Jiangyu Li, University of Washington
Lecture 9Strain and Elasticity
Textbook: Mechanical Behavior of Materials Sec. 6.6, 5.3, 5.4
Jiangyu LiUniversity of Washington
Mechanics of Materials Lab
Strain:Fundamental Definitions
• "Strain" is a measure of the deformation of a solid body
• There are two "types" of strain; normal strain (ε) and shearstrain (γ)
ε = (change in length)(original length)
; units = in / in, m / m, etc
γ = (change in angle ) ; units = radians 28
Prof. M. E. TuttleUniversity of Washington
Strain Within a Plane
+x
+y
a
bc
lx
ly
∠ abc = π/2 radians
Original Shape
+x
+ya
b
c
lx + ∆lx
ly + ∆ly
∠ abc < π/2 radians
Deformed Shapeεxx = ∆lx/lx
εyy = ∆ly/ly
γxy = ∆(∠ abc)
Prof. M. E. TuttleUniversity of Washington
29
Strain Sign Convention• A positive (tensile) normal strain is associated with an
increase in length
• A negative (compressive) normal strain is associated with adecrease in length
• A shear strain is positive if the angle between two positivefaces (or two negative faces) decreases
32
+x
+y
+x
+y
All Strains Positive ε Positiveε and γ Negative
xx
yy xy
3-Dimensional Strain States
• In the most general case, six components of strain exist "ata point": εxx, εyy, εzz, γxy, γxz, γyz
• Strain is a 2nd-order tensor; the numerical values of theindividual strain components which define the "state ofstrain" depend on the coordinate system used
• This presentation will primarily involve strains which existwithin a single plane
29
Prof. M. E. TuttleUniversity of Washington
Strain Within a Plane
• We are often interested in the strains induced within asingle plane; specifically, we are interested in two distinctconditions:
• "Plane stress", in which all non-zero stresses lie withina plane. The plane stress condition induces fourstrain components: εxx, εyy, εzz, and γxy
• "Plane strain", in which all non-zero straincomponents lies within a plane. By definition then,the plane strain condition involves three straincomponents (only): εxx, εyy, and γxy 30
Prof. M. E. TuttleUniversity of Washington
Strain Transformations
• Given strain components in the x-y coordinate system (εxx, εyy,γxy), what are the corresponding strain components in the x'-y'coordinate system?
lx + ∆lx
+x
+y
ly + ∆ly
εxx = ∆lx/lx
εyy = ∆ly/ly
γxy = ∆(∠ abc)
+x'+y'
εx'x' = ?εy'y' = ?γx'y' = ?
θ?
35
Prof. M. E. TuttleUniversity of Washington
Strain TransformationEquations
• Based strictly on geometry, it can be shown:
ε x' x ' =ε xx + ε yy
2+
ε xx − ε yy
2cos2θ +
γ xy
2sin2θ
ε y' y' =ε xx + ε yy
2−
ε xx − ε yy
2cos2θ −
γ xy
2sin2θ
γ x'y'
2= −
ε xx − ε yy
2sin 2θ +
γ xy
2cos2θ
36
Prof. M. E. TuttleUniversity of Washington
"Constitutive Models"
• The structural engineer is typically interested in the state ofstress induced in a structure during service
• The state of stress cannot be measured directly...
• The state of strain can be measured directly...
• Hence, we must develop a relation between stress andstrain...this relationship is called a "constitutive model," andthe most common is "Hooke's Law"
39
Prof. M. E. TuttleUniversity of Washington
3-D Form of Hooke's Law:Apply Principle of Superposition
• The total strain εxx caused by all stresses appliedsimultaneously is determined by "adding up" the strain εxxcaused by each individual stress component
ε xx = σxxE
+
−νσ yy
E
+
−νσ zzE
+ 0[ ] + 0[ ] + 0[ ]
or
ε xx = 1E
σ xx − νσ yy − νσ zz[ ]51
Prof. M. E. TuttleUniversity of Washington
Hooke's Law:3-Dimensional Stress States
• Following this procedure for all six strain components:
ε xx = 1E
σ xx − νσ yy − νσ zz[ ] γ yz =2(1+ ν)τ yz
E
ε yy = 1E
−νσ xx + σ yy − νσ zz[ ] γ zx = 2(1+ ν)τ zxE
ε zz = 1E
−νσ xx − νσ yy + σzz[ ] γ xy =2(1+ ν)τ xy
E
52
Prof. M. E. TuttleUniversity of Washington
Hooke's Law:3-Dimensional Stress States
54
Prof. M. E. TuttleUniversity of Washington
σ xx = E(1+ ν)(1− 2ν)
(1− v)εxx + νεyy + νεzz[ ] τyz =Eγ yz
2(1+ ν)
σyy = E(1+ ν)(1− 2ν )
vε xx + (1− v)εyy + νε zz[ ] τ zx = Eγ zx2(1 + ν)
σ zz = E(1+ ν )(1− 2ν)
vεxx + νε yy + (1− v)ε zz[ ] τxy =Eγ xy
2(1 + ν )
Hooke's Law:Plane Stress
• For thin components (e.g., web or flange of an I-beam,automobile door panel, airplane "skin", etc) the "in-plane"stresses are much higher than "out-of-plane" stresses:(σxx,σyy,τxy) >> (σzz,τxz,τyz)
• It is convenient to assume the out-of-plane stresses equalzero...this is called a state of "plane stress"
+x
+y
+z
57
Prof. M. E. TuttleUniversity of Washington
Hooke's Law:Plane Stress
σxx
τxy
σyy
εxx = 1E
(σxx − νσyy )
εyy = 1E
(σyy − νσxx )
γ xy = τxy
G=
2(1+ ν)τxy
E
εzz = −νE
(σxx + σyy )
γ xz = γ yz = 0 58
Prof. M. E. TuttleUniversity of Washington
Hooke's Law:Plane Stress
σxx
τxy
σyy
σxx = E(1- ν2 )
(εxx + νεyy )
σyy = E(1- ν2 )
(εyy + νεxx )
τxy = Gγ xy = E2(1+ ν)
γ xy
σzz = τ xz = τ yz = 059
Prof. M. E. TuttleUniversity of Washington
Hooke's Law:Plane Strain
• For thick or very long components (e.g., thick-walled pressurevessels, buried pipe, etc) the "in-plane" strains are muchhigher than "out-of-plane" strains: (εxx,εyy,γxy) >> (εzz,γxz,γyz)
• It is convenient to assume the out-of-plane strains equalzero...this is called a state of "plane strain"
+x
+y
+z
62
Prof. M. E. TuttleUniversity of Washington
Hooke's Law:Plane Strain
σ xx
τxy
σyy
63
Prof. M. E. TuttleUniversity of Washington
σ xx = E(1 + ν)(1− 2ν)
(1− ν )ε xx + νε yy[ ]σyy = E
(1+ ν)(1− 2ν)νε xx + (1− ν)ε yy[ ]
σzz = νE(1+ ν)(1− 2ν)
ε xx + ε yy[ ] = ν σ xx + σyy[ ]
τ xy = Gγ xy =Eγ xy
2(1+ ν )τ xz = τyz = 0
Hooke's LawPlane Strain
66
σ xx
τxy
σyy
ε xx = 1 − ν2
Eσxx − ν
1− ν
σ yy
ε yy =1− ν 2
Eσ yy −
ν1− ν
σ xx
γ xy = 2(1+ ν)E
τxy
ε zz = γ xz = γ yz = 0
....and: σzz = ν σ xx + σ yy[ ]
Hooke's Law:Uniaxial Stress
• Truss members are designed to carry axial loads only (i.e.,uniaxial stress)
• In this case we are interested in the axial strain only (eventhough transverse strains are also induced)
67
Prof. M. E. TuttleUniversity of Washington
Hooke's Law:Uniaxial Stress
• For: σyy = σzz = τyz = τxz = τzy = 0,
"Hooke's Law" becomes:
• "Shorthand" notation:
σ xx = Eε xx
σ{ } = D[ ] ε{ } where D[ ] = E[ ]68
Prof. M. E. TuttleUniversity of Washington
Jiangyu Li, University of Washington
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Jiangyu Li, University of Washington
Assignment
• Textbook: Mechanical behavior of materials
HW 5.11, 5.12, 5.37