Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

or
σx + σ y σx − σ y
σ n = + cos2θ + τxy
sin2θ
2 2
σx + σ y σx − σ y
σ t = − cos2θ − τxy
sin2θ
2 2
σx
− σ y
τ nt =− sin2θ + τxy
cos2θ
2
Principal stress magnitudes
σ
p1, p2
σx + σ y ⎛ σx − σ y⎞
= ±
τ
2 ⎝
⎜
2 ⎠
⎟ +
Orientation of principal planes
2τ
xy
tan2θ
p =
σ − σ
x
Maximum in-plane shear stress magnitude
τ
max
y
2
2
2
xy
⎛ σx
− σ y⎞
σ
2
=±
τxy
or τmax
⎝
⎜
2 ⎠
⎟ + =
σx
+ σ y
σ avg =
2
σx
− σ y
tan2θ
s =−
2τ
xy
note: θ = θ ± 45°
Absolute maximum shear stress magnitude
σmax
− σmin
τ abs max =
2
Normal stress invariance
σ + σ = σ + σ = σ + σ
x y n t p1 p2
Plane strain transformations
Strain in arbitrary directions
2 2
ε = ε cos θ + ε sin θ + γ sinθcosθ
or
n x y xy
2 2
ε = ε sin θ + ε cos θ − γ sinθcosθ
t x y xy
2 2
γ =−2( ε − ε )sinθcos θ + γ (cos θ − sin θ)
nt x y xy
εx + εy εx − εy γ xy
εn
= + cos2θ
+
2 2
2 sin2 θ
εx + εy εx − εy γ xy
εt
= − cos2θ
−
2 2
2 sin2 θ
γ =−( ε − ε )sin2θ + γ cos2θ
nt x y xy
Principal strain magnitudes
ε
p1, p2
εx + εy ⎛ εx − εy⎞
γ xy
= ±
2 ⎝
⎜
2 ⎠
⎟ + ⎛ ⎝ ⎜ ⎞
2 ⎠
⎟
Orientation of principal strains
γ xy
tan2θ
p =
ε − ε
x
y
s
p
2 2
− σ
p1 p2
2
Maximum in-plane shear strain
γ
max
=± 2
εx
+ εy
ε avg =
2
Normal strain invariance
2 2
⎛ εx − εy⎞
γ xy
or γ εp
ε p
⎝
⎜
2 ⎠
⎟ + ⎛ ⎝ ⎜ ⎞
2 ⎠
⎟ = −
ε + ε = ε + ε = ε + ε
x y n t p1 p2
Generalized Hooke’s law
Normal stress/normal strain relationships
1
εx = [ σx − νσ ( y + σ z )]
E
1
εy = [ σ y − νσ ( x + σ z )]
E
1
εz = [ σz − νσ ( x + σ y )]
E
E
σ x =
[(1 − νε ) x + νε ( y + ε z)]
(1 + ν)(1−
2 ν) E
σ y =
[(1 − νε ) y + νε ( x + ε z)]
(1 + ν)(1−
2 ν) E
σ z =
[(1 − νε ) z + νε ( x + ε y)]
(1 + ν)(1−
2 ν) Shear stress/shear strain relationships
1
1
1
γ = τ ; γ = τ ; γ = τ
G G G
where
E
G =
2(1 + ν)
xy xy yz yz zx zx
Volumetric strain or Dilatation
max 1 2
∆ V
1−
2 ν
e = = εx + εy + ε z = ( σ x + σ y + σ z )
V
E
Bulk modulus
E
K =
3(1−
2 ν)
Normal stress/normal strain relationships for plane stress
1
εx = ( σx −νσ
y )
E
1
εy = ( σ y −νσ
x )
E
ν
ε z =− ( σ x + σ y )
E
ν
ε z =− ( x y )
1 ν ε + ε −
or
E
σ x = ( εx + νε y )
2
1 − ν
E
σ y = ( εy + νε x )
2
1 − ν
Shear stress/shear strain relationships for plane stress
1
γ = τ or τ = Gγ
G
xy xy xy xy
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