Matrix 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 vector u can be denoted as u x ,u y ,u z , 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,andzdirections; the displacement vector can therefore be written equivalently as u 1 ,u 2 ,u 3 . A common and useful shorthand is simply to write the displacement vector as u i ,wherethe isubscript 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 equation u i = a implies three separate scalar equations: u 1 = a u 2 = a u 3 = 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 = u i = u 1 u 2 u 3 = u x u y u z 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]= σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 Here the first subscript index denotes the row and the second the column. The indices also have a physical meaning, for instance σ 23 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 σ ij ,where both the i and the j range from 1 to 3; this gives the nine components listed explicitly above. 1 (Since the stress matrix is symmetric, i.e. σ ij = σ ji , 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: σ kk = 3 k=1 σ kk = σ 11 + σ 22 + σ 33 The multiplication rule for matrices can be stated formally by taking A =(a ij )tobean (M×N)matrixandB=(b ij )tobean(R×P) matrix. The matrix product AB is defined only when R = N,andisthe(M×P)matrixC=(c ij )givenby c ij = N k=1 a ik b kj = a i1 b 1j + a i2 b 2j + ···+a iN b Nk Using the summation convention, this can be written simply c ij = a ik b kj where the summation is understood to be over the repeated index k.Inthecaseofa3×3 matrix multiplying a 3 × 1 column vector we have a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 b 1 b 2 b 3 = a 11 b 1 + a 12 b 2 + a 13 b 3 a 21 b 1 + a 22 b 2 + a 23 b 3 a 31 b 1 + a 32 b 2 + a 33 b 3 = a ij b j The comma convention uses a subscript comma to imply differentiation with respect to the variable following, so f ,2 = ∂f/∂y and u i,j = ∂u i /∂x j . For instance, the expression σ ij,j =0 uses all of the three previously defined index conventions: range on i, sum on j, and differentiate: ∂σ xx ∂x + ∂σ xy ∂y + ∂σ xz ∂z =0 ∂σ yx ∂x + ∂σ yy ∂y + ∂σ yz ∂z =0 ∂σ zx ∂x + ∂σ zy ∂y + ∂σ zz ∂z =0 The Kroenecker delta is a useful entity is defined as δ ij = 0,i=j 1,i=j This is the index form of the unit matrix I: δ ij = I = 100 010 001 So, for instance 2 σ kk δ ij = σ kk 00 0σ kk 0 00σ kk where σ kk = σ 11 + σ 22 + σ 33 . 3 Modules in Mechanics of Materials List of Symbols A area, free energy, Madelung constant A transformation matrix A plate extensional stiffness a length, transformation matrix, crack length a T time-temperature shifting factor B design allowable for strength B matrix of derivatives of interpolation functions B plate coupling stiffness b width, thickness 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 e ij deviatoric strain F force f s form factor for shear G shear modulus G viscoelastic shear stiffness operator G c critical strain energy release rate g acceleration of gravity GF gage factor for strain gages H Brinell hardness h depth of beam I moment of inertia, stress invariant I identity matrix 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 length, beam span L matrix of differential operators 1 M bending moment N crosslink or segment density, moire fringe number, interpolation function, cycles to failure N traction per unit width on plate N A Avogadro’s number N viscoelastic Poisson operator n refractive index, number of fatigue cycles ˆn unit normal vector P concentrated force P f fracture load, probability of failure P s probability of survival p pressure, moire gridline spacing Q force resultant, first moment of area q distributed load R radius, reaction force, strain or stress rate, gas constant, electrical resistance R Reuter’s matrix r radius, area reduction ratio S entropy, moire fringe spacing, total surface energy, alternating stress S compliance matrix s Laplace variable, standard deviation SCF stress concentration factor T temperature, tensile force, stress vector, torque T g glass transition temperature t time, thickness t f time to failure U strain energy U ∗ strain energy per unit volume UTS ultimate tensile stress ˜u approximate displacement function V shearing force, volume, voltage V ∗ activation volume v velocity W weight, work u, v, w components of displacement x, y, z rectangular coordinates X standard normal variable α, β curvilinear coordinates α L coefficient of linear thermal expansion γ shear strain, surface energy per unit area, weight density δ deflection δ ij Kroenecker delta normal strain strain pseudovector ij strain tensor T thermal strain η viscosity θ angle, angle of twist per unit length κ curvature λ extension ratio, wavelength 2 ν Poisson’s ratio ρ density, electrical resistivity Σ ij distortional stress σ normal stress σ stress pseudovector σ ij stress tensor σ e endurance limit σ f failure stress σ m mean stress σ M Mises stress σ t true stresss σ Y yield stresss τ shear stress, relaxation time φ Airy stress function ξ dummy length or time variable Ω configurational probability ω angular frequency ∇ gradient operator 3 Modules in Mechanics of Materials Unit Conversion Factors Density 1 Mg/m 3 = 1 gm/cm 3 = 62.42 lb/ft 3 = 0.03613 lb/in 3 = 102.0 N/m 3 Energy 1 J = 0.2390 calorie =9.45×10 −4 Btu =10 7 erg = 0.7376 ft-lb = 6.250×10 18 ev Force 1 N = 10 5 d (dyne) = 0.2248 lbf = 0.1020 kg = 3.597 oz = 1.124×10 −4 ton (2000lb) Length 1 m = 39.37 in = 3.281 ft =10 10 ˚ A Mass 1 kg = 2.205 lb = 35.27 oz = 1.102×10 −3 ton (2000lb) Power 1 W = 1 J/s = 0.7378 ft-lb/s =1.341 × 10 −3 hp Stress 1 Pa = 1 N/m 2 =10d/cm 2 = 1.449×10 −4 psi = 1.020×10 −7 kg/mm 2 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 N A =6.022 × 10 23 /mol Acceleration of gravity g =9.805 m/s 2 1 . vector can therefore be written equivalently as u 1 ,u 2 ,u 3 . A common and useful shorthand is simply to write the displacement vector as u i ,wherethe isubscript is an index that is assumed to. grain size E modulus of elasticity, electric field E ∗ activation energy E viscoelastic stiffness operator e electronic charge e ij deviatoric strain F force f s form factor for shear G shear modulus G. spacing, total surface energy, alternating stress S compliance matrix s Laplace variable, standard deviation SCF stress concentration factor T temperature, tensile force, stress vector, torque T g glass