Notation Reference

Notation Reference Guide

This section provides a comprehensive glossary of mathematical symbols used throughout the course.

Geometric and Kinematic Quantities

Symbol Meaning Chapter(s)
\(\vec{r}\) Position vector All
\(\vec{u}(\vec{r})\) Displacement field 4, 5, 6, 11
\(u_x, u_y, u_z\) Displacement components 4, 5, 6
\(w(x)\) Beam deflection (transverse displacement) 4, 10
\(\theta(x)\) Rotation angle (slope) 4, 10
\(\kappa(x)\) Curvature of beam 4, 10
\(\varepsilon_0(x)\) Axial strain at beam axis 4

Stress and Force Quantities

Symbol Meaning Chapter(s)
\(\underline{\sigma}(\vec{r})\) Cauchy stress tensor (3×3 matrix) 5, 8, 9, 12
\(\sigma_{ij}\) Stress tensor component 5, 8, 9, 12
\(\sigma_{xx}, \sigma_{yy}, \sigma_{zz}\) Normal stresses (diagonal) 5, 8, 9, 12
\(\tau_{xy}, \tau_{yz}, \tau_{xz}\) Shear stresses (off-diagonal) 5, 8, 9, 10, 11
\(\sigma_1, \sigma_2, \sigma_3\) Principal stresses 9, 12
\(\sigma_m = I_1/3\) Mean stress (hydrostatic pressure) 9, 12
\(\underline{s}\) Deviatoric stress tensor 12
\(\sigma_e\) Equivalent (von Mises) stress 12
\(\sigma_y\) Yield stress (uniaxial) 12
\(\sigma_h\) Hydrostatic stress 5, 12

Internal Force and Moment Quantities (Beams)

Symbol Meaning Chapter(s)
\(N(x)\) Normal (axial) force 3, 4, 10
\(Q(x)\) Shear force 3, 4, 10
\(M(x)\) Bending moment 3, 4, 10
\(q(x)\) Distributed load (force per unit length) 3, 4, 10
\(I_y\) Second moment of area (moment of inertia) 4, 10, 11
\(A\) Cross-sectional area 3, 4

Strain Quantities

Symbol Meaning Chapter(s)
\(\underline{\varepsilon}\) Strain tensor (small strain) 5, 6, 8, 9
\(\varepsilon_{ij}\) Strain tensor component 5, 6, 8, 9
\(\varepsilon_{xx}, \varepsilon_{yy}, \varepsilon_{zz}\) Normal strains 5, 6, 8
\(\gamma_{xy}, \gamma_{yz}, \gamma_{xz}\) Engineering shear strains 8
\(\varepsilon_h\) Volumetric strain 6, 8

Material Parameters

Symbol Meaning Chapter(s)
\(E\) Young’s modulus (elastic modulus) 4, 6, 8, 10
\(\nu\) Poisson’s ratio 6, 8, 11
\(\mu = G\) Shear modulus 6, 8
\(\lambda\) Lamé’s first constant 8
\(K\) Bulk modulus 8
\(\alpha, k\) Drucker-Prager material parameters 12
\(c\) Cohesion 12
\(\phi\) Friction angle 12

Operators and Special Symbols

Symbol Meaning Usage
\(\nabla\) Nabla (del) operator Gradient, divergence, curl
\(\nabla \cdot\) Divergence operator \(\nabla \cdot \underline{\sigma}\) = stress divergence
\(\nabla^2\) Laplacian operator Second derivative
\(\partial_x\) or \(f_{,x}\) Partial derivative w.r.t. \(x\) Same notation, different styles
\(\underline{I}\) or \(\delta_{ij}\) Identity tensor Kronecker delta
\(\times\) Cross product \(\vec{a} \times \vec{b}\)
\(\otimes\) Outer product (tensor product) \(\vec{a} \otimes \vec{b}\)
\(:\) Double contraction \(\underline{A} : \underline{B} = A_{ij}B_{ij}\)

Coordinate Systems and Transformation

Symbol Meaning Chapter(s)
\((x, y, z)\) Cartesian coordinates All
\(\hat{e}_i\) Unit basis vectors 2, 5
\(\hat{n}\) Unit normal vector 5
\(\underline{R}(t)\) Rotation matrix 2
\(\Omega(t)\) Skew-symmetric angular velocity matrix 2
\(\vec{\omega}\) Angular velocity vector 2

Boundary Conditions and Geometry

Symbol Meaning Chapter(s)
\(L\) Length of beam 3, 4, 10
\(h\) Height of cross-section 4, 10, 11
\(EI\) Flexural rigidity 4, 10, 11
\(K\) Effective length factor 10
\(\lambda = L/r\) Slenderness ratio 10

Force and Moment Quantities (Rigid Bodies)

Symbol Meaning Chapter(s)
\(\vec{F}\) Force vector 1, 2, 3
\(\vec{M}\) Moment (torque) vector 1, 2, 3
\(\vec{R}\) Resultant force 1