introduction to mechanics-i
TRANSCRIPT
![Page 1: Introduction to Mechanics-I](https://reader031.vdocument.in/reader031/viewer/2022021419/58a17df41a28abf0458b60d0/html5/thumbnails/1.jpg)
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
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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
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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
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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
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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
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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
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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
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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
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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)
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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)
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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
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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
=
=
=
∆∆∆=∆∆∆
,
)()(
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Prof. Ramesh Singh
Plane Stress
00
===
zxyx
zz
ττσ
Only three components of stress
http://www.shodor.org/~jingersoll/weave4/tutorial/Figures/sc.jpg
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Prof. Ramesh Singh
2-D Stresses at an Angle
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Prof. Ramesh Singh
Shear Stress on Inclined Plane
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Prof. Ramesh Singh
Transformation Equations
Use θ and 90 +θ for x and y directions, respectively
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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
+
−
−
±=
+
−±=
−=
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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
−=
−
−=
=
+
−=
′
+
−±
+=
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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
+
−=
′
±=
±=
−=
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Prof. Ramesh Singh
Equations of Mohr’s Circle
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Prof. Ramesh Singh
Mohr’s Circle
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Prof. Ramesh Singh
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Prof. Ramesh Singh
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Prof. Ramesh Singh