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