48
3 Geometrically Nonlinear Theories
ε αβ =
0
ε αβ + Θ
3 1
ε αβ + (Θ
3
)
2 2
ε αβ ,
(3.68)
ε α3 =
0
ε α3 + Θ
3 1
ε α3 ,
(3.69)
ε 33 =
0
ε 33 ,
(3.70)
where the strain terms in the above equations are
2
0
ε αβ = ¯
a α · ¯
a β − a α · a β ,
(3.71)
2
1
ε αβ = ¯
a α · ¯
a 3,β + ¯
a 3,α · ¯
a β − a α · a 3,β − a 3,α · a β ,
(3.72)
2
2
ε αβ = ¯
a 3,α · ¯
a 3,β − a 3,α · a 3,β ,
(3.73)
2
0
ε α3 = ¯
a α · ¯
a 3 ,
(3.74)
2
1
ε α3 = ¯
a 3,α · ¯
a 3 ,
(3.75)
2
0
ε 33 = ¯
a 3 · ¯
a 3 − 1 .
(3.76)
Here, (
0
ε 11 ,
0
ε 22 ) represent the in-plane longitudinal strains, (
0
ε 12 ,
0
ε 21 ) are the in-plane
shear strains, (
1
ε 11 ,
1
ε 22 ) denote the bending strains, (
1
ε 12 ,
1
ε 21 ) are the torsional strains,
(
0
ε 13 ,
0
ε 23 ) are the transverse shear strains,
0
ε 33 denotes the transverse normal strain.
Additionally,
2
ε αβ ,
1
ε α3 are corrections respectively for the in-plane and shear strains.
Considering the relations given in (3.34) and (3.35), the Green-Lagrange strain
components can be obtained in terms of the base vectors and displacement vectors
in the undeformed configuration as
2
0
ε αβ = a α ·
0
u ,β +
0
u ,α · a β +
0
u ,α ·
0
u ,β ,
(3.77)
2
1
ε αβ = a α ·
1
u ,β +
0
u ,α ·
1
u ,β +
0
u ,α · n ,β
+
1
u ,α · a β +
1
u ,α ·
0
u ,β + n ,α ·
0
u ,β ,
(3.78)
2
2
ε αβ =
1
u ,α ·
1
u ,β +
1
u ,α · n ,β + n ,α ·
1
u ,β ,
(3.79)
2
0
ε α3 = a α ·
1
u + a α · n +
0
u ,α ·
1
u +
0
u ,α · n ,
(3.80)
2
1
ε α3 =
1
u ,α ·
1
u +
1
u ,α · n + n ,α ·
1
u + n ,α · n ,
(3.81)
2
0
ε 33 =
1
u ·
1
u +
1
u · n + n ·
1
u + n · n − 1 .
(3.82)
Substituting Eqs. (3.36), (3.37) and (3.48) into (3.77)–(3.82) yields the straindisplacement relations in terms of six parameters as