Glossary of Symbols

Scalars

²

A

Area, cross-section [ m² ].

B

Bandwidth of matrix [- ].

B

Boundary [ m² ].

C

Concentration.

C

Damping coefficient.

C

Hardening constant.

C_c

Compression index.

D

Interface stiffness modulus [ N/m³ ].

D

Linear stiffness modulus [ N/m² ].

D^II

Crack shear stiffness [ N/m² ].

E

Young's modulus [ N/m² ].

E_har

Hardening modulus [ N/m² ].

E_p

Hardening modulus [ N/m² ].

F

Concentrated force [ N] .

F

R.m.s. wavefront of matrix [ - ].

F^R

Reaction force [ N ].

F^r

Residual force [ N ].

G

Shear modulus [ N/m² ].

G

Energy release rate (LEFM).

G_c

Compressive fracture energy [ N/m ].

G_f

Fracture energy [ N/m² ].

H

Enthalpy [ J/m³ ].

H

Hardening parameter.

I

Moment of inertia [ m⁴ ].

J

Creep function [ m²/N ].

K

Bulk or compression modulus [ N/m² ].

K

Conduction coefficient [ W/(m^{2 .} K) ].

K

Spring stiffness [ N/m ].

K

Stress intensity factor (LEFM) [ - ].

K₀

Lateral pressure ratio [ - ].

K_D

Drained compression modulus [ N/m² ].

K_f

Fluid compression modulus [ N/m² ].

K_s

Solid compression modulus [ N/m² ].

L

Area coordinates.

M

Bending moment (Bernoulli) [ N ^. m ].

M

Concentrated mass [ kg ].

M

Concentrated moment [ N ^. m ].

M

Maturity.

M^R

Reaction moment [ N ^. m ].

M^r

Residual moment [ N ^. m ].

N

Interpolation polynomial [ - ].

N

Normal force (Bernoulli) [ N ].

N

Number of nodes [ - ].

N

Order of matrix [ - ].

OCR

Overconsolidation ratio [ - ].

P

Perimeter [ m ].

P

Potential.

P

Profile of matrix.

P_ex

External potential (LEFM).

Q

Discharge [ m³/s ].

Q

Produced heat [ J/m³ ].

Q

Second Biot material parameter [ N/m² ].

Q

Shear force (Bernoulli) [ N ].

RH

Relative humidity [ - ].

Re

Reynolds number [ - ].

S

Boundary [ - ].

S

Degree of saturation [ - ].

S

Second Piola-Kirchhoff stress [ N/m² ].

S

Shear stress correction factor [ - ].

S

Surface area [ m² ].

S_A

Spectral acceleration [ m/s² ].

S_D

Spectral displacement [ m ].

T

Period of time [ s ].

T

Temperature [ K ].

T

Transmissivity.

V

Volume [ m³ ].

W

Maximum wavefront of matrix [ - ].

W

Moment of rigidity [ m³ ].

W

Section modulus.

W

Strain energy function [ J ].

W_in

Elastic energy (LEFM) [ J ].

X

First global Cartesian coordinate [ m ], or axis.

Y

Second global Cartesian coordinate [ m ], or axis.

Z

Third global Cartesian coordinate [ m ], or axis.

a

Damping parameter.

a

Mode amplitude.

a

Crack length (LEFM) [m].

b

Damping parameter.

b

Width [ m ].

c

Capacitance [ J/(m^{3 .} K) ].

c

Cohesion [ N/m² ].

c

Incompressibility penalty factor.

c

Wave speed [ m/s ].

c_A

Arrhenius constant [ K ].

c_e

Elastic storativity.

c_p

Phreatic storativity [ - ].

co

Reinforcement coverage [ m ].

d

Diameter [ m ].

e

Deviatoric strain [ - ].

e

Element number [ - ].

e

Emissivity [ - ].

e

Void ratio [ - ].

f

Distributed force [ N/m , N/m² ].

f

Natural frequency [ 1/s ].

f

Volume source.

f_c

Compressive strength [ N/m² ].

f_cc

Cube compressive strength [ N/m² ].

f_ck

Characteristic strength [ N/m² ].

f_cm

Mean compressive strength [ N/m² ].

f_k

Characteristic strength [ N/m² ].

f_t

Tensile strength [ N/m² ].

g

Acceleration of gravity [ m/s² ].

h

Height [ m ].

h

Phreatic level [ m ].

h_cr

Crack bandwidth [ m ].

i

Counter [ - ].

j

Counter [ - ].

k

Conductivity [ W/(m ^. K) ].

k

Creep factor [ - ].

k

Interface stiffness modulus [ N/m³ ].

k

Permeability [ m² ].

k'

Modified permeability [ m⁴/(N ^. s) ].

l

Length [ m ].

m

Distributed moment [ N ^. m/m ].

m'

Reinforcement moment.

n

Degree of nonlinear elasticity [ - ].

n

Distributed in-plane force [ N/m ].

n

Node number [ - ].

n

Order of system matrix [ - ].

n

Porosity [ - ].

n

Traction axis, normal.

p

Isotropic strain invariant [ - ].

p

Polynomial.

p

Pore pressure [ N/m² ].

p

Pressure [ N/m² ].

p'

Effective pressure [ N/m² ].

p'_c

Preconsolidation stress [ N/m² ].

p_e

Excess pore pressure [ N/m² ].

p_head

Pressure head [ m ].

q

Deviatoric strain invariant [ - ].

q

Deviatoric stress [ N/m² ].

q

Distributed shear force [ N/m ].

q

Flux.

q

Specific discharge, distributed source.

r

Degree of reaction [ - ].

r

Distance from crack tip (LEFM) [m].

r

Maturity variable [ - ].

r

Radius [ m ].

s

Deviatoric stress [ N/m² ].

s

Length along crack front (LEFM) [m].

s

Traction axis.

t

Thickness [ m ].

t

Time [ s ].

t

Traction axis.

t

Traction [ N/m² ].

t_eq

Equivalent age of concrete [ s ].

u

Translational displacement [ m ].

$\dot{{u}}$

Velocity [ m/s ].

$\ddot{{u}}$

Acceleration [ m/s² ].

v

Speed (wind, water) [ m/s ].

v₀

Specific volume [ - ].

v_c

Convection velocity [ m/s² ].

x

First local Cartesian coordinate [ m ], or axis.

y

Second local Cartesian coordinate [ m ], or axis.

z

Third local Cartesian coordinate [ m ], or axis.

$\Delta$t

Increment of t [ s ].

$\Gamma$

Boundary (surface).

$\Gamma$

Euler gamma function.

$\Omega$

Domain.

$\Omega$

Excitation frequency [ Hz ].

$\Omega$

Rotation speed [ rad /s ].

$\Pi$

Energy potential [ J ].

$\Psi$

Generalized strain.

$\alpha$

First Biot material parameter [ - ].

$\alpha$

General diffusivity.

$\alpha$

Generalized modal displacement.

$\alpha$

Preconditioning parameter.

$\alpha$

Shape factor.

$\alpha$

Thermal expansion coefficient [ 1/K ].

$\alpha$

Time integration parameter.

$\beta$

Shear retention factor [ - ].

$\gamma$

Compressibility parameter.

$\gamma$

Concentration expansion coefficient.

$\gamma$

Decaying factor.

$\gamma$

Deviator strain [ - ].

$\gamma$

Green-Lagrange strain [ - ].

$\gamma$

Iteration parameter.

$\gamma$

Shape factor.

$\gamma$

Shear deformation [ - ].

$\gamma$

Structural damping factor.

$\gamma$

Modal participation factor.

$\gamma$

Volumetric weight [ N/m³ ].

$\gamma_{{\mathrm{f}}}^{}$

Volumetric fluid weight [ N/m³ ].

$\gamma^{{\mathrm{p}}}_{}$

Deviatoric plastic strain.

$\delta$

Extension, elongation [ m ].

$\delta$

Time integration parameter.

$\delta_{{ij}}^{}$

Kronecker delta [ - ].

$\delta$t

Change in $\Delta$t [ s ].

$\epsilon$

Convergence criterion [ - ].

$\epsilon$

Logarithmic strain [ - ].

$\epsilon$

Tolerance.

$\varepsilon$

Engineering strain [ - ].

$\varepsilon^{{\mathrm{C}}}_{}$

Concentration strain [ - ].

$\varepsilon^{{\mathrm{T}}}_{}$

Thermal strain [ - ].

$\varepsilon^{{\mathrm{cr}}}_{}$

Crack strain [ - ].

$\varepsilon^{{\mathrm{c}}}_{}$

Creep strain [ - ].

$\varepsilon^{{\mathrm{e}}}_{}$

Elastic strain [ - ].

$\varepsilon^{{\mathrm{p}}}_{}$

Plastic strain [ - ].

$\varepsilon^{{\mathrm{sh}}}_{}$

Shrinkage strain [ - ].

$\varepsilon$

Emission coefficient [ - ].

$\zeta$

Third parametric coordinate.

$\eta$

Second parametric coordinate.

$\eta$

Viscosity.

$\kappa$

Compression modulus.

$\kappa$

Curvature (Bernoulli) [ 1/m ].

$\kappa$

Equivalent plastic strain [ - ].

$\kappa$

Hardening parameter.

$\kappa$

Swelling index.

$\lambda$

Buckling value.

$\lambda$

Compression index.

$\lambda$

Eigenvalue.

$\lambda$

Loading parameter.

$\lambda$

Plastic multiplier [ - ].

$\lambda$

Relaxation time [ s ].

$\mu$

Dynamic viscosity [ N ^. s/m² ].

$\mu$

Friction coefficient (Coulomb).

$\mu$

Shifting factor [ - ].

$\mu$

Spring stiffness [ N/m ].

$\nu$

Kinematic viscosity [ m²/s ].

$\nu$

Poisson's ratio [ - ].

$\xi$

Damping ratio [ - ].

$\xi$

First parametric coordinate [ - ].

$\rho$

Mass density [ kg/m³ ].

$\rho_{{\mathrm{f}}}^{}$

Fluid density [ kg/m³ ].

$\sigma$

Stefan-Boltzmann constant [ J/(m^{2 .} s ^. K⁴) ].

$\sigma$

Stress (Cauchy) [ N/m² ].

$\sigma{^\prime}$

Effective stress [ N/m² ].

$\sigma{^\prime}$

In-situ stress [ N/m² ].

$\sigma^{{\mathrm{cr}}}_{}$

Crack stress [ N/m² ].

$\sigma_{{\mathrm{eq}}}^{}$

Equivalent stress (Von Mises) [ N/m² ].

$\sigma_{{\mathrm{y}}}^{}$

Yield stress [ N/m² ].

$\tau$

Dimensionless time [ - ].

$\upsilon$

Curvature (Bernoulli).

$\phi$

Friction angle.

$\phi$

Pore pressure potential [ N/m² ].

$\phi$

Potential.

$\phi$

Rotational displacement.

$\phi_{{\mathrm{head}}}^{}$

Hydraulic head [ m ].

$\phi_{{\mathrm{p}}}^{}$

Pressure component of hydraulic head.

$\varphi$

Phase angle.

$\psi$

Dilatancy angle.

$\omega$

Natural angular frequency.

$\omega$

Reinforcement percentage.

$\omega$

Rotational velocity (spin).

Vectors, Tensors

a

Pseudo-acceleration vector [ m/s² ].

f

Distributed forces.

f

Equivalent element forces.

f

Load vector.

f_w

Dead weight load.

f_$\phi$

Discharge vector [ m³/s ].

f_u

Force vector [ N ].

g

Body forces per unit volume.

g

Gravity acceleration vector [ m/s² ].

g

Out-of-balance forces.

i

Unity vector.

k

Conductivity tensor.

m

Distributed moments.

n

Normal vector.

n

Generalized element forces.

p

Pressure vector.

q

Flux vector [ m/s ].

r

Internal forces, reactions.

r

Residual vector.

t

Traction [ N/m² ].

u

Displacement vector [ m ].

$\dot{{\mathbf{u}}}$

Velocity vector [ m/s ].

$\ddot{{\mathbf{u}}}$

Acceleration vector [ m/s² ].

x

Position vector [ m ].

y

Result of forward substitution.

E

Strain tensor.

$\boldsymbol\beta$

Convective velocity field.

$\boldsymbol\gamma$

Green-Lagrange strains.

$\boldsymbol\varepsilon$

Strain vector.

$\boldsymbol\sigma$

Stress vector [ N/m² ].

$\boldsymbol\tau$

2nd Piola-Kirchhoff stresses.

$\boldsymbol\phi$

Buckling modes.

$\boldsymbol\phi$

Eigenvector.

$\boldsymbol\phi$

Pressure potentials vector [ N/m² ].

$\boldsymbol\varphi$

Phase angles.

Matrices

B

Interpolation matrix.

B

Strain-displacement relation.

C

Capacity matrix.

C

Compliance matrix.

C

Coupling matrix.

C

Damping matrix.

C

Right Cauchy-Green stretch tensor.

D

Diagonal matrix.

D

Elasticity matrix.

D

Material stiffness matrix.

D

Rigidity matrix.

E

Green-Lagrange strain tensor.

F

Deformation gradient.

H

Modified elastic stiffness matrix.

I

Identity (unity) matrix.

J

Jacobian matrix.

K

Conductivity matrix.

K

Permeability matrix.

K

Stiffness matrix.

L

Differential operator.

L

Lower triangular matrix.

M

Mass matrix.

N

Interpolation matrix.

P

Preconditioning matrix.

Q

Preconditioning matrix.

R

Rotation matrix.

S

Second Piola-Kirchhoff stress tensor.

S

Stress matrix.

S

System directions.

T

Element directions.

T

Transformation matrix.

U

Upper triangular matrix.

$\boldsymbol\Lambda$

Eigenvalue diagonal matrix.

Accents, sub- and superscripts

$\bar{{u}}$

Average of ...

$\ddot{{u}}$

Second time derivative of ...

$\dot{{u}}$

First time derivative of ...

$\hat{{u}}$

Amplitude of ...

$\tilde{{\mathbf{M}}}$

Added ...

K_B

...for the boundary.

K_F

...for the fluid domain.

K_I

...for the interface.

K_S

...for the structural domain.

$\sigma_{{{123}}}^{{}}$

Principal ...

u_XYZ

...in global orientation.

u_xyz

...in local orientation.

u^(e)

...in element orientation.

u^(s)

...in system orientation.

$\sigma_{{{n}}}^{{}}$

Normal ...

$\sigma_{{{t}}}^{{}}$

Tangential ...

K_e

...for an element.

S_n

...for a node.

f_dy

Dynamic ...

f_ext

External ...

f_int

Internal ...

f_st

Static ...

K ^*

Effective ...

$\sigma^{{0}}_{}$

Initial ....

_$\Im$u

Imaginary part of ...

_$\Re$u

Real part of ...

_$\varphi$u

Phase angle of ...