Validation of Class of Applications Using Progressive Failure And Discrete Cohesive Zone Model for Line

And Surface Cracks

Mohit Garg, Galib H. Abumeri and Dade Huang Alpha STAR Corporation, Suite 410, Long Beach, CA 90804, USA

Combination of multi-scale and multi-physics Progressive Failure Analysis (PFA) methodology and linear elastic fracture mechanics is introduced. The approach combined approach detects crack-path or multi-site crack path and predicts the complete load-displacement curve correctly. Though more accurate, LEFM based methodologies of Discrete Cohesive Zone Modeling (DCZM) and Virtual Crack Closure Technique (VCCT) alone seem to be test duplication methodologies rather than test prediction for several complex problems. This is primarily because the structural finite element (FE) models need pre-defined crack path information from the test, in addition, to fracture toughness and cohesive material properties. This seriously limits the capability of both DCZM and VCCT approaches. The proposed approach in this paper focuses mainly to use DCZM and VCCT approach as test prediction methods rather than test duplication methodology. The methodology works in two stages: first, analyzing an FE model with the strength-/strain-based PFA; second, performing crack propagation analysis with either DCZM or VCCT approach. The approach is designed as a test prediction/reduction strategy for analyzing crack initiation and propagation problems. The first step of simulating a problem with PFA helps determine the expected crack-path initiation and propagation information that both DCZM and VCCT methodologies require solely based on material stiffness and strength properties. The second step helps improve the simulation results and overcome the stress singularity issues predominate in crack propagation fracture analysis problems (associated with removal of elements in FE simulations). The methodology was validated by simulating the delamination process (initiation and propagation) in composite structural components such as joints and z-pinned reinforced composite beam structures. The two step strategy has been applied to a skin/stringer structural joint debonding analysis subjected to a tension and a three-point bending load. The FE models are a three dimensional surface crack problem analyzed using PFA and DCZM approach consisting predominantly of Mode II (tension) and Modes I and II (3 point bending) failure. All simulation results are in close agreement with the test data. It is noted that strength-based PFA approach alone gave results which are in good agreement with the test data and was able to show secondary failure mechanisms not visible when DCZM alone was used to analyze the joint. On the other hand, the usefulness of the DCZM methodology is apparent for simulating the crack propagation in z-pinned unidirectional composite double cantilever beam (here crack path is well defined). The results are within reasonable agreement with the test data.

Keywords: Discrete Cohesive Zone Modeling, Virtual Crack Closure Technique, Progressive Failure Analysis, Z-pins, Composites Joints

I. Introduction

Fracture mechanics based Discrete Cohesive Zone Modeling (DCZM) and Virtual Crack Closure

Technique (VCCT) techniques are usually recommended for simulating crack propagation and delamination in composite structures.1-6 The VCCT approach is used for mostly linear elastic materials while DCZM for mostly nonlinear elastic materials. The VCCT and DCZM methodologies are gaining popularity because they are very effective and computationally efficient in simulating the failure modes in

1

50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference <br4 - 7 May 2009, Palm Springs, California

AIAA 2009-2563

Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

composites and metals by overcoming the stress singularity issues posed by finite element (FE) models containing sharp cracks.

Progressive Failure Analysis (PFA), on the other hand, is a widely accepted methodology for simulating composite materials. Unlike metals, which usually fail homogenously due to their isotropic material properties, composites show failure heterogeneously (ply level) before catastrophic failure. For this PFA has proved to be an ideal approach. PFA uses micro-mechanics approach to assess damage at the fiber/matrix level and progressively damages the plies until the laminate fails. To account for the damaged plies in FEM, the damaged elements (representing damaged plies) are either removed from the FE model or are assigned low (~1%) stiffness of the original material stiffness. Based on the type of failure (matrix or fiber), the stiffness is reduced in that direction. This sometimes induces stress concentration issues and may result in over or under estimating the strength values for the composite structure during the simulation when plies are modeled discretely in an FE model.

Both DCZM and VCCT methodologies carry the potential to minimize the stress concentration issues posed by damaged or sharp cracks in FE models. However, an engineer needs to know the crack path and other cohesive material properties for implementation in the FE model before the simulation. This additional information comes mostly from test data, which in turn renders the linear elastic fracture mechanics based approaches test duplication methodologies rather than test prediction approach. This is usually the case in complex problems, such as joints or skin/stringer panels. However, for simple problems like double cantilever beam (DCB) test, the crack path is well defined and DCZM can be used as a test prediction approach rather than test duplication. In this paper, a two step strategy is proposed that involves the use of both PFA and DCZM or PFA and VCCT approach; PFA for predicting the crack path and DCZM or VCCT for crack propagation for improved results.

II. Methodology A. Progressive Failure Analysis (PFA)

Progressive Failure Analysis is the approach widely used for failure analysis in composite materials. In this approach, failure in each ply constituting the laminate, under investigation, is assessed based on the ply limit strength/strain properties, as shown in Figure 1. Some failure criterias are maximum stress- or strain-based and some interactive. The damaged plies are then either removed from the FE model or given a negligible stiffness to simulate a broken ply and study its effect on overall laminate. In this paper, once all the plies in an element are damaged, the element is removed. The stresses in individual plies are obtained using classical laminate theory, which are then compared against the limit stresses or strains parameters using default failure criterias.7 Micro-mechanics is usually used to find the ply strength properties if fiber/matrix constituent properties are used as an input to the code. A commercial damage and durability tolerance software, GENOA,8 is used in conjunction with ABAQUS FE solver. ABAQUS is used to evaluate the stresses, strains and displacement values at the element level and GENOA is used to evaluate damage at the ply level constituting the element in the FE model. The process is iterative and goes back and forth between GENOA and ABAQUS. Once GENOA assesses that the element is damaged, it automatically controls the material stiffness matrix which is fed as an input to ABAQUS for further stress calculations. The process repeats until the structural component fails. GENOA has about 25 strength- and strain-based failure criterias and has option to define ply properties as fiber/matrix or ply properties. The software is an extension to the CODSTRAN software originally developed at NASA.9 Details about the PFA can be found elsewhere.7,8

2

Unit cell at node

2D Woven

Laminate

Sliced unit cell

Component FEM

Vehicle

Micro-Scale

Traditional FEM stops hereGENOA goes down to micro-scale

Lamina

3D Fiber

FEM results carried down to micro scale Reduced properties propagated up to vehicle scale

Unit cell at each element

Figure 1: Progressive Failure Analysis: Micro-macro scale interaction in GENOA.7

B. Discrete Cohesive Zone Modeling (DCZM)

The DCZM is a widely used approach for simulating fracture initiation and subsequent growth when material non-linearity effects are significant.2,3,6. However, it can be used for linear materials as well. Again GENOA and ABAQUS are used for this study. In GENOA, DCZM uses spring type elements and applies the cohesive law as the spring internal force versus nodal separation. The DCZM is different from widely used continuum cohesive zone model (CCZM). In CCZM the cohesive zone model is implemented within continuum type elements (usually shell and solid elements) and the cohesive law is applied at each integral point. DCZM uses a triangle type cohesive law for mixed mode (Figure 2) and is very similar to the methodology used in CCZM with the exception of continuum elements. Cohesive strength and stiffness define the triangle in cohesive law. The cohesive strength refers to the strength where the stiffness of the virtual spring type elements drops to simulate the non-linear response of adhesives and cohesive stiffness is the initial stiffness of these spring elements until the spring elements reaches its cohesive strength (Figure 3). The cohesive strength usually is assumed to be same as the material tensile and shear strength for Modes-I, II, and III type failures. Cohesive stiffness is taken usually to be large number (~stiffness of the material) and may need to be calibrated for improved results.

(a) Triangle type cohesive law for mixed mode failure analysis

(b) Virtual Discrete spring elements

Figure 2: Discrete Cohesive Zone Model (DCZM).6

In the chosen software, using DCZM technique with FE analysis explicitly determines the fracture

mode separation. Using this technique, for crack propagation analysis, the crack advancement is controlled via sequentially releasing the nodes along the crack path. The nodes in an FE model are initially tied together with highly stiff virtual spring elements and are released when mode I, II, and III components of strain energy release rate exceed the mixed-mode fracture criteria. The fracture criteria compares the instantaneous fracture energy release rate with the critical fracture energy release rate values of the material

3

under consideration using either B-K (Benzeggagh-Kenane) or Power law. The instantaneous fracture energy release rates are calculated based on the mesh size at the crack front, nodal forces at the crack tip, relative nodal displacements behind the crack tip and crack edge or surface.1,5 Further details of the approach can be found elsewhere.2,3,6

In the next section, simulation results are presented for different types of problems. First, skin/stringer

(lap type) joint subjected to tension and three-point bending load is analyzed via PFA approach alone. Once the crack path is known from PFA predictions, the DCZM technique is used to simulate the joint again under tension and three-point bending loads. Finally, DCZM approach is used to simulate mode-I delamination for z-pin reinforced composite DCB test.

III. Results A. Progressive Failure Analysis (PFA)

Using PFA approach a skin/stringer (lap type) joint, as shown in Figure 3, is analyzed under tension and three-point bending. The joint consists of a bonded skin and flange assembly (Figures 3 and 4). The skin lay-up consists of 14 plies ([0/45/90/-45/45/-45/0]s) and the flange consists of 10 plies ([45/90/-45/0/90]s). The flange and the skin are bonded together with CYTEC 1515,10 a grade-5 film adhesive, with a final thickness of 0.102 mm.11 A three-dimensional FE model (Figure 4) is prepared using 8-node solid elements in ABAQUS according to the dimensions (Figure 3) given in experimental work presented by Krueger et al.,11. The adhesive layer of 0.102 mm in thickness was also represented with additional elements in between the elements representing the skin and the flange of the joint.

Figure 3: Co-cured Skin/Stringer Joint.1

4

(a) Ply lay-up setup (side view).

(b) Three dimensional 8-node solid element FE model

Figure 4: Co-cured Skin/Stringer Joint FE Model Setup in GENOA.

Each ply in the flange and the skin is made of IM6/3501-6 graphite epoxy prepreg tape with a nominal thickness of 0.188 mm. The stiffness and Poisson’s ratio of both the unidirectional tape and adhesive are taken from Krueger et al.,11 and are listed in Table 1 below. Since PFA approach works based on material strength properties, the IM6/3501-6 strength properties were obtained from Camanho et al.,1 and UCSD test data12 and adhesive 1515 from CYTEC product sheet.10 Since the coefficient of thermal expansion (CTE) for the adhesive is not available, the effect of thermal residual stresses due to curing the joint is not considered here (∆T=0۫ C).

Table 1: Material properties for IM6/3501-6 unidirectional graphite epoxy tape. E11 E22=E33 ν12= ν13 ν23 G12=G13 G23

[GPa] [GPa] [-] [-] [GPa] [GPa]144.7† 9.65† 0.3† 0.45† 5.2† 3.4†

S11T S11C S22T S22C S33T S33C S12S S23S[MPa] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa]

2563.0* 1864.0* 61.0† 105.2* 61.0† 105.2* 68.0† 68.0†

† (Ref.1,11) * (Ref. 12)

Table 2: Material properties for CYTEC 1515 film adhesive. Em νm SmT SmC SmS

[GPa] [-] [MPa] [MPa] [MPa]1.72† 0.3† 42.12* 105.2* 37.0**

† (Ref. 11) * (Ref. 12) ** (Ref. 10)

5

A.1 Tension of skin/stringer joint The skin/stringer joint is subjected to tension loading as shown in Figure 3. Figure 5 below shows the

load versus displacement results obtained from PFA simulation and comparison with test data.1,11 Using the PFA approach, the predicted delamination load is 22.2 kN and is within the upper and lower bounds of the test data.1,11 The simulated load displacement curve matches the test curves with reasonable accuracy. Figure 6 show damage initiation and final delamination of the flange from the skin because of the failure of the adhesive. As can be expected, first the 90 degree plies fail near the flange edge (Figure 6e) which is a matrix dominated failure. Immediately after that the adhesive fails and the flange delaminates, as shown in Figure 6f. This is consistent with the experimental observations presented in Krueger et al.,.11 The software indicated that the adhesive failed because of higher in-plane shear stresses in the adhesive. The delamination is indicated (removal of damaged elements) by the first drop in the load displacement curve which then continues to increase further as the skin caries the load again also a typical behavior shown by composite structures until skin fails as well.

0

5

10

15

20

25

30

0 0.05 0.1 0.15 0.2Extensometer Displacement [mm]

Load

[kN

]

Test [ref 11]Test [ref 1]Simulation (PFA)

Test Bounds

Figure 5: PFA predicted and experimental load-extensometer measurement curves for skin/stringer

specimens under tension1,11; ∆T=0۫ C

(a) Damage initiation at 22.2 kN due to damage in 90 deg skin plies near the flange edges

6

(c) Damage propagation at

22.6kN

(b) Damage Propagation at 22.6 kN

(d) Fracture Initiation (delamination) at 22.85kN

(e) transverse tension in 90 degree ply

constituting the skin (f) PFA predicted delamination of the flange

from the skin

Figure 6: PFA prediction for damage Initiation and propagation in the adhesive and 90 degree ply in the skin near the flange at peak ~22.2 kN for skin/stringer specimens under tension; ∆T=0۫ C

A.2 Three-point bending of skin/stringer joint

The skin/stringer joint is later subjected to out-of-plane (3-point bending) loading, as shown in Figure 3. Figure 7 below shows the load versus displacement results obtained from PFA simulation and comparison with test data.1,11 PFA predicts the delamination load to be 417.0 N—again very close to the test data (~440 N). The simulated overall load displacement curve slope (stiffness) matches the test curves with reasonable accuracy. Note that the tension strength value of the adhesive is taken as the tensile strength of 3501-6 epoxy,12 as it is not provided in CYTEC product sheet for film adhesive 1515. The adhesive is subjected to both high shear (Mode-II) and tension stresses (Mode-I) during the simulation and results in the failure of the adhesive. Figure 8 show damage initiation (Figure 8a), propagation (Figure 8b) and final delamination (Figure 8c) of the flange from the skin because of the failure of the adhesive. Unlike the tension loading case in previous section, both the skin and the flange material does not show any damage before the flange delaminates from the skin. The simulation indicated that the flange delaminates almost catastrophically. The delamination is indicated by the first drop in the load displacement curve which then continues to increase further as the skin caries the bending load again.

7

0

100

200

300

400

500

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Displacement [mm]

Forc

e [N

]

Test [ref 11]Test [ref 1]Simulation (PFA)

Test Bounds

Figure 7: PFA predicted and experimental load-extensometer measurement curves for skin/stringer

specimens under three-point bending1,11; ∆T=0۫ C

(a) Damage initiation at 417 N

8

(b) Damage (delamination) propagation

(c) Delamination (adhesive failure)

Figure 8: PFA prediction for damage Initiation and propagation in the adhesive at 417 N for

skin/stringer specimens under three-point bending; ∆T=0۫ C

PFA results above clearly show the power of micro-mechanics-based methodology that, not only captures secondary failure modes in the skin, but also predict the crack path and propagation in joint subjected to different loading conditions. Next, the skin/stringer joint analysis is repeated incorporating the crack path information predicted by PFA simulation runs above using the DCZM methodology. B. Discrete Cohesive Zone Modeling (DCZM)

The validation of the DCZM begins via simulation of the skin/stringer joint analysis performed in the previous section via PFA approach. Here the adhesive layer is replaced with virtual spring elements that connect node pairs, in the skin and the flange, whose stiffness and strength are defined via cohesive triangular law (Figure 2). B.1 Skin/Stiffener Joint Analysis

One advantage of using DCZM approach is that coarser mesh can be used compared to the PFA approach. However, cohesive stiffness, strength and fracture toughness parameters need to be determined, in addition, to the crack path. Another three dimensional FE model was generated similar to the one in Ref 1. The FE model consists of 828 8-node solid elements. Note that the adhesive layer is not explicitly modeled like FE model used for PFA approach. Here the two components, flange and the skin, are connected to each other via virtual spring elements (shown with red and green color in Figure 9) in GENOA software.

9

Figure 9: FE model for skin/stringer used with DCZM spring elements in GENOA (shown in red and green color).

During the simulation same ply schedule lay-up and material properties (stiffness and Poisson’s ratio) were used as shown and listed in Figure 4a and Table 1, respectively. Note that the all strength-based failure criterias were off for the DCZM methodology. Table 3 below lists the fracture toughness (G1C, GIIC, GIIIC), cohesive strength (σ1C, σIIC, σIIIC) and stiffness (K1C, KIIC, KIIIC) values used for the simulation. The values were taken from Camanho et al.,1 in order to be consistent with the test data and their calibration for B-K Law. The main difference in simulation work of Camanho et al.,1 and current work is CCZM versus DCZM.

Table 3: Cohesive material properties for the DCZM springs for the interface.1

GIC GIIC=GIIIC σIC σIIC= σIIIC KIC KIIC=KIIIC η (B-K Law) [N/mm] [N/mm] [MPa] [MPa] [N/mm3] [N/mm3] [-] 0.075 0.547 61 68 (10.0)6 (10.0)6 1.45

Next, the model is subjected to tension and three-point bending loading conditions. Subsections below

present the results and comparison with test data.1,11

B.1.1 Tension of skin/stringer joint

The skin/stringer joint is subjected to tension loading as shown in Figure 3. Figure 10 below shows the load versus displacement results obtained from DCZM simulation and comparison with test data.1,11 The simulated load displacement curve matches the test curves with reasonable accuracy. Using the DCZM approach, the predicted delamination load is also 22.2 kN and is within the upper and lower bounds of the test data.1,11 The post-delamination load-displacement curve is also within the range of test data. Figure 11 shows the Von Mises stresses in the joint at the peak load (22.2 kN) while Figures 12a and 12b show damage initiation and final delamination of the flange from the skin because of the failure of the interface, respectively. Notice the similarity in the delamination patterns in Figure 6a and Figure 12b. Unlike PFA approach, the secondary damage modes that appeared in the skin transverse plies are not accounted in the DCZM approach which could result in a stiffer response in the post-delamination point. The software indicates that the interface failed because of higher in-plane shear stresses (Mode II failure). For this simulation curing effect is taken into account since for plies the CTE values were available in Ref. 1. The CTE for unidirectional plies are: α11 = -2.4*10-8 1/۫C and α22 = 3.7*10-5 1/۫C.

0

5

10

15

20

25

30

0 0.05 0.1 0.15 0.2Extensometer Displacement [mm]

Load

[kN

]

Test [ref 11]Test [ref 1]Simulation (DCZM)

Test Bounds

10

Figure 10: DCZM predicted/duplicated and experimental load-extensometer measurement curves for skin/stringer specimens under tension1,11; ∆T=-150۫ C

Figure 11: Von Mises stress at peak load (22.2 kN) just before delamination initiation for

skin/stringer specimens under tension; ∆T=-150۫ C

(a) 23.9 kN (b) 24.0 kN

Figure 12: DCZM prediction/duplication for initiation (a) and propagation (b) of delamination in the interface for skin/stringer specimens under tension; ∆T=-150۫ C

B.1.2 Three-point bending of skin/stringer joint

The skin/stringer joint is subjected to three-point bending load as shown in Figure 4. Figure 14 shows the load versus displacement results obtained from DCZM simulation and comparison with test data [1,11]. The simulated load displacement curve matches the test curves with reasonable accuracy. Using the DCZM approach, the predicted delamination load is 403.2 N and is within the upper and lower bounds of the test data.1,11 Figure 15 shows the Von Mises stresses in the joint at the peak load (403.2 N) while Figures 16a, 16b and 16c show damage initiation and final delamination of the flange from the skin because of the failure of the interface, respectively. Notice the similarity in the delamination patterns in Figure 9c and

11

Figure 16b and 16c. The software indicates that the interface failed because of higher in-plane shear and out-of-plane tensile stresses (Modes I and II failure). For this simulation curing effect is taken into account since for plies the CTE values were available in Ref. 1.

0

100

200

300

400

500

600

700

0.0 1.0 2.0 3.0 4.0 5.0

Displacement [mm]

Forc

e [N

]

Test [ref 7]Test [ref 1]Simulation (DCZM)

Test Bounds

Figure 13: DCZM predicted/duplicated and experimental load-extensometer measurement curves for

skin/stringer specimens under three-point bending1,11; ∆T=-150۫ C

Figure 14: Von Mises stress at peak load (403.2 N) just before delamination initiation for

skin/stringer specimens under three-point bending; ∆T=-150۫ C

12

(a) 403.2 N (b) 544.8 N

(c) 571.0 N

Figure 15: DCZM prediction/duplication for initiation (a), propagation (b) and final (c) delamination in the interface for skin/stringer specimens under three-point bending; ∆T=-150۫ C

Next, DCZM capabilities are explored for another complex composite problem—a z-pinned reinforced double cantilever composite beam (DCB test). Simulating a simple and basic test like this can help calibrate the cohesive strength and stiffness properties which can further be used to simulate more complex examples like skin/stringer joint analysis above. Camanho et al.,1 used similar approach to calibrate the power term (η=1.45) for B-K critical criteria. Notice, unlike complex problems, the crack path is well defined in DCB test. B.2 Double Cantilever Beam (DCB) Analysis of a Z-pinned unidirectional composite material

A method for enhancing fracture toughness of laminated composite structures uses pins inserted normal to the laminate plane to increase delamination resistance.13,14 These pins are usually referred to as Z-pins. To estimate the effect of adding Z-pins on delamination resistance of the composite laminate, DCB tests are performed that fails under Mode-I failure mode and where the crack is well defined.

Here a three dimensional FE model (Figure 16) was prepared in ABAQUS using thick shell elements (Reissner-Mindlin). The DCB coupon is 18 mm wide, 153 mm in length, and total 6 mm in height (Figure 16). The initial crack is 73 mm running along the length of the coupon. They both were attached using virtual spring elements via DCZM feature in GENOA. Each spring follows the cohesive law defined via parameters such as cohesive strength and stiffness and fracture toughness for all three failure modes. Since DCB is primarily a Mode-I failure; therefore, only mode-I values were entered, as shown in Table 4, and the remaining direction parameters were defined via large value to negate their response. The cohesive properties were obtained from Ratcliff et al.,14 and Appendix A shows the detailed calculations to show how they were obtained. Figure 17 shows the virtual spring elements (in red) assigned to the FE model. The virtual springs also have areal density of 1.5% implying each spring represents each z-pin in the laminate. The mesh was generated such that each node represents the Z-pin location. Z-fiber details are given in Table 5. The upper and lower arms of the DCB model were assigned unidirectional ply schedules

13

with [0]48 lay-up where each ply has a thickness of 0.0625 mm and is made of IM7/8552 fiber/matrix material. The material properties used are listed in Table 6.

Figure 16: FE model of IM7/8552 DCB specimen containing 1.5% areal density of Z-fibers13,14; σ1C=

6.25 MPa; k1C =873.52 MPa/mm; G1C = 6.250 N/mm

Figure 17: DCZM virtual spring elements are visible in the DCB specimen containing 1.5% areal

density of Z-fibers13,14; σIC= 6.25 MPa; kIC =873.52 MPa/mm; GIC = 6.250 N/mm.

Table 4: Cohesive material properties for the DCZM springs for the z-pins (areal density is 1.5% and FE model in Figure 17).14

G IC G IIC =G IIIC σIC σ IIC =σ IIIC K IC K IIC =K IIIC

[N/mm] [N/mm] [MPa] [MPa] [N/mm 3 ] [N/mm 3 ]6.25 (10.0)4 6.25 (10.0)8 873.52 (10.0)6

Table 5: Z-fiber Details.14

Diameter (d) Length (Lp) Axial Modulus (Ep) Spacing (a)[mm] [mm] [GPa] [mm]0.28 4 227 2

Areal Density (a.b)[%]1.5

14

Table 6: Material properties for IM7/8552 unidirectional graphite epoxy tape.15

E11 E22=E33 ν12= ν13 ν23 G12=G13 G23[GPa] [GPa] [-] [-] [GPa] [GPa]

160 9 0.34 0.5 5.6 2.8

S11T S11C S22T S22C S33T S33C S12S S23S[MPa] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa]2560.0 1590.0 73.0 185.0 63.0 185.0 90.0 57.0

Figure 18: Longitudial-out-of-plane stresses at peak load of 290.0 N for DCB specimen containing

1.5% areal density of Z-fibers13,14; σIC= 6.25 MPa; kIC =873.52 MPa/mm; GIC = 6.250 N/mm

0.0 N Peak load 286.5 N

250.0 N

Figure 19: Estimated load displacement response of IM7/8552 DCB specimen containing 1.5% areal density of Z-fibers13,14; σIC= 6.25 MPa; kIC =873.52 MPa/mm; GIC = 6.250 N/mm

15

050

100150

200250

300350

0.0 10.0 20.0 30.0

Displacement [mm]

Load

[N]

Test [ref 11]Simulation [ref 12]Simulation

Figure 20: Estimated load displacement response of IM7/8552 DCB specimen containing 1.5% areal

density of Z-fibers13,14; σIC= 6.25 MPa; kIC =873.52 MPa/mm; GIC = 6.250 N/mm

Figures 18 shows the stresses induced at the peak load of about 290 N. On the other hand, Figure 19 shows the delamination process predicted by the DCB approach. The load-displacement curve in Figure 20 agrees well with both test data from Ref. 13 and simulation results from Ref. 14. In contrast to the approach taken in Ref. 13 and Ref. 14, full anisotropy of the material is considered and the FE model is reduced to thick shell elements further simplifying the FEM capability in conjunction with DCZM approach. If the Z-pin density is changed then the methodology requires the mesh is to be changed and the cohesive strength and stiffness parameters will have to be recalculated according to the formulation in Appendix A.

IV. Conclusions Combination of micro-mechanics, damage-mechanics and fracture mechanics is introduced. Damage-

mechanics formulation determines five stages of critical damage events: 1) damage initiation, 2) damage propagation, 3) fracture initiation, 4) fracture propagation and 5) residual strength. Damage mechanics formulation determines accurately stages 1 to 3 and may capture 4 and 5 depending on the crack size and FE models. Combined failure theory assists in determining the fracture propagation (stage 4 and onwards) for blunt and sharp notches and predict more accurate load-displacement relationship after the peak load where residual strength becomes important and acts as a double check to the PFA approach.

PFA prediction of subjecting the skin/stringer joint to tension and 3-point bending load matched reasonably well with the test data and predicted the delamination of the flange from the skin due to adhesive failure. The simulation was then repeated again using DCZM technique with crack path predicted by PFA to improve the stage 4 and 5 of the analysis. The results were again in close agreement with the test data from literature.1,11 DCZM approach potential is then highlighted by simulating a DCB test for Z-fiber reinforced composite beam problem. Here each Z-fiber is represented via a equivalent and discrete cohesive spring element. The simulated results were again in good agreement with the test data and other simulations using similar methods. Except considering full anisotropy of the problem and with simplified FE model using shell elements.

16

In brief, PFA results clearly show the potential of the micro-mechanics strength-based approach, as a methodology to predict the delamination process and crack path in composites material and other advance materials including sandwich structures based on similar principles. More accurate results can be generated if material strength properties are known. Composite materials are known for a wide variation in their stiffness and strength properties from batch to batch; therefore, in order to use PFA as a predictive tool with much higher accuracy, proper and calibrated material with basic 5 ASTM standard unidirectional static tests properties should be used as an input for simulating structural components for damage and durability type analysis. Similarly, in order to use the DCZM to its full potential, the cohesive properties should be calibrated with basic DCB tests before using them to simulate complex problem. Without proper validation of material properties in PFA or cohesive properties in DCZM, predicting the composite material failure behavior may result in unreasonable predictions more than the industry certification accepted ±10% error process.

Appendix The initial Z-fiber row stiffness, ko, is calculated using the following equation:14

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

p

ppo L

AErk (1)

Where Ep, Ap (=π/4d2), and Lp are the axial tensile modulus, cross-sectional area and length of a single Z-fiber, respectively. r (= width / b) is the number of Z-fibers per row. b and d are the width-wise spacing and diameter of the Z-fibers. The Z-fiber row displacement corresponding to the onset of damage, zo, is calculated from the row stiffness, ko, and the peak load, Fc, measured from a single Z-fiber pull-out test, such that:

⎥⎦

⎤⎢⎣

⎡=

o

Co k

Frz (2)

The Z-fiber displacement corresponding to complete removal from the specimen, zf, is equal to the insertion depth into one of a DCB specimen arms. The cohesive strength, σIC, and stiffness, KIC, can then be calculated using the following equations:

⎥⎦⎤

⎢⎣⎡=

abFC

ICσ ; (3)

⎥⎦

⎤⎢⎣

⎡=

o

ICIC z

Kσ

(4)

Areal Density ⎥⎦

⎤⎢⎣

⎡=

abd )4/( 2π

(5)

In Equations 3 to 5, a, is the length-wise spacing of the Z-pins. Now using data from Table 5 and Equations 1 to 5 above, we get the following Test data from fiber pull-out test14 and bridging law gave ko = 31446 N/mm; zf = 2 mm; zo=0.007 mm; r = 9 From Equation 2 and above parameters, we get Fc = 25.0 N Then using Equation 3, and recalling that a = b = 2.0 mm, we get σIC = 6.25 N/mm2

17

KIC = 873.52 N/mm3

Acknowledgments The Authors of this paper are grateful and highly appreciate guidance and financial support of Dr. Frank Abdi.

References

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