弹塑性力学2-Stress-

x yx zx
x x y z x xy xz yx y yz z x z y z
Stress state of a point depends on 9 stress components
i 1
3
3
ai bi ai bi a1b1 a 2b2 a3b3
i 1
应力分析
a ij b j a ij b j a i1b1 a i 2 b2 a i 3 b3
a ij bi c j aijbi c j
i 1 j 1
3
j 1 3
3
a11b1c1 a12b1c2 a13b1c3 a21b2 c1 a22b2 c2 a23b2 c3 a31b3c1 a32b3c2 a33b3c3
弹塑性力学
Chapter 2 Stress Analysis
应力分析
2-1 Volume forces and surface forces
• Force is an action of one body to another
• Volume forces act over body volume • Surface forces act on body surface
In which
应力分析
2-5 Principal stresses and stress invariants
• For every point in a loaded body, we can find 3 planes perpendicular each other, on which there are only normal
应力分析
Stress state of a point
• There are infinite planes passing through a point
• Stresses in different plane passing through a point are called stress state of a point
z y y z
z x x z
y x x y
应力分析
Tensor expression
ij lii l jj ij
(i , j x , y , z )
l xx lij l yx l zx l xy l yy l zy l xz l yz l zz
应力分析
In turn
y x l y x y l y y z l y z 2 ( xy l y x l y y yz l y y l y z zxl y z l y x )
2 2 2
z x l z x y l z y z l z z 2 ( xy l z x l z y yz l z y l z z zxl z z l z x )
应力分析
a、notation of tensor
• 1 subscript，3 components，ui（i=1，2，3） • 2 ，9，ij （i,j=1，2，3） • 3，27，ijk （i,j,k=1，2，3）
• 4，81，ijkl （i,j,k,l=1，2，3）
应力分析
b、Plus convention for tensor
l xz cos( x, z ) l yz cos( y , z ) l zz cos( z , z )
应力分析
Expressions for stresses on inclined plane
x T x l x x T y l x y T z l x z
ABC 1
OBC 1 cos( n , x )
C
z
n
Ty
Tz
OAC 1 cos( n , y )
OAB 1 cos( n , z )
A
Tx
O
B
y
x Tx x cos n, x yx cos n, y zx cos n, z
应力分析
Stress tensor
x ij yx zx i, j x, y ,
xy y zy
z
xz yz z
2 order tensor: 9 physical measures which obey certain conversion relation between two different coordinates
Tx Tl1 n l1 Ty Tl2 n l2 Tz Tl3 n l3
x y T x l y x T y l y y T z l y z
x z T x l z x T y l z y T z l z z
Tx x cos x, x yx cos x, y zx cos x, z Tz zx cos z , x yz cos z , y z cos z , z
If a subscript appears twice in a production of tensor expression, it means this production is a plus for that the subscript equals 1,2 and 3.
aii aii a11 a 22 a33
Divide body as part A and B with plane C p z lim S S 0
Decompose stress alone nominal and tangent direction
p n n lim S 0 S pt lim S 0 S
stress.
• 3 planes-Principal planes • Normal stress-Principal stress • Normal direction- principal directions for principal stresses.
应力分析
Note 3 direction cosines of principal directions as l1，l2， l3
Ty yx cos y , x y cos y , y yz cos y , z
应力分析
We find
x x l x x y l x y z l x z 2 ( xy l x x l x y yz l x y l x z zxl x z l x x )
p
n
S
n
B
A
C
x
y
应力分析
If normal of cross section C is the direction of coordinate y
z
y
yz
C

y
yx
x
y
yx y yz
Subscripts of shear stress： • first indicating the direction of the normal to the plane • the second indicating the direction of the component of the stress
2 2 2
y z x l y x l z x xy (l z x l y y l y x l z y ) y l y y l z y yz (l z y l y z l y y l z z )
z l y z l z z zx (l z z l y x l y z l z x )
2 2 2
xy x l xx l yx xy (l yx l xy l xx l yy ) y l xy l yy yz (l yy l xz l xy l yz )
z l xz l yz zx (l yz l xx l xz l yx )
2
应力分析
2-4 Stresses in different coordinates
The direction cosine of two coordinates
l xx cos( x, x ) l yx cos( y , x ) l zx cos( z , x )
l xy cos( x, y ) l yy cos( y , y ) l zx cos( z , y )
z x x l z x l xx xy (l xx l z y l z x l xy ) y l z y l xy yz (l xy l z z l z y l xz )
z l z z l xz zx (l xz l z x l z z l xx )
Ti ij l jn
l yn cos n , y l zn cos n , z
(i, j x, y , z )
应力分析
Stress resultant
Ti ij l jn
2 2
z
C
n
Ty
(i , j x , y , z )
Tx
2
Tz
T Tx T y Tz
应力分析
Surface force components
Volume force
lim
S 0
T T S
T
Tx , Ty , Tz
z
S
components
lim Fx , Fy , Fz
V 0
F F V
y
x
应力分析
2-2 stress and stress state
For equilibrium Ty yx cos n, x y cos n, y yz cos n, z
Tz zx cos n, x yz cos n, y z cos n, z
l xn cos n , x
O
B
y
x A n Tx cosn, x Ty cosn, y Tz cosn, z
x l xn y l yn z l zn
2 2 2
2 xy l xn l yn yz l yn l zn zxl znl xn
2
n T n
z
yx y yz
x
xy xz xz yx xy x zx zy y
z
z zy zx x yz
Stress components
x y yx zx
xy y zy
xz yz z
应力分析
Change of coordinates
应力分析
ai, i
a i a1 a 2 a 3 xi x1 x 2 x3
i i i i 2 2 2 x j x j x1 x 2 x3
2 2 2 2
i , jj
应力分析
2-3 Stresses in a inclined plane