**Introduction****: **Sheet metal shaping procedures are innovatively among the most critical metal working forms. Products made by sheet-forming process include a very large variety of different geometrical shapes and sizes, like simple bend to double curvatures even with deep recesses and very complex shapes. Typical examples are automobile bodies, aircraft panels, appliance bodies, kitchen utensils, and beverage cans.

Sheet metal forming processes are widely used in the manufacturing industry. It is usually involved in developing and building tools namely “Die” and “Punch”. Usually, tools are costly and the cycle time for building them is long. However, once die and punch are built, the tools can be used to produce a large amount of products. Therefore Sheet metal forming is a simple and efficient manufacturing process.

Great productivity and low production cost can be expected for commercial scale production. Deep drawing analysis provides a great challenge because of the complex physical phenomenon to be simulated. In such analysis for example large deformation, large strain anisotropic material behavior, contact conditions including friction need to be modeled.

The basic tools for deep drawing process are blank, punch, die and blank holder.

**Problem Statement****: **An Aluminum alloy cup with an inside radius of 30mm and a thickness of 3mm is to be drawn from a blank of radius 40mm.

- Determine the drawing force, assuming that the coefficient of friction ‘µ’=0.1 and ‘β’=0.05.
- Determine the minimum possible radius of the cup which can be drawn from the given blank without causing a fracture.

**Geometry and Details****: **

The Geometry was modeled by using the following dimensions:

Punch radius (r_{p}) |
30 mm |

Punch corner radius | 6 mm |

Die radius (r_{d}) |
33 mm |

Die corner radius | 6 mm |

Blank radius (r_{j}) |
40 mm |

Blank thickness (t) | 3 mm |

Table: 1

**Material Properties****:** The material assigned for blank is ‘Aluminum Alloy’ with given material properties:

Density (ρ) |
2770 kg/m |

Modulus of Elasticity (E) |
7.1E10 Pa |

Poisson’s Ratio (ʋ) |
0.33 |

Ultimate Tensile Strength |
310 MPa |

Shear Yield Stress (K) |
201.5 N/mm |

Maximum Allowable Stress |
300 N/mm |

Table: 2

**Calculations used in Analysis Process****: **The following calculations were performed to find out the required parameters in the Analysis Process:

**I. ****Blank holding Force (F _{h})**

**:**Firstly the blank holding force ‘F

_{h}’ will be calculated from the given formula:

Therefore,

**II. Radial Stress (σ _{r}): **Next we find the value of ‘σ

_{r}’ at r = r

_{d }by using the following equation:

Therefore,

As the job slides along the die corner, the radial stress increases to ‘σ_{z}’ due to frictional forces.

This increment of ‘σ_{z}’ can be estimated by the following equation:

Therefore,

**III. ****Drawing Force****: **Drawing force is to be calculated by the following equation:

Therefore,

Now we will find the minimum possible radius of the cup which can be drawn from the blank without causing a fracture.

In this case ‘σ_{z}’ = 300 N/mm^{2}

Therefore, σ_{r}**|**r=r_{d} = 256.39 N/mm^{2}

Now using radial stress formula and taking ‘F_{h}’= 50642.5 N, we get:

**Methodology Adopted****: **This analysis was carried out by using Explicit Dynamics module in Ansys Workbench. As the analysis is highly non-linear the time integration method will be used explicitly which is very efficient for each time step but the run time would be too enormous to be accepted.

**Explicit Dynamics****: **Explicit Dynamics is used to perform high-speed impact simulation i.e. collision of two objects. It is also used to perform drop test simulation i.e. an object falls down from a certain height onto floor.

A time integration method used in Explicit Dynamics analysis system. It is so named because the method calculates the response at current time using explicit information.

Once the body is meshed properly, the next step is to define initial conditions or boundary conditions. At least one initial condition is required to complete the set up.

After defining the initial conditions (initial velocity, Angular velocity), analysis setting has to be maintained as per the problem requirement. In analysis setting, time step has to be defined explicitly. The solution time is depends on the time steps.

The time steps include:-

- Initial time step
- Minimum time steps
- Maximum time step
- Time step safety factor

Deep drawing analysis is also a highly non-linear problem in which the collision takes place in between ‘punch’ and ‘blank’. Punch moves down with certain velocity and exerts force on blank. Therefore it is prescribed to use explicit dynamics module in Ansys Workbench for Deep Drawing Analysis.

**Meshed Model****:**

The problem was solved to find out the deformation of the blank in Y- direction and the shear stress induced in it.

**Boundary Conditions****: **As the problem was solved analytically, the value of Drawing force and Blank holding force was taken from the calculation. The complete boundary conditions and its details are shown in fig. 5.

After setting up the boundary conditions the problem was solved explicitly, it took a total of ‘21368’ cycles and a run time of total 8.8 secs to execute the solution. The problem was solved to find out the maximum deformation of the blank in –Y- Direction without any shear.

**Results****:**

**Directional Deformation****:**

**Shear Stress****:**

If we allow Blank to deform beyond 18.75 mm and increase the time step from 0.00055 secs to 0.0006 secs then the shear stress will go beyond the value of maximum allowable stress and material starts to shear.

**Directional Deformation****:**

**Conclusion****: **

After analyzing the above results for deformation of the blank in –Y – Direction, we observed that the blank can maximum deform till 18.754 mm beyond this value of deformation material starts to shear and the shear stress value crossed the Maximum allowable stress value of the prescribed material.

**Final Deduction****: **

Explicit Dynamic analysis for the Deep Drawing setup was carried out successfully, and it is observed that the maximum deformation of the blank under the safer limit of the shear stress is around 18.75 mm which also matches with the analytical solution. The percentage of error in between analytical and computational deformations was only 0.5 %.