A coupled temperature and strain rate dependent yield function for dynamic deformations of bcc metals

A coupled temperature and strain rate microstructure physically based yield function is proposed in this work. It is incorporated along with the Clausius–Duhem inequality and an appropriate free energy definition in a general thermodynamic framework for deriving a three-dimensional kinematical model for thermo-viscoplastic deformations of body centered cubic (bcc) metals. The evolution equations are expressed in terms of the material time derivatives of the elastic strain, accumulated plastic strain (isotropic hardening), and the back stress conjugate tensor (kinematic hardening). The viscoplastic multipliers are obtained using both the Consistency and Perzyna viscoplasticity models. The athermal yield function is employed instead of the static yield function in the case of the Perzyna viscoplasticity model. It is found that the static strain rate value, at which the material shows rate-independent behavior, varies with the material deformation temperature. Computational aspects of the proposed model are addressed through the finite element implementation with an implicit stress integration algorithm. Finite element simulations are performed by implementing the proposed viscoplasticity constitutive models in the commercial finite element program ABAQUS/Explicit [ABAQUS, 2003. User Manual, Version 6.3. Habbitt, Karlsson and Sorensen Inc., Providence, RI] via the user material subroutine coded as VUMAT. Numerical implementation for a simple compression problem meshed with one element is used to validate the proposed model implementation with applications to tantalum, niobium, and vanadium at low and high strain rates and temperatures. The analysis of a tensile shear banding is also investigated to show the effectiveness and the performance of the proposed framework in describing the strain localizations at high velocity impact. Results show mesh independency as a result of the viscoplastic regularization used in the proposed formulation.     

Constitutive modeling for anisotropic/asymmetric hardening behavior of magnesium alloy sheets: Application to sheet springback

The constitutive model for the unusual asymmetric hardening behavior of magnesium alloy sheet presented in a companion paper (Lee, M.G., Wagoner, R.H., Lee, J.K., Chung, K., Kim, H.Y., 2008. Constitutive modeling for anisotropic/asymmetric hardening behavior of magnesium alloy sheet, Int. J. Plasticity 24(4), 545–582) was applied to the springback prediction in sheet metal forming. The implicit finite element program ABAQUS was utilized to implement the developed constitutive equations via user material subroutine. For the verification purpose, the springback of AZ31B magnesium alloy sheet was measured using the unconstrained cylindrical bending test of Numisheet (Numisheet ’2002 Benchmark Problem, 2002. In: Yang, D.Y., Oh, S.I., Huh, H., Kim, Y.H. (Eds.), Proceedings of 5th International Conference and Workshop on Numerical Simulation of 3D Sheet Forming Processes, Jeju, Korea) and 2D draw bend test. With the specially designed draw bend test the direct restraining force and long drawn distance were attainable, thus the measurement of the springback could be made with improved accuracy comparable with conventional U channel draw bend test. Besides the developed constitutive models, other models based on isotropic constitutive equations and the Chaboche type kinematic hardening model were also considered. Comparisons were made between simulated results by the finite element analysis and corresponding experiments and the newly proposed model showed enhanced prediction capability, which was also supported by the simple bending analysis adopting asymmetric stress–strain response.     

A ferroelectric and ferroelastic 3D hexahedral curvilinear finite element

An isoparametric 3D electromechanical hexahedral finite element integrating a 3D phenomenological ferroelectric and ferroelastic constitutive law for domain switching effects is proposed. The model presents two internal variables which are the ferroelectric polarization (related to the electric field) and the ferroelastic strain (related to the mechanical stress). An implicit integration technique of the constitutive equations based on the return-mapping algorithm is developed. The mechanical strain tensor and the electric field vector are expressed in a curvilinear coordinate system in order to handle the transverse isotropy behavior of ferroelectric ceramics. The hexahedral finite element is implemented into the commercial finite element code Abaqus® via the subroutine user element. Some linear (piezoelectric) and non linear (ferroelectric and ferroelastic) benchmarks are considered as validation tests.     

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.     

延性单晶非均匀变形的三维有限元模拟

对延性单晶在拉伸载荷作用下的应变局域化和颈缩等非均匀变形过程进行了三维有限元数值模拟。将相关晶体塑性本构模型及一种新的数值积分方法补充到ABAQUS6.1商用有限元软件中。该方法的特点是，利用晶体塑性的动力学方程，获得一个关于晶体弹性变形梯度的演化方程，采用半隐式积分方案进行求解。本文推导出一种新的应力变本构矩阵。按此方式更新本构矩阵，计算速度和计算稳定性大大提高。加载方式，边界条件和变形程度等因素影响着滑移系的启动状况，这是平面模型所不能预测的。本文利用三维有限元方法模拟了不同取向下滑移系的启动状况，全面地考虑了FCC单晶材料12个可能滑移系在变形过程中的启动状况，合理地模拟了FCC面心立方单晶沿不同取向加载时晶轴旋转导致的应变局域化和颈缩等非均匀变形过程。     

Characterization of yielding behavior of polycrystalline metals with single crystal plasticity based on representative characteristic length

Single crystal plasticity based on a representative characteristic length is proposed and introduced into a homogenization approach based on finite element analyses, which are applied to characterization of distinctive yielding behaviors of polycrystalline metals, yield-point elongation, and grain size strengthening. The computational manner for an implicit stress update is derived with the framework of a standard multi-surface plasticity at finite strain, where the evolution of the characteristic lengths are numerically converted from the accumulated slips of all of slip systems by exploiting the mathematical feature of the characteristic length as the intermediate function of the plastic internal variables. Furthermore, a constitutive model for a single crystal reproduces the stress–strain curve divided into three parts. Using two-scale finite element analysis, the macroscopic stress–strain response with yield-point elongation under a situation of low dislocation density is reproduced. Finally, the grain size effect on the yield strength is analyzed with modeling of the grain boundary in the context of the proposed constitutive model and is discussed from both macroscopic and microscopic views.     

An implicit consistent algorithm for the integration of thermoviscoplastic constitutive equations in adiabatic conditions and finite deformations

The so-called viscoplastic consistency model, proposed by Wang, Sluys and de Borst, is extended here to the integration of a thermoviscoplastic constitutive equation for J₂ plasticity and adiabatic conditions. The consistency condition in this case includes not only strain rate but also the effect of temperature on the yield function. Using the backward Euler integration scheme to integrate the constitutive equations, an implicit algorithm is proposed, leading to generalized expressions of the classical return mapping algorithm for J₂ plasticity, both for the iterative calculation of the plastic multiplier increment and for the consistent tangent operator when strain rate and temperature are considered also as state variables of the hardening equation. The model was implemented in a commercial finite element code and its performance is demonstrated with the numerical simulation of four Taylor impact tests.     

A nonlinear finite element model for the stress analysis of soft solids with a growing mass

The paper presents a new finite element (FE) model for the stress analysis of soft solids with a growing mass based on the work of Lubarda and Hoger (2002). Contrary to the traditional numerical methods emphasizing on the influence of growth on constitutive equations, an equivalent body force is firstly detected, which is resulted from the linearization of the nonlinear equation and acts as the driver for material growth in the numerical aspect. In the algorithm, only minor correction on the traditional tangent modulus is needed to take the growth effects into consideration and its objectivity could be guaranteed comparing with the traditional method. To solve the resulted equation in time domain, both explicit and implicit integration algorithms are developed, where the growth tensor is updated as an internal variable of Gauss point. The explicit updating scheme shows higher efficiency, while the implicit one seems to be more robust and accurate. The algorithm validation and its good performance are demonstrated by several two-dimensional examples, including free growth, constrained growth and stress dependent growth.     

Finite element implementation of a generalised non-local rate-dependent crystallographic formulation for finite strains

This work describes the finite element implementation of a generalised strain gradient and rate-dependent crystallographic formulation for finite strains and general anisothermal conditions based on a multiplicative decomposition of the deformation gradient. The implementation involved the development of both a novel finite element formulation to determine the spatial slip rate gradients at each material point, and an implicit numerical integration scheme at the constitutive level to update the stresses and solution dependent variables. The time-integration procedure uses a Newton–Raphson scheme with a single level of iteration to solve the incremental non-linear equations associated with the non-local constitutive formulation. Closed-form solutions for the relevant fourth-order Jacobian tensors are given. The proposed numerical scheme is formulated in a general form and hence should be applicable to most existing crystallographic models. The crystallographic formulation is then used to investigate the effect of the morphology and volume fraction of the reinforcing phase of a two-phase single crystal on its macroscopic behaviour.     

Comparison of diagonal-implicit, linear-implicit and half-explicit Runge–Kutta methods in non-linear finite element analyses

The method of vertical lines in the case of quasi-static solid mechanics applying constitutive models of evolutionary-type yields after the spatial discretization by means of finite elements a system of differential-algebraic equations. It is of substantial interest how fast, accurate, and stable such computations can be carried out. Moreover, the questions are how simple the implementation can be done and how susceptible a procedure is to programming errors. In this article, this is investigated for half-explicit Runge–Kutta methods that are applied to small and finite strain viscoelasticity. The advantage of the method is given by a non-iterative scheme on element level. Additionally, it turns out that for models where linear elasticity is one ingredient in the constitutive model, the method leads to only one required LU-decomposition at the beginning of the entire computation, and in each time step, only one back-substitution step has to be carried out. This outperforms current finite element computations. Order investigations of various integration schemes and the automatic step-size behavior are studied. This new proposal is compared to a classical Backward-Euler implementation, high-order stiffly accurate diagonally implicit Runge–Kutta, and recently proposed Rosenbrock-type methods. Advantages and disadvantages of the applied schemes are discussed.     

Mean-field homogenization of elasto-viscoplastic composites based on a general incrementally affine linearization method

We propose a general formulation – which we believe to be new – for the mean-field homogenization of inclusion-reinforced elasto-viscoplastic composites assuming small strains. Our proposal is based on an interplay between constitutive equations and numerical algorithms, and the key ideas behind it are the following. The evolution equations for inelastic strain and internal variables at the beginning of each time interval are linearized around the ending time of the same interval. The linearized equations are then numerically integrated using a fully implicit backward Euler scheme. The obtained algebraic equations lead to an incrementally affine stress–strain relation which involves two important terms. The first one is the algorithmic tangent operator, obtained by consistent linearization of the time discretized constitutive equations. The second term is a new one and called an affine strain increment. The proposal leads to thermoelastic-like relations directly in the time domain, and not in the Laplace–Carson (L–C) one. There is no need for viscoelastic-type integral rewriting of the evolution equations, for L–C transforms, or for numerical inversion back from L–C to time domains. The proposed method can be readily applied to sophisticated elasto-viscoplastic models with an arbitrary set of scalar or tensor internal variables, and is valid for multi-axial, non-monotonic and non-proportional loading histories. The theory is applied in detail to a well-known constitutive model, and verified against finite element simulations of representative volume elements or unit cells, for a number of composite materials.     

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.     

Experiments and Material Parameter Identification Using Finite Elements. Uniaxial Tests and Validation Using Instrumented Indentation Tests

In this article, we focus our attention on the relation between instrumented indentation tests and the prediction by means of finite element calculations. To this end, a finite strain viscoplasticity model of Perzyna-type with non-linear isotropic and kinematic hardening is calibrated at experimental data of steel S690QL. A particular concept for conducting uniaxial tensile and compression tests is taken up in order to represent the basic rate-dependent material behavior. In this respect, an algorithmic framework of material parameter identification using finite elements is proposed leading to a two-stage procedure in the case of the underlying rate-dependent constitutive model. On the basis of the termination points of relaxation the rate-independent equilibrium stress state can be identified and all viscous parts of the model are obtained using rate-dependent loading paths. Finally, use is made of finite elements for predicting indentation experiments, which results in a critical view on modeling and parameter identification on the basis of experimental results occurring in instrumented indentation tests.     

A new integration algorithm for finite deformation of thermo-elasto-viscoplastic single crystals

An algorithm for integrating the constitutive equations in thermal framework is presented, in which the plastic deformation gradient is chosen as the integration variable. Compared with the classic algorithm, a key feature of this new approach is that it can describe the finite deformation of crystals under thermal conditions. The obtained plastic deformation gradient contains not only plastic defor- mation but also thermal effects. The governing equation for the plastic deformation gradient is obtained based on ther- mal multiplicative decomposition of the total deformation gradient. An implicit method is used to integrate this evo- lution equation to ensure stability. Single crystal 1 100 aluminum is investigated to demonstrate practical applications of the model. The effects of anisotropic properties, time step, strain rate and temperature are calculated using this integration model.     

Characteristic Galerkin method for convection-diffusion equations and implicit algorithm using precise integration

This paper presents a finite element procedure for solving transient, multidimensional convection-diffusion equations. The procedure is based on the characteristic Galerkin method with an implicit algorithm using precise integration method. With the operator splitting procedure, the precise integration method is introduced to determine the material derivative in the convection-diffusion equation, consequently, the physical quantities of material points. An implicit algorithm with a combination of both the precise and the traditional numerical integration procedures in time domain in the Lagrange coordinates for the characteristic Galerkin method is formulated. The stability analysis of the algorithm shows that the unconditional stability of present implicit algorithm is enhanced as compared with that of the traditional implicit numerical integration procedure. The numerical results validate the presented method in solving convection-diffusion equations. As compared with SUPG method and explicit characteristic Galerkin method, the present method gives the results with higher accuracy and better stability. The project sponsored by the State Scientific and Technological Commission of China through “China State Key Project: the Theory and Methodology for Scientific and Engineering Computations with Large Scale”, the National Natural Science Foundation of China and the European Commission Research Project CI1^*CT94-0014.     

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.     

On the use of a reduced enhanced solid-shell (RESS) element for sheet forming simulations

A recently proposed reduced enhanced solid-shell (RESS) element [Alves de Sousa, R.J., Cardoso, R.P.R., Fontes Valente, R.A., Yoon, J.W., Grácio, J.J., Natal Jorge, R.M., 2005. A new one-point quadrature enhanced assumed strain (EAS) solid-shell element with multiple integration points along thickness: Part I – Geometrically Linear Applications. International Journal for Numerical Methods in Engineering 62, 952–977; Alves de Sousa, R.J., Cardoso, R.P.R., Fontes Valente, R.A., Yoon, J.W., Grácio, J.J., Natal Jorge, R.M., 2006. A new one-point quadrature enhanced assumed strain (EAS) solid-shell element with multiple integration points along thickness: Part II – Nonlinear Applications. International Journal for Numerical Methods in Engineering, 67, 160–188.] is based on the enhanced assumed strain (EAS) method with a one-point quadrature numerical integration scheme. In this work, the RESS element is applied to large-deformation elasto-plastic thin-shell applications, including contact and plastic anisotropy. One of the main advantages of the RESS is its minimum number of enhancing parameters (only one), which when associated with an in-plane reduced integration scheme, circumvents efficiently well-known locking phenomena, leading to a computationally efficient performance when compared to conventional 3D solid elements. It is also worth noting that the element accounts for an arbitrary number of integration points through thickness direction within a single element layer. This capability has proven to be efficient, for instance, for accurately describing springback phenomenon in sheet forming simulations. A physical stabilization procedure is employed in order to correct the element’s rank deficiency. A general elasto-plastic model is also incorporated for the constitutive modelling of sheet forming operations with plastic anisotropy. Several examples including contact, anisotropic plasticity and springback effects are carried out and the results are compared with experimental data.     

Finite element computation of torsional plastic waves in a thin-walled tube

Summary  A finite element technique is presented for the analysis of one-dimensional torsional plastic waves in a thin-walled tube. Three different nonlinear consitutive relations deduced from elementary mechanical models are used to describe the shear stress–strain characteristics of the tube material at high rates of strain. The resulting incremental equations of torsional motion for the tube are solved by applying a direct numerical integration technique in conjunction with the constitutive relations. The finite element solutions for torsional plastic waves in a long copper tube subjected to an imposed angular velocity at one end are given, and a comparison with available experimental results to assess the accuracy of the constitutive relations considered is conducted. It is demonstrated that the strain-rate dependent solutions show a better agreement with the experimental results than the strain-rate independent solutions. The limitations of the constitutive equations are discussed, and some modifications are suggested. Received 9 February 1999; accepted for publication 28 March 2000