Stress Tensor. Normal Stress  A stress measures the surface force per unit area. Elastic for small changes  A normal stress acts normal to a surface.

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Presentation transcript:

Stress Tensor

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 A xx

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 A xx A (goes into screen) L

Volume Stress  Fluids exert a force in all directions. Same force in all directions  The force compared to the area is the pressure. A P VV V A (surface area)

Surface Force  Any area in the fluid experiences equal forces from each direction. Law of inertiaLaw of inertia All forces balancedAll 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.

Stress Tensor  Represent the stress function by a tensor. Normal vector n = dSNormal vector n = dS T ij component acts on surface elementT ij component acts on surface element  The components transform like a tensor. Transformation lTransformation l Dummy subscript changesDummy subscript changes

Symmetric Form  The stress tensor includes normal and shear stresses. Diagonal normalDiagonal normal Off-diagonal shearOff-diagonal shear  An ideal fluid has only pressure. Normal stressNormal stress IsotropicIsotropic  A viscous fluid includes shear. SymmetricSymmetric 6 component tensor6 component tensor

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 next