forces
DESCRIPTION
Forces. Normal Stress. A stress measures the surface force per unit area. Elastic for small changes A normal stress acts normal to a surface. Compression or tension. A. D x. A. Deformation is relative to the size of an object. The displacement compared to the length is the strain e. - PowerPoint PPT PresentationTRANSCRIPT
Forces
Normal Stress
A stress measures the surface force per unit area. Elastic for small changes
A normal stress acts normal to a surface. Compression or tension
F
A
A
Ft
F
A
x
Strain
Deformation is relative to the size of an object.
The displacement compared to the length is the strain .
L
L
LL
Shear Stress
A shear stress acts parallel to a surface. Also elastic for small changes
Ideal fluids at rest have no shear stress. Solids Viscous fluids
F
A
x
A (goes into screen)
L
F
A
Ft
Volume Stress
Fluids exert a force in all directions. Same force in all directions
The force compared to the area is the pressure.
F
A
A
FP
F
F
P
V
V
A (surface area)
Surface Force
Any area in the fluid experiences equal forces from each direction. Law of inertia All forces balanced
Any arbitrary volume in the fluid has balanced forces.
Force Prism
Consider a small prism of fluid in a continuous fluid. Stress vector t at any point Normal area vectors S form a
triangle
The stress function is linear.
1Sd
2Sd
21 SdSd
)( 2Sdt
)( 1Sdt
)( 21 SdSdt
)()( SdtcScdt
)()( SdtSdt
)()()( 2121 SdSdtSdtSdt
Stress Function
The stress function is symmetric with 6 components.
To represent the stress function requires something more than a vector. Define a tensor
If the only stress is pressure the tensor is diagonal.
The total force is found by integration.
SdSdP
P)(
SdpSdSdP
1P)(
S
SdF
P
Transformation Matrix
A Cartesian vector can be defined by its transformation rule.
Another transformation matrix T transforms similarly.
x1
x2
x3
jiji xlx
iijj xlx
1x
2x
3x
ijjqippq TllT
pqjqipij TllT
Order and Rank
For a Cartesian coordinate system a tensor is defined by its transformation rule.
The order or rank of a tensor determines the number of separate transformations. Rank 0: scalar Rank 1: vector Rank 2 and up: Tensor
The Kronecker delta is the unit rank-2 tensor.
ss Scalars are independent of coordinate system.
iijj xlx
ijjqippq TllT
nnnn iipipipp TllT 1111
ijjqippq ll
Direct Product
A rank 2 tensor can be represented as a matrix.
Two vectors can be combined into a matrix. Vector direct product Old name dyad Indices transform as separate
vectors
332313
322212
312111
bababa
bababa
bababa
BABA T
C
C
333231
232221
131211
CCC
CCC
CCC
C
Tensor Algebra
Tensors form a linear vector space. Tensors T, U Scalars f, g
Tensor algebra includes addition and scalar multiplication. Operations by component Usual rules of algebra
TT1
TT )()( fggf
UTUT fff )(
ijij UT UT
TUUT
TTT gfgf )(
Contraction
The summation rule applies to tensors of different ranks. Dot product Sum of ranks reduce by 2
A tensor can be contracted by summing over a pair of indices. Reduces rank by 2 Rank 2 tensor contracts to the trace
iiij TT
jiji bAc
iibas
jijkik vT
3
1
tri
iiijij TTT
Symmetric Tensor
The transpose of a rank-2 tensor reverses the indices. Transposed products and
products transposed
A symmetric tensor is its own transpose. Antisymmetric is negative
transpose
All tensors are the sums of symmetric and antisymmetric parts.
jiijij TTT ~T
jiij SS
AST
TTT)( TUTU
jiij AA
TTTTT~~
21
21
Stress Tensor
Represent the stress function by a tensor. Normal vector n = dS Tij component acts on surface
element
The components transform like a tensor. Transformation l Dummy subscript changes
),(),,( txTnSdtxt ijij
ijij Tnt
ijqqippj Tnltl
ijqqiipipj TnlTnl
ijqqirjipipjrj TnllTnll
ijqqirjiriipirp TnllTnTn
ijrjqiqqrq TllnTn ijrjqiqr TllT
Symmetric Form
The stress tensor includes normal and shear stresses.
Diagonal normal Off-diagonal shear
An ideal fluid has only pressure. Normal stress Isotropic
A viscous fluid includes shear. Symmetric 6 component tensor
ijij PT
33231
23221
13121
T
jiij TT
32313
23212
13121
T
Force Density
The total force is found by integration. Closed volume with Gauss’
law Outward unit vectors
A force density due to stress can be defined from the tensor. Due to differences in stress as
a function of position
S
dSnF ˆT
V
dVF T
PSf
S
SdF
T