A Modified Arrhenius‐Type Constitutive Model and its Implementation by Means of the Safe Version of Newton–Raphson Method

Abstract

Arrhenius‐type constitutive models are widely used in thermomechanical processing due to their ease of parameter calibration and ability to track the deformation behavior with various materials. Herein, a modified Arrhenius model is proposed, and the model is implemented in the finite element (FE) simulation to improve the simulation accuracy. The VUMAT Fortran subroutine of the model is established for the commercial nonlinear FE software Abaqus/Explicit based on an implicit time integration scheme and the radial return mapping algorithms. A root‐finding method called the safe version of Newton–Raphson method is applied to calculate the plastic corrector term. The efficiency and accuracy of both analytical and numerical methods are evaluated for calculating the derivatives in this algorithm. The implementation of the user‐defined modified Arrhenius‐type material model is successfully applied to a thermal–mechanical coupling FE model for the hot rolling of heavy steel plate. The goal optimization function is used to calibrate the material parameters for the model. The plastic strain field and temperature field are in accordance with the theoretical derivation and experimental results. It provides valuable references for characterizing and optimizing thermomechanical processing.

Development and Implementation of an ANN Based Flow Law for Numerical Simulations of Thermo-Mechanical Processes at High Temperatures in FEM Software

“…As introduced in Section 1, implementing a flow law in a FE code requires both the computation of the flow stress σ y as a function of the input data, performed using the previous Equations ( 5)-( 11), but also the evaluation of the three derivatives of σ y with respect to the input data to use a Newton-Raphson algorithm within the stress integration scheme, as proposed by many authors [24,28,45,46] based on the radial return algorithm in the Abaqus FE code. It is, therefore, necessary to perform a numerical evaluation of these three derivatives based on the ANN to obtain these quantities.…”

“…ε and temperatures T. From the observations made, we can define two main classes of behavior laws: the flow laws based on physics and the empirical flow laws. From the mechanics of continuous media and experimental tests and depending on the materials used, several flow models have been developed in the past, including the Johnson-Cook flow law [8,9], the Zerilli-Armstrong flow law [10] and their respective derived forms [11][12][13][14][15][16][17][18][19], the Hansel-Spittle [20,21] or the Arrhenius [22][23][24] flow laws, to name only a few of the most widely used in the metal-forming processes at high temperature. As an example, and because it is widely used in numerical simulation of metal forming processes, the equation that describes the Johnson-Cook flow law [8] is given as follows:…”

“…Thus, a user of the Abaqus FE code will turn more particularly to a Johnson-Cook [8] flow law, where it is natively implemented in this software. The choice of another form of flow law, Zerilli-Armstrong, or Arrhenius, for example, obliges the user to program himself the computation of the flow stress σ y of the material through a VUMAT subroutine in FORTRAN 77 as proposed by Gao et al [27], Ming et al [28] for a Johnson-Cook flow law, or Liang et al [24] for an Arrhenius type flow law with the following expression:…”

“…Implementing the flow law as a VUMAT FORTRAN subroutine requires the computation of the derivatives ∂σ y /∂ε p , ∂σ y /∂ . ε and ∂σ y /∂T of the flow stress σ y , which can quickly become relatively complex as the complexity of the flow law increases, i.e., the relative complexity of the Arrhenius flow law defined by Equations ( 3) and ( 4), regarding the relative simplicity of the Johnson-Cook model defined by Equation ( 2), one can refer to the work proposed by Liang et al [24] for details concerning this implementation using the safe version of the Newton-Raphson method proposed by Ming et al [28]. The choice of the flow law to use for a problem is therefore doubly guided by the behavior of the material on the one hand, but a more important aspect is the list of flow laws implemented natively in the FE code we plan to use for the numerical simulation.…”

“…Only the modified Johnson-Cook and Arrhenius models can describe correctly the behavior of the material during the compression process. Unfortunately, and this is not part of their study, if these last two models are satisfactory from a theoretical point of view, from a practical point of view for the user of a FE code such as Abaqus, it will be necessary to carry out a numerical implementation in a FORTRAN 77 VUMAT subroutine of the modified Johnson-Cook flow law or the Arrhenius law as carried out by a few authors [24,27,28] to use these laws for numerical simulation. This requires a certain expertise in the development and implementation of flow laws, which is not available to all users of the Abaqus FE code.…”