We define the geometrically exact beam as a structural element that has a special configuration, that is a reference configuration, where the beam is a three-dimensional body with the position field

- (11)

In this configuration, as depicted in Figure 2, an appropriate chart *φ* gives us the body coordinates , which are referred to as convected coordinates. The space curve *ϕ*_{0}(*s*) = **X**(0,0,*s*) is the reference curve of the beam and is bounded by its ends *s* = *s*_{1} and *s* = *s*_{2} for *s*_{2} > *s*_{1}. At every material point *s* of the reference curve *ϕ*_{0}, we have attached a positively oriented orthonormal director triad {**d**_{k}}. The two directors **d**_{α} span the plane cross section of the beam. The area of the cross section is parametrized by the coordinates . The director triad {**d**_{k}} can be related to an orthonormal basis {**e**_{k}} fixed in space by introducing the rotation tensor **R**_{0}(*s*) ∈ *SO*(3) such that

- (12)

According to Section 2, the beam is in its reference configuration, a three-dimensional body. This implies bijectivity of the position field **X**(*θ*^{α},*s*). Because of the kinematical constraints (11), it can happen that interpenetration occurs when the beam is curved too much or the cross sections are too large, compare with Chapter 5 of Rubin [18]. As in the three-dimensional theory, we exclude for the reference configuration interpenetration that is expressed mathematically as **X**,_{1} ⋅ (**X**,_{2} × **X**,_{3}) > 0 for all . That is the reason why beam theories are generally related to slender bodies.

The motion of the geometrically exact beam is the restricted position field

- (13)

We call the space curve *ϕ*(*s*,*t*) = **x**(0,0,*s*,*t*), which is a bijective function of *s* and time *t*, the center line of the beam. At each material point of the center line *ϕ*, we have attached a positively oriented orthonormal director triad {**d**_{k}}, which is related to the basis {**e**_{k}} by introducing the rotation tensor **R**(*s*,*t*) ∈ *SO*(3) such that

- (14)

The directors **d**_{α} describe the current state of the cross section . In the current configuration, we merely ask the center line *ϕ* to be a bijective function. That means, interpenetration of the cross sections may occur. Insofar, the beam is strictly speaking not a three-dimensional body anymore but a one-dimensional body. Because every material point can be addressed by the convected coordinates *θ*^{k} and the variations *δ***x** have to comply with the restricted kinematics (13), the dynamics of the constrained system can be derived by the principle of virtual work (7).

In the following, we introduce the effective curvature, the angular velocity, and the virtual rotation. All three objects describe the change of the directors when changing a single parameter as, for example, the parameter *s*. Using (14), the effective curvature is obtained by

- (15)

Because the present finite element formulation (see Section 4) relies on the nodal interpolation of the director frame, the directors generally do not span an orthornormal basis anymore. That is, on element level the metric coefficients *d*^{ij} = **d**^{i} ⋅ **d**^{j} and *d*_{ij} = **d**_{i} ⋅ **d**_{j} in general do not coincide with the Kronecker delta *δ*_{ij}. In other words, on element level, the tensor **R** introduced in (14) generally does not belong to the special orthogonal group. Inspired by this observation, we relax the orthonormality condition on the director frame and consider metric coefficients that are merely assumed to be constant. Under this condition, the tensor still remains skew-symmetric. This is shown in the Appendix. Hence, the skew-symmetric effective curvature has an associated axial vector so that

- (16)

Later in this paper, we are interested in the contravariant components of the effective curvature, which can be written as (cf. Schade & Neemann [19])

- (17)

where *ϵ*_{ijk} are the alternating symbols and

- (18)

Analogously to (16), we introduce the angular velocity and its associated axial vector *ω* as

- (19)

In the same manner, as for the effective curvature and the angular velocity, we obtain the virtual rotation when changing the variational parameter *ϵ*.

- (20)

The last expression of the virtual rotation vector is motivated by the following identity and is used in the sequel for the director formulation.

- (21)

The velocity and acceleration fields are introduced by taking the total time derivative of the position field (13) and the kinematical relation introduced in (19).

- (22)

- (23)

With regard to (13) and (16), the partial derivatives of the position field assume the form

- (24)

Using (13), (20) and (24), the variation of the position field and its partial derivatives are given by

- (25)

In the Appendix, we verify the important identity

- (26)

which is obtained by the fact that the derivative with respect to *s* and the variation commute, that is, (*δ***d**_{k}),_{s} = *δ*(**d**_{k}),_{s}.