A finite element approach to anisotropic damage of ductile materials in large deformations Part II: finite element formulation and applications

A finite element analysis model for material and geometrical non-linearities due to large plastic deformations of ductile materials is presented using the continuum damage mechanics approach. To overcome limitations of the conventional plastic analysis, a fourth-order tensor damage, defined in Part I of this paper to represent the stiffness degradation in the finite strain regime, is incorporated. General forms of an updated Lagrangian (U.L.) finite element procedure are formulated to solve the governing equations of the coupled elastic–plastic-damage analysis, and a computer program is developed for two-dimensional plane stress/strain problems. A numerical algorithm to treat the anisotropic damage is proposed in addition to the non-linear incremental solution algorithm of the U.L. formulation. Selected examples, compared with published results, show the validity of the presented finite element approach. Finally, the necking phenomenon of a plate with a hole is studied to explore plastic damage in large strain deformations. This revised version was published online in August 2006 with corrections to the Cover Date.     

Analysis of three‐dimensional crack initiation and propagation using the extended finite element method

An Erratum has been published for this article in International Journal for Numerical Methods in Engineering 2005, 63(8): 1228. We present a new formulation and a numerical procedure for the quasi‐static analysis of three‐dimensional crack propagation in brittle and quasi‐brittle solids. The extended finite element method (XFEM) is combined with linear tetrahedral elements. A viscosity‐regularized continuum damage constitutive model is used and coupled with the XFEM formulation resulting in a regularized ‘crack‐band’ version of XFEM. The evolving discontinuity surface is discretized through a C⁰ surface formed by the union of the triangles and quadrilaterals that separate each cracked element in two. The element's properties allow a closed form integration and a particularly efficient implementation allowing large‐scale 3D problems to be studied. Several examples of crack propagation are shown, illustrating the good results that can be achieved. Copyright © 2005 John Wiley & Sons, Ltd.     

A general linear method to evaluate the hardening behaviour of metals at large strain with full‐field measurements

The hardening behaviour of metals is generally described in terms of a stress‐strain curve derived from experiments. In this paper, a linear method to identify the stress‐strain curve starting from full‐field measurement data is presented. This method can be applied to any stress state using a generic yield function, the only requirement is that the full‐field measurement is extended up to the border of the specimen. The method is presented and validated using a finite element model of a notched specimen. Moreover, experiments were performed on specimens cut from a BH340 steel sheet to illustrate the viability to actual cases. Two geometries were considered, a standard uniaxial test, where the method was used to evaluate the post‐necking behaviour, and a notched specimen with a heterogeneous strain field. The proposed method, named linear stress‐strain curve identification (LSSCI), can be a useful tool in combination with inverse methods to identify the constitutive behaviour of metals in large strain plasticity.     

Analysis of adhesively bonded joints: a finite element method and a material model with damage

This paper deals with the derivation of a finite element (FE) method for an adhesively bonded joint which consists of two relatively thin bodies, joined by an even thinner adhesive layer. It is based on a model of the compound joint where the three bodies involved are described as material surfaces. A geometrically two‐dimensional model, where the middle surfaces of the upper and lower body are represented as geometrically coinciding surfaces, is obtained. An elastic–plastic material model with damage is used for the adhesive layer, and an important implication is that the (quasi‐static) propagation of the local failure zone in the adhesive layer in a structure can be simulated. Consequently, the failure load is obtained as a computational result and no failure criterion is needed. The problem is discretized, and a surface model, where only a single surface needs to be FE‐meshed, is obtained. A single‐lap joint is analysed and good agreement is obtained when compared to an analysis using a fine mesh with brick element. Furthermore, the failure load is computed and compared with experiments. The derived FE method opens up the possibility to efficiently model and analyse the mechanical behaviour of large bonded structures. Copyright © 2005 John Wiley & Sons, Ltd.     

Progressive damage analyses of angle‐ply laminates exhibiting free edge effects using continuum damage mechanics with layer‐wise finite element method

Capability of continuum damage mechanics (CDM) to predict the damage mechanism evolution of composite laminates has rarely been carried out, and most of the previous CDM works mainly focused on the overall response of the laminates. In this paper, progressive damage and overall response of the composite laminates under quasi‐static, monotonic increasing loading are investigated using three‐dimensional (3D) CDM implementation in a finite element method that is based on the layer‐wise laminate plate theory. In the damage formulation, each composite ply is treated as a homogeneous orthotropic material exhibiting orthotropic damage in the form of distributed microscopic cracks that are normal to the three principal material directions. The progressive damage of different angle‐ply composite laminates under quasi‐static loading that exhibit free edge effects is investigated. It is shown that using CDM global behaviour and various damage mechanisms affected by the complex nature of free edges can be qualitatively well predicted.     

Numerical issues in the virtual fields method

This paper deals with the direct identification of parameters governing anisotropic elastic constitutive equations. These parameters are identified from heterogeneous strain fields with the virtual fields method. This method is based on a relevant use of the principle of virtual work. Different numerical aspects of the implementation of the method are discussed in the paper, mainly in terms of stability of the identified parameters when noisy data are processed. It is shown that the sensitivity of the method to noisy data is compatible with a practical use during experiments. Copyright © 2004 John Wiley & Sons, Ltd.     

A mixed finite element for the nonlinear analysis of in-plane loaded masonry walls

A mixed membrane eight-node quadrilateral finite element for the analysis of masonry walls is presented. Assuming that a nonlinear and history-dependent 2D stress-strain constitutive law is used to model masonry material, the element derivation is based on a Hu-Washizu variational statement, involving displacement, strain, and stress fields as primary variables. As the behavior of masonry structures is often characterized by strain localization phenomena, due to strain softening at material level, a discontinuous, piecewise constant interpolation of the strain field is considered at element level, to capture highly nonlinear strain spatial distributions also within finite elements. Newton's method of solution is adopted for the element state determination problem. For avoiding pathological sensitivity to the finite element mesh, a novel algorithm is proposed to perform an integral-type nonlocal regularization of the constitutive equations in the present mixed formulation. By the comparison with competing serendipity displacement-based formulation, numerical simulations prove high performances of the proposed finite element, especially when coarse meshes are adopted.     

A 3D domain decomposition approach for the identification of spatially varying elastic material parameters

The post‐treatment of (3D) displacement fields for the identification of spatially varying elastic material parameters is a large inverse problem that remains out of reach for massive 3D structures. We explore here the potential of the constitutive compatibility method for tackling such an inverse problem, provided an appropriate domain decomposition technique is introduced. In the method described here, the statically admissible stress field that can be related through the known constitutive symmetry to the kinematic observations is sought through minimization of an objective function, which measures the violation of constitutive compatibility. After this stress reconstruction, the local material parameters are identified with the given kinematic observations using the constitutive equation. Here, we first adapt this method to solve 3D identification problems and then implement it within a domain decomposition framework which allows for reduced computational load when handling larger problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Application of continuum damage mechanics to vibration fatigue life prediction

It is pivotal to predict the multiaxial vibration fatigue life during mechanical structural dynamics design. An algorithm of the finite element method implementation for multiaxial high cycle fatigue life evaluation is proposed, on the basis of elastic evolution model of continuum damage mechanics. By considering structural dynamic characteristics, namely, resonant frequencies and mode shapes, this algorithm includes a modal analysis and harmonic analysis, which makes this different from existing fatigue life prediction methods. A 10% decrease in the resonant frequency is regarded as the failure criterion. A critical damage value was obtained, which indicates mesocrack initiation fulfilment. To validate the effectiveness of the algorithm, auto‐phase sine resonance track‐and‐dwell experiments were conducted on notched cantilever beams made of Ti‐6Al‐4V alloy. The life predictions are conservative and in good agreements with the experimental results, which are mainly distributed within a scatter band of 2. This investigation could provide technical support for structural dynamics design and the analysis of reusable spacecraft.     

A stabilized finite element formulation for monolithic thermo‐hydro‐mechanical simulations at finite strain

An adaptively stabilized monolithic finite element model is proposed to simulate the fully coupled thermo‐hydro‐mechanical behavior of porous media undergoing large deformation. We first formulate a finite‐deformation thermo‐hydro‐mechanics field theory for non‐isothermal porous media. Projection‐based stabilization procedure is derived to eliminate spurious pore pressure and temperature modes due to the lack of the two‐fold inf‐sup condition of the equal‐order finite element. To avoid volumetric locking due to the incompressibility of solid skeleton, we introduce a modified assumed deformation gradient in the formulation for non‐isothermal porous solids. Finally, numerical examples are given to demonstrate the versatility and efficiency of this thermo‐hydro‐mechanical model. Copyright © 2015 John Wiley & Sons, Ltd.     

A revised time domain force identification method based on Bayesian formulation

Time domain force identification problems are generally ill-posed. Regularization techniques are widely adopted to well-condition the problem. Traditional regularization such as Tikhonov regularization is mathematically equivalent to assuming a zero-mean prior distribution on the unknown force. This assumption could be unreasonable for problems where notable trend components exist in force histories, such as vehicle-bridge moving force identification and cutting tool force identification. In this paper, a revised method is proposed to address this issue. The proposed method formulates the force identification problem within the Bayesian framework. The trend components are considered as low-order polynomials and as the mean term in the prior distribution of the force history. The joint maximum a posterior estimate of unknown variables is derived. The solution algorithm is given based on conditional maximization. A mass-spring system and a cutting tool under forces containing various types of trend components and vehicle-bridge systems under moving interaction forces are simulated to validate the proposed method.     

A Dissipation Gap Method for full‐field measurement‐based identification of elasto‐plastic material parameters

Using enriched data such as displacement fields obtained from digital image correlation is a pathway to the local identification of material parameters. Up to now, most of the identification techniques for nonlinear models are based on Finite Element Updating Methods. This article explains how an appropriate use of the Dissipation Gap Method can help in this context and be an interesting alternative to these classical techniques. The Dissipation Gap Methods rely on the concept of error in dissipation that has been used mainly for the verification of finite element simulations. We provide here an original application of these founding developments to the identification of material parameters for nonlinear behaviors. The proposed technique and especially the main technical keypoint of building the admissible fields are described in detail. The approach is then illustrated through the identification of heterogeneous isotropic elasto‐plastic properties. The basic numerical features highlighted through these simple examples demonstrate this approach to be a promising tool for nonlinear identification.Copyright © 2012 John Wiley & Sons, Ltd.     

An orthotropic variant of the equilibrium gap method applied to the analysis of a biaxial test on a composite material

In this paper, it is proposed to apply the equilibrium gap method to orthotropic composite materials to retrieve damage laws from measured displacement fields. A finite difference implementation method is first proposed. A linear system is formed, for which the unknowns are piecewise constant orthotropic rigidities, while the measured displacements are input (known) data. In this example, a cruciform specimen is considered for biaxial test. It is shown that, by referring to FE computed displacement fields a prescribed contrast map can be identified. Corrupted artificial displacement fields obtained through non-linear simulations are also used. When considering shear damage, a procedure using estimated contrast maps to identify a damage law is validated. An experimental biaxial test on a 2.5 C/C composite is finally analysed following the proposed approach. For each unloading step, a contrast map for all moduli is obtained from full-field measurements. By assuming that the shear moduli contrasts result from a damage mechanism, one subsequently obtains damage maps, and therefore, a growth law. The results are first validated by comparing measured and FE reconstructed displacement fields, and by comparing the identified damage fields with post-processed ones.     

A non‐iterative finite element method for inverse heat conduction problems

A non‐iterative, finite element‐based inverse method for estimating surface heat flux histories on thermally conducting bodies is developed. The technique, which accommodates both linear and non‐linear problems, and which sequentially minimizes the least squares error norm between corresponding sets of measured and computed temperatures, takes advantage of the linearity between computed temperatures and the instantaneous surface heat flux distribution. Explicit minimization of the instantaneous error norm thus leads to a linear system, i.e. a matrix normal equation, in the current set of nodal surface fluxes. The technique is first validated against a simple analytical quenching model. Simulated low‐noise measurements, generated using the analytical model, lead to heat transfer coefficient estimates that are within 1% of actual values. Simulated high‐noise measurements lead to h estimates that oscillate about the low‐noise solution. Extensions of the present method, designed to smooth oscillatory solutions, and based on future time steps or regularization, are briefly described. The method's ability to resolve highly transient, early‐time heat transfer is also examined; it is found that time resolution decreases linearly with distance to the nearest subsurface measurement site. Once validated, the technique is used to investigate surface heat transfer during experimental quenching of cylinders. Comparison with an earlier inverse analysis of a similar experiment shows that the present method provides solutions that are fully consistent with the earlier results. Although the technique is illustrated using a simple one‐dimensional example, the method can be readily extended to multidimensional problems. Copyright © 2003 John Wiley & Sons, Ltd.     

Elasto‐plastic finite element analysis of shells with damage due to microvoids

This paper presents a non‐linear finite element analysis for the elasto‐plastic behaviour of thick/thin shells and plates with large rotations and damage effects. The refined shell theory given by Voyiadjis and Woelke (Int. J. Solids Struct. 2004; 41 :3747–3769) provides a set of shell constitutive equations. Numerical implementation of the shell theory leading to the development of the C⁰ quadrilateral shell element (Woelke and Voyiadjis, Shell element based on the refined theory for thick spherical shells. 2006, submitted) is used here as an effective tool for a linear elastic analysis of shells. The large rotation elasto‐plastic model for shells presented by Voyiadjis and Woelke (General non‐linear finite element analysis of thick plates and shells. 2006, submitted) is enhanced here to account for the damage effects due to microvoids, formulated within the framework of a micromechanical damage model. The evolution equation of the scalar porosity parameter as given by Duszek‐Perzyna and Perzyna (Material Instabilities: Theory and Applications, ASME Congress, Chicago, AMD‐Vol. 183/MD‐50, 9–11 November 1994; 59–85) is reduced here to describe the most relevant damage effects for isotropic plates and shells, i.e. the growth of voids as a function of the plastic flow. The anisotropic damage effects, the influence of the microcracks and elastic damage are not considered in this paper. The damage modelled through the evolution of porosity is incorporated directly into the yield function, giving a generalized and convenient loading surface expressed in terms of stress resultants and stress couples. A plastic node method (Comput. Methods Appl. Mech. Eng. 1982; 34 :1089–1104) is used to derive the large rotation, elasto‐plastic‐damage tangent stiffness matrix. Some of the important features of this paper are that the elastic stiffness matrix is derived explicitly, with all the integrals calculated analytically (Woelke and Voyiadjis, Shell element based on the refined theory for thick spherical shells. 2006, submitted). In addition, a non‐layered model is adopted in which integration through the thickness is not necessary. Consequently, the elasto‐plastic‐damage stiffness matrix is also given explicitly and numerical integration is not performed. This makes this model consistent mathematically, accurate for a variety of applications and very inexpensive from the point of view of computer power and time. Copyright © 2006 John Wiley & Sons, Ltd.     

Analysis on the fatigue damage evolution of notched specimens with consideration of cyclic plasticity

This paper presents a damage mechanics method applied successfully to assess fatigue life of notched specimens with plastic deformation at the notch tip. A damage‐coupled elasto‐plastic constitutive model is employed in which nonlinear kinematic hardening is considered. The accumulated damage is described by a stress‐based damage model and a plastic strain‐based damage model, which depend on the cyclic stress and accumulated plastic strain, respectively. A three‐dimensional finite element implementation of these models is developed to predict the crack initiation life of notched specimens. Two cases, a notched plate under tension‐compression loadings and an SAE notched shaft under bending‐torsion loadings including non‐proportional loadings, are studied and the predicted results are compared with experimental data.     

A computational model for the finite element analysis of thermoplasticity coupled with ductile damage at finite strains

An updated Lagrangian implicit FEM model for the analysis of large thermo‐mechanically coupled hyperelastic‐viscoplastic deformations of isotropic porous materials is considered. An appropriate framework for constitutive modelling is introduced that includes a stress‐free thermally expanded configuration and a plastically deformed unstressed damaged configuration. A two‐level iterative scheme is employed at each time increment to solve the field equations governing the conservation of momentum (mechanical step) and the conservation of energy (thermal step) for the coupled thermo‐mechanical problem. Exact linearizations for the calculation of the tangent stiffness are performed in each of these solution steps. A fully implicit, thermo‐mechanically coupled and incrementally objective Euler‐backward radial return based map is developed for the time integration of the constitutive equations. The present model is used to analyse a number of benchmark examples including metal forming processes wherein temperature and the accumulated damage play an important role in influencing the deformation mechanism and the nature of the deformed workpiece. Copyright © 1999 John Wiley & Sons, Ltd.     

Towards Material Testing 2.0. A review of test design for identification of constitutive parameters from full‐field measurements

Full‐field optical measurements like digital image correlation or the grid method have brought a paradigm shift in the experimental mechanics community. While inverse identification techniques like finite element model updating or the virtual fields method have been the object of significant developments, current test methods, inherited from the age of strain gauges or linear variable displacement transducers, are generally not well adapted to the rich information provided by these new measurement tools. This paper provides a review of the research dealing with the design and optimization of heterogeneous mechanical tests for the identification of material parameters from full‐field measurements, christened here Material Testing 2.0 (MT2.0).     

Iterative solution of the random eigenvalue problem with application to spectral stochastic finite element systems

A new algorithm for the computation of the spectral expansion of the eigenvalues and eigenvectors of a random non‐symmetric matrix is proposed. The algorithm extends the deterministic inverse power method using a spectral discretization approach. The convergence and accuracy of the algorithm is studied for both symmetric and non‐symmetric matrices. The method turns out to be efficient and robust compared to existing methods for the computation of the spectral expansion of random eigenvalues and eigenvectors. Copyright © 2006 John Wiley & Sons, Ltd.     

A stabilized finite element formulation for the transonic small‐disturbance system

A stabilized finite element formulation for the transonic small‐disturbance system of equations is developed and used to solve a variety of problems in transonic aerodynamics. An adaptive mesh refinement technique and a common discontinuity capturing operator are used to resolve regions with large gradients in the velocity field. The scheme works well in both flow regimes, subsonic and supersonic, and captures shocks naturally. Agreement with available experimental observations and theoretical approximations is very good. Copyright © 2001 John Wiley & Sons, Ltd.