4.12. cohesive zone material (czm) model

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4.12.CohesiveZoneMaterial(CZM)Model

Fracture or delamination along an interface between phases plays a major role in limiting the toughness and the ductility of the multi-phase materials, such as matrix-matrix composites and laminated composite structure. This has motivated considerable research on the failure of the interfaces. Interface delamination can be modeled by traditional fracture mechanics methods such as the nodal release technique. Alternatively, you can use techniques that directly introduce fracture mechanism by adopting softening relationships between tractions and the separations, which in turn introduce a critical fracture energy that is also the energy required to break apart the interface surfaces. This technique is called the cohesive zone material (CZM) model . The interface surfaces of the materials can be represented by a special set of interface elements or contact elements, and a CZM model can be used to characterize the constitutive behavior of the interface.

The CZM model consists of a constitutive relation between the traction T acting on the interface and the corresponding interfacial separation (displacement jump across the interface). The definitions of traction and separation depend on the element and the material model.

The following related topics are available:

Interface ElementsContact Elements

4.12.1.InterfaceElements

For interface elements, the interfacial separation is defined as the displacement jump, (that is, the difference of the displacements of the adjacent interface surfaces):

The definition of the separation is based on local element coordinate system, Figure4.33:Schematicof Interface Elements. The normal of the interface is denoted as local direction n, and the local tangent direction is denoted as t. Thus:

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Figure4.33:SchematicofInterfaceElements

The following related topics are available:

Material Model - Exponential BehaviorMaterial Model - Bilinear Behavior

4.12.1.1.MaterialModel-ExponentialBehavior

An exponential form of the CZM model (input via TB,CZM), originally proposed by Xu and Needleman([363]), uses a surface potential:

where:() = surface potential

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e = 2.7182818

max = maximum normal traction at the interface (input on TBDATA command as C1 using

The traction is defined as:

or

and

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From equations Equation4364 and Equation4365, we obtain the normal traction of the interface

and the shear traction

The normal work of separation is:

and shear work of separation is assumed to be the same as the normal work of separation,

defined as:

For the 3-D stress state, the shear or tangential separations and the tractions have two components, t1 and t2 in the element's tangential plane, and we have:

The traction is then defined as:

and

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(In POST1 and POST26 the traction, T, is output as SS and the separation, , is output as SD.)

The tangential direction t1 is defined along ij edge of element and the direction t2 is defined along

direction perpendicular to the plane formed by n and t1. Directions t1, t2, and n follow the righthand

side rule.

4.12.1.2.MaterialModel-BilinearBehavior

The bilinear CZM model (input via TB,CZM with TBOPT = BILI) can be used with interface elements. The model is based on the model proposed by Alfano and Crisfield [365].

Mode I Dominated Bilinear CZM Model

The Mode I dominated bilinear CZM model assumes that the separation of the material interfaces is dominated by the displacement jump normal to the interface, as shown in the following figure:

Figure4.34:ModeIDominatedBilinearCZMLaw

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The relation between normal cohesive traction Tn and normal displacement jump n can be expressed

as:

Where:

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For Mode I dominated cohesive law, the tangential cohesive traction and tangential displacement jump behavior is assumed to follow the normal cohesive traction and normal displacement jump behavior and is expressed as:

Were:

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Mode II Dominated Bilinear CZM Model

The Mode II dominated bilinear CZM model assumes that the separation of the material interfaces is dominated by the displacement jump that is tangent to the interface, as shown in the following figure:

Figure4.35:ModeIIDominatedBilinearCZMLaw

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The relation between tangential cohesive traction Tt and tangential displacement jump

expressed as:

Where:

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For Mode II dominated cohesive law, the normal cohesive traction and normal displacement jump behavior is assumed to follow the tangential cohesive traction and tangential displacement jump behavior and is expressed as:

Where:

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Mixed-Mode Bilinear Cohesive Zone Material Model

For bilinear cohesive law under the mixed-mode fracture, the separation of material interfaces depends on both the normal and tangential components of displacement jumps. To take into account the difference in their contributions to the separation of material interfaces, a non-dimensional effective displacement jump for mixed-mode fracture is defined as

(4376)

where the non-dimensional parameter (input via the TBDATA command as C6 using assigns different weights to the tangential and normal displacement jumps.

The normal and tangential components of the cohesive tractions are expressed as:

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The damage parameter Dm associated with mixed mode bilinear cohesive law is defined as:

Where:

Fracture Mode Identification of a CZM Model

Determining the fracture mode of a CZM model is based on the input data, as follows:

Case Input on the TBDATA command as follows:

Mode I Dominated C1, C2, C3, C4, C5 (where C3 = -max)

Mode II/III Dominated C1, C2, C3, C4, C5 (where C1 = -max)

Mixed-Mode C1, C2, C3, C4, C5, C6 (where C1 = max and C3 = max

4.12.2.ContactElements

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Delamination with contact elements is referred to as debonding. The interfacial separation is defined in terms of contact gap or penetration and tangential slip distance. The computation of contact and tangential slip is based on the type of contact element and the location of contact detection point. The cohesive zone model can only be used for bonded contact (KEYOPT(12) = 2, 3, 4, 5, or 6) with the augmented Lagrangian method (KEYOPT(2) = 0) or the pure penalty method (KEYOPT(2) = 1). See CONTA174 - 3-D 8-Node Surface-to-Surface Contact for details.

4.12.2.1.MaterialModel-BilinearBehavior

The bilinear cohesive zone material model (input using TB,CZM) is based on the model proposed by Alfano and Crisfield [365].

Mode I Debonding

Mode I debonding defines a mode of separation of the interface surfaces where the separation normal to the interface dominates the slip tangent to the interface. The normal contact stress (tension) and contact gap behavior is plotted in Figure4.36:NormalContactStressandContactGapCurveforBilinear Cohesive Zone Material. It shows linear elastic loading (OA) followed by linear softening (The maximum normal contact stress is achieved at point A. Debonding begins at point completed at point C when the normal contact stress reaches zero value; any further separation occurs without any normal contact stress. The area under the curve OAC is the energy released due to debonding and is called the critical fracture energy. The slope of the line OA determines the contact gap at the maximum normal contact stress and, hence, characterizes how the normal contact stress decreases with the contact gap, i.e., whether the fracture is brittle or ductile. After debonding has been initiated it is assumed to be cumulative and any unloading and subsequent reloading occurs in a linear elastic manner along line OB at a more gradual slope.

Figure4.36:NormalContactStressandContactGapCurveforBilinearCohesiveZoneMaterial

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The equation for curve OAC can be written as:

where:P = normal contact stress (tension)

Kn = normal contact stiffness

un = contact gap

= contact gap at the maximum normal contact stress (tension)

= contact gap at the completion of debonding (input on TBDATA command as C2 using CZM)

dn = debonding parameter

The debonding parameter for Mode I Debonding is defined as:

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with dn = 0 for n 1 and 0 < dn 1 for n > 1.

where:

The normal critical fracture energy is computed as:

where:max = maximum normal contact stress (input on TBDATA command as C1 using TB

For mode I debonding the tangential contact stress and tangential slip behavior follows the normal contact stress and contact gap behavior and is written as:

where:t = tangential contact stress

Kt = tangential contact stiffness

ut = tangential slip distance

Mode II Debonding

Mode II debonding defines a mode of separation of the interface surfaces where tangential slip dominates the separation normal to the interface. The equation for the tangential contact stress and tangential slip distance behavior is written as:

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where:

= tangential slip distance at the maximum tangential contact stress

= tangential slip distance at the completion of debonding (input on TBDATA command as C4 using TB,CZM)dt = debonding parameter

The debonding parameter for Mode II Debonding is defined as:

with dt = 0 for t 1 and 0 < dt 1 for t > 1.

where:

For the 3-D stress state an "isotropic" behavior is assumed and the debonding parameter is computed using an equivalent tangential slip distance:

where:u1 and u2 = slip distances in the two principal directions in the tangent plane

The components of the tangential contact stress are defined as:

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and

The tangential critical fracture energy is computed as:

where:max = maximum tangential contact stress (input on TBDATA command as C3 using

The normal contact stress and contact gap behavior follows the tangential contact stress and tangential slip behavior and is written as:

Mixed-Mode Debonding

In mixed-mode debonding the interface separation depends on both normal and tangential components. The equations for the normal and the tangential contact stresses are written as:

and

The debonding parameter is defined as:

with dm = 0 for m 1 and 0 < dm 1 for m > 1, and m and are defined below.

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where:

and

The constraint on that the ratio of the contact gap distances be the same as the ratio of tangential slip distances is enforced automatically by appropriately scaling the contact stiffness values.

For mixed-mode debonding, both normal and tangential contact stresses contribute to the total fracture energy and debonding is completed before the critical fracture energy values are reached for the components. Therefore, a power law based energy criterion is used to define the completion of debonding:

where:

and

are, respectively, the normal and tangential fracture energies. Verification of satisfaction of energy criterion can be done during postprocessing of results.

Identifying Debonding Modes

The debonding modes are based on input data:

1. Mode I for normal data (input on TBDATA command as C1, C2, and C5).2. Mode II for tangential data (input on TBDATA command as C3, C4, and C5).3. Mixed mode for normal and tangential data (input on TBDATA command as C1

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C5 and C6).

Artificial Damping

Debonding is accompanied by convergence difficulties in the Newton-Raphson solution. Artificial damping is used in the numerical solution to overcome these problems. For mode I debonding the normal contact stress expression would appear as:

where:

= damping coefficient (input on TBDATA command as C5 using TB,CZM).

The damping coefficient has units of time, and it should be smaller than the minimum time step size so that the maximum traction and maximum separation (or critical fracture energy) values are not exceeded in debonding calculations.

Tangential Slip Under Normal Compression

An option is provided to control tangential slip under compressive normal contact stress for mixed-mode debonding. By default, no tangential slip is allowed for this case, but it can be activated by setting the flag (input on TBDATA command as C6 using TB,CZM) to 1. Settings on are: = 0 (default) no tangential slip under compressive normal contact stress for mixed-mode debonding

= 1 tangential slip under compressive normal contact stress for mixed-mode debonding

Post Separation Behavior

After debonding is completed the surface interaction is governed by standard contact constraints for normal and tangential directions. Frictional contact is used if friction is specified for contact elements.

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Results Output for POST1 and POST26

All applicable output quantities for contact elements are also available for debonding: normal contact stress P (output as PRES), tangential contact stress t (output as SFRIC) or its components

(output as TAUR and TAUS), contact gap un (output as GAP), tangential slip ut (output as SLIDE) or

its components u1 and u2 (output as TASR and TASS), etc. Additionally, debonding specific output

quantities are also available (output as NMISC data): debonding time history (output as DTSTART), debonding parameter dn , dt or dm (output as DPARAM), fracture energies Gn and Gt (output as

DENERI and DENERII).

Release 14.5 - SAS IP, Inc. All rights reserved.

4.12. cohesive zone material (czm) model

Documents

interface input

interface delamination

schematicof interface

adjacent interface surfaces

czm model input

shear work of separation

maximum normal traction

traction t