Chap' 7

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Chap. 7 Stress and Strain
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7.1 Basic Loading Conditions

• Deformation
• relative displacement of any two points
• The extent of the shape change may depend on the
  magnitude, direction, and duration of the applied
  forces, material properties of the body and
  environmental conditions such as heat and
  humidity.

Bar AB tension by F1 bending by F2 torsion by
F3
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7.2 Uniaxial tension test

• uniaxial or simple tension test
• relative displacement of any two points

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7.3 Load and Elongation Diagrams
For the same material
A is stiffer than B.
?,?,? lie on a single curve.
A
?,?,?
B
Stress-strain curve
Stress-strain curve
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7.4 Simple Stress

• Stress force per unit area
• ML-1T-2

shear stress (tangential)
normal stress (perpendicular)
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7.5 Simple Strain

• normal strain

• shear strain

d
C
B
C
B
A
D
- tensile compressive
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7.5 Simple Strain

• Effects of loading

tensile
compressive
shear
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7.6 Stress-Strain Diagrams

• stress-strain diagram for axial loading

U
Y
P
E
R
Y
Offset method
P
0.002
P proportionality limit E elastic limit Y
yield point, U highest point, R rupture or
failure point

• After yield point, considerable elongation
  occurs.
• Offset method
• In engineering application, yield point is 0.2
  of strain.
• Beyond the ultimate strength, necking occurs.

yield strength
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7.7 Elastic Deformations

• linearly elastic material

• nonlinearly elastic material

Y
P

• Complete recovery after the load is removed.
• Youngs modulus, elastic modulus or stiffness, E
• Stiffer material vs. more compliant material

• soft tissue,bone,
• not a single Youngs modulus
• elastic region at low stress levels

• Hookes law

Thus, we have
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7.7 Elastic Deformations

• linearly elastic material

• shear modulus or modulus of rigidity, G
• larger G, more rigid

G

• relationship between E and G constitutive
  equations

1
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7.8 Hookes Law

• spring (elastic materials)

• Analogy

k
1

• k spring constant, or stiffness of the spring

• potential energy

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7.9 Plastic Deformations

• Plasticity permanent deformation
• Elasticity

U
R
Y
P
elastic strain (recoverable)
plastic strain or permanent strain
(unrecoverable)
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7.10 Necking

• Necking
• increased rate of deformation beyond the
  ultimate strength
• conventional stress
• true or actual stress
• considering actual area due to the rapid
  elongation
• increasing strain with decreasing stress

Necking
Conventional and actual stress-strain curve
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7.11 Work and Strain Energy

• Work done

Y

• elastic strain energy
• Internal work done elastic strain energy
• plastic strain energy dissipated as heat while
  deforming the body

Internal work done and elastic strain energy per
unit volume
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7.12 Strain Hardening

• Strain hardening

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7.13 Hysteresis Loop

• Hysteresis

Hysteresis loop O?A?B?C?O
Area total strain energy dissipated as heat
to deform the body in tension and
compression
A
tensile
unloading
O
B
unloading
compressive
C
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7.14 Properties Based on Stress-Strain Diagram

• Stiffness

Material 1 is stiffer than material 2.
Slope
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Ductile material
Brittle material
engineering strain
true strain
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7.15 Idealized Models of Material Behavior
Rigid
Perfectly plastic (non-strain- hardening)
Elastic-perfectly plastic (non-strain-hardening)
Elastic-plastic (strain-hardening)
Rigid-plastic (strain- hardening)
Linearly elastic
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7.16 Mechanical Properties of Materials
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7.17 Example Problems

• Example 7.1 tension test of a circular
  cylindrical rod

Tensile strain and average tensile stress?
Deformation is
Thus, tensile strain is
Tensile stress is
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7.17 Example Problems

• Example 7.2 Uniaxial tension test

aluminum
steel
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7.17 Example Problems

• Example 7.3 Uniaxial tension test

A
2
B
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7.17 Example Problems

• Example 7.4 bone fixation device

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7.17 Example Problems

• Example 7.5 simple tension test of human
  cortical bone

O(0,0), A(85, 0.005), B(114, 0.010), C(128, 0.026)
O-A linearly elastic
B-C linearly plastic
For
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7.17 Example Problems

• Example 7.6

Static analysis
h
A
B
C
Geometric compatibility (deflection)
for small
Stress-strain relationship
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7.17 Example Problems

• Example 7.6 (continued)

h
Since
,
A
B
C
Thus, stress and strain for each steel bar