Author Topic: Computer Simulation of Forging Using the Slab Method Analysis  (Read 1900 times)

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Computer Simulation of Forging Using the Slab Method Analysis
« on: August 20, 2011, 10:37:40 am »
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Author : S. B. Mehta, D. B. Gohil
International Journal of Scientific & Engineering Research Volume 2, Issue 6, June-2011
ISSN 2229-5518
Download Full Paper : PDF

Abstract— Forging is a very complex process and the measurement of actual forces for real material is difficult and cumbersome. The main objective of this document is to use the analytical methods for  measuring parameters such as load, and stress distribution of forging process and use them to make simple, reliable, fast and non-expensive simulation tools, contrary to the commercial software’s which require much means, time and a perfect knowledge of the process. Of the various methods used for analysing forging operations, the most often used SLAB method techniques are described here.
   
Index terms—analysis, simulation, modelling, forging, closed-die forging, open die forging, process.

Introduction
THE developments in forging technology have increased the range of shapes, sizes and properties of the forged products enabling them to have various design and performance requirements. Also Closed-Die forging is an extremely complex forming process from the point of view of deformation mechanics. The non-steady state and non-uniform material flow, the considerable interface friction, and the heat transfer between the deforming material and the tooling are very difficult to analyze. To ensure the quality of the final product, even a highly experienced engineer spends a lot of time on optimizing the design of the process through a time-consuming trial and error method. Sometimes, the design engineers take the help of FE analysis to fine-tune the process and avoid the costly physical trials. In this context, it is worthwhile to mention that the FE analysis of metal forming is a time-consuming process, even on a powerful PC. Interpretation of the results of FE analysis requires in-depth knowledge and experience of both forging process and FE method. Other than the forging process parameters, the results of FE analysis depend on proper selection of a large number of FE parameters, e.g., element type and size, mesh topology, node numbering, and others. A proper selection of these FE parameters requires in-depth knowledge [1]. A feasible solution is the use of analytical methods, of which the  most widely used, ‘Sachs’ or ‘Slab’ method, is used. The slab method here is used to divide the work piece into various slabs. The parameters such as load and stress are then easily calculated for these shapes and then added to get the value of final forging load. 

THE SLAB METHOD
The work piece being deformed is decomposed into several slabs. For each slab, simplifying assumptions are made mainly with respect to stress distributions. The resulting approximate equilibrium equations are solved with imposition of stress compatibility between slabs and boundary tractions. The final result is a reasonable load prediction with an approximate stress distribution [Kobayashi et al .1989]. The following assumptions are made in using the slab method of analysis [2]:

● the deforming material is isotropic and      incompressible,
● the elastic deformations of the deforming material and         
          tool are neglected,     
      ● the inertial forces are small and are neglected,
      ● the frictional shear stress, τ, is constant at the,           
         die/material interface,
      ● the material flows according to the von Mises rule,
      ● the flow stress and the temperature are constant        within   the analyzed portion of the deforming material.

   Open die forging

   Plane strain
In applying slab analysis to plane strain upsetting, a slab of infinitesimal thickness is selected perpendicular to the direction of metal flow (Fig. 1). Assuming a depth of “1” or unit length, a force balance is made on this slab. Thus, a simple equation of static equilibrium is obtained [Thomsen et al., 1965] [Hoffman et al., 1953].

Summation of forces in the X direction is zero or

∑▒F_x = σ_x h-(σ_x+ dσ_x )h-2τdx

Or

dσ_x= -2τdx/h

Thus, by integration one gets:
 
Fig 1. Equilibrium of forces in plane strain homogenous upsetting

σ_x=-2τ/h  x+C

From the flow rule of plane strain, it follows that:

σ_x= -2τ/h  x+C+|2/√3 σ|


The constant C is determined from the boundary condition at x= l/2, where σ_x=0, and from equation


σ_x=|2/√3 σ|

Thus,

σ_x= -2τ/h (l/2-x)-2/√3 σ


Equation illustrates that the vertical stress linearly increases from the edge(x=l/2) of the figure towards the centre (x=0) In equation, the frictional shear stress, τ is equal to mσ ̅/√3. Thus, integration of the equation over the entire width l of the strip of unit depth gives the upsetting load per unit depth [2]:


L=2σ/√3(1+ ml/4h)l


   Axisymetric

The flow rule for Axisymetric deformation is obtained by using a derivation similar to that used in plane strain deformation. The equilibrium of forces in the r direction gives [Thomsen et al., 1986][Hoffman et all., 1953] :

∑▒F_r = σ_r (dθ)rh-(σ_r+ dσ_r )(r+dr)hdθ+2σ_θ sin dθ/2 hdr-2τrdθdr

Further simplification using appropriate boundary conditions gives the result as

σ_z= 2τ/h (r-R)- σ

The equation says that stress increases linearly from the edge towards the centre. The upsetting load can now be found out as,

L=σπR^2 (1+ 2mR/(2h√3))

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