고체역학
(Mechanics of Materials)
Prof. BAEK Jong Dae
School of Automotive Engineering
Yeungnam University, Republic of Korea
1
Normal stress and strain
Normal Stress
•
Symbol: σ (sigma)
•
Equation:
σ=
A: Cross-sectional area
P: Axial force
Units: force per unit of area
•
•
•
2
P
A
Imperial unit (IP unit):

Pounds per square inch (psi)

Kips per square inch (ksi)
International system unit (SI unit):

Newton per square meter (N/m2)

Pascal (Pa)
Normal stress and strain
Normal Strain
• Symbol: ε (epsilon)
• Equation:
ε=
δ
L
• Units: none (dimensionless quantity)
• Usually very small ( For structural steel, 10-5~10-6 range)
Requirements for Normal Stress and Strain
3
•
The bar is prismatic.
•
The loads act through the centroids of the cross section.
•
The material is homogeneous.
Example
4
Example
* Weight density of steel: 77 kN/m3
6
Tensile test process
8
Stress-Strain Diagrams
• The stress is plotted against strain to achieve a stress-strain diagram.
•
Proportional Limit: The point in which the stress and strain are no longer proportional to one another.
•
Modulus of Elasticity (E): The slope of the straight line the plotting of stress and strain produces within the
linear region of the curve.
Yield Stress (σy): The corresponding magnitude of stress at which point the material “yields” under the
•
loading and allows for rapid increase of stress.
Ultimate Stress (σu): The corresponding magnitude of stress at which point the material begins to “neck,”
•
thereby allowing a decrease in cross-sectional area.
9
Ductility
• A measure of the amount of permanent strain the material undergoes before failure
occurs.
• Can be quantified by “percent elongation” and “percent reduction in area”.
• Percent Elongation: % Elongation =
L1 − Lo
(100)
Lo
• Percent Reduction in Area: % Reduction =
10
Ao − A1
(100)
Ao
Fracture shape
Ductile
Cup-and-cone fracture of aluminum bar
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Brittle
Fracture of Mild steel
Offset Method
•
Used to determine an arbitrary yield stress.
•
A straight line is drawn on the stress-strain diagram parallel to the initial linear part of the curve at an
offset of 0.002 (or 0.2%) strain.
-The intersection of the offset line and the stress-strain curve (point A) defines the “offset yield stress”.
12
Linear Elasticity and Hooke’s Law
• Linear Elastic: When a material exhibits a linear relationship between stress and
strain. Typically, engineers design structures and machines to function within this
region to avoid permanent deformations due to yielding.
Hooke’s Law
• Expression for the linear relationship between stress and strain.
•
Young’s Modulus
• Equation: σ = Eε
• Limitations– Relates only to the longitudinal stresses and strains in simple tension or
compression of a bar (i.e. uniaxial stresses).
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Poisson’s Ratio
• Lateral contraction accompanies axial elongation and Poisson’s Ratio relates these two strains to one
another.
• The ratio of these two strains = Poisson’s ratio
• Symbol: ν (nu)
• Equation:
lateral strain
ε'
ν=
−
=
−
ε
axial strain
• Units: none (dimensionless quantity)
ε y = ε z = −νε x
• Limitations:
14
•
The material must be homogeneous.
•
Elastic properties must be the same in all directions perpendicular to the longitudinal axis.