introduction to mechanics of materialsocw.snu.ac.kr/sites/default/files/note/chapter 3_2.pdf ·...
TRANSCRIPT
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445.204
Introduction to Mechanics of Materials
(재료역학개론)
Myoung-Gyu Lee, 이명규
Tel. 880-1711; Email: [email protected]
TA: Seong-Hwan Choi, 최성환
Lab: Materials Mechanics lab.
Office: 30-521
Email: [email protected]
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Chapter 3
3.1. Concept of Strain
3.2. Material (Mechanical) Properties
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Outline
• Deformation• Definition of Strain• Components of Strain• Stress–Strain Diagrams• True Stress and True Strain• Elastic versus Plastic
Behavior• Hooke’s Law
• Poisson’s Ratio• Generalized Hooke’s Law• Strain Energy• (optional) Impact Strength• (optional) Fatigue• Permanent Deformation• General Properties of
Materials
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In this chapter ...• Concept of deformation and strain• Small deformation theory
- Hooke´s law- Principle of superposition
• Material properties in 1D - Tension/compression or axial strains- Shear strains
• Hooke‘s law• Various mechanical properties
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3.1. Concept of Strain
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Deformation• Rigid body displacement
- Translation- Rotation- Combination of transloation and rotation
• Elastic deformation- Occurs due to a change in shape by small enough external load- No permanent shape change if the load is removed - By elastic lattice distortions (in microscale)- Example: axial, bending or torsion (angle of twist), general...
• Plastic deformation- Occurs due to a change in shape by large enough external load- Permanent shape change if the load is removed - Results in elastic lattice distortions + shear of lattice by
microscopic defects such as dislocations, twin, phase transformation etc...
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Deformation - elastic
Elastic means reversible!
2. Small load
F
d
bonds stretch
1. Initial 3. Unload
return to initial
F
d
Linear-elastic
Non-Linear-elastic
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Deformation - plastic
Plastic means permanent!
1. Initial 2. Large load 3. Unload
planesstill sheared
F
δelastic + plastic
bonds stretch & planes shear
δplastic
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Strain• Strain (ε or γ)
• Change in the relative position of two points within a body
- Axial strain, ε (elongation (+) or contraction (-))- Bending strain, ε (can be zero, positive or negative)- Torsional strain, γ (can be positive or negative)- Combination of the above
axial Bending Torsion
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Displacements and strains in a body.
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Principle of superposition
• Small deformation theory- Valid in the linear elasticity - Hooke‘s law or linear relationship between stress and strain
diagram
• Small displacement (or deformation) assumption and linear behavior of materials lead to the principle of superposition.
• This rule is valid whenever the following conditions are satisfied:
- The quantity (displacement or stress) to be determined is directly proportional to the loads that produce it (linear elasticity)
- The loading does not significantly change the size and shape of the member
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Normal Strain• Strain in the axial direction
ε = Change in length (δ)Original length (L)
ε= δ / L
Normal strain is positive when there is elongation, otherwise, negative
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Example
• A stepped round bar is subjected to a load P that produces axial deformation in each portion of the bar.
Q: Calculate the strain in portions AB and BC and the total strain in the bar.
P
1.5 m0.5 m
0.45 cm1.3 cm
A B C
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εAB = 0.0045 m / 0.5 m = 0.009εBC = 0.013 m / 1.5 m = 0.0087
Using the superposition principle:Total strain, εAC = εAB + εBC
= 0.009 + 0.0087= 0.0177 (or 1.77%)
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On the otherhand, if we use the definition of strain (change in length / original length) as
applied to the entire bar,
εAC = (0.0045 m + 0.013 m) / (1.5 m + 0.5 m) = 0.00875 (or 0.875%)
Why is this result different than the previous one?
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Example 3.1.• See Figure 3.4 (Next slide)
A pin-connected frame ABCD consists of three bars and a wire. After a horizontal load P is applied at joint B, joint C moves 10 mm to the right.
What is the normal strain in the wire?
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Figure 3.4: Displacements of a plane truss
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Shear Strain
• Shearing strain is positive if the right angle between the reference lines decrease (see Figure 3.3)
γnt = π/2 – θ′≅ tan (π/2 – θ′)
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Example 3.2
• A metallic, rectangular plate of length L and width L / 2. When the plate is subjected to stresses acting along the edge faces, it distorts into a parallelogram as illustrated in figure.
Find shearing strain between the edges AB and AD.
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Example 3.3
• A thin, triangular plate ABC is uniformly deformed into a shape ABC′, as shown by the dashed lines. (Figure 3.6)
• Find:(a) The normal strain along the centerline OC.(b) The normal strain along the edge AC.(c) The shear strain between the edges AC and BC.
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COMPONENTS OF STRAIN
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EXAMPLE 3.5
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3.2. Material (Mechanical) Properties
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Engineering Materials
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Mechanical properties of materials
• Mechanical properties of engineering materials loaded in tension and compression are determined using a standard testing methods
• Example: ASTM standards, JIS, KS, DIN etc..
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Ductile vs. brittle materials
• Ductile materials can undergo large inelastic strains prior to fracture.
• For example, structural steel and many alloys of other metals, and nylon, are characterized by their ability to yield at normal temperatures.
• Percentage elongation is 5 or more.
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Ductile vs. brittle materials
• Brittle materials (for example, cast iron or concrete) exhibits little deformation before rupture and, as a result, fails suddenly without visible warning.
• Percentage elongation is less than around 5.
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Ductile vs. brittle materials
Ductile Brittle
Ductile
Brittle
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Homogeneous material
• A homogeneous solid displays identical properties throughout.
• For example, if a bar made of the same material is divided in to several small pieces (of any shape and size) and the density (weight or mass / volume) of each piece is determined to be same then the material is homegeneous.
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• If the properties of a material are identical in all directions at a point, the material is said to be isotropic
• A nonisotropic, or anisotropic, material displays direction-dependent properties
• An orthotropic material is a special case of anisotropic material in which the material properties differ in two mutually perpendicular directions
Isotropic vs. anisotropic
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• An orthotropic material has at least 2 orthogonal planes of symmetry, where material properties are independent of direction within each plane.
• In contrast, a material without any planes of symmetry is fully anisotropic.
• A material with an infinite number of symmetry planes (i.e. every plane is a plane of symmetry) is isotropic.
Orthotropic, anisotropic, isotropic
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Orthotropic, anisotropic, isotropic
Isotropic
Transversely isotropic Orthotropic
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Stress strain diagram
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Strain gage or Extensometer
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Digital Image Correlation
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Digital Image Correlation
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Stress-strain diagram**
Ductile materials
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Typical stress-strain diagram
Ultimate tensile strength
At failure
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Conventional stress-strain diagram
Measurement of strain or alternative method
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How to define yield stress – offset method
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Stress-strain diagram: brittle materials
Eexhibits no “Necking“
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What happens in compression?
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Mechanical properties in shear
• The properties of a material can also be determined from direct-shear tests ortorsion tests (discussed in Chapter 5).
• Shear stress–shear strain diagrams may be obtained from the results of torque (T) versus shear strain (γ) that are analogous to the tension test results.
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True stress-strain relationship **
True strain
True stress
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Elastic vs. Plastic Behavior
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Hooke’s Law**• Simple way to correlate between stress-strain
for 1D linear isotropic elastic materials
- Hookean materials: materials that obey the Hooke’s law- Hookean materials = Linear elastic materials
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Young‘s modulus (E)
• Young‘s modulus or modulus of elasticity is denoted by E.
• The value of E for steel is roughly 3 times that for aluminum.
Esteel ≈ 30 x 106 psi (English units)Esteel ≈ 210 GPa (SI units)
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Poisson’s ration
ν = - [lateral strain, εw / axial strain, εx]
This expression is valid for uniaxial loading
PP w w′
L
L′
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• Typical values of ν for steel ranges from about 0.25 to 0.3, while for aluminum materials it is of the order of 0.33
εL
ε
-ν
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Shear modulus (G)• Shear modulus or modulus of rigidity is denoted by G.
The value of G for steel is a little over 1/3rd the value of E for steel.
Gsteel ≈ 11.5 x 106 psi (English units)Gsteel ≈ 77 GPa (SI units) τ
Gγ
( )v
EG
+=
12
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Volumetric strain under tension
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Volumetric strain under tension
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Generalized Hooke’s law
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Three dimensional stress
Generalized Hooke’s law
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Unit volume change
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Unit volume change
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Strain energy uniaxial stress
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Strain energy (shear stress)