Modeling fracture in large scale shell structures

We consider a problem of modeling fracture and failure preceded by large scale yielding of ductile shells from the point of view of large-scale structural analysis. We place a special emphasis on the computational efficiency of the constitutive formulation. In this context, we seek the formulation embedded in the shell mechanics framework, which is both theoretically sound and easily implementable into a large-scale explicit dynamic finite element code without precluding vectorization or parallelization. This is achieved through the elasto-plastic damage constitutive model for finite-element analysis of plates and shells. The proposed damage model is purely phenomenological with a scalar damage parameter, which has no physical interpretation, except that it represents on a global scale the micromechanical changes the material undergoes during the process of necking and fracture. The localization leading to softening and fracture is represented by the damage calibration function with exponential damage growth after the onset of necking. The proposed phenomenological damage model uses a general plasticity and shell mechanics frameworks which makes it general and easily implementable into existing finite element codes. The proposed formulation has been implemented into the explicit dynamic finite element software code EPSA (Atkatsh et al., 1980, Atkatsh et al., 1983).     

A micro-macro approach to rubber-like materials. Part III: The micro-sphere model of anisotropic Mullins-type damage

A micromechanically based non-affine network model for finite rubber elasticity and viscoelasticity was discussed in Parts I and II [Miehe, C., Göktepe, S., Lulei, F., 2004. A micro-macro approach to rubber-like materials. Part I: The non-affine micro-sphere model of rubber elasticity. J. Mech. Phys. Solids 52, 2617-2660; Miehe, C., Göktepe, S., 2005. A micro-macro approach to rubber-like materials. Part II: Viscoelasticity model for polymer networks. J. Mech. Phys. Solids, published on-line, doi:10.1016/j.jmps.2005.04.006.] of this work. In this follow-up contribution, we further extend the micro-sphere network model such that it incorporates a deformation-induced softening commonly referred to as the Mullins effect. To this end, a continuum formulation is constructed by a superimposed modeling of a crosslink-to-crosslink (CC) and a particle-to-particle (PP) network. The former is described by the non-affine elastic network model proposed in Part I. The Mullins-type damage phenomenon is embedded into the PP network and micromechanically motivated by a breakdown of bonds between chains and filler particles. Key idea of the constitutive approach is a two-step procedure that includes (i) the set up of micromechanically based constitutive models for a single chain orientation and (ii) the definition of the macroscopic stress response by a directly evaluated homogenization of state variables defined on a micro-sphere of space orientations. In contrast to previous works on the Mullins effect, our formulation inherently describes a deformation-induced anisotropy of the damage as observed in experiments. We show that the experimentally observed permanent set in stress-strain diagrams is achieved by our model in a natural way as an anisotropy effect. The performance of the model is demonstrated by means of several numerical experiments including the solution of boundary-value problems.     

Micromorphic approach for finite gradient-elastoplasticity fully coupled with ductile damage: Formulation and computational aspects

It is well established that the use of inelastic constitutive equations accounting for induced softening, leads to pathological space (mesh) and time discretization dependency of the numerical solution of the associated Initial and Boundary Value Problem (IBVP). To avoid this drawback, many less or more approximate solutions have been proposed in the literature in order to regularize the IBVP and to obtain numerical solutions which are, at convergence, much less sensitive to the space and the time discretization. The basic idea behind these regularization techniques is the formulation of nonlocal constitutive equations by introducing some effects of characteristic lengths representing the materials microstructure. In this work, using the framework of generalized nonlocal continua, a thermodynamically-consistent micromorphic formulation using appropriate micromorphic state variables and their first gradients, is proposed in order to extend the classical local constitutive equations by incorporating appropriate characteristic internal lengths. The isotropic damage, the isotropic and the kinematic hardenings are supposed to carry the targeted micromorphic effects. First the theoretical aspects of this fully coupled micromorphic formulation is presented in details and the proposed generalized balance equations as well as the fully coupled micromorphic constitutive equations deduced. The associated numerical aspects in the framework of the classical Galerkin-based FE formulation are briefly discussed in the special case of micromorphic damage. Specifically, the formulation of 2D finite elements with additional degrees of freedom (d.o.f.), the dynamic explicit global resolution scheme as well as the local integration scheme, to compute the stress tensor and the state variables at each integration point of each element, are presented. Application is made to the typical uniaxial tension specimen under plane strain conditions in order to chow the predictive capabilities of the proposed micromorphic model, particularly against its ability to give (at convergence) a mesh independent solution even for high values of the ductile damage (i.e., the macroscopic cracks).     

Cohesive–frictional interface constitutive model

In the framework of numerical analysis of joined bodies, the present paper is devoted to the constitutive modeling, via an interface kinematic formulation, of mechanical behaviour of internal adhesive layers. The proposed interface constitutive model couples a cohesive behaviour, based on the damage mechanics theory, with a frictional one, defined in a non-associative plasticity framework. Namely, the interface formulation follows the transition of the adhesive material from the sound elastic condition to the fully cracked one. This formulation is able to model, by means of a specific interpretation of the damage variable and in a relevant mathematical setting, the interface intermediate mechanical properties, during the microcracks spreading process up to the discontinuity surface formation (macrocrack). The constitutive modeling is performed in fully compliance with the thermodynamic principles, in order to ensure the thermodynamic consistency requirement. In the present work, various monotonic and cyclic loading conditions are examined in order to show the main features of the constitutive formulation as well as several significant differences with respect to other existing models. Computational efficiency of the interface constitutive model is tested in a numerical application by FEM resolution strategy approach.     

Micromechanics-based elastic-damage analysis of laminated composite structures

Based on the micromechanics-based constitutive model, derived in our preceding work [Lee, H.K., Pyo, S.H., 2009. A 3D-damage model for fiber-reinforced brittle composites with microcracks and imperfect interfaces. Journal of Engineering Mechanics-ASCE, in press, doi:10.1061/(ASCE)EM.1943.7889.0000039.], incorporating a multi-level damage model and a continuum damage model, the overall elastic behavior and damage evolution of laminated composite structures are studied in detail. The constitutive model is implemented into the finite element program ABAQUS using a user-subroutine UMAT to solve boundary value problems of the composite structures. The validity of the implemented constitutive model is assured by comparing the predicted stress–strain curves with experimental data available in literature under uniaxial tension with various fiber orientations. The results show that the proposed micromechanics-based constitutive model accurately predict the overall elastic-damage behavior of laminated composite structures having different material compositions, thicknesses and boundary conditions.     

An elastoplastic constitutive relation of whisker-reinforced composite for meso damage: Part I – formulation

An elastoplastic constitutive relation is developed for meso damage of whisker-reinforced composites. A model is constructed that includes orientation distribution of whiskers and slip systems as well as interface and crystal sliding. Evolution of damage will be addressed. Given in Part I is the formulation while examples will be illustrated in Part II.     

Quasi-micromechanical constitutive theory for brittle damaged materials under tension

One of fundamental but difficult problems in damage mechanics is the formulation of the effective constitutive relation of microcrack-weakened brittle or quasi-brittle materials under complex loading, especially when microcrack interaction is taken into account. The combination of phenomenological and micromechanical damage mechanics is a promising approach to constructing an applicable damage model with a firm physical foundation. In this paper, a quasi-micromechanical model is presented for simulating the constitutive response of microcrack-weakened materials under complex loading. The microcracking damage is characterized in terms of the orientation domain of microcrack growth (DMG) as well as a scalar microcrack density parameter. The DMG describes the complex damage and its evolution associated with microcrack growth, while the scalar microcrack density factor defining the isotropic magnitude of damage yields an easy calculation of the effects of microcrack interaction on effective elastic moduli. Project supported by the National Natural Science Foundation of China (19891180).     

A symmetric over-nonlocal microplane model M4 for fracture in concrete

This paper analyzes the effectiveness of a nonlocal integral-type formulation of a constitutive law such as microplane model M4 in which the yield limits soften as a function of the total strain for prediction of fracture propagation. For a correct regularization of the mathematical problems caused by the softening behavior, an “over-nonlocal” generalization of the type proposed by Vermeer and Brinkgreve [Vermeer, P.A., Brinkgreve, R.B.J., 1994. A new effective non-local strain measure for softening plasticity. In: Chambon, R., Desrues, J., Vardoulakis, I. (Eds.), Localization and Bifurcation Theory for Soil and Rocks, Balkema, Rotterdam, pp. 89–100.] is adopted. Moreover, the symmetric weight function, proposed by Borino et al. [Borino, G. Failla, B., Parrinello, F., 2003. A symmetric nonlocal damage theory. International Journal of Solids and Structure 40, 3621–3645.] for damage mechanics, is introduced for the calculation of the nonlocal averaging of the total strain upon which the yield limits depend. The capability of the proposed model for reproducing the stress and strain fields in the vicinity of a notch is also investigated. Finally, the symmetric over-nonlocal generalization of microplane model M4 has been applied for the simulation of a mixed-mode fracture test such as the four-point-shear test and the test of axial tension at constant shear force [Nooru-Mohamend, M.B., 1992. Mixed-mode fracture of concrete: an experimental approach. Doctoral Thesis Delft University of Thechnology, Delft, The Netherlands.]     

A constitutive model for the in-plane mechanical behavior of nonwoven fabrics

A constitutive model is presented for the in-plane mechanical behavior of nonwoven fabrics. The model is developed within the context of the finite element method and provides the constitutive response for a mesodomain of the fabric corresponding to the area associated to a finite element. The model is built upon the ensemble of three blocks, namely fabric, fibers and damage. The continuum tensorial formulation of the fabric response rigorously takes into account the effect of fiber rotation for large strains and includes the nonlinear fiber behavior. In addition, the various damage mechanisms experimentally observed (bond and fiber fracture, interfiber friction and fiber pull-out) are included in a phenomenological way and the random nature of these materials is also taken into account by means of a Monte Carlo lottery to determine the damage thresholds. The model results are validated with recent experimental results on the tensile response of smooth and notched specimens of a polypropylene nonwoven fabric.     

Rate-dependent transition from tensile damage to discrete fracture in dynamic brittle failure

Based on the work for a combined damage/plasticity model of geologic materials and the bifurcation analysis of material failure, an analytical framework is established to study the rate-dependent transition from continuum damage to discrete fracture in dynamic brittle failure. Because of the simple formulation, a vectorized constitutive model solver can be designed for large-scale computer simulation. A continuum tangent stiffness tensor is invoked for the tensile damage evolution such that the bifurcation analysis can be performed to identify the initiation and orientation of tensile failure. It is shown that the orientation of tensile failure is rate-independent although the limit state is rate-dependent for the rate-dependent tensile damage model. Sample problems are considered to demonstrate the features of the proposed approach.     

A thermodynamics based damage mechanics model for particulate composites

A micro-mechanical damage model is proposed to predict the overall viscoplastic behavior and damage evolution in a particle filled polymer matrix composite. Particulate composite consists of polymer matrix, particle fillers, and an interfacial transition interphase around the filler particles. Yet the composite is treated as a two distinct phase material, namely the matrix and the equivalent particle-interface assembly. The CTE mismatch between the matrix and the filler particles is introduced into the model. A damage evolution function based on irreversible thermodynamics is also introduced into the constitutive model to describe the degradation of the composite. The efficient general return-mapping algorithm is exploited to implement the proposed unified damage coupled viscoplastic model into finite element formulation. Furthermore, the model predictions for uniaxial loading conditions are compared with the experimental data.     

Constitutive model for flake graphite cast iron automotive brake discs: induced anisotropic damage model under complex loadings

The present paper details an elasto-viscoplastic constitutive model for automotive brake discs made of flake graphite cast iron. In a companion paper (Augustins et al. in Contin Mech Thermodyn, 2015), the authors proposed a one-dimensional setting appropriate for representing the complex behavior of the material (i.e., asymmetry between tensile and compressive loadings) under anisothermal conditions. The generalization of this 1D model to 3D cases on a volume element and the associated challenges are addressed. A direct transposition is not possible, and an alternative solution without unilateral conditions is first proposed. Induced anisotropic damage and associated constitutive laws are then introduced. The transition from the volume element to the real structure and the numerical implementation require a specific basis change. Brake disc simulations with this constitutive model show that unilateral conditions are needed for the friction bands. A damage deactivation procedure is therefore defined.     

Formulation DNLR intégrale des lois de comportement : généralisation du principe de superposition de BoltzmannDNLR integral formulation of behaviour laws: generalisation of Boltzmann's superposition principle.

A thermodynamic approach of relaxation phenomena called DNLR (Distribution of Non Linear Relaxations) has been employed for about fifteen years to describe the behaviour of materials. In this Note, we show that different DNLR constitutive equations, initially written in a rate formulation, can be put in a time-integrated (hereditary) form which generalises the superposition principle to non linear fields like viscoplasticity or damage behaviours. To cite this article: B. Martin et al., C. R. Mecanique 332 (2004).     

一个描述脆性材料非线性行为的损伤力学模型

提出了一个分析和描述脆性材料各向异性损伤的宏观力学模型。在小变型的情况下，利用损伤面和损伤势的概念以及材料性能和材料受损后的弹性响应取决于损伤变量而与加载路径无关的假设，文中导出材料弹性损伤的一般本构关系及联系材料系数与损伤面和损伤势的表达式。当损伤面和损伤势的构造确定后，建立了损伤变量的演变方程和材料性能的变化规律。文中以两种具体材料为例说明该理论模型的应用．     

Strain gradient plasticity by internal-variable approach with normality structure

This paper presents a constitutive formulation for materials with strain gradient effects by internal-variable approach with normality structure. Specific micro-structural rearrangements are assumed to account for the inelasticity deformations for this class of materials, and enter the constitutive formulations in form of internal variables. It is further assumed that the kinetic evolution of any specific micro-structural rearrangement may be fully determined by the thermodynamic forces associated with that micro-structural rearrangement, by normality relations via a flow potential. Macroscopic gradient-enhanced inelastic behaviours may then be predicted in terms of the microscopic internal variables and their conjugate forces, and thus a micro–macro bridging formulation is available for strain-gradient-characterised materials. The obtained formulations are first applied to crystallographic materials, and a crystal gradient plasticity model is developed to account for the influence of microscopic slip rearrangements on the macroscopic gradient-dependent mechanical behaviour for this class of materials. Micro-cracked geomaterials are also treated with these formulations and a gradient-enhanced damage constitutive model is developed to address the impacts of the evolutions of micro-cracks on the macroscopic inelastic deformations with strain gradient effects for these materials. The available formulations are further compared with other thermodynamic approaches of constitutive developing.     

Development of plasticity and damage in vibrating structural elements performing guided rigid-body motions

Summary  A numerical algorithm for studying the development of plastic and damaged zones in a vibrating structural element with a large, guided rigid-body motion is presented. Beam-type elements vibrating in the small-strain regime are considered. A machine element performing rotatory motions, similar to an element of a slider-crank mechanism, is treated as a benchmark problem. Microstructural changes in the deforming material are described by the mesolevel variables of plastic strain and damage, which are consistently included into a macroscopic analysis of the overall beam motion. The method is based on an eigenstrain formulation, considering plastic strain and damage to contribute to an eigenstrain loading of a linear elastic background structure. Rigid-body coordinates are incorporated into this beam-type structural formulation, and an implicit numerical scheme is presented for iterative computation of the eigenstrains from the mesolevel constitutive behavior. Owing to the eigenstrain formulation, any of the existing constitutive models with internal variables could in principle be implemented. Linear elastic/perfectly plastic behavior is exemplarily treated in a numerical study, where plastic strain is connected to the Kachanov damage parameter by a simple damage law. Inelastic effects like plastic shakedown and damage-induced low-cycle rupture are shown to occur in the examplary problems. Received 1 September 1999; accepted for publication 9 March 2000     

A thermo-viscoelastic–viscoplastic–viscodamage constitutive model for asphaltic materials

A temperature-dependent viscodamage model is proposed and coupled to the temperature-dependent Schapery’s nonlinear viscoelasticity and the temperature-dependent Perzyna’s viscoplasticity constitutive model presented in Abu Al-Rub et al., 2009, Huang et al., in press in order to model the nonlinear constitutive behavior of asphalt mixes. The thermo-viscodamage model is formulated to be a function of temperature, total effective strain, and the damage driving force which is expressed in terms of the stress invariants of the effective stress in the undamaged configuration. This expression for the damage force allows for the distinction between the influence of compression and extension loading conditions on damage nucleation and growth. A systematic procedure for obtaining the thermo-viscodamage model parameters using creep test data at different stress levels and different temperatures is presented. The recursive-iterative and radial return algorithms are used for the numerical implementation of the nonlinear viscoelasticity and viscoplasticity models, respectively, whereas the viscodamage model is implemented using the effective (undamaged) configuration concept. Numerical algorithms are implemented in the well-known finite element code Abaqus via the user material subroutine UMAT. The model is then calibrated and verified by comparing the model predictions with experimental data that include creep-recovery, creep, and uniaxial constant strain rate tests over a range of temperatures, stress levels, and strain rates. It is shown that the presented constitutive model is capable of predicting the nonlinear behavior of asphaltic mixes under different loading conditions.     

温度变化对热压电介质裂纹尖端力电损伤的影响分析

由连续介质损伤力学的基本理论出发,引入力电损伤变量并建立了一个热压电介质断裂的损伤本构模型．再由虚功原理导出了求解这类含损伤的关于热力电耦合问题的有限元方程．通过数值计算,分析了温度改变对裂纹尖端力电损伤的影响规律．     

A thermodynamic consistent damage and healing model for self healing materials

Thermodynamics of the damage and the healing processes for viscoplastic materials is discussed in detail and constitutive equations for coupled inelastic-damage-healing processes are proposed in a thermodynamic consistent framework. Small deformation state is utilized and the kinematic and the isotropic hardening effects for the damage and healing processes are introduced into the governing equations. Two new yield surfaces for the damage and healing processes are proposed that take into account the isotropic hardening effect. The computational aspect for solving the coupled elasto-plastic-damage-healing problem is investigated, and the mechanical behavior of the proposed polymeric based self healing system is obtained. Uniaxial compression tests are implemented on a shape memory polymer based self healing system and the damage and the healing are captured by measurement of the changes in the modulus of elasticity. It is concluded that the proposed constitutive equations can model the damage and healing effectively and the mechanical behavior of a shape memory polymer based self healing system can be precisely modeled using this formulation.