Theory of Elasticity ????

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Theory of Elasticity????

• Chapter 5
• Elastic stress-strain relations
• (????????)

???? ???
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Content (??)

• Introduction(??)
• Mathematical Preliminaries (????)
• Stress and Equilibrium(?????)
• Displacements and Strains (?????)
• Material Behavior- Linear Elastic
  Solids(????????)
• Formulation and Solution Strategies(????????)
• Two-Dimensional Formulation (????????)
• Two-Dimensional Solution (???????????)
• Two-Dimensional Solution (??????????)
• Three-Dimensional Problems(??????)
• Bending of Thin Plates (????)
• Plastic deformation Introduction(??????)
• Introduction to Finite Element Mechod(???????)

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Material Behavior- Linear Elastic Solids

• 5.1 Material Characterization(????)
• 5.2 Linear Elastic MaterialsHookes Law(??????)
• 5.3 Physical Meaning of Elastic Moduli
• (?????????)
• Vocabularies (????)
• Homework(??)

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5.1 Material Characterization
Constitutive Equation (????)
Relations that characterize the physical
properties of materials are called constitutive
equations. (????????)
Normally defined by constitutive stress- strain
relations.(??-????)
.
Generally
Scopelinear elastic solid (????)
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5.1 Material Characterization

• Elastic solid(???)

recovers its original configuration when the
loadings are removed , does not include rate or
history effects(?????)
linear elastic solid (???,?????)
linear elasticity predictions have shown good
agreement with experimental data.
(????????,???????????)
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5.1 Material Characterization
Testing machines(??????)
Room Temperature(??)
High Temperature(??)
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5.1 Material Characterization
Cylindrical or flat stock sample(??)
Load cell and clip gage(????,?????)
Definition of stress and stress
sP/A , e?l/l
Small strain
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5.1 Material Characterization
Typical stress-strain curves(????-????)
proportional limit(????)
elastic limit(????)
yield point(???)
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5.2 Linear Elastic Materials-Hookes Law
Hookes Law(????)
Cijkl elastic moduli (????) Units Stress
(force/area). How many components?
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5.2 Linear Elastic Materials-Hookes Law
Cijkl fourth-order elasticity tensor(??????)
81 Components
symmetry of the stress and strain tensors
36 Components
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5.2 Linear Elastic Materials-Hookes Law
36 Components of Elastic Moduli (36?????)
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5.2 Linear Elastic Materials-Hookes Law
isotropic homogenous materials, Need 36
?????????, 36??)
For isotropic materials the influence of ex on
sx is the same as ey on sy (??????) The
influence of e y, ez on s x is the same.
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5.2 Linear Elastic Materials-Hookes Law
isotropic homogenous materials(??????)
only 2 independent elastic constants are needed
to describe the behavior of isotropic materials.
(2?????)
Lames constant ????
shear modulus ????
??????3?,????
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5.2 Linear Elastic Materials-Hookes Law
In individual scalar equations(??????????)
Express the strain in terms of the
stress. (???????)
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5.2 Linear Elastic Materials-Hookes Law
E the modulus of elasticity or Youngs
modulus ????
?Poissons ratio. ???
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5.2 Linear Elastic Materials-Hookes Law
Hookes Law(????)
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5.3 Physical Meaning of Elastic Moduli
E and ?
E is called the modulus of elasticity or Yangs
modulus, and ? is referred to as Poissons ratio.
(????(????)????)
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5.3 Physical Meaning of Elastic Moduli
G or µ
G or µ is called the shear modulus.(????)
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5.3 Physical Meaning of Elastic Moduli
K and sm and ?
sm is called average stress. (????)
?is called the bulk strain, related to the
change in volume of material element.(???)
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5.3 Physical Meaning of Elastic Moduli
K and sm and ?
K is called the bulk modulus of elasticity (????.)
??,P52,???
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5.3 Physical Meaning of Elastic Moduli
Pure Shear(???)
this modulus µor G is simply the slope of the
shear stress-shear strain curve
a thin-walled cylinder torsional loading
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5.3 Physical Meaning of Elastic Moduli

• Hydrostatic Compression (or Tension)(????/?)

the bulk modulus of elasticity(??????)
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5.3 Physical Meaning of Elastic Moduli

• for isotropic materials,led to the definition of
  five constants.(5???)

• Only 2 of these are needed to characterize the
  material. (2?????)

• if any two are given, the remaining three can be
  determined by using simple formulae.
  (??2???????3?)

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4.3 Physical Meaning of Elastic Moduli
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5.3 Physical Meaning of Elastic Moduli
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5.3 Physical Meaning of Elastic Moduli
The forms of Hookes law in curvilinear
coordinates?
For isotropic materials, the elasticity tensor
Cijkl is the same in all coordinate frames, and
thus the structure of Hookes law remains the
same in any orthogonal curvilinear system.
(???????????,?????????)
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Vocabulary (????)
constitutive equation ???? Independent
??? isotropic
???? Hydrostatic ??? Pure shear
??? Bulk modulus
?? ?? ductile materials
???? brittle material
???? Bulk strain
???
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Homework
???4-1 ? ? 4-3
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