Gépészet Budapest 4-5.Ma. G--Section-o ITERFACE CRACK I ORTHOTROPIC KIRCHHOFF PLATES András Szekrénes Budapest Universit of Technolog and Economics Department of Applied Mechanics Budapest Műegetem rkp. 5 Building MM Abstract: The classical laminated plate theor is applied to calculate the stresses and energ release rate function in smmetricall delaminated orthotropic plates. The governing equation sstem of the double-plate model consists of ten equations. As an eample an orthotropic simpl-supported delaminated plate subjected to a point force is analzed. The 3D finite element model of the plate was also created. Results indicate a reasonabl good agreement between analtical and numerical models. Kewords: composite plate theor J-integral interfacial stress.. ITRODUCTIO Laminated composite plates have man industrial applications e.g. the fields of pressure vessels bridge and bodwork construction aeroplanes and finall but not least ship construction can be mentioned. It is well-known that delamination is one of the major damage modes in laminated fiber-reinforced composite materials []. Mechanicall the formation of cracks and delamination surfaces can be characterized b the energ based principles of fracture mechanics []. The energ release rate (ERR) is the basic parameter to dimensionize the structures against crack initiation and propagation. The limit value of the ERR is called the critical ERR (CERR) which can be determined using standard (or nonstandard) test methods. Although for the mode-i mode-ii and mied-mode I/II fractures there is a consensus to use simple beam tests for mode-iii there is not an internationall accepted test method. In general it is thought that the practical significance of mode-iii fracture is little in spite of that in the last decades the attention was subsequentl focused on this fracture mode. It will be shown in this paper that the bending of delaminated plates involves significant mode-iii contribution; however the mode-ii and mode-iii fracture take place simultaneousl leading to a mied-mode II/III problem.. CLASSICAL PLATE MODEL WITH FLEXIBLE JOIT In this section we analze elastic laminated plates with smmetric delamination. The classical laminated plate model is completed with the effect if interface deformation [] in a mied-mode II/III plate problem. We consider the differential plate element shown in Fig. which represents the uncracked plate portion. The equilibrium of forces in the and directions leads to: = τ + = τ z () where are the in-plane forces and shear force τ τ z are the interfacial shear stresses. Moreover equilibrium of bending moments about aes and results in: M M t + = τ M + M = τ t z () / 6
Gépészet Budapest 4-5.Ma. G--Section-o where M M M are the bending and twisting moments respectivel furthermore t is the thickness (see Fig..). For laminated plates the relationship between the in-plane forces and strains (ε ε γ ) can be epressed as [3]: A A ε ε a a A A ε ε a a = =. (3) A 66 γ γ a 66 The relationship between moments and curvatures is [3]: M D D w M D D = w (4) M D 66 w where w = w() is the plate deflection. Moreover in Eqs. (3)-(4) A ij and D ij (ij = 6) are the components of the etensional and bending stiffness matrices a ij (compliance matri) is the inverse of A ij [3]: ( k) ( k) 3 3 A= C ( zk+ zk) D= C ( zk+ zk). (5) 3 k= k= Fig.. Equilibrium of the top (a) and bottom (b) plate elements. The net step is the formulation of displacement continuit in the interface plane of the double-plate sstem. There are three different sources of in-plane displacements: in-plane normal forces bending and shear deformation i.e.: t w 55 u = ( a / a ) d k z t shτ = + = (6) t w 44 v = ( a / a ) d k z t shτ = + z = (7) where k sh is the shear compliance and can be defined as the generalization of that in []: 55 zk+ zk 44 zk+ zk ksh = k ( k) sh = ( k) k= 3C55 k= 3C44 (8) where is the number of laers moreover the shear stiffnesses are [3]: C55 = C44 sin θ + C55 cos θ C44 = C44 cos θ + C55 sin θ. (9) For plates there is a third condition formulated with respect to the in-plane shear strain: w 55 τ 44 τ z γ = a / 66 t ksh ksh z= t + =. () Compatibilit of the displacement field requires the following: / 6
Gépészet Budapest 4-5.Ma. G--Section-o γ u v = + z = t / z = t /. () It can be seen that the second and third terms in Eqs.(6)-(7) satisf automaticall the compatibilit condition. On the contrar among the first terms the following relation can be established: = a + a a +. () a 66 Eqs. ()()(4)(6)(7) and () define a boundar value problem including ten equations with ten parameters: τ τ z M M M w/ and w/. Combining Eqs.() and (4) it is possible to derive the governing equation of the plate deflection: 4 4 4 w w w t τ τ z D + ( D 4 + D66) + D = 4 + (3) where the inhomogeneit is caused b the interface shear stresses. Also b combining the equations a PDE sstem can be obtained for the in-plane forces: 4 4 A 4 3 4 4 5 + (4) 4 4 6 4 7 8 9 4 = and similarl: 4 4 B 4 3 4 4 5 + (5) 4 4 6 4 7 8 9 4 = where A i and B i are constants depending on the stiffness and compliance parameters as well as the thickness of the plate. Boundar conditions are necessar to solve this boundar-value problem a specific case is presented in the net section. 3. EXAMPLE A SIMPLY-SUPPORTED DELAMIATED PLATE In this section we adopt Lév plate formulation to solve the PDE sstem presented in section. Fig. shows a simpl-supported delaminated orthotropic plate. The problem is solved in two steps: problem (a) is a traditional plate bending problem problem (b) improves the former with the effect of crack front deformation which was presented onl for beams []. Here onl a brief description is given. The deflections in the cracked and uncracked part are approimated as (both for (a) and (b)): w( ) = W( )sin β w( ) = W ( )sin β (6) I where β = nπ/b. The interfacial shear stresses for problem (b) are: In II τ ( ) = T( )sin β τ ( ) = R( )cos β (7) and finall the in-plane forces (problem (b)) can be written as: n z ( ) = n ( )sin β ( ) = n ( )sin β ( ) = n ( )cos β. (8) n n n IIn n 3 / 6
Gépészet Budapest 4-5.Ma. G--Section-o Problem (a) can be solved in the usual wa [3]. Problem (b) involves ten constants from which eight can be determined based on the kinematic and dnamic boundar conditions with respect to the plate deflection. Further two conditions can be formulated forτ : it vanishes at the free end (= -c) and it is the highest at the delamination front. The distribution of the shear stress along the front can be obtained b the aial equilibrium of shear tractions over the midplane of the delaminated and uncracked portions []. Fig.. A simpl-supported delaminated plate subjected to a point force. 4. EERGY RELEASE RATE J-ITEGRAL The energ release rate distribution over the delamination front can be calculated b the 3D J-integral []. The 3D J-integral is defined as []: Jk = ( Wnk σijuik n j) ds + ( Wδk3 σi3 uik )3dA k = J3 = W3n σ 3 ju3 n j) ds (9) C A where n i is the outward normal vector of the contour C δ ij is the Kronecker tensor σ ij is the stress tensor u i is the displacement vector A is the area enclosed b contour C. The contour C contains the crack tip and the integration is carried out in the counterclockwise direction []. For the problem (a) the mode-ii and mode-iii integrals become: ( a a a a ) a a a a JIIa = MIwI MII wii J ( =+ = IIIa = MIwI M II wii ) () =+ = where subscript I and II refers to the delaminated and uncracked parts in Fig.. For problem (b) we obtain (without details): zk + b b b b ( k) b bk ( ) JIIb = { ( MI wi M ) ( ) } IIwII + ε + τ wii uii z dz =+ = = = () J ( M w M w ) ( v ) v dz b b b b b ( k) bk ( ) { γ τ } = + +. () IIIb I I =+ II II = II z II z = = k= zk 4. RESULTS AD DISCUSSIO The properties of the analzed simpl-supported plate were (refer to Fig. ) the following: a=5 mm (crack length) c=45 mm (uncracked length) b= mm (plate width) t= mm (plate thickness) Q = (point force magnitude) =3 mm =5 mm (point of action coordinates of Q ). The plate is made of a carbon/epo material the la-up of the plate was [±45 f ; ; ±45 f ] for the delaminated and [±45 f ; ; ±45 f ] S for the uncracked part. The superscript "f" refers to the fact that the cross-pl laminate is a woven fabric panel. The properties of the individual orthotropic laminae are E =E =E 3 =6.39 GPa G =6.4 G 3 =G 3 =5.46 GPa ν =.3 ν 3 =ν 3 =.5 for the ±45 f laminate and E =48 GPa E =E 3 =9.65 GPa G =3.7 GPa G 3 =4.66 GPa G 3 =4.9 GPa ν =.3 ν 3 =.5 ν 3 =.7 for the laminate. The finite element (FE) model of the plate was also constructed the ERR k= zk Cε zk + 4 / 6
Gépészet Budapest 4-5.Ma. G--Section-o was calculated b the virtual crack-closure technique (VCCT). Fig. 3 shows the distribution of the shear stresses [3] τ τ z over the thickness at given points of the crack front i.e. at =. It is seen that τ changes significantl b 4 % compared to problem (a). Also τ z is improved b more than twice of its original value (increase b 65 %). Thus the shear deformation of the crack front results in significant changes in the interlaminar shear stress distributions. Fig. 3. Through thickness distribution of the shear stresses at certain points in the delamination front. Fig. 4. Distribution of the interface shear stresses: τ (a) τ z (b). Fig.4 demonstrates the distribution of the shear stresses over the interface of the undelaminated region of the top plate the stresses deca to zero behind the crack front. The mode-ii mode-iii ERRs and the mode ratio along the crack front are shown b Fig.5. The smbols show the result of the VCCT [4] while the curves represent the purel analtical plate theor solutions. The solution of problem (a) provides the major part of the ERRs. It is seen that the mode-ii component is significantl underestimated b the plate theor solution. On the contrar the mode-iii component is overestimated. Problem (b) provides a reasonable improvement for the mode-ii ERR which is 56% for G II at =b/. Also it is clear that plate theor (a+b) underestimates the mode-ii component in the midpoint the difference is about 4% compared to the VCCT result. In contrast the mode-iii ERR is slightl smaller compared to the plate theor solution. The difference between maimum of the mode-iii ERR b VCCT and the plate model is % the curve crosses over the maimum point smbol. 5 / 6
Gépészet Budapest 4-5.Ma. G--Section-o While the VCCT predicts that the mode-iii ERR decas suddenl near the edges there is not an deca in accordance with the plate solution. In other words edge effects are not captured properl b the plate model. Fig. 5. Energ release rate distributions (a) and mode ratios (b) b laminated plate theor (Lév solution) and FE analsis. The mode ratio (G II /G III ) is depicted in Fig.5b. The plate theor solution slightl underestimates the mode ratio however the nature of the curves matches well with the numerical result. Also it is clear that the improvement related to the crack front shear deformation is reasonable. 5. COCLUSIO A purel analtical plate theor approach has been presented to calculate the shear stresses and the mode-ii and mode-iii ERR distributions along the crack front of smmetricall delaminated laered composite plates subject to bending. The overall agreement between the VCCT and plate theor methods is fairl good. The difference between the VCCT and plate theor solution can be attributed to the transverse shear effect however it should be kept in mind that the VCCT method is also mesh sensitive. ACKOWLEDGMETS This work is connected to the scientific program of the "Development of qualitoriented and harmonized R+D+I strateg and functional model at BME" project. This project is supported b the ew Hungar Development Plan (Project ID: TÁMOP-4../B- 9//KMR--). REFERECES [] Wang J Qiao P. ovel beam analsis of end notched fleure specimen for mode-ii fracture. Engineering Fracture Mechanics 4;7:9-3. [] Rigb RH Aliabadi MH. Decomposition of the mied-mode J-integral - revisited. International Journal of Solids and Structures 998;35(7): 73-99. [3] Redd J. Mechanics of laminated composite plates and shells Theor and Application. Boca Raton London ew York Washington: CRC Press 4. [4] Marat-Mendes RM Freitas MM. Failure criteria for mied mode delamination in glass fibre epo composites. Composite Structures ;9(9): 9-98. 6 / 6