Integration Of Constitutive Equations Research Articles

Numerical simulation and experimental investigation on friction stir welding of AZ31 magnesium alloy

Integrated numerical simulations and experimental investigations were employed to scrutinize the thermal, mechanical, and microstructural transformations of the AZ31 magnesium alloy during the friction stir welding (FSW) process. Especially, the primary focus was on the influence of process parameters such as rotational speed and welding speed on the temperature distribution, grain refinement, and mechanical properties of welded joints in alloys. By employing Deform-3D coupled with the integration of constitutive equations and dynamic recrystallization (DRX) models, the FSW process was investigated. The investigation revealed a significant increase in temperature when the tool’s shoulder made contact with the weld, resulting in the substantial accumulation of heat during FSW. Distinctions became apparent between the advancing side (AS) and the receding side (RS), with the AS exhibiting slightly elevated levels of temperature, equivalent stress, strain, and grain size. Specifically, adjustments in the rotational speed of the stirring tool and a reduction in welding speed resulted in larger grain sizes within the alloy. For example, when the rotational speed was set at 1200 rpm and the travel rate was 200 mm min−1, the initial grain size of the weld experienced a substantial decrease from 57.8 μm to 8.2 μm. Subsequent experimental verification, considering grain size and microhardness, was carried out to optimize FSW parameters for achieving the desired material properties. The accuracy of simulation results was validated through a meticulous comparison with experimental findings, underscoring the potential of numerical simulation in comprehending and predicting FSW processes.

Computationally efficient viscoelastic flow simulation using a Lagrangian-Eulerian method and GPU-acceleration

A recently proposed Lagrangian-Eulerian method for viscoelastic flow simulation is extended to high performance calculations on the Graphics Processing Unit (GPU). The two most computationally intensive parts of the algorithm are implemented for GPU calculation, namely the integration of the viscoelastic constitutive equation at the Lagrangian nodes and the interpolation of the resulting stresses to the cell centers of the Eulerian grid.In the original CPU method, the constitutive equations are integrated with a second order backward differentiation formula, while with the proposed GPU method the implicit Euler method is used. To allow fair comparison, the latter is also implemented for the CPU. The methods are validated for two flows, a planar Poiseuille flow of an upper-convected Maxwell fluid and flow past a confined cylinder of a four-mode Phan Thien Tanner fluid, with identical results.The calculation times for the methods are compared for a range of grid resolutions and numbers of CPU threads, revealing a significant reduction of the calculation time for the proposed GPU method. As an example, the total simulation time is roughly halved compared to the original CPU method. The integration of the constitutive equation itself is reduced by a factor 50 to 250 and the unstructured stress interpolation by a factor 15 to 60, depending on the number of CPU threads used.

Solution for the general integral‐type nonlocal plasticity model with Tikhonov regularization

SummaryA new solution approach, based on Tikhonov regularization on the Fredholm integral equations of the first kind, is proposed to find the approximate solutions of the strain softening problems. In this approach, the consistency condition is regularized with the Tikhonov stabilizers along with a regularization parameter, and the internal variable increments are solved from the resulting Euler's equations. It is shown that, as the regularization parameter is increased, the solutions converge to a unique one. A nonlocal yield condition and a nonlocal return mapping algorithm are proposed to carry out the integration of constitutive equations in the time and spatial domains. A global plastic dissipation principle is proposed to relax the classical local plastic dissipation postulate. Numerical examples show that the proposed approach leads to objective, mesh‐independent solutions of the softening‐induced localization problems. A comparison of the results from the proposed approach with those from the gradient‐dependent plasticity model shows that the two models give close solutions.

Integration of constitutive equations with a new explicit scheme

Precise integration of a constitutive law is required whenever a boundary value problem is to be solved. In this paper a new explicit scheme for the integration of elasto-plastic constitutive models, that consist of algebraic yield condition and differential evolution equations, is presented. The scheme was primarily developed for the integration of constitutive models in boundary value problems which are solved with a direct solving technique (like in Abaqus/Explicit). Although the scheme is as fast and as simple for implementation as unstable explicit forward-Euler scheme, it can reach the same level of accuracy as commonly used backward-Euler scheme. Two examples are shown in the paper, the first one being simple for demonstration purposes only, whereas in the second one the scheme is also implemented in the Abaqus/Explicit user subroutine VUMAT. The obtained results confirm that the scheme is accurate and stable. It can be concluded in consequence, that the scheme is very handy for implementation of complex elasto-plastic constitutive laws into finite element software.

Hyper Reduction of finite strain elasto-plastic models

The aim of this work is to propose a new method to compute the reduced state variables related to a known POD basis. The classical Galerkin formulation of the reduced governing equation related to a POD basis is not efficient for medium size elasto-plastic problems (about 20000 Degrees of freedom). In such problems the computational cost related to the local integration of the nonlinear constitutive equations is very important. It can constitute 80% of the total computational cost. The classical POD basis applied to primal variables (the displacements) does not affect the efficiency of the integration of the constitutive equations. To overcome this drawback we propose a Petrov-Galerkin formulation related to a Reduced Integration Domain (RID). This RID is created by selecting few elements of the mesh to perform the local integration of the constitutive equations. This is a generic approach coined Hyper Reduction, which have been successfully applied to simplify nonlinear thermal transient models. The novelty of this work is the extension of the Hyper Reduction method to the problems involving internal variables. A 3D finite strain elasto-plastic model is considered to illustrate the capabilities of the Hyper reduction method.

Lifetime Predictions For High-Temperature Low-Alloy Ferritic Steel Weldments

The paper reports the finite element creep continuum damage mechanics (CDM) analysis of the creep rupture behaviour of thin-section butt-welded, internally pressurized pipes, and of uniaxial cross-welded tension testpieces made of Cr-Mo-V steels. The CDM analyses are shown to predict accurately lifetimes and failure mechanisms observed from metallographic analyses of sectioned components. The paper then addresses the lifetime prediction of components by comparing predictions made using British Standards BSi PD 5500 with the results of the experiments. Predictions have been based on uniaxial plain bar data for both parent and Type IV materials; both of these have been obtained by integration of constitutive equations. Comparison of predicted and experimental weldment lifetimes has been made using a weld strength reduction factor. It is found that the required factors are very dependent on temperature and the ratio of axial to hoop pipe stresses. Finally, the paper examines the British Energy Generation Limited R5 assessment procedure and compares lifetime predictions with those made using the British Standards BSi PD 5500 approach. It is noted that the stresses in both approaches are in good agreement for the external/internal radius ratio of 1.25 considered in the butt-welded pipe experiments.

A hybrid finite element approach to the solution of creep problems

A new general procedure for constructing numerical algorithms for creep analysis is presented. Based on the simultaneous use of the principles of virtual displacements and virtual stresses, the new method treats both displacements and stresses as independent variables. A well established hybrid element analysis is expanded to account for creep. As a result, a numerically stable algorithm for the integration of constitutive equations in time has been developed. The algorithm does not involve the subincrementation process and hence allows large variations in the magnitude of the time step.

FE-formulation of a nonlocal plasticity theory

A nonlocal continuum plasticity theory is presented. The nonlocal field introduced here is defined as a certain weighted average of the corresponding local field, taken over all the material points in the body. Hereby, a quantity with the dimension of length occurs as a material parameter. When this so-called internal length is equal to zero, the local classical plasticity theory is regained. In the present model, the yield function will depend on a nonlocal field. The consistency condition and the integration algorithm result in integral equations for determination of the field of plastic multipliers. The integral equations are classified as Fredholm equations of the second kind and the existence of a solution will be commented upon. After discretization, a matrix equation is obtained, and an algorithm for finding the solution is proposed. For a generalized von Mises material, a plane boundary value problem is solved with a FE-method. Since the nonlocal quantities are integrals, C 0-continuous elements are sufficient. The solution strategy is split into a displacement estimate for equilibrium and the integration of constitutive equations. In the numerical simulations shear band formation is analysed and the results display mesh insensitivity.

Integration of Constitutive Equations in Soil Plasticity

Explicit and implicit integration of elastoplastic constitutive relations subjected to the constraint of undrained behavior is investigated. The effect on numerical efficiency of various modes in terms of total stress and strain components is considered. The suggested explicit integration algorithm employs the appropriate tangent relation that incorporates the incompressibility condition and requires no iterations. The fully implicit algorithm used in this paper, on the other hand, is based on complete strain control and satisfies, in an iterative manner, the incompressibility constraint as well as equilibrium pertinent to any prescribed total stress component. The constitutive equation that is considered is a three-invariant generalized cam-clay model with volumetric hardening/softening as well as deviatoric hardening. Both the explicit and implicit methods seem reliable and efficient for reasonable load steps, even for nonconventional control paths.

Exact integration of constitutive equations of kinematic hardening materials and its extended applications

In this paper, an exact formula for the integration of the constitutive equations of kinematic hardening material is presented. Its algorithms are simple and clear. For isotropic hardening or mixed hardening material, the formula is still an exact solution for the case of radial loading, and it is an approximate solution with reasonable accuracy for the case of non-radial loading. The computation results show that the procedure proposed in this paper improves both accuracy and efficiency of numerical integration schemes adopted widely in elastic-plastic finite element analysis.

An Improved Method for Numerical Integration of Constitutive Equations of the Work Hardening-Recovery Type

An Improved Method for Numerical Integration of Constitutive Equations of the Work Hardening-Recovery Type