A simple orthotropic finite elasto–plasticity model based on generalized stress–strain measures

 In this paper we present a formulation of orthotropic elasto-plasticity at finite strains based on generalized stress–strain measures, which reduces for one special case to the so-called Green–Naghdi theory. The main goal is the representation of the governing constitutive equations within the invariant theory. Introducing additional argument tensors, the so-called structural tensors, the anisotropic constitutive equations, especially the free energy function, the yield criterion, the stress-response and the flow rule, are represented by scalar-valued and tensor-valued isotropic tensor functions. The proposed model is formulated in terms of generalized stress–strain measures in order to maintain the simple additive structure of the infinitesimal elasto-plasticity theory. The tensor generators for the stresses and moduli are derived in detail and some representative numerical examples are discussed. Received: 2 April 2002 / Accepted: 11 September 2002     

Application of the material force method to isotropic continuum damage

 The objective of this work is the exploitation of the notion of material forces in computational continuum damage mechanics. To this end we consider the framework of isotropic geometrically non–linear continuum damage and investigate the spatial and material settings that lead to either spatial or material forces, respectively. Thereby material forces essentially represent the tendency of material defects to move relative to the ambient material. In this work we combine an internal variable approach towards damage mechanics with the material force method. Thus the appearance of distributed material volume forces that are conjugated to the damage field necessitates the discretization of the damage variable as an independent field in addition to the deformation field. Consequently we propose a monolithic solution strategy for the corresponding coupled problem. The underlying kinematics, strong and weak forms of the coupled problem will be presented and implemented within a standard Galerkin finite element procedure. As a result in particular global discrete nodal quantities, the so–called material node point (surface) forces, are obtained and are studied for a number of computational examples. Received: 19 August 2002 / Accepted: 16 October 2002     

Re-analysis procedure for laminated plates using FSDT finite element model

 A new method is proposed for effective analysis of laminated plates incorporating accurate through-the-thickness distribution of displacements, strains and stresses in the finite element formulation. It is a two-step analysis procedure. In the first step, displacements are obtained using a post-processing procedure based on the three-dimensional stress equilibrium equations and the thermoelasticity equations, from the results of FSDT finite element analysis. In the second step, the higher-order through-the-thickness distribution of displacements are reflected on the subsequent finite element analysis. The effectiveness of the present approach for the analysis of laminated plates is shown by numerical examples. Received: 13 September 2001 / Accepted: 23 May 2002     

On finite strain poroplasticity with reversible and irreversible porosity laws. Formulation and computational aspects

The main purpose of this paper is the formulation of a poroplastic framework suitable for finite strain and high pore pressure in saturated porous media. Here we make a distinction between poroplasticity with totally reversible porosity and poroplasticity with the occurrence of irreversible porosity. For this latter, an important key point is that the total porosity is not additively decomposed as usual into reversible and irreversible parts. As shown, the permanent porosity is embedded into the definition of the total porosity itself. The approach is built around the physical restriction that the actual Eulerian porosity is bounded in the interval [0, 1] for any admissible process. Elementary considerations motivate the modeling throughout the paper and the formulation is integrated within the unified continuum thermodynamics of open media, which is crucial in setting the convenient forms of the state laws and evolution equations for the flux variables to fully characterize the behavior of porous materials. On the numerical side, the algorithmic design is described in detail for an easy implementation within the context of the finite element method. Finally, we present a set of numerical simulations to illustrate the effectiveness of the proposed framework.     

Computational modelling of static indentation-induced damage in glass

The present paper is focused on the numerical simulation of a glass plate subjected to static indentation by a spherical indenter. For this purpose, a combined approach of continuum damage mechanics (CDM) and fracture mechanics is performed. Results provided by an axisymmetric finite element model were compared with analytical solutions. A CDM based constitutive model with an anisotropic damage tensor was selected and implemented into a finite element code to study the damage of glass. The numerical results were analysed through the framework of the stress and damage distribution. Various regions with critical damage values were therefore predicted in good agreement with the experimental observations in the literature. In these regions, the directions of crack propagation, including both cracks initiating on the surface as well as in the bulk, were predicted using the strain energy density factor. Predicted directions were found in good agreement with those experimentally obtained in the literature results.     

A finite element model of skeletal muscles

The present paper surveys recent developments in constitutive and computational modelling of skeletal muscles, concerning mainly the generalization to two- and three-dimensional (2D, 3D) continuum deformation analysis of typical one-dimensional (1D) Hill-type muscle models. Extending our previous work in the field and recent contributions by other authors, we describe a constitutive model for skeletal muscles that incorporates all the features of the 3 typical elements (parallel elastic, series elastic and contractile elements) in Hill's muscle model. In particular the proposed incompressible transversely isotropic model incorporates: a multiplicative split of the fibre stretch into contractile and (series) elastic stretches; the possibility of energy storage in the series elastic element; the dependence of the contractile stress on the strain rate; the governing equation of activation dynamics, so that general histories of neural stimulation may be taken as input data. The resulting 2D or 3D constitutive equations are implemented as user subroutines in the large deformation finite element software package ABAQUS. Simple numerical tests are presented and discussed, as well as an example that involves passive or active deformations of a pelvic floor muscle using shell finite elements.     

Parallel 3-d simulations for porous media models in soil mechanics

 Numerical simulations in 3-d for porous media models in soil mechanics are a difficult task for the engineering modelling as well as for the numerical realization. Here, we present a general numerical scheme for the simulation of two-phase models in combination with an abstract material model via the stress response with a specialized parallel saddle point solver. Therefore, we give a brief introduction into the theoretical background of the Theory of Porous Media and constitute a two-phase model consisting of a porous solid skeleton saturated by a viscous pore-fluid. The material behaviour of the skeleton is assumed to be elasto-viscoplastic. The governing equations are transfered to a weak formulation suitable for the application of the finite element method. Introducing an abstract formulation in terms of the stress response, we define a clear interface between the assembling process and the parallel solver modules. We demonstrate the efficiency of this approach by challenging numerical experiments realized on the Linux Cluster in Chemnitz. Received 15 February 2002 / Accepted 12 April 2002     

Computational applications of a coupled plasticity-damage constitutive model for simulating plain concrete fracture

A coupled plasticity-damage model for plain concrete is presented in this paper. Based on continuum damage mechanics (CDM), an isotropic and anisotropic damage model coupled with a plasticity model is proposed in order to effectively predict and simulate plain concrete fracture. Two different damage evolution laws for both tension and compression are formulated for a more accurate prediction of the plain concrete behavior. In order to derive the constitutive equations and for the easiness in the numerical implementation, in the CDM framework the strain equivalence hypothesis is adopted such that the strain in the effective (undamaged) configuration is equivalent to the strain in the nominal (damaged) configuration. The proposed constitutive model has been shown to satisfy the thermodynamics requirements. Detailed numerical algorithms are developed for the finite element implementation of the proposed coupled plasticity-damage model. The numerical algorithm is coded using the user subroutine UMAT and then implemented in the commercial finite element analysis program Abaqus. Special emphasis is placed on identifying the plasticity and damage model material parameters from loading-unloading uniaxial test results. The overall performance of the proposed model is verified by comparing the model predictions to various experimental data, such as monotonic uniaxial tension and compression tests, monotonic biaxial compression test, loading-unloading uniaxial tensile and compressive tests, and mixed-mode fracture tests.     

A finite element model of skeletal muscles

The present paper surveys recent developments in constitutive and computational modelling of skeletal muscles, concerning mainly the generalization to two- and three-dimensional (2D, 3D) continuum deformation analysis of typical one-dimensional (1D) Hill-type muscle models. Extending our previous work in the field and recent contributions by other authors, we describe a constitutive model for skeletal muscles that incorporates all the features of the 3 typical elements (parallel elastic, series elastic and contractile elements) in Hill's muscle model. In particular the proposed incompressible transversely isotropic model incorporates: a multiplicative split of the fibre stretch into contractile and (series) elastic stretches; the possibility of energy storage in the series elastic element; the dependence of the contractile stress on the strain rate; the governing equation of activation dynamics, so that general histories of neural stimulation may be taken as input data. The resulting 2D or 3D constitutive equations are implemented as user subroutines in the large deformation finite element software package ABAQUS. Simple numerical tests are presented and discussed, as well as an example that involves passive or active deformations of a pelvic floor muscle using shell finite elements.     

Probabilistic analysis of multi-site damage in aircraft fuselages

 Most aircraft fleets nowadays are operating under the concept of damage tolerance, which requires an aircraft to have sufficient residual strength in the presence of damage in one of its principal structural elements (PSE) during the interval of service inspections. The residual strength however is significantly reduced due to multi site damage (MSD). In the present paper, a probabilistic framework for the computation of the failure probability is developed. The MSD problem of a PSE is considered, where the uncertainties in crack initiation and crack growth as well as yield stress and fracture toughness are described by random variables. For the crack growth calculations the finite element alternating method [1], which avoids a remeshing of the finite element problem, is used. After specifying link up and failure criteria, importance sampling is employed to obtain the probability of failure of the PSE due to MSD. Received: 29 July 2002 / Accepted: 18 December 2002     

Multi-objective optimization of tire carcass contours using a systematic aspiration-level adjustment procedure

 While forming a basic tire configuration and supporting most static and dynamic loads of automobiles, tire carcass influences major tire performances according to its contour. Among significant tire performances, we in this study intend to improve the automobile maneuverability and the tire durability by optimizing the sidewall carcass contour. In order to effectively maximize these multi-objectives, we refine the conventional satisficing trade-off methods (STOM) which were proposed originally for the multi-objective structural optimization, by introducing a systematic aspiration-level adjustment procedure. According to the systematic procedure, we perform the sidewall contour optimization that ideally distributes the sidewall carcass tension and minimizes strain-energy density at the belt edge. Since the tire analysis is highly nonlinear problem we employ an incremental analysis scheme, together with the finite-difference sensitivity scheme. Through the numerical experiment, we confirmed that the refined multi-objective optimization technique systematically leads to a final optimum sidewall contour, together with the stable and rapid convergence. Received: 20 August 2001 / Accepted: 29 July 2002 This work was supported by Kumho Industries Co., Ltd in Korea and by the Ministry of Science and Technology under the NRL program (M10203000017-02J0000-00910).     

A well-conditioned technique for solving the inverse problem of boundary traction estimation for a constrained vibrating structure

 Estimation of the frequency and spatial dependent boundary traction vector from measured vibration responses in a vibrating structure is addressed. This problem, also referred to as the inverse problem, may in some circumstances be ill-conditioned. Here a technique to overcome the ill-conditioning is proposed. A subset of a set of available eigenmodes is chosen such that the problem becomes well-conditioned enough. It is shown that the ill-conditioning originates from the fact that not all eigenmodes are orthogonal over the surface where the traction vector is sought. Consequently, by choosing a set of eigenmodes orthogonal over the surface of interest, the problem becomes well-conditioned. The calculated traction vector is shown to converge to the true one in the sense of a L₂-norm on the boundary of the body. The proposed technique is verified, using numerical simulation of measured responses, with good agreement. Received: 10 January 2002 / Accepted: 24 October 2002 This work was performed under contract from the Swedish Defence Material Administration (FMV). The funding provided is gratefully acknowledged. The author would like to thank all staff at the Structural Dynamics research group, Department of Structures and Materials, Aeronautics Division, Swedish Defence Research Agency.     

Simulation of strain localization with gradient enhanced damage models

The incorporation of higher order strain gradients into the constitutive equations of continuum damage mechanics is presented. Thereby, not only scalar-valued isotropic damage models but also anisotropic damage models allowing for directional dependent stiffness degradation are elaborated. An elegant possibility of describing anisotropic material behavior based on the microplane theory is demonstrated. Its conceptual simplicity originates from the idea of modeling the material behavior through uniaxial stress–strain laws on several individual material planes. For each plane individual damage loading functions are introduced allowing for different failure modes. In order to account for long ranging microstructural mechanisms, second-order gradients of the strains are incorporated in each of these damage loading functions. The overall response can be determined by an integration of the resulting microplane laws over the solid angle. The features of gradient enhanced continuum damage are demonstrated by means of several selected examples.     

A meshfree formulation of local radial point interpolation method (LRPIM) for incompressible flow simulation

 The LRPIM method is adopted to simulate the two-dimensional natural convection problems within enclosed domain of different geometries. In this paper, the vorticity-stream function form of N-S equations is taken as the governing equations. It was observed that the obtained results agreed very well with others available in the literatures, and with the same nodal density, the accuracy achieved by the LRPIM method is much higher than that of the finite difference (FD) method. The numerical examples show that the present LRPIM method can successfully deal with incompressible flow problems on randomly distributed nodes. Received: 2 April 2002 / Accepted: 6 January 2003 The authors would like to thank Mr. Y. T. Gu for his contribution to this work.     

An updated continuum damage model to investigate fracture process of structures in DBTT region

In the Ductile–Brittle Transition Temperature (DBTT) region, it is not realistic to take unique fracture stress or fracture strain as the fracture criterion to investigate the fracture properties. In this paper, an updated continuum damage model was proposed, in which the fracture energy density, a function of the stress triaxiality, temperature and strain rate in the transition region was taken as the critical damage factor. Uniaxial tension tests were carried out to get the basic material properties at different temperatures, to calibrate the fracture model constants and verify the validity of the damage model. The fracture behaviour of pipes with penetrating cracks under the internal pressure was experimentally investigated with the load–deflection curves and the crack propagation length captured from tests. The J–R curves were obtained from the testing results for different temperatures. Based on the Finite Element Analyses (FEA) with the proposed fracture criterion of the updated continuum damage model, the loading level of pipes with penetrating cracks were estimated and compared with the experimental results. Meanwhile the fracture processes of the pipeline structures in the transition region were reproduced. The experimental and numerical results agreed very well in present calculations. It has been shown that the fracture process in the transition region strongly depends on both the stress and strain states, and could be effectively predicted using the continuum damage model.     

A new constitutive model of austenitic stainless steel for cryogenic applications

A series of uniaxial tensile test under cryogenic temperature was carried out for AISI 304 and 316 austenitic stainless steels (ASS) in this study. Typical non-linear hardening phenomena under the cryogenic environment, such as transformation induced strain hardening and threshold strain for the 2nd hardening, has been observed in a quantitative manner.The important factors affecting the non linear material behavior of austenitic stainless steel including phase transformation, discontinuous yielding and micro-damage are modeled using constitutive equations system based on strain decomposition at the small strain formulation. A strong nonlinearity of strain hardening is described using the coupling of modified Bodner’s plasticity model and phase transformation induced strain model. The strain (threshold strain for onset of 2nd hardening) dependent plasticity model was proposed in the hardening function of Bodner’s model. In order to explicitly express the phase transformation induced strain, TI model (Tomita and Iwamoto model [Y. Tomita, T. Iwamoto, Constitutive modeling of TRIP steel and its application to the improvement of mechanical properties, International Journal of Mechanical Sciences 37 (1995) 1295–1305.]), which is a function of accumulated plastic strain and volume fraction factor of martensite, is selected in this study.Also the unified damage model, which can be connected with elasto-plastic constitutive equation developed in this study, is suggested, and the utility of proposed model was validated by the comparison between experiments and numerical evaluations.     

Damage evolution in experiments and simulation in a construction steel

Results from the modelling of the initiation and the evolution of microvoids governed by triaxiality and effective plastic strain in heterogeneous materials are presented. In particular, the damage evolution in a microalloyed thermomechanically treated steel consisting of brittle hard phases embedded in a ductile matrix is studied. Results obtained by numerical simulations using the finite element method are compared with experimental investigations of a notched cylindrical tensile bar. For this, two kinds of continuum damage models are used: the model of Rousselier, which assumes the yield stress as function of damage, and the concept of effective strains and stresses proposed by Lemaitre.     

A composite ply failure model based on continuum damage mechanics

A material model including the failure behaviour is derived for a thin unidirectional (UD) composite ply. The model is derived within a thermodynamic framework and the failure behaviour is modelled using continuum damage mechanics. The following features describe the model: (i) The ply is assumed to be in a plane state of stress. (ii) Three damage variables associated with the stress in the fibre-, transverse and shear directions, respectively, are used. (iii) The plastic behaviour of the matrix material is modelled. (iv) The difference in the material response in tensile and compressive loading is modelled. (v) Rate dependent behavior of plasticity and damage (i.e. strength) is modelled.     

Continuum damage mechanics approach to composite laminated shallow cylindrical/conical panels under static loading

Static response characteristics and failure load of laminated composite shallow cylindrical and conical panels subjected to internal/external lateral pressure are investigated using continuum damage mechanics approach considering geometric nonlinearity and damage evolution. The damage model is based on a generalized macroscopic continuum theory within the framework of irreversible thermodynamics and enables to predict the progressive damage and failure load. Damage variables are introduced for the phenomenological treatment of the state of defects and its implications on the degradation of the stiffness properties. The analysis is carried out using finite element method based on the first order shear deformation theory. The nonlinear governing equations are solved using Newton–Raphson iterative technique coupled with the adaptive displacement control method to efficiently trace the equilibrium path. The detailed parametric study is carried out to investigate the influences of geometric nonlinearity, evolving damage, span-to-thickness ratio, lamination scheme and semi-cone angle on the static response and failure load of laminated cylindrical/conical panels. It is revealed that the membrane forces due to geometric nonlinearity significantly influence the damage distribution and failure load.