Roman Vodička

A Model of Rupturing Lithospheric Faults with Reoccurring Earthquakes

An isothermal small-strain model based on the concept of generalized standard materials is devised, combining Maxwell-type rheology, damage, and perfect plasticity in the bulk. An interface analogue of the model is prescribed at the lithospheric faults, exploiting concepts of adhesive contacts with interfacial plasticity. The model covers simultaneously features such as rupturing of the fault zone accompanied with weakening/healing effects and also seismic waves emission and propagation connected with the sudden ruptures of the fault or a fluid-like aseismic response between the ruptures. A stable numerical strategy based on semi-implicit discretization in time is devised, and its convergence is shown. Numerical simulations documenting the capacity of the model to simulate earthquakes with repeating occurrence are performed, too.

Comparing the conventional displacement BIE and the BIE formulations of the first and second kind in frictionless contact problems

There are several formulations of boundary integral equations (BIEs) used in the general numerical procedure known as boundary element method (BEM). There are also several approaches to deal with contact problems using BEM. In this paper, a comparison between the following procedures: the conventional discretization of the displacement BIE by collocations, the Galerkin discretizations of the symmetric BIE formulation of the first kind and the non-symmetric BIE formulation of the second kind, is performed. Although several aspects of these procedures are discussed, the emphasis is put on the accuracy of the results obtained with identical meshes. The comparison is carried out including problems with analytical solutions or in the presence of singularities, covering conforming, advancing and receding contact problems. Linear elements, conforming discretizations of surfaces in contact and absence of friction define the frame where the study is performed.

An asymptotic property of degenerate scales for multiple holes in plane elasticity

The solution of an exterior Dirichlet boundary value problem of plane isotropic elasticity by the boundary integral equation of the first kind obtained from the Somigliana identity is considered. The logarithmic function appearing in the integral kernel may cause that the operator is non-invertible. Such a situation occurs if the size of the boundary coincides with a degenerate scale for a certain form of the fundamental solution used. A technique for the evaluation of the degenerate scales for an asymptotic case of an exterior domain with holes placed far one from another is discussed and analyzed. The examples provide results of particular cases and assess suitability of the proposed technique in relation to numerical calculation of degenerate scales by the Boundary Element Method.

The first-kind and the second-kind boundary integral equation systems for solution of frictionless contact problems

An original approach to the numerical solution of displacement boundary integral equation (BIE) and traction hypersingular boundary integral equation (HBIE) by the boundary element method (BEM) for contact problems is given. The main point is to show, how the contact conditions are used to formulate the first-kind and the second-kind BIE systems in the case of frictionless two-body elastic contact. The solution of the first-kind BIE is performed by symmetric Galerkin BEM; the second-kind BIE is solved by an appropriate collocation BEM. The contact problem in itself is solved by the method of subsequent approximations of contact region. Both forms of BIE system are compared in several numerical examples. This comparison is made for different kinds of contact problem. The major emphasis is put on the evaluation of contact pressure. The obtained results are compared with referenced numerical and with the analytical ones.

Quasistatic normal-compliance contact problem of visco-elastic bodies with Coulomb friction implemented by QP and SGBEM

The quasistatic normal-compliance contact problem of isotropic homogeneous linear visco-elastic bodies with Coulomb friction at small strains in Kelvin-Voigt rheology is considered. The discretization is made by a semi-implicit formula in time and the Symmetric Galerkin Boundary Element Method (SGBEM) in space, assuming that the ratio of the viscosity and elasticity moduli is a given relaxation-time coefficient. The obtained recursive minimization problem, formulated only on the contact boundary, has a nonsmooth cost function. If the normal compliance responds linearly and the 2D problems are considered, then the cost function is piecewise-quadratic, which after a certain transformation gets the quadratic programming (QP) structure. However, it would lead to second-order cone programming in 3D problems. Finally, several computational tests are presented and analysed, with an additional discussion on numerical stability and convergence of the involved approximated Poincare-Steklov operators.The quasistatic normal-compliance contact problem of isotropic homogeneous linear visco-elastic bodies with Coulomb friction at small strains in Kelvin-Voigt rheology is considered. The discretization is made by a semi-implicit formula in time and the Symmetric Galerkin Boundary Element Method (SGBEM) in space, assuming that the ratio of the viscosity and elasticity moduli is a given relaxation-time coefficient. The obtained recursive minimization problem, formulated only on the contact boundary, has a nonsmooth cost function. If the normal compliance responds linearly and the 2D problems are considered, then the cost function is piecewise-quadratic, which after a certain transformation gets the quadratic programming (QP) structure. However, it would lead to second-order cone programming in 3D problems. Finally, several computational tests are presented and analysed, with additional discussion on numerical stability and convergence of the involved approximated Poincaré-Steklov operators.

A Comparative Study of Three Systems of Boundary Integral Equations in the Potential Theory

Abstract A numerical study of Laplace equation solution by the direct BEM in 2D is presented. The following three basic systems of BIEs: the conventional BIE of potential, the first kind symmetric BIE system and the second kind BIE system, are compared. Error behaviour along the boundary and error convergence in L2 and maximum norms for h-refinements of BEM meshes are studied and discussed.

Interfacial debonds of layered anisotropic materials using a quasi-static interface damage model with Coulomb friction

A new quasi-static and energy based formulation of an interface damage model which includes Coulomb friction at the interface between anisotropic solids is provided. The interface traction-relative displacement response is based on an assumption of a thin adhesive layer whose behaviour is analogous to cohesive zone models. The damaged interface is considered, if exposed to a pressure, as a contact zone where Coulomb friction law is also taken into account. As the contacting solids are generally anisotropic, the friction may exhibit some anisotropic behaviour, too, which is included into the proposed model. The solution of the problem is sought numerically by a semi-implicit time-stepping procedure which uses recursive decoupled double minimisation in displacements and damage variables. The spatial discretisation is based on the symmetric Galerkin boundary-element method of a multidomain problem, where the interface variables are calculated by sequential quadratic programming, being a tool for resolving each partial minimisation in the proposed recursive scheme. Sample numerical examples demonstrate applicability of the described model.

Revisiting the Problem of Debond Initiation at Fibre-Matrix Interface under Transversal Biaxial Loads - A Comparison of Several Non-Classical Fracture Mechanics Approaches

Understanding matrix failure in LFRP composites is one of the main challenges when developing failure criteria for these materials. This work aims to study the influence of the secondary transverse load on the crack initiation at micro-scale. Four non-classical approaches of fracture mechanics are used to model the onset of fibre-matrix interface debonds: Linear Elastic Brittle Interface Model (LEBIM), an Energetic Approach for the Linear Elastic Brittle Interface Model (EA-LEBIM), an Energetic Approach for the bilinear Cohesive Zone Model (EA-CZM) and the Coupled Criterion of the Finite Fracture Mechanics (CC-FFM). Results obtained by these approaches predict that, for brittle fibre-matrix configurations, a secondary transverse compression reduces the critical value of the main transverse tension leading to the debond onset. This fact is not taken into account by the currently used failure criteria

A Numerical Approach to an Interface Damage Model Under Cyclic Loading

A model for numerical analysis of interface damage is presented which leads to interface crack initiation and propagation in a multi-domain structure exposed to a cyclic loading. The proposed numerical approach is physically based on energy evolution, which includes stored elastic energy, energy of the external load and also dissipation due to damage. The solution evolution is approximated by a semi-implicit time stepping algorithm. In this algorithm, the variables in the energy functionals, related to deformation and damage, are separated in order to recursively solve the problem with respect to only one variable. The separation provides a variational structure to the solved problem followed by two separated minimisations in one time step. The variables of the elastic state are calculated by Symmetric Galerkin BEM which has the advantage that the minimisations, implemented by (sequential) quadratic programming solvers, can be performed only with respect to the interface displacement gap.

A Quasi-Static Delamination Model with Rate-Dependent Interface Damage Exposed to Cyclic Loading

A model for numerical analysis of interface damage which leads to interface crack initiationand propagation in multi-domain structures under cyclic loading is considered. Modelling of damagetakes into account various relations between interface stresses and displacement gaps providing theresponse of a cohesive zone model, additionally equipped by a kind of viscosity associated to theevolution of the interface damage. Together with repeating loading-unloading conditions, it makesthis damage process to have a fatigue-like character, where the crack appears for smaller magnitudeof the cyclic load than for pure uploading.