On Numerical Simulation of Interlaminar Failure of Composites
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PAMM · Proc. Appl. Math. Mech. 8, 10221 – 10222 (2008) / DOI 10.1002/pamm.200810221 On Numerical Simulation of Interlaminar Failure of Composites Krzysztof Kula 1,2, ∗ , Rainer Schlebusch 1 , Bernd W. Zastrau 1 , and Mieczyslaw Kuczma 2 1 Technische Universit¨ at Dresden, Institut f¨ ur Mechanik und Fl¨ achentragwerke 2 University of Zielona G´ ora, Institute of Structural Engineering This paper deals with one of the most dangerous failure modes in fiber reinforced composites, namely delamination. The lami- nated structure is modeled by a solid-shell finite element. The mechanical modeling of delamination onset and propagation is based upon a cohesive zone model implemented into a cohesive element located between laminae of a composite structure. The numerical simulations are accomplished within the commercial software package ABAQUS by the implementation of a user-written finite element. c 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction For an appropriate numerical prediction of the composite structures behavior subjected by load, several failure modes have to be taken into account. One of the most dangerous and dominant failure mode often leading to the loss of structural integrity is delamination. This failure mode can be considered as being caused by the state of stress, material defects such as transversal matrix cracking, and different assembly imperfections. The realistic prediction and simulation of the initiation and propagation of this dangerous phenomenon is very important for designing process of composite structures. 2 Solid-Shell Finite Element The nonlinear analysis of the thin layers is accomplished by a solid-shell finite element. The basis of this finite element is the standard eight-node brick element with tri-linear shape functions. The element provides a accurate description of the interlaminar normal and shear stresses only for a very fine mesh in thickness direction of the composite structure. The pure displacement solid-shell element is improved by the assumed natural strain method (ANS) and the enhanced assumed strain method (EAS). The enrichments are necessary, since the pure element behaves too stiff and tends to locking. The ANS method being applied to transverse shear strains is able to reduce transverse shear locking, see e.g. [2, 5]. To avoid the artificial thickness strains, i. e. curvature thickness locking, it is essential to use additionally a specific interpolation of thickness strains [3]. The volumetric locking and locking effects concerned with the membrane and bending behavior are reduced by the EAS method which introduces additional enhanced strains into the finite element formulation [4, 8]. Five strain parameters are utilized within the enhancement. The material of a typical layer is modeled by a hyperelastic orthotropic material formulation and is defined in a local material coordinate system. A suitable transformation matrix can be found e. g. in [7]. 3 Delamination: Model and Numerical Example In literature one can find several techniques for the prediction, i. e. the onset, and the propagation of cracks, see review [1,9]. The cohesive zone technique is one of them. It assumes a cohesive zone between two laminae in a layered structure. The cohesive layer has a very small but finite thickness relatively to the thickness of the whole layer. The ratio between the two thicknesses is assumed to be of order 1:100. The model relates stresses to relative displacements in the interface. This stress- separation model assumes initially a linear elastic behavior of the interface followed by an initiation and an evolution of the failure. The used stress-separation-based model assumes three different types of separation well-known from the fracture mechanics, whereby one is normal to the interface (opening mode) and the two others are parallel to it (sliding or shearing modes). The failure initiation depends on the interfacial strength and can be estimated by a quadratic stress interaction function, originally proposed by HASHIN [6]. This criterion can be represented as follows: σ zz R zz 2 + σ yz R yz 2 + σ xz R xz 2 ≤ 1, (1) where σ zz is the interlaminar normal stress and σ yz and σ xz are the corresponding shear stresses in the interface. The stresses are calculated and evaluated at integration points of a cohesive element. The HASHIN criterion is presented in terms of stresses. The quantities R zz , R xz and R yz are the tensile strength in the thickness direction, i. e. in the normal direction and the shear ∗ Corresponding author E-mail: [email protected] c 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Transcript of On Numerical Simulation of Interlaminar Failure of Composites
PAMM · Proc. Appl. Math. Mech. 8, 10221 – 10222 (2008) / DOI 10.1002/pamm.200810221
On Numerical Simulation of Interlaminar Failure of Composites
Krzysztof Kula1,2,∗, Rainer Schlebusch1, Bernd W. Zastrau1, and Mieczysław Kuczma2
1 Technische Universitat Dresden, Institut fur Mechanik und Flachentragwerke2 University of Zielona Gora, Institute of Structural Engineering
This paper deals with one of the most dangerous failure modes in fiber reinforced composites, namely delamination. The lami-nated structure is modeled by a solid-shell finite element. The mechanical modeling of delamination onset and propagationis based upon a cohesive zone model implemented into a cohesive element located between laminae of a composite structure.The numerical simulations are accomplished within the commercial software package ABAQUS by the implementation ofa user-written finite element.
c© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction
For an appropriate numerical prediction of the composite structures behavior subjected by load, several failure modes have tobe taken into account. One of the most dangerous and dominant failure mode often leading to the loss of structural integrity isdelamination. This failure mode can be considered as being caused by the state of stress, material defects such as transversalmatrix cracking, and different assembly imperfections. The realistic prediction and simulation of the initiation and propagationof this dangerous phenomenon is very important for designing process of composite structures.
2 Solid-Shell Finite Element
The nonlinear analysis of the thin layers is accomplished by a solid-shell finite element. The basis of this finite elementis the standard eight-node brick element with tri-linear shape functions. The element provides a accurate description ofthe interlaminar normal and shear stresses only for a very fine mesh in thickness direction of the composite structure. Thepure displacement solid-shell element is improved by the assumed natural strain method (ANS) and the enhanced assumedstrain method (EAS). The enrichments are necessary, since the pure element behaves too stiff and tends to locking. TheANS method being applied to transverse shear strains is able to reduce transverse shear locking, see e. g. [2, 5]. To avoidthe artificial thickness strains, i. e. curvature thickness locking, it is essential to use additionally a specific interpolation ofthickness strains [3]. The volumetric locking and locking effects concerned with the membrane and bending behavior arereduced by the EAS method which introduces additional enhanced strains into the finite element formulation [4, 8]. Fivestrain parameters are utilized within the enhancement. The material of a typical layer is modeled by a hyperelastic orthotropicmaterial formulation and is defined in a local material coordinate system. A suitable transformation matrix can be found e. g.in [7].
3 Delamination: Model and Numerical Example
In literature one can find several techniques for the prediction, i. e. the onset, and the propagation of cracks, see review [1, 9].The cohesive zone technique is one of them. It assumes a cohesive zone between two laminae in a layered structure. Thecohesive layer has a very small but finite thickness relatively to the thickness of the whole layer. The ratio between the twothicknesses is assumed to be of order 1:100. The model relates stresses to relative displacements in the interface. This stress-separation model assumes initially a linear elastic behavior of the interface followed by an initiation and an evolution of thefailure. The used stress-separation-based model assumes three different types of separation well-known from the fracturemechanics, whereby one is normal to the interface (opening mode) and the two others are parallel to it (sliding or shearingmodes). The failure initiation depends on the interfacial strength and can be estimated by a quadratic stress interactionfunction, originally proposed by HASHIN [6]. This criterion can be represented as follows:
( 〈σzz〉Rzz
)2
+(
σyz
Ryz
)2
+(
σxz
Rxz
)2
≤ 1, (1)
where σzz is the interlaminar normal stress and σyz and σxz are the corresponding shear stresses in the interface. The stressesare calculated and evaluated at integration points of a cohesive element. The HASHIN criterion is presented in terms of stresses.The quantities Rzz , Rxz and Ryz are the tensile strength in the thickness direction, i. e. in the normal direction and the shear
∗ Corresponding author E-mail: [email protected]
c© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
10222 Sessions of Short Communications 03: Damage and fracture
p
p
x
y
z
cohesive elements (COH3D8)
initial crack
(l-a)=70 mm a=30mmb=
20m
m
h
Ex [GPa] Ey [GPa] Ez [GPa]135.3 9.0 9.0
Gxy [GPa] Gxz [GPa] Gyz [GPa]5.2 5.2 3.08
νxy νxz νyz
0.24 0.24 0.46
Fig. 1 Geometry and loading of the double cantileverbeam (top). Material parameters of the layers (table).
0
10
20
30
40
50
60
70
0 2.00 4.00 6.00 8.00 10.00
EPP3D8EAS5+COH3D8VCCT [ALFANO & CRISFIELD 2001]
pb [N]
w [mm]
Fig. 2 Vertical reaction force vs. prescribed relative displacement.
strengths, respectively. The symbol 〈•〉 denotes the MACAULAY brackets, i. e. 〈•〉 = 12 (|• |+•). The propagation is described
by the fracture toughness (critical fracture energy GC) and if the toughness is reached, the stresses in the interface are reducedto zero while the crack propagates. The details concerning the delamination propagation are described e.g. in [1, 9].
To verify the presented method a classical benchmark – the double cantilever beam (DCB) – is studied. The DCB representsan example with a pure opening mode (mode I). The geometrical and material data for the considered DCB are taken from [1].The DCB is clamped at one end and has an initial crack at the other end. The geometrical data are also given in Fig. 1, where ais the length of the initial crack, b is the width of the beam and l is the length of the specimen. The beam consists of twounidirectional composite laminae, each having a thickness of 1.5 mm. The material data for the layers are presented in thetable, see Fig. 1. The reinforcement is oriented along the x-direction. Between the layers, i. e. in the part where the beam isbonded (0 ≤ x ≤ (l− a)), the cohesive elements with finite thickness hc = 0.01 mm are included. Hence, the total thicknessof the double cantilever beam is h = 3.01 mm. The composite layers are simulated with solid-shell finite elements. For thediscretization of the cohesive zone, the ABAQUS cohesive element COH3D8 is used. The delamination starts from the initialcrack tip and is analyzed under displacement control. Additionally, some experimental data are needed for a more realisticmodeling. Particularly in the applied approach, the critical fracture energy (GI,C = 0.28 N/m) and the strength parameterin the normal direction to the cohesive surface (Rzz = 57.0 MPa) are necessary. The results of the numerical simulationare presented in Fig. 2 and are compared with the results from [1] obtained by the VCCT (virtual crack closure technique).The curves show the correlation between the vertical reaction force pb and the opening w, i. e. the separation between the twolayers at the and of the specimen. The crack propagation process starts when the transverse tensile strength Rzz is reached.The presented approach provides the results which, as can be seen in Fig. 2, are in good agreement with the results from [1]in both phases of this deformation process: the elastic and the softening phase.
Acknowledgements The authors gratefully acknowledge the financial support from DFG (German Research Foundation) within the Son-derforschungsbereich SFB 528 Textile Reinforcements for Structural Strengthening and Repair at Technische Universitat Dresden.
References
[1] G. Alfano and M. A. Crisfield, International Journal for Numerical Methods in Engineering 50, 1701–1736 (2001).[2] K.-J. Bathe and E. N. Dvorkin, International Journal for Numerical Methods in Engineering 21, 367–383 (1985).[3] P. Betsch and E. Stein, Communications in Numerical Methods in Engineering 11, 899–909 (1995).[4] N. Buchter and E. Ramm, Computational Methods in Applied Sciences, 55–62 (1992).[5] E. N. Dvorkin and K.–J. Bathe, Engineering Computations, 1, 77–88 (1984).[6] Z. Hashin, Journal for Applied Mechanics 47, 329–334 (1980).[7] R. Schlebusch, Theorie und Numerik einer oberflachenorientierten Schalenformulierung (Ph. D. thesis, TU Dresden, 2005).[8] J. C. Simo and M. S. Rifai, International Journal for Numerical Methods in Engineering, 29, 1595–1638 (1990).[9] A. Turon, P. P. Camanho, J. Costa and C. G. Davila, An interface damage model for the simulation of delamination under variable-mode
ratio in composite materials (NASA/TM-2004-213277, 2004).
c© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.gamm-proceedings.com
