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Stress Tensor

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. D x. A. A shear stress acts parallel to a surface. Also elastic for small changes

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Stress Tensor

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  1. Stress Tensor

  2. 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 Dx A

  3. A shear stress acts parallel to a surface. Also elastic for small changes Ideal fluids at rest have no shear stress. Solids Viscous fluids Shear Stress A Dx L A(goes into screen)

  4. 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 DV V A (surface area)

  5. 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. Surface Force

  6. 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.

  7. 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 Stress Tensor

  8. 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 Symmetric Form

  9. 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

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