1/61/6 m.chrzanowski: strength of materials sm1-08: continuum mechanics: stress distribution...
DESCRIPTION
3/63/6 M.Chrzanowski: Strength of Materials SM1-08: Continuum Mechanics: Stress distribution x2x2 x1x1 x3x3 Volume V Surface S Volume V 0 Surface S 0 Stress vector Volumetric force GGO theorem Surface traction (loading)TRANSCRIPT
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
stop