Prof. Ramesh Singh

What is Bulk Deformation?

• Imparting changes in material (dM~0):– Geometry

• Force• Die

– Material properties and Mechanics• Strength• Hardness• Toughness• Yield Criteria• Theory of plasticity

Prof. Ramesh Singh

Part 1- Mechanics Review• Concept of stress and strain, True and engineering• Stresses in 2D/3D, Mohr’s circle (stress and strain) for

2D/3D• Elements of Plasticity• Material Models• Yielding criteria, Tresca and Von Mises• Invariants of stress and strain• Levy-Mises equations• Strengthening mechanisms• Basic formulation of slip line in plane strain

Prof. Ramesh Singh

Part 2- Bulk Deformation

• Process description, videos and first order mathematical analysis– Plane strain forging– Cylindrical forging– Closed die forging– Rolling– Drawing– Tube drawing– Extrusion

Prof. Ramesh Singh

Outline

• Stress and strain– Engineering stresses/strain– True stresses/strain

• Stress tensors and strain tensors– Stresses/strains in 3D– Plane stress and plane strain

• Principal stresses• Mohr’s circle in 2D/3D

Prof. Ramesh Singh

#1 - Match

a) Force equilibriumb) Compatibility of

deformationc) Constitutive

equation

1) δT = δ1 + δ2

2) ΣF=0; ΣM=0

3) σ = E e

Prof. Ramesh Singh

Steps of a Mechanics Problem

• O Read and understand problem• 1 Free body diagram• 2 Equilibrium of forces

– e.g. ΣF=0, ΣM=0.• 3 Compatibility of deformations

– e.g. δT = δ1 + δ2

• 4 Constitutive equations– e.g. σ = E e

• 5 Solve

F F

A

Prof. Ramesh Singh

Key ConceptsLoad (Force), P, acting over area,

A, gives rise to stress, σ.

Engineering stress: σ = P/Ao

(Ao = original area)

True stress: σt = P/A(A = actual area)

P PA

Prof. Ramesh Singh

Deformation

• Quantified by strain, e or ε

• Engineering strain: e = (lf-li)/li

• True Strain: ε = ln(lf/li)

• Shear strain: γ = a/b

lilf

a

b

Prof. Ramesh Singh

True and Engineering strains

e = (lf-li)/lie = (lf/li) – 1(lf/li) = e + 1ln(lf/li) = ln(e + 1)ε = ln(e + 1)

Prof. Ramesh Singh

3-D Stress State in Cartesian Plane

There are two subscripts in any stress component:

Direction of normal vector of the plane (first subscript)

σ x x and τ x y

Courtesy: http://www.jwave.vt.edu/crcd/kriz/lectures/Anisotropy.html

Direction of action (second subscript)

Prof. Ramesh Singh

Stress Tensor

zzzyzx

yzyyyx

xzxyxx

στττστττσ

333231

232221

131211

σσσσσσσσσ

Mechanics Notation Expanded Tensorial Notation

It can also be written as σij in condensed form and is a second order tensor, where i and j are indicesThe number of components to specify a tensor

• 3n , where n is the order of matrix

Prof. Ramesh Singh

Symmetry in Shear Stress • Ideally, there has to be nine components• For small faces with no change in stresses• Moment about z-axis,

• Tensor becomes symmetric and have only six components, 3 normal and 3 shear stresses

xzzx

zyyz

yxxy

yxxy

Similarly

yzxxzy

ττ

ττ

ττ

ττ

=

=

=

∆∆∆=∆∆∆

,

)()(

Prof. Ramesh Singh

Plane Stress

00

===

zxyx

zz

ττσ

Only three components of stress

http://www.shodor.org/~jingersoll/weave4/tutorial/Figures/sc.jpg

Prof. Ramesh Singh

2-D Stresses at an Angle

Prof. Ramesh Singh

Shear Stress on Inclined Plane

Prof. Ramesh Singh

Transformation Equations

Use θ and 90 +θ for x and y directions, respectively

Prof. Ramesh Singh

Principal StressesFor principal stresses τxy=0

= 0

22

22

2

22cos

2

2sin

2

2tan

xyyx

yx

p

xyyx

xyp

yx

xyp

τσσ

σσ

θ

τσσ

τθ

σστ

θ

+

−

−

±=

+

−±=

−=

Prof. Ramesh Singh

Principal Stresses• Substituting values of sin2θp and cos2θp in

transformation equation we get principal stresses

2

2tan

plane, stress principal maximum The

22tan

02cos2sin2

stress,shear maximum of Angle

222

2

2,1

yx

xyp

xy

yx

s

xyxyxy

s

xyyxyx

dd

dd

σστ

θ

τ

σσ

θ

θτϑσσ

ϑϑτ

θ

τσσσσ

σ

−=

−

−=

=

+

−=

′

+

−±

+=

Prof. Ramesh Singh

Maximum Shear Stresses

22

max 2

in of valuengsubstituti

45

9022

2cot2tan

xyyx

xys

ps

ps

ps

τσσ

τ

τθ

θθ

θθ

θθ

+

−=

′

±=

±=

−=

Prof. Ramesh Singh

Equations of Mohr’s Circle

Prof. Ramesh Singh

Mohr’s Circle

Prof. Ramesh Singh

Prof. Ramesh Singh

Prof. Ramesh Singh