1/6M.Chrzanowski: Strength of Materials

SM1-08: Continuum Mechanics: Stress distribution

CONTINUUM MECHANICS(STRESS DISTRIBUTION)

2/6M.Chrzanowski: Strength of Materials

SM1-08: Continuum Mechanics: Stress distribution

rpnnrpp const

;

npnrrpp const

;State of stress

Stress distribution

Stress vector

constn

rp

constrn

n

r

nrpp ,

3/6M.Chrzanowski: Strength of Materials

SM1-08: Continuum Mechanics: Stress distribution

x2

x1

x3

Volume VSurface S

ip

iqq

Volume V0Surface S0

Stress vector

Volumetric force

iPP

dSpdVPSV

00

0

000

dSdVP jS

ijV

i

000

dSdVPS

iV

i

000

dVx

dVPV j

ij

Vi

00

dVx

PV j

iji

jiji

0

j

iji x

P

,,, 321 xxxijij

GGO theorem

Surface traction(loading) q

4/6M.Chrzanowski: Strength of Materials

SM1-08: Continuum Mechanics: Stress distribution

0

j

iji x

P

On the body surface stress vector has to be balanced by the traction vector

q

pjijiiq

Stress on the body surface

Coordinates of vector normal to the surface

jijiq

This equation states statics boundary conditions to comply with the solution of the equation:

This equation (Navier equation) reflects internal equilibrium and has to be fulfilled in any point of the body (structure).

5/6M.Chrzanowski: Strength of Materials

SM1-08: Continuum Mechanics: Stress distribution

0

j

iji x

P

We have to deal with the set of 3 linear partial differential equations.

Navier equation

in coordintes reads:

0

0

0

3

33

2

32

1

313

3

23

2

22

1

212

3

13

2

12

1

111

xxxP

xxxP

xxxP

There are 6 unknown functions which have to fulfil static boundary conditions (SBC):

jijiq

We need more equations to determine all 6 functions of stress distribution. To attain it we have to consider deformation of the body.

6/6M.Chrzanowski: Strength of Materials

SM1-08: Continuum Mechanics: Stress distribution

Comments

1. Equation is derived from one of two

equilibrium equations, i.e. that the sum of forces acting over the body has to vanish.

0

j

iji x

P

2. The other equilibrium equation – sum of the moments equals zero – yield already assumed symmetry of stress matrix, σij= σji

3. Navier equation is the special case of the motion equation i.e. uniform motion (no inertia forces involved). The inertia effects can be included by adding d’Alambert forces to the right hand side of Navier equation.

7/6M.Chrzanowski: Strength of Materials

SM1-08: Continuum Mechanics: Stress distribution

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