Ch3b Slip Line f

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Analysis of large plastic deformation of

elasto-plastic solids

Friction involves large plastic deformation.

There are different ways of solving thedeformation of elasto-plastic solids.

are approximate solutions, can be very usefulin engineering.

One of the methods used is the slip-line fieldmethod, which gives a physical feel for thedeformation process. It is an exact analysis fordeformation of rigid-perfectly plastic solids.

Also upper- and lower-bound solutions, which


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Classification of partial differential

equations

Three different types of pd equations:

Elliptic (elastic deformation)

Parabolic (heat transfer, mass transfer)

Hyperbolic equations (wave propagation, plastic

deformation


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equations

Consider the following second-order partialdifferential equation:

a2z

x 2+b 2z

xy+c 2z

y2=e

Boundary conditions :

z,z

x

z

yare specified., and


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equations

Then the variation ofz

xand

z

xmay be expressed as

d(z

x) =2z

x2dx + 2z

xydy

d(z

y) = 2z

xydx+2z

y2dy

We have three equations and three unknowns,

2z

x2,

2z

xy

2z

y2

., and


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equations

Solving for

2z

x2

,

2z

x2= N

D

Depending on whether D equal to orgreater than 0,

the pde represents different physical phenomena.

b 2

b 2

b

2

=

- 4ac < 0, elliptic equation.

- 4ac > 0, hyperbolic equation.

- 4ac 0, parabolic equation.


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We want to get approximation solutions for

large deformation of rigid-perfectly plastic

solid in plane strain.

We have to satisfy

Equilibrium condition (F=ma)

Geometric compatibility

Stress-strain relationship (constitutive relationship)

Yield condition

Boundary conditions

The Upper- and the Lower-Bound Solutions


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Lower bound solutions are obtained if we

satisfy

Equilibrium condition

Yield condition

Boundary conditions on stress

Consider the punch indentation problem. The

lower-bound can be obtained as

The Upper- and the Lower-Bound Solutions


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The Lower-Bound Solutions

Figure by MIT OCW.

0 A

2k 2k

4k

4k

px2

x1

B

2k 2k Rigid


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The Upper-Bound Solutions

The upper-bound-solutions are obtained by

satisfying the following for an assumed

displacement field:

1. Incompressibility condition

2. Geometric compatibility

3. Velocity boundary conditions.


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The Upper-Bound Solutions

Figure by MIT OCW.

a

x2

q = displacement

of indentation

p

(all angles 45o)

x1

B C

A


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The lowerbound solutionp 4k

The upperbound solution

p 6k

The Lower- and the Upper-Bound Solutions


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The Slip-line Field Solutions

The stresses can be represented in terms of two

invariants, p and k,as

III =2k

Graph removed for copyright reasons.

Tribophysics. Englewood

The Tresca yield condition:

See Figure 3.A1 - bottom in [Suh 1986]: Suh, N. P.Cliffs NJ: Prentice-Hall, 1986. ISBN: 0139309837.


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The Slip-line Field Solution for Asperity

Deformation

Diagram removed for copyright reasons.

Tribophysics. EnglewoodSee Figure 3.A1 - top in [Suh 1986]: Suh, N. P.Cliffs NJ: Prentice-Hall, 1986. ISBN: 0139309837.


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The Slip-line Field Solution for Asperity

Deformation

Figure by MIT OCW. After Suh, N.P., and H.C. Sin. "The Genesis of Friction." Wear69 (1981): 91-114.

0

0.5

1.0

15

15o

20o

=

10o

5o

0o

'

30 45

q

q

a

m

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Ch3b Slip Line f