A 3D stress block: Stresses are shown on positive x,y and z faces only

Concept of Generalized Hooke’s law

EEEzyx

xσυ

συσε −−=

Generalized Hooke’s law in 3D

(Stress-strain relationships)

EEEzxy

yσυσυ

σε −−=

EEEyxz

zσ

υσυσε −−=

Gxy

xyτ

γ =

Gyz

yzτ

γ =

Gzx

zxτγ =

( ) ( )[ ]zyxxE εευευ

υυσ ++−

−+= 1

)21)(1(

( )[ ]zxyyE εευευ

υυσ ++−

−+= )1(

)21)(1(

( )[ ]yxzzE εευευ

υυσ ++−

−+= )1(

)21)(1(

xyxy Gγτ =

yzyz Gγτ =

zxzx Gγτ =

E= elastic modulus

ν = poisson’s ratio

)1(2 υ+= EG

G= shear modulus

Hooke’s law in 2DPlane Stress Problem

EEyx

xσ

υσε −=

EExy

yσυ

σε −=

Gxy

xyτ

γ =

Assumptions:

0;0;0 === yzxzz ττσ

EEyx

zσ

υσυε −−=

[ ]yxxE υεευ

σ +−

=)1( 2

[ ]xyyE υεευ

σ +−

=)1( 2

xyxy Gγτ =

A thin plate subject to in plane (x-y) loading

Other stress and strain quantities are non-zero

Hooke’s law in 2D

Plane Strain Problem

Gxy

xyτ

γ =

Assumptions:

0;0;0 === yzxzz γγε

xyxy Gγτ =

EEEzyx

xσυ

συσε −−=

EEEzxy

yσυσυ

σε −−=

EEEyxz σ

υσυσ −−=0

Other stress and strain quantities are non-zero

( )[ ]yxxE υεευ

υυσ +−

−+= 1

)21)(1(

( )[ ]xyyE υεευ

υυσ +−

−+= 1

)21)(1(

( )[ ]yxzE εευ

υυσ +

−+=

)21)(1(

t

L

2r

Fig. 1

p

zx

y

A cylindrical pressure tube constrained between rigid walls

Volume Change

Original Volume

abcVo =

)1(

)1)(1)(1(

))()((1

zyxo

zyx

zyx

V

abc

ccbbaaV

εεεεεε

εεε

+++≈

+++=

+++=

)(1 zyxoo VVVV εεε ++=−=Δ

Final Volume

Ignoring square terms of strain, since they are small

Volume change:

( )zyx

zyxo

E

VVe

σσσυ

εεε

++−=

++=Δ=

21Dilatation (e): Change in Unit Volume