(a)Beam under large tensile deforma-tion due to axial forceF
(b) Nonlinear stress-strain relation in the case of the beam made of a nearly incompressible neo-Hookean material displayed in Fig. 1.5a. The beam’s normal strain componentεxx
associated with the axial direction is plotted against the corresponding stress componentσxx.
Figure 1.5. Illustration of material nonlinearity when using a nearly incompressible neo-Hookean material model
1.2 Geometrically nonlinear beam finite elements
In this section, the major focus lies on the geometrically nonlinear spatial beam formulation, the ANCF which has been well suited for the modelling of complicated systems of highly flexible structures in the context of multibody system dynamics.
This thesis illustrates that the ANCF can be widely used to solve problems within broader scope, in the area of general nonlinear finite element. To justify employing
1.2 Geometrically nonlinear beam finite elements 29
the ANCF as an underlying beam formulation and its capability in conjunction with the introduced finite element based approaches in this work, a brief review over the existing beam finite elements is presented as follows. Most broadly speaking, beam theories can be categorized into three major groups: induced beam theories [27],intrinsic beam theories [31] andsemi-induced beam theories.
The so-calledinducedbeam theories can be interpreted as reduced one-dimensional continuum theories consistently derived from the three-dimensional theory of elasticity with respect to the general continuum mechanics. With theinduced beams, motion and deformation can be described in the three-dimensional space on the basis of proper kinematic, kinetic and constitutive resultant quantities in the context of classical continuum mechanics. The aforementioned resultant constitutive quantities can for example be obtained via integration of three-dimensional stress measures over the beam cross-section. The three-three-dimensional stress measures typically emanates from the standard three-dimensional strain measures and constitutive relations. In this context, the cross-section of a beam represents the collection of all material points sharing the same beam length coordinate. These material points constituting the cross-section geometry, depending on the beam formulation employed, can be shear-free (e.g., Kirchhoff-Love beam theories [43, 50, 51]), shear-deformable (e.g., Simo-Reissner beam theories [77, 80]) or deformation-dependent i.e. fully deformable (e.g., ANCF [74, 73, 75]).
On the contrary, the intrinsic beam theories directly postulate the one-dimensional quantities associated with constitutive laws and are distinguished from the three-dimensional theory of elasticity. In the intrinsic beam theories, constitutive constants (tensor of material properties) relating stress and strain measures are experimentally measured, while the material properties of the induced beam theo-ries are directly obtained from the corresponding three-dimensional constitutive laws [54]. An application of the intrinsic beam lies for example within the scope of linear elastic fracture mechanics. Nonetheless, these measured quantities fulfill the essential mechanical principles such as equilibrium of forces and moments, the conservation of energy or existence of work conjugated corresponding to stress-strain pairs, frame indifference (objectivity) or path-independence of conservative problems.
Ultimately, a compromise between the induced and intrinsic theories considered so far is denotedsemi-inducedbeam theories. Therein, only the constitutive law is postulated and all the remaining kinetic and kinematic equations are consistently obtained from the three-dimensional elasticity known from the general continuum mechanics [54].
Based on the Stephen Timoshenko’s hypothesis of the shear deformable beam’s cross-sections, Reissner in [67] for the two-dimensional case and in [68] for the
general three-dimensional case, completed the theory by introducing two additional deformation measures representing the shear deformation of the beam. While the three-dimensional problem statement of Reissner was still based on some additional approximations, Simo [77] extended the work of Reissner to yield a formulation that is truly consistent in the sense of a semi-induced beam theory.
Thus, starting from a basic kinematic assumption, all kinetic and kinematic quantities are consistently derived from the three-dimensional continuum theory of elasticity, while the constitutive law has been postulated. Originally, theory characterised above has been denoted as Geometrically Exact Beam (GEB) theory.
Currently, it is also referred to as Simo-Reissner beam theory.
A beam theory is recognised as geometrically exact, if the relationships between the configuration and the strain measures remain consistent in terms of the principle of variational work and the equations of equilibrium (Cauchy’s equations of motion of the first kind) at a deformed state regardless of the magnitude of the translational and rotational displacements, and strains [77]. For that reason, the notation “finite-strain beams” has also been applied in the original work [77].
However, as further justified in [19], in the sense of the (fully) induced beam theories, the consistency of the GEB theory and the three-dimensional theory of elasticity known from the general continuum mechanics is only valid within the small strains regime. The enforcement of the basic kinematic constraints of rigid cross-sections underlying the GEB theory requires pointwise six translational and rotational degrees-of-freedom in order to uniquely describe the centreline position and orientation of the cross-sections along the beam’s longitudinal direction.
Theoretically, this beam theory can be identified as the one-dimensional Cosserat continuum beam [18, 3, 2], which in turn derived from a three-dimensional Boltzmann continuum with pointwise three translational (analogous to three-dimensional solid continua) degrees-of-freedom. On the other hand, in the ANCF, as well as all the kinetic and kinematic quantities and their relations, the constitutive relations and the resulted stress measures can be derived with respect to the continuum mechanics theory and nothing is postulated. Consequently, ANCF can be taken into account among the fully induced beam theories.
This is the point where the GEB theory is distinguished from the ANCF. The other substantial feature of ANCF beam or plate element is to employ spatial shape functions as known from solid finite element formulations in order to interpolate the three-dimensional displacement field within the beam. Instead of introducing a kinematic constraint and deriving a resultant one-dimensional model, different polynomial degrees are typically applied for the interpolation in beam length direction and in transverse directions. In the ANCF beam or plate, an element’s degree-of-freedom is prescribed with respect to what is known from the classic continuum mechanics in the sense that in spite of GEBs whose rotational degrees-of-freedom are derived frompoint-based (pointwise) one-dimensional Cosserat
1.3 Nonlinear finite elements for beam contact 31