A finite element formulation based on non-associated plasticity for sheet metal forming

In the present paper, a finite element formulation based on non-associated plasticity is developed. In the constitutive formulation, isotropic hardening is assumed and an evolution equation for the hardening parameter consistent with the principle of plastic work equivalence is introduced. The yield function and plastic potential function are considered as two different functions with functional form as the yield function of Hill [Hill, R., 1948. Theory of yielding and plastic flow of anisotropic metals. Proc. Roy. Soc. A 193, 281–297] or Karafillis–Boyce associated model [Karafillis, A.P. Boyce, M., 1993. A general anisotropic yield criterion using bounds and a transformation weighting tensor. J. Mech. Phys. Solids 41, 1859–1886]. Algorithmic formulations of constitutive models that utilize associated or non-associated flow rule coupled with Hill or Karafillis–Boyce stress functions are derived by application of implicit return mapping procedure. Capabilities in predicting planar anisotropy of the Hill and Karafillis–Boyce stress functions are investigated considering material data of Al2008-T4 and Al2090-T3 sheet samples. The accuracy of the derived stress integration procedures is investigated by calculating iso-error maps.     

A return-map algorithm for general associative isotropic elasto-plastic materials in large deformation regimes

The present paper addresses a flexible solution algorithm for associative isotropic elasto-plastic materials, i.e. for materials whose elastic and plastic behaviors are described through an isotropic free-energy function, an isotropic yield function and an associative flow rule. The discussion is relative to a large deformation regime, while no hardening mechanisms are included. The algorithm is based on a combination of the operator split method and a return map scheme. Both the algorithm linearization and the requirements for the yield criterion convexity are discussed in detail. Finally, to show the algorithm flexibility and performance, the discussion is specialized to three yield criteria and some test problems are studied.     

Gradient-enhanced softening material models

Classical constitutive models exhibit strong mesh dependency during softening and the numerical responses tend towards perfectly brittle behavior upon mesh refinements. Such sensitivity can be avoided by adopting the gradient-enhanced formulation. The implicit approach incorporates the gradient contributions indirectly via an additional Helmholtz equation and requires only C⁰ continuity. The explicit approach computes the gradient terms directly from the local field variables. Assuming a weak satisfaction of the yield function, C¹ continuity or C⁰ continuity with additional degrees of freedoms in the penalty approach is required. This makes the explicit method less attractive computationally. However, the explicit approach is able to fully regularize some material models where the standard implicit method fails to perform. Drawing analogy to the over-nonlocal integral formulation, the over-implicit-gradient framework is proposed. In addition, an alternative framework for the explicit gradient method requiring only C⁰ continuity is proposed. The regularizing effects of the abovementioned two gradient frameworks show promising applications to strain-softening materials.     

Nonlinear constitutive models from nanoindentation tests using artificial neural networks

This paper presents a new approach using Artificial Neural Networks (ANNs) models to simulate the response during nanohardness tests of a variety of materials with nonlinear behavior. The ANNs continuous input and output variables usually include material parameters, indentation deflection, and resisting force. Different ANN models, including dimensionless input/output variables, are generated and trained with discrete finite-element (FE) simulations with different geometries and nonlinear material parameters. Only the monotonic loading part of the load–displacement indentation response is used to generate the trained ANN models. This is a departure from classical indentation simulations or tests where typically the unloading portion is used to determine the stiffness and hardness. The experimental part of this study includes nanoindentation tests performed on a silicon (Si) substrate with and without a nanocrystalline copper (Cu) film. The new ANN models are used to back-calculate (inverse problem) the in situ nonlinear material parameters for different copper material systems. The results are compared with available data in the literature. The proposed FE–ANN modeling approach is very effective and can be used in calibrating and predicting the in situ inelastic material properties using the monotonic part of the indentation response and for depths above 50 nm where the overall resisting force represents a continuum response.     

FE-analysis of sheet metal forming processes using continuous contact treatment

Discrete meshes cause stepwise propagation of the contact nodes of a sheet despite the fact that the contact region in the actual forming process is altered very smoothly. This can cause problems of convergence and accuracy in contact-sensitive processes, such as a bending process. In this study, a scheme for a continuous contact treatment is proposed in order to consider the more realistic behavior of the contact phenomena during the forming process. For verification of the proposed method, the contact pressures and forming load are evaluated during the compression forming of a tube. The analysis of a hemispherical dome formed without a blank holder is also presented in order to investigate the effects of the proposed algorithm. The results show that the precise deformation mode is predicted by the utilization of the proposed method.     

Embedded polycrystal plasticity and adaptive sampling

A simulation capability for multi-scale embedded polycrystal plasticity is demonstrated with over two orders of magnitude wall-clock speedup compared to direct embedding. In the coarse-scale material model, the visco-plastic part of the material response is based on parameters determined from polycrystal level fine-scale calculations. Polycrystal plasticity parameters are approximated from fine-scale calculations using adaptive sampling to substantially reduce the total number of expensive fine-scale calculations which must be performed. The adaptive sampling method uses Kriging models for local interpolation of the fine-scale plasticity parameters and a metric-tree database for storage and retrieval of the fine-scale response models. Efficacy of the method is demonstrated through a variety of example problems involving both quasi-static and dynamic loading scenarios.     

Evaluation of advanced anisotropic models with mixed hardening for general associated and non-associated flow metal plasticity

The main objective of this paper is to develop a generalized finite element formulation of stress integration method for non-quadratic yield functions and potentials with mixed nonlinear hardening under non-associated flow rule. Different approaches to analyze the anisotropic behavior of sheet materials were compared in this paper. The first model was based on a non-associated formulation with both quadratic yield and potential functions in the form of Hill’s (1948). The anisotropy coefficients in the yield and potential functions were determined from the yield stresses and r-values in different orientations, respectively. The second model was an associated non-quadratic model (Yld2000-2d) proposed by Barlat et al. (2003). The anisotropy in this model was introduced by using two linear transformations on the stress tensor. The third model was a non-quadratic non-associated model in which the yield function was defined based on Yld91 proposed by Barlat et al. (1991) and the potential function was defined based on Yld89 proposed by Barlat and Lian (1989). Anisotropy coefficients of Yld91 and Yld89 functions were determined by yield stresses and r-values, respectively. The formulations for the three models were derived for the mixed isotropic-nonlinear kinematic hardening framework that is more suitable for cyclic loadings (though it can easily be derived for pure isotropic hardening). After developing a general non-associated mixed hardening numerical stress integration algorithm based on backward-Euler method, all models were implemented in the commercial finite element code ABAQUS as user-defined material subroutines. Different sheet metal forming simulations were performed with these anisotropic models: cup drawing processes and springback of channel draw processes with different drawbead penetrations. The earing profiles and the springback results obtained from simulations with the three different models were compared with experimental results, while the computational costs were compared. Also, in-plane cyclic tension–compression tests for the extraction of the mixed hardening parameters used in the springback simulations were performed for two sheet materials.     

A general theory for nonlocal softening plasticity of integral-type

This paper deals with a comparison of several models, proposed in the literature, of softening plasticity with internal variables regularized by nonlocal averaging of integral type.     

A non-associated constitutive model with mixed iso-kinematic hardening for finite element simulation of sheet metal forming

In this paper an anisotropic material model based on non-associated flow rule and mixed isotropic–kinematic hardening was developed and implemented into a user-defined material (UMAT) subroutine for the commercial finite element code ABAQUS. Both yield function and plastic potential were defined in the form of Hill’s [Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. A 193, 281–297] quadratic anisotropic function, where the coefficients for the yield function were determined from the yield stresses in different material orientations, and those of the plastic potential were determined from the r-values in different directions. Isotropic hardening follows a nonlinear behavior, generally in the power law form for most grades of steel and the exponential law form for aluminum alloys. Also, a kinematic hardening law was implemented to account for cyclic loading effects. The evolution of the backstress tensor was modeled based on the nonlinear kinematic hardening theory (Armstrong–Frederick formulation). Computational plasticity equations were then formulated by using a return-mapping algorithm to integrate the stress over each time increment. Either explicit or implicit time integration schemes can be used for this model. Finally, the implemented material model was utilized to simulate two sheet metal forming processes: the cup drawing of AA2090-T3, and the springback of the channel drawing of two sheet materials (DP600 and AA6022-T43). Experimental cyclic shear tests were carried out in order to determine the cyclic stress–strain behavior and the Bauschinger ratio. The in-plane anisotropy (r-value and yield stress directionalities) of these sheet materials was also compared with the results of numerical simulations using the non-associated model. These results showed that this non-associated, mixed hardening model significantly improves the prediction of earing in the cup drawing process and the prediction of springback in the sidewall of drawn channel sections, even when a simple quadratic constitutive model is used.     

Finite deformation pseudo-elasticity of shape memory alloys – Constitutive modelling and finite element implementation

In this paper we suggest a new phenomenological material model for shape memory alloys. In contrast to many earlier concepts of this kind the present approach includes arbitrarily large deformations. The work is motivated by the requirement, also expressed by regulatory agencies, to carry out finite element simulations of NiTi stents. Depending on the quality of the numerical results it is possible to circumvent, at least partially, expensive experimental investigations. Stent structures are usually designed to significantly reduce their diameter during the insertion into a catheter. Thereby large rotations combined with moderate and large strains occur. In this process an agreement of numerical and experimental results is often hard to achieve. One of the reasons for this discrepancy is the use of unrealistic material models which mostly rely on the assumption of small strains. In the present paper we derive a new constitutive model which is no longer limited in this way. Further its efficient implementation into a finite element formulation is shown. One of the key issues in this regard is to fulfil “inelastic” incompressibility in each time increment. Here we suggest a new kind of exponential map where the exponential function is suitably computed by means of the spectral decomposition. A series expansion is completely avoided. Finite element simulations of stent structures show that the new concept is well appropriate to demanding finite element analyses as they occur in practically relevant problems.     

A Finite Element approach with patch projection for strain gradient plasticity formulations

Several strain gradient plasticity formulations have been suggested in the literature to account for inherent size effects on length scales of microns and submicrons. The necessity of strain gradient related terms render the simulation with strain gradient plasticity formulation computationally very expensive because quadratic shape functions or mixed approaches in displacements and strains are usually applied. Approaches using linear shape functions have also been suggested which are, however, limited to regular meshes with equidistanced Finite Element nodes. As a result the majority of the simulations in the literature deal with plane problems at small strains. For the solution of general three dimensional problems at large strains an approach has to be found which has to be computationally affordable and robust.     

Three dimensional combined fracture–plastic material model for concrete

This paper describes a combined fracture–plastic model for concrete. Tension is handled by a fracture model, based on the classical orthotropic smeared crack formulation and the crack band approach. It employs the Rankine failure criterion, exponential softening, and it can be used as a rotated or a fixed crack model. The plasticity model for concrete in compression is based on the Menétrey–Willam failure surface, the plastic volumetric strain as a hardening/softening parameter and a non-associated flow rule based on a nonlinear plastic potential function. Both models use a return-mapping algorithm for the integration of constitutive equations. Special attention is given to the development of an algorithm for the combination of the two models. The suggested combination algorithm is based on a recursive substitution, and it allows for the two models to be developed and formulated separately. The algorithm can handle cases when failure surfaces of both models are active, but also when physical changes such as crack closure occur. The model can be used to simulate concrete cracking, crushing under high confinement and crack closure due to crushing in other material directions. The model is integrated in a general finite element package ATENA and its performance is evaluated by comparisons with various experimental results from the literature.     

A three-dimensional crystal plasticity model for duplex Ti–6Al–4V

A rate dependent crystal plasticity model for the α/β Ti–Al alloy Ti–6Al–4V with duplex microstructure is developed and presented herein. Duplex Ti–6Al–4V is a dual-phase alloy consisting of an hcp structured matrix primary α-phase and secondary lamellar α + β domains that are composed of alternating layers of secondary α laths and bcc structured residual β laths. The model accounts for distinct three-dimensional slip geometry for each phase, anisotropic and length scale dependent slip system strengths, the non-planar dislocation core structure of prismatic screw dislocations in the primary α-phase, and crystallographic texture. The model is implemented in the general purpose finite element code (ABAQUS, 2005. Ver 6.5, Hibbitt, Karlsson, and Sorensen, Inc., Pawtucket, RI) via a UMAT subroutine.     

Path-dependence of the forming limit stresses in a sheet metal

The effect of changing strain paths on the forming limit stresses of sheet metals is investigated using the Marciniak–Kuczyński model and a phenomenological plasticity model with non-normality effects [Kuroda, M., Tvergaard, V., 2001. A phenomenological plasticity model with non-normality effects representing observations in crystal plasticity. J. Mech. Phys. Solids 49, 1239–1263]. Forming limits are simulated for linear stress paths and two types of combined loading: a combined loading consisting of two linear stress paths in which unloading is included between the first and second loadings (combined loading A), and combined loading in which the strain path is abruptly changed without unloading (combined loading B). The forming limit stresses calculated for combined loading A agree well with those calculated for the linear stress paths, while the forming limit curves in strain space depend strongly on the strain paths. The forming limit stresses calculated for the combined loading B do not, however, coincide with those calculated for the linear stress paths. The strain-path dependence of the forming limit stress is discussed in detail by observing the strain localization process.     

Well-posedness of a model of strain gradient plasticity for plastically irrotational materials

The initial boundary value problem corresponding to a model of strain gradient plasticity due to [Gurtin, M., Anand, L., 2005. A theory of strain gradient plasticity for isotropic, plastically irrotational materials. Part I: Small deformations. J. Mech. Phys. Solids 53, 1624–1649] is formulated as a variational inequality, and analysed. The formulation is a primal one, in that the unknown variables are the displacement, plastic strain, and the hardening parameter. The focus of the analysis is on those properties of the problem that would ensure existence of a unique solution. It is shown that this is the case when hardening takes place. A similar property does not hold for the case of softening. The model is therefore extended by adding to it terms involving the divergence of plastic strain. For this extended model the desired property of coercivity holds, albeit only on the boundary of the set of admissible functions.     

A multiscale method for dislocation nucleation and seamlessly passing scale boundaries

In the concurrent multiscale analysis, it is difficult to have truly seamless transition between the atomistic and continuum scale. This situation is even worse when defects pass through the boundary between different scales. For example, there is a lack of effective methods to handle the dislocation passing through scale boundaries which is important to investigate plasticity at the nanoscale. In this paper, the generalized particle (GP) method proposed by the first author is further developed so that a seamless transition and dislocation passing between different scales can be realized. Specifically, the linkage between different scales is through material neighbor-link cells (NLC) with scale duality. This indicates that material elements can be high-scale particles through a lumping process and can also be atoms via decomposition depending on the needs of the simulation. At the interface, the information transfer from bottom scale-up or from top scale-down is through the particles or atoms in the NLC. They are with the same material structure, all possess nonlocal constitutive behavior; thus, the smooth transition at the interface between different scales can be attained and validated to avoid non-physical responses. To save degrees of freedom, atoms are lumped together into a generalized particle in the domain in which the deformation gradient is near homogeneous. On the other hand, when defects such as dislocations in the atomistic domain are near the particle domain, the particles along dislocation propagation path and its surrounding region will be decomposed into atoms so dislocations can freely pass through the scale boundary and propagate inside the model just as it propagates in the deformed atomistic crystal structure. The method is verified first for seamless transition of variables at the scale boundary by a one-dimensional model and then verified for dislocation nucleation and propagation passing through scale boundaries in two cases, one is near the free surface and the other is inside of the copper nanowire. All the validations are through comparisons with fully atomistic analyses under same conditions. The comparison is satisfactory.     

Necking in glassy polymers: Effects of intrinsic anisotropy and structural evolution kinetics in their viscoplastic flow

Two recently proposed developments of the Glass–Rubber constitutive model for glassy polymers treat the viscoplastic deformation as intrinsically anisotropic, and incorporate the kinetics of structural evolution. These features enable the model to capture better the distinctive features of glassy polymers’ constitutive response: post-yield strain-softening and strain-hardening and effects of pre-existing molecular orientation. They have been combined to form a new variant of the model, and the consequences for necking have been explored. Uniaxial extension of prismatic bars was simulated using the finite element method, employing a numerical implementation of the new model, with material parameters of polystyrene. Strain localization predicted with the new model was found to be systematically retarded as compared to predictions with the original (intrinsically isotropic) version of the model, for the same conditions. In particular, the effect of frozen-in molecular orientation was examined. This was found to retard strain localization for stretching parallel to the orientation direction, for both models. But the localization predicted with the new model was always significantly less pronounced than with the original model. Indeed, for sufficiently high pre-orientation (e.g. a uniaxial stretch of 2.2), localization could be effectively prevented with the new model, under conditions when otherwise failure by necking is predicted. Such results can all be explained in terms of a linear stability analysis. They suggest that all previous simulations of necking in glassy polymers made using intrinsically isotropic representations of polymer viscoplasticity may have over-predicted the rate of strain localization.