiÞ�3

vi

�x;y;zðiÞ

�¼ v0ðx;yÞþ v0iðx;yÞþ qðyÞiðx;yÞzðiÞ þfðyÞiðx;yÞ

�zðiÞ�2

þlðyÞiðx;yÞ�zðiÞ�3

wiðx;yÞ ¼wiðx;yÞ(1)

where i is the index of the actual ESL, z(i) is the local throughthickness coordinate of the ith ESL (refer to Fig. 1), u0 and v0 are theglobal, u0i and v0i are the local membrane displacements, moreover,q means the rotations of the cross sections about the X and Y axes(refer to Fig. 1), f denotes the second-order, l represents the third-order terms in the displacement functions. The displacementfunctions of SSDTcan be obtained by reducing Eq. (1) and taking l(x)i ¼ l(y)i ¼ 0 (Izadi and Tahani, 2010; Baddour, 2011; Petrolito, 2014).

3. The system of exact kinematic conditions e SEKC

The system of exact kinematic conditions has been developed in(Szekr�enyes, 2013b) for first-order Mindlin plates and it was sub-sequently extended to second- (Szekr�enyes, 2014b) and third-orderplates (Szekr�enyes, 2014d). In the mentioned papers the plate wasdivided into two ESLs in the undelaminated portion. Among theseproposed solutions, Reddy's theory provided the best agreementwith the finite element (FE) results (based on the virtual crackclosure technique (VCCT, (Bonhomme et al., 2010; Valvo, 2012)) forasymmetrically delaminated composite plates. In spite of that, ifthe delamination is very close to the free surface of the plate, theneven Reddy's theory leads to inaccurate results. The aim of thispaper is to model the delaminated plate by 4ESLs and to introducethe necessary conditions to obtain the best possible solutions underassumed plane stress state.

We assume that a plate consisting of orthotropic plies contains asingle through-width delamination. The delamination front isparallel to the Y axis. A displacement field given by Eq. (1) isassociated to each ESLs. The kinematic continuity between theadjacent ESLs is established by the SEKC requirements. The first setof conditions formulates the continuity of the in-plane displace-ments between the neighboring plies as:

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�uðiÞ; vðiÞ;wðiÞ

��zðiÞ¼ti=2

¼�uðiþ1Þ; vðiþ1Þ;wðiþ1Þ

��zðiþ1Þ¼�tiþ1=2

(2)

where t is the thickness of the specified layer. The second set ofconditions defines the global membrane displacements (u0, v0) atthe reference plane of the actual region. If the location of thereference plane is zðkÞR and is located in the kth layer, then therelevant conditions become:

uðkÞ��zðkÞ¼zðkÞR

� u0 ¼ 0; vðkÞ��zðkÞ¼zðkÞR

� v0 ¼ 0 (3)

The two sets of conditions given by Eqs. (2)e(3) are sufficient todevelop semi-layerwise models using the FSDT. If the SSDT or TSDTis applied, then we can impose the shear strain continuity at the

interface planes. In accordance with Figs. 1b and 2b these condi-tions are formulated as:�gxzðiÞ;gyzðiÞ

��zðiÞ¼ti=2

¼�gxzðiþ1Þ;gyzðiþ1Þ


(4)

For the TSDT theory two more sets of conditions are reasonableto introduce. The imposition of continuous shear strain derivativesand curvatures prevent the unwanted oscillations in the stressdistributions (see Figs. 1b and 2b):

�vgxzðiÞvzðiÞ

;vgyzðiÞvzðiÞ

��zðiÞ¼ti=2

¼�vgxzðiþ1Þvzðiþ1Þ ;

vgyzðiþ1Þvzðiþ1Þ


(5)

Fig. 1. Cross sections and deformation of the top and bottom plate elements of a delaminated plate in the XeZ plane (a). Distribution of the transverse shear strains by SSDT andTSDT (b).

Fig. 2. Cross sections and deformation of the top and bottom plate elements of a delaminated plate in the YeZ plane (a). Distribution of the transverse shear strains by SSDT andTSDT (b).

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and: v2gxzðiÞv�zðiÞ�2; v2gyzðiÞ

v�zðiÞ�2!��

zðiÞ¼ti=2¼ v2gxzðiþ1Þv�zðiþ1Þ�2; v

2gyzðiþ1Þv�zðiþ1Þ�2

!��zðiþ1Þ¼�tiþ1=2

(6)An important addition compared to previous papers is the so-

called shear strain control condition (SSCC). In accordance withReddy theory the top and bottom surfaces of the plate are traction-free (zero shear stresses). If two ESLs are applied this conditioninvolves a little over-stiffening of the system. Moreover, if thesystem is modelled by 4ESLs the traction-free conditions leads tooverconstraining the model and wrong results are obtained.Therefore, instead of imposing zero stresses at the free surfaces weimpose the identical shear strain values at the boundary planes.Figs. 1a and 2a show the so-called control line and the controlledshear strain distributions. The set of conditions applied is:�gxzðlÞ;gyzðlÞ

��zðlÞ¼�tl=2

¼�gxzðmÞ;gyzðmÞ

��zðmÞ¼�tm=2

(7)

where l and m denote the boundaries, where the shear strains areequal to each other. The SEKC can be applied for the undelaminatedand delaminated portions of the plate. Moreover these conditionscan be implemented into any plate theory.

4. Development of kinematically admissible displacementfields

The SEKC is applied to the problem shown in Figs. 1 and 2. Usingthe conditions defined by Eqs. (2)e(7) we can eliminate certainparameters from Eq. (1), which involves 34 parameters altogetherplus the deflections (wi(x,y)). This step is called parameter elimi-nation. The parameters to be eliminated are chosen in order toobtain a system of equations, which contain linearly independentequations. However, the global membrane parameters should beremaining or primary parameters, the local membrane displace-ments are typically secondary parameters, that should be eliminatedfrom the displacement field. In the subsequent sections the unde-laminated and delaminated regions are discussed separately. In thefirst step, the TSDT solution is presented, then in the second stepthe SSDT field equations are obtained by the reduction of the TSDTequations.

4.1. Undelaminated plate region

The transition zone around the delamination front in the X�Zplane of the composite plate is shown in Fig. 1a. The distribution ofthe in-plane displacement functions is quadratic in the case of theSSDT and cubic for the TSDT. The corresponding shear strain dis-tributions are shown in Fig. 1b: it is piecewise linear by SSDT andpiecewise quadratic by TSDT with continuous derivatives and cur-vatures in the latter case. In accordance with Figs. 1a and 2a, thefollowing conditions are formulated between the four ESLs (con-tinuity of in-plane displacement at the interface planes):

ðu1; v1;w1Þ��zð1Þ¼t1=2

¼ ðu2; v2;w2Þ��zð2Þ¼�t2=2


¼ ðu3; v3;w3Þ��zð3Þ¼�t3=2


¼ ðu4; v4;w4Þ��zð4Þ¼�t4=2

(8)

The reference plane belongs to the second ESL, therefore, thefollowing condition is imposed:

ðu2; v2Þ��zð2Þ¼zð2ÞR

¼ ðu0ðx; yÞ; v0ðx; yÞÞ (9)

where the zð2ÞR ¼ 1=2ðt3 þ t4 � t1Þ in accordance with Fig. 1a andactually coincides with the global midplane of the model (Reddy,2004). The next set of conditions imposes the continuous shearstrains at the interface planes:

�gxzð1Þ;gyzð1Þ

��zð1Þ¼t1=2

¼�gxzð2Þ;gyzð2Þ

��zð2Þ¼�t2=2�

gxzð2Þ;gyzð2Þ��

zð2Þ¼t2=2¼�gxzð3Þ;gyzð3Þ

��zð3Þ¼�t3=2�


zð3Þ¼t3=2¼�gxzð4Þ;gyzð4Þ

��zð4Þ¼�t4=2

(10)

As discussed previously, the oscillations in the shear straindistribution can be reduced by ensuring continuous shear strainderivatives at the interface planes 1e2 and 3e4:

�vgxzð1Þvzð1Þ

;vgyzð1Þvzð1Þ

��zð1Þ¼t1=2

¼�vgxzð2Þvzð2Þ

;vgyzð2Þvzð2Þ

��zð2Þ¼�t2=2�

vgxzð3Þvzð3Þ

;vgyzð3Þvzð3Þ

��zð3Þ¼t2=2

¼�vgxzð4Þvzð4Þ

;vgyzð4Þvzð4Þ

��zð4Þ¼�t4=2

(11)

furthermore, by imposing continuous shear strain curvatures in thesame planes by using the conditions below:

v2gxzð1Þ

v�zð1Þ�2; v2gyzð1Þ

v�zð1Þ�2!��

zð1Þ¼t1=2

¼ v2gxzð2Þ


v�zð2Þ�2!��

zð2Þ¼�t2=2 v2gxzð3Þ


v�zð3Þ�2!��

zð3Þ¼t3=2

¼ v2gxzð4Þ


v�zð4Þ�2!��

zð4Þ¼�t4=2

(12)

To further reduce the number of parameters in the displacementfield and to obtain more accurate results, the SSCC is applied at thetop and bottom boundaries:�gxzð1Þ;gyzð1Þ

��zð1Þ¼�t1=2


��zð4Þ¼t4=2

(13)

In Eq. (1) the displacement functions are modified in order tosatisfy Eqs. (8)e(13). In the general sense, by applying the SSDT andTSDT theories the in-pane displacement functions can be writtenas:

ui ¼ u0 þ�Kð0Þij þKð1Þ

ij zðiÞ þ Kð2Þij

�zðiÞ�2 þKð3Þ

ij

�zðiÞ�3�

jðxÞj

vi ¼ v0 þ�Kð0Þij þKð1Þ

ij zðiÞ þKð2Þij

�zðiÞ�2 þKð3Þ

ij

�zðiÞ�3�

jðyÞj(14)

where the matrices denoted by Kij are related to the geometry (ESLthicknesses), i refers to the ESL number, j defines the component inj, which is the vector of primary parameters, finallywi(x,y)¼w(x,y)for each ESLs, i.e. the transverse normals of each ESL are inex-tensible (Reddy, 2004).

4.1.1. Third-order plate theoryUsing the conditions above (Eqs. (8)e(13)) we can eliminate

twentyfour parameters from Eq. (1), the secondary parameters are:u0i, v0i, q(x)i, q(y)i, f(x)i, f(y)i for i ¼ 1,3, l(x)i, l(y)i for i ¼ 1…4. Theprimary parameters are: u0, v0, q(x)i, q(y)i for i ¼ 2,4 and f(x)i, f(y)i for

i ¼ 2,4. The nonzero elements of the matrices Kð0Þij , Kð1Þ

ij , Kð2Þij and

Kð3Þij are defined in Appendix A. The vector of primary parameters

is: j(p) ¼ (q(p)2 f(p)2 q(p)4 f(p)4)T with p ¼ x or y.


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4.1.2. Second-order plate theoryIn this case l(x)i ¼ 0 and l(y)i ¼ 0 in Eq. (1). Eqs. (8)e(9) apply

together with Eq. (10) (shear strain continuity), however Eqs. (11)and (12) are omitted. Therefore we can eliminate sixteen parame-ters from Eq. (1), the secondary parameters are: u0i, v0i, q(x)i, q(y)i,f(x)i, f(y)i for i ¼ 1,3. The primary parameters are: u0, v0, q(x)i, q(y)i fori ¼ 2,4, f(x)i and f(y)i for i ¼ 2,4. The nonzero elements of the

matrices Kð0Þij , Kð1Þ

ij and Kð2Þij are defined in Appendix B. Obviously

Kð3Þij ¼ 0 in this case. The vector of primary parameters becomes:

j(p) ¼ (q(p)2 f(p)2 q(p)4 f(p)4)T with p ¼ x or y, i.e. thesame as that for the TSDT.

4.2. Delaminated plate portion

In the delaminated portion (refer to Figs. 1b and 2b) the top andbottom plates are modelled by two ESLs, and thus the first and thirdof Eq. (8) still hold. It is important to note that in accordance withEq. (8) the transverse deflections of the top and bottomplates of thedelaminated region are identical (constrained mode model,(Szekr�enyes, 2014d)). In other words, the crack opening is elimi-nated in the plate, and the problem provides essentially mixed-mode II/III fracture without the presence of mode-I. The aim is topredict the mechanical fields in the plate as accurately as possiblecompared to FE calculations. If this is done for mixed-mode II/III,then the model can be extended to general mixed-mode I/II/III casein the course of the further research work. The definition of the topand bottom reference planes involve:

ðu1; v1Þjzð1Þ¼t2=2 ¼ ðu0bðx; yÞ; v0bðx; yÞÞðu3; v3Þjzð3Þ¼t4=2 ¼ ðu0tðx; yÞ; v0tðx; yÞÞ (15)

where u0b and u0t are the global membrane displacements of thebottom and top layers in accordance with Figs. 1 and 2. Further-more, the first and third of Eq. (10) apply again, as well as Eqs. (11)and (12) leading to eleven conditions altogether. Three moreequations are formulated by using the shear strain controlconditions:�gxzð1Þ;gyzð1Þ

��zð1Þ¼�t1=2


��zð2Þ¼t2=2�


zð3Þ¼�t3=2¼�gxzð4Þ;gyzð4Þ

��zð4Þ¼t4=2�


zð1Þ¼�t1=2¼�gxzð4Þ;gyzð4Þ

��zð4Þ¼t4=2

(16)

i.e., instead of imposing traction-free boundaries we control thestrain distribution by having equal values at the boundaries.

4.2.1. Third-order plate theoryThe first and third in Eq. (8) hold, moreover Eq. (14) is implied,

again the first and third of Eq. (10) are utilized together with Eqs.(11) and (12) and Eqs. (15) and (16) leading to 26 conditions alto-gether. The secondary parameters are: u0i, v0i, f(x)i, f(y)i for i¼ 1…3,l(x)i, l(y)i for i ¼ 1…4. The primary parameters are: u0t, v0t, u0b, v0b,q(x)i, q(y)i for i ¼ 2,4 and f(x)i, f(y)i for i ¼ 4. The modified displace-ment field has the same form as that given by Eq. (14), the co-efficients denoted by K are placed in Appendix A and: j(p) ¼ (q(p)2q(p)4 f(p)4)T, where p ¼ x or y.

4.2.2. Second-order plate theoryIn this case l(x)i ¼ 0 and l(y)i ¼ 0 in Eq. (1). The first and third in

Eq. (8) hold, moreover Eq. (14) is implied, again the first and third ofEq. (10) is utilized, however Eqs. (11) and (12) are omitted. Thestrain control conditions are applied through Eq. (16). Therefore wecan eliminate eighteen parameters from Eq. (14), the secondary

parameters are: u0i, v0i, f(x)i, f(y)i for i ¼ 1,3 and 4, and q(x)i, q(y)i fori¼ 1,3 respectively. The primary parameters are: u0t, v0t, u0b, v0b, q(x)i, q(y)i for i¼ 2,4, f(x)i and f(y)i for i¼ 2. The elements of the matrices

Kð0Þij , Kð1Þ

ij and Kð2Þij are defined in Appendix B. Apparently Kð3Þ

ij ¼ 0 in

this case. The vector of primary parameters takes the form:j(p) ¼ (q(p)2 f(p)2 q(p)4)T, where p ¼ x or y.

5. Equilibrium equations

The strain field in an elastic body in terms of the displacementfield is obtained by the following equation (Chou and Pagano,1967):

εij ¼12�ui;j þ uj;i

�(17)

where εij is the strain tensor, ui is the displacement vector field. Byassuming plane stress state in the plate the vector of in-planestrains becomes (Reddy, 2004):

8<:εxεygxy

9=;ðiÞ

¼

8><>:εð0Þx

εð0Þy

gð0Þxy

9>=>;ðiÞ

þ zðiÞ$

8><>:εð1Þx

εð1Þy

gð1Þxy

9>=>;ðiÞ

þhzðiÞi2$

8><>:εð2Þx

εð2Þy

gð2Þxy

9>=>;ðiÞ

þhzðiÞi3$

8><>:εð3Þx

εð3Þy

gð3Þxy

9>=>;ðiÞ

(18)

The vector of transverse shear strains is:

gxzgyz

ðiÞ

¼(gð0Þxz

gð0Þyz

)ðiÞ

þ zðiÞ$

(gð1Þxz

gð1Þyz

)ðiÞ

þhzðiÞi2$

(gð2Þxz

gð2Þyz

)ðiÞ

(19)

By using the constitutive equations (Koll�ar and Springer, 2003;Reddy, 2004) the stress resultants are calculated by integratingthe stresses over the thicknesses of each ESL:

8>>><>>>:Nab

Mab

LabPab

9>>>=>>>;ðiÞ

¼Zti=2

�ti=2

sab

8>>><>>>:1z

z2

z3

9>>>=>>>;ðiÞ

dzðiÞ;

8><>:Qa

RaSa

9>=>;ðiÞ

¼Zti=2

�ti=2

saz

8><>:1z

z2

9>=>;ðiÞ

dzðiÞ

(20)

where a and b takes x or y. The relationship between the strain fieldand the stress resultants is (Szekr�enyes, 2014d):

8>><>>:fNgfMgfLgfPg

9>>=>>;ðiÞ

¼

2664½A� ½B� ½D� ½E�½B� ½D� ½E� ½F�½D� ½E� ½F� ½G�½E� ½F� ½G� ½H�

3775ðiÞ

8>><>>:�εð0Þ��εð1Þ��εð2Þ��εð3Þ�

9>>=>>;ðiÞ

(21)

8<: fQgfRgfSg

9=;ðiÞ

¼24 ½A�� ½B�� ½D��½B�� ½D�� ½E��½D�� ½E�� ½F��

35ðiÞ

8<:�gð0Þ

��gð1Þ

��gð2Þ

�9=;

ðiÞ(22)

where:


1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465

66676869707172737475767778798081828384858687888990919293949596979899

100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130



½::�� ¼ ð:Þ55 0

0 ð:Þ44

�(23)

furthermore: f Ng TðiÞ ¼

�Nx Ny Nxy

�ðiÞ is the vector of in-plane

forces, f Mg TðiÞ ¼

�Mx My Mxy

�ðiÞ is the vector of bending and

twisting moments, f Qg TðiÞ ¼

�Qx Qy

�ðiÞ is the vector of trans-

verse shear forces, and finally f Lg TðiÞ ¼

�Lx Ly Lxy

�ðiÞ ,

f Pg TðiÞ ¼

�Px Py Pxy

�ðiÞ and f Rg T

ðiÞ ¼�

Rx Ry�

ðiÞ,

f Sg TðiÞ ¼

�Sx Sy

�ðiÞ are the vectors of higher-order stress re-

sultants. In Eq. (21) Aij is the extensional, Bij is coupling, Dij is thebending, Eij, Fij, Fij and Hij are higher-order stiffnesses (Szekr�enyes,2014d):

�Aij;Bij;Dij;Eij;Fij;Gij;Hij

�ðiÞ ¼

XNl

k¼1

Zzkþ1

zk

CðkÞij

�1;z;z2;z3;z4;z5;z6

�ðiÞdzðiÞ

(24)

The stiffnesses above have to be calculated with respect to thelocal reference planes of each ESL. The equilibrium equations of theplate system can be obtained by using the virtual work principle(Reddy, 2004). Since this step was shown many times in previouspapers (Szekr�enyes, 2013b, 2014d,b), here only the final results aregiven. We define the following vectors in order to give the equi-librium equations in compact form:

Nðx;xyÞi ¼ �Nx Nxy

�TðiÞ; Nðxy;yÞ

i ¼ �Nxy Ny�TðiÞ

Mðx;xyÞi ¼ �Mx Mxy

�TðiÞ; Mðxy;yÞ

i ¼ �Mxy My�TðiÞ

(25)

The higher-order stress resultants become:

Lðx;xyÞi ¼ � Lx Lxy�TðiÞ; Lðxy;yÞi ¼ � Lxy Ly

�TðiÞ

Pðx;xyÞi ¼ � Px Pxy

�TðiÞ; Pðxy;yÞ

i ¼ � Pxy Py�TðiÞ

(26)

Finally, the vectors of shear and higher-order forces become:

Q i ¼�Qx Qy

�TðiÞ;Ri ¼

�Rx Ry

�TðiÞ; Si ¼

�Sx Sy

�TðiÞ (27)

In the sequel the equilibrium equations are derived separatelyfor the undelaminated and delaminated parts.

5.1. Undelaminated portion

Formulating the total potential energy of the undelaminatedplate portion based on the displacement field satisfying the SEKCrequirements (Eq. (14)) the equilibrium equations can be obtainedby variational calculus (Reddy, 2004). The equilibrium of the in-plane forces involves the equations above independently of theapplied theory (SSDT or TSDT):

du0 :X4i¼1

V$Nðx;xyÞi ¼ 0; dv0 :

X4i¼1

V$Nðxy;xÞi ¼ 0 (28)

where V is the Hamilton differential operator (Chou and Pagano,1967). In the general sense (using SSDT or TSDT) the number ofprimary parameters (ignoring the global membrane displace-ments) in the displacement field is k (which is equal to the numberof elements in j(p)), the first variation of the total potential energyresults in the following equations:

djðxÞk :djðyÞk :

) X4i¼1

Kð0Þik

0@V$Nðx;xyÞi

V$Nðxy;yÞi

1Aþ Kð1Þik

0@V$Mðx;xyÞi

V$Mðxy;yÞi

1Aþ Kð2Þ

ik

0@V$Lðx;xyÞi

V$Lðxy;yÞi

1Aþ Kð3Þik

0@V$Pðx;xyÞi

V$Pðxy;yÞi

1Aþ Kð1Þik

Qix

Qiy

!

� 2Kð2Þik

RixRiy

!� 3Kð3Þ

ik

SixSiy

!¼ 0

(29)

where j(x)k and j(y)k denote the primary parameters. In the TSDTformulation there are four primary parameters (apart from u0 andv0), refer to Section 4.1.1. The variation of the total potential energywith respect to the w(x,y) plate deflection leads to:

dw :X4i¼1

V$Q i � q ¼ 0 (30)

where q is the parameter of the external load. Apparently, thedifferences among the equilibrium equations of SSDT and TSDT arethe matrices defined in Appendices A and B.

5.2. Delaminated portion

The delaminated region consists of a top and a bottom plate.Each subplate is modelled by two ESLs. Therefore, the globalmembrane displacements u0, v0 are replaced by u0b, v0b for ESL1and ESL2, moreover by u0t, v0t for ESL3 and ESL4 in accordance withFigs.1 and 2. Thus, the equilibrium equations of in-plane forces takethe form below:

du0b :X2i¼1

V$Nðx;xyÞi ¼ 0; du0t :

X4i¼3

V$Nðx;xyÞi ¼ 0

dv0b :X2i¼1

V$Nðxy;xÞi ¼ 0; dv0t :

X4i¼3

V$Nðxy;xÞi ¼ 0

(31)

The form of the other equilibrium equations are the same asthose given by Eqs. (29) and (30). The number of primary param-eters in j(p) is three, refer to Section 4.2.1. In the case of the SSDTtheory, the constants denoted by Kð3Þ

ij are zero.

6. Examples e L�evy plates

In this paper we consider laminated orthotropic plates withdifferent (L�evy type) boundary conditions and asymmetricdelamination shown in Fig. 3a and b (with simply supported edges).The plates are loaded by a concentrated force. Some of the prob-lems have been solved in some previous papers (Szekr�enyes, 2013b,2014d,b) by using the method of two ESLs. In accordance with L�evyplate formulation (Bodaghi and Saidi, 2010; Hosseini-Hashemiet al., 2011; Thai and Kim, 2012; Kapuria and Kumari, 2012) theprimary displacement parameters and the external load parameter,q in Eq. (30) are expressed by trial functions:8>><>>:

qxðx; yÞqyðx; yÞfxðx; yÞfyðx; yÞ

9>>=>>; ¼X∞n¼1

8>><>>:XnðxÞsinbyYnðxÞcosbyTxnðxÞsinbyTynðxÞcosby

9>>=>>;;

8>><>>:u0ðx; yÞv0ðx; yÞqðx; yÞwðx; yÞ

9>>=>>;¼X∞n¼1

8>><>>:U0nðxÞsinbyV0nðxÞcosbyQnðxÞsinbyWnðxÞsinby

9>>=>>; (32)

where b ¼ np/b. By taking back the solution in Eq. (32) into theequilibrium equations given by Eqs. (28)e(31) it is possible to


1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465

66676869707172737475767778798081828384858687888990919293949596979899

100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130



reduce the system of PDEs to system of ODEs, which can be solvedby the state-space approach (Jianqiao, 2003). The stress resultantscan be expressed in terms of the displacement parameters throughEqs. (21) and (22). The state-space model takes the form below(Jianqiao, 2003; Reddy, 2004):

Z0 ¼ TZþ F (33)

where Z is the state vector, T is the systemmatrix, F is the vector ofparticular solutions. The general solution of Eq. (33) becomes(Jianqiao, 2003):

ZðxÞ ¼ eTx

0@KþZxx�

e�TxFðxÞdx1A ¼ GðxÞKþHðxÞ (34)

where K is the vector of constants. The parameters of thedisplacement field can be expressed through:

ZðdÞi ¼Xrj¼1

GðdÞij KðdÞ

j þ HðdÞj ; ZðudÞi ¼

Xsj¼1

GðudÞij KðudÞ

j þ HðudÞj (35)

where subscript (d) refers to the delaminated, while (ud) refers theundelaminated plate portion, r ¼ s ¼ 22 are the size of vectors andmatrices, respectively. The state-space model is available in manypapers (Szekr�enyes, 2013a, 2014e,d,c), therefore the vectors andmatrices in Eqs. (33)e(35) are discussed here only briefly.

6.1. Undelaminated region

In the case of both the SSDT and TSDT the state vector containsthe parameters of vector j (refer to Sections 4.1.1e4.1.3), the globalmembrane parameters u0 and v0, the deflection w and the firstderivatives of all these parameters leading to (using the L�evy so-lution): ZT ¼ (U0n, U0

0n, V0n, V 00n, Xin, X0

in, Yin, Y0in, Txin,

T0xin, Tyin, T0yin, Wn, W 0n) for i ¼ 2,4, i.e. the vector Z contains 22 el-

ements, while the system matrix size is 22 � 22.

6.2. Delaminated region

The state vector of both the SSDT and TSDT model of thedelaminated part contains the following elements ZT ¼ (U0bn, U0

0bn,V0bn, V 0

0bn, U0tn, U00tn, V0tn, V 0

0tn, Xin, X0in, Yin, Y

0in, Tx4, T

0x4, Ty4, T 0y4, Wn,

W 0n) for i¼ 2,4, i.e. the size of vector Z is 22, the systemmatrix size is

22 � 22 (similarly to the undelaminated portion).In the next sections the boundary and continuity conditions are

given for the TSDT model, the conditions for the SSDT are the sameby taking Kð3Þ

ij ¼ 0 in all of the equations.

7. Boundary conditions

The B.C.s of the problem in Fig. 3a at x ¼ a in the delaminatedportion are determined through the displacement parameters:�w; v0b; v0t ; qy2; qy4;fy4

�ð1aÞ��x¼a ¼ 0 (36)

and the stress resultants:

ðNx1 þ Nx2;Nx3 þ Nx4;Mx1 þMx2;Mx3 þMx4; Lx4Þð1aÞ��x¼a ¼ 0

(37)

At x ¼ �c we have:�w; v0; qy2; qy4;fy2;fy4

�ð2Þ��x¼�c ¼ 0 (38)

and:

ðMx1þMx2;Mx3þMx4;Lx2;Lx4;Nx1þNx2þNx3þNx4Þð2Þ��x¼�c ¼ 0

(39)

Fig. 3. Simply supported delaminated composite plates subjected to a concentrated force.


1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465

66676869707172737475767778798081828384858687888990919293949596979899

100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130



8. Continuity conditions between regions (1) and (2)

The conditions between regions (1) and (2) (refer to Fig. 3a)involves the continuity of the displacement parameters and stressresultants. In the sequel, these conditions are discussed separately.

8.1. Continuity of stress resultants

To define the continuity conditions the equivalent stress re-sultants can be defined based on the equilibrium equations (Eqs.(28) and (29)) and (31) and the vectors given by Eqs. (25) and (26):

cMðx;xyÞð1Þ12 ¼

Xj¼1;2

��Kð0Þj1 þ Kð0Þ

j2

�Nðx;xyÞj þ

�Kð1Þj1 þ Kð1Þ

j2

�Mðx;xyÞ

j

þ�Kð2Þj1 þ Kð2Þ

j2

�Lðx;xyÞj þ


j2

�Pðx;xyÞj

�ð1Þ(40)


Xj¼3;4


j2

�Nðx;xyÞj þ


j2

�Mðx;xyÞ

j


j2

�Lðx;xyÞj þ


j2

�Pðx;xyÞj

�ð1Þ(41)

bLðx;xyÞð1Þ1234 ¼

Xj¼1::4

�Kð0Þj3 Nðx;xyÞ

j þ Kð1Þj3 Mðx;xyÞð1Þ

j þ Kð2Þj3 Lðx;xyÞj

þ Kð3Þj3 Pðx;xyÞ

j

�ð1Þ(42)

for the delaminated portion (1), wherecM12 andcM34 are the vectorsof equivalent bending/twisting moments, bL1234 is the vector ofequivalent higher-order stress resultants. Moreover, for the unde-laminated region (2) we have:


Xj¼1;2


j3

�Nðx;xyÞj þ


j3

�Mðx;xyÞ

j


j3

�Lðx;xyÞj þ


j3

�Pðx;xyÞj

�ð2Þ(43)


Xj¼3;4


j3

�Nðx;xyÞj þ


j3

�Mðx;xyÞ

j


j3

�Lðx;xyÞj þ


j3

�Pðx;xyÞj

�ð2Þ(44)


Xj¼1;2


j4

�Nðx;xyÞj þ


j4

�Mðx;xyÞ

j


j4

�Lðx;xyÞj þ


j4

�Pðx;xyÞj

�ð2Þ(45)


Xj¼3;4


j4

�Nðx;xyÞj þ


j4

�Mðx;xyÞ

j


j4

�Lðx;xyÞj þ


j4

�Pðx;xyÞj

�ð2Þ(46)

The continuity conditions using the equivalent stress resultantsare: cMðx;xyÞ

12 ;cMðx;xyÞ34 ; bLðx;xyÞ1234 ;

Xi¼1::4

Nðx;xyÞi

!��ð1Þ

x¼�0

¼ cMðx;xyÞ

12 ;cMðx;xyÞ34 ; bLðx;xyÞ

12 þ bLðx;xyÞ34 ;

Xi¼1::4

Nðx;xyÞi

!��ð2Þ

x¼þ0

(47)

8.2. Continuity of displacement parameters

In the case of the general TSDT the continuity of the in-planedisplacement is ensured only if the constant, linear, quadratic andcubic terms are exactly the same in the delamination front (x ¼ 0).Because of the parameter elimination based on the SEKC it is notpossible to match directly the constant, quadratic and cubic termsin the displacement function from layer by layer. Only the conti-nuity of primary parameters can be defined between each ESL. Inspite of that the continuity of the remaining membrane, linear,quadratic and cubic terms can be ensured indirectly (automatically)if certain conditions are met. The requirements of automatic con-tinuity is formulated in the form of a theorem. We define thefollowing set of parameters:

ga ¼�wn;w

0n; qx2; qy2; qx4; qy4;fx4;fy4

�(48)

which contains the deflection, slope and the mutual primary pa-rameters. The total continuity of the displacement functions in-volves the following necessary conditions:

gð1Þa

��x¼þ0 ¼ gð2Þa

��x¼�0 (49)

However these are not sufficient. The sufficient conditions arepresented through a theorem.

8.2.1. The theorem of autocontinuity (AC theorem)Theorem: If the displacement field in the form of Eq. (14) in a

laminated plate with delamination is developed by using the SEKCrequirements and Nd2ℕ and Nud2ℕ are the numbers of eliminatedparameters in the delaminated and undelaminated parts, respec-tively, and Nd s Nud, then the total continuity of the first-, second-and third-order terms in the in-plane displacement functions of eachESL in the delaminated and undelaminated plate parts e apart fromthose imposed by Eq. (48) (mutual primary parameters) e can beensured by imposing the continuity of jNd � Nudj2ℕ number ofparameters. These parameters are the autocontinuity (or simply AC)parameters, which are at the same time primary parameters too. Theautocontinuity is satisfied only if along interfaces 1e2 and 3e4 andat the plate boundaries (Figs. 1 and 2) the same conditions areimposed in the delaminated and undelaminated portions. Along thedelamination plane (interface 2e3) different conditions can beapplied. Fig. 4a shows the casewhen the autocontinuity between thedelaminated and undelaminated parts is satisfied, Fig. 4b indicates acase when dissimilar conditions are imposed at interface 1e2 lead-ing to discontinuous displacement field in the bottom plates.

Proof: In the case of the TSDT model Nd ¼ 26, Nud ¼ 24 (refer toSections 4.1.1 and 4.2.1), thus the number of AC parameters isjNd � Nudj ¼ 2. The AC parameters can be assigned based on thevector of primary parameters: the comparison of the j(p) vectors inSections 4.1.1 and 4.2.1 reveals that the AC parameters are fx2 andfy2 in the undelaminated region (2). The comparison of thedisplacement field (Eq. (14)) for the undelaminated and delami-nated regions (Appendix A) reveals the following sufficientconditions:


1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465

66676869707172737475767778798081828384858687888990919293949596979899

100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130



fp2

��ð2Þx¼�0

¼Xj¼1::3

Kð2Þ2j jðpÞj

��ð1Þx¼þ0

; p ¼ x; y (50)

The former conditions ensure the total continuity of thedisplacement functions, i.e., the linear, quadratic and cubic terms inthe displacement fields of regions (1) and (2) at x ¼ 0 (Fig. 3a) willbe continuous apart from those imposed by Eq. (49). Consideringthe fact that the parameters in gb are continuous between regions(1) and (2) and by using the matrix elements given in Appendix A(TSDT) it is possible to have the following expression for fp2 atx ¼ �0:

fp2

��ð2Þx¼�0

¼ t1

2t3�qp4 � qp2

�þ fp4t4ð2t3 þ t4Þð2t1 þ t2Þt2t3

!��ð1Þ

x¼þ0

(51)

Taking the former condition back into the displacement func-tions given by Eq. (14) of each ESL of the delaminated part (1) yieldsthe following at x ¼ þ0:Xj¼1::3

�Kð1Þ1j jðpÞj

��ð1Þx¼þ0

¼�

1t2ð2t1 þ t2Þ

t1ð2t1 þ t2Þqp2 � ðt1 � t2Þ

� ðt1 þ t2Þqp4 �t42t3

ðt1 � t2Þðt1 þ t2Þ

� ð2t3 þ t4Þfp4

��ð1Þx¼þ0

(52)

Xj¼1::3

�Kð1Þ3j jðpÞj

��ð1Þx¼þ0

¼ �12fp4

t23 � t24t3

� qp4

��ð1Þx¼þ0

(53)

Xj¼1::3

�Kð2Þ1j jðpÞj

��ð1Þx¼þ0

¼ �2�qp2 þ qp4

�ð2t1 þ t2Þ

� 2t4ðt3 þ t4Þt3ð2t1 þ t2Þ

fp4

��ð1Þx¼þ0

(54)

Xj¼1::3

�Kð2Þ3j jðpÞj

��ð1Þx¼þ0

¼ �t4t3fp4

��ð1Þx¼þ0

(55)

Xj¼1::3

�Kð3Þ1j jðpÞj

��ð1Þx¼þ0

¼Xj¼1::3

�Kð3Þ2j jðpÞj

��ð1Þx¼þ0

¼ �4�qp2 � qp4

�3ð2t1 þ t2Þ

þ 23

t4ð2t3 þ t4Þt2t3ð2t1 þ t2Þ

fp4

��ð1Þx¼þ0

(56)

Xj¼1::3

�Kð3Þ3j jðpÞj

��ð1Þx¼þ0

¼Xj¼1::3

�Kð3Þ4j jðpÞj

��ð1Þx¼þ0

¼ 23t3

fp4

��ð1Þx¼þ0

(57)

At the same time, by taking back Eq. (51) into the displacementfunctions of every ESLs of the undelaminated part (2) defined by Eq.(14) we have at x ¼ �0:

Fig. 4. Illustration of the theorem of autocontinuity: similar (a) and dissimilar (b) conditions are imposed at interface planes 1e2 and 3e4 of the delaminated and undelaminatedparts.


1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465

66676869707172737475767778798081828384858687888990919293949596979899

100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130



Xj¼1::4

�Kð1Þ1j jðpÞj

��ð2Þx¼�0

¼�

1t2ð2t1 þ t2Þ

t1ð2t1 þ t2Þqp2 � ðt1 � t2Þ

� ðt1 þ t2Þqp4 �t42t3

ðt1 � t2Þðt1 þ t2Þ

� ð2t3 þ t4Þfp4

��ð2Þx¼�0

(58)

Xj¼1::4

�Kð1Þ3j jðpÞj

��ð2Þx¼�0

¼ �12fp4

t23 � t24t3

� qp4

��ð2Þx¼�0

(59)

Xj¼1::4

�Kð2Þ1j jðpÞj

��ð2Þx¼�0

¼ �2�qp2 þ qp4

�ð2t1 þ t2Þ

� 2t4ðt3 þ t4Þt3ð2t1 þ t2Þ

fp4

��ð2Þx¼�0

(60)

Xj¼1::4

�Kð2Þ3j jðpÞj

��ð2Þx¼�0

¼ �t4t3fp4

��ð2Þx¼�0

(61)

Xj¼1::4

�Kð3Þ1j jðpÞj

��ð2Þx¼�0

¼Xj¼1::3

�Kð3Þ2j j

�ðpÞj��ð2Þ

x¼�0

¼ �4�qp2 � qp4

�3ð2t1 þ t2Þ

þ 23


fp4

��ð2Þx¼�0

(62)

The left-hand sides of Eqs. (52)e(62) define the values of q(p)1,q(p)3, f(p)1, f(p)3, l(p)1, l(p)2, l(p)3 and l(p)4, i.e. the secondary param-eters. Obviously, The right-hand sides of Eqs. (52)e(57) and Eqs.(58)e(62) are the same in pairs, and more importantly determinedin terms of the primary parameters. Considering the continuity ofthe parameters in Eq. (48) by Eq. (49) it can be seen that the con-tinuity of the linear, quadratic and cubic terms in the displacementfunctions of regions (1) and (2) are automatically satisfied if Eq. (51)is utilized.

Consequence: If the continuity of linear terms (q rotations) in thedisplacement field in each ESL are continuous, moreover the con-tinuity of quadratic and cubic terms of each ESL are imposed usingthe AC parameters, then the continuity of the membrane displace-ment components between the top plates (as well as the bottomplates) of the delaminated and undelaminated regions can beensured by imposing the equality between the membrane (con-stant) displacement terms of only a single ESL in the delaminatedpart and a single one in the undelaminated part, but not in everyESLs. The ESLs can be chosen optionally, however the chosen ESLsshould be in the same through-thickness position in the delami-nated and undelaminated plate regions. In this case the continuityof the membrane parts in the other ESLs is satisfied automatically.We choose the first (in the bottom layer) and third (in top layer)ESLs to impose the continuity of the membrane displacementsusing the equations below:�u0bv0b

�þ P

j¼1::3Kð0Þ1j

�jðxÞjjðyÞj

��ð1Þx¼þ0

¼�u0v0

�þXj¼1::4

Kð0Þ1j

�jðxÞjjðyÞj

��ð2Þx¼�0�

u0tv0t

� Pj¼1::3

Kð0Þ3j

�jðxÞjjðyÞj

��ð1Þx¼þ0

¼�u0v0

�þXj¼1::4

Kð0Þ3j

�jðxÞjjðyÞj

��ð2Þx¼�0

(63)

The autocontinuity theorem is also valid for the SSDT. The onlydifference is that the Kð3Þ

ij constants are zero.

9. Continuity between regions (1)-(1q) and (1q)-(1a)

The continuity between regions (1)-(1q) and (1q)-(1a) (seeFig. 3a) can be imposed by defining the sets of parameters below:

gb ¼�u0b;u0t ; v0b; v0t ;wn;w

0n; qx2; qy2; qx4; qy4;fx4;fy4

�(64)

gg ¼ cMðx;xyÞ

12 ;cMðx;xyÞ34 ; bLðx;xyÞ1234 ;

Xi¼1::2

Nðx;xyÞi ;

Xi¼3::4

Nðx;xyÞi

!(65)

The continuity conditions are (refer to Fig. 3a):

gð1Þb

��x¼x0�d0

¼ gð1qÞb

��x¼x0�d0

; gð1Þg

��x¼x0�d0

¼ gð1qÞg

��x¼x0�d0

(66)

gð1qÞb

��x¼x0þd0

¼ gð1aÞb

��x¼x0þd0

; gð1qÞg

��x¼x0þd0

¼ gð1aÞg

��x¼x0þd0

(67)

The summary of the equations results in: Eqs. (36)e(39) mean22 B.C.s, Eqs. (49), (51) and (63) yield 22 conditions between re-gions (1) and (2). Eqs. (66) and (67) provide 2 � 22 conditions. Thatmeans 22 þ 22þ22 þ 22 ¼ 88 conditions altogether in the case ofproblem a in Fig. 3. Problem b in Fig. 3 can be solved similarly,therefore the details are not given. The B.C.s and the C.C.s for theSSDT model can be defined similarly, in these cases the constantsKð3Þij ¼ 0 in Eq. (14) and in the stress resultants.It has to be mentioned that although mathematically it is

possible to impose traction-free conditions to the plate boundaries(similarly to Reddy theory), but this makes the model over-con-strained. Over-constraining means, that mechanically the modelbecomes too stiff and the bad estimation of the stress field in thetransition zone between (1) and (2) takes place. The reason for thebad description of the mechanical problem is that there are notenough conditions to impose the continuity of the primary pa-rameters, the membrane displacements and the correspondingstress resultants. In this respect the models proposed in this paperare well-constrained models.

10. J-integral

The J-integral (Rice, 1968; Cherepanov, 1997) for SSDT and TSDThas already been derived for plates with symmetric (Szekr�enyes,2013a, 2014e, 2014a, 2014c) and asymmetric (Szekr�enyes, 2013b,2014d, 2014b) lay-up. Therefore, in this work the details are notdiscussed. The mode-II and mode-III J-integrals (JII ¼ GII, JIII ¼ GIII)are given in Appendix C.

11. Results and discussions

To demonstrate the performance of the analytical models theexamples shown in Fig. 3 are solved. The data of the problemdepicted in Fig. 3a are: a ¼ 105 mm (delamination length),c ¼ 55 mm (undelaminated length), b ¼ 160 mm (plate width),tt þ tb ¼ 4.5 mm (plate thickness), Q0 ¼ 1000 N, xQ ¼ 31 mm,yQ ¼ 50 mm and yQ ¼ 80 mm (point of action coordinates of Q0),d0 ¼ 0.1 mm. For the problem in Fig. 3b the data are: a ¼ 55 mm(delamination length), c ¼ 35 mm (undelaminated length),b ¼ 90 mm (plate width), tt þ tb ¼ 4.5 mm (plate thickness),Q0 ¼ 10000 N, xQ ¼ 11 mm, yQ ¼ 30 mm and yQ ¼ 45 mm (point ofaction coordinates of Q0), d0 ¼ 0.1 mm. The problem was solved byreplacing the concentrated load by distributed force on the distance2d0. The details can be found in (Szekr�enyes, 2014d). Thematerial ofthe plates is carbon/epoxy. The lay-up of the plate is½±45f =0=±45f2=0�S, the material properties can be found in(Szekr�enyes, 2014d). Finite element models were also constructed


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to validate the analytical results. The details of the FE models arepresented in a recent paper (Szekr�enyes, 2014d). The delaminationtip elements and the global mesh resolution were chosen inaccordance with recommendations of the literature (Raju et al.,1988; Johnston et al., 2014; Mehrabadi, 2014) The position of thedelamination was varied in the through thickness direction, thesewere assigned as cases I, II, III and IV (Szekr�enyes, 2014d). Theproblems are solved for some selected cases and different boundaryconditions.

11.1. Displacement and stress distributions

In this section the solution of the problems in Fig. 3 are dis-cussed including cases II and IV (refer to Szekr�enyes, 2013b, 2014d).The in-plane displacements and the normal stresses are evaluatedat cross sections on the delamination front. Apart from the present4ESL solutions a previously published 2ESL model based on Reddythird-order theory (Szekr�enyes, 2014d) is used to assess the per-formance of the shear strain controlled SSDT and TSDT. Fig. 5 showsthe distributions of the in-plane displacements (u and v) and thenormal stresses (sx and sy) for problem a in Fig. 3 in case II. In thecase of the stresses the average of the values at x ¼ �0 (undela-minated) and x ¼ þ0 (delaminated) part are calculated.

The three different solutions agree well in this case for u, vand sy. Considering sx the SSDT seems to agree very well with

the FE solution, while Reddy theory and TSDT are a little bitworst. The shear stresses are plotted in Fig. 6. For txz the threesolutions are reasonable and approximate well the FE solution.On the contrary, tyz is badly estimated by the TSDT, at the sametime the SSDT and Reddy theory are still good. Because of thefact that the SSDT solution is based on the triangle shape shearstrain distribution, this is also called as “triangle” solution in thesequel.

Case IV is documented in Fig. 7. In the case of the in-planedisplacement u, the TSDT indicates some oscillations in the toppart of the plate. The curve obtained by Reddy theory shows a littlebit higher slope than that of the FE solution. Apparently the straincontrolled SSDT agrees the best with the FE results. These conclu-sions are equally true for the v displacement and the normalstresses: the SSDT seems to be the most accurate solution.

The shear stress distributions evaluated over the thickness atthe delamination front are shown in Fig. 8. It is surprising that theTSDT approximates badly the distribution of shear stresses,although it is also clear that in the plate theories the shear force isequivalent to the area under the distribution. In this respect theresult is correct, but the control condition leads to too large oscil-lations in the shear stress. In contrast, the SSDT (piecewise linearshear strain approximation) approximates much better the set ofpoints from FE analysis and the piecewise second-order distribu-tion by Reddy theory for both components.

Fig. 5. Distribution of the in-plane displacements (u and v) and normal stresses (sx and sy) over the plate thickness for problem a in Fig. 3, case II, b ¼ 160 mm. Note: the Reddysolution is available in (Szekr�enyes, 2014d).


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100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130



Fig. 6. Distribution shear stresses (txz and tyz) over the plate thickness for problem a in Fig. 3, case II, b ¼ 160 mm. Note: the Reddy solution is available in (Szekr�enyes, 2014d).

Fig. 7. Distribution of the in-plane displacements (u and v) and normal stresses (sx and sy) over the plate thickness for problem a in Fig. 3, case IV, b ¼ 160 mm. Note: the Reddysolution is available in (Szekr�enyes, 2014d).


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Fig. 8. Distribution shear stresses (txz and tyz) over the plate thickness for problem a in Fig. 3, case IV, b ¼ 160 mm. Note: the Reddy solution is available in (Szekr�enyes, 2014d).

Fig. 9. Distribution of the in-plane displacements (u and v) and normal stresses (sx and sy) over the plate thickness for problem b in Fig. 3, case II, b ¼ 90 mm. Note: the Reddysolution is available in (Szekr�enyes, 2014d).


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The results of problem b (Fig. 3b) are documented in Figs. 9e12.This example involves smaller plate dimensions and since the pointof action of the force is closer to the delamination front, higherinfluence of the load on the fields is experienced. In case II (Fig. 9) it

is apparent that the displacement component u is perturbatedmore intensively than in problem a. The highest oscillations areprovided by the TSDT, while the SSDT and Reddy theory give betteragreement with the FE results. The stresses are approximated with

Fig. 10. Distribution of the shear stresses (txz and tyz) over the plate thickness for problem b in Fig. 3, case II, b ¼ 90 mm. Note: the Reddy solution is available in (Szekr�enyes, 2014d).

Fig. 11. Distribution of the in-plane displacements (u and v) and normal stresses (sx and sy) over the plate thickness for problem b in Fig. 3, case IV, b ¼ 90 mm. Note: the Reddysolution is available in (Szekr�enyes, 2014d).


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66676869707172737475767778798081828384858687888990919293949596979899

100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130



similar accuracy by each theory. In Fig. 10 the shear stresses areplotted. The SSDT and Reddy theory is better again than the TSDTfor txz. By each solution the shear stress tyz disagrees with the FEresults.

Fig. 11 presents the results of case IV in problem b. Based on thecomparison of analytical results to those by FE computations, forboth the in-plane displacement components and the stresses theSSDT performs better than the strain controlled TSDT and Reddytheory (Fig. 11). A similar behavior of TSDT is observed in the shearstress distributions in Fig. 12 to that by Fig. 8. On the contrary theSSDT and Reddy theory lead to more correct results.

To further assess the accuracy of the different theories the shearstresses are also evaluated behind and ahead of the delaminationfront. In Fig. 13 problem a, case IV is evaluated in this respectshowing that the shear strain controlled TSDT and SSDT theoriesapproximate very well the FE distributions in both shear stresscomponents, while the Reddy theory underestimates txz, but fol-lows well tyz at the chosen locations. Fig. 14 demonstrates the shearstresses obtained from the solution of problem b in Fig. 3 for case IV.

This time the difference between the three different solutions isvery small and the agreement with the FE results is established. InTable 1 the displacements at the point of action of Q0 are collectedby SSDT, TSDT, Reddy theory and FE solution for case IV. While theSSDT and TSDT agree excellently with the FE results, the modelbased on Reddy theory seems to be too stiff in case IV. This can beexplained by the imposition of traction-free boundaries. For theother cases similar behavior was experienced.

12. J-integral and mode ratio

The J-integral is equivalent to the ERR under quasi-static con-ditions and linear elastic material behavior. The solution of problema is shown in Fig. 15 in case II. In fact each theory provides goodresults compared to the numerical solution of the ERRs and themode ratio. The solution of case IV is plotted in Fig. 16. Reddy third-order theory approximates excellently the mode-III ERR (Fig. 16),however, the mode-II ERR and so the mode ratio differs signifi-cantly from the VCCT results. The TSDT solution is moderately

Fig. 12. Distribution of the shear stresses (txz and tyz) over the plate thickness for problem b in Fig. 3, case IV, b ¼ 90 mm. Note: the Reddy solution is available in (Szekr�enyes,2014d).



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100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130



coarse for GIII and extremely wrong for GII in the middle of the plate(y ¼ 60…100 mm). On the contrary, the triangle solution (SSDT)provides the best overall agreement with the VCCT results. In spiteof that it has to be mentioned the in the middle part of thedelamination (y ¼ 60…100 mm) a very small overprediction of theFE results is observed. The mode ratio is again accurately predictedby the controlled SSDT. For problem b, case II the results are plotted

in Fig. 17. The Reddy theory is the best this time, however the SSDTis also good in the estimation. Problem b in case IV is documentedin Fig. 18. As a summary, neither the presented theories is able toaccurately follow the FE results, however the best choice is theSSDT among the models presented. Especially the overprediction ofthe mode II ERR by Reddy theory in problem a, case IV is high-lighted. It should be noted that problem b involves smaller platedimensions and the stress concentration effect in the FE model ismore significant in this case than in problem a. This could be one ofthe reasons for the bad agreement between FEM and analysis.

Some more examples were solved under different boundaryconditions in Appendix D. Fig. D.20 depicts the ERR and mode ratiodistributions for problem b, case I in Fig. 3 with simply supportededge at x ¼ �c and free edge at x ¼ a (SF). Apparently the threemethods predict the same results in this case, with underprediction

Table 1Comparison of the displacement at the point of action of Q0 from the differenttheories for case IV.

Deflection at x ¼ x0 b [mm] SSDT TSDT Reddy FEM

w0 [mm] 160 (problem a) �1.379 �1.377 �1.234 �1.423w0 [mm] 90 (problem b) �4.519 �4.519 �3.768 �4.699


Fig. 15. Distribution of the energy release rates and mode mixity along the delamination front for problem a in Fig. 3, case II. Note: the Reddy solution is available in (Szekr�enyes,2014d).


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of the mode-III component. In Fig. D.21 the same problem is solvedin case III, which shows that Reddy theory is very good, the SSDT isacceptable, while the TSDT is the worst. In spite of that the moderatios approximate reasonably the FE results. Fig. D.23 was createdbased on problem a in Fig. 3 with clampedeclamped (CC) condi-tions at x ¼ �c and x ¼ a. Although in case II Reddy theory is thebest again, in case IV plotted in Fig. D.24 it is the SSDT that providesthe best overall performance. It should be mentioned that eachsolution provides negative mode-II ERR in the middle of the plate,however this is very moderate in the SSDT solution but significantfor the other two ones.

To investigate the behavior of the system under crack propa-gation the normalized distributions of the ERRs by SSDT withdifferent crack lengths are displayed in Fig. 19 for case IV andb ¼ 100 mm. Each edge was simply supported. The point of actionof the force was in themiddle of the plate for each crack length. Thelength of the plate was a þ c ¼ 150 mm in each model. It has to bementioned that the crack propagation requires some experimentaldata, and the delamination front does not remain straight undercrack propagation, which was assumed in these computations.Fig. 19 shows that the self-similarity (Anderson, 2005) condition is

Fig. 16. Distribution of the energy release rates and mode mixity along the delamination front for problem a in Fig. 3, case IV. Note: the Reddy solution is available in (Szekr�enyes,2014d).

Fig. 17. Distribution of the energy release rates and mode mixity along the delamination front for problem b in Fig. 3, case II. Note: the Reddy solution is available in (Szekr�enyes,2014d).


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violated significantly under mixed-mode II/III case, viz. the distri-bution is not the same if the delamination length increases.

13. Conclusions

The method of four equivalent single layers is presented in thispaper for the modelling of delaminated orthotropic compositeplates. The in-plane displacement functions were captured by thesecond- and third-order plate theories. The problem of a plate withstraight delamination front was considered, each region wascaptured by four equivalent single layers. The kinematic continuitybetween the equivalent single layers was established by the systemof exact kinematic conditions. It is important to note that the set ofconditions was complemented with the continuity of the derivativeand the curvature of the shear strains between the adjacent layersof third-order plates. A so-called shear strain control condition was

also introduced for second- and third-order plate theory. With theaid of the kinematic conditions the number of parameters in thedisplacement functions was reduced significantly. The strain andstress fieldswere derived using the basic equations of elasticity. Theequilibrium equations of the delaminated and undelaminated partswere derived based on variational calculus. To exemplify thedeveloped models layered plates with different boundary condi-tions were considered with concentrated load. The problems weresolved by the state-space approach.

An important contribution of this paper compared to similarprevious developments is the introduction of the theorem ofautocontinuity (AC theorem) for second- and third-order plates.Because of the parameter elimination the number of parameters inthe displacement field that continuity is required against is higherthan the available constants in the state-space model. Thus, thereare no free constants for the matching of certain first-, second- and

Fig. 18. Distribution of the energy release rates and mode mixity along the delamination front for problem b in Fig. 3, case IV. Note: the Reddy solution is available in (Szekr�enyes,2014d).

Fig. 19. Normalized energy release rate distributions along the delamination front for different crack lengths in case IV using SSDT (b ¼ 100 mm).


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third-order displacement parameters. However, in accordance withthe AC theorem the continuity of these parameters can be achievedby imposing the continuity of certain autocontinuity (AC) param-eters. An important requirement is that the autocontinuity issatisfied only if in each interface plane between the adjacentequivalent single layers and on the plate boundaries the sameconditions are imposed in the delaminated and undelaminatedplate regions, except for the plane of delamination, where differentconditions are necessary to be specified because of the presence ofthe delamination front. Using the proposed conditions and the ACtheorem the examples were solved and the mechanical fields werecompared to the results of 3D finite element calculations. A previ-ous solution based on Reddy third-order theory was also used forcomparison purposes.

Several cases were investigated with respect to the position ofthe delamination in the through the thickness direction, and itwas shown that in problem a the second-order plate theoryprovides the highest accuracy compared to the finite elementcalculations. It was elaborated that the shear strain control con-dition leads to large oscillations in the third-order plate theorysolution. In problem b the more intense perturbation because ofthe load and geometry led to the fact that the analytical modelsgave inaccurate results. However, among the theories applied thesecond-order plate theory with shear strain control condition wasfound to be the closest to the finite element calculations in mostof the cases. The updated system of exact kinematic conditionsand the theorem of autocontinuity work well and can be imple-mented into more accurate plate and also beam and shell the-ories. Also, it was discussed that the dynamic boundary conditionleads to a slight stiffening in the case of the model based onReddy third-order theory. However, in the method of fourequivalent single layers these conditions make the model over-constrained leading to continuity problems between thedelaminated and undelaminated portions, and thus these werenot considered in this work.

The next step could be the application of the system of exactkinematic conditions, the AC theorem and the shear strain controlconditions to models built by six equivalent single layers, or a fulllayerwise model, where the system parameters can be reducedsignificantly to obtain state-space models with the same di-mensions as those of the method of four layers.

Acknowledgments

This work was supported by the J�anos Bolyai Research Schol-arship of the Hungarian Academy of Sciences and the HungarianNational Scientific Research Fund (OTKA) under grant No. 44615-066-15 (108414).

Appendix A. TSDT constants

In this Appendix the constants in Eq. (14) denoted by Kð0Þij , Kð1Þ

ij ,

Kð2Þij and Kð3Þ

ij are collected.

Appendix A.1. Undelaminated region

In accordance with Section 4.1.1 the following constants can beobtained:

Kð0Þ11 ¼ a2

a1

�a1z

ð2ÞR þ a2

�zð2ÞR

�2 þ a3t21 þ a4t

22 þ a5

�(A.1)

a1 ¼ �13t3ðt3 þ t4Þðt1 þ t2Þ; a2 ¼ 2

3t3ðt3 þ t4Þ;

a3 ¼ �116

t3ðt3 þ t4Þ �12t24 ; a4 ¼ �1

3t3ðt3 þ t4Þ;

a5 ¼ �53t1t2t3ðt3 þ t4Þ �

12t1t2t

24

(A.2)

Kð0Þ12 ¼ a2

a1

�b1�zð2ÞR

�2 þ b2t21 þ b3t

22 þ b4t

31 þ b5t

32 þ b6

�(A.3)

b1 ¼ �23t2t3ðt3 þ t4Þ �

13t24ðt1 þ t2Þ �

43t1t3ðt3 þ t4Þ;

b2 ¼ �13zð2ÞR

�4t23 þ 4t3t4 þ t24

�þ 16t2

�7t23 þ 7t3t4 þ

32t24

�(A.4)

b3 ¼ �16zð2ÞR

�t23 þ t3t4 þ t24

�þ 712

t1t3ðt3 þ t4Þ; b4 ¼ 16ð2t3 þ t4Þ2

b5 ¼ 112

�t23 þ t3t4 � t24

�;b6 ¼ �zð2ÞR t1t2

�t23 þ t3t4 þ

16t24

�(A.5)

Kð0Þ13 ¼ � a2

6a1t3ðt3 þ t4Þ

�ðt1 þ t2Þ2 � 2zð2ÞR

�t1 þ t2 � 2zð2ÞR

��(A.6)

Kð0Þ14 ¼ � a2

12a1t4ðt3 þ t4Þð2t3 þ t4Þ

�ðt1 þ t2Þ2 � 2zð2ÞR

��t1 þ t2 � 2zð2ÞR

�� (A.7)

Kð0Þ21 ¼ 1

a1

�c1�zð2ÞR

�3 þ c2t21 þ c3t

22 þ c4

�(A.8)

c1 ¼ 43t3ðt3 þ t4Þ; c2 ¼ �zð2ÞR ð2t3 þ t4Þ2; c3 ¼ �zð2ÞR t3ðt3 þ t4Þ

c4 ¼ �zð2ÞR t1t2�4t23 þ 4t3t4 þ t24

�(A.9)

Kð0Þ22 ¼ 1

a1

�d1�zð2ÞR

�2 þ d2t21 þ d3t

22 � d4

�(A.10)

d1 ¼ �43t23ð2t1 þ t2Þ �

23t24ðt1 þ t2Þ �

43t3t4ðt1 þ t2Þ;

d2 ¼ ��zð2ÞR

�2ðt4 þ 2t3Þ2; d3 ¼ ��zð2ÞR

�2t3ðt3 þ t4Þ;

d4 ¼ �zð2ÞR t1t2�4t3t4 þ 4t23 þ t24

� (A.11)

Kð0Þ23 ¼ � 4

3a1

�zð2ÞR

�3t3ðt3 þ t4Þ;Kð0Þ

24

¼ � 23a1

�zð2ÞR

�3t4ðt3 þ t4Þð2t3 þ t4Þ (A.12)

Kð0Þ31 ¼ 1

a1

�e1z

ð2ÞR þ e2

�zð2ÞR

�3 þ e3t21 þ e4t

22 þ e5t

32 þ e6

�(A.13)


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66676869707172737475767778798081828384858687888990919293949596979899

100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130



e1 ¼ a1; e2 ¼ 43t3ðt3 þ t4Þ;

e3 ¼ �zð2ÞR

�4t23 þ 4t3t4 þ t24

�þ t23

�76t3 þ

32t4 þ 2t2

�þ t4

�12t3t4 þ

12t2t4 þ 2t2t3

� (A.14)

e4 ¼ �zð2ÞR

�t23 þ t3t4

�þ t1

�12t24 þ 2t3t4 þ 2t23

�;

e5 ¼ 13t3ðt3 þ t4Þ;

e6 ¼ zð2ÞR t21�4t23 þ 4t3t4 þ t24

�þ zð2ÞR t22t3ðt3 þ t4Þ

þ t3t1

�32t3t4t2 þ

12t24t2 þ

76t3t2

�(A.15)

Kð0Þ32 ¼ 1

a1

�g1z

ð2ÞR þ g2

�zð2ÞR

�3 þ g3t22 þ g4t

21 þ g5t

33 þ g6t

42 þ g7

�(A.16)

g1 ¼ a1; g2 ¼ �43t2t3ðt3 þ t4Þ �

83t1t3ðt3 þ t4Þ �

23t24ðt2 þ t3Þ

g3 ¼ 14t3t4ð3t3 þ t4Þ þ

43t1t3ðt3 þ t4Þ þ

13t1t

24 þ 7

12t33 ;

g4 ¼ ��zð2ÞR

�2�4t23 þ 4t3t4 þ t34

�þ t2t3t4

�32t3 þ

12t4 þ t2

�þt2

�14t2t

24 þ t2t

23 þ 7

6t33

�(A.17)

g5 ¼ 712

t2ðt1 þ t2Þð2t1 þ t2Þ; g6 ¼ 112

�5t23 þ 5t3t4 þ t24

�g7 ¼

�zð2ÞR

�2t21�4t23 þ 4t3t4 þ t34

�þ t1t2t3t4

�34t2t4 þ

94t2t3

�þ76t2t

33

�t21 þ t22

�(A.18)

Kð0Þ33 ¼ 1

a1

�h1�zð2ÞR

�3 þ h2t32 þ h3t

21 þ h4t

33 þ h5

�(A.19)

h1 ¼ �43t3ðt3 þ t4Þ; h2 ¼ 1

6t3ðt3 þ t4Þ;h3 ¼ 1

6t23ð5t3 þ 3t4Þ;

h4 ¼ 56t21 þ 1

2t22 þ 5

6t1t2; h5 ¼ 1

2t23

�� 53t3t

21 þ t22t4 þ t1t2t4

�(A.20)

Kð0Þ34 ¼ 1

a1

�i1�zð2ÞR

�3 þ i2t32 þ i3t

21 þ i4t

33 þ i5t

23 þ i6

�(A.21)

i1 ¼�23t4ðt3 þ t4Þð2t3 þ t4Þ; i2 ¼

112

t4ðt3 þ t4Þð2t3 þ t4Þ;

i3 ¼� 112

t23ð4t3 þ3t4Þð2t3 þ t4Þ;

i4 ¼� 112

t4�10t21 � t22 þ10t1t2

� (A.22)

i5 ¼ � 112

t3ðt3 þ t4Þ�� 3t24 � 3t3t4 þ 2t23

�i6 ¼ �t23t1t2

�14t24 þ 2

3t23

�þ t23t4

�56t21 � 1

12t22

� (A.23)

Kð0Þ41 ¼ 1

a1

�j1�zð2ÞR

�3 þ j2t32 þ j3t

21 þ j4t

33 þ j5t

22 þ j6

�(A.24)

j1 ¼ 43t3ðt3 þ t4Þ; j2 ¼ 1

3t3ðt3 þ t4Þ;

j3 ¼ 16t23ð2t3 þ t4Þ2

�2t3 � 6zð2ÞR þ 3t2 þ t4

�;

j4 ¼ 43t1ðt1 þ t2Þ; j5 ¼ �zð2ÞR

�t23 þ t3t4

�þ t1

�12t24 þ 2t23 þ t3t4

�j6 ¼ �zð2ÞR t1t2

�t24 þ 4t23 þ 4t3t4

�þ t1t2t3t4ð2t3 þ t4Þ

�t1

�43t1t

33 � 1

6t2t

34

�(A.25)

Kð0Þ42 ¼ 1

a1

�k1�zð2ÞR

�2 þ k2�zð2ÞR

�3 þ k3t32 þ k4t

21 þ k5t

33 þ k6t

34

þ k7t42 þ k8

�(A.26)

k1 ¼ a1; k2 ¼ �43t2t3ðt3 þ t4Þ �

23t24ðt1 þ t2Þ �

83t1t3ðt3 þ t4Þ;

k3 ¼ 112

ð2t3 þ t4Þ2ð2t3 þ 4t1 þ t4Þ;

k4 ¼ 112

ð2t3 þ t4Þ2�4t2t3 þ 3t22 þ 2t2t4 � 12

�zð2ÞR

�2�(A.27)

k5 ¼ 23t2ðt1 þ t2Þð2t1 þ t2Þ; k6 ¼ 1

12t2ðt1 þ t2Þð2t1 þ t2Þ;

k7 ¼ 112

�t24 þ 5t23 þ 4t3t4

�k8 ¼ �zð2ÞR t21

�t24 þ 4t23 þ 4t3t4

�þ t1t

22t3t4

�32t4 þ 3t3

��t21t2

�16t34 þ 4

3t33

�� t32

�23t33 þ 1

12t34

�(A.28)

Kð0Þ43 ¼ 1

a1

�l1�zð2ÞR

�3 þ l2t32 þ l3t

21 þ l4t

33 þ l5t

34 þ l6t

22 þ l7

�(A.29)

l1 ¼ 43t3ðt3 þ t4Þ; l2 ¼ 1

6t3ðt3 þ t4Þ; l3 ¼ 1

3ð2t3 þ t4Þ2;

l4 ¼ 83t1ðt1 þ t2Þ þ t22

l5 ¼ 13t1ðt1 þ t2Þ; l6 ¼ 1

2t3ðt3 þ t4Þð2t3 þ t4Þ

l7 ¼ t23

�� t3t

22 þ 4t1t2t4 �

83t3t

21

�þ t1t

24

�� 13t1t4 þ 2t2t3

�(A.30)

Kð0Þ44 ¼ 1

a1

�m1

�zð2ÞR

�3þm2t32 þm3t

21 þm4t

33 þm5t

34 þm6t

22 þm7

�(A.31)


1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465

66676869707172737475767778798081828384858687888990919293949596979899

100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130



m1 ¼ �23t4ðt3 þ t4Þð2t3 þ t4Þ;m2 ¼ 1

12t4ðt3 þ t4Þð2t3 þ t4Þ;

m3 ¼ � 112

ð2t3 þ t4Þ4;m4 ¼ �13t4ð8t1ðt1 þ t2Þ þ t22

(A.32)

m5 ¼ 13t3�t22 � 2t21 � 2t1t2

�� 23

�zð2ÞR

�3 þ 112

t32 ;

m6 ¼ � 112

ð2t3 þ t4Þ2ðt3 � t4Þðt3 þ t4Þ

m7 ¼ 13t34

�� t3

�t22 � 2t21

�þ 2

�zð2ÞR

�3�þ 13t4t

23

�8t21 þ t22

��t1t2t

24

�2t23 þ 1

12t24

�� 112

t32t34 � 4

3t1t2t

43

(A.33)

Kð1Þ11 ¼ t1

a1

��t23 þ t3t4

�ð3t1 þ 2t2Þ þ t24ðt1 þ t2Þ

�Kð1Þ12 ¼ �ðt1 þ t2Þ

2a1

�t21�4t23 þ 4t3t4 þ t24

�þt2

�2t1t

23 þ 2t1t3t4 � t2t

24

��(A.34)

Kð1Þ13 ¼ t3

a1ðt1 þ t2Þ2ðt3 þ t4Þ;Kð1Þ

14 ¼ t42a1

ðt1 þ t2Þ2ðt3 þ t4Þð2t3 þ t4Þ

Kð1Þ21 ¼ 1;Kð1Þ

22 ¼ Kð1Þ23 ¼ Kð1Þ

24 ¼ 0

(A.35)

Kð1Þ31 ¼ t1

a1ðt2 þ t3Þ2ðt1 þ t2Þ;Kð1Þ

32 ¼ t22a1

ðt1 þ t2Þðt3 þ t4Þ2ð2t1 þ t2Þ

Kð1Þ33 ¼ t3

a1

��t21 þ t1t2

�ð3t3 þ 2t4Þ þ t22ðt3 þ t4Þ

�(A.36)

Kð1Þ34 ¼ �ðt3 þ t4Þ

2a1

�t23�4t21 þ 4t1t2 þ t22

�þt4

�2t3t

21 þ 2t1t2t3 � t4t

22

��Kð1Þ41 ¼ Kð1Þ

42 ¼ 0;Kð1Þ43 ¼ 1;Kð1Þ

44 ¼ 0

(A.37)

Kð2Þ11 ¼ 2t3

a1ðt3 þ t4Þðt1 þ t2Þ;

Kð2Þ12 ¼ � t2

a1

�t2�t23 þ t3t4 þ t24

�þ t1

�2t23 þ 2t2t3 � t24

��Kð2Þ13 ¼ �2t3


Kð2Þ14 ¼ � t4

a1ðt3 þ t4Þð2t3 þ t4Þðt1 þ t2Þ

(A.38)


22 ¼ 1;Kð2Þ23 ¼ Kð2Þ

24 ¼ 0

Kð2Þ31 ¼ �2t1


Kð2Þ32 ¼ � t2

a1ðt1 þ t2Þð2t1 þ t2Þðt3 þ t4Þ

(A.39)

Kð2Þ33 ¼ 2t1


Kð2Þ34 ¼ � t4

a1

�t4�t21 þ t1t2 þ t22

�þ t3

�2t21 þ 2t1t2 þ t22

��Kð2Þ41 ¼ Kð2Þ

42 ¼ Kð2Þ43 ¼ 0;Kð2Þ

44 ¼ 1

(A.40)

Kð3Þ11 ¼ Kð3Þ

21 ¼�4t33a1

ðt3þ t4Þ;Kð3Þ13 ¼ Kð3Þ

23 ¼ 4t33a1

ðt3þ t4Þ

Kð3Þ12 ¼ Kð3Þ

22 ¼ 23a1

�t4ð2t3t4þ t1t4þ4t1t3Þþ t2

�2t23 þ2t3t4þ t24

��(A.41)

Kð3Þ14 ¼ Kð3Þ

24 ¼ 2t43a1

ð2t3 þ t4Þðt3 þ t4Þ;Kð3Þ31 ¼ Kð3Þ

41 ¼ 4t13a1

ðt1 þ t2Þ

Kð3Þ32 ¼ Kð3Þ

42 ¼ 2t23a1

ð2t1 þ t2Þðt1 þ t2Þ;Kð3Þ33 ¼ Kð3Þ

43 ¼ �4t13a1

ðt1 þ t2Þ

(A.42)

Kð3Þ34 ¼ Kð3Þ

44 ¼ 23a1

�t4�2t21 þ 2t1t2 þ t22

�þ t3

�t22 þ 4t1t2 þ 4t21

��(A.43)

where:

a1 ¼ ðt1t4 þ t2t4 þ 2t1t3 þ t2t3Þðt1t4 þ 2t1t3 þ t2t3Þ;a2 ¼ t1 þ t2 þ 2zð2ÞR

(A.44)

Appendix A.2. Delaminated region

In accordance with Section 4.2.1 the following constants can beobtained for the delaminated region:

Kð0Þ11 ¼ �1

6

�3t21 þ 6t1t2 þ 2t22

�2t1 þ t2

;Kð0Þ12 ¼ 1

63t21 � t22ð2t1 þ t2Þ

;

Kð0Þ13 ¼ 1

12

t4�3t21 � t22

�ð2t3 þ t4Þ

t3ð2t1 þ t2Þ

Kð0Þ21 ¼ 1

6

t1�2t21 þ 6t1t2 þ 3t22

�t2ð2t1 þ t2Þ

;Kð0Þ22 ¼ �1

3t31

t2ð2t1 þ t2Þ;

Kð0Þ23 ¼ �1

6t4t

23ð2t3 þ t4Þ

t2t3ð2t1 þ t2Þ

(A.45)


32 ¼ �14t4;K

ð0Þ33 ¼ 1

12

t4�3t23 � t24

�t3

;Kð0Þ41 ¼ 0;

Kð0Þ42 ¼ 1

3t3;K

ð0Þ43 ¼ �1

6t23

Kð1Þ11 ¼ t1ðt1 þ 2t2Þ

t2ð2t1 þ t2Þ;Kð1Þ

12 ¼ �ðt1 � t2Þðt1 þ t2Þt2ð2t1 þ t2Þ

;

Kð1Þ13 ¼ �1

2t4ðt1 � t2Þðt1 þ t2Þð2t3 þ t4Þ

t2t3ð2t1 þ t2Þ(A.46)


1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465

66676869707172737475767778798081828384858687888990919293949596979899

100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130




22 ¼ 0;Kð1Þ23 ¼ 0;Kð1Þ

31 ¼ 0;Kð1Þ32 ¼ 1;

Kð1Þ33 ¼ �1

2ðt3 � t4Þðt3 þ t4Þ

t3


42 ¼ 1;Kð1Þ43 ¼ 0;Kð2Þ

11 ¼ 22t1 þ t2

;

Kð2Þ12 ¼ � 2

2t1 þ t2;Kð2Þ

13 ¼ �12t4ð2t3 þ t4Þt3ð2t1 þ t2Þ

(A.47)


22 ¼ 0;Kð1Þ23 ¼ 0;Kð1Þ

31 ¼ 0;Kð1Þ32 ¼ 1;

Kð1Þ33 ¼ �1

2ðt3 � t4Þðt3 þ t4Þ

t3


42 ¼ 1;Kð1Þ43 ¼ 0;Kð2Þ

11 ¼ 22t1 þ t2

;

Kð2Þ12 ¼ � 2

2t1 þ t2;Kð2Þ

13 ¼ �12t4ð2t3 þ t4Þt3ð2t1 þ t2Þ

(A.48)

Kð2Þ21 ¼ � 2t1

t2ð2t1 þ t2Þ;Kð2Þ

22 ¼ 2t1t2ð2t1 þ t2Þ

;

Kð2Þ23 ¼ t1t4ð2t3 þ t4Þ

t2t3ð2t1 þ t2Þ;Kð2Þ

31 ¼ 0;Kð2Þ32 ¼ 0

Kð2Þ33 ¼ �t4

t3;Kð2Þ

41 ¼ 0;Kð2Þ42 ¼ 0;Kð2Þ

43 ¼ 1;

Kð3Þ11 ¼ Kð3Þ

12 ¼ Kð3Þ21 ¼ Kð3Þ

22 ¼ 43t2ð2t1 þ t2Þ

(A.49)

Kð3Þ13 ¼ Kð3Þ

23 ¼ 23


;Kð3Þ31 ¼ Kð3Þ

32 ¼ Kð3Þ41 ¼ Kð3Þ

42 ¼ 0;

Kð3Þ33 ¼ Kð3Þ

43 ¼ 2t3

(A.50)

Appendix B. SSDT constants

This Appendix contains the constants of Eq. (14) denoted by Kð0Þij ,

Kð1Þij and Kð2Þ

ij for the SSDT solution. Obviously Kð3Þij ¼ 0 in this case.

Appendix B.1. Undelaminated region

According to Section 4.1.2 the following constants are obtained:

Kð0Þ11 ¼ �3

8t1 � zð2ÞR � 1

2t2;K

ð0Þ12 ¼ 3

8t1t2 �

�zð2ÞR

�2 þ 12t22 ;

Kð0Þ13 ¼ �1

8t1K

ð0Þ14 ¼ �1

8t1t4;K

ð0Þ22 ¼ �zð2ÞR ;Kð0Þ

22 ¼ ��zð2ÞR

�2;

Kð0Þ23 ¼ Kð0Þ

24 ¼ 0

(B.1)

Kð0Þ31 ¼ 3

8t3 � zð2ÞR þ 1

2t2;K

ð0Þ32 ¼ 3

8t3t2 �

�zð2ÞR

�2 þ 14t22 ;

Kð0Þ33 ¼ 1

8t3;K

ð0Þ34 ¼ �1

8t3t4K

ð0Þ41 ¼ 1

2t3 � zð2ÞR þ 1

2t2;

Kð0Þ42 ¼ 1

2t3t2 �

�zð2ÞR

�2 þ 14t22 ;K

ð0Þ43 ¼ 1

2t3 þ

12t4;

Kð0Þ44 ¼ �1

4t4ðt4 þ 2t3Þ

(B.2)

Kð1Þ11 ¼ 1

2;Kð1Þ

12 ¼ �12t2;K

ð1Þ13 ¼ 1

2;Kð1Þ

14 ¼ 14t4;K

ð1Þ21 ¼ 1;


23 ¼ 0;Kð1Þ24 ¼ 0

Kð1Þ31 ¼ 1

2;Kð1Þ

32 ¼ 12t2;K

ð1Þ33 ¼ 1

2;Kð1Þ

34 ¼ �14t4;K

ð1Þ41 ¼ 0;


43 ¼ 1;Kð1Þ44 ¼ 0

(B.3)

Kð2Þ11 ¼ 1

2t1;Kð2Þ

12 ¼ �12t2t1;Kð2Þ

13 ¼ � 12t1

;Kð2Þ14 ¼ �1

2t4t1;


22 ¼ 1


24 ¼ 0;Kð2Þ31 ¼ � 1

2t3;Kð2Þ

32 ¼ �12t2t3;Kð2Þ

33 ¼ 12t3

;

Kð2Þ34 ¼ �1

2t4t3


42 ¼ 0;Kð2Þ43 ¼ 0;Kð2Þ

44 ¼ 1

(B.4)

Appendix B.2. Delaminated region

Following Section 4.2.2 the contants of the delaminated regionbecome:

Kð0Þ11 ¼ �1

4t2ð2t1 þ t2Þ

t1;Kð0Þ

32 ¼ 14t22t1;Kð0Þ

33 ¼ 14t22t4t1

;

Kð0Þ21 ¼ 1

4

�3t21 � t22 þ t1t2

�t1

Kð0Þ22 ¼ �1

4

�t21 � t22 þ t1t2

�t1

;Kð0Þ23 ¼ �1

4

t4�t21 � t22 þ t1t2

�t1

(B.5)


32 ¼ �12t4;K

ð0Þ33 ¼ 1

4t34t3


42 ¼ 12t3;K

ð0Þ43 ¼ �1

4

t4�t23 � t24 þ t3t4

�t3

(B.6)


12 ¼ Kð1Þ13 ¼ 0;Kð1Þ

21 ¼ 1;Kð1Þ22 ¼ Kð1Þ

23 ¼ 0


32 ¼ 1;Kð1Þ33 ¼ 0;Kð1Þ

41 ¼ 0;Kð1Þ42 ¼ 1;Kð1Þ

43 ¼ 0(B.7)

Kð2Þ11 ¼ 1

t1;Kð2Þ

12 ¼ �1t1;Kð2Þ

13 ¼ �t4t1;Kð2Þ

21 ¼ �1t2;Kð2Þ

22 ¼ 1t2;Kð2Þ

23 ¼ t4t2


32 ¼ 0;Kð2Þ33 ¼ �t4

t3;Kð2Þ

41 ¼ 0;Kð2Þ42 ¼ 0;Kð2Þ

43 ¼ 1

(B.8)


1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465

66676869707172737475767778798081828384858687888990919293949596979899

100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130



Appendix C. J-integral e mode-II and mode-III ERRs

The J-integral has already been derived in previous papersSzekr�enyes (2014d, 2013b, 2014b) for similar problems to thoseconsidered in this paper. The mode-II and mode-III integrals are:

where the notations can be found in the former papers.

Appendix D. ERR and mode ratio distribution for SF and CCplates

For problem b with simply supported edge at x ¼ �c the con-ditions are given by Eqs. (38)e(39), at x ¼ a the free edge involvesthe following conditions instead of those given by Eqs. (36)e(37):

bMðx;xyÞ12 ; bMðx;xyÞ

34 ; bLðx;xyÞ12

þ bLðx;xyÞ34 ;Xi¼1::2

Nðx;xyÞi ;

Xi¼3::4

Nðx;xyÞi ;

Xi¼1::4

Qxi

��ð1Þx¼a

(D.1)

The corresponding distributions for cases I and III are shown inFigs. D.20 and D.21.

The clampedeclamped conditions for problem a in Fig. 3 at x¼ aand x ¼ �c involve:�w;u0b;u0t ; v0b; v0t ; qx2; qx4;fx4; qy2; qy4;fy4

�ð1aÞ��x¼a ¼ 0 (D.2)

�w;u0; v0; qx2; qx4;fx2;fx4; qy2; qy4;fy2;fy4

�ð2Þ��x¼�c ¼ 0 (D.3)

JII ¼12

Xi¼1::4

(�Nx1ðiÞε

ð0Þx1ðiÞ

��x¼þ0 � Nx2ðiÞε

ð0Þx2ðiÞ

��x¼�0

��Ny1ðiÞε

ð0Þy1ðiÞ

��x¼þ0 � Ny2ðiÞε

ð0Þy2ðiÞ

��x¼�0

�þ

�Mx1ðiÞε

ð1Þx1ðiÞ

��x¼þ0 �Mx2ðiÞε

ð1Þx2ðiÞ

��x¼�0

��My1ðiÞε

ð1Þy1ðiÞ

��x¼þ0 �My2ðiÞε

ð2Þy2ðiÞ

��x¼�0

�þ�

Lx1ðiÞεð2Þx1ðiÞ

��x¼þ0 � Lx2ðiÞε

ð2Þx2ðiÞ

��x¼�0

��Ly1ðiÞε

ð2Þy1ðiÞ

��x¼þ0 � Ly2ðiÞε

ð2Þy2ðiÞ

��x¼�0

�þ�

Px1ðiÞεð3Þx1ðiÞ

��x¼þ0 � Px2ðiÞε

ð3Þx2ðiÞ

��x¼�0

��Py1ðiÞε

ð3Þy1ðiÞ

��x¼þ0 � Py2ðiÞε

ð3Þy2ðiÞ

��x¼�0

�o(C.1)

JIII ¼12

Xi¼1::4

8<:�Nxy1ðiÞbgð0Þ

xy1ðiÞ��x¼þ0 � Nxy2ðiÞbgð0Þ

xy2ðiÞ��x¼�0

�þ�Mxy1ðiÞbgð1Þ

xy1ðiÞ��x¼þ0 �Mxy2ðiÞbgð1Þ


�þ�

Lxy1ðiÞbgð2Þxy1ðiÞ

��x¼þ0 � Lxy2ðiÞbgð2Þ


�þ�Pxy1ðiÞbgð3Þ

xy1ðiÞ��x¼þ0 � Pxy2ðiÞbgð3Þ


� 9=; (C.2)

Fig. 20. Distribution of the energy release rates and mode mixity along the delamination front for problem b in Fig. 3, case I. Note: the Reddy solution is available in (Szekr�enyes,2014d).


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The distributions ot the ERRs and mode ratios are shown inFigs. D.22 and D23.

Fig. 23. Distribution of the energy release rates and mode mixity along the delamination front for problem a in Fig. 3, case IV. Note: the Reddy solution is available in (Szekr�enyes,2014d).

Fig. 21. Distribution of the energy release rates and mode mixity along the delamination front for problem b in Fig. 3, case III. Note: the Reddy solution is available in (Szekr�enyes,2014d).

Fig. 22. Distribution of the energy release rates and mode mixity along the delamination front for problem a in Fig. 3, case II. Note: the Reddy solution is available in (Szekr�enyes,2014d).

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