Pre-midterm Flashcards
Molecular scale
Length of a cube with the volume of a molecule
Density field
ρ(x)= lim(l->Lmicro) Ml(x)/Vl
l is small but large enough for V to contain many molecules
Relative size of density fluctuations
Δρ/ρ = 1/sqrt(N)
More molecules = smaller changes in density due to molecular motion
Micro scale
Length at which density fluctuations are less than ε
Macro scale
Length over which density changes due to fluid
Continuum approximation
Valid for L»_space; Lmacro
Conservation laws (def)
Related cinematic and dynamic quantities
Conservation laws (examples)
dP/dt = F P = momentum (vector), F=force
dL/dt = M L= angular momentum, M=moment of force
dK/dt = P
K=kinetic energy, P=power (scalar)
Centre of mass
xm = 1/m * sum(N, i=1)mixi
Moment of force
M = sum(N, i=1) xi x(cross) fi
Angular momentum
L = sum(N, i=1) xi x(cross) mi*dxi/dt
Power
P=sum(N, i=1) fi*dxi/dt
Body force
External force that acts over the entire volume of the object
Contact force
Force that acts at contact of surfaces of objects
Tension
Normal component of force in same direction as unit outward normal
Shear
Tangential component of force
Pressure
Normal component of force in opposite direction of unit outward normal
Stress vector t
dF = t(x, t, n)dS
F=force, x=position, t=time, n=normal, S=surface
Part of “stress principle of Euler and Cauchy”
Fluid at rest cannot have shear stresses (t parallel to n)
Pascal’s law
The pressure in a fluid is the same in all directions
Hydrostatic equilibrium
Global: triple integral over V (ρg-nabla(p)) = 0
Local: nabla(p) = ρg
Barotropic
Independent of temperature
Density and pressure are functions of each other
Isothermal
Constant temperature
Polytropic and ideal gas density
ρ=(RT/C)^(1/γ-1)
Pressure potential
Gravity is conservative: g=-nabla(Φ)
Flat earth
g= (0, 0, -g0)
Φ=g0z
Hydrostatic eqn with constant density
Nabla(Φ*)=0
Φ*=Φ+p/ρ0
Hydrostatic eqn with barotropiv fluid
Nabla(Φ*)=0
Φ*=Φ+w(p)
w(p)=integral(p,p0) dp/ρ(p)
Specific heat at constant pressure
Polytropic
FP=γ/(γ-1) * R
p/ρ = RT
Bulk modulus
Large: small changes in volume
Bulk modulus, isothermal
K = p (ideal gas)
Bulk modulus, isentropic
K = γp (polytropic)
Bulk modulus, polytropic water
K = K0 + γ(p-p0)
Adiabatic
No heat transfer with environment
Isentropic
Adiabatic and reversible (delta(S)=0)
Homentropic
Constant entropy
Homentropic ideal gas
pρ^-γ=p0/ρ0^-γ
Polytropic with C=p0/ρ0^γ
Direct product
ab^T=(axbx axby axbz
aybx ayby aybz
azbx azby azbz)
Volume product
a x b • c = det(cx cy cz
ax ay az
bx by bz)
Volume of parallelepiped spanned by a, b, c
Tensor product (indicial)
(ab^T)ij = aibj
Cross product (indicial notation)
(a x b)i = εijkajbk
Volume product (indicial notation)
a x b • c = εijkaibj*ck
Rotation matrices
Orthogonal (B^-1 = B^T)
(BB^T)ij = δij
Reflections
Convert right handed coord sys to left handed
Vector transformation
u’ = Au
Matrix transformation
T’ = ATA^T
Scalar field
S’=S
Rank 0 tensor
Vector field
Ui’ = aij*Uj
Rank 1 tensor
Tensor field
T’=ATA^T
Tij’ = aikajlTkl
Rank 2 tensor
Contractions (indicial notation)
If two indices of a rank r tensor are equal (summed over), the result is an r-2 rank tensor
Tensor products
Product of rank r and rank S tensor is rank r+S tensor
Zero tensor: all components 0 in one coord sys, then they’re 0 in all rotations
Quotient rule (indicial notation)
Tij satisfies Ai=Tij*Bj in all coord systems and Ai and Bj are vectors
Then Tij is a rank 2 tensor
Stress field
- Local and can be normal or tangential
- units of pressure
- sigma = F/A
- sigma is shear stress (either internal or external)
External stress
Stress on interface between body and environment
Internal stress
Act at imaginary surfaces inside the body
Tensile strength
Maximum tension a material can sustain before it breaks
Yield stress
Stress beyond which solids don’t bounce back
Cauchy’s stress tensor
- a stress tensor field (rank 2)
- σij: i shows force, j shows surface
- dFi = sigma ij*dSj
- ti = σij*nj (stress vector)
Stress tensor: diagonal components
Normal stresses
Stress tensor: off-diagonal components
Shear stresses
Stress vectors (not t vector, but with sigma)
Sigma x = dFx/dSx (similar for y and z)
Sigma x = (sigma xx, sigma yx, sigma zx)
Mechanical pressure
P = -1/3 Tr(sigma)
Rank 0 tensor
Cauchy’s equilibrium equation
fi + dσij/dxj = 0
f + divergence(sigma T) = 0
Constitutive equation
Defines the behaviour of a continuum
Normal and shear stresses (formulas with stress vector)
tnormal = (t*n)n tshear = t - tnormal t = σ*n = tnormal + tshear
n is unit outward normal
Symmetric stress tensor
Gives no torque (bc moment of force is 0)
Is the stress tensor is symmetric, what do we know?
- eigenvalues are real
Stress tensor eigenvalues
Principle stresses
Stress tensor eigenvectors
Principle directions of stress
In eigenvector basis, stress tensor is diagonal (no shear stresses)
Stress vector boundary conditions
Must be continuous across a surface
Eulerian displacement
u(x) = x - X(x)
Lagrangian displacement
u(X) = x(X) - X
Linear displacement
x = aX + b
a not singular
Linearize needle
δa = a - a0 such that δaj = ajdui/dxj
Alt: δa = (nabla(u))^Ta
δ|a| = |a| - |a0| δ(|a|^2) = 2|a|*δ|a|
Displacement gradient
Nabla(u) = dui/dxj
Rank 2 tensor field
Slowly varying
|ui| «_space;L
L is length of domain
Cauchy’s strain tensor
uij = 1/2 * (duj/dxi + dui/dxj)
Symmetric
u~0 gives no deformations
Contains all local geometry info
Eigenvalues and eigenvectors for cauchy’s strain tensor
Eigenvectors: principle axes of strain
Eigenvalues: how much material is being deformed along principle axes
Diagonal elements of uij
Fractional rate of change in that index’s direction
Off-diagonal elements of uij
Fractional rate of change in angle between the two needles a and b
Projection of uij onto vectors a and b: uab=aub
Curve element
δ(dli) = δlj*dui/dxj
Volume element
δ(dV) = (divergence(u))dV
Divergence if u sets spread in volume
dρ/ρ = - divergence(u)
Young’s modulus
Displacement parallel to force
E = kL/A, k as in hooke’s law
Units of pressure
Poisson’s ratio
Dimensionless and positive (except for crystals)
Displacement orthogonal to force
Isotropic materials
Uniform / homogenous in all directions
No preferred direction of matter
Lamé coefficients
Units of pressure
μ is shear modulus (if it’s 0, no shear stress)
Mechanical pressure and bulk modulus
K = E/3(1+ν)
dp = -1/3 σii = -(λ+2/3 μ)uii change in mechanical pressure
Anisotropic materials
Elasticity tensor: σij = Eijkl*ukl
Rank 4 tensor
Norm of a matrix
|A| = sqrt(sum(i,j) aij^2)
