A finite element approach to anisotropic damage of ductile materials in large deformations Part I: an anisotropic ductile elastic-plastic-damage model

During past decades, many material models using the continuum damage mechanics (CDM) approach have been proposed successfully in the small deformation regime to describe inelastic behaviors and fracturing phenomena of a material. For ductile materials, large deformation takes place at the level of damage appearance. Damage is anisotropic in nature. In this paper, the ductile damage at finite deformations is modeled as an anisotropic tensor quantity. Then, a fourth-order symmetric stress correction tensor is proposed for computationally efficient and easy implementation in the finite element formulations. Consequently, an explicit form of the fourth-order constitutive equations of the proposed elastic-plastic-damage model is derived. Both isotropic and kinematic hardening effects are included in the formulation. The new constitutive model can predict not only the elastic-plastic behaviors, but also the sequential variations of ductile materials. An evaluation of the constitutive and damage evolution equations is presented. This revised version was published online in August 2006 with corrections to the Cover Date.     

Ductile damage modelling with locking‐free regularised GTN model

The major goal of this work is to develop a robust modelling strategy for the simulation of ductile damage development including crack initiation and subsequent propagation. For that purpose, a Gurson‐type model is used. This model class, as many other damage models, leads to significant material softening and must be used within a large deformation framework due to the ductile character of the materials. This leads to 2 main difficulties that should be dealt with carefully: mesh dependency and volumetric locking. In this work, a logarithmic finite strain framework is adopted in which the Gurson‐Tvergaard‐Needleman constitutive law is reformulated. Then a nonlocal formulation with regularisation of hardening variable is applied so as to solve mesh dependency and strain localization problem. In addition, the nonlocal model is combined with mixed “displacement‐pressure‐volume variation” elements to avoid volumetric locking. Thereby, a mesh‐independent and locking‐free finite strain framework suitable for the modelling of ductile rupture is established. Attention is paid to mathematical properties and numerical performance of the model. Finally, the model parameters are identified on an experimental database for a nuclear piping steel. Simulations of standard test specimens (notched tensile bars and compact tension and single edge notched tensile cracked specimens) are performed and compared to experimental results.     

Microstructural effects on ductile fracture in heterogeneous materials. Part I: Sensitivity analysis with LE-VCFEM

Ductile failure of heterogeneous materials, such as cast aluminum alloys and discretely reinforced aluminums or DRA’s, initiates with cracking, fragmentation or interface separation of inclusions, that is followed by propagation in the matrix by a ductile mechanism of void nucleation and growth. Damage localizes in bands of intense plastic deformation between inclusions and coalesces into a macroscopic crack leading to overall failure. Ductile fracture is very sensitive to the local variations of the microstructure morphology. This is the first of a two part paper on the effect of microstructural morphology and properties on the ductile fracture in heterogeneous ductile materials. In this paper the locally enhanced Voronoi cell finite element method (LE-VCFEM) for rate-dependent porous elastic–viscoplastic materials is used to investigate the sensitivity of strain to failure to loading rates, microstructural morphology and material properties. A model is also proposed for strain to failure, incorporating the effects of important morphological parameters.     

R–S ADAPTED ARBITRARY LAGRANGIAN–EULERIAN FINITE ELEMENT METHOD FOR METAL-FORMING PROBLEMS WITH STRAIN LOCALIZATION

In this paper, an adaptive Arbitrary Lagrangian–Eulerian (ALE) finite element method is developed for solving large deformation problems with applications in metal-forming simulation. The ALE mesh movement is coupled with r-adaptation of automatic node relocation, to minimize element distortion during the process of deformation. Strain localization is considered in this study through the constitutive relations for ductile porous materials. Prediction of localized deformation is achieved through a multilevel mesh superimposition method, termed as s-adaptation. The model is validated by comparison with established results and codes, and a few metal-forming problems are simulated to test its effectiveness.     

Locally enhanced Voronoi cell finite element model (LE‐VCFEM) for simulating evolving fracture in ductile microstructures containing inclusions

Ductile heterogeneous materials such as cast aluminum alloys undergo catastrophic failure that initiates with particle fragmentation, which evolves with void growth and coalescence in localized bands of intense plastic deformation and strain softening. The Voronoi cell finite element model (VCFEM), based on the assumed stress hybrid formulation, is unable to account for plastic strain‐induced softening. To overcome this shortcoming of material softening due to plastic strain localization, this study introduces a locally enhanced VCFEM (LE‐VCFEM) for modeling the very complex phenomenon of ductile failure in heterogeneous metals and alloys. In LE‐VCFEM, finite deformation displacement elements are adaptively added to regions of localization in the otherwise assumed stress‐based hybrid Voronoi cell finite element to locally enhance modeling capabilities for ductile fracture. Adaptive h‐refinement is used for the displacement elements to improve accuracy. Damage initiation by particle cracking is triggered by a Weibull model. The nonlocal Gurson–Tvergaard–Needleman model of porous plasticity is implemented in LE‐VCFEM to model matrix cracking. An iterative strain update algorithm is used for the displacement elements. The LE‐VCFEM code is validated by comparing with results of conventional FE codes and experiments with real materials. The effect of various microstructural morphological characteristics is also investigated. Copyright © 2008 John Wiley & Sons, Ltd.     

Determination of deformation and failure properties of ductile materials by means of the small punch test and neural networks

This paper describes an approach to identify plastic deformation and failure properties of ductile materials. The experimental method of the small punch test is used to determine the material response under loading. The resulting load displacement curve is transferred to a neural network, which was trained using load displacement curves generated by finite element simulations of the small punch test and the corresponding material parameters. The simulated material behavior of the specimen is based on the ductile elastoplastic damage theory of Gurson, Tvergaard and Needleman. During a training process the neural network generates an approximated function for the inverse problem relating the material parameters to the shape of the load displacement curve of the small punch test. This technique was tested for three different materials (ductile steels). The identified parameters are verified by testing and simulating notched tensile specimens.     

Computational mesomechanics of particle-reinforced composites

Numerical models of deformation, damage and fracture in particle-reinforced composite materials, based on the method of multiphase finite elements (MPFE) and element elimination technique (EET), are presented in this paper. The applicability of these techniques for different materials and different levels of simulation was studied. The simulation of damage and crack growth was conducted for several groups of composites: WC/Co hard metal alloys, Al/Si and Al/SiC composites on macro- and mesolevel. It is shown that the used modern techniques of numerical simulation (MPFE and EET) are very efficient in understanding deformation and damage evolution in heterogeneous brittle/ductile materials with inclusions.     

Micromechanical modelling of ductile crack growth in the binder phase of WC–Co

This study focuses on numerical simulation of ductile failure in the Co binder phase of WC–Co hardmetal. The growth of edge cracks under mode I loading is considered. A computational micromechanics approach is taken where the Co binder ligaments are explicitly represented in finite element models. An embedding technique is employed. Crystal plasticity theory is used to represent plastic deformation in the Co ligaments. Crack propagation in the binder is simulated using an element removal technique based on a modified Rice and Tracey model for ductile void growth, and fracture resistance curves are generated. Parameter studies are performed for variations in microstructrual parameters such as numbers of Co ligaments ahead of the crack tip and local Co volume fraction. The importance of thermal residual stresses and finite element mesh density are also investigated.     

Micropolar hyperelasticity: constitutive model, consistent linearization and simulation of 3D scale effects

This study describes a computational framework for three-dimensional finite strain and finite curvature micropolar hyperelasticity. The model is based on the non-linear kinematic setting and features an appropriate hyperelastic material law which is derived within the thermodynamically consistent framework. The material tangent operator is obtained by consistent linearization. An implicit finite element method with a Newton-Raphson procedure is employed for the computation of the nodal displacements and rotations. A number of numerical examples is presented. The results demonstrate (i) that the methodology is capable of capturing 3D length scale effects in finite deformation, (ii) that it is robust and computationally efficient and (iii) that the proposed micropolar element tangent renders asymptotically quadratic convergence of the Newton-Raphson procedure. It is shown that the classical Neo-Hooke type material behaviour is recovered as a special case within the proposed micropolar setting.     

Application of triangular element invariants for geometrically nonlinear analysis of functionally graded shells

This paper is an attempt to construct a computationally effective curved triangular finite element for geometrically nonlinear analysis of elastic shear deformable shells fabricated from functionally graded materials. The focus is on the concise finite-element formulation under the demand of accuracy-simplicity trade-off. To this end, a nonconventional approach based on the invariants of the natural strains of fibers parallel to the element edges is used. The approach allows one to obtain algorithmic formulas for computing the stiffness matrix, gradient, and Hessian of the total strain energy of the finite element. Transverse shear deformation effects are taken into account using the first order shear deformation theory with the shear correction factor dependent on the material property distribution across the shell thickness. The performance of the proposed finite element is demonstrated using problems of functionally graded plates and shells under mechanical and thermal loads.     

基于微观机制的复杂应力状态下钢材韧性断裂行为研究

韧性断裂是钢材最常见的破坏形式,研究钢材韧性断裂机理并准确预测钢材韧性断裂行为具有重要的理论意义和工程实用价值。基于微观机制的断裂预测方法对研究钢材韧性断裂行为有较好的适用性。该文基于体胞模型空穴演化机理改进了现有的韧性断裂模型,校核了Q345钢材断裂模型参数。此外,在韧性断裂模型中引入损伤因子,以考虑应力状态在加载过程中的变化,使断裂模型能准确描述每一加载时刻的累积损伤值。文末采用Fortran语言将断裂模型编写USDFLD子程序,并将其植入有限元程序ABAQUS,对一组十字型刚节点试件单轴拉伸试验进行数值模拟。结果表明,该断裂模型在拉-剪复合应力状态下具有良好的预测精度,且能够准确捕捉钢材断裂起始位置及裂缝扩展路径。该文改进的韧性断裂模型也可用于其它韧性金属材料断裂预测分析。     

Damage models for studying ductile matrix failure in composites

The contribution aims at assessing different computational models for their capabilities in describing the initiation and growth of ductile cracks in inhomogeneous materials at the constituent level. The main emphasis is put on the ductile damage models of Gurson and Rousselier and on a ductile damage indicator triggered model. The models were implemented via user subroutines into the finite element code ABAQUS/Standard, a nonlocal approach being used to overcome their well-known mesh-dependence.     

Investigation of anisotropy problems in sheet metal forming using finite element method

This paper discusses the results of the numerical study of rectangular cup drawing of steel sheets using finite element methods. To be able to verify the results of the numerical solutions, an experimental study was done where the material behavior under deformation was analyzed. A 3D parametric finite element (FE) model was built using the commercial FE-package ABAQUS/Standard. ABAQUS allows analyzing physical models of real processes putting special emphasis on geometrical non-linearities caused by large deformations, material non-linearities and complex friction conditions. Friction properties of the deep drawing quality steel sheet were determined by using the pin-on-disc tribometer. The results show that the friction coefficient depends on the measured angle from the rolling direction and corresponds to the surface topography. A quadratic Hill anisotropic yield criterion was compared with von Mises yield criterion having isotropic hardening. The sensitivity of constitutive laws to the initial data characterizing material behavior is also presented. It is found out that plastic anisotropy of the matrix in ductile sheet metal has influence on deformation behavior of the material. When the material and friction anisotropy are taken into account in the finite element analysis, this approach gives better approximate numerical results for real processes.     

A micromechanical model of distributed damage due to void growth in general materials and under general deformation histories

We develop a multiscale model of ductile damage by void growth in general materials undergoing arbitrary deformations. The model is formulated in the spirit of multiscale finite element methods (FE ²), that is, the macroscopic behavior of the material is obtained by a simultaneous numerical evaluation of the response of a representative volume element. The representative microscopic model considered in this work consists of a space‐filling assemblage of hollow spheres. Accordingly, we refer to the present model as the packed hollow sphere (PHS) model. A Ritz–Galerkin method based on spherical harmonics, specialized quadrature rules, and exact boundary conditions is employed to discretize individual voids at the microscale. This discretization results in material frame indifference, and it exactly preserves all material symmetries. The effective macroscopic behavior is then obtained by recourse to Hill's averaging theorems. The deformation and stress fields of the hollow spheres are globally kinematically and statically admissible regardless of material constitution and deformation history, which leads to exact solutions over the entire representative volume under static conditions. Excellent convergence and scalability properties of the PHS model are demonstrated through convergence analyses and examples of application. We also illustrate the broad range of material behaviors that are captured by the PHS model, including elastic and plastic cavitation and the formation of a vertex in the yield stress of porous metals at low triaxiality. This vertex allows ductile damage to occur under shear‐dominated conditions, thus overcoming a well‐known deficiency of Gurson's model. Copyright © 2012 John Wiley & Sons, Ltd.     

Eulerian formulation using stabilized finite element method for large deformation solid dynamics

This paper describes an Eulerian formulation for large deformation solid dynamics. In the present Eulerian approach, an advective equation is solved using the Stream‐Upwind/Petrov–Galerkin finite element method. The Eulerian finite element method is applied to path‐dependent solid analyses such as impact bar and ductile necking problems. These computational results using the Eulerian finite element method are compared with the results obtained from using the Lagrangian finite element method and an Eulerian formulation based on a finite difference method. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical analysis of a new Eulerian–Lagrangian finite element method applied to steady‐state hot rolling processes

A finite element code for steady‐state hot rolling processes of rigid–visco‐plastic materials under plane–strain conditions was developed in a mixed Eulerian–Lagrangian framework. This special set up allows for a direct calculation of the local deformations occurring at the free surfaces outside the contact region between the strip and the work roll. It further simplifies the implementation of displacement boundary conditions, such as the impenetrability condition. When applied to different practical hot rolling situations, ranging from thick slab to ultra‐thin strip rolling, the velocity–displacement based model (briefly denoted as vu‐model) in this mixed Eulerian–Lagrangian reference system proves to be a robust and efficient method. The vu‐model is validated against a solely velocity‐based model (vv‐model) and against elementary methods based on the Kármán–Siebel and Orowan differential equations. The latter methods, when calibrated, are known to be in line with experimental results for homogeneous deformation cases. For a massive deformation it is further validated against the commercial finite‐element software package Abaqus/Explicit. It is shown that the results obtained with the vu‐model are in excellent agreement with the predictions of the vv‐model and that the vu‐model is even more robust than its vv‐counterpart. Throughout the study we assumed a rigid cylindrical work roll; only for the homogeneous test case, we also investigated the effect of an elastically deformable work roll within the frame of the Jortner Green's function method. The new modelling approach combines the advantages of conventional Eulerian and Lagrangian modelling concepts and can be extended to three dimensions in a straightforward manner. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical study via total Lagrangian smoothed particle hydrodynamics on chip formation in micro cutting

Numerical simulation is an effective approach in studying cutting mechanism. The widely used methods for cutting simulation include finite element analysis and molecular dynamics. However, there exist some intrinsic shortcomings when using a mesh-based formulation, and the capable scale of molecular dynamics is extremely small. In contrast, smoothed particle hydrodynamics (SPH) is a candidate to combine the advantages of them. It is a particle method which is suitable for simulating the large deformation process, and is formulated based on continuum mechanics so that large scale problems can be handled in principle. As a result, SPH has also become a main way for the cutting simulation. Since some issues arise while using conventional SPH to handle solid materials, the total Lagrangian smoothed particle hydrodynamics (TLSPH) is developed. But instabilities would still occur during the cutting, which is a critical issue to resolve. This paper studies the effects of TLSPH settings and cutting model parameters on the numerical instability, as well as the chip formation process. Plastic deformation, stress field and cutting forces are analyzed as well. It shows that the hourglass coefficient, critical pairwise deformation and time step are three important parameters to control the stability of the simulation, and a strategy on how to adjust them is provided.The full text can be downloaded at https://link.springer.com/article/10.1007/s40436-020-00297-z     

A Self-Learning Data-Driven Development of Failure Criteria of Unknown Anisotropic Ductile Materials with Deep Learning Neural Network

This paper first proposes a new self-learning data-driven methodology that can develop the failure criteria of unknown anisotropic ductile materials from the minimal number of experimental tests. Establishing failure criteria of anisotropic ductile materials requires time-consuming tests and manual data evaluation. The proposed method can overcome such practical challenges. The methodology is formalized by combining four ideas: 1) The deep learning neural network (DLNN)-based material constitutive model, 2) Self-learning inverse finite element (SELIFE) simulation, 3) Algorithmic identification of failure points from the self-learned stress-strain curves and 4) Derivation of the failure criteria through symbolic regression of the genetic programming. Stress update and the algorithmic tangent operator were formulated in terms of DLNN parameters for nonlinear finite element analysis. Then, the SELIFE simulation algorithm gradually makes the DLNN model learn highly complex multi-axial stress and strain relationships, being guided by the experimental boundary measurements. Following the failure point identification, a self-learning data-driven failure criteria are eventually developed with the help of a reliable symbolic regression algorithm. The methodology and the self-learning data-driven failure criteria were verified by comparing with a reference failure criteria and simulating with different materials orientations, respectively.