Theory of elasticity and plasticity (Equations sheet part 01)

1. Theory of elasticity and plasticity
Equations sheet, Part 1
Theory of stress
Cauchy stress formula
σT
· n = t, or


σxx σyx σzx
σxy σyy σzy
σxz σyz σzz




nx
ny
nz

 =


tx
ty
tz


where tx, ty and tz are the traction components; nx = cos(n, x), ny = cos(n, y) and nz = cos(n, z) are
the direction cosines of the outward normal vector.
Normal and shear components of the stress (traction) vector
tn = t · ˆn = tini, ts = |t|2 − t2
n
Hint: Unite normal to the plane
ˆn =
φ
| φ|
Principal stresses
(σ − λI)n = 0
Characteristic equation
λ3
− I1λ2
+ I2λ − I3 = 0
Stress invariants
I1 = σxx + σyy + σzz, I2 =
σxx σxy
σyx σyy
+
σxx σxz
σzx σzz
+
σyy σyz
σzy σzz
, I3 =
σxx σxy σxz
σyx σyy σyz
σzx σzy σzz
Principal directions
(σ − λiI)ni = 0
1

2. Equilibrium equations
∂σxx
∂x
+
∂σyx
∂y
+
∂σzx
∂z
+ fx = 0
∂σxy
∂x
+
∂σyy
∂y
+
∂σzy
∂z
+ fy = 0
∂σxz
∂x
+
∂σyz
∂y
+
∂σzz
∂z
+ fz = 0
Displacements and strains
Displacement gradient tensor
u =



∂u
∂x
∂u
∂y
∂u
∂z
∂v
∂x
∂v
∂y
∂v
∂z
∂w
∂x
∂w
∂y
∂w
∂z



where = e1
∂
∂x + e2
∂
∂y + e3
∂
∂z and u is the displacement vector
Strain tensor
ε =
1
2
u + ( u)
T
Strain components
εxx =
∂u
∂x
, εyy =
∂v
∂y
, εzz =
∂w
∂z
εxy =
1
2
∂u
∂y
+
∂v
∂x
, εxz =
1
2
∂u
∂z
+
∂w
∂x
, εyz =
1
2
∂v
∂z
+
∂w
∂y
Saint-Venant compatibility equations
εxx,yy + εyy,xx = 2εxy,xy, εyy,zz + εzz,yy = 2εyz,yz, εzz,xx + εxx,zz = 2εzx,zx
εxy,xz + εxz,xy = εxx,yz + εyz,xx, εyz,yx + εyx,yz = εyy,zx + εzx,yy, εzx,zy + εzy,zx = εzz,xy + εxy,zz
2

3. Constitutive equations- Hooke’s law
Generalized Hooke’s law- isotropic material








εxx
εyy
εzz
2εxy
2εyz
2εzx








=
1
E








1 −ν −ν 0 0 0
1 −ν 0 0 0
1 0 0 0
2(1 − ν) 0 0
sym. 2(1 − ν) 0
2(1 − ν)
















σxx
σyy
σzz
σxy
σyz
σzx
















σxx
σyy
σzz
σxy
σyz
σzx








=
E
(1 + ν)(1 − 2ν)








1 − ν ν ν 0 0 0
1 − ν ν 0 0 0
1 − ν 0 0 0
1−2ν
2 0 0
sym. 1−2ν
2 0
1−2ν
2
















εxx
εyy
εzz
2εxy
2εyz
2εzx








• Representation by the Lamé coefficients








σxx
σyy
σzz
σxy
σyz
σzx








=








2µ + λ λ λ 0 0 0
2µ + λ λ 0 0 0
2µ + λ 0 0 0
2µ 0 0
sym. 2µ 0
2µ
















εxx
εyy
εzz
εxy
εyz
εzx








where µ = G = E
2(1+ν) and λ = νE
(1+ν)(1−2ν) are Lamé constants.
Boundary conditions
Stress and displacement BCs (see Cauchy stress formula)
σT
· n = ¯t, u = ¯u
Energy principles
Total potential energy functional
ΠT P E =
1
2 V
σ : εdV −
V
f · udV −
S
t · udS
Principle of minimum potential energy
δΠT P E(u) = 0
3

4. Classical beam and bar theories
TPE functionals for bars and beams
ΠT P E(u) =
0
EA
2
du
dx
2
dx −
0
qxudx
ΠT P E(w) =
0
EI
2
d2
w
dx2
2
dx −
0
qwdx
BCs for beams
• Fixed end- w = 0 and dw
dx = 0
• Free end- M = −EI d2
w
dx2 = M0 and Q = −EI d3
w
dx3 = Q0
• Pinned end- w = 0 and M = −EI d2
w
dx2 = 0
Ritz method
Displacement approximation functions
u = u0 +
n
i=1
aiui, v = v0 +
n
i=1
bivi, w = w0 +
n
i=1
ciwi
where the terms of u0, v0 and w0 are chosen to satisfy any non-homogeneous displacement BCs and ui,
vi and wi satisfy the corresponding homogeneous displacement BCs.
The TPE functional
ΠT P E = ΠT P E(ai, bi, ci)
Minimum conditions
∂ΠT P E
∂ai
= 0,
∂ΠT P E
∂bi
= 0,
∂ΠT P E
∂ci
= 0
4