Matrix and Index NotationDavid Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, MA 02139 September 18, 2000 A vector can be descr
Trang 1Matrix and Index Notation
David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology
Cambridge, MA 02139 September 18, 2000
A vector can be described by listing its components along the xyz cartesian axes; for in-stance the displacement vectoru can be denoted as ux, uy, uz, using letter subscripts to indicate the individual components The subscripts can employ numerical indices as well, with 1, 2, and 3 indicating the x, y, and z directions; the displacement vector can therefore be written equivalently as u1, u2, u3
A common and useful shorthand is simply to write the displacement vector asui, where the
i subscript is an index that is assumed to range over 1,2,3 ( or simply 1 and 2 if the problem is
a two-dimensional one) This is called the range convention for index notation Using the range convention, the vector equationui=a implies three separate scalar equations:
u1 =a
u2 =a
u3 =a
We will often find it convenient to denote a vector by listing its components in a vertical list enclosed in braces, and this form will help us keep track of matrix-vector multiplications a bit more easily We therefore have the following equivalent forms of vector notation:
u = ui =
u1
u2
u3
=
ux
uy
uz
Second-rank quantities such as stress, strain, moment of inertia, and curvature can be de-noted as 3×3 matrix arrays; for instance the stress can be written using numerical indices as
[σ] =
σ11 σ12 σ13
σ21 σ22 σ23
σ31 σ32 σ33
Here the first subscript index denotes the row and the second the column The indices also have
a physical meaning, for instanceσ23 indicates the stress on the 2 face (the plane whose normal
is in the 2, or y, direction) and acting in the 3, or z, direction To help distinguish them, we’ll use brackets for second-rank tensors and braces for vectors
Using the range convention for index notation, the stress can also be written as σij, where both the i and the j range from 1 to 3; this gives the nine components listed explicitly above
Trang 2(Since the stress matrix is symmetric, i.e σij = σji, only six of these nine components are independent.)
A subscript that is repeated in a given term is understood to imply summation over the range
of the repeated subscript; this is the summation convention for index notation For instance, to indicate the sum of the diagonal elements of the stress matrix we can write:
σkk=
3
X
k=1
σkk=σ11+σ22+σ33
The multiplication rule for matrices can be stated formally by taking A = (aij) to be an (M × N) matrix and B = (bij) to be an (R × P ) matrix The matrix product AB is defined only whenR = N, and is the (M × P ) matrix C = (cij) given by
cij =
N
X
k=1
aikbkj =ai1b1j +ai2b2j +· · · + aiNbNk Using the summation convention, this can be written simply
cij =aikbkj where the summation is understood to be over the repeated index k In the case of a 3 × 3 matrix multiplying a 3× 1 column vector we have
a11 a12 a13
a21 a22 a23
a31 a32 a33
b1
b2
b3
=
a11b1+a12b2+a13b3
a21b1+a22b2+a23b3
a31b1+a32b2+a33b3
=aijbj The comma convention uses a subscript comma to imply differentiation with respect to the variable following, so f,2 = ∂f/∂y and ui,j = ∂ui/∂xj For instance, the expression σij,j = 0 uses all of the three previously defined index conventions: range on i, sum on j, and differentiate:
∂σxx
∂σxy
∂σxz
∂z = 0
∂σyx
∂σyy
∂σyz
∂z = 0
∂σzx
∂σzy
∂σzz
∂z = 0 The Kroenecker delta is a useful entity is defined as
δij =
(
0, i 6= j
1, i = j This is the index form of the unit matrix I:
δij =I =
10 01 00
So, for instance
Trang 3σkkδij =
whereσkk =σ11+σ22+σ33
Trang 4Modules in Mechanics of Materials
List of Symbols
A area, free energy, Madelung constant
A plate extensional stiffness
a length, transformation matrix, crack length
aT time-temperature shifting factor
B design allowable for strength
B matrix of derivatives of interpolation functions
B plate coupling stiffness
C stress optical coefficient, compliance
C viscoelastic compliance operator
c numerical constant, length, speed of light
C.V coefficient of variation
D stiffness matrix, flexural rigidity of plate
D plate bending stiffness
d diameter, distance, grain size
E modulus of elasticity, electric field
E∗ activation energy
E viscoelastic stiffness operator
e electronic charge
eij deviatoric strain
fs form factor for shear
G viscoelastic shear stiffness operator
Gc critical strain energy release rate
g acceleration of gravity
GF gage factor for strain gages
I moment of inertia, stress invariant
J polar moment of inertia
K bulk modulus, global stiffness matrix, stress intensity factor
K viscoelastic bulk stiffness operator
k spring stiffness, element stiffness, shear yield stress, Boltzman’s constant
L matrix of differential operators
Trang 5M bending moment
N crosslink or segment density, moire fringe number, interpolation function, cycles to failure
N traction per unit width on plate
NA Avogadro’s number
N viscoelastic Poisson operator
n refractive index, number of fatigue cycles
ˆn unit normal vector
Pf fracture load, probability of failure
Ps probability of survival
p pressure, moire gridline spacing
Q force resultant, first moment of area
R radius, reaction force, strain or stress rate, gas constant, electrical resistance
r radius, area reduction ratio
S entropy, moire fringe spacing, total surface energy, alternating stress
s Laplace variable, standard deviation
SCF stress concentration factor
T temperature, tensile force, stress vector, torque
Tg glass transition temperature
tf time to failure
U∗ strain energy per unit volume
UTS ultimate tensile stress
˜
u approximate displacement function
V shearing force, volume, voltage
V∗ activation volume
u, v, w components of displacement
x, y, z rectangular coordinates
X standard normal variable
α, β curvilinear coordinates
αL coefficient of linear thermal expansion
γ shear strain, surface energy per unit area, weight density
δij Kroenecker delta
strain pseudovector
ij strain tensor
T thermal strain
θ angle, angle of twist per unit length
λ extension ratio, wavelength
Trang 6ν Poisson’s ratio
ρ density, electrical resistivity
Σij distortional stress
σ stress pseudovector
σij stress tensor
σe endurance limit
σf failure stress
σm mean stress
σM Mises stress
σt true stresss
σY yield stresss
τ shear stress, relaxation time
φ Airy stress function
ξ dummy length or time variable
Ω configurational probability
Trang 7Modules in Mechanics of Materials
Unit Conversion Factors
= 62.42 lb/ft3
= 0.03613 lb/in3
= 9.45×10−4 Btu
= 0.7376 ft-lb
= 6.250×1018 ev
= 1.124×10−4 ton (2000lb)
= 1.102×10−3 ton (2000lb)
= 0.7378 ft-lb/s
= 1.341× 10−3 hp
= 1.449×10−4 psi
= 1.020×10−7 kg/mm2
Toughness 1 MPa√
m = 0.910 ksi√
in Physical constants:
Boltzman constant k = 1.381× 10−23 J/K
Gas constant R = 8.314 J/mol-K
Avogadro constant NA= 6.022× 1023 /mol
Acceleration of gravity g = 9.805 m/s2