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A numerical procedure for the simulation of skin–stringer debonding growth in stiffened composite panels
Aerospace Science and Technology 39 (2014) 307–314
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Aerospace Science and Technology www.elsevier.com/locate/aescte
A numerical procedure for the simulation of skin–stringer debonding growth in stiffened composite panels A. Riccio, A. Raimondo ∗ , G. Di Felice, F. Scaramuzzino Second University of Naples, Department of Industrial and Information Engineering, via Roma n 29, 81031 Aversa, Italy
a r t i c l e
i n f o
Article history: Received 14 March 2014 Received in revised form 11 September 2014 Accepted 13 October 2014 Available online 16 October 2014 Keywords: Skin–stringer debonding Damage growth FEM Cohesive elements
a b s t r a c t In this paper, a numerical study on the skin–stringer debonding growth in composite panels under compressive load is presented. A novel numerical procedure, for the selection of proper material parameters governing the traction–separation law in Cohesive Zone Model (CZM) based elements, is introduced and demonstrated. Indeed, the proposed procedure uses Virtual Crack Closure Technique (VCCT) based FEM analyses on Double Cantilever Beam (DCB) and End Notched Flexure (ENF) specimen to characterize the traction–separation law, respectively, for fracture mode I and mode II. The established traction–separation laws are then applied to composite structures containing inter-laminar damages modeled by cohesive elements. To validate the proposed approach, a single stringer panel under compression with an artificial debonding between skin and stringer, has been considered. The numerical results, in terms of displacements and debonding size as a function of applied compressive load, have been compared to experimental data available in literature providing a good numerical–experimental correlation. © 2014 Elsevier Masson SAS. All rights reserved.
1. Introduction In recent years, Carbon Fibers Reinforced Plastics (CFRP) composites, due to their high specific strength and stiffness have become of common use for aerospace structural applications. Nevertheless, the difficulties in predicting and controlling their complex failure mechanisms, have considerably slowed down their integration in primary aeronautical structure and have contributed to increase their certification and the maintenance costs. Indeed, among the several composites failure mechanisms, inter-laminar damages, such as delaminations, arising from manufacturing or low velocity impacts with foreign objects, can be considered particularly critical since they can rapidly grow and lead to the catastrophic failure of composite components. Furthermore the lack of robust numerical tools able to predict the onset and the evolution of such failure mechanisms leads to over-conservative designs, not fully realizing the promised economic benefits of composite materials. This is the main reason why the mechanical behavior of composite structure with inter-laminar damages has been widely investigated in literature both experimentally and numerically [7, 9–14,21–25,33].
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Corresponding author. Tel.: +39 3389535218. E-mail address: [email protected] (A. Raimondo).
http://dx.doi.org/10.1016/j.ast.2014.10.003 1270-9638/© 2014 Elsevier Masson SAS. All rights reserved.
The delamination growth predictions, obtained with numerical models developed based on the Virtual Crack Closure Technique (VCCT) and on the fail release approach [7,11–14,23,24,33], have been demonstrated [9,10,21,22,25,26] to be highly dependent on the finite elements size at the delamination front and on the load step size; these analysis parameters can be correctly set in VCCT based models only with the aid of ad-hoc experimental data. Although in literature [15,19] exist procedures based on VCCT able to avoid comparison with experimental results, these procedures require a major programming effort, while the technique proposed in this work use only the standard capabilities of the commercial FEM codes. On the other hand, Cohesive Zone Model (CZM) based elements [6,8,16], whilst having the advantage, over Virtual Crack Closure Technique VCCT based elements, of being mesh and timestep independent, are characterized by a traction–separation law based on almost arbitrary and empirical parameters. VCCT models, instead, uses only G Ic and G IIc as material parameters which are toughness properties usually available from a standard experimental characterization campaign on a material system. The issue of mesh size dependency exists also for cohesive elements [30,31], however according to [8] it is possible to estimate the length of cohesive zone and a minimum number of elements to obtain accurate results. On the other hand, when dealing with VCCT approach does not exists a way to know a priori the correct number of elements to use in the FEM model and even using elements much
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smaller than the length of cohesive zone it does not ensure the accuracy of the solution. In this paper a practical procedure is proposed allowing to properly select the parameters governing the traction–separation law of CZM based elements, avoiding arbitrariness and empirical choices. Compared to the approach in [4,5,32], the added value of this work is to provide a simple method to determine a priori the CZM parameters in order to numerical simulate the skin–stringer debonding phenomenon as faithfully as possible. This procedure, implemented in the ABAQUSTM FEM code, uses VCCT based models to set the CZM traction–separation law parameters for fracture mode I and mode II. Indeed, starting from the load–displacement curves of Double Cantilever Beam (DCB) and End Notched Flexure (ENF) experimental tests [3] for a generic material system, the proper mesh and time step of DCB and ENF FEM models, adopting Virtual Crack Closure Technique interface elements, are selected to fit the experimental curves. Starting from the DCB and EBF models, it is possible to set up the CZM parameters by comparison with VCCT numerical data for other material systems, without the need of additional experimental data. The resulting traction–separation law can be used in composite structures with inter-laminar damage modeled by CZM elements. In Section 2 the theory behind the proposed novel numerical procedure and the numerical implementation in ABAQUSTM are described. In Section 3, the numerical applications on a stiffened composite panels, used in aircraft fuselage or empennage, with an artificial skin–stringer debonding, taken from literature [20], are introduced. In order to prove the effectiveness of the developed numerical procedure, the numerical results have been compared with experimental data available in literature [20]. 2. Theoretical background In this section, the theoretical background of the Virtual Crack Closure Technique (VCCT) and the Cohesive Zone Model (CZM) are briefly resumed in order to clearly introduce the novel numerical approach which is illustrated in detail.
Fig. 1. Linear elastic traction–separation behavior.
In Eq. (3) F j is the force at crack tip, u j the opening displacement one element behind the crack and A the crack surface created by the crack opening. As demonstrated in [21,22,25] the results of the standard VCCT approach are, generally, highly dependent on the mesh at the delamination front and on the load step size. It is not possible to properly setting these parameters in VCCT based FEM models without comparison with experimental data. 2.2. Cohesive Zone Model (CZM) Cohesive Zone Model based element can be used, as an effective alternative to VCCT based elements, to predict the initiation and growth of inter-laminar damages in finite element codes. Cohesive elements are used to model the bonded interface between two components. The constitutive response is characterized by a traction–separation law with an initial damage phase, a damage evolution phase and a full separation phase. The bilinear softening model for the traction–separation law, shown in Fig. 1, is often used for its simplicity and effectiveness. An initial linear elastic behavior is followed by the initiation and the evolution of damage until the complete separation. The area under the traction–separation curve is the critical energy release rate for each fracture mode. The cohesive element is considered completely failed when the Linear Law criterion is satisfied:
GI 2.1. Virtual Crack Closure Technique (VCCT)
G Ic
Generally in FEM software the inter-laminar damage (delamination) growth is modeled by means of pairs of nodes with identical coordinates which are kept together by constraints and are released, when prescribed conditions are satisfied. In ABAQUSTM [1] the general fracture criterion is defined as:
f =
G equiv G equivC
≥ 1.0
(1)
where G equiv is the equivalent strain energy release rate calculated on the delamination front, and G equivC is the critical equivalent strain energy release rate. The ratio in Eq. (1) is evaluated according to a user-specified mode-mix growth criterion. One of the most used growth criteria is the Linear Law:
G equiv G equivC
=
GI G Ic
+
G II G IIc
+
G III G IIIc
(2)
where G j is the Energy Release Rate associated to the fracture mode j and G jc is the critical value of the Energy Release Rate associated to the fracture mode j that can be evaluated from standard experimental procedures [2,3,27]. In case of 8-noded threedimensional solid element the VCCT equations for the calculation of the Energy Release Rate can be written as follows:
Gj =
F j u j 2 A
with j = I , II, III
(3)
+
G II G IIc
+
G III G IIIc
=1
(4)
where G j is the Energy Release Rate associated to the fracture mode j and G jc is the critical value of the Energy Release Rate associated to the fracture mode j. Usually, from standard experimental characterization campaigns on material systems, only the value of G c is available while the other properties such as the penalty stiffness (K p ), the damage initiation displacement (δ0 ) and nominal strength – in the pure normal mode I, in the first shear direction and in the second shear direction – ( N , T , S ) have to be selected in an arbitrary way. For the most, a trial and error procedure is followed. The final displacements (δ F ), for the bilinear softening model, can be computed according to the following relations:
δ FI = 2
G Ic N
,
δ IIF = 2
G IIc T
and
δ III F =2
G IIIc S
(5)
The problem of mesh size and time step sensitivity is mitigated with cohesive elements, respect to VCCT based elements, however, CZM based elements requires the selection of several material parameters not easily obtainable from standard experimental tests. 2.3. Novel numerical procedure The numerical procedure, proposed in the present paper, suggests an alternative way to set the cohesive elements parameters by combining experimental data and VCCT based numerical analysis. The procedure is schematically represented in Fig. 2.
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Fig. 2. Schematic representation of the proposed numerical procedure.
Fig. 3. DCB test specimen.
Fig. 4. Load versus applied displacements (DCB test).
According to Fig. 2, starting from the experimental curves from Double Cantilever Beam (DCB) and End Notched Flexure (ENF) tests for a reference material, it is possible to determine the proper mesh size and time-step for the VCCT based FEM models to achieve the best possible match between numerical and experimental results. These tuned FEM models can be used to find numerical load–displacement curves for DCB and ENF specimen made of a generic material for which only the fracture toughness is known. The proposed numerical procedure suggests a practical preliminary estimation of the CZM based elements parameters via a VCCT approach. The use of the same discretization for the VCCT FEM model is possible only under the assumption of small variations of the fracture toughness properties when using materials different from the one adopted to set up the model. This is surely verified within the same class of prepreg carbon fiber reinforced composites used for aerospace applications (considered within this paper). The numerical load–displacement curves can be used to tune the material parameters characterizing the cohesive elements.
Once the optimal CZM parameters have been found for fracture mode I (DCB model) and mode II (ENF model) they can be applied directly to a generic FEM model with inter-laminar damage modeled with cohesive elements without the need of further experimental data. This procedure provides physically based material parameters for CZM elements (derived by G Ic and G IIc material properties), which can be successfully used in FEM models avoiding mesh and time-step dependency. In the next subsections the VCCT based FEM models of DCB and ENF specimen, with mesh and time-step set to fit the experimental load–displacement curve available in literature [3,6,20], are described in detail. 2.3.1. Double Cantilever Beam (DCB) According to the ASTM standard D5528 [16], the Double Cantilever Beam (DCB) experimental test is used to determine the inter-laminar fracture toughness for the fracture mode I (namely G Ic ). Experimental data available in Refs. [8,16], in terms of load
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Fig. 5. ENF test specimen.
Fig. 6. Load versus applied deflection (ENF test).
Fig. 7. Geometrical description of stiffened panel configuration.
displacement curves, obtained by means of ASTM standard specimen for the determination of fracture toughness of mode I, have been used as a reference to set the correct combination of finite element size and time step for the numerical model. The geometrical characteristics of the unidirectional specimen are shown in Fig. 3 together with the material properties of graphite/epoxy. The finite element model shown in Fig. 3 has been generated by adopting the commercial FEM software ABAQUSTM which allows the simulation of inter-laminar damage growth both by VCCT and CZM based elements. The VCCT based DCB model has been shaped by superimposing two layers of elements connected each other by means of node-to-surface contacts with the option “Virtual Crack Closure Technique” [1]. C3D8I elements have been adopted for their superior bending behavior. The anticlastic effects and the
distribution of Energy Release Rate were neglected and only one layer of elements have been used across the width. A value of 0.2, suggested by the ABAQUS manual [1], has been assumed for the VCCT release tolerance while no stabilization or damping have been used. A displacement-controlled analysis has been carried out by applying an opening displacement to the nodes on the tip of the DCB. In Fig. 4, the numerical results, in terms of load versus applied end displacements, obtained with different element sizes and load increments are compared to experimental data. From Fig. 4 it can be noted that the VCCT results are strongly influenced by the mesh size and load increment. Indeed, the load at which the delamination starts to growth and the slope of the curve after the beginning of the growth can change considerably depending on the mesh size and load increment. The combination
A. Riccio et al. / Aerospace Science and Technology 39 (2014) 307–314
Table 1 Material properties of the composite lamina used in the stiffened panel. Property
IM7/8552
Description
E 11 E 22 G 12 = G 13 G 23
147 GPa 11.8 GPa 6.0 GPa 4.0 GPa 0.3 0.45 243 J/m2
Longitudinal Young’s modulus Transverse Young’s modulus
ν12 = ν13 ν23 G Ic
G IIc = G IIIc
514 J/m
Tp
0.125 mm
2
Shear modulus Poisson’s ratio Critical strain energy release rate for mode I Critical strain energy release rate for mode II/mode III Ply thickness
Fig. 8. Geometrical model of the stiffened-panel defined in ABAQUSTM .
of elements size and load increment that better fits the experimental data, is 2.5 mm/0.5 mm. These values have been considered for the definition of the final VCCT based FEM model of the DCB specimen used for the selection of cohesive parameters for the fracture mode I. 2.3.2. End Notched Flexure (ENF) According to the ASTM working document WK22949 [2], the End Notched Flexure (ENF) experimental test is used to determine the inter-laminar fracture toughness for the fracture mode II (namely G IIc ). The geometrical characteristics of the unidirectional specimen are shown in Fig. 5 together with and the material properties of AS4/PEEK carbon-fiber reinforced taken from Ref. [6] are shown in Fig. 5. Similarly to the DCB specimen, the model has been obtained by means of two layers of elements, superimposed and connected each other by means of node-to-surface contact with the option “Virtual Crack Closure Technique” [1]. Four elements C3D8I have been used across the width to increase the contact behavior and avoid convergence problems. Again, a value of 0.2, suggested by the ABAQUS manual [1], has been assumed for the VCCT release tolerance while no stabilization or damping have been used. The numerical and experimental results in terms of load versus applied displacements curves are shown in Fig. 6. Fig. 6 shows that the VCCT results, in terms of delamination growth initiation load, are strongly affected only by the element size; indeed, the load increment has no influence on the delamination growth initiation. The size of the elements that better fit the experimental data, is 0.7 mm. Although all the results in Fig. 6 exhibit the same trend, the value of 0.7 mm has been chosen because it better fits the experimental data in terms of load and displace-
311
ment of propagation initiation. This value of the element size has been considered for the definition of the final VCCT based FEM model of the ENF specimen used for the selection of cohesive parameters for the fracture mode II. 3. Numerical application 3.1. Test case description As numerical application, used for the preliminary validation of the proposed procedure, a literature [20] test case, consisting in a flat skin with a single stiffener with an artificial skin– stringer debonding under compressive load, has been considered. The debonded region, located centrally at the skin–stringer interface, has been created by replacing the bonding epoxy adhesive with a Teflon strip. The panel edges have been immersed in a potting to ensure an even distribution of the applied load. Further details on the structure under investigation are summarized in Fig. 7, while the properties of the material system IM7/8552 are reported in Table 1. The stacking sequence of the skin is [(90/+45/−45/0)]s while the stacking sequences of the stringer foot and stringer web are respectively [(+45/−45)3 , 06 ] and [(+45/−45)3 , 06 ]s . 3.2. FEM implementation The commercial FEM software ABAQUSTM has been used to define the FEM model of the stiffened panel. The geometry has been modeled using eight node continuum shell elements (SC8R) available in ABAQUSTM [1] with three degrees of freedom for each node, instead of the C3D8I used previously to obtain reasonable computational time. The complete geometrical model is shown in Fig. 8. Two different volumes have been defined for the skin and the stringer. These volumes have been coupled by means of a layer of cohesive elements (COH3D8) [1] in the bonded region potentially interested by debonding propagation (see Fig. 9.a). The artificial debonding (Fig. 9.b) has been simulated by interrupting the layer of cohesive elements and by inserting contact elements able to avoid penetration between skin and stringer. The finite element representation of the analyzed stiffened panel is reported in Fig. 10 together with the boundary condition adopted to replicate the experimental test. One edge of the structure has been clamped, while the other edge has been subjected to an applied displacement. In the potting region only the lateral and the out-of-plane displacement have been constraint. In the next subsection the settings of cohesive element parameters are illustrated and commented. 3.3. Setting up the cohesive elements Applying the material properties of Table 1 to the calibrated VCCT based FEM models of the DCB and ENF specimen, introduced in Sections 2.3.1 and 2.3.2, it has been possible to obtain reference load displacement curves allowing to set the CZM parameters that regulate the interface behavior of the cohesive elements. CZM based DCB and ENF FEM models have been developed
Fig. 9. Solid model representing: a) Cohesive elements in the bonded region; b) the initial bonded region.
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Table 2 Cohesive parameters.
Mode I Mode II Mode III
Fig. 10. FEM model with boundary conditions of the stiffened panel.
in ABAQUSTM to set up the CZM parameters (namely the penalty stiffness ( K p ) and the nominal strength ( N , T , S )) by a trial and error procedure aimed to fit the VCCT based DCB and ENF models and CZM based DCB and ENF models. The results of the performed analyses are illustrated in Fig. 11.a and Fig. 11.b respectively for DCB and ENF specimen. From Fig. 11, it is clear that the behavior of cohesive elements can be strongly influenced by the nominal strengths and the penalty stiffness. The values selected, to fit the VCCT base models numerical curves, consistent with literature numerical results, obtained with cohesive element in Ref. [8] are illustrated in Table 2. The choice of equals parameters for the opening modes II and III also made in [17,18,28,29] is due to the fact that, for the composite laminates under investigation, the mechanism of mode II crack growth is presumed to be essentially identical to that of mode III crack growth as observed in [17]. 3.4. Results The nonlinear deformed shapes and the debonding growth shapes obtained at different values of the applied displacement (U x = 0.47 mm, 0.79 mm and 1.23 mm) are shown in Fig. 12. The status of the cohesive elements is also reported and compared to experimental data [18] at failure in Fig. 13. The closed cohesive elements are represented in blue while the detached cohesive elements are represented in red. From Fig. 13 an excellent agreement can be observed in terms of debonding size with experimental data on one side of the panel, while on the other one the numerical model tends to overestimate the debonding area. The global compressive behavior of the stiffened panel can be also represented as shown in Fig. 14 by the numerical applied displacement as a function of the compressive load curve. In Fig. 14 this curve is compared to data form a number of experimental tests.
Nominal strength
Penalty stiffness
26.5 MPa 38 MPa 38 MPa
26 500 MPa/mm 38 000 MPa/mm 38 000 MPa/mm
In Fig. 14, the agreement between the numerical results and the experimental data, in terms of global panel stiffness, is very good. The scatter of experimental data observed in Fig. 14, probably depends on the difficulties to properly positioning the Teflon insert and its interaction with the failure mechanisms of the stiffened panel. Whatever are the differences among the experimental load/displacement curves, similar curves’ slope changes can be observed. These slope changes depends on the propagation of the skin–stringer debonding. The use of incorrect CZM parameters leads to significant differences in the propagation of the debonding leading to significant differences between numerical and experimental load/displacement curves’ slopes whatever is the experimental scatter on the final failure on the stiffened panels. The behavior of the numerical curve is almost linear up to debonding buckling load (8 kN). Beyond this load level, relevant changes of stiffness can be observed in the numerical curve, up to the panel failure, which are, again, in excellent agreement with the experimental results. The complete loss of load carrying capability of a stiffened panel (structural failure) in general can occur due to the complete separation of skin from the stringer or before, due to global instability of the structure. The latter is what has been observed during the numerical analyses presented in this paper. As can be seen from the figure, the applied displacement at which the structural collapse occurs (1.5 mm) is in good accord with the experimental values, especially if the relevant scatter in the experimental data is considered. 4. Conclusions In this paper the results of a numerical study on interlaminar damages propagation in composite structures are presented. A novel numerical procedure based on the simultaneous use of Virtual Crack Closure Technique (VCCT) and Cohesive Zone Model (CZM) has been introduced. The novel numerical procedure, able to overcome the difficulties in setting the cohesive elements parameters, has been implemented in a commercial FEM code and adopted here to investigate the skin–stringer debonding growth in a stiffened composite panel subjected to compressive load. Comparison between the numerical results and the experimental data, in terms of load–displacement curves and debonding size at failure, demonstrates that the proposed CZM parameters setting offer
Fig. 11. Cohesive elements – load versus applied deflection: a) DCB model, b) ENF model.
A. Riccio et al. / Aerospace Science and Technology 39 (2014) 307–314
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Fig. 12. Nonlinear deformed shape and debonding growth shape at different value of applied compressive displacement. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
Fig. 14. Compressive load vs. applied displacements.
References Fig. 13. Debonding growth shape – comparison between numerical results and experimental data. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
physically consistent results for the skin–stringer debonding phenomenon, providing a first validation of the proposed numerical procedure. As future work, the numerical results could be further validated by comparison with experimental data on more complex structures with different defects shapes. Conflict of interest statement The authors declare that there is no conflict of interests regarding the publication of this article.
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Aerospace Science and Technology www.elsevier.com/locate/aescte
A numerical procedure for the simulation of skin–stringer debonding growth in stiffened composite panels A. Riccio, A. Raimondo ∗ , G. Di Felice, F. Scaramuzzino Second University of Naples, Department of Industrial and Information Engineering, via Roma n 29, 81031 Aversa, Italy
a r t i c l e
i n f o
Article history: Received 14 March 2014 Received in revised form 11 September 2014 Accepted 13 October 2014 Available online 16 October 2014 Keywords: Skin–stringer debonding Damage growth FEM Cohesive elements
a b s t r a c t In this paper, a numerical study on the skin–stringer debonding growth in composite panels under compressive load is presented. A novel numerical procedure, for the selection of proper material parameters governing the traction–separation law in Cohesive Zone Model (CZM) based elements, is introduced and demonstrated. Indeed, the proposed procedure uses Virtual Crack Closure Technique (VCCT) based FEM analyses on Double Cantilever Beam (DCB) and End Notched Flexure (ENF) specimen to characterize the traction–separation law, respectively, for fracture mode I and mode II. The established traction–separation laws are then applied to composite structures containing inter-laminar damages modeled by cohesive elements. To validate the proposed approach, a single stringer panel under compression with an artificial debonding between skin and stringer, has been considered. The numerical results, in terms of displacements and debonding size as a function of applied compressive load, have been compared to experimental data available in literature providing a good numerical–experimental correlation. © 2014 Elsevier Masson SAS. All rights reserved.
1. Introduction In recent years, Carbon Fibers Reinforced Plastics (CFRP) composites, due to their high specific strength and stiffness have become of common use for aerospace structural applications. Nevertheless, the difficulties in predicting and controlling their complex failure mechanisms, have considerably slowed down their integration in primary aeronautical structure and have contributed to increase their certification and the maintenance costs. Indeed, among the several composites failure mechanisms, inter-laminar damages, such as delaminations, arising from manufacturing or low velocity impacts with foreign objects, can be considered particularly critical since they can rapidly grow and lead to the catastrophic failure of composite components. Furthermore the lack of robust numerical tools able to predict the onset and the evolution of such failure mechanisms leads to over-conservative designs, not fully realizing the promised economic benefits of composite materials. This is the main reason why the mechanical behavior of composite structure with inter-laminar damages has been widely investigated in literature both experimentally and numerically [7, 9–14,21–25,33].
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Corresponding author. Tel.: +39 3389535218. E-mail address: [email protected] (A. Raimondo).
http://dx.doi.org/10.1016/j.ast.2014.10.003 1270-9638/© 2014 Elsevier Masson SAS. All rights reserved.
The delamination growth predictions, obtained with numerical models developed based on the Virtual Crack Closure Technique (VCCT) and on the fail release approach [7,11–14,23,24,33], have been demonstrated [9,10,21,22,25,26] to be highly dependent on the finite elements size at the delamination front and on the load step size; these analysis parameters can be correctly set in VCCT based models only with the aid of ad-hoc experimental data. Although in literature [15,19] exist procedures based on VCCT able to avoid comparison with experimental results, these procedures require a major programming effort, while the technique proposed in this work use only the standard capabilities of the commercial FEM codes. On the other hand, Cohesive Zone Model (CZM) based elements [6,8,16], whilst having the advantage, over Virtual Crack Closure Technique VCCT based elements, of being mesh and timestep independent, are characterized by a traction–separation law based on almost arbitrary and empirical parameters. VCCT models, instead, uses only G Ic and G IIc as material parameters which are toughness properties usually available from a standard experimental characterization campaign on a material system. The issue of mesh size dependency exists also for cohesive elements [30,31], however according to [8] it is possible to estimate the length of cohesive zone and a minimum number of elements to obtain accurate results. On the other hand, when dealing with VCCT approach does not exists a way to know a priori the correct number of elements to use in the FEM model and even using elements much
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smaller than the length of cohesive zone it does not ensure the accuracy of the solution. In this paper a practical procedure is proposed allowing to properly select the parameters governing the traction–separation law of CZM based elements, avoiding arbitrariness and empirical choices. Compared to the approach in [4,5,32], the added value of this work is to provide a simple method to determine a priori the CZM parameters in order to numerical simulate the skin–stringer debonding phenomenon as faithfully as possible. This procedure, implemented in the ABAQUSTM FEM code, uses VCCT based models to set the CZM traction–separation law parameters for fracture mode I and mode II. Indeed, starting from the load–displacement curves of Double Cantilever Beam (DCB) and End Notched Flexure (ENF) experimental tests [3] for a generic material system, the proper mesh and time step of DCB and ENF FEM models, adopting Virtual Crack Closure Technique interface elements, are selected to fit the experimental curves. Starting from the DCB and EBF models, it is possible to set up the CZM parameters by comparison with VCCT numerical data for other material systems, without the need of additional experimental data. The resulting traction–separation law can be used in composite structures with inter-laminar damage modeled by CZM elements. In Section 2 the theory behind the proposed novel numerical procedure and the numerical implementation in ABAQUSTM are described. In Section 3, the numerical applications on a stiffened composite panels, used in aircraft fuselage or empennage, with an artificial skin–stringer debonding, taken from literature [20], are introduced. In order to prove the effectiveness of the developed numerical procedure, the numerical results have been compared with experimental data available in literature [20]. 2. Theoretical background In this section, the theoretical background of the Virtual Crack Closure Technique (VCCT) and the Cohesive Zone Model (CZM) are briefly resumed in order to clearly introduce the novel numerical approach which is illustrated in detail.
Fig. 1. Linear elastic traction–separation behavior.
In Eq. (3) F j is the force at crack tip, u j the opening displacement one element behind the crack and A the crack surface created by the crack opening. As demonstrated in [21,22,25] the results of the standard VCCT approach are, generally, highly dependent on the mesh at the delamination front and on the load step size. It is not possible to properly setting these parameters in VCCT based FEM models without comparison with experimental data. 2.2. Cohesive Zone Model (CZM) Cohesive Zone Model based element can be used, as an effective alternative to VCCT based elements, to predict the initiation and growth of inter-laminar damages in finite element codes. Cohesive elements are used to model the bonded interface between two components. The constitutive response is characterized by a traction–separation law with an initial damage phase, a damage evolution phase and a full separation phase. The bilinear softening model for the traction–separation law, shown in Fig. 1, is often used for its simplicity and effectiveness. An initial linear elastic behavior is followed by the initiation and the evolution of damage until the complete separation. The area under the traction–separation curve is the critical energy release rate for each fracture mode. The cohesive element is considered completely failed when the Linear Law criterion is satisfied:
GI 2.1. Virtual Crack Closure Technique (VCCT)
G Ic
Generally in FEM software the inter-laminar damage (delamination) growth is modeled by means of pairs of nodes with identical coordinates which are kept together by constraints and are released, when prescribed conditions are satisfied. In ABAQUSTM [1] the general fracture criterion is defined as:
f =
G equiv G equivC
≥ 1.0
(1)
where G equiv is the equivalent strain energy release rate calculated on the delamination front, and G equivC is the critical equivalent strain energy release rate. The ratio in Eq. (1) is evaluated according to a user-specified mode-mix growth criterion. One of the most used growth criteria is the Linear Law:
G equiv G equivC
=
GI G Ic
+
G II G IIc
+
G III G IIIc
(2)
where G j is the Energy Release Rate associated to the fracture mode j and G jc is the critical value of the Energy Release Rate associated to the fracture mode j that can be evaluated from standard experimental procedures [2,3,27]. In case of 8-noded threedimensional solid element the VCCT equations for the calculation of the Energy Release Rate can be written as follows:
Gj =
F j u j 2 A
with j = I , II, III
(3)
+
G II G IIc
+
G III G IIIc
=1
(4)
where G j is the Energy Release Rate associated to the fracture mode j and G jc is the critical value of the Energy Release Rate associated to the fracture mode j. Usually, from standard experimental characterization campaigns on material systems, only the value of G c is available while the other properties such as the penalty stiffness (K p ), the damage initiation displacement (δ0 ) and nominal strength – in the pure normal mode I, in the first shear direction and in the second shear direction – ( N , T , S ) have to be selected in an arbitrary way. For the most, a trial and error procedure is followed. The final displacements (δ F ), for the bilinear softening model, can be computed according to the following relations:
δ FI = 2
G Ic N
,
δ IIF = 2
G IIc T
and
δ III F =2
G IIIc S
(5)
The problem of mesh size and time step sensitivity is mitigated with cohesive elements, respect to VCCT based elements, however, CZM based elements requires the selection of several material parameters not easily obtainable from standard experimental tests. 2.3. Novel numerical procedure The numerical procedure, proposed in the present paper, suggests an alternative way to set the cohesive elements parameters by combining experimental data and VCCT based numerical analysis. The procedure is schematically represented in Fig. 2.
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Fig. 2. Schematic representation of the proposed numerical procedure.
Fig. 3. DCB test specimen.
Fig. 4. Load versus applied displacements (DCB test).
According to Fig. 2, starting from the experimental curves from Double Cantilever Beam (DCB) and End Notched Flexure (ENF) tests for a reference material, it is possible to determine the proper mesh size and time-step for the VCCT based FEM models to achieve the best possible match between numerical and experimental results. These tuned FEM models can be used to find numerical load–displacement curves for DCB and ENF specimen made of a generic material for which only the fracture toughness is known. The proposed numerical procedure suggests a practical preliminary estimation of the CZM based elements parameters via a VCCT approach. The use of the same discretization for the VCCT FEM model is possible only under the assumption of small variations of the fracture toughness properties when using materials different from the one adopted to set up the model. This is surely verified within the same class of prepreg carbon fiber reinforced composites used for aerospace applications (considered within this paper). The numerical load–displacement curves can be used to tune the material parameters characterizing the cohesive elements.
Once the optimal CZM parameters have been found for fracture mode I (DCB model) and mode II (ENF model) they can be applied directly to a generic FEM model with inter-laminar damage modeled with cohesive elements without the need of further experimental data. This procedure provides physically based material parameters for CZM elements (derived by G Ic and G IIc material properties), which can be successfully used in FEM models avoiding mesh and time-step dependency. In the next subsections the VCCT based FEM models of DCB and ENF specimen, with mesh and time-step set to fit the experimental load–displacement curve available in literature [3,6,20], are described in detail. 2.3.1. Double Cantilever Beam (DCB) According to the ASTM standard D5528 [16], the Double Cantilever Beam (DCB) experimental test is used to determine the inter-laminar fracture toughness for the fracture mode I (namely G Ic ). Experimental data available in Refs. [8,16], in terms of load
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Fig. 5. ENF test specimen.
Fig. 6. Load versus applied deflection (ENF test).
Fig. 7. Geometrical description of stiffened panel configuration.
displacement curves, obtained by means of ASTM standard specimen for the determination of fracture toughness of mode I, have been used as a reference to set the correct combination of finite element size and time step for the numerical model. The geometrical characteristics of the unidirectional specimen are shown in Fig. 3 together with the material properties of graphite/epoxy. The finite element model shown in Fig. 3 has been generated by adopting the commercial FEM software ABAQUSTM which allows the simulation of inter-laminar damage growth both by VCCT and CZM based elements. The VCCT based DCB model has been shaped by superimposing two layers of elements connected each other by means of node-to-surface contacts with the option “Virtual Crack Closure Technique” [1]. C3D8I elements have been adopted for their superior bending behavior. The anticlastic effects and the
distribution of Energy Release Rate were neglected and only one layer of elements have been used across the width. A value of 0.2, suggested by the ABAQUS manual [1], has been assumed for the VCCT release tolerance while no stabilization or damping have been used. A displacement-controlled analysis has been carried out by applying an opening displacement to the nodes on the tip of the DCB. In Fig. 4, the numerical results, in terms of load versus applied end displacements, obtained with different element sizes and load increments are compared to experimental data. From Fig. 4 it can be noted that the VCCT results are strongly influenced by the mesh size and load increment. Indeed, the load at which the delamination starts to growth and the slope of the curve after the beginning of the growth can change considerably depending on the mesh size and load increment. The combination
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Table 1 Material properties of the composite lamina used in the stiffened panel. Property
IM7/8552
Description
E 11 E 22 G 12 = G 13 G 23
147 GPa 11.8 GPa 6.0 GPa 4.0 GPa 0.3 0.45 243 J/m2
Longitudinal Young’s modulus Transverse Young’s modulus
ν12 = ν13 ν23 G Ic
G IIc = G IIIc
514 J/m
Tp
0.125 mm
2
Shear modulus Poisson’s ratio Critical strain energy release rate for mode I Critical strain energy release rate for mode II/mode III Ply thickness
Fig. 8. Geometrical model of the stiffened-panel defined in ABAQUSTM .
of elements size and load increment that better fits the experimental data, is 2.5 mm/0.5 mm. These values have been considered for the definition of the final VCCT based FEM model of the DCB specimen used for the selection of cohesive parameters for the fracture mode I. 2.3.2. End Notched Flexure (ENF) According to the ASTM working document WK22949 [2], the End Notched Flexure (ENF) experimental test is used to determine the inter-laminar fracture toughness for the fracture mode II (namely G IIc ). The geometrical characteristics of the unidirectional specimen are shown in Fig. 5 together with and the material properties of AS4/PEEK carbon-fiber reinforced taken from Ref. [6] are shown in Fig. 5. Similarly to the DCB specimen, the model has been obtained by means of two layers of elements, superimposed and connected each other by means of node-to-surface contact with the option “Virtual Crack Closure Technique” [1]. Four elements C3D8I have been used across the width to increase the contact behavior and avoid convergence problems. Again, a value of 0.2, suggested by the ABAQUS manual [1], has been assumed for the VCCT release tolerance while no stabilization or damping have been used. The numerical and experimental results in terms of load versus applied displacements curves are shown in Fig. 6. Fig. 6 shows that the VCCT results, in terms of delamination growth initiation load, are strongly affected only by the element size; indeed, the load increment has no influence on the delamination growth initiation. The size of the elements that better fit the experimental data, is 0.7 mm. Although all the results in Fig. 6 exhibit the same trend, the value of 0.7 mm has been chosen because it better fits the experimental data in terms of load and displace-
311
ment of propagation initiation. This value of the element size has been considered for the definition of the final VCCT based FEM model of the ENF specimen used for the selection of cohesive parameters for the fracture mode II. 3. Numerical application 3.1. Test case description As numerical application, used for the preliminary validation of the proposed procedure, a literature [20] test case, consisting in a flat skin with a single stiffener with an artificial skin– stringer debonding under compressive load, has been considered. The debonded region, located centrally at the skin–stringer interface, has been created by replacing the bonding epoxy adhesive with a Teflon strip. The panel edges have been immersed in a potting to ensure an even distribution of the applied load. Further details on the structure under investigation are summarized in Fig. 7, while the properties of the material system IM7/8552 are reported in Table 1. The stacking sequence of the skin is [(90/+45/−45/0)]s while the stacking sequences of the stringer foot and stringer web are respectively [(+45/−45)3 , 06 ] and [(+45/−45)3 , 06 ]s . 3.2. FEM implementation The commercial FEM software ABAQUSTM has been used to define the FEM model of the stiffened panel. The geometry has been modeled using eight node continuum shell elements (SC8R) available in ABAQUSTM [1] with three degrees of freedom for each node, instead of the C3D8I used previously to obtain reasonable computational time. The complete geometrical model is shown in Fig. 8. Two different volumes have been defined for the skin and the stringer. These volumes have been coupled by means of a layer of cohesive elements (COH3D8) [1] in the bonded region potentially interested by debonding propagation (see Fig. 9.a). The artificial debonding (Fig. 9.b) has been simulated by interrupting the layer of cohesive elements and by inserting contact elements able to avoid penetration between skin and stringer. The finite element representation of the analyzed stiffened panel is reported in Fig. 10 together with the boundary condition adopted to replicate the experimental test. One edge of the structure has been clamped, while the other edge has been subjected to an applied displacement. In the potting region only the lateral and the out-of-plane displacement have been constraint. In the next subsection the settings of cohesive element parameters are illustrated and commented. 3.3. Setting up the cohesive elements Applying the material properties of Table 1 to the calibrated VCCT based FEM models of the DCB and ENF specimen, introduced in Sections 2.3.1 and 2.3.2, it has been possible to obtain reference load displacement curves allowing to set the CZM parameters that regulate the interface behavior of the cohesive elements. CZM based DCB and ENF FEM models have been developed
Fig. 9. Solid model representing: a) Cohesive elements in the bonded region; b) the initial bonded region.
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Table 2 Cohesive parameters.
Mode I Mode II Mode III
Fig. 10. FEM model with boundary conditions of the stiffened panel.
in ABAQUSTM to set up the CZM parameters (namely the penalty stiffness ( K p ) and the nominal strength ( N , T , S )) by a trial and error procedure aimed to fit the VCCT based DCB and ENF models and CZM based DCB and ENF models. The results of the performed analyses are illustrated in Fig. 11.a and Fig. 11.b respectively for DCB and ENF specimen. From Fig. 11, it is clear that the behavior of cohesive elements can be strongly influenced by the nominal strengths and the penalty stiffness. The values selected, to fit the VCCT base models numerical curves, consistent with literature numerical results, obtained with cohesive element in Ref. [8] are illustrated in Table 2. The choice of equals parameters for the opening modes II and III also made in [17,18,28,29] is due to the fact that, for the composite laminates under investigation, the mechanism of mode II crack growth is presumed to be essentially identical to that of mode III crack growth as observed in [17]. 3.4. Results The nonlinear deformed shapes and the debonding growth shapes obtained at different values of the applied displacement (U x = 0.47 mm, 0.79 mm and 1.23 mm) are shown in Fig. 12. The status of the cohesive elements is also reported and compared to experimental data [18] at failure in Fig. 13. The closed cohesive elements are represented in blue while the detached cohesive elements are represented in red. From Fig. 13 an excellent agreement can be observed in terms of debonding size with experimental data on one side of the panel, while on the other one the numerical model tends to overestimate the debonding area. The global compressive behavior of the stiffened panel can be also represented as shown in Fig. 14 by the numerical applied displacement as a function of the compressive load curve. In Fig. 14 this curve is compared to data form a number of experimental tests.
Nominal strength
Penalty stiffness
26.5 MPa 38 MPa 38 MPa
26 500 MPa/mm 38 000 MPa/mm 38 000 MPa/mm
In Fig. 14, the agreement between the numerical results and the experimental data, in terms of global panel stiffness, is very good. The scatter of experimental data observed in Fig. 14, probably depends on the difficulties to properly positioning the Teflon insert and its interaction with the failure mechanisms of the stiffened panel. Whatever are the differences among the experimental load/displacement curves, similar curves’ slope changes can be observed. These slope changes depends on the propagation of the skin–stringer debonding. The use of incorrect CZM parameters leads to significant differences in the propagation of the debonding leading to significant differences between numerical and experimental load/displacement curves’ slopes whatever is the experimental scatter on the final failure on the stiffened panels. The behavior of the numerical curve is almost linear up to debonding buckling load (8 kN). Beyond this load level, relevant changes of stiffness can be observed in the numerical curve, up to the panel failure, which are, again, in excellent agreement with the experimental results. The complete loss of load carrying capability of a stiffened panel (structural failure) in general can occur due to the complete separation of skin from the stringer or before, due to global instability of the structure. The latter is what has been observed during the numerical analyses presented in this paper. As can be seen from the figure, the applied displacement at which the structural collapse occurs (1.5 mm) is in good accord with the experimental values, especially if the relevant scatter in the experimental data is considered. 4. Conclusions In this paper the results of a numerical study on interlaminar damages propagation in composite structures are presented. A novel numerical procedure based on the simultaneous use of Virtual Crack Closure Technique (VCCT) and Cohesive Zone Model (CZM) has been introduced. The novel numerical procedure, able to overcome the difficulties in setting the cohesive elements parameters, has been implemented in a commercial FEM code and adopted here to investigate the skin–stringer debonding growth in a stiffened composite panel subjected to compressive load. Comparison between the numerical results and the experimental data, in terms of load–displacement curves and debonding size at failure, demonstrates that the proposed CZM parameters setting offer
Fig. 11. Cohesive elements – load versus applied deflection: a) DCB model, b) ENF model.
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Fig. 12. Nonlinear deformed shape and debonding growth shape at different value of applied compressive displacement. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
Fig. 14. Compressive load vs. applied displacements.
References Fig. 13. Debonding growth shape – comparison between numerical results and experimental data. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
physically consistent results for the skin–stringer debonding phenomenon, providing a first validation of the proposed numerical procedure. As future work, the numerical results could be further validated by comparison with experimental data on more complex structures with different defects shapes. Conflict of interest statement The authors declare that there is no conflict of interests regarding the publication of this article.
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