Prof. Ramesh Singh

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

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 equilibrium

b) Compatibility of

deformation

c) Constitutive

equation

1) δT = δ1 + δ2

2) SF=0; SM=0

3) s = 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, SM=0.

• 3 Compatibility of deformations

– e.g. δT = δ1 + δ2

• 4 Constitutive equations

– e.g. s = E e

• 5 Solve

F F

A

Prof. Ramesh Singh

Key Concepts

Load (Force), P, acting over area, A,

gives rise to stress, s.

Engineering stress: s = P/Ao

(Ao = original area)

True stress: st = P/A

(A = actual area)

P PA

Prof. Ramesh Singh

Deformation

• Quantified by strain, e or e

• Engineering strain: e = (lf-li)/li

• True Strain: e = ln(lf/li)

• Shear strain: g = a/b

lilf

a

b

Prof. Ramesh Singh

True and Engineering strains

e = (lf-li)/li

e = (lf/li) – 1

(lf/li) = e + 1

ln(lf/li) = ln(e + 1)

e = 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)

s x x and t 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

stt

tst

tts

333231

232221

131211

sss

sss

sss

Mechanics Notation Expanded Tensorial Notation

It can also be written as sij in condensed form and is a second

order tensor, where i and j are indices

The 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

tt

tt

tt

tt

=

=

=

=

,

)()(

Prof. Ramesh Singh

Plane Stress

Only three components of stress

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

𝜎𝑧𝑧 = 0𝜏𝑥𝑧 = 𝜏𝑦𝑧 = 0

Prof. Ramesh Singh

Plane Strain (1)

One pair of faces has NO strain

– each cross-section has the same

strain

Material in a groove

𝜀𝑧𝑧 = 0𝜏𝑥𝑧 = 𝜏𝑦𝑧 = 0

Prof. Ramesh Singh

2-D Stresses at an Angle

Prof. Ramesh Singh

Shear Stress on Inclined Plane

Prof. Ramesh Singh

Transformation Equations

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

Prof. Ramesh Singh

Principal StressesFor principal stresses txy=0

= 0

2

2

2

2

2

22cos

2

2sin

2

2tan

xy

yx

yx

p

xy

yx

xy

p

yx

xy

p

tss

ss

q

tss

tq

ss

tq

+

−

−

=

+

−=

−=

Prof. Ramesh Singh

Principal Stresses• Substituting values of sin2qp and cos2qp in

transformation equation we get principal stresses

2

2tan

plane, stress principal maximum The

22tan

02cos2sin2

stress,shear maximum of Angle

22

2

2

2,1

yx

xy

p

xy

yx

s

xy

xyxy

s

xy

yxyx

d

d

d

d

ss

tq

t

ss

q

qtss

t

q

tssss

s

−=

−

−=

=

+

−=

+

−

+=

Prof. Ramesh Singh

Maximum Shear Stresses

2

2

max2

in of valuengsubstituti

45

9022

2cot2tan

xy

yx

xys

ps

ps

ps

tss

t

tq

qq

qq

qq

+

−=

=

=

−=

Prof. Ramesh Singh

Equations of Mohr’s Circle

Prof. Ramesh Singh

Mohr’s Circle

Prof. Ramesh Singh

Prof. Ramesh Singh

Prof. Ramesh Singh