Cohesive zone model

The cohesive zone model (CZM) is a model in fracture mechanics in which fracture formation is regarded as a gradual phenomenon in which separation of the surfaces involved in the crack takes place across an extended crack tip, or cohesive zone, and is resisted by cohesive tractions. The origin of this model can be traced back to the early sixties by Barenblatt (1962)[1] and Dugdale (1960)[2] to represent nonlinear processes located at the front of a pre-existent crack.[3] [4]

The major advantages of the CZM over the conventional methods in fracture mechanics like those including LEFM (Linear Elastic Fracture Mechanics), CTOD (Crack Tip open Displacement) are:[3]

• It is able to adequately predict the behaviour of uncracked structures, including those with blunt notches.
• Size of non-linear zone need not be negligible in comparison with other dimensions of the cracked geometry in CZM, while in other conventional methods, it is not so.
• Even for brittle materials, the presence of an initial crack is needed for LEFM to be applicable.

Another important advantage of CZM falls in the conceptual framework for interfaces.

The Cohesive Zone Model does not represent any physical material, but describes the cohesive forces which occur when material elements are being pulled apart.

As the surfaces (known as cohesive surfaces) separate, traction first increases until a maximum is reached, and then subsequently reduces to zero which results in complete separation. The variation in traction in relation to displacement is plotted on a curve and is called the traction-displacement curve. The area under this curve is equal to the energy needed for separation. CZM maintains continuity conditions mathematically; despite physical separation. It eliminates singularity of stress and limits it to the cohesive strength of the material.

The traction-displacement curve gives the constitutive behavior of the fracture. For each material system, guidelines are to be formed and modelling is done individually. This is how the CZM works. The amount of fracture energy dissipated in the work region depends on the shape of the model considered. Also, the ratio between the maximum stress and the yield stress affects the length of the fracture process zone. The smaller the ratio, the longer is the process zone. The CZM allows the energy to flow into the fracture process zone, where a part of it is spent in the forward region and the rest in the wake region.

Thus, the CZM provides an effective methodology to study and simulate fracture in solids.

Dugdale model for thin elastic-perfectly plastic plate

The Dugdale model assumes thin plastic strips of length, ${\displaystyle r_{p}}$, (sometimes referred to as the strip yield model[5]) are at the forefront of two Mode I crack tips in a thin elastic-perfectly plastic plate. [6][7]

Derivation of Dugdale model through superposition

Dugdale's model can be derived using the complex stress functions, but is derived below using superposition. A traction, ${\displaystyle \sigma _{yy}}$, exists along the plastic region and is equal to the yield stress, ${\displaystyle \sigma _{y}}$, of the material. This traction results in a negative stress intensity factor, ${\displaystyle K''_{I}}$. ${\displaystyle K''_{I}=-\sigma _{ys}{\sqrt {\pi (a+r_{p})}}+2\sigma _{ys}{\sqrt {\frac {a+r_{p}}{\pi }}}sin^{-}1\left({\frac {a}{a+p}}\right)}$ If the traction were zero, a positive stress intensity factor, ${\displaystyle K_{I}}$, is produced assuming the plate is infinitely large. ${\displaystyle K'_{I}=-\sigma ^{\infty }{\sqrt {\pi (a+r_{p})}}}$ For the stress to be bounded at ${\displaystyle x=a+r_{p}}$, the following is true through superposition: ${\displaystyle K'_{I}+K''_{I}=0}$ The length of the inelastic zone can be estimated by solving for ${\displaystyle r_{p}}$: ${\displaystyle {\frac {r_{p}}{a}}=sec\left({\frac {\pi \sigma ^{\infty }}{2\sigma _{y}}}\right)-1}$

In the case where ${\displaystyle \sigma ^{\infty }<<\sigma _{y}}$,and therefore ${\displaystyle r_{p}<<a}$, the plastic zone size is: [5][6][7]

${\displaystyle r_{p}={\frac {\pi }{8}}\left({\frac {K_{I}}{\sigma _{y}}}\right)}$

which is similar to, but slightly smaller than Irwin's predicted plastic zone diameter.

References

1. G.I. Barenblatt (1962). The mathematical theory of equilibrium cracks in brittle fracture. Advances in Applied Mechanics. 7. pp. 55–129. doi:10.1016/S0065-2156(08)70121-2. ISBN 9780120020072.
2. Donald S. Dugdale (1960). "Yielding of steel sheets containing slits". Journal of the Mechanics and Physics of Solids. 8 (2): 100–104. Bibcode:1960JMPSo...8..100D. doi:10.1016/0022-5096(60)90013-2.
3. Znedek P. Bazant; Jaime Planas (1997). Fracture and size effect in concrete and other quasibrittle materials. 16. CRC Press.
4. Kyoungsoo Park; Glaucio H. Paulino (2011). "Cohesive zone models: a critical review of traction-separation relationships across fracture surfaces". Applied Mechanics Reviews. 64 (6): 06802. CiteSeerX 10.1.1.654.839. doi:10.1115/1.4023110.
5. Janssen, Micheal (2004). "3.3 The Plastic Zone Size According to Dugdale: The Strip Yield Model". Fracture mechanics. Zuidema, J. (Jan), Wanhill, R. J. H. (2nd ed.). London: Spon Press. pp. 65–70. ISBN 0-203-59686-2. OCLC 57491375.
6. Suresh, Subra (1998). "9.5.2 The Dugdale Model". Fatigue of materials (2nd ed.). Cambridge: Cambridge University Press. pp. 303–304. ISBN 978-0-511-80657-5. OCLC 817913181.
7. "Dugdale-Barenblatt Model". Springer handbook of experimental solid mechanics. Sharpe, William N. Boston, MA: Springer Science+Business Media. 2008. pp. 132–133. ISBN 978-0-387-30877-7. OCLC 289032317.CS1 maint: others (link)