Effect of rise, peak and fall characteristics of CZM in predicting fracture processes
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Effect of rise, peak and fall characteristics of CZM in predicting fracture processes. Motivation. Possible micromechanisms active in wake and forward regions cohesive energy includes most of the dissipations in the FPZ Does cohesive energy includes plastic dissipation.

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Motivation predicting fracture processes

  • Possible micromechanisms active in wake and forward regions

  • cohesive energy includes most of the dissipations in the FPZ

  • Does cohesive energy includes plastic dissipation.

  • Is there a link between micro- mechanic processes of the material and curve,


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Motivations predicting fracture processes

  • What is the relationship between the physics/mechanics of the separation process and shape of CZM? (There are as many shapes/equations as there are number of interface problems solved!)

    • What is the physical significance of

    • - Shape of the curve C

    • - tmax and interface strength

    • - Separation distance sep

    • - Initial slope

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Terminology predicting fracture processes


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Different CZM Models predicting fracture processes

  • To study the shape effect, three CZMs with different shapes were considered

    • Exponential Model

    • Trapezoidal Model

    • Bilinear Model

  • To study the effect of initial rising slope and the tail effect, parameters of trapezoidal model and of bilinear model are varied.


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Exponential model predicting fracture processes

Following the work of Xu and Needleman (1993), the interface potential is taken as


are some characteristic distance

Normal displacement after shear separation under the condition

Of zero normal tension

  • Normal and shear traction are given by

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  • The interfacial energy variation due to normal and shear tractions

    are shown below


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Trapezoidal model are shown below

Following the work of Trevegaard and Huchinson (1992), the interface tractions are given by


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Bilinear model are shown below

Bilinear model is obtained from trapezoidal model by making


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Cohesive zone parameters are shown below

  • Material Al 2024-T3 alloy

  • The input energy in the exponential cohesive zone model are related to the interfacial stress and characteristic displacement as

  • The input energy is equated

    to material parameter

  • For the material considered


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Cohesive zone parameters (contd.) are shown below

For trapezoidal model

For bilinear model


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Cohesive zone parameters Material model for the bounding material

E=72 GPa, =0.33,

  • Material considered are aluminum alloy Al 2024-T3

Stressstrain curve is given by



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Geometry and boundary/loading conditions material

a = 0.025m, b = 0.1m, h = 0.1m


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Finite element mesh material

24340 plane strain 4 node elements

7300 cohesive elements (width of element along the crack plan is ~7E-7m

28189 nodes


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  • (very small scale plasticity), material

    plastic energy ~ 15% of total dissipation.

    • Plasticity induced at the initial stages

    • of the crack growth

    • plasticity ceases during crack

    • propagation.

    • Very small error is induced by ignoring

    • plasticity.

  • plastic work increases

    considerably, ~100 to 200% as that of

    cohesive energy.

  • For large scale plasticity problems the

    amount of total dissipation (plastic and

    cohesive) is much higher than 8000

  • Plastic dissipation very sensitive to

    ratio beyond 2 till 3

  • Crack cannot propagate beyond

    and completely elastic below

Effect of smax/sy on plastic dissipation(exponential model)

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Effect of materialsmax/sy on plastic dissipation


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Effect of Shape of CZM on plastic work and cohesive work material

Exponential Model

Bilinear Model

Trapezoidal Model

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  • A set of patch of elements (each having app. 50 elements) were selected in the bounding material.

  • The patches are approximately squares (130 ). They are spaced equally from each other.

  • Adjoining these patches, patches of cohesive elements are considered to record the cohesive energies.

Local/spatial Energy Distribution


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  • The cohesive energy in the patch increases up to point C (corresponding to inset Figure ) after which the crack tip is presumed to advance.

  • The energy consumed by the cohesive elements at this stage is approximately 1/7 of the total cohesive energy for the present CZM.

  • Once the point C is crossed, the patch of elements fall into the wake region.

  • The rate of cohesive zone energy absorption depends on the slope of the curve and the rate at which elastic unloading and plastic dissipation takes place in the adjoining material.

  • The curves flattens out once the entire cohesive energy is dissipated within a given zone.

  • Similar curves were obtained even for trapezoidal and bilinear models

Variation of Cohesive Energy

The variation of Cohesive Energy in the Wake and Forward region as the crack propagates. The numbers indicate the Cohesive Element Patch numbers Falling Just Below the binding element patches

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Variation of Elastic Energy (corresponding to inset

  • Considerable elastic energy is built up till

    the peak of curve is reached after

    which the crack tip advances.

  • After passing C, the cohesive elements near

    the crack tip are separated and the elements

    in this patch becomes a part of the wake.

  • At this stage, the values of normal traction

    reduces following the downward slope of

    curve following which the stress in the

    patch reduces accompanied by reduction in

    elastic strain energy.

  • The reduction in elastic strain energy is used up in dissipating cohesive energy to those cohesive elements adjoining this patch.

  • The initial crack tip is inherently sharp leading to high levels of stress fields due to which higher energy for patch 1

  • Crack tip blunts for advancing crack tip leading to a lower levels of stress, resulting in reduced energy level in other patches.

Variation of Elastic Energy in Various Patch of Elements as a Function of Crack Extension. The numbers indicate Patch numbers starting from Initial Crack Tip

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Variation of Plastic Work ( ) (corresponding to inset

  • plastic energy accumulates considerably along with elastic energy, when the local stresses bounding material exceeds the yield

  • After reaching peak point C on curve traction reduces and plastic deformation ceases. Accumulated plastic work is dissipative in nature, it remains constant after debonding.

  • The accumulated plastic work decreases up to patch 4 from that of 1 as a consequence of reduction of the initial sharpness of the crack.

  • Mechanical work is increased to propagate the crack, during which the does not increase resulting in increased plastic work. That increase in plastic work causes the increase in the stored work in patches 4 and beyond

  • Plastic dissipation is significantly higher at the crack initiation

  • Small amount of plastic dissipation continues to take place during crack growth.

Variation of dissipated plastic energy in various patched as a function of crack extension. The number indicate patch numbers starting from initial crack tip.

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Schematic of crack initiation and propagation (corresponding to inset

process in a ductile material

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  • plastic energy accumulates considerably along with elastic energy, when the local stresses bounding material exceeds the yield

  • After reaching peak point D on curve traction reduces and plastic deformation ceases. Accumulated plastic remains constant after debonding.

  • Plastic dissipation takes place only at crack initiation.

  • No plastic dissipation takes place during crack growth.

  • Elastic energy curve repeats its feature even during crack growth


Variation of Elastic & Plastic Work ( )

For trapezoidal model


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Effect of Shape of CZM on plastic work and cohesive work (contd.)

  • Both bilinear and trapezoidal models predict considerably less amount of plasticity when compared to exponential models

  • Exponential model shows plastic energy is dissipated not only at crack initiation but also during crack growth.

  • Both bilinear and trapezoidal models shows plastic dissipation only at crack initiation and during crack growth plastic dissipation does not take place.

  • Plastic dissipation decreases with with reduction of initial slope (with increase in dmax1 for trapezoidal model anddmax in case of bilinear model)

  • Exponential model shows a slightly higher stress field near the crack tip when compared to that of bilinear and trapezoidal models. This also contributes to increased plasticity in exponential models.


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Size of plastic zone (contd.)


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Effect of Tail of CZMs (contd.)


0.1, 0.2, 0.4, 0.6, 0.8

  • Tail part of trapezoidal CZM is varied keeping constant at 0.1 and varying from 0.1 to 0.8.

  • Plastic dissipation decreases with longer tail in CZM


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Effect of Initial slope of CZMs (contd.) (contd.)

Exponential Model

Bilinear Model

Trapezoidal Model

  • Exponential model shows slightly higher stress field in the bounding material

  • The intensity of stress field decreases with increase in initial slope


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Effect of Initial slope of CZMs (contd.) (contd.)

Trapezoidal Model

  • Trapezoidal model shows slightly higher initial stiffness when compared to bilinear model

  • In both the models stiffness reduces as the initial slope of CZMs are increased.


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Effect of Shape and Initial slope of CZMs on crack initiation loads

  • From LEFM (small scale yielding) the crack initiation load obtained by limiting SIF to is 80MPa


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Conclusion initiation loads

  • smax/sy has considerable effect on plasticity in case of exponential model and has little effect in case of trapezoidal and bilinear model

  • Shape of CZM has considerable effect of amount of plasticity induced in the bounding material. Exponential model induces higher level of plasticity when compared to trapezoidal and bilinear models.

  • Exponential model induces plasticity at both crack initiation and crack propagation: applicable for ductile metals

  • Trapezoidal and bilinear models induces plasticity only at crack initiation (even at higher smax/sy ) and not during crack propagation: applicable for brittle or metals with low ductility

  • Shape of CZM has considerable effect on the size of plastic zone. Exponential model predict better size of plastic zone than other models.


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Conclusion(contd) initiation loads

  • Initial slope has several effect on fracture process

    • By increasing the initial slope (reducing value for dmax1 and dmax) stiffness of CZ elements increases, there by it increases plasticity.

    • By decreasing the slope, stiffness of elements reduces and hence plasticity.

    • Stiff initial slope induces higher stress field in the bounding material.

    • Higher initial slope makes the wake length longer.

    • Higher initial slope results in lesser cohesive energy consumptionin the forward region, hence crack initiates at lower load.

  • Crown shape (flat in trapezoidal, sharp point in bilinear, curved in exponential model) of CZMs has no significant effect except for trapezoidal model where more number of elements experience constant peak stress, which may favor in plastic dissipation.

    • Flat crown also reduces critical displacement for both crack initiation and complete separation

  • Tail part of CZM also effects in plasticity growth. When trapezoidal model is used with longer tail amount of plasticity is reduced.