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 A A xx

Strain Deformation is relative to the size of an object. The displacement compared to the length is the strain . LL L

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 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.

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.

Transformation Matrix A Cartesian vector can be defined by its transformation rule. Another transformation matrix T transforms similarly. x1x1 x2x2 x3x3

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. Scalars are independent of coordinate system.

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

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

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

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.

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

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

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