A review of Engineering Mechanics of Materials

A review of Mechanics

1. 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

2. 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

3. Prof. Ramesh Singh
#1 -Match
a)Force equilibrium
b)Compatibility of
deformation
c)Constitutive
equation
1)δ
T= δ
1+ δ
2
2)SF=0;SM=0
3)s= E e

4. Prof. Ramesh Singh
Steps of a Mechanics Problem
•O Read and understand problem
•1Free body diagram
•2Equilibrium of forces
–e.g. ΣF=0, SM=0.
•3Compatibility of deformations
–e.g. δ
T= δ
1+ δ
2
•4Constitutive equations
–e.g. s= E e
•5Solve
F F
A

5. Prof. Ramesh Singh
Key Concepts
Load (Force), P, acting over area, A,
gives rise to stress, s.
Engineering stress: s= P/A
o
(A
o= original area)
True stress: s
t= P/A
(A = actual area)
P
P
A

6. Prof. Ramesh Singh
Deformation
•Quantified by strain, eor e
•Engineering strain: e = (l
f-l
i)/l
i
•True Strain: e= ln(l
f/l
i)
•Shear strain: g= a/b
l
il
f
a
b

7. Prof. Ramesh Singh
True and Engineering strains
e = (l
f-l
i)/l
i
e = (l
f/l
i) –1
(l
f/l
i) = e + 1
ln(l
f/l
i) = ln(e + 1)
e = ln(e + 1)

8. 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)

9. 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 ass
ijin condensed form and is a second
order tensor, where i and j are indices
The number of components to specify a tensor
•3
n
, where n is the order of matrix

10. 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 stressesxzzx
zyyz
yxxy
yxxy
Similarly
yzxxzy
tt
tt
tt
tt
=
=
=
=
,
)()(

11. Prof. Ramesh Singh
Plane Stress
Only three components of stress
http://www.shodor.org/~jingersoll/weave4/tutorial/Figures/sc.jpg
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��=0
??????
��=??????
��=0

12. Prof. Ramesh Singh
Plane Strain (1)
One pair of faces has NOstrain
–each cross-section has the same
strain
Material in a groove
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��=0
??????
��=??????
��=0

13. Prof. Ramesh Singh
2-D Stresses at an Angle

14. Prof. Ramesh Singh
Shear Stress on Inclined Plane

15. Prof. Ramesh Singh
Transformation Equations
Use qand 90 +q for x and y directions, respectively

16. Prof. Ramesh Singh
Principal Stresses
For principal stresses t
xy=0
= 02
2
2
2
2
2
2cos
2
2sin
2
2tan
xy
yx
yx
p
xy
yx
xy
p
yx
xy
p
t
ss
ss
q
t
ss
t
q
ss
t
q
+







−







−
=
+







−
=







−
=

17. Prof. Ramesh Singh
Principal Stresses
•Substituting values of sin2q
pand cos2q
pin
transformation equation we get principal stresses2
2tan
plane, stress principal maximum The
2
2tan
02cos2sin
2
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
t
q
t
ss
q
qt
ss

t
q
t
ssss
s
−
=
−
−=
=








+
−
=

+







−

+
=

18. Prof. Ramesh Singh
Maximum Shear Stresses2
2
max
2
in of valuengsubstituti
45
9022
2cot2tan
xy
yx
xys
ps
ps
ps
t
ss
t
tq
qq
qq
qq
+







−
=

=
=
−=

19. Prof. Ramesh Singh
Equations of Mohr’s Circle

20. Prof. Ramesh Singh
Mohr’s Circle

21. Prof. Ramesh Singh

22. Prof. Ramesh Singh

23. Prof. Ramesh Singh