Modeling Fracture in Elasticplastic Solids Using Cohesive Zone

Modeling Fracture in Elastic-plastic Solids Using Cohesive Zone CHANDRAKANTH SHET Department of Mechanical Engineering FAMU-FSU College of Engineering Florida State University Tallahassee, Fl-32310 Sponsored by US ARO, US Air Force

Fracture/Damage theories to model failure q Fracture Mechanics - Ø Linear solutions leads to singular fieldsdifficult to evaluate Ø Fracture criteria based on Ø Non-linear domain- solutions are not unique Ø Additional criteria are required for crack initiation and propagation q Basic breakdown of the principles of mechanics of continuous media q Damage mechanicsØ can effectively reduce the strength and stiffness of the material in an average sense, but cannot create new surface

CZM is an Alternative method to Model Separation Ø CZM can create new surfaces. Maintains continuity conditions mathematically, despite the physical separation. Ø CZM represent physics of fracture process at the atomic scale. Ø 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. Ø Uses fracture energy(obtained from fracture tests) as a parameter and is devoid of any ad-hoc criteria for fracture initiation and propagation. Ø Eliminates singularity of stress and limits it to the cohesive strength of the material. Ø Ideal framework to model strength, stiffness and failure in an integrated manner. Ø Applications: geomaterials, biomaterials, concrete, metallics, composites…

Conceptual Framework of Cohesive Zone Models for interfaces

Interface in the undeformed configuration

Interface in the deformed configuration

Constitutive Model for Bounding Domains W 1, W 2

Constitutive Model for Cohesive Zone W*

Development of CZ Models-Historical Review Figure (a) Variation of Cohesive traction (b) I - inner region, II - edge region q Barenblatt (1959) was first to propose the concept of Cohesive zone model to brittle fracture Ø Molecular force of cohesion acting near the edge of the crack at its surface (region II ). Ø The intensity of molecular force of cohesion ‘f ’ is found to vary as shown in Fig. a. Ø The interatomic force is initially zero when the atomic planes are separated by normal intermolecular distance and increases to high maximum after that it rapidly reduces to zero with increase in separation distance. E is Young’s modulus and is surface tension (Barenblatt, G. I, (1959), PMM (23) p. 434)

Phenomenological Models q The theory of CZM is based on sound principles. q However implementation of model for practical problems grew exponentially for practical problems with use of FEM and advent of fast computing. q Model has been recast as a phenomenological one for a number of systems and boundary value problems. q The phenomenological models can model the separation process but not the effect of atomic discreteness. Hillerborg etal. 1976 Ficticious Grujicic, 1999, fracture behacrack model; concrete vior of polycrystalline; bicrystals Bazant etal. 1983 crack band Costanzo etal; 1998, dynamic fr. theory; concrete Ghosh 2000, Interfacial debo. Morgan etal. 1997 earthquake nding; composites rupture propagation; geomaterial Rahulkumar 2000 viscoelastic Planas etal, 1991, concrete fracture; polymers Eisenmenger, 2001, stone fragm. Liechti 2001 Mixed-mode, timeentation squeezing" by evanescent depend. rubber/metal debonding waves; brittle-bio materials Ravichander, 2001, fatigue Amruthraj etal. , 1995, composites Tevergaard 1992 particle-matrix interface debonding Tvergaard etal 1996 elasticplastic solid : ductile frac. ; metals Brocks 2001 crack growth in sheet metal Camacho &ortiz; 1996, impact Dollar; 1993 Interfacial debonding ceramic-matrix comp Lokhandwalla 2000, urinary stones; biomaterials

Fracture process zone and CZM q CZM essentially models fracture process zone Mathematical crack tip by a line or a plane ahead of the crack tip Material crack tip subjected to cohesive traction. q The constitutive behavior is given by traction y displacement relation, obtained by defining potential function of the type x where are normal and tangential displacement jump The interface tractions are given by

Following the work of Xu and Needleman (1993), the interface potential is taken as where are some characteristic distance Normal displacement after shear separation under the condition Of zero normal tension §Normal and shear traction are given by

Numerical Formulation • The numerical implementation of CZM for interface modeling with in implicit FEM is accomplished developing cohesive elements • Cohesive elements are developed either as line elements (2 D) or planar elements (3 D)abutting bulk elements on either side, with zero thickness 1 2 • The virtual work due to cohesive zone traction in a given cohesive element can be written as 3 4 5 6 7 8 Continuum elements Cohesive element The virtual displacement jump is written as Where [N]=nodal shape function matrix, {v}=nodal displacement vector J = Jacobian of the transformation between the current deformed and original undeformed areas of cohesive surfaces Note: is written as d{T}- the incremental traction, ignoring time which is a pseudo quantity for rate independent material

Numerical formulation contd The incremental tractions are related to incremental displacement jumps across a cohesive element face through a material Jacobian matrix as For two and three dimensional analysis Jacobian matrix is given by Finally substituting the incremental tractions in terms of incremental displacements jumps, and writing the displacement jumps by means of nodal displacement vector through shape function, the tangent stiffness matrix takes the form