B. Kröplin

Fragmentation of Shells

We present a theoretical and experimental study of the fragmentation of closed thin shells made of a disordered brittle material. Experiments were performed on brown and white hen egg shells under two different loading conditions: impact with a hard wall and explosion by a combustible mixture. Both give rise to power law fragment size distributions. A three-dimensional discrete element model of shells is worked out. Based on simulations of the model, we give evidence that power law fragment mass distributions arise due to an underlying phase transition which proved to be abrupt for explosion and continuous for impact. We demonstrate that the fragmentation of closed shells defines a new universality class of fragmentation phenomena.

A FINITE ELEMENT MODEL FOR DELAMINATION IN COMPOSITE PLATES

SUMMARY A finite element model for the calculation of the static load-deflection behaviour of plates containing a delamination is presented. The introduction of a process layer, in which the connection between two laminate parts can be fixed or controlled, allows the use of one element across the laminate thickness in both the undamaged and delaminated area. To avoid very high numerical costs, the model is based on the simplifying Reissner-Mindlin plate theory and not on the general three-dimensional elasticity theory. The element formulation leads to the possibility of considering contact in a very simple way and applying both fracture and damage mechanics to estimate delamination growth. Although generally, a fracture mechanical mode separation as well as an exact determination of the stress state along the delamination front is not possible, the simplified model allows an efficient calculation of the load-deflection behaviour for stationary delaminations and an estimation of delamination growth with Gr...

Scaling behavior of fragment shapes

We present an experimental and theoretical study of the shape of fragments generated by explosive and impact loading of closed shells. Based on high speed imaging, we have determined the fragmentation mechanism of shells. Experiments have shown that the fragments vary from completely isotropic to highly anisotropic elongated shapes, depending on the microscopic cracking mechanism of the shell. Anisotropic fragments proved to have a self-affine character described by a scaling exponent. The distribution of fragment shapes exhibits a power-law decay. The robustness of the scaling laws is illustrated by a stochastic hierarchical model of fragmentation. Our results provide a possible improvement of the representation of fragment shapes in models of space debris.

Breakup of shells under explosion and impact

A theoretical and experimental study of the fragmentation of closed thin shells made of a disordered brittle material is presented. Experiments were performed on eggshells under two different loading conditions: fragmentation due to an impact with a hard wall and explosion by a combustion mixture giving rise to power law fragment size distributions. For the theoretical investigations a three-dimensional discrete element model of shells is constructed. Molecular dynamics simulations of the two loading cases resulted in power law fragment mass distributions in satisfactory agreement with experiments. Based on large scale simulations we give evidence that power law distributions arise due to an underlying phase transition which proved to be abrupt and continuous for explosion and impact, respectively. Our results demonstrate that the fragmentation of closed shells defines a universality class, different from that of two- and three-dimensional bulk systems.

Finite element implementation of Puck’s failure theory for fibre-reinforced composites under three-dimensional stress:

The objectives of this article are to apply Puck’s failure criteria to predict the failure of 12 test problems, proposed in Part A of the second World-Wide Failure Exercise. These problems include a polymer material, various unidirectional laminae and three multi-directional laminates under a variety of 3D stress loadings. The implementation was carried out through a commercial finite element code where material nonlinearities, due to material behaviour under shear and transverse and through-thickness loadings and due to post failure damage, were taken into account. This is the first time where the critertion has been stretched to its limits. Some of the challenges found include the need to determine the fracture angle of action plane under 3D stresses and the treatment of the strengthening effects on the nonlinear stress strain curves when a lamina is subjected to combined compressive stresses in both the transverse and through-thickness directions. The successful methodology developed here will be used to analyse the effects of boundary conditions in Part B of the second World-Wide Failure Exercise to improve correlation with experimental results for the test problems.

A computational method for the analysis of delamination growth in composite plates

Abstract This paper presents a finite element model based on the Reissner-Mindlin plate theory to calculate the static load-deflection behavior of delaminated plates. Delamination growth is taken into account by applying a fracture mechanics criterion which can only be checked along the delamination front. Therefore, a so-called front control method has to be used for the delamination growth simulation which is realized with a moving mesh technique adapting the mesh successively to the delamination front. Although the simplifications of the model compared to the three-dimensional elasticity theory restrict the model to Griffith-type growth criteria, qualitative delamination growth phenomena can be described rather well.

Algorithmic implementation of a generalized cohesive crack model

Smeared fictitious crack models can be regarded as generalized cohesive crack models. The classic fictitious crack models, i.e. the fixed crack, multiple fixed crack, rotating crack and microplane model, are based on different assumptions for the orientation of developing cracks. A smooth transition between the extreme cases, the fixed crack and the rotating crack model, is provided by the adaptive fixed crack model. In this approach, the critical direction of failure is uniquely identified based on Mohrs hypothesis. Thus, the critical direction depends on the character of the failure criterion and the type of loading. The numeric implementation of the adaptive fixed crack model has given rise to some subtle questions. It is shown that even for a classical fixed crack concept, the algorithmic tangent stiffness may have to include components of crack rotation, depending on the imposed strategy for the global equilibrium iteration scheme.

Generalized Cohesive Zone Model: incorporating triaxiality dependent failure mechanisms

Abstract The present work has been inspired by a presentation given in the preceding conference of this series (T. Siegmund, W. Brocks, Int. J. Fracture, in print). In this study, a modified Gurson Model has been adopted as a reference solution and the response of classic Cohesive Zone Models (CZM) has been evaluated. It has been shown that the conventional CZM in general is not able to predict the influence of the triaxiality on the failure initiation, and that it is not possible to reproduce the expected reference behaviour with a single set of calibration parameters. In the presented framework of modelling, the feature of mode I failure and its transition to mixed mode failure is incorporated within a Generalized Cohesive Zone Model (GCZM; S. Weihe, Dissertation, 1995). Fracture is initiated by a strength criterion while progressive material degradation is based on energy criteria in analogy to Fracture Mechanics. Complete separation due to fracture is obtained when the critical fracture toughness G f c has been dissipated, where the actual value of G f c is dependent on the predicted failure mode. It is shown that the transition to mixed mode failure allows the GCZM to reproduce the varying contributions of modes I and II over the triaxiality regime realistically.

A treatment of mixed mode fracture in debonding

Abstract A strength based fracture criterion is introduced to describe progressive cracking. Separate internal state variables for each principal failure mode ensure consistent treatment of Mode I, Mode II and mixed mode failure. In numerical examples mixed mode debonding is observed at a curved material interface whereas Mode I separation dominates the response in a homogeneous material.

Classical and Advanced Models for Laminated Plates with Piezoelectric Layers Actuated in Shear Mode.

We discuss the two-dimensional modelling of laminated plates with embedded piezoelectric layers actuated in shear mode. Refined approaches are necessary to capture all relevant phenomena characterizing the electro-mechanical interactions. The framework of our approach is a previously established Unified Formulation, which permits the use of Equivalent Single Layer as well as Layer-Wise descriptions in conjunction with higher-order thickness approximations. Two distinct model types are proposed: classical models based on Hamiltons principle and advanced models based on a novel partially mixed four-field formulation. Advanced models are capable of a priori fulfilling the interlaminar continuity of all transverse fluxes. In this paper, we adopt an analytical Navier-type closed-form solution method to solve the resulting system of coupled partial differential equations governing the static and the free-vibration problems. Different interlaminar conditions and electrode configurations are addressed. The proposed 2D models recover accurately the global and local response of exact 3D solutions available in literature. An assessment of various 2D models confirms the superiority of the advanced models with respect to classical ones. By virtue of their accuracy, the analytical solutions obtained from our higher-order advanced models could be employed for providing reference solutions for more general numerical approximation schemes like the FEM which are more suitable for practically relevant applications.