Husnu Dal

A large number of methods to describe fracture mechanical features of structures on basis of computational algorithms have been developed in the past due to the importance of the topic. In this paper, current and promising numerical approaches for the characterization of fracture processes are presented. A fracture phenomenon can either be depicted by a continuum formulation or a discrete notch. Thus, starting point of the description is a micromechanically motivated formulation for the development of a local failure situation. A current, generalized method without any restriction to material modelling and loading situation in order to describe an existing crack in a structure is available through the material force approach. One possible strategy to simulate arbitrary crack growth is based on an adaptive implementation of cohesive elements in combination with the standard discretization of the body. In this case, crack growth criteria and the determination of the crack propagation direction in combination with the modification of the finite element mesh are required. The nonlinear structural behaviour of a fibre reinforced composite material is based on the heterogeneous microstructure. A two-scale simulation is therefore an appropriate and effective way to take into account the scale differences of macroscopic structures with microscopic elements. In addition, fracture mechanical structural properties are far from being sharp and deterministic. Moreover, a wide range of uncertainties influence the ultimate load bearing behaviour. Therefore, it is evident that the deterministic modelling has to be expanded by a characterization of the uncertainty in order to achieve a reliable and realistic simulation result. The employed methods are illustrated by numerical examples.

The oxidation reactions responsible for physical ageing are thermally activated processes and yield cross-linking similar to the vulcanization process and chain-scission reactions. Chain scission, is responsible for the relaxation behaviour under constant stretch, whereas cross-linking is responsible for hardening of the material. The decrease in the stretch at break upon cross-linking motivates the so called network alteration proposed in this contribution. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Rubbery polymers are modelled with hyperelastic constitutive laws by which the material can be stretched indefinitely or up to a certain locking stretch value. However, no material can sustain indefinite stretch and store infinite energy. Failure occurs after a threshold value for stress and/or energy is attained. Within this context, an approach for failure analysis of rubber–like materials based on the constitutive description of material is presented. The model is based on a hyperelastic material law representing the energy stored in rubber network due to the chain conformations connected in series to an interatomic bond potential representing the energy stored in the polymer chain due to the interatomic displacement. For the representation of the micro–macro transition in terms of non–affine kinematics, micro–sphere model is utilized. Morse potential is used for the description of the interatomic bond, which describes the energetic contribution to rubber–like materials and governs the failure of the polymer chain. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

One of the successful approaches to model the time-dependent behaviour of elastomers is proposed by Bergström and Boyce (JMPS 46:931–954, 1998). The model is micromechanically inspired from the relaxation of a single entangled chain in a polymer gel matrix. Although the theory of inelasticity based on multiplicative decomposition of the deformation gradient is well established, the complexity of the nonlinear evolution law as well as the nonlinear equilibrium and non-equilibrium material response necessitates a precise description of the algorithmic setting. This contribution presents for the first time a novel numerical implementation of the Bergström–Boyce model in the context of finite element analysis and elaborates theoretical aspects of the model. The thermodynamical consistency of the evolution law is proven and a parameter study with respect to the material parameters has been carried out. The agreement of the model with the recent experimental data is investigated.

This work introduces a novel, unconditionally stable and fully coupled finite element method for the bidomain system of equations of cardiac electrophysiology. The transmembrane potential and the extracellular potential are treated as independent variables. To this end, the respective reaction-diffusion equations are recast into weak forms via a conventional isoparametric Galerkin approach. The resultant nonlinear set of residual equations is consistently linearised. The method results in a symmetric set of equations, which reduces the computational time significantly compared to the conventional solution algorithms. The proposed method is inherently modular and can be combined with phenomenological or ionic models across the cell membrane. The efficiency of the method and the comparison of its computational cost with respect to the simplified monodomain models are demonstrated through representative numerical examples.

This work introduces a novel, unconditionally stable and fully coupled finite element method for the bidomain system of equations of cardiac electrophysiology. The transmembrane potential and the extracellular potential are treated as independent variables. To this end, the respective reaction-diffusion equations are recast into weak forms via a conventional isoparametric Galerkin approach. The resultant nonlinear set of residual equations is consistently linearised. The method results in a symmetric set of equations, which reduces the computational time significantly compared to the conventional solution algorithms. The proposed method is inherently modular and can be combined with phenomenological or ionic models across the cell membrane. The efficiency of the method and the comparison of its computational cost with respect to the simplified monodomain models are demonstrated through representative numerical examples.

Rubbery polymers are subjected to severe environmental conditions under service. As a consequence of various ageing mechanisms, the outer surface of rubber components hardens in time and cracking occurs as a result of combined mechanical and chemical processes. Conventional phenomenological hyperelastic constitutive models do not account for material softening. Consequently, the stored energy and stresses tend to infinity as stretch increases. In this contribution, a network alteration for the ageing mechanism of rubber-like materials is introduced along with a micromolecular description of material failure. The proposed micro-continuum material model is based on a serial construction of a Langevin-type spring representing the energy storage owing to conformational changes induced by deformation, to a bond potential representing the energy stored in the polymer chain due to the interatomic displacement. For the representation of the micro–macro transition, the non-affine kinematics of the micro-sphere model is used. The Morse potential is utilized for the interatomic bond, which describes the energetic contribution to rubber-like materials and governs the failure of the polymer chain in terms of bond rupture. A novel numerical scheme for the FE implementation of the proposed model is demonstrated. The hardening phenomenon as a result of diffusion limited oxidation of rubber is explained by the principle of mass conservation which dictates simultaneous modulus hardening along with decrease in ultimate stretch observed in aged rubbery polymers.

The nature of elastomeric material demands the consideration of finite deformations, nonlinear elasticity including damage as well as rate-dependent and rate-independent dissipative properties. While many models accounting for these effects have been refined over time to do better justice to the real behavior of rubber-like materials, the realistic simulation of the elastoplastic characteristics for filled rubber remains challenging.The classical elastic-ideal-plastic formulation exhibits a distinct yield-surface, whereas the elastoplastic material behavior of filled rubber components shows a yield-surface free plasticity. In order to describe this elastoplastic deformation of a material point adequately, a physically based endochronic plasticity model was developed and implemented into a Finite Element code. The formulation of the ground state elastic characteristics is based on Arruda and Boyce (1993) eight-chain model. The evolution of the constitutive equations for the nonlinear endochronic elastoplastic response are derived in analogy to the Bergström–Boyce finite viscoelasticity model discussed by Dal and Kaliske (2009).