Algorithmic Tangent Stiffness Research Articles

A robust explicit integration of elasto-plastic constitutive models, based on simple subincrement size estimation

PurposeThe purpose of this study, which is designed for the implementation of models in the implicit finite element framework, is to propose a robust, stable and efficient explicit integration algorithm for rate-independent elasto-plastic constitutive models.Design/methodology/approachThe proposed automatic substepping algorithm is founded on an explicit integration scheme. The estimation of the maximal subincrement size is based on the stability analysis.FindingsIn contrast to other explicit substepping schemes, the algorithm is self-correcting by definition and generates no cumulative drift. Although the integration proceeds with maximal possible subincrements, high level of accuracy is attained. Algorithmic tangent stiffness is calculated in explicit form and optionally no analytical second-order derivatives are needed.Research limitations/implicationsThe algorithm is convenient for elasto-plastic constitutive models, described with an algebraic constraint and a set of differential equations. This covers a large family of materials in the field of metal plasticity, damage mechanics, etc. However, it cannot be directly used for a general material model, because the presented algorithm is convenient for solving a set of equations of a particular type.Practical implicationsThe estimation of the maximal stable subincrement size is computationally cheap. All expressions in the algorithm are in explicit form, thus the implementation is simple and straightforward. The overall performance of the approach (i.e. accuracy, time consumption) is fully comparable with a default (built-in) ABAQUS/Standard algorithm.Originality/valueThe estimated maximal subincrement size enables the algorithm to be stable by definition. Subincrements are much larger than those in conventional substepping algorithms. No error control, error correction or local iterations are required even in the case of large increments.

Stress‐driven integration strategies and m‐AGC tangent operator for Perzyna viscoplasticity and viscoplastic relaxation: application to geomechanical interfaces

SummaryThe paper proposes a stress‐driven integration strategy for Perzyna‐type viscoplastic constitutive models, which leads also to a convenient algorithm for viscoplastic relaxation schemes. A generalized trapezoidal rule for the strain increment, combined with a linearized form of the yield function and flow rules, leads to a plasticity‐like compliance operator that can be explicitly inverted to give an algorithmic tangent stiffness tensor also denoted as the m‐AGC tangent operator. This operator is combined with the stress‐prescribed integration scheme, to obtain a natural error indicator that can be used as a convergence criterion of the intra‐step iterations (in physical viscoplasticity), or to a variable time‐step size in viscoplastic relaxation schemes based on a single linear calculation per time step. The proposed schemes have been implemented for an existing zero‐thickness interface constitutive model. Some numerical application examples are presented to illustrate the advantages of the new schemes proposed. Copyright © 2016 John Wiley & Sons, Ltd.

Dynamic Analysis of Mooring Cables with Application to Floating Offshore Wind Turbines

Floating offshore wind turbines are recently being considered widely for adoption in the wind power industry, attracting interest of several researchers and calling for the development of appropriate computational models and techniques. In the present work, a nonlinear finite-element formulation is proposed and applied to the static and dynamic analysis of mooring cables. Numerical examples are presented, and in particular, a mooring cable typically used for floating offshore wind turbines is analyzed. Hydrodynamic effects on the cable are accounted for using the Morison approach. A key enabling development here is an algorithmic tangent stiffness operator including hydrodynamic coupling. Numerical results also suggest that previous empirical hydrodynamic coefficients could be obtained by fully coupled fluid–structure interaction. Convergence-rate and energy-balance calculations have been used to demonstrate the accuracy of computed solutions. The introduction of the developed cable model in a framework for the study of the global behavior of floating offshore wind turbines is the subject of the current work. Source code developed for this work is available as online supplemental material with the paper.

Algorithmic tangent stiffness in elastoplasticity and elastoviscoplasticity: A geometric insight

The algorithmic, or consistent, tangent stiffness was introduced to improve the asymptotic convergence rate of the iterative correction algorithm for the evolutive analysis of elastoplastic structures. The original approach is based on a formulation of the elastoplastic law in terms of a plastic multiplier with an analysis which, in general, requires an operator inversion. A geometric description of the method, based on hypersurface theory, is proposed here to provide a clear picture of the algorithmic properties. An estimate of the tangent stiffness associated with finite step elastoplastic and elastoviscoplastic constitutive models is given. It is based on the properties of the projection operator on the elastic domain and avoids operator inversions retaining the beneficial properties of the original one.

Nonlinear shell theory: a duality approach

The nonlinear kinematics of thin shells is developed in full generality according to a duality approach in which kinematics plays the basic role in the definition of the model. The Kirchhoff‐Love shell model is the central issue and is discussed in detail but shear deformable and polar models are also considered and critically reviewed. The analysis is developed with a coordinate-free approach which provides a direct geometrical picture of the shell model. The finite and tangent Green strains of the foliated continuum are explicitly expressed in terms of middle surface kinematics. The new expressions contributed here do not require the splitting of the velocity into parallel and normal components to the middle surface, and provide a computationally convenient context. Finite strain measures for the shell and their tangent and secant rates are analyzed and consistency and nonredundancy properties are discussed. The relations between the finite Green strain, its tangent and secant rates and the corresponding shell strains, are provided. The differential and boundary equilibrium equations of the shell are given in variational terms, both in unsplit and split form. A new expression of the boundary equilibrium equations is contributed and its mechanical soundness with respect to the classical one is emphasized. Equilibrium in a reference placement for the shell model is briefly discussed. The theory of shells is intimately connected with the geometry of Riemannian manifolds and elements of the relevant theory are summarized in the paper. The middle surface of a shell model is a two-dimensional submanifold of the three-dimensional euclidean space (the classical physical space). The treatment of large configuration changes of the Kirchhoff‐Love shell is conveniently carried out by considering the shell as a foliation of the middle surface induced by a distance function. The description of the shell kinematics in terms of that of the middle surface is carried out by considering the nonlinear projection of the points of the foliation onto the middle surface and its derivative which is a symmetric linear operator expressible in terms of the linear projector onto the tangent plane and of the middle surface shape operator. Remarkably, this result is also useful in elastoplasticity for conveniently estimating the algorithmic tangent stiffness [Romano et al. 2005a; Romano et al. 2006]. The properties of consistency and nonredundancy to be fulfilled by a well defined strain measure are briefly illustrated to provide a basis for the analysis of shell models. The nonlinear analysis of shell deformation is developed in detail with a coordinate-free exposition. The treatment is founded on a variational approach in which kinematics plays a primary role while statics is introduced by duality. Stress fields are envisaged as Lagrange multipliers of the implicit representation of the rigidity constraint provided by a strain measure. In the classical three-dimensional continuum mechanics the Green strain measure is the most natural one. Accordingly,

RVE computations with error control and adaptivity: the power of duality

The goal of computational homogenization is to obtain the macro-scale response, normally in terms of macro-scale stress for given macro-scale deformation, via RVE-computations. In this paper we investigate, in a systematic manner, the effects of Dirichlet and Neumann boundary conditions on the RVE. Adaptive computations are carried out with respect to, in particular, control of the error in the macro-scale stress tensor. This requires the corresponding dual solutions. As a new result, it is shown how the same dual solutions can be conveniently used in computing the algorithmic tangent stiffness tensor, thereby demonstrating the “power of duality”.

Implicit algorithm for finite deformation hypoelastic–viscoplasticity in fcc metals

An implicit objective stress update algorithm is proposed for a hypoelastic–viscoplastic model. A thermal/dynamic yield function, which is derived based on the thermal activation analysis and dislocation interaction mechanisms, is used, along with the Consistency approach and the framework of additive viscoplasticity, in deriving the proposed model for fcc metals. The corotational formulation approach is utilized in developing the proposed model in the finite deformation field. For the case of the Newton–Raphson iteration method, a new expression for the consistent (algorithmic) tangent stiffness matrix of rate-dependent metals is derived by direct linearization of the stress update algorithm. Finite element simulations are performed by implementing the proposed viscoplasticity constitutive models in the commercial finite element program ABAQUS. Numerical implementation for a simple tensile problem is used for validating the material parameters of the OFHC Copper under low and high strain rates and temperatures. The numerical results of the adiabatic true stress–true strain curves compare very well with the experimental data. The effectiveness of the present approach is tested by studying strain localization in a simple plane strain problem. Results indicate excellent performance of the present framework in describing the strain localization problem and in obtaining mesh-independent results. Copyright © 2006 John Wiley & Sons, Ltd.

On analysis of the elasto‐viscoplastic response of single crystals with anisotropic damage: constitutive modelling and computational aspects

AbstractBased on the concept of continuum damage mechanics, an anisotropic damage model for single crystals under the theory of crystal plasticity is presented. Damage and inelastic deformations are incorporated in the proposed model which is developed within the framework of thermodynamics with internal state variables. The dependence of the plastic anisotropy on the damage evolution has been considered. The anisotropic damage is characterized kinematically here through a second‐order damage tensor which is physically based. The proposed model can successfully describe the interaction between the evolution of micro‐structure of single crystals such as lattice orientation and the hardness development of each slip system and the process of material degradation. The Newton–Raphson iterative scheme is used to integrate the constitutive equations that work directly with the evolution equations for the elastic deformation gradient. The consistent algorithmic tangent stiffness for the present algorithm is formulated. The prescribed algorithm together with the consistent algorithmic tangent stiffness has been implemented into the ABAQUS finite element code by using user subroutine. Using the loading processes with homogeneous deformations and simulation of the classical tensile test of a notched bar illustrate the basic aspects of the model described. Numerical simulations show the validation and performance of the present model and algorithm. Copyright © 2004 John Wiley & Sons, Ltd.

Time Integration Algorithm for a Cyclic Damage Coupled Thermo-Viscoplasticity Model for 63Sn-37Pb Solder Applications

In this paper, a semi-implicit time integration scheme has been developed for a damage-coupled constitutive model to characterize the mechanical behavior of 63Sn-37Pb solder material under thermo-mechanical fatigue (TMF) loading. The scheme is developed to provide an efficient numerical procedure of integration and iteration for calculating stress and other associated state variables within a strain-driven format. In particular, a novel Newton-Raphson iteration algorithm for the damage coupled constitutive material model involving von Mises viscoplastic potential function with nonlinear mixed hardening is formulated. An algorithmic tangent stiffness tensor is derived and the model is implemented numerically into a commercial finite element (FE) code ABAQUS through its user-defined material subroutine. Several numerical simulations are conducted for validation of the proposed algorithm.

Numerical finite element formulation of the Schapery non‐linear viscoelastic material model

AbstractThis study presents a numerical integration method for the non‐linear viscoelastic behaviour of isotropic materials and structures. The Schapery's three‐dimensional (3D) non‐linear viscoelastic material model is integrated within a displacement‐based finite element (FE) environment. The deviatoric and volumetric responses are decoupled and the strain vector is decomposed into instantaneous and hereditary parts. The hereditary strains are updated at the end of each time increment using a recursive formulation. The constitutive equations are expressed in an incremental form for each time step, assuming a constant incremental strain rate. A new iterative procedure with predictor–corrector type steps is combined with the recursive integration method. A general polynomial form for the parameters of the non‐linear Schapery model is proposed. The consistent algorithmic tangent stiffness matrix is realized and used to enhance convergence and help achieve a correct convergent state. Verifications of the proposed numerical formulation are performed and compared with a previous work using experimental data for a glassy amorphous polymer PMMA. Copyright © 2003 John Wiley & Sons, Ltd.

Anisotropic damage coupled to plasticity: Modelling based on the effective configuration concept

AbstractA model framework for anisotropic damage, that is kinetically coupled to inelastic deformations, is outlined. Key ingredients are the concept of the effective undamaged configuration and the assumption of energy equivalence. The pertinent rate equations for the chosen internal variables are integrated using the standard fully implicit scheme, and the Jacobian matrix of the resulting non‐linear incremental constitutive problem is used to compute the algorithmic tangent stiffness in straightforward fashion. Numerical results showing the effect of deformation‐induced anisotropy conclude the paper. Copyright © 2002 John Wiley & Sons, Ltd.

A class of thermo-hyperelastic–viscoplastic models for porous materials: theory and numerics

Modelling and numerical issues in conjunction with large strain analysis of thermohyperelastic–viscoplastic models are discussed with emphasis placed on porous materials (metal powders). In order to account for the proper temperature effect on the rate-dependent response of metal powders, the concept of “viscoplastic admissibility” is employed as part of a model framework that includes a dynamic yield surface. A generic class of pressure-sensitive models is considered; a particular model based on quasistatic and dynamic yield surfaces, that are elliptic in the meridian planes in the principal stress space, is used for the subsequent numerical evaluation. The generic model also includes kinematic hardening, thus introducing non-coaxiality between the deformation, the back-stress and the stress tensors (in any given reference configuration). Implicit (backward Euler) integration is used and the corresponding algorithmic tangent stiffness (ATS) tensor is established in a setting that is valid for complete noncoaxiality. For the prototype problem of simple shear (using constant strain approximation), quadratic convergence in the equilibrium iterations is demonstrated. The paper is concluded by an investigation of a Hot-Isostatic-Pressing (HIP) problem, whereby experimental results from the literature are used for calibration of the model parameters.

A return-mapping algorithm for plastic-damage models: 3-D and plane stress formulation

Three-dimensional and plane stress formulations of the return-mapping algorithm for a class of plastic-damage models are derived using the spectral decomposition form of the stress. An efficient plane stress computation scheme based on the spectral return-mapping algorithm is developed. The consistent algorithmic tangent stiffness for the present algorithm is formulated. The validation and performance of the present return-mapping algorithm is demonstrated by numerical examples. Copyright © 2001 John Wiley & Sons, Ltd.

Algorithmic implementation of a generalized cohesive crack model

Smeared fictitious crack models can be regarded as generalized cohesive crack models. The classic fictitious crack models, i.e. the fixed crack, multiple fixed crack, rotating crack and microplane model, are based on different assumptions for the orientation of developing cracks. A smooth transition between the extreme cases, the fixed crack and the rotating crack model, is provided by the adaptive fixed crack model. In this approach, the critical direction of failure is uniquely identified based on Mohr's hypothesis. Thus, the critical direction depends on the character of the failure criterion and the type of loading. The numeric implementation of the adaptive fixed crack model has given rise to some subtle questions. It is shown that even for a classical fixed crack concept, the algorithmic tangent stiffness may have to include components of crack rotation, depending on the imposed strategy for the global equilibrium iteration scheme.

A model for elastic-plastic pressure sensitive materials subjected to large deformations

Development of a finite deformation elasto-plastic model for pressure sensitive materials is presented. The chosen model, which has its roots in the MRS-Lade material model is influenced by recent developments. The thermodynamic consequences of introducing non-associative yielding (both deviatoric and volumetric) and hardening⧸softening characteristics are assessed. The consistently linearized Algorithmic Tangent Stiffness (ATS) tensor is presented. This tensor is used in the constitutive driver as a key feature of the efficient iterative procedure for satisfying equilibrium in the case of stress (or mixed) control. The chosen model is calibrated using data from experiments conducted in a Directional Shear Cell (DSC) , which has been used extensively at the University of Colorado at Boulderto investigate the behavior of pressure sensitive materials under deformations of large magnitude.

Implicit integrations in elastoplastic geotechnics

The paper presents the derivation of a fully implicit Newton algorithm for direct integration of constitutive equations, in extended stress-internal variable space, involving hardening or softening of a general dilatant isotropic elastoplastic geomaterial. All relevant derivatives are provided in tensor notation, thus facilitating implementation. The consistent, algorithmic tangent stiffness tensor is derived. The relative accuracy of a template algorithm is assessed on a number of examples by means of iso-error maps. We present a rather simple, one-increment example concerning convergence properties of the Newton iterative scheme at the global, finite element level, associated with the consistent tangent stiffness tensor for integrating the weak form of the equilibrium equations. © 1997 by John Wiley & Sons, Ltd. Mech. Cohesive-frictional mater. 2, 165–183 (1977)

Fracture and Slip of Interfaces in Cementitious Composites. II: Implementation

This paper focuses on the numerical treatment of an interface model that was presented in the companion paper. A generalized trapezoidal rule is proposed for integrating the slip rule. The new model is used in finite element predictions, whereby Newton iterations to achieve equIlibrium require the algorithmic tangent stiffness matrix that is pertinent to the incremental problem. In order to ensure convergence to a physically acceptable (stable) solution, one should employ a modified tangent stiffness that is always positive definite. The analysis of a simple composite microstructure confirms that the model is capable of predicting interface failure. Application of the interface model as a generalized fictitious crack model concludes the paper. Preliminary results show that it is possible Do obtain a crack pattern that extends from the Aggregate‐mortar boundary into the mortar itself along predefined interelement boundaries.

Implicit integration in soil plasticity under mixed control for drained and undrained response

AbstractAn algorithm is outlined for the implicit integration of isotropic plasticity models for an arbitrary choice of mixed stress and strain control variables. Drained as well as undrained behaviour is considered. The closest‐point‐projection method in conjunction with a completely strain‐driven format is used in a core algorithm. In the drained case strain response variables are determined via iterations to satisfy equilibrium of prescribed and calculated stresses that correspond to the strain response variables. In the undrained case, on the other hand, strains and pore pressure are determined via simultaneous iterations to satisfy equilibrium and the incompressibility condition. The algorithm is applied to a new generalized cam‐clay model, and various iteration techniques are assessed. In particular, Newton iterations which employ the matrix of algorithmic tangent stiffness moduli are shown to compete favourably with more conventional methods.

Computational coupled non-associative thermo-plasticity

Computationally oriented formulation of the isothermal, rate-independent theory of non-associative elasto-plasticity is extended in this paper to describe coupled thermo-elastic-plastic and thermo-elastic-visco-plastic behaviour of materials. This is done by additionally considering thermal strains, assumming all material properties to be temperature dependent and accounting for the mechanical coupling terms in the non-stationary heat conduction equation. The finite deformation effects are included in the analysis. The theory is employed for the analysis of thermo-mechanical response of ductile metals with damage effects modelled by a generalization of the so-called Gurson approach. This constitutive model is known to generate equations typical of non-associative plasticity and hence it can be consistently incorporated into the present more general considerations. The finite element assessment of combined thermal and damage effects on the axisymmetric necking process illustrates the paper. Numerical aspects such as a ‘tangent’ stiffness for rate-dependent thermo-plasticity and the algorithmic (or consistent) tangent stiffness matrix for non-associative plasticity are discussed as well.