University of California, Berkeley Spring Semester 2005
Department of Mechanical Engineering Instructor: P. Papadopoulos
ME280B - Finite Element Methods in Nonlinear Continua
Notation and list of symbols
General scheme of notation Roman and italic letters Lower-case bold letters Upper-case bold letters Calligraphic upper-case letters Scalars (or scalar fields) Vectors and tensors (or associated fields) Tensors (or tensor fields) Sets
Please note that some exceptions apply.
List of frequently used symbols [L] [M ] [T ] ˜ f ¯ f ˆ f f˙ eijk h m p r t E E3 H I1 , I 2 , I 3 J K R S W P Physical dimension of length Physical dimension of mass Physical dimension of time Spatial (Eulerian) form of function f Material form of function f Referential (Lagrangian) form of function f Material time derivative of function f Permutation symbol Heat flux per unit area Mass Pressure Heat supply per unit mass Time Young’s modulus of elasticity Three-dimensional Euclidean vector space Rate of heating Principal invariants of a tensor Jacobian of the motion Kinetic energy Rate of externally applied forces Stress power Strain energy per unit volume Particle label
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ME280B
2
N R
δij ε λ µ ν ρ ρ0 Ψ da ds dv df dA dS dV B E3 P ∂P R0 R ∂R S a b e ei g n m p q t u v w x
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The set of natural numbers The set of real numbers Kronecker symbol Internal energy per unit mass Stretch Shear modulus of elasticity Poisson’s ratio Mass density in the current configuration Mass density in the reference configuration Strain energy function per unit mass Differential area element in the current configuration Differential line element in the current configuration Differential volume element in the current configuration Differential force applied on area da Differential area element in the reference configuration Differential line element in the reference configuration Differential volume element in the reference configuration Body Three-dimensional Euclidean point space Subset of a region occupied by a body Boundary of a closed region P Region occupied by a body in the reference configuration Region occupied by a body in the current configuration Boundary of a closed region R Subset of a body Acceleration vector Body force vector Relative Eulerian (Almansi) strain tensor Cartesian basis vectors in current configuration Gravitational force vector Outward unit normal in the current configuration Unit vector in the direction dx Stress vector measured in the reference area Heat flux vector per unit area Stress vector Displacement vector Velocity vector Vorticity vector Position vector in the current configuration ME280B
3
B C D E EA F H I L M N P R S T U V W X ε κ0 κR κ σ τ χ ω Ω curl det div Div grad Grad tr · × ⊗
Left Cauchy-Green deformation tensor Right Cauchy-Green deformation tensor Rate-of-deformation tensor Relative Green-Lagrange strain tensor Cartesian basis vectors in reference configuration Deformation gradient tensor Displacement gradient tensor Identity tensor Velocity gradient tensor Unit vector in the direction dX Outward unit normal in the reference configuration First Piola-Kirchhoff stress tensor Rotation tensor Second Piola-Kirchhoff stress tensor Cauchy stress tensor Right stretch tensor Left stretch tensor Vorticity (or spin) tensor Position vector in the reference configuration Infinitesimal strain tensor Initial configuration Reference configuration Current configuration Infinitesimal stress tensor Kirchhoff stress tensor Motion Angular velocity vector Angular velocity tensor Curl of a vector Determinant of a tensor Divergence (or spatial divergence) of a vector or tensor Material divergence of a vector or tensor Gradient (or spatial gradient) of a scalar or vector Material gradient of a scalar or vector Trace of a tensor Inner product of two vectors or tensors Cartesian product of sets, cross product of two vectors Tensor product in E 3
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ME280B
4
T−1 TT T∗ symT skwT A∪B A∩B A−B A⊂B A⊆B A×B x∈A
∅
Inverse of a tensor T Transpose of a tensor T Adjugate of a tensor T Symmetric part of a tensor T Skew-symmetric part of a tensor T Union of sets A and B Intersection of sets A and B Difference of sets A and B Set A is a proper subset of set B Set A is a subset of set B Cartesian product of sets A and B Element x belongs to set A Empty set
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ME280B