The_Cambridge_Handbook_of_Physics_Formulas

3.8 Elasticity
81
Isotropic elastic solids
Lamé coefficients
µ = E
2(1+σ)
Eσ
λ =
(1+σ)(1−2σ)
(3.239)
(3.240)
µ,λ
E
σ
Lamé coefficients
Young modulus
Poisson ratio
Longitudinal
modulus a M l = E(1−σ)
(1+σ)(1−2σ) = λ+2µ (3.241) M l longitudinal elastic
modulus
e ii = 1 E [τ ii −σ(τ jj +τ kk )] (3.242)
Diagonalised
[
equations b τ ii = M l e ii +
σ ]
1−σ (e jj +e kk ) (3.243)
t =2µe+λ1tr(e) (3.244)
Bulk modulus
(compression
modulus)
E
K =
3(1−2σ) = λ+ 2 3 µ (3.245)
1
= − 1 ∂V
∣ (3.246)
K T V ∂p T
−p = Ke v (3.247)
e ii strain in i direction
τ ii stress in i direction
e strain tensor
t stress tensor
1 unit matrix
tr(·) trace
K
K T
V
p
T
bulk modulus
isothermal bulk
modulus
volume
pressure
temperature
3
Shear modulus
(rigidity modulus)
Young modulus E = 9µK
µ+3K
µ = E
2(1+σ)
(3.248)
τ T = µθ sh (3.249)
3K −2µ
Poisson ratio σ =
2(3K +µ)
a In an extended medium.
b Axes aligned along eigenvectors of the stress and strain tensors.
(3.250)
(3.251)
e v volume strain
µ shear modulus
τ T transverse stress
θ sh shear strain
τ T
θ sh
Torsion
Torsional rigidity
(for a
homogeneous
rod)
Thin circular
cylinder
Thick circular
cylinder
Arbitrary
thin-walled tube
G = C φ l
(3.252)
C =2πa 3 µt (3.253)
C = 1 2 µπ(a4 2 −a 4 1) (3.254)
C = 4A2 µt
P
(3.255)
G twisting couple
C torsional rigidity
l rod length
φ twist angle in
length l
a radius
t wall thickness
µ shear modulus
a 1
a 2
A
P
inner radius
outer radius
cross-sectional
area
perimeter
l
A
G
a
φ
t
Long flat ribbon C = 1 3 µwt3 (3.256)
w
cross-sectional
width
w
t