A numerical approach for the simulation of ductile fracture with embedded discontinuities

In the present study, a computational approach for the numerical simulation of ductile fracture within the framework of the finite element method is proposed. In the developed macroscopic formulation, the inelastic behavior in the bulk of the material is described by the finite elasto‐plastic material model proposed in [4]. The failure process is modeled by introducing discontinuities when a special local fracture criterion is satisfied. The discontinuities are incorporated via special triangular finite elements with embedded interfaces following the line of [2]. Finally, the numerical procedure is evaluated for a twodimensional representative test problem. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Life Prediction of FRP Composites by a Direct Method

A direct computational approach for lifetime prediction of fibre-reinforced polymer (FRP) composites is presented. The approach is based on a direct method which allows predicting the fatigue life from the stabilised damage state. The classical direct method is generalised to the case of coupled plasticity with damage mechanics of the UD-FRP composite materials [1]. The constitutive model is based on a continuous damage meso-scale approach [2]. By analysing damage variables and thermodynamical forces associated with damage at the stabilised state, fatigue life prediction law is proposed as a power law function of stabilised thermodynamic forces. The obtained numerical results have been validated by experimental test results on standard glass-fibre/epoxy angle-ply and cross-ply laminate plates. The proposed approach could serve as a useful tool for the design of FRP composites. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase-field approach to fracture for finite-deformation contact problems

The present contribution deals with a variationally consistent Mortar contact algorithm applied to a phase-field fracture approach for finite deformations, see [4]. A phase-field approach to fracture allows for the numerical simulation of complex fracture patterns for three dimensional problems, extended recently to finite deformations (see [2] for more details). In a nutshell, the phase-field approach relies on a regularization of the sharp (fracture-) interface. In order to improve the accuracy, a fourth-order Cahn-Hilliard phase-field equation is considered, requiring global C¹ continuity (see [1]), which will be dealt with using an isogeometrical analysis (IGA) framework. Additionally, a newly developed hierarchical refinement scheme is applied to resolve for local physical phenomena e.g. the contact zone (see [3] for more details). The Mortar method is a modern and very accurate numerical method to implement contact boundaries. This approach can be extended in a straightforward manner to transient phase-field fracture problems. The performance of the proposed methods will be examined in a representative numerical example. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase Field Modeling of Brittle and Ductile Fracture

The phase field modeling of brittle fracture was a topic of intense research in the last few years and is now well-established. We refer to the work [1-3], where a thermodynamically consistent framework was developed. The main advantage is that the phase-field-type diffusive crack approach is a smooth continuum formulation which avoids the modeling of discontinuities and can be implemented in a straightforward manner by multi-field finite element methods. Therefore complex crack patterns including branching can be resolved easily. In this paper, we extend the recently outlined phase field model of brittle crack propagation [1-3] towards the analysis of ductile fracture in elastic-plastic solids. In particular, we propose a formulation that is able to predict the brittle-to-ductile failure mode transition under dynamic loading that was first observed in experiments by Kalthoff and Winkler [4]. To this end, we outline a new thermodynamically consistent framework for phase field models of crack propagation in ductile elastic-plastic solids under dynamic loading, develop an incremental variational principle and consider its robust numerical implementation by a multi-field finite element method. The performance of the proposed phase field formulation of fracture is demonstrated by means of the numerical simulation of the classical Kalthoff-Winkler experiment that shows the dynamic failure mode transition. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Smoothed Particle Hydrodynamics modelling of poroelastic media

Numerical modelling of poroelastic properties in porous media allows widely varied investigations at low costs and relatively short times. The study of porous media is of high interest at different scales, in this case we focus our analysis at meso- and macroscale which is highly relevant e.g. in geothermal explorations. We model a biphasic poroelastic media assuming incompressible fluid and solid grains and a large solid-fluid density ratio. Meshfree methods are nowadays more widely used due to the advantages that present in the simulation of large deformations. In this case we choose to employ the Smoothed Particle Hydrodynamics method (SPH), a Lagrangian method where the domain is discretized in particles. The solution is computed in parallel, which allows to simulate large domains more representative of the scale of our study cases. We validate our implementation with a classical consolidation problem and compare the simulated diffusion process with Terzaghi's analytical solution. Future work includes simulation of fractures initiation and propagation in the porous media at reservoir scale. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical modeling of brittle fracture in porous CFC-materials

Both energy and stress criteria are necessary conditions for fracture but neither the one nor the other is sufficient. A combination of these criteria is proposed in [1]. This combined criterion is used for numerical simulation of crack propagation by the 4-point bending test in porous materials. Examples of such materials are carbon-carbon composites (CFC) [2, 3]. Micrographs of the cross-sections of these materials are used for FEM modeling of the crack propagation on the basis of the proposed criterion. Results of the numerical modeling are compared with experimental results. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

沿两条内部平行裂纹受压时材料的断裂问题

顺着平面内的平行裂纹受压材料的断裂问题并不能在线性断裂力学的框架内进行描述，Grif-fith－Irvin型或COC型的断裂判据，虽然可以用来处理经典的线性断裂力学，但对本题则完全不适用，这是因为这些压力对应力强度系数没有影响，与裂缝孔隙值也没有关系[1，2]，这一类问题只能采用新的方法，本文的第一作者曾建议过一种新方法，在这一方法中仍然使用了线性关系，但这种线性关系是从变形固体力学中的非线性方程导出的[3，4，5]．这里必须指出，这种方法曾在变形体稳定性问题中广泛地采用过。作为断裂开始的判据，我们采用了裂缝缺陷附近的局部失稳的判据，在这类情况下，我们认为是从失稳过程引发断裂过程的。     

The phase-field model with an auto-calibrated degradation function based on general softening laws for cohesive fracture

Phase-field models have become popular to simulate cohesive failure problems because of their capability of predicting crack initiation and propagation without additional criteria. In this paper, a new phase-field damage model coupled with general softening laws for cohesive fracture is proposed based on the unified phase-field theory. The commonly used quadratic geometric function in the classical phase-field model is implemented in the proposed model. The modified degradation function related to the failure strength and length scale is used to obtain the length scale insensitive model. Based on the analytical solution of a 1-D case, general softening laws in cohesive zone models can be considered. Parameters in the degradation function can be calibrated according to different softening curves and material properties. Numerical examples show that the results obtained by the proposed model have a good agreement with experimental results and the length scale has a negligible influence on the load-displacement curves in most cases, which cannot be observed in classical phase-field model.     

A Finite Element Approach for the Simulation of Quasi-Brittle Fracture

In the context of a strong discontinuity approach, we propose a finite element formulation with an embedded displacement discontinuity. The basic assumption of the proposed approach is the additive split of the total displacement field in a continuous and a discontinuous part. An arbitrary crack splits the linear triangular finite element into two parts, namely a triangular and a quadrilateral part. The discontinuous part of the displacement field in the quadrilateral portion is approximated using linear shape functions. For these purposes, the quadrilateral portion is divided into two triangular parts which is in this way similar to the approach proposed in [5]. In contrast, the discretisation is different compared to formulations proposed in [1] and [3], where the discontinuous part of the displacement field is approximated using bilinear shape functions. The basic theory of the underlying finite element formulation and a cohesive interface model to simulate brittle fracture are presented. By means of representative numerical examples differences and similarities of the present formulation and the formulations proposed in [1] and [3] are highlighted. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Study of the Stressed-Deformed State of Rock during Explosive Fracture

The dynamic loading of a rock mass during explosion of a borehole explosive is studied using a continuum mechanics approach in two-dimensional plane and axially symmetric formulations with the aid of a modified finite element method [1, 2]. This numerical technique makes it possible to study wave processes in a rock mass owing to explosions of single charges as well as those of systems of borehole explosives under different conditions. These include varying the site at which the charge is initiated and accounting for the propagation velocity of detonations in the explosive, so it is possible to calculate the shape of the stress field created by a charge with a given design. Numerical simulation of the explosion process for multiple borehole explosive charges with delays relative to one another can be used to obtain the optimum delay time for initiation and the distances between the charges. These results can also extend our concepts of the processes taking place in a rock mass during explosive fracture.     

On the efficient evaluation of ruin probabilities for completely monotone claim distributions

In this paper we propose a highly accurate approximation procedure for ruin probabilities in the classical collective risk model, which is based on a quadrature/rational approximation procedure proposed in [2]. For a certain class of claim size distributions (which contains the completely monotone distributions) we give a theoretical justification for the method. We also show that under weaker assumptions on the claim size distribution, the method may still perform reasonably well in some cases. This in particular provides an efficient alternative to a related method proposed in [3]. A number of numerical illustrations for the performance of this procedure is provided for both completely monotone and other types of random variables.     

On the numerical modelling of initiation of sediment transport using Smoothed Particle Hydrodynamics

Mobilization of solid particles at the interface between a porous and a free flow domain is a relevant subject in many fields of mechanical, civil and environmental engineering. One example is the initiation of sediment transport as it appears in river beds. To approximate this initiation state, various theoretical models exist. Common approaches use two-domain formulations as in [1] or one-domain formulations as in [6]. The named approaches were compared with Direct Numerical Simulations (DNS) using Smoothed Particle Hydrodynamics (SPH) in [3]. The results of these simulations showed that the theoretical models often underestimate the occurring velocities at the interface and therefore critical velocities to initialize the motion of single grains can be lower than predicted by theoretical approaches. In our numerical simulations, we study creeping flow in a free flow domain coupled to flow in a porous media applying various porous structures. To investigate velocities and shear stresses at the interface more intensively we then compare our numerical results to data from experiments that were performed on equivalent microstructures. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Impact of heterogeneously distributed solid particles in the transition zone between porous and free flow domain

We present a direct numerical simulation approach for the simulation of shallow water flow using the particle based meshfree Smoothed Particle Hydrodynamics (SPH) method. Simulations of single phase flow are done to characterize the occurring flow parameters on both macro-scale and pore-scale. More precisely, we examine initiation of motion and sediment transport as appearing at the interface between a free flow and porous flow domain under parallel flow conditions. Therefore we evaluate three theoretical models presenting analytical solutions for this coupled problem. Moreover, we discuss the influence of heterogeneities at the interface on forces on single grains by implementing and testing various microstructures into our numerical model. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Error-controlled homogenization for a class of linear elastic composite problems

The research field of model adaptivity is well established, aiming at adaptive selection of mathematical models from a well defined class of models (model hierarchy) to achieve a preset level of accuracy, see e.g. [4, 5]. The present work addresses its application to a class of linear elastic composite problems. In order to adaptively control both macro model and macro discretization errors, we present a coupled adaptive scheme, which is driven by a goal-oriented error estimator based on duality techniques. For efficient computation of the dual solution, we make use of a patch-based recovery technique proposed in [6]. As a possibility for model adaptivity, we propose a model hierarchy based on the classical bounding theories [2, 3], which is established in a natural and theoretically consistent manner, without combination of different methods using a priori knowledge. For illustration, a numerical example is presented. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling Abrasive Wear Caused by Small Solid Particles of Different Sizes

Abrasive wear is one of the mechanisms which cause the decrease of efficiency of hydraulic machines. The working fluid of a hydraulic machine, e.g., a turbine of a hydroelectric power plant, transports small solid particles of different sizes. Those small particles damage the surface of the hydraulic machine when contacting. In contrast to classical approaches in fluid dynamics, here, we present an approach where only mesh-free methods are applied. The Smoothed Particle Hydrodynamics (SPH) method is used for modeling the fluid in this study. The SPH method is a mesh-free method which has its advantages in describing transient fluid flows with free surfaces and large motions. The loading of the fluid consists of small solid particles of different sizes. A coupled approach for describing the loading is used. For the larger abrasive particles the Discrete Element Method and for smaller ones a transport equation is utilized. In doing so it is possible to model a loading of the fluid consisting of small particles of different sizes. The abrasive wear is described with an abrasive wear model. The wear model takes into account different parameters like the size, the velocity of the abrasive particles, and of course material parameters of both the target and the particles. On impact of an abrasive particle, the amount of removed material is stored at the boundary and in doing so the removed material over time is identified. In this work, a representative numerical example is presented. The simulations were performed with the code Pasimodo, developed at the Institute of Engineering and Computational Mechanics. It is the aim of this work to point out that it is possible to model abrasive wear due to abrasive particles with different sizes with a mesh-free approach. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efficient and accurate numerical modeling of a micro–macro nonlinear optics model for intense and short laser pulses

In this work we are interested in the fast simulation of ultrashort and intense laser pulses propagating in macroscopic nonlinear media. In this goal, we consider the numerical micro–macro Maxwell–Schrödinger-Plasma model originally presented by Lorin et al. [9], [10]. Although this model is, in theory, applicable to large domains, due to its computational complexity, only short distances of propagation could be considered (less than 1 mm so far, see [9]). In the present paper, we explore some simple, but fast and accurate techniques allowing to reduce the computational complexity by a large factor (up to 60) and then to consider larger domains. This reduction is naturally essential to make this model relevant to study realistic laser–matter interactions at a macroscopic scale. Numerical simulations are proposed to illustrate the chosen approach.     

Configurational-Force-Based Adaptive FE Solver for a Phase Field Model of Fracture

The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by diffusive crack modeling, based on the introduction of a crack phase field as outlined in [1, 2]. Following these formulations, we outline a thermodynamically consistent framework for phase field models of crack propagation in elastic solids, develop incremental variational principles and, as an extension to [1, 2], consider their numerical implementations by an efficient h-adaptive finite element method. A key problem of the phase field formulation is the mesh density, which is required for the resolution of the diffusive crack patterns. To this end, we embed the computational framework into an adaptive mesh refinement strategy that resolves the fracture process zones. We construct a configurational-force-based framework for h-adaptive finite element discretizations of the gradient-type diffusive fracture model. We develop a staggered computational scheme for the solution of the coupled balances in physical and material space. The balance in the material space is then used to set up indicators for the quality of the finite element mesh and accounts for a subsequent h-type mesh refinement. The capability of the proposed method is demonstrated by means of a numerical example. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A coupled NMM-SPH method for fluid-structure interaction problems

The main challenges in the numerical simulation of fluid–structure interaction (FSI) problems include the solid fracture, the free surface fluid flow, and the interactions between the solid and the fluid. Aiming to improve the treatment of these issues, a new coupled scheme is developed in this paper. For the solid structure, the Numerical Manifold Method (NMM) is adopted, in which the solid is allowed to change from continuum to discontinuum. The Smoothed Particle Hydrodynamics (SPH) method, which is suitable for free interface flow problem, is used to model the motion of fluids. A contact algorithm is then developed to handle the interaction between NMM elements and SPH particles. Three numerical examples are tested to validate the coupled NMM-SPH method, including the hydrostatic pressure test, dam-break simulation and crack propagation of a gravity dam under hydraulic pressure. Numerical modeling results indicate that the coupled NMM-SPH method can not only simulate the interaction of the solid structure and the fluid as in conventional methods, but also can predict the failure of the solid structure.     

An improved mixed finite element based on a modified least-squares formulation for hyperelasticity

The present work deals with the solution of geometrically nonlinear elastic problems solved by the least-squares finite element method (LSFEM). The main goal is to obtain an improved performance and an accurate approximation in particular for lower-order elements. Basis for the mixed element is a first-order stress-displacement formulation resulting from a classical least-squares method. Similar to the ideas in SCHWARZ ET AL. [1] a modified weak form is derived by the introduction of an additional term controlling the stress symmetry condition. The approximation of the unknowns follows the same procedures as for a conventional least-squares method, see e.g. CAI & STARKE [2]. The proposed modified formulation is compared to recently developed classical LSFEMs, in order to show the improvement of performance and accuracy. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)