Forces

1. Forces

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. Deformation is relative to the size of an object. The displacement compared to the length is the straine. Strain L DL

4. 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)

5. 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)

6. 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

7. 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.

8. 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.

9. Transformation Matrix • A Cartesian vector can be defined by its transformation rule. • Another transformation matrix T transforms similarly. x3 x2 x1

10. 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. Order and Rank Scalars are independent of coordinate system.

11. 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 Direct Product

12. Tensors form a linear vector space. Tensors T, U Scalarsf, g Tensor algebra includes addition and scalar multiplication. Operations by component Usual rules of algebra Tensor Algebra

13. 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 Contraction

14. 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. Symmetric Tensor

15. 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

16. 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

17. 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