Consistent Tangent Operator Research Articles

Numerical modeling of thermo‐mechanical failure processes in granitic rock with polygonal finite elements

AbstractThis paper considers numerical modeling of intensive heating induced thermo‐mechanical failure processes in granitic rock. For this end, a numerical method based on polygonal finite elements and a damage‐plasticity model is developed. A staggered scheme is employed to solve the global thermo‐mechanical problem. The rock failure is described by a Rankine‐Mohr‐Coulomb plasticity model with separate scalar damage variables for tension and compression. Consistent tangent operator is derived for this model. Special attention is given to the temperature dependence of the thermo‐mechanical material properties of heterogeneous rock. In the numerical examples, the method is first verified with an analytical solution of thermal stresses in a hollow cylinder, and then qualitatively validated with the problems of thermal cracking of concentric cylinders and uniaxial compressive test on rock under elevated temperatures. Finally, the method is applied in novel simulations inspired by the degradation of sauna stones under slow heating‐rapid cooling and the comminution by rapid heating‐cooling cycles.

On the efficient enforcement of uniform traction and mortar periodic boundary conditions in computational homogenisation

In this contribution, the enforcement of uniform traction (UT) and periodic boundary (PB) conditions on arbitrarily generated meshes is analysed in detail, and the derivation of the associated macroscopic consistent tangent operators is described. Two different approaches to impose these boundary conditions are examined: the saddle point solution (SPS) resulting from the Lagrange multiplier method and the condensation method (CM). The numerical treatment required to implement the UT and the PB conditions with both methods is presented for rectangular and cuboidal representative volume elements (RVEs). It is shown that the UT condition only needs rigid body motion to be prescribed since it intrinsically restricts rigid body rotations in the finite strain setting. The application of the SPS to the UT condition leads to a formulation where the Lagrange multipliers are the homogenised stresses, and the macroscopic consistent tangent operator for FE2 simulations can be directly derived. The enforcement of PB conditions on non-conforming meshes with the mortar approach is also described for the CM and SPS. The need to modify the Lagrange multiplier dual interpolation functions is emphasised to avoid over-constraints. The proposed implementation recovers the conventional PB condition solution when conforming meshes are used. The computational performance of both methods is compared in terms of time and memory requirements. As anticipated, the results obtained for each boundary condition do not depend on the enforcement method employed. With the direct solvers employed, the SPS is more efficient despite increasing the number of unknowns in the linear system of equations.

A nonuniform transformation field analysis for composites with strength difference effects in elastoplasticity

Many composites and their constituents exhibit strength difference (SD) effects. To reduce the computational cost associated to twoscale simulations, the theory of nonuniform transformation field analysis (NTFA, originally proposed by Michel and Suquet, 2003) is extended to consider a general case of SD effects in elastoplasticity, containing several well-known plasticity models like von Mises or Drucker and Prager as special cases. A space-time decomposition is done separately for volumetric and deviatoric inelastic strain fields, thus leading to two sets of plastic modes with different charateristics. Localization rules are deduced from superposition principles for the microscopic strain and stress fields, which are then homogenized to obtain the effective response. A coupled model governing the evolution of the resulting reduced variables is proposed based on some dissipative considerations. A convenient matrix formulation for an FE-based implementation is presented in detail. The proposed coupled model is shown to be exact for homogeneous cases and sufficiently accurate for three-dimensional heterogeneous microstructures. Finally, an efficient structural analysis is performed by using corresponding algorithmic consistent tangent operators.

Application of closest point projection method to unified hardening model

This paper presents a framework of the closest point projection method (CPPM) based on the unified hardening (UH) model. The UH model based on the Cam-clay model and the concept of the subloading yield surface was rewrote as a concise form of classical elasto-plastic model. Then the CPPM algorithm combining the exact elastic property and the spatially mobilized plane (SMP) strength criterion by the transformed stress method could be conveniently derived for the numerical implementation of the UH model. According to this algorithm, the consistent tangent operator corresponding to the final stress state of the integration increment was deduced formally. And the mean stress probed by the elastic predictor was avoided to be negative or zero for the usage of exact elastic property. Eventually, the accuracy, robustness and efficiency of the proposed integration scheme were demonstrated by the numerical examples including the boundary value problems of the single integral point and multi-elements of geotechnical engineerings, which was respectively compiled as the single integral point program and the user defined subroutine (UMAT) of the commercial finite element program ABAQUS.

Enhancement of Drucker-Prager yield model by adding corner points for pressure-dependent materials

To flexibly describe the pressure-dependent behaviors of materials, Drucker-Prager yield model is enhanced by introducing corner points. We call it piecewise Drucker-Prager (pDP) yield model. Constitutive model is derived by considering strain hardening with non-associative yield flows. Integration algorithm includes the return mapping to corner point. Consistent tangent operator is also formulated, and user material subroutine is written. It is demonstrated that the pDP model is effective in describing the pressure-dependent yield behavior of polymers.

A multisurface kinematic hardening model for the behavior of clays under combined static and undrained cyclic loading

AbstractIn dynamic geotechnical problems, soils are often subjected to a combination of sustained static and fast cyclic loading. Under such loading conditions, saturated and normally consolidated clays generally experience a build‐up of excess pore water pressure along with a degradation of stiffness and strength. If the strength of the soil falls below the static stress demand, a self‐driven failure is triggered. In this paper, a constitutive model is presented for the analysis of such problems, based on a general multisurface plasticity framework. The hardening behavior, the initial arrangement of the surfaces, and the nonassociated volumetric flow rule are defined to capture important aspects of cyclic clay behavior. This includes nonlinear hysteretic stress‐strain behavior, the effect of anisotropic consolidation, and the generation of excess pore water pressure during undrained cyclic loading along with a degradation of stiffness and strength. The model requires nine independent parameters, which can be derived from standard laboratory tests. A customized experimental program has been performed to validate the model performance. The model predictions show a good agreement with test results from monotonic and cyclic undrained triaxial tests, in particular with respect to the strain‐softening response and the number of loading cycles to failure. A procedure for a general stress‐space implicit numerical implementation for undrained, total stress‐based finite element analyses is presented, including the derivation of the consistent tangent operator. Finally, a simulation of the seismic response of a submarine slope is shown to illustrate a possible application of the presented model.

Hydromechanical modeling of solid deformation and fluid flow in the transversely isotropic fissured rocks

Geomaterials containing fissures such as some sedimentary rocks often exhibit a bimodal pore size distribution, and they are also inherently anisotropic due to the distinct bedding planes. Hydromechanical modeling of solid deformation and fluid flow of such geomaterials remains a significant challenge. In this paper, we have developed a unique anisotropic double porosity elastoplastic framework to describe such processes. Furthermore, for the solid constitutive model, because of the loss of stress tensor coaxiality between the trial state and the final state, we have derived a new implicit return mapping algorithm to obtain the updated effective stress, history parameters and consistent tangent operator for any given strain increment efficiently, followed by a uniaxial strain point simulation to provide benchmark results. Subsequently, 3D stress point simulations are carried out to calibrate the projection and plasticity parameters using triaxial experimental data as well as to illustrate the strain-softening phenomenon. Initial boundary value problem simulations have been conducted to analyze the impacts of fluid flow and solid constitutive model on the resulting geomaterials’ responses. The overarching goal of this paper is to better understand the coupled solid deformation-fluid flow in the transversely isotropic fissured rocks.

A 2D BEM formulation considering dissipative phenomena and a full coupled multiscale modelling

A full coupled multi-scale modelling using the Boundary Element Method for analysing the 2D problem of stretched plates composed of heterogeneous materials, where dissipative phenomena can be considered, is presented. Both the macro-scale and the micro-scale are modelled by BEM formulations where the consistent tangent operator (CTO) is used to achieve the equilibrium of the iterative procedures. The equilibrium equation of the plate (macro-continuum) is written in terms of in-plane strains while the equilibrium problem of the microstructure, which is defined by the RVE (Representative Volume Element), is solved in terms of displacements fluctuations. In this kind of modelling, the mechanical behaviour of the material is governed by the homogenized response of the RVE, obtained after solving its equilibrium problem. As this kind of modelling is expensive computationally, it is important to investigate other numerical methods to have faster formulations, but which are still accurate. To validate the presented model, the numerical results are compared to the ones where the material microstructure (RVE) is modelled by the FEM.

Numerical implementation of modified Chaboche kinematic hardening model for multiaxial ratcheting

For simulating multiaxial ratcheting behavior, the modified Chaboche kinematic hardening model was numerically implemented by using the framework of a small-strain elastic-plastic theory. Unlike early models, this improved multiaxial model is difficult to implement using finite element methods owing to its complicated constitutive relations, such as radial evanescence terms and the fourth hardening rule with a threshold. We present an effective procedure for numerical implementation using Voigt notations and the implicit radial return method with Newton-Raphson iterations. All the equations of constitute numerical integration and consistent tangent operator (CTO) are simply solved using matrix operations. The integration algorithm is validated by using both numerical examples and analytical solutions. The CTO is verified by additional stress calculations. The model detects variations in the cyclic indentation response with changes in a multiaxial-dependent parameter. The numerical implementation allows simulations of both biaxial and general multiaxial ratcheting behaviors.

A gradient-extended two-surface damage-plasticity model for large deformations

This work is concerned with the theoretical development, algorithmic implementation and numerical investigation of a novel thermodynamically consistent ‘two-surface’ gradient-extended damage-plasticity model for arbitrarily large deformations. It can be considered the geometrically nonlinear version of a corresponding model for small deformations which was presented recently (Brepols et al., (2017; 2018)). The terminology ‘two-surface’ stems from the fact that damage and plasticity are treated in the model as truly distinct but coupled dissipative mechanisms, meaning that individual damage loading and plastic yield criteria as well as appropriate loading/unloading conditions are used in the formulation, respectively. Such an approach makes the model flexibly adaptable to various situations in which the material under consideration shows either a more brittle- or ductile-like damaging behavior. Nonlinear Armstrong-Frederick kinematic hardening, nonlinear Voce isotropic hardening as well as nonlinear damage hardening are accounted for in the presented formulation that relies exclusively on symmetric internal variables, being beneficial, e.g., from the computational point of view. The model is formulated in the framework of Continuum Damage Mechanics and its gradient-extension is derived on the basis of the micromorphic approach according to Forest (2009, 2016). As such, it allows for the computation of mesh-insensitive results in finite element simulations involving material softening which is practically confirmed by means of several structural example problems. Furthermore, many other interesting aspects are highlighted, as, e.g., the model's implementation into finite elements, the computation of the algorithmically consistent tangent operators necessary to retain a quadratic rate of convergence in a global Newton-Raphson scheme, or a suitable time-integration of the model's evolution equations at the integration point level.

Damaged plasticity model for concrete using scalar damage variables with a novel stress decomposition

In the presented work, a new plastic damage model for concrete is proposed with a novel stress decomposition, to account for shear induced damage. A consistent thermodynamic approach is used to derive the constitutive model. With the classical stress decomposition into positive and negative components, the novel stress decomposition is developed to further decompose tensile and compressive parts into pure biaxial shear and pure tensile/compressive biaxial stresses. This decomposition introduces four additional scalar damage parameters (ϕ±S, ϕ±EB). The additional damage parameters with classical damage parameters (ϕ+, ϕ−) are responsible for the different damages induced under loading. The two traditionally used, damage criteria (tensile/compressive) are further decomposed into four damage criteria (tensile/compressive shear, tensile/compressive pure) depending upon the formation of novel decomposition. The delayed damage growth, reduction in damage evolution and ductile behavior under triaxial confining stresses are captured by suppression of damage evolution and retardation of plastic hardening depending upon the confining stresses and minimum principal strain. The plasticity yield criteria and non-associative plastic flow rule with multiple hardening functions are presented in the effective stress space. Strain equivalency hypothesis is utilized for the transformation from effective configuration to damaged configuration. A Helmholtz free energy elastic-plastic function is described to define the relationship between the elastic-plastic-damage constitutive model and internal state variables. The damage-elastic-plastic consistent tangent operator is also derived.

Large deformation analysis of plane-stress hyperelastic problems via triangular membrane finite elements

A finite-element formulation based on triangular membranes of any order is proposed to analyze problems involving highly deformable hyperelastic materials under plane-stress conditions. The element kinematics is based on positional description and the degrees of freedom are the current plane coordinates of the nodes. Two isotropic and nonlinear hyperelastic models have been selected: the compressible neo-Hookean model and the incompressible Rivlin–Saunders model. The constitutive relations and the consistent tangent operator are condensed to the compact 2D forms imposing plane-stress conditions. The resultant algorithm is implemented in a computer code. Three benchmark problems are numerically solved to assess the formulation proposed: the Cook’s membrane, involving bending, shear, and a singularity point; a partially loaded membrane, which presents severe mesh distortion and large compression levels; and a rubber sealing, which is a more realistic problem. Convergence analysis in terms of displacements, applied forces, and stresses is performed for each problem. It is demonstrated that mesh refinement avoids locking problems associated with incompressibility condition, bending-dominated problems, stress concentration, and mesh distortion. The processing times are relatively small even for fifth-order elements.

Nonlinear equilibrium of partially liquid-filled tanks: A finite element/level-set method to handle hydrostatic follower forces

This paper deals with the nonlinear finite element computation of the prestressed state of structures partially filled with an incompressible inviscid liquid. The fluid is modeled by hydrostatic follower forces such that no volumetric fluid mesh is needed. Large deformations are taken into account and lead to the fluid height variation of the wetted surface to satisfy the fluid volume conservation. The main originality of this work lies on the use of a level-set approach to handle numerical integration on the fluid–structure interface. The method is developed on 3D problems considering a finite element quadratic mesh. Various numerical examples are computed using a Newton–Raphson algorithm and a quadratic convergence rate is reached by using consistent tangent stiffness operators.

Implementation of the Tresca yield criterion in finite element analysis of burst capacity of pipelines

The Tresca and von Mises criteria are two classic yield criteria for metals. While the von Mises criterion has been widely implemented in commercial finite element analysis (FEA) software package, the Tresca yield criterion in the context of the associated-plastic flow rule is not available in any well-known FEA packages such as ABAQUS and ANSYS. In this study, the Tresca yield criterion as well as the associated plastic flow and hardening rules are implemented in a user-defined material model (UMAT) in commercial FEA package ABAQUS based on the return-mapping technique. The consistent tangent operator for the Tresca criterion is also implemented in the UMAT model to facilitate the computation of the tangent stiffness matrix during the Newton-Raphson iteration for solving the nonlinear global equilibrium equations in the plastic domain. The constitutive integration algorithm for the Tresca UMAT is written in the principal stress space, which markedly simplifies the implementation. The implemented Tresca UMAT model is validated by theoretical solution and full-scale burst tests of both pristine and defected pipes. Furthermore, based on the Tresca-based and von Mises-based FEA predictions, a new formula is proposed to accurately predict burst pressure of thin-walled corroded pipes.

Non associated-anisotropic plasticity model fully coupled with isotropic ductile damage for sheet metal forming applications

This paper presents an implementation of a fully coupled non-associated anisotropic plasticity-ductile damage model, including a mixed nonlinear isotropic- kinematic hardening. With the non-associated anisotropic plasticity assumption, the yielding and plastic potential are determined independently. The quadratic Hill’48 function is considered for yield and plastic potential to describe the anisotropic plastic behavior of DD13 sheets. A three non-linear local scalar equation problem is solved using the Newton–Raphson method. The numerical resolution of the proposed algorithm is implemented into ABAQUS using the user interface material subroutines (UMAT and VUMAT). A consistent tangent operator is developed to preserve the quadratic rate of asymptotic convergence that characterizes Newton's method. The evaluation of the proposed implementation ability to predict the real behavior of DD13 steel material during forming is achieved using some experimental setups performed by the authors.

Implementation of a Phase Mixture Model for Rate‐Dependent Inelasticity

AbstractPower plant components have to withstand complex thermo‐mechanical loads, i.e. frequent start‐ups and shut‐downs in combination with long holding times at high temperatures induce creep‐fatigue loads. Tempered martensitic steels with high chromium content feature high creep strength and corrosion resistance among other properties such that these alloys are often used for power plant components.This contribution focuses on the simulation of the complex mechanical behavior of the alloy X20CrMoV12‐1 based on a multi‐axial phase mixture model. Within this model, we distinguish two phases: a hard and a soft phase. Both phases are connected via an iso‐strain approach, and nonlinear kinematic hardening as well as microstructural softening effects are taken into account by introducing a backstress and a softening variable as internal variables. The model is implemented into the commercial finite element code ABAQUS with a user material subroutine. For this purpose, the stress update algorithm and the consistent tangent operator are implemented based on the backward EULER method in combination with NEWTON‐RAPHSON iteration. Next, a wide range of benchmarks for uni‐axial as well as multi‐axial stress states is taken into account in order to verify the numerical implementation. Furthermore, a thermo‐mechanical fatigue test based on a typical sequence of start‐ups and shut‐downs of power plants is simulated.

The Benefits of Using a Consistent Tangent Operator for Viscoelastoplastic Computations in Geodynamics

AbstractStrain localization is ubiquitous in geodynamics and occurs at all scales within the lithosphere. How the lithosphere accommodates deformation controls, for example, the structure of orogenic belts and the architecture of rifted margins. Understanding and predicting strain localization is therefore of major importance in geodynamics. While the deeper parts of the lithosphere effectively deform in a viscous manner, shallower levels are characterized by an elastoplastic rheological behavior. Herein we propose a fast and accurate way of solving problems that involve elastoplastic deformations based on the consistent linearization of the time‐discretized elastoplastic relation and the finite difference method. The models currently account for the pressure‐insensitive Von Mises and the pressure‐dependent Drucker‐Prager yield criteria. Consistent linearization allows for resolving strain localization at kilometer scale while providing optimal, that is, quadratic convergence of the force residual. We have validated our approach by a qualitative and quantitative comparison with results obtained using an independent code based on the finite element method. We also provide a consistent linearization for a viscoelastoplastic framework, and we demonstrate its ability to deliver exact partitioning between the viscous, the elastic, and the plastic strain components. The results of the study are fully reproducible, and the codes are available as a subset of M2Di MATLAB routines.

A thermo-elasto-viscoplastic constitutive model for polymers

In this study, a thermo-elasto-viscoplastic model is developed for a low density cross-linked polyethylene (XLPE) in an attempt to describe the combined effects of temperature and strain rate on the stress-strain response and the self-heating of the material at elevated strain rates. The proposed model consists of two parts. On the one side, Part A models the thermo-elastic and thermo-viscoplastic response, and incorporates an elastic Hencky spring in series with two Ree-Eyring dashpots. The two Ree-Eyring dashpots represent the effects of the main α relaxation and the secondary β relaxation processes on the plastic flow. Part B, on the other side, consists of an eight chain spring capturing the entropic strain hardening due to alignment of the polymer chains during deformation.The constitutive model was implemented in a nonlinear finite element (FE) code using a semi-implicit stress update algorithm combined with sub-stepping and a numerical scheme to calculate the consistent tangent operator. After calibration to available experimental data, FE simulations with the constitutive model are shown to successfully describe the stress-strain curves, the volumetric strain, the local strain rate and the self-heating observed in the tensile tests. In addition, the FE simulations adequately predict the global response of the tensile tests, such as the force-displacement curves and the deformed shape of the tensile specimen.

A 2D boundary element formulation to model the constitutive behavior of heterogeneous microstructures considering dissipative phenomena

A boundary element formulation to obtain the constitutive response of heterogeneous microstructures is proposed, considering a RVE-based multiscale theory. The sub-region technique is adopted to model the RVE (Representative Volume Element), where voids and inclusions can be defined inside the matrix and different elastic properties and constitutive models can be assumed for each phase. Triangular cells are adopted to approximate the dissipative forces over the RVE domain, where they are assumed constant. After a strain vector is imposed to the microstructure boundary, its elastic forces field is obtained and the respective strain field computed. Then, according to the RVE-based multiscale theory, the RVE equilibrium problem must be solved what requires an iterative procedure to find the displacement fluctuations field that auto-equilibrates the microstructure stress field. After the RVE solution, the boundary tractions must be recalculated to consider the displacement fluctuations field as well as the dissipative forces. Then, from the boundary tractions the homogenized stress vector can be computed. The homogenized constitutive tensor is evaluated considering the constitutive tensors of all cell nodes and also the consistent tangent operator. To validate the proposed model, the homogenized responses of different microstructures are compared to the responses obtained via finite element model.

Return Mapping Algorithms (RMAs) for Two-Yield Surface Thermoviscoplastic Models Using the Consistent Tangent Operator

Abstract The return mapping algorithms (RMAs) presented here are designed for use with pressure-dependent thermoviscoplastic constitutive models involving irreversible effects associated with solid–solid phase transformations. During the volume solid–solid phase transformations occurring under mechanical loads, an “anomalous” plasticity, the so-called “TRansformation Induced Plasticity” (TRIP), is generated at much lower stress levels than those related to the yield stress of the material in the context of the classical plasticity. TRIP mechanisms are superimposed on the classical plasticity which is liable to occur in the case of metallic materials. Based on a non-standard generalized material framework, two different models are presented in which an “associative” plastic flow is introduced in the context of classical plasticity and a “non-associative” flow rule in the context of TRIP-like plasticity. In this paper, a complete algorithmic treatment of these two rate-dependent constitutive models is therefore proposed with the associated consistent tangent operator for dealing the quasi-surface irreversible solid–solid transformations which can appear in metal alloys during specific thermomechanical solicitations. The predictive abilities of the presented numerical procedure for modelling this kind of the irreversible solid–solid transformations involving two plasticity processes are tested and assessed by performing a two-dimensional finite-element analysis on some numerical examples.