Spacetime damage-based cohesive model for elastodynamic fracture with dynamic contact
Dynamic material failure is important in a number of scientific and engineering applications and a variety of numerical methods for its modeling have been proposed. This thesis presents the formulation and implementation of an interfacial-damage, cohesive-fracture model, including contact and friction effects, for dynamic failure of brittle materials. The model is implemented within a spacetime discontinuous Galerkin (SDG) finite element method. An adaptive meshing procedure generates %When implemented on suitable spacetime grids that satisfy a special causality constraint to enable an efficient patch-by-patch, advancing-front solution scheme with O(N) computational complexity. Per-element balance properties, local adaptive operations, and the use of Riemann fluxes provide to the SDG method the extreme accuracy and efficiency required to solve multiscale fracture problems. A dimensional analysis of linear elastodynamics, with extensions to fracture models based on cohesive traction--separation laws, supports the formulation. The problem is formulated and analyzed using differential forms and the exterior calculus in spacetime. The analysis demonstrates that the velocity scalings implied by the spatial and temporal coordinate scalings and by the scalings of the material properties must be identical to obtain a self-similar scaling of an elastodynamic process. The use of differential forms reveals intrinsic structure and relations between the spacetime mechanics fields which are otherwise obscured by conventional tensorial analysis. For example, only four distinct scalings are required to define a scaled elastodynamic process when we work directly with forms, while eight are required when tensorial analysis is used. In the context of dynamic cohesive fracture, the analysis shows that, among the nondimensional variables, the ratio of the stress-loading scale to the cohesive strength is proportional to the ratio of the radius of the singularity-dominant zone from Linear Elastodynamic Fracture Mechanics (LEFM), to the cohesive-process-zone size. These ratios are, in turn, useful indicators of whether the small-scale-yielding caveat of LEFM is satisfied. A novel continuum formulation of the linear elastodynamic contact problem also supports the SDG finite element model. In contrast to previous contact models that invoke quasi-static contact conditions, the proposed model enforces dynamic contact conditions by prescribing momentum flux and compatibility conditions obtained from the local Riemann problems for bonded, separation, contact--stick, and contact--slip modes. This approach preserves the characteristic structure of the underlying equations at the contact interface, a property that is lacking in previous formulations. The fully-bonded and contact--stick conditions are identical, as expected, so the non-penetration and tangential slip constraints are treated exactly in the new continuum formulation. Furthermore, the direction of the tangential contact traction (friction) is shown to be continuous through transitions between contact--stick and contact-slip modes. These favorable properties, which improve the accuracy of and facilitate numerical implementations of the proposed model, are not obtained in many existing models which, for example, replace the non-penetration constraint with a large interfacial stiffness in the normal direction. %For example, the required continuity conditions are replaced with large penetration stiffness values in penalty methods. % Furthermore, it is shown that the relative tangential velocity in slip mode is aligned with the tangential traction that would have resulted under the stick mode. It is well documented that the determination of slip traction from slip velocity is discontinuous and poses several difficulties in numerical methods. The choice of stick tranction, on the otherhand, provides a continuous representation for the direction of slip traction. % The direction of slip traction is determined from the slip velocity in Coulomb law of friction. However, the discontinuous nature of this representation causes several difficulties in the numerical methods. It is shown that the orientation derived from a tangential traction that would have acted in stick mode eliminates the discontinuity. The transition between separation and contact modes retains its physically discontinuous character, and a regularization of this transition is introduced to facilitate and reduce the cost of numerical implementations. A discretization and numerical implementation within the adaptive SDG framework demonstrate the effectiveness of the new contact model in a numerical setting. A new two-scale cohesive fracture model replaces the usual traction-separation law with a damage model that represents mesoscale processes of void growth and coalescence. The evolution of a single damage parameter D, which represents the debonded area fraction on cohesive interfaces, is governed by an irreversible, time-delay evolution law characterized by a cohesive strength and a relaxation time that determines the maximum damage rate. Riemann fluxes for the fully-bonded condition are enforced in the undamaged area fraction (1-D) of the cohesive interface, while the Riemann fluxes for the contact--stick, contact--slip or separation conditions determine the fluxes in the debonded area fraction. These mesoscale Riemann values are averaged to derive macroscopic cohesive conditions. % The damage-based cohesive model is implemented within the adaptive SDG finite element framework to produce a numerical model that efficiently and accurately resolves the multi-scale response associated with dynamic fracture and transitions between contact, separation, stick and slip conditions in the event of crack closure. Beyond ensuring solution accuracy, the model uses the SDG scheme's adaptive meshing capabilities to freely nucleate and extend cohesive interfaces to capture solution-dependent crack paths. The SDG adaptive meshing aligns the boundaries of spacetime elements with crack-path trajectories having arbitrary position and orientation, and two adaptive error indicators ensure the accurate rendering of both the cohesive model and the bulk solution. Thus, the present model does not suffer the limited resolution and mesh-dependent effects encountered in most other numerical fracture models. Numerical results obtained with the proposed model demonstrate crack propagation, microcrack formation and crack branching phenomena.
fracture, finite element methods, spacetime discontinuous, cohesive models, interfacial damage, Riemann solution, dimensional analysis, Linear Elastodynamic Fracture Mechanics (LEFM)
University of Illinois - Urbana-Champaign
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University of Illinois at Urbana-Champaign