Review (2 nd order tensors): Tensor – Linear mapping of a vector onto another vector Tensor components in a Cartesian basis (3x3 matrix): Basis change formula for tensor components Dyadic vector product General dyadic expansion

Routine tensor operations Addition Vector/Tensor product Tensor product

Routine tensor operations Transpose Trace Inner product Outer product Determinant Inverse Invariants (remain constant under basis change) Eigenvalues, Eigenvectors (Characteristic Equation – Cayley-Hamilton Theorem)

Recipe for computing eigenvalues of symmetric tensor

Special Tensors Symmetric tensors Have real eigenvalues, and orthogonal eigenvectors Skew tensors Have dual vectors satisfying Proper orthogonal tensors Represent rotations – have Rodriguez representation Polar decomposition theorem

Polar Coordinates Basis change formulas

Gradient operator

Review: Deformation Mapping Eulerian/Lagrangian descriptions of motion Deformation Gradient

Review Sequence of deformations Lagrange Strain

Review Volume Changes Area Elements Infinitesimal Strain Approximates L-strain Related to ‘Engineering Strains’

Review Principal values/directions of Infinitesimal Strain Infinitesimal rotation Decomposition of infinitesimal motion

Left and Right stretch tensors, rotation tensor U,V symmetric, so principal stretches Review Left and Right Cauchy-Green Tensors

Review Generalized strain measures Eulerian strain

Review Velocity Gradient Stretch rate and spin tensors

Vorticity vector Spin-acceleration-vorticity relations Review

Review: Kinetics Surface traction Body Force Internal Traction Resultant force on a volume

Restrictions on internal traction vector Review: Kinetics Newton II Newton II&III Cauchy Stress Tensor

Other Stress Measures Kirchhoff Nominal/ 1 st Piola-Kirchhoff Material/2 nd Piola-Kirchhoff

Review – Reynolds Transport Relation

Review – Mass Conservation Linear Momentum Conservation Angular Momentum Conservation

Rate of mechanical work done on a material volume Conservation laws in terms of other stresses Mechanical work in terms of other stresses

Review: Thermodynamics Specific Internal Energy Specific Helmholtz free energy Temperature Heat flux vector External heat flux First Law of Thermodynamics Second Law of Thermodynamics Specific entropy

Conservation Laws for a Control Volume R is a fixed spatial region – material flows across boundary B Mass Conservation Linear Momentum Conservation Angular Momentum Conservation Mechanical Power Balance First Law Second Law

Review: Transformations under observer changes Transformation of space under a change of observer All physically measurable vectors can be regarded as connecting two points in the inertial frame These must therefore transform like vectors connecting two points under a change of observer Note that time derivatives in the observer’s reference frame have to account for rotation of the reference frame

Some Transformations under observer changes

Constitutive Laws General Assumptions: 1.Local homogeneity of deformation (a deformation gradient can always be calculated) 2.Principle of local action (stress at a point depends on deformation in a vanishingly small material element surrounding the point) Restrictions on constitutive relations: 1. Material Frame Indifference – stress-strain relations must transform consistently under a change of observer 2. Constitutive law must always satisfy the second law of thermodynamics for any possible deformation/temperature history. Equations relating internal force measures to deformation measures are known as Constitutive Relations