Figure 1 shows the reference and the current configurations of the element, and the kinematics in curvilinear coordinates. We describe the element kinematics through a linear combination of a pair of material points at the top and at the bottom surfaces of the element. Each point at the top or at the bottom surface of an element in the original configuration is labeled by its position vectors: **X**_{t} and **X**_{b}, respectively. The variables *ξ* and *η* are the local curvilinear coordinates in the two independent in-plane directions, and *ζ* is the local curvilinear coordinate in the thickness direction. The position of a material point in the undeformed configuration is written as a function of the three curvilinear coordinates:

- (1)

where **X**^{0}(*ξ*,*η*) is the projection of the point on the mid-surface of shell and **D** is the thickness director at this point. In conventional solid-like shell elements, they are obtained as:

- (2)

In an isogeometric formulation, these quantities are computed directly, as we will show in the remainder of this paper. The position of the material point in the deformed configuration **X**(*ξ*,*η*) is related to **X**(*ξ*,*η*) via the displacement field *ϕ*(*ξ*,*η*,*ζ*) as:

- (3)

where

- (4)

In this relation, **u**^{0} and **u**^{1} are the displacement of **X**^{0} on the shell mid-surface and the motion of the thickness director **D**, respectively. The projection of the material point onto the mid-surface leads to:

- (5)

and:

- (6)

Conventionally, the displacements **u**^{0} and **u**^{1} are calculated as:

- (7)

- (8)

which is convenient for applying the boundary conditions, although not strictly necessary in an isogeometric approach. In Equation (4), **u**^{2} is the internal stretching of the element, which is colinear with the shell thickness director **D** in the deformed configuration. This quantity is expressed in terms of stretch degree of freedom, *w*, through:

- (9)

In any material point, a local reference triad can be established. The covariant base vectors are then obtained as the partial derivatives of the position vectors with respect to the curvilinear coordinates *Θ* = [*ξ*,*η*,*ζ*]. In the undeformed configuration, they are defined as:

- (10)

where (.)_{,α} denotes the partial derivative with respect to Θ^{α}. **E**_{α} is the covariant base vector defined on the mid-surface:

- (11)

Similarly, in the deformed configuration, we have:

- (12)