Pearson eText for Manufacturing Processes for Engineering Materials -- Instant Access (Pearson+)
Pearson eText for Manufacturing Processes for Engineering Materials -- Instant Access (Pearson+)
6th Edition
ISBN: 9780137503520
Author: Serope Kalpakjian, Steven Schmid
Publisher: PEARSON+
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Chapter 2, Problem 2.82P
To determine

The expression for force versus piston travel d up to d = 10 mm.

Expert Solution & Answer
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Answer to Problem 2.82P

The expression for Force F versus piston travel d is

  F=8835.732d+294522ln(1+d25)0.4

And the value of force F at d =10 mm is 127996.71 N

Explanation of Solution

Given data:

  Diamter of cylindrical slug, D1=25mmHeight of cylindrical slug, d=25mm,Diamterof cavity in a rigid die, D2=25mm,Pressureacting of matrix is, pm=150ΔVVomMPa,Stress of the slug material, σ=600ε0.4.

A cylindrical is surrounded by a compressible matrix. Cylindrical slug is compressed vertically by a force F which is to be determined. When the force F is acted then there will be the generation of stress inside the cylindrical slug and pressure is also generated inside the compressible matrix. The force acted will be equal to the force generated inside the cylindrical slug plus the force generated inside the matrix.

Calculation:

Drawing the free body diagram of the given system,

  Pearson eText for Manufacturing Processes for Engineering Materials -- Instant Access (Pearson+), Chapter 2, Problem 2.82P

From the free body diagram, it can be written as,

  F=Fmatrix+Fcylinder...........(1)

Area of the matrix and cylinder can be given as −

  Area of the matrix,Am=π4(D12D22)Am=π4(502252)Am=1472.622mm2

  Area of the cylinder,Ac=π4D22Ac=π4×252Ac=490.87mm2

  Change in the volume for matrix can be given as,ΔV=Am[25(25d)]ΔV=Am×dΔV=1472.622×d

Calculating the force exerted by the matrix:-

  Fmatrix=pm×AreaFmatrix=150×ΔVVom×AreaVolume of the matrix will be, Vom=Am×25Fmatrix=150×ΔVAm×25×AmFmatrix=150×ΔV25Fmatrix=150×1472.622×d25Fmatrix=8835.732d...............(2)

The strain developed inside the cylinder for deformation of height d is −

  Strain,e=d25

True strain inside the cylinder would be −

  True strain, ε=ln(1+e)ε=ln(1+d25)

Writing true stress-true strain relation,

  σ=600ε0.4σ=600ln(1+d25)0.4

The force will be given as,

  Fc=σAcFc=600ln(1+d25)0.4×490.87Fc=294522ln(1+d25)0.4...................(3)

From equation (1), (2) and (3),

  F=Fmatrix+FcylinderBut,Fmatrix=8835.732dFc=294522ln(1+d25)0.4Therefore,F=8835.732d+294522ln(1+d25)0.4

The above is the required expression for force versus d .

Putting d = 10 mm we get,

  F=8835.732d+294522ln(1+d25)0.4F=8835.732×10+294522ln(1+1025)0.4F=88357.32+39639.39F=127996.71N

Hence, the expression for Force F versus piston travel d is

  F=8835.732d+294522ln(1+d25)0.4

And the value of force F at d =10 mm is 127996.71 N

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Chapter 2 Solutions

Pearson eText for Manufacturing Processes for Engineering Materials -- Instant Access (Pearson+)

Ch. 2 - Prob. 2.11QCh. 2 - Prob. 2.12QCh. 2 - Prob. 2.13QCh. 2 - Prob. 2.14QCh. 2 - Prob. 2.15QCh. 2 - Prob. 2.16QCh. 2 - Prob. 2.17QCh. 2 - Prob. 2.18QCh. 2 - Prob. 2.19QCh. 2 - Prob. 2.20QCh. 2 - Prob. 2.21QCh. 2 - Prob. 2.22QCh. 2 - Prob. 2.23QCh. 2 - Prob. 2.24QCh. 2 - Prob. 2.25QCh. 2 - Prob. 2.26QCh. 2 - Prob. 2.27QCh. 2 - Prob. 2.28QCh. 2 - Prob. 2.29QCh. 2 - Prob. 2.30QCh. 2 - Prob. 2.31QCh. 2 - Prob. 2.32QCh. 2 - Prob. 2.33QCh. 2 - Prob. 2.34QCh. 2 - Prob. 2.35QCh. 2 - Prob. 2.36QCh. 2 - Prob. 2.37QCh. 2 - Prob. 2.38QCh. 2 - Prob. 2.39QCh. 2 - Prob. 2.40QCh. 2 - Prob. 2.41QCh. 2 - Prob. 2.42QCh. 2 - Prob. 2.43QCh. 2 - Prob. 2.44QCh. 2 - Prob. 2.45QCh. 2 - Prob. 2.46QCh. 2 - Prob. 2.47QCh. 2 - Prob. 2.48QCh. 2 - Prob. 2.49PCh. 2 - Prob. 2.50PCh. 2 - Prob. 2.51PCh. 2 - Prob. 2.52PCh. 2 - Prob. 2.53PCh. 2 - Prob. 2.54PCh. 2 - Prob. 2.55PCh. 2 - Prob. 2.56PCh. 2 - Prob. 2.57PCh. 2 - Prob. 2.58PCh. 2 - Prob. 2.59PCh. 2 - Prob. 2.60PCh. 2 - Prob. 2.61PCh. 2 - Prob. 2.62PCh. 2 - Prob. 2.63PCh. 2 - Prob. 2.64PCh. 2 - Prob. 2.65PCh. 2 - Prob. 2.66PCh. 2 - Prob. 2.67PCh. 2 - Prob. 2.68PCh. 2 - Prob. 2.69PCh. 2 - Prob. 2.70PCh. 2 - Prob. 2.71PCh. 2 - Prob. 2.72PCh. 2 - Prob. 2.73PCh. 2 - Prob. 2.74PCh. 2 - Prob. 2.75PCh. 2 - Prob. 2.76PCh. 2 - Prob. 2.78PCh. 2 - Prob. 2.79PCh. 2 - Prob. 2.80PCh. 2 - Prob. 2.81PCh. 2 - Prob. 2.82PCh. 2 - Prob. 2.83PCh. 2 - Prob. 2.84PCh. 2 - Prob. 2.85PCh. 2 - Prob. 2.86PCh. 2 - Prob. 2.87PCh. 2 - Prob. 2.88PCh. 2 - Prob. 2.89PCh. 2 - Prob. 2.90PCh. 2 - Prob. 2.91PCh. 2 - Prob. 2.92PCh. 2 - Prob. 2.93PCh. 2 - Prob. 2.94PCh. 2 - Prob. 2.95PCh. 2 - Prob. 2.96PCh. 2 - Prob. 2.97PCh. 2 - Prob. 2.98PCh. 2 - Prob. 2.99PCh. 2 - Prob. 2.100PCh. 2 - Prob. 2.101P
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