Stress TensorStress Tensor

Normal StressNormal Stress

A stress measures the A stress measures the surface force per unit area.surface force per unit area.• Elastic for small changesElastic for small changes

A normal stress acts normal A normal stress acts normal to a surface.to a surface.• Compression or tensionCompression or tension

F

A

A

Ft

F

A

x

Shear StressShear Stress

A shear stress acts parallel A shear stress acts parallel to a surface.to a surface.• Also elastic for small Also elastic for small

changeschanges

Ideal fluids at rest have no Ideal fluids at rest have no shear stress.shear stress.• SolidsSolids

• Viscous fluidsViscous fluids

F

A

x

A (goes into screen)

L

F

A

Ft

Volume StressVolume Stress

Fluids exert a force in all Fluids exert a force in all directions.directions.• Same force in all directionsSame force in all directions

The force compared to the The force compared to the area is the pressure.area is the pressure.

F

A

A

FP

F

F

P

V

V

A (surface area)

Surface ForceSurface Force

Any area in the fluid Any area in the fluid experiences equal forces experiences equal forces from each direction.from each direction.• Law of inertiaLaw of inertia

• All forces balancedAll forces balanced

Any arbitrary volume in the Any arbitrary volume in the fluid has balanced forces.fluid has balanced forces.

Force PrismForce Prism

Consider a small prism of Consider a small prism of fluid in a continuous fluid.fluid in a continuous fluid.• Stress vector Stress vector tt at any point at any point

• Normal area vectors Normal area vectors SS form form a trianglea triangle

The stress function is linear.The stress function is linear.

1Sd

2Sd

21 SdSd

)( 2Sdt

)( 1Sdt

)( 21 SdSdt

)()( SdtcScdt

)()( SdtSdt

)()()( 2121 SdSdtSdtSdt

Stress TensorStress Tensor

Represent the stress Represent the stress function by a tensor.function by a tensor.• Normal vector Normal vector n = dSn = dS

• TTijij component acts on component acts on

surface elementsurface element

The components transform The components transform like a tensor.like a tensor.• Transformation Transformation ll

• Dummy subscript changesDummy subscript changes

),(),,( txTnSdtxt ijij

ijij Tnt

ijqqippj Tnltl

ijqqiipipj TnlTnl

ijqqirjipipjrj TnllTnll

ijqqirjiriipirp TnllTnTn

ijrjqiqqrq TllnTn ijrjqiqr TllT

Symmetric FormSymmetric Form

The stress tensor includes The stress tensor includes normal and shear stresses.normal and shear stresses.• Diagonal normalDiagonal normal• Off-diagonal shearOff-diagonal shear

An ideal fluid has only pressure.An ideal fluid has only pressure.• Normal stressNormal stress• IsotropicIsotropic

A viscous fluid includes shear.A viscous fluid includes shear.• Symmetric Symmetric • 6 component tensor6 component tensor

ijij PT

33231

23221

13121

T

jiij TT

32313

23212

13121

T

Force DensityForce Density

The total force is found by The total force is found by integration.integration.• Closed volume with Gauss’ Closed volume with Gauss’

lawlaw

• Outward unit vectorsOutward unit vectors

A force density due to stress A force density due to stress can be defined from the can be defined from the tensor.tensor.• Due to differences in stress Due to differences in stress

as a function of positionas a function of position

S

dSnF ˆT

V

dVF T

PSf

S

SdF

T

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