Strain and Strain Tensor (Elastic)

1. Elastic Strain Tensor : ε and εij - (section 2.8)

2. Elastic σ−ε Relations (section 2.11)

3. Elastic Stresses (σ) from Strains (ε) – (section 2.12)

4. Plane Stress (Eq. 2-77) and Plane Strain (Eq. 2-78)

5. Elastic Strain Energy (section 2-13 and Eq. 2-83)

Text : p.38

1-D:

x

udx

dxdxx

udx

AB

AB'B'A

L

Lex

∂∂

=

−∂∂+

=

−=

∆=

Generalize for 3-D :

jiji

zzzyxz

yzyyyx

xzxyxx

xeu

zeyexew

zeyexev

zeyexeu

=

++=

++=

++=

(2-34)

Or, u = ex x (2-33) Or, j

ixu

ije∂∂= , etc.

Principal Strains :

• Similar in concept to principal stresses • Can identify, principal axes along which there are no shear strains or

rotations, only pure extension or contraction • For isotropic solids, principal strain axes coincide with the principal

stress axes • Definition of principal strain axes: Three mutually perpendicular

directions in the body which remain mutually perpendicular during deformation

• Remain unchanged if and only if ϖ ij=0

Dilatation, ∆∆ • Volume change or dilatation :

∆ = (1+ε1) (1+ε2) (1+ε3) – 1 = ε1+ε2+ε3 for ε’s<<1 • Note ∆ is the first invariant of the strain tensor • Mean Strain, εm = ∆/3

• Strain deviator, 'ijε , is the part of the strain tensor that represents

shape change at constant volume : )( ij3ijmij'ij δ−ε=ε−ε=ε ∆

Engineering Shear Strain, γ = a/h = tanθ ~ θ xy

yxxy

yxxyxy

2

ee

ε=

ε+ε=

+=γ

Simple Shear + Rotation = Pure Shear (Fig. 2-15)

Elasticity (for isotropic solids) Equations that relate stresses to strains are known as “Constitutive Equations” --------- Hooke’s law: σx=Eεx and Poisson’s Relation: εy = εz = -ν εx = −νσx/E

(p. 49) so that

(2-64)

zxzxyzzyxyxy G;G;G γ=τγ=τγ=τ (2-65) Or, εxy = τxy/2G etc. Need only two elastic constants, E and ν

since G, E and κ are related through ν (recall 1st class on Elasticity)

Bulk Modulus : ∆−

=∆

σ=

pK m

and )21(3E

Kν−

= (2-67)

Strains in terms of Stresses : ijkkijijEE

1δσ

ν−σ

ν+=ε (2-69)

Stresses in terms of Strains :

(Inversion of Eq. 2-69) : ijkkij1E

ij δλε+ε=συ+ (2-73)

where λ is Lame’s constant : )21)(1(

E

υ−υ+υ

=λ

Or

kkij31'

ijij σδ+σ=σ where Distortion: 'ij

'ij G2 ε=σ (2-75)

and Dilatation: σii = 3κεkk (2-76) Special Cases : Plane Stress (σ3=0) and Plane Strain (ε3=0) ----- Eqs. (2-77) and (2-78) Strain Energy: Elastic strain energy, U = energy spent by the external forces in deforming an elastic body (= ½ Pδ --- area under the load-displacement curve)

2

E

E22

1U

2x

2x

xx0ε

=σ

=εσ= for simple uniaxial loading

For generalized stresses and strains Uo = ½ σij εij (Eq. 2-83) or Eq. 2-82 in expanded form

( ) ( )

( )2zx

2yz

2xy

xzzyyx2z

2y

2x0

G2

1EE2

1U

τ+τ+τ+

σσ+σσ+σσν

−σ+σ+σ=

(2-84)

Note that the derivative of Uo with respect to any strain component equals the corresponding stress component :

xxx

o G2U

σ=ε+∆λ=ε∂

∂ and similarly x

x

oUε=

σ∂∂

Eq. 2-86

Generalized Hooke’s Law : εij = Sijkl σkl Here Sijkl is Elastic Compliance Tensor (4th rank) σij = Cijkl εkl Cijkl is Elastic Stiffness (or Elastic Constants) Crystal Symmetry reduces the number of independent terms : Cubic – 3 Hexagonal – 5 etc. (see table, p.58) --- are related to E and G ************************* Next ---- Ch. 3 on Plasticity