R
position vector for an arbitrary point in the shell space
r
position vector for an arbitrary point at the mid-surface
g i
covariant base vectors in the shell space
g
i
contravariant base vectors in the shell space
g ij
covariant metric tensor in the shell space
g
ij
contravariant metric tensor in the shell space
a i
covariant base vectors at the mid-surface
a
i
contravariant base vectors at the mid-surface
a ij
covariant metric tensor at the mid-surface
a
ij
contravariant metric tensor at the mid-surface
d
j
i
Kronecker delta
C
‚
ab
Christoffel symbols of the second kind
b ab
covariant components of the curvature tensor
b
a
b
mixed components of the curvature tensor
e
ijk
permutation symbol
u
displacement vector for an arbitrary point in the shell space
u
0
translational displacement vector at the mid-surface
u
1
rotational vector of the £
3 -line
v i
translational displacements in the shell space
v i
0
translational displacements at the mid-surface
v i
1
rotational displacements at the mid-surface
l
shifter tensor
l
i
j
components of the shift tensor
H
mean curvature of the surface
K
Gaussian curvature of the surface
V
volume of the parallelepiped spanned by the covariant base vectors
F
deformation gradient tensor
C
right Cauchy-Green tensor
G
Riemannian metric tensor
q
mass density
Y
Young’s modulus
t
Poisson’s ratio
G
shear modulus
"
Green-Lagrange strain vector
r
second Piola-Kirchhoff stress vector
c
elasticity constant matrix
e; d
piezoelectric constant matrix
dielectric constant matrix
D
electric displacement vector
E
electric field vector
T
kinetic energy
W int internal work
xiv
Acronyms