Stress and deformation analysis of the metal extrusion process

This paper presents a complete derivation and implementation of the Arbitrary Lagrangian Eulerian (ALE) formulation for the simulation of large deformation quasi-static and dynamic problems. While most of the previous work done on ALE for dynamic applications was mainly based on operator split and explicit calculations, this work derives the quasi-static and dynamic ALE equations using a fully coupled implicit approach. Full expression for the ALE virtual work equations and finite element matrices are given. Time integration relations for the dynamic equations are also derived. Several quasistatic and dynamic large deformation applications are solved and presented. IntroductionThe Arbitrary Lagrangian Eulerian (ALE) formulation has emerged in recent years as a technique that can alleviate many of the drawbacks of the traditional Lagrangian and Eulerian formulations [2,5,7,11,16]. Using ALE, the computational grid need not adhere to the material nor be fixed in space but can be moved arbitrarily. The grid is continuously moved to optimize element shapes independently from material deformation. A proper ALE formulation should reduce to both the Lagrangian and Eulerian formulations at any degree of freedom as desired. Combining the merits of both the Lagrangian and Eulerian formulations, ALE can easily describe different types of boundary conditions and prevent mesh distortion.The ALE equations are derived by substituting the relationship between the material time derivative and grid time derivative into the continuum mechanics governing equations. This substitution gives rise to convective terms in the ALE equations which account for the transport of material through the grid. ALE is usually termed a coupled formulation since material deformation and convective effects are coupled in the same equations. A survey of the ALE literature [17], however, shows that the majority of ALE analyses, whether quasi-static or dynamic, are based on the computationally convenient operator split technique. In this approach, material deformation and convective effects are treated separately. Thus each time step may be divided into two steps: a regular Lagrangian step followed by an Eulerian step. The main advantage of this technique over the fully coupled approach is the reduction in the cost of implementation of ALE to current Lagrangian codes as the Lagrangian step is unchanged and only the Eulerian step algorithm needs to be added. Moreover, the decoupling of the Lagrangian and Eulerian steps results in simpler equations to be solved. However, from the theoretical point of view, the fully coupled ALE approach represents a true kinematic description in which material deformation is described relative to a moving reference configuration.In this work, a complete treatment for the fully coupled implicit ALE formulation is presented. Virtual work equations are first derived from the basic principles of continuum mechanics. Next, finite element descritization of the virtual work equations is performed. Full expression for the resulting...

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“…Substituting by (36), (37) and (29) into (12), the principle of virtual displacements at time t þ Dt referred to time t can be written as Z…”

Section: Treatment Of Convective Termsmentioning

confidence: 99%

“…The length of the die is taken as 1.2a. The same problem was solved using the updated Lagrangian approach in [12]. The material properties for the aluminum billet are taken as: Young's modulus = 10 4 ksi, Poisson's ratio = 0.3, initial yield stress = 57 ksi and hardening parameter = 165 ksi.…”

Section: Sheet Metal Extrusionmentioning

confidence: 99%

A complete finite element treatment for the fully coupled implicit ALE formulation

Bayoumi

Gadala

2004

Computational Mechanics

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“…Besides the die, the process includes a conveying system, involving the rotation of a screw (or several screws), such that the material is transported by an Archimedean screw action through the device. Nowadays, more and more forming processes use this technique, and the products that can be extruded are numerous: plastics [2,3], drugs [4][5][6][7], food [8][9][10], metals [11,12], tissue engineering scaffolds using electrospinning [13][14][15], porous items using supercritical fluid [16], etc.…”

Section: Introductionmentioning

confidence: 99%

A Review of Dynamic Models of Hot-Melt Extrusion

2016

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Hot-melt extrusion is commonly applied for forming products, ranging from metals to plastics, rubber and clay composites. It is also increasingly used for the production of pharmaceuticals, such as granules, pellets and tablets. In this context, mathematical modeling plays an important role to determine the best process operating conditions, but also to possibly develop software sensors or controllers. The early models were essentially black-box and relied on the measurement of the residence time distribution. Current models involve mass, energy and momentum balances and consists of (partial) differential equations. This paper presents a literature review of a range of existing models. A common case study is considered to illustrate the predictive capability of the main candidate models, programmed in a simulation environment (e.g., MATLAB). Finally, a comprehensive distributed parameter model capturing the main phenomena is proposed.

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“…Traditionally, steady-state processes with history-dependent material behaviour are analysed with Lagrangian kinematic descriptions in which the displacement is the primary response ÿeld [1][2][3]. This method requires a transient analysis to determine the evolution of the history-dependent material variables.…”

Section: Introductionmentioning

confidence: 99%