Theory of elasticity and plasticityEquations sheet, Part 1

Theory of stress

Cauchy stress formula

σT · n = t, or

σxx σyx σzxσxy σyy σzyσxz σyz σzz

nxnynz

=

txtytz

where tx, ty and tz are the traction components; nx = cos(n, x), ny = cos(n, y) and nz = cos(n, z) arethe direction cosines of the outward normal vector.

Normal and shear components of the stress (traction) vector

tn = t · n = tini, ts =√|t|2 − t2n

Hint: Unite normal to the plane

n =∇φ

|∇φ|

Principal stresses

(σ − λI)n = 0

Characteristic equation

λ3 − I1λ2 + I2λ− I3 = 0

Stress invariants

I1 = σxx + σyy + σzz, I2 =

∣∣∣∣ σxx σxyσyx σyy

∣∣∣∣+

∣∣∣∣ σxx σxzσzx σzz

∣∣∣∣+

∣∣∣∣ σyy σyzσzy σzz

∣∣∣∣ , I3 =

∣∣∣∣∣∣σxx σxy σxzσyx σyy σyzσzx σzy σzz

∣∣∣∣∣∣Principal directions

(σ − λiI)ni = 0

1

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

Constitutive equations- Hooke’s law

Generalized Hooke’s law- isotropic material

εxxεyyεzz

2εxy2εyz2εzx

=1

E

1 −ν −ν 0 0 0

1 −ν 0 0 01 0 0 0

2(1− ν) 0 0sym. 2(1− ν) 0

2(1− ν)

σxxσyyσzzσxyσyzσzx

σxxσyyσzzσxyσyzσzx

=E

(1 + ν)(1− 2ν)

1− ν ν ν 0 0 0

1− ν ν 0 0 01− ν 0 0 0

1−2ν2 0 0

sym. 1−2ν2 0

1−2ν2

εxxεyyεzz

2εxy2εyz2εzx

• Representation by the Lamé coefficients

σxxσyyσzzσxyσyzσzx

=

2µ+ λ λ λ 0 0 0

2µ+ λ λ 0 0 02µ+ λ 0 0 0

2µ 0 0sym. 2µ 0

2µ

εxxεyyεzzεxyεyzεzx

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

ΠTPE =1

2

∫V

σ : εdV −∫V

f · udV −∫S

t · udS

Principle of minimum potential energy

δΠTPE(u) = 0

3

Classical beam and bar theoriesTPE functionals for bars and beams

ΠTPE(u) =

∫ `

0

EA

2

(du

dx

)2

dx−∫ `

0

qxudx

ΠTPE(w) =

∫ `

0

EI

2

(d2w

dx2

)2

dx−∫ `

0

qwdx

BCs for beams

• Fixed end- w = 0 and dwdx = 0

• Free end- M = −EI d2wdx2 = M0 and Q = −EI d

3wdx3 = Q0

• Pinned end- w = 0 and M = −EI d2wdx2 = 0

Ritz methodDisplacement 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

ΠTPE = ΠTPE(ai, bi, ci)

Minimum conditions

∂ΠTPE

∂ai= 0,

∂ΠTPE

∂bi= 0,

∂ΠTPE

∂ci= 0

4