Review (2nd order tensors):

1
Review (2nd 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
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Routine tensor operations
Addition
Vector/Tensor product
Tensor product
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Routine tensor operations
Transpose
Trace Inner product Outer product Determinant Inv
erse Invariants (remain constant under basis
change) Eigenvalues, Eigenvectors
(Characteristic Equation Cayley-Hamilton
Theorem)
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Recipe for computing eigenvalues of symmetric
tensor
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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
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Polar Coordinates
Basis change formulas
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Gradient operator
8
Review
Deformation Mapping
Eulerian/Lagrangian descriptions of motion
Deformation Gradient
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Review
Sequence of deformations
Lagrange Strain
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Review
Volume Changes
Area Elements
Infinitesimal Strain
Approximates L-strain
Related to Engineering Strains
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Review
Principal values/directions of Infinitesimal
Strain
Infinitesimal rotation
Decomposition of infinitesimal motion
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Review
Left and Right stretch tensors, rotation tensor
U,V symmetric, so
principal stretches
Left and Right Cauchy-Green Tensors
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Review
Generalized strain measures
Eulerian strain
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Review
Velocity Gradient
Stretch rate and spin tensors
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Review
Vorticity vector


Spin-acceleration-vorticity relations
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Review Kinetics
Surface traction
Body Force
Internal Traction
Resultant force on a volume
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Review Kinetics
Restrictions on internal traction vector
Newton II
Newton IIIII
Cauchy Stress Tensor
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Other Stress Measures
Kirchhoff
Nominal/ 1st Piola-Kirchhoff
Material/2nd Piola-Kirchhoff
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Review Reynolds Transport Relation
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Review
Mass Conservation
Linear Momentum Conservation
Angular Momentum Conservation
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Rate of mechanical work done on a material volume
Conservation laws in terms of other stresses
Mechanical work in terms of other stresses
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Review Thermodynamics
Temperature
Specific Internal Energy Specific Helmholtz free
energy
Heat flux vector
External heat flux
Specific entropy
First Law of Thermodynamics
Second Law of Thermodynamics
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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
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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 observers
reference frame have to account for rotation of
the reference frame
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Some Transformations under observer changes
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Constitutive Laws
Equations relating internal force measures to
deformation measures are knownas Constitutive
Relations

• General Assumptions
• Local homogeneity of deformation
• (a deformation gradient can always be
  calculated)
• 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.