Historically, the first hypothesis on the bending behavior of elastic rods was devised by the Swiss mathematician and physicist J. Bernoulli in 1691, who postulated that the cross section of a beam remains straight and normal to its center line during deformation, which is denoted as normal hypothesis.
The first mathematically correct theory on the structural behavior of plates accounting for Bernoulli’s normal hypothesis was derived by G. Kirchhoff in 1850, which states that normals to the mid-surface in the undeformed configuration remain normal and unstretched in the deformed configuration (Kirchhoff(1850)).
Based on the work of Kirchhofffor plate problems, A. E. H. Love derived a general theory for both curved and plane surfaces (Love(1888)). The normality hypothesis for shell structures is therefore frequently associated with the term Kirchhoff-Love hypothesis.
In the 20th century, a multitude of Kirchhoff-Love type shell theories were developed (Reissner(1942),Green and Zerna(1954),Wunderlich(1966),Koiter(1960), Koiter(1961)), to name but a few. It lasted, however, until 1963, when the first consistent Kirchhoff-Love type shell theory was derived byNaghdi(1963).
Kinematics
In this section, the linearized Green-Lagrange strain tensorε3pis derived in curvilinear convective coordinates, as required for the virtual work expression of the Kirchhoff-Love
shell model. The derivation is based on the concept of degeneration discussed in Sec-tion 4.1.3, the differential shell geometry and kinematic specificaSec-tions of SecSec-tion 4.2 and common assumptions and approximations for thin shells to be established in the follow-ing.
The deformation behavior of the thin, elastic and homogeneous shells in this thesis is physically dominated by membrane and bending action. The constraint to be imposed on the director of staying normal to the mid-surface, also in the deformed configuration, yields vanishing transverse shear deformations. Rotations of cross-sectional fibers are therefore not independent, but equal to the gradient of the mid-surface displacement field. Additionally, extensibility of the thin shell in thickness direction and transverse normal strains and stresses are disregarded.
By summarizing these assumptions and simplifications, the kinematics of thin shells with surface parametrization, as displayed in Figure 4.1, can be described with three independent parameters (3-parameter formulation (3p)), which correspond to the mid-surface displacement componentsviof a material point.
The computation of the linearized Green-Lagrange strain tensor with Kirchhoff-Love kinematics requires the specification of several additional variables which have not been introduced so far.
First, the position vectorx3p to an arbitrary material point of the shell body in the deformed configuration is defined as
x3p&θ1,θ2,θ3'=r&θ1,θ2'+θ3a3p3
&
θ1,θ2'. (4.5)
The directora3p3 of the deformed configuration may be derived from a linearized rotation ofA3, as shown in the principal sketch of Figure 4.2 and described for the general case of a linearized vector rotation, for instance, inBelytschko et al.(2008)
a3p3 =A3+Φ×A3. (4.6)
Φ×A3represents the difference vector between the director of the undeformed config-urationA3and the rotated director of the current configurationa3p3. The orthogonality ofΦ×A3to the undeformed directorA3ensures satisfaction of the inextensibility con-straint of the shell in thickness direction during deformation by assuming linearized kinematics.
The rotation ofA3is defined by the rotation vectorΦ, which is a function ofAαand associated rotation anglesϕα. Φrepresents an element of the tangent space, which is spanned by the in-plane base vectorsAα.
e1
e2
e3
A1
A2
A3
a1
a2 a3p3
Φ×A3
θ1
θ2 θ3= 0
r x3p
Figure 4.2:Deformed configuration – 3-parameter model.
Φ=ϕ1A1+ϕ2A2 (4.7)
The associated rotation anglesϕαread
ϕ1 = 1
∥A1×A2∥(a2−A2)·A3 = 1
∥A1×A2∥v,2·A3,
ϕ2 = − 1
∥A1×A2∥(a1−A1)·A3 = − 1
∥A1×A2∥v,1·A3.
(4.8)
aαare the covariant base vectors of the current configuration andv,αrepresent the par-tial derivatives of the mid-surface displacement fieldvof the shell with respect to the in-plane convective coordinatesθα. ∥A1×A2∥denotes the Euclidean norm ofA1×A2. The computation of the rotation anglesϕαis therefore related to the projection of the rotated in-plane base vectorsaαin the direction of the undeformed normalA3. The difference of the position vectorsx3pandXyields the displacement fieldu3pat any point of the shell body. For the 3-parameter Kirchhoff-Love shell model, this results in
u3p =x3p−X
=r+θ3a33p−R−θ3A3
=v+θ3(Φ×A3).
(4.9)
The derivation of the Green-Lagrange strain tensor coefficientsε3pij requires the partial derivatives of the displacement fieldu3pwith respect to the convective coordinatesθi
u3p,α =v,α+θ3(Φ,α×A3+Φ×A3,α),
u3p,3 =a3p3 −A3=Φ×A3. (4.10)
The covariant base vectors of the shell body have already been defined in Equation (4.2).
The contravariant form is obtained from Equation (2.7), such thatε3p and its coeffi-cients can be finally computed according to Equation (2.16)
ε3p =ε3pij Gi⊗Gj, with
ε3pij =1 2
&
u3p,i ·Gj+u3p,j ·Gi '.
(4.11)
With the kinematic assumptions made so far, the strain tensor consists of constant, linear and quadratic components. The constant part ofε3p represents the membrane strains and the linear contributions are related to changes in curvature, i.e. bending. Quadratic terms in Equation (4.11) will be neglected in accordance with common assumptions of classical shell theories, which in general do not consider higher-order contributions for the stresses and strains or their resultants, respectively.
The choice of working with stresses and strains as kinematic and static variables in this work, however, does not preclude the use of quadratic components in general. The re-sulting error of neglecting quadratic terms remains acceptably small for thin shells with little curvature according to investigations performed inBaşar and Krätzig(1985) orBüchter(1992), for instance.
The individual strain tensor components for the presented 3-parameter Kirchhoff-Love shell model, which account for both constant and linear contributions, are
ε3p11 = v,1·A1
+θ3(v,1·A3,1+Φ,1×A3·A1), 2ε3p12= v,1·A2+v,2·A1
+θ3(v,1·A3,2+Φ,1×A3·A2+v,2·A3,1+Φ,2×A3·A1), ε3p22 = v,2·A2
+θ3(v,2·A3,2+Φ,2×A3·A2), ε3pi3 = 0.
(4.12)
All strain contributionsΦ×A3,α·Aβ in Equation (4.12) vanish: The vectorial quan-tities obtained from the cross products ofΦ and A3,α both lie in the tangent space spanned by the vectorsAαand are normal to the mid-surface. By subsequent scalar multiplication with the in-plane base vectorsAβ, the final result becomes zero.
Constitutive law
The linear elastic constitutive law for three-dimensional continua was defined in Sec-tion 2.4 according to EquaSec-tion (2.22). For the shell models of this chapter, the deriva-tion of the material tensor relies on the base vectors of Equaderiva-tion (4.2).
The inextensibility constraint requires zero transverse normal strains, i.e.ε3p33= 0. Zero transverse normal strains, however, do not automatically ensure zero transverse normal stresses σ3p,33, i.e. for nonzero Poisson’s ratio. Asymptotic correctness of the model therefore necessitates the modification of the constitutive law by implementing the stress assumptionσ3p,33= 0 to eliminateε3p33 via static condensation. Equations with regard to transverse shear are automatically equal to zero and therefore do not have to be considered for the Kirchhoff-Love model. Thus, the modified material tensorC3p only relates the in-plane stress componentsσ3p,αβto the in-plane strain componentsε3pαβ. Further details on the definition of the constitutive law for the Kirchhoff-Love shell model can be found, for instance, inBaşar and Krätzig(1985),Bischoff et al.
(2004) orBischoff(2011a).
Love’s first approximation, which simplifies the “true” shape of an infinitesimal cross-sectional area element by neglecting contributions with regard to curvature is not consid-ered in the definition of the material tensorC3p. Consequently, membrane and bending action are coupled due to nonzero off-diagonal blocks in the constitutive matrix.
Numerical integration of the discrete finite element equations is performed with three in-dependent nested loops, two for the in-plane directions (θα) and one across the thickness (θ3).
Internal virtual work
With the strain tensorε3pof Equation (4.12), its first variation and the material tensor C3pthe internal virtual work of the Kirchhoff-Love shell can be defined
δΠ3p, intPvw =.
Ω
δ&ε3p'T:C3p:ε3pdΩ. (4.13)
With regard to the formulation of Kirchhoff-Love-type shell finite elements in the sub-sequent chapter, the existence of second derivatives in the kinematics with respect to curvature requires at leastC1-continuity of the displacement field. The satisfaction of this condition is rather challenging when discretizations with standard Lagrange finite element basis functions are applied, but can be naturally satisfied with subdivision tech-niques or the higher-continuity NURBS discretizations of Chapter 3.