Investigation of a Phase Field Model for Elasto-plastic Fracture

Phase field modelling of brittle fracture is very well understood today. However, the attempts of investigation of elasto-plastic fracture by the phase field approach are limited. This contribution deals with the investigation of a phase field model for elasto-plastic fracture. Based on a free energy density comprising elastic, fracture and plastic contributions, the model describes an extension of the linear elastic model towards von Mises plasticity. In this work it is analyzed numerically to which extend analytical findings concerning the interpretation of the model parameters in 1D are transferable to 2D scenarios. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of Atomic Force Microscopy for investigating BaTiO3 and LiMn2O4 nanostructures based on Phase Field Approach

Atomic Force Microscopy (AFM) probes the surface features of specimens using an extremely sharp tip scanning the sample surface while the force is applied. AFM is also widely used for investigating the electrically non-conductive materials by applying an electric potential on the tip. Piezoresponse Force Microscopy (PFM) and Electrochemical Strain Microscopy (ESM) are variants of AFM for different materials. Both PFM and ESM signals are obtained by observing the displacement of the tip when applying electric fields during the scanning process. The PFM technique is based on converse piezoelectric effect of ferroelectrics and the ESM technique is based on electrochemical coupling in solid ionic conductors. In this work, two continuum-mechanical formulations for simulation of PFM and ESM are discussed. In the first model, for PFM simulation, a phase field approach based on the Allen-Cahn equation for non-conserved order parameters is employed for ferroelectrics. Here, the polarization vector is chosen as order parameter. Since ferroelectrics have highly anisotropic properties, this model accounts for transversely isotropic symmetry using an invariant formulation. The polarization switching behavior under the electric field will be discussed with some numerical examples. In the simulation of ESM, we employ a constitutive model based on the work of Bohn et al. [8] for the modeling of lithium manganese dioxide LiMn₂O₄ (LMO). It simulates the deformation of the LMO particle according to an applied voltage and the evolution of lithium concentration after removing a DC pulse. The modeling results are compared to experimental data. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Prediction of fracture toughness of embrittled steels by means of the Small Punch Test

To characterise the randomly distributed strength and fracture toughness of brittle steels, many specimens have to be destroyed. Since the Small Punch Test (SPT) needs only little material, it is a well suited experiment, when only a small volume of material is available. In this study the cleavage fracture of a ferritic steel at low temperature was investigated using the Beremin model. The failure probability is described with a 2-parameter Weibull distribution in terms of the so-called Weibull stress, which is calculated using an elastic-plastic finite element stress analysis. While the transfer of Weibull parameters works well between similar geometries and loading conditions, it works bad in more general cases. Modifications of the Beremin model are necessary to overcome this problem. Recent publications consider a lower threshold value of the Weibull stress, which leads to a lower Weibull modulus and therefore to a stronger volume size effect of strength. The suitability of this approach to transfer cleavage fracture results from SPT to fracture mechanics specimens was investigated. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Enhancement of the micro mechanical basis for local approach cleavage models

The present study is concerned with the investigation of the micro mechanisms of micro defect nucleation in bainitic steels in order to provide an enhanced basis for probabilistic cleavage models. By a micro mechanical modelling of the cleavage initiation process the effects and the interactions of the relevant parameters can be identified. For this purpose Representative Volume Elements (RVE) of the micro structure are utilised, accounting for both, the grain structure as well as the brittle particles at the grain boundaries. The RVE's are loaded based on the local mechanical field quantities determined numerically at the cleavage origins of different fracture mechanics specimens. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase Field Approximation of Dynamic Brittle Fracture

The numerical assessment of fracture has gained importance in fields like the safety analysis of technical structures or the hydraulic fracturing process. The modelling technique discussed in this work is the phase field method which introduces an additional scalar field. The smooth phase field distinguishes broken from undamaged material and thus describes cracks in a continuum. The model consists of two coupled partial differential equations - the equation of motion including the constitutive behaviour of the material and a phase field evolution equation. The crack growth follows implicitly from the solution of this system of PDEs. The numerical solution with finite elements can be accelerated with an algorithm that performs computationally extensive tasks on a graphic processing unit (GPU). A numerical example illustrates the capability of the model to reproduce realistic features of dynamic brittle fracture. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A strong discontinuity based adaptive refinement approach for the modeling of crack branching

In the current work, the physical phenomena of dynamic fracture of brittle materials involving crack growth, acceleration and consequent branching is simulated. The numerical modeling is based on the approach where the failure in the form of cracks or shear bands is modeled by a jump in the displacement field, the so called ‘strong discontinuity’. The finite element method is employed with this strong discontinuity approach where each finite element is capable of developing a strong discontinuity locally embedded into it. The focus in this work is on branching phenomena which is modeled by an adaptive refinement method by solving a new sub-boundary value problem represented by a finite element at the growing crack tip. The sub-boundary value problem is subjected to a certain kinematic constraint on the boundary in the form of a linear deformation constraint. An accurate resolution of the state of material at the branching crack tip is achieved which results in realistic dynamic fracture simulations. A comparison of resulting numerical simulations is provided with the experiment of dynamic fracture from the literature. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

韧脆转变的一种细观随机模糊统计分析

对不同温度和应力状态下40Cr材料进行大子样宏观试验和细观观测,提出了一种新的材料断裂韧脆转变统计随机模糊模型。该模型认为,在统计意义上,材料的韧性断裂为空穴机制,临界空穴扩张比参数可以作为韧性断裂的判据;材料的脆性断裂可以用内嵌币状裂纹的脆性断裂模型来模拟,为此测量断裂特征长度,提出并具体计算了控制币状裂纹失稳扩张的细观临界应力强度因子;在韧脆转变区域内,这两种机理并存并相互竞争,为此提出了模糊准则。对模型参数进行了测量和统计分析,给出分布规律,给出了计算断裂特征的概率模型。计算了韧脆转变区域内的细观机制变化和宏观响应。结果表明,该模型及分析方法可以很好地模拟应力状态及温度对韧脆转变的影响。     

Three-dimensional cell model analysis of ductile damage under dynamic loading condition

The paper deals with the damage and fracture behavior of ductile metals under dynamic loading conditions. The in [1–3] presented phenomenological continuum damage and fracture model, which takes into account the rate- and temperature-dependence of the material, provides reasonable results of experiments with high strain rates while the identification of the corresponding material parameters results difficult from the available experimental data. This lack of information can be resolved by micro-mechanical numerical simulations of void containing unit-cells. In this context results of dynamic micro-mechanical simulations are presented which can be used to study the damage effects on the micro-scale and to validate the rate-dependent continuum damage model. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A coupled phase-field model for ductile fracture in crystal plasticity

The objective of this work is to present a simplified, nonetheless representative first stage of a phenomenological model to predict the crack evolution of ductile fracture in single crystals. The proposed numerical approach is carried out by merging a conventional well- stablished elasto-plastic crystal plasticity model and a well-known phase-field model (PFM) modified to predict ductile fracture. A two-dymensional initial boundary-value problem of ductile fracture is introduced considering a single crystal Nickel-base superalloy material. the model is implemented into the finite element context subjected to a one-dimensional tension test (displacement-controlled). (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling of Type I fracture network: Objective function formulation by fuzzy sensitivity analysis

This paper advances the fundamental understanding in mathematical and computational modelling of discrete fracture networks (Type I). It presents a systematic procedure to solve the most important problem in modelling by global optimization — objective function formulation, which negates guesswork in objective function formulation by automatic selection of highly ranked components and their corresponding weighting factors. The procedure starts from real data to identify potential components of the objective function. The components are then ranked by fuzzy sensitivity analysis, based on their effects on the final objective function value and simulation convergence. The final fracture network inversion is subsequently realized and validated. Results of the study provide an explanation why previous methods such as stochastic simulations are not sufficiently reliable, compared to global optimization methods.     

Material forces in continuum damage material

The material force method is present state of the art in computational fracture mechanics. Material forces yield more information than standard approaches like the J‐integral in combination with a highly efficient evaluation in a separate post‐processing step. In this research, the material force method will be applied in tire durability simulations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Formulation of the material force approach for elastic and inelastic materials

Fracture mechanical investigations are important for the durability of fracture sensitive products. Fracture mechanical simulations help to shorten the product development cycle and to decrease product costs. The so-called material force approach as an efficient tool is used to determine fracture mechanical parameters for elastic and inelastic materials. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of Hertzian cone cracks using a phase field description for fracture

The present contribution focuses on fracture caused by indentation loading on the surface of a brittle solid. Its theoretical prediction is a challenging task due to the fact that crack nucleation is not geometrically induced, but is caused by the stress concentration in the contact near-field. The application of the phase field model requires constitutive assumptions to ensure a tension-compression asymmetric material response and prevent damage in compressed regions. This is achieved at the cost of giving up the variational concept of brittle fracture. We simulate the indentation of a cylindrical flat-ended punch on brittle materials like silicate glass. In order to reduce the numerical effort, we exploit axisymmetric conditions for the finite element formulation. After crack initiation stable propagation of a cone crack can be observed in good agreement with experiments. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Long-time strength of polyethylene in biaxial tension

The long-time strength of high-density polyethylene (HDPE) in biaxial tension has been experimentally investigated at ratios of the principal stress components =₁/₂=0, 0.5, 1, 4. The maximum duration of the experiments was 2500 h. The limit surface for HDPE has been constructed. It is shown that the limit surface for this material varies with time as the mode of fracture changes from quasibrittle at medium stresses to brittle at low stresses. In the case of quasibrittle fracture the condition of equivalence of simple and plane states of stress is satisfactorily described by the Malmeister and Gol'denblat-Kopnov criteria, and in the case of brittle fracture by the maximum normal stress criterion.All-Union Scientific-Research Institute of Hydromechanization, Sanitary Engineering, and Special Construction, Leningrad. Translated from Mekhanika Polimerov, No. 3, pp. 401–408, May–June, 1976.     

Conditions of the ductile-brittle transition in failure of polymer materials

Conclusions The polymer materials are characterized by the transition from ductile to brittle fracture with increasing loading rate and decreasing temperature. The brittle fracture susceptibility of the material can be determined on the basis of the critical size of the defect/ crack. The measure of the cracking resistance of plastics can often be represented by the material scale of the crack length. The quality of the critical size of the defect/crack to the material scale of the crack length can be used as a criterion determining the conditions of transition from ductile to brittle fracture.Translated from Mekhanika Kompozitnykh Materialov, No. 5, pp. 779–785, September–October, 1988.     

Fracture Mechanical Investigation of Elastomeric Material

The increasing use of elastomeric components in advanced engineering applications requires a thorough understanding of the complex material properties and a reliable assessment of the quality and durability of the products. This contribution concentrates on the computational determination of fracture mechanical parameters for rubber material using the material force method. For dissipative, inelastic material, a distinction between two fracture mechanical parameters is presented. The time-dependent behaviour of these fracture mechanical parameters is illustrated by an application to the dwell-effect. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling and Simulation of Degassing Process

This work deals with the modelling and simulation of a degassing process mainly used for extruders in polymer industry. The numerical simulations are done with finite-volume method using OpenFOAM for a 2D single screw extruder. The material parameters have been all chosen for a PDMS-Pentane polymer mixture so that the results could be compared with the available experiments already performed for this mixture [1]. In addition to experiments, the numerical results will be compared with an analytical solution derived from Danckwerts' model [2][3]. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On iterative techniques for computing flow in large two-dimensional discrete fracture networks

Computation of flow in discrete fracture networks often involves solving for hydraulic head values at all intersection points of a large number of stochastically generated fractures inside a bounded domain. For large systems, this approach leads to the generation of problems involving highly sparse matrices which must be solved iteratively. Distributions of fracture lengths spanning over several orders of magnitude, and the randomness of fracture orientations and locations, lead to coefficient matrices that are devoid of any regular structure in the sparsity pattern. In addition to the rapid increase in computational effort with increase in the size of the fracture network, the spread in the distribution of fracture parameters, such as length and transmissivity, dramatically influences the convergence behavior of the system of linear equations. An overview of the discrete fracture network (DFN) methodology for computation of flow is presented along with a comparative study of various Krylov subspace iterative methods for the resulting class of sparse matrices. The rate of convergence of the iterative techniques is found to exhibit a systematic pattern with respect to changes in statistical parameters of the stochastically generated fracture networks. Salient features of the observed trends in the convergence pattern are discussed and guidelines for design of DFN algorithms are provided.     

Experimental Validation of RVE Based Failure Simulations of Macro-Porous Materials

Connections between inhomogeneities and the failure behavior of brittle material may be investigated by finite element simulations of representative volume elements. Representative volume elements are typically subjected to periodic boundary conditions. Moreover, representative volume elements are often chosen as planar, i. e., two dimensional in order to reach reasonable statistics with regard to random distributions of inhomogeneities. The significance of such strongly simplified simulations needs to be validated, especially if the matrix failure is potentially dominated by defects, as is the case, e. g., in macro-porous ceramics. We propose a quasi-periodic concept to design specimens with cylindrical pores, which reproduce the stress state in a two dimensional representative volume element. This is achieved by a partial periodic replication of the region of interest. We suggest that material models used in simulations can be assessed by comparison between simulated and experimentally observed failure. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)