MSE 1103 012 Cohesive Zone Modeling to Predic Failure Processes

Canadian Journal on Mechanical Sciences & Engineering Vol. 2, No. 3, March 2011

42

COHESIVE ZONE MODELING TO PREDICT FAILURE PROCESSES

E. A. Bonifaz Universidad San Francisco de Quito, Casilla Postal: 17-12-841 Cumbay, Quito-Ecuador

Email: [email protected] Tel.: 593 2 2971700 Fax: 593 2 2890070

ABSTRACT

A two dimensional finite element asymptotic crack tip model (ABAQUS code) to simulate crack tip opening displacement (CTOD) has been developed. The finite element mesh is constructed using both, plane strain solid and cohesive elements. CTOD profiles and damage variation along cohesive element path is clearly observed. Results showing material separation are calculated for different ratios of solid/cohesive Youngs modulus. Evolution of CTOD with time for various ratios of solid/cohesive Youngs Modulus is also presented. For ratios of solid/cohesive Youngs Modulus: 10, 8 and 6, the analysis had completed successfully in a total time of 1, i.e. =1. On the other hand, the analysis had not completed successfully for ratios: 4, 2, and 1, i.e., computations are partially unstable in the numerical solution at stress intensity factor ranges larger than = 0.5 for the used aluminum alloys parameters. K is an applied stress intensity factor range, K0 is the fracture toughness. A comparison of stress vs. true distance along cohesive path for two different stress-strain curves is also shown. Results of force vs. displacement show that for equal load conditions, the stiffness degradation process is observed in materials with lower stress-strain curves.

Keywords: Asymptotic crack tip model, cohesive energy, finite element modelling, CTOD, Cohesive layer modulus.

1. INTRODUCTION

Predicting the properties and performance of materials is central to the success of major industries producing a vast range of consumer goods and to research programs at laboratories and universities around the world [1]. In recent years, the advent of powerful, massively parallel computers, coupled with spectacular advances in the theoretical framework that describes materials, has enabled the development of new concepts and algorithms for the computational modeling of materials. As the field of computational materials science develops and matures, the notion is taking hold in the community that modeling efforts should be an integral part of interdisciplinary materials research and must include experimental validation [1].

In multiscale modeling, the goal is to predict the performance and behavior of complex materials across all relevant length and time scales, starting from fundamental physical principles and experimental data. The challenge is tremendous. At the atomic (nanometer) scale, electrons govern the interactions among atoms in a solid, and therefore quantum mechanical descriptions are required to characterize the collective behavior of atoms in a material. However, at the engineering scale, forces arising from macroscopic stresses and/or temperature gradients may be


43

the controlling elements of materials performance. At scales in between, defects such as dislocations control mechanical behavior on the microscale (tens of micrometers), while large collections of such defects, including grain boundaries and other microstructural elements, govern mesoscopic properties (hundreds of micrometers). The net outcome of these interactions can be described as a constitutive law that ultimately governs continuum behavior on the macroscale (centimeters) [1].

There is not yet a complete answer to fatigue cracking in ductile materials where the crack may arrest before it severs the solid into separate pieces. The reason is because the specific physical mechanisms are material-dependent and the responses becomes nonlinear, and in general, involve a broad range of length scales. The challenge lies in the fact that all scales are connected, and all may contribute to the total fracture energy. The nonlinear response is attributed to observed plasticity phenomena, which display a size effect whereby the smaller is the size the stronger is the response [2]. In an effort to extend plasticity theory to small scales, we focus on room-temperature crack growth in polycrystalline materials under circumstances where dislocation glide is the dominant mechanism of plastic dissipation. The steep gradients of plastic strain that appear in the plastic zone at the crack tip will be much better predicted if we include the concept of cohesive zone modeling (CZM). CZM is an alternative method to model separation. It is an ideal framework to model strength, stiffness and failure in an integrated manner. In the following paragraphs we will concentrate on the normal stress component across the cohesive surface and normal separation.

In numerical simulations, surface elements are introduced at the boundaries of solid elements along a pre-defined crack path. The constitutive relation, T(), of the interface elements represents the effective mechanical behavior due to the physical processes of micro-void nucleation, growth and coalescence in a ductile material. Commonly, the cohesive law is defined by two parameters, a cohesive strength (CS),

c = max,0, and a critical separation, sep=

c, or,

alternatively, a separation energy (SE), c, which simply represents the area under the traction-

separation curve (Fig. 1).

nnnc dTc

= )(0

(1)

The constitutive equation relating Tn to n for a reversible interface is constructed as follows:

The traction-displacement relation is most conveniently characterized by a scalar inter-planar potential (n) by setting

n

nT

= (2)


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The value of represents the work done per unit area in separating the interface by (the separation between two surfaces of the cohesive zone). A number of different functions are used to approximate. For instance, Equation (3) is the simplified version of a potential suggested by Xu and Needleman [3].

+=

n

n

n

n

nnn exp1)( (3)

Here, n denotes the normal separation between two surfaces of the cohesive zone, n (the cohesive energy) and n (the characteristic cohesive length) are material properties. The cohesive energy is related to the cohesive strength (max,0) through the equation (4):

nn 0max,)1exp(= (4) By differentiating the potential (n), the continuum normal traction Tn is determined:

=

n

n

n

n

nT 1exp0max, (5)

Under purely normal tensile loading, the interface has work of separation n, and the normal traction reaches a value of max,0 at an interface separation n = n (Fig. 1)

Fig. 1. Exponential traction-separation curve

Normal separation,

Trac

tion

, Tn

c = max,0

n c = sep

c = n

nANo material separation

C

B D

E


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Figure 2 shows a schematic of the concept of wake and forward region in the cohesive process zone. Note the correspondence of terminology between Figures 1 and 2.

Figure 2. Concept of wake and forward region in the cohesive process zone.

2. COHESIVE ZONE MODELING

Material separation is assumed to occur through atomic scale separation, i.e. cleavage of atomic layers [4]. A cohesive zone model prescribes a potential function relating material separation and cohesive energy per surface area along an assumed crack path.

Constitutive Behavior

In a typical ABAQUS [5] cohesive zone model, a damage evolution law describes the rate at which the material stiffness is degraded once the corresponding initiation criterion is reached. A scalar damage variable, D, represents the overall damage in the material and captures the combined effects of all the active mechanisms. It initially has a value of 0. If damage evolution is modeled, D monotonically evolves from 0 to 1 upon further loading after the initiation of damage. The dependence of the fracture energy on the mode mix can be defined based on a power law fracture criterion. The power law criterion states that failure under mixed-mode conditions is governed by a power law interaction of the energies required to cause failure in the individual (normal and two shears) modes. It is given by

+

+

= 1 (6)

with the mixed-mode fracture energy Gc = Gn + Gs + Gt when the above condition is satisfied. In the expression above the quantities Gn, Gs, and Gt, refer to the work done by the traction and its conjugate relative displacement in the normal, the first, and the second shear directions, respectively. The user specifies the quantities Gnc, Gsc, and Gtc, which refer to the critical fracture energies required to cause failure in the normal, the first, and the second shear directions, respectively. In ABAQUS, the option *DAMAGE EVOLUTION, TYPE=ENERGY, MIXED MODE BEHAVIOR=POWER LAW, POWER=, Gnc, Gsc, Gtc, is used to provide material properties that


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define the evolution of damage based on a power law fracture criterion. In the present study, this approach was used for all the calculations. However, an exponential traction-separation curve (Fig. 1) determined from the cohesive potential represents much better the non-linear elastic constitutive behavior. It is for this reason that cohesive zone modeling for normal separation based on an exponential traction-separation curve (Fig. 1) is currently underway and will be published later. The exponential traction-separation curve will be incorporated in the analysis through the development of an ABAQUS-UMAT subroutine.

Material parameters

To test the model, material properties representative for Aluminium alloys were used. As we were interested in obtaining results showing material separation for different ratios of solid/cohesive Youngs modulus, the following values were included in the model: Solid Material Youngs Modulus ESOL= 60E3 MPa, Cohesive Youngs Modulus EMOD = 6E3, 7.5E3, 10E3, 15E3, 30E3, and 60E3.

We employ isotropic hardening with the stress-strain curve shown on Fig. 3 and initial yield strength , 250 . The material separation processes is characteristic of atomistic separation and cleavage processes. Consequently, the cohesive strength is related to Youngs modulus , = !/20 [4]. For a value of EMOD = 60E3 MPa, , = 3000 and the ratio $%&',($),( = 12. According our preliminary results, no crack propagation would occur at higher ratios in a framework of conventional plasticity. The reason is because under cyclic loading and without incorporation of any stress enhancement of the cohesive strength due to the presence of dislocations (strain gradient plasticity), the tractions would never exceed the cohesive zone strength and the model would shakedown, i.e. lead to infinite fatigue life.

Fig. 3. Aluminum alloy stress-strain curve used in the finite element analysis.

The cohesive length of n= 1 nm approximates the characteristic separation data of the Morse potential of Al [4]. The resulting cohesive energy is nn 0max,)1exp(= which corresponds to fracture toughness in terms of a reference mode I stress intensity factor of

= *+, (1 /0) (7)

050

100150200250300350400

0 0.05 0.1 0.15 0.2

Stre

ss, M

Pa

Strain

Aluminium alloy Stress-Strain Curve


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The Asymptotic Finite Element Model

In conventional FE modeling, the asymptotic crack tip geometry is modelled by means of a FE mesh using plane strain solid elements [6]. Pure Mode I is considered with asymptotic displacements ux (r,) and uy (r, ) applied on the outer boundary of the mesh (See Fig 4 and Eqs. 9). The plastic regime is modelled by isotropic hardening and associated flow rule. The constitutive model is normally expressed in terms of stress-strain relations. When the material exhibits strain-softening behavior, leading to strain localization, this formulation results in a strong mesh dependency of the finite element results in that the energy dissipated decreases upon mesh refinement.

To determine the size of the FEM mesh, we begin by defining for the material under consideration, a characteristic strength and a characteristic distance. For the prediction of static (i.e. monotonic) strength, the characteristic strength, o, is simply the strength of a plain specimen, which is defined as a specimen containing no defects large enough to reduce its strength [6]. In practice such specimens may be difficult to obtain since critical defect sizes may be as small as 1m in some materials; however this is not a problem because the characteristic strength can be deduced from the strength of specimens containing cracks of known size obtained from standard fatigue experiments. The characteristic distance, L, is defined in terms of o and the materials fracture toughness, K0:

2

0

0.

1

=

pi

KL (8)

In this work, the characteristic distance L that defines the outer boundary of the elastic singularity zone (Figures 2 and 4) it will be used to construct the finite element mesh. The reason is because we consider that only beyond the distance L (or in its proximities) the elastic stress field and/or the asymptotic displacements (Eqs.9) apply. Even though the material is brittle, a plastic zone should spans at some critical length before crack. This is true since in many cases the initiation of a crack is not the critical stage, but its propagation beyond some critical length. Different to ductile materials, in brittle solids, once a crack begins to run, it runs very fast.

Fig. 4. FE mesh for crack tip problem using ABAQUS solid elements CPE4I. Mesh radius L = 0.1 mm.

outer boundary

crack face

crack tip

Symmetry


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Current Impact

Introduction of cohesive elements into an asymptotic Finite Element (FE) mesh eliminates singularity of stress and limits it to the cohesive strength of the material. All of the available damage evolution models use a formulation intended to alleviate the mesh dependency. CZM can create new surfaces and maintains continuity conditions mathematically, despite the physical separation. CZM represents physics of the fracture process at the atomic scale [7]. It can also be perceived at the meso- scale as the effect of energy dissipation mechanisms, energy dissipated both in the forward and the wake regions of the crack tip (See Fig. 2)[8].

The inclusion of cohesive elements in a finite element model consists of: choosing the appropriate cohesive element type, including the cohesive elements in a finite element model, connecting them to other components, understanding typical issues that arise during modeling using cohesive elements, defining the initial geometry (Figs. 4 and 5) and the mechanical constitutive behavior of the cohesive elements. The mechanical constitutive behavior of the cohesive elements was defined by using the above ABAQUS constitutive model specified directly in terms of traction versus separation description. For these models, the tractions are a function only of the relative displacement of the material planes adjacent to the interface, and are independent of the history or rate of loading. This means that the traction-displacement relation for the interface is reversible. The above described ABAQUS traction-separation model assumes initially linear elastic behavior followed by the initiation and evolution of damage.

Fig. 5. Asymptotic FE mesh using ABAQUS solid CPE4I and cohesive COH2D4 elements. Mesh radius 0.1 mm . Cohesive element thickness 1 nm. Shown with the plastic zone around the tip.


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The following program shows the FORTRAN user subroutine DISP developed to input the asymptotic displacements ux (r,) and uy (r, ) applied on the outer boundary of the mesh (Figs. 4 and 5) for mode I (Eqs. 9).

+

+

=

2cos

21)43(

2sin

2sin

21)43(

2cos

22 2

2

2

1

pi

r

GK

u

u I (9)

SUBROUTINE DISP(U,KSTEP,KINC,TIME,NODE,NOEL,JDOF,COORDS) C INCLUDE 'ABA_PARAM.INC' C DIMENSION U(3),TIME(2),COORDS(3) C R in mm, U in mm, GIc in MPa mm^0.5, EMOD and G in MPa TOL=0.0001 PI=3.1416 C EMOD ==> Cohesive layer modulus N/mm^2 EMOD=60.E3 V=0.3 G=EMOD/(2*(1+V)) DELTAN=1.0E-6 C ULTI ==> Ultimate cohesive strength in tensile mode I (N/mm^2) ULTI=EMOD/20. C Cohesive Energy (N/mm^2)*mm: COHENERGY=2.7183*ULTI*DELTAN C Fracture toughness (N/mm^2)*mm^0.5: GIc=((EMOD*COHENERGY)/(1-V**2))**0.5 R=0.1 5 THETA=ANGULO*PI/180. X=R-R*COS(PI-THETA) X2=X-TOL IF (COORDS(1).GE.X2.AND.JDOF.EQ.1) THEN U(1)=(GIc/(2*G))*SQRT(2*R/PI)*(COS(THETA/2)*(((3-4*V-1)/ 12)+SIN(THETA/2)**2)) U(1)=U(1)*TIME(2) ELSE IF (COORDS(1).GE.X2.AND.JDOF.EQ.2) THEN U(1)=(GIc/(2*G))*SQRT(2*R/PI)*(SIN(THETA/2)*(((3-4*V+1)/ 22)-COS(THETA/2)**2)) U(1)=U(1)*TIME(2) ELSE IF(ANGULO.LE.175) THEN ANGULO=ANGULO+5. GOTO 5 ENDIF RETURN END


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Fig. 6. Resultant of asymptotic displacements ux (r,) and uy (r, ) applied on the outer boundary of the mesh calculated with Eqs. 9 and material parameters described in subroutine DISP.

3. RESULTS AND DISCUSSIONS

Results showing material separation are calculated for different ratios of solid/cohesive Youngs modulus. In Fig. 7a, a zoom of the FEM mesh showing CTOD profile and damage (in red) along cohesive path is presented. In Fig 7b, the displacement U2 vs. distance along cohesive path for two ratios of solid/cohesive Youngs modulus, showing material separation is plotted. Fig 7c shows the damage (D) variation along the cohesive path.


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Fig. 7. a) FEM mesh showing CTOD profile and damage (in red) along cohesive path; b) Displacement U2 vs. distance along cohesive path for two ratios of solid/cohesive Youngs modulus, showing material separation; c) Damage variation along the cohesive path.

The material parameters used here are representative of FCC aluminum alloys. A comparison of stress vs. true distance along cohesive path for two different stress-strain curves is shown in Fig. 8a. In Fig 8b, the evolution of CTOD with time for various ratios of solid/cohesive Youngs Modulus is also presented. For ratios of solid/cohesive Youngs Modulus: 10, 8 and 6, the analysis had completed successfully in a total time of 1, i.e. = 1. On the other hand, the analysis had not completed successfully for ratios: 4, 2, and 1, i.e., computations are partially unstable in the numerical solution at stress intensity factor ranges larger than = 0.5 for the used material constants. K is an applied stress intensity factor range and K0 is the fracture toughness.


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Fig. 8. a) Stress S22 vs. True distance along cohesive path for two stress-strain curves; b) Evolution of CTOD with time for various ratios of solid/cohesive Youngs Modulus.

Figure 9 shows the results of force vs. displacement for node 3697 located in the outer boundary of the mesh (See Fig. 6). In this figure is clearly seen that for equal load conditions, the stiffness degradation process is observed in materials with lower stress-strain curves.

Fig. 9. Force vs. displacement for node 3697 (See Fig. 6). For equal load conditions, the stiffness degradation process is observed in materials with lower stress-strain curves.


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4. CONCLUSIONS

A two dimensional finite element asymptotic crack tip model (ABAQUS code) to simulate crack tip opening displacement (CTOD) has been developed. The finite element mesh is constructed using both, plane strain solid and cohesive elements.

For ratios of solid/cohesive Youngs Modulus: 10, 8 and 6, the analysis had completed successfully in a total time of 1, i.e. = 1. On the other hand, the analysis had not completed successfully for ratios: 4, 2, and 1, i.e., computations are partially unstable in the numerical solution at stress intensity factor ranges larger than = 0.5 for the used aluminum alloys parameters. K is an applied stress intensity factor range, K0 is the fracture toughness.

CTOD profiles and damage variation along cohesive element path is clearly observed. Results of force vs. displacement show that for equal load conditions, the stiffness degradation process is observed in materials with lower stress-strain curves.

This is a new effort for extending plasticity theory to small scales. The formulation of a physically correct continuum theory to connect the physics of the actual material separation process to the driving force for fracture is currently underway and will be published later.

REFERENCES

1. Toms Daz de la Rubia and Vasily V. Bulatov. Materials Research by Means of Multiscale Computer Simulation. MRS Bulletin/march 2001, Volume 26, No.3.

2. Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W., 1994. Strain gradient plasticity: theory and experiment. Acta Metall. Mater. 42 (2), 475487.

3. Xu and Needleman, (1995), J Mech. Phys. Solids 42 p. 1397 4. Brinckmann S. and Siegmund T. (2008). Modelling Simul. Mater. Sci. Eng. 16, 065003

(19pp) 5. ABAQUS v. 6.9 documentation 6. Bonifaz and Gil Sevillano (2000). PhD Thesis, Universidad de Navarra, Spain. 7. Namas Chandra. Department of Mechanical Engineering FAMU-FSU College of

Engineering Florida State University Tallahassee, Fl-32310. 8. Allan Bower, Applied Mechanics of Solids. CRC Press, Taylor & Francis Group 2010.

BIOGRAPHY


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Edison A. Bonifaz Birth Date: 06-25-1961 Place: Riobamba-Ecuador

Education: -PhD, Materials Engineering, Cum Laude. Universidad de Navarra, Spain. -Master of Science, Metallurgy. University of Illinois at Chicago, U.S.A. -Bachelor of Engineering, Mechanics. Escuela Superior Politcnica de Chimborazo, Ecuador.

Academia Appointments: -Professor, Mechanical Engineering Department, Universidad San Francisco de Quito, Quito-Ecuador. -Research Associate (Post Doc Fellow), Mechanical and Manufacturing Engineering Department, University of Manitoba-Canada. - Fulbright Research Visiting Professor, Department of Civil and Materials Engineering, University of Illinois at Chicago. -Associate Professor, Mechanical Engineering Department, ESPOCH, Riobamba-Ecuador.

Research Objectives: -Finite Element Modeling of Materials Processing (Laser Welding, GMAW, Functionally Graded Thermal Barrier Coatings (FG-TBC), Casting, Heat Treatments, Repair of Gas Turbine Components including CFD modeling). -Multiscale, Thermomechanical Modelling of Advanced, high Temperature Materials for Power Generation. Development of user material (UMAT) subroutines in ABAQUS to analyze thermo-mechanical deformation and associated continuum damage mechanics. -Cohesive zone models based on the micromechanics of dislocations (A unified treatment of fatigue crack initiation, fatigue crack growth, and stable tearing crack growth). -Thermo-mechanical fatigue analysis using the FE-SAFE software, www.safetechnology.com.

Review and Consulting for Publishers: -International Journal of Thermal Sciences -Journal of Materials Processing Technology

Honors and Awards: Korean Government Scholar United Nations Scholar Fulbright-Laspau Scholar Organization of American States Scholar AECI-MUTIS Scholar Fulbright Visiting Scholar

MSE 1103 012 Cohesive Zone Modeling to Predic Failure Processes

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