An isogeometric finite element formulation for geometrically exact Timoshenko beams with extensible directors

https://doi.org/10.1016/j.cma.2021.113993Get rights and content

Highlights

  • The extensible directors belong to the space R3, and describe in-plane cross-sectional deformations.

  • Enhanced assumed strain method is employed to alleviate Poisson locking.

  • A straightforward interface to general three-dimensional constitutive law is obtained.

  • The significance of considering correct surface loads rather than equivalent central axis loads is shown.

  • The developed beam formulation is verified by comparison with the results from brick elements.

Abstract

An isogeometric finite element formulation for geometrically and materially nonlinear Timoshenko beams is presented, which incorporates in-plane deformation of the cross-section described by two extensible director vectors. Since those directors belong to the space R3, a configuration can be additively updated. The developed formulation allows direct application of nonlinear three-dimensional constitutive equations without zero stress conditions. Especially, the significance of considering correct surface loads rather than applying an equivalent load directly on the central axis is investigated. Incompatible linear in-plane strain components for the cross-section have been added to alleviate Poisson locking by using an enhanced assumed strain (EAS) method. In various numerical examples exhibiting large deformations, the accuracy and efficiency of the presented beam formulation is assessed in comparison to brick elements. We particularly use hyperelastic materials of the St. Venant-Kirchhoff and compressible Neo-Hookean types.

Introduction

A rod (or rod-like body) can be regarded as a spatial curve, to which two deformable vectors, called directors are assigned. This curve is also called directed or Cosserat curve. The balance laws can be stated directly in terms of the curve velocity and director velocity vectors, and their work conjugate force and director force vectors, which eventually yields the equations of motion in the one-dimensional (curve) domain [1]. Since we actually deal with a three-dimensional continuum, one can consistently derive the equations of motion of the rod from those of the full three-dimensional continuum. This dimensional reduction, or degeneration procedure is based on a suitable kinematic assumption, and this dimensionally reduced theoretical model is referred to as beam model. An exact expansion of the position vector of any point of the beam at time t is given as [2] xt=p=0q=0p(ξ1)pq(ξ2)qd(pq,q)(ξ3,t),where ξγ (γ=1,2) denote the two coordinates in transverse (principal) directions of the cross-section plane, ξ3 denotes the coordinate along the central axis, and d(pq,q)(ξ3,t)1(pq)!q!pxt(ξ1)pq(ξ2)q.Using the full conservation laws of a three-dimensional continuum as a starting point, applying the kinematics in Eq. (1) offers an exact reparameterization of the three-dimensional theory into the one-dimensional one [2], [3]. However, this theory has infinite number of equations and unknowns, which makes it intractable for a finite element formulation and computation. The first order theory assumes the position vector to be a linear function of the coordinates ξγ, i.e. [2], [4] xt=φ(ξ3,t)+γ=12ξγdγ(ξ3,t),where φ(ξ3,t)d(0,0)(ξ3,t) denotes the position of the beam central axis, and two directors are denoted by d1(ξ3,t)d(1,0)(ξ3,t) and d2(ξ3,t)d(0,1)(ξ3,t). This approximation simplifies the strain field; it physically implies that planar cross-sections still remain planar after deformation, but allows for constant in-plane stretching and shear deformations of the cross-section. This implies that the linear in-plane strain field in the cross-section due to the Poisson effect in bending mode cannot be accommodated in the first order theory,1 which consequently increases the bending stiffness. This problem is often referred to as Poisson locking, and the resulting error does not reduce with mesh refinement along the central axis since the displacement field in the cross-section is still linear [6]. One may extend the formulation in Eq. (3) to quadratic displacement field in the cross-section by adding the second order terms about the coordinates ξγ in order to allow for a linear in-plane strain field. There are several theoretical works on this second order theory including the work by Pastrone [7] and on even higher Nth order theory by Antman and Warner [2]. Since shell formulations have only one thickness direction, higher-order formulations are simpler than for beams. Several works including Parisch [8], Brank et al. [9], and Hokkanen and Pedroso [10] employed second order theory in shell formulations. In beam formulations, several previous works considering the extensible director kinematics, which allows in-plane cross-section deformations, can be found. A theoretical study to derive balance equations and objective strain measures based on the polar decomposition of the in-plane cross-sectional deformation can be found in [11]. Further extension to initially curved beams was proposed in [12], where unconstrained quaternion parameters were utilized to represent both in-plane stretching and rotation of cross-sections. In those works, constitutive models are typically simplified to the form of quadratic strain energy density function. Durville [13] also employed a first order theory in frictional beam-to-beam contact problems, where the constitutive law was simplified to avoid Poisson locking. Coda [14] employed second order theory combined with an additional warping degree-of-freedom. However, it turns out that the linear in-plane strain field for the cross-section is not complete, so that the missing bilinear terms may lead to severe Poisson locking. In order to have a linear strain field in the cross-section with the increase of the number of unknowns minimized, one may extend the kinematics of Eq. (3) to x=φ(ξ3)+ξ11+a1ξ1+a2ξ2d1(ξ3)+ξ21+b1ξ1+b2ξ2d2(ξ3),where four additional unknown coefficient functions aγ=aγ(ξ3) and bγ=bγ(ξ3) (γ=1,2) are introduced. Here and hereafter, the dependence of variables on time t is usually omitted for brevity of expressions. This enrichment enables additional modes of the cross-sectional deformation (see Fig. 1 for an illustration), which are also induced in bending deformation due to the Poisson effect.

Eq. (4) recovers the kinematic assumption2 in [14] if a2=b1=0, which means the absence of bilinear terms, so that the trapezoidal cross-section deformation, shown in Fig. 1b, cannot be accomodated. Therefore, Poisson locking cannot be effectively alleviated. In this paper, we employ the enhanced assumed strain (EAS) method to circumvent Poisson locking in the first order theory. In order to verify the significance of those bilinear terms in Eq. (4), in a numerical example of Section 6.3, we compare two different EAS formulations based on five and nine enhanced strain parameters, respectively. The formulation of five enhanced strain parameters is obtained by ignoring the incompatible modes of trapezoidal cross-section deformation, i.e., it considers only the incompatible modes of Fig. 1a. The other one with nine enhanced strain parameters considers the whole set of incompatible linear cross-section modes, i.e., it considers both of the incompatible modes of Figs. 1a and 1b.

The enhanced assumed strain (EAS) method developed in [15] is based on the three-field Hu–Washizu variational principle. As the independent stress field is eliminated from the variational formulation by an orthogonality condition, it becomes a two-field variational formulation in terms of displacement and enhanced strain fields. Further, the enhanced strain parameters can be condensed out on the element level; thus the basic features of a displacement-based formulation are preserved. This method was generalized in the context of nonlinear problems in [16] in which a multiplicative decomposition of the deformation gradient into compatible and incompatible parts is used. One can refer to several works including Büchter et al. [17], Betsch et al. [18], Bischoff and Ramm [6], and Brank et al. [9] for EAS-based shell formulations. In this paper, we apply the EAS method to the beam formulation. Beyond previous beam formulations based on the kinematics of extensible directors, our work has the following highlights:

  • Consistency in balance equations and boundary conditions: The director field as well as the central axis displacement field satisfy the momentum balance equations and boundary conditions consistently derived from those of the three-dimensional continuum body. In the formulation of Coda [14] and Durville [13], there are no detailed expressions of balance equations, beam strains, and stress resultants. To the best of our knowledge, in those works, the finite element formulation can be obtained by substituting the beam kinematic expression of the current material point position into the deformation gradient of three-dimensional elasticity. This solid-like formulation yields an equivalent finite element formulation through a much more simplified derivation process. However, in addition to the possibility of applying mixed variational formulations in future works, the derivation of balance equations, beam strains, and stress resultants turns out to be significant in the interpretation of coupling between different strain components (for examples, see Sections 6.2.1 Coupling between bending and axial strains, 6.2.2 Coupling between bending and through-the-thickness stretch.)

  • We employ the EAS-method, where the additional strain parameters are statically condensed out, so that the same number of nodal degrees-of-freedom is used as in the pure displacement-based formulation. Each of the enhanced in-plane transverse normal strain components is linear in both of ξ1 and ξ2, which is in contrast to the strains obtained from the kinematic assumption in [14]. In the numerical example of Section 6.3, it is verified that this further enrichment alleviates Poisson locking more effectively.

  • Significance of correct surface loads: The consistently derived traction boundary condition shows that considering the correct surface load leads to an external director stress couple term that turns out to play a significant role in the accuracy of analysis.

  • Incorporation of general hyperelastic constitutive laws: As we consider the complete six stress components without any zero stress condition, our beam formulation naturally includes a straightforward interface for general three-dimensional constitutive laws.

  • Verification by comparison with brick element solution: We verify the accuracy and efficiency of our beam formulation by comparison with the results from brick elements.

It turns out that if linear shape functions are used to interpolate the director field, an artificial thickness stretch arises in bending deformations due to parasitic strain terms, and it eventually increases the bending stiffness. This effect is called curvature thickness locking. Since the parasitic terms vanish at the nodal points, the assumed natural strain (ANS) method interpolates the transverse normal (through-the-thickness) stretch at nodes instead of evaluating it at Gauss integration points [6], [19]. For membrane and transverse shear locking, there are several other existing methods, for examples, selective reduced integration method in [20], Greville quadrature method in [21], and mixed variational formulation in [22], [23]. However, since curvature-thickness, membrane, and transverse shear locking issues become less significant by mesh refinement or higher-order basis functions, especially in low to moderate slenderness ratio of our interests, no special treatment is implemented in this paper (see the investigation on those locking issues in Section 6.2.5). Further investigation on the application of existing method remains future work.

If we restrict the two directors in Eq. (3) to be orthonormal, which physically means that the cross-section is rigid, large rotations of the cross-section can be described by an orthogonal transformation. In planar static problems, Reissner [24] derived the force and moment balance equations, from which the strain–displacement relation is obtained via the principle of virtual work and work conjugate relations. Since this approach poses no assumption on the magnitude of deformations, it is often called geometrically exact beam theory. This work was extended to three-dimensional dynamic problems by Simo [25], which was followed by the finite element formulation of static problems in [26]. An additional degree-of-freedom related to torsion-warping deformation was added in [27], and this work was extended by Gruttmann et al. [28] to consider eccentricity with arbitrary cross-section shapes. There have been a number of works on the parameterization of finite rotations, and the multiplicative or additive configuration update process. One may refer to the overviews on this given by Meier et al. [29] and Crisfield and Jelenić [30]. In [30], it was pointed out that the usual spatial discretization of the strain measures in [26] leads to non-invariance of the interpolated strain measures in rigid body rotation, even though the strain measures in continuum form are objective. This non-objectivity stems from the non-commutativity, i.e., non-vectorial nature of the finite rotation. To retain the objectivity of strain measures in the underlying continuum formulation, the isoparametric interpolation of director vectors is used instead of interpolating the rotational parameters (see for example [31], [32], [33]), and the subsequent weak form of finite element formulation is reformulated. As those beam formulations still assume rigid cross-sections, the orthonormality condition of the director vectors should be satisfied. Several methods to impose the constraint can be found in the literature, examples are the Lagrange multiplier method [31], [33], and the introduction of nodal rotational degrees-of-freedom [31], [32]. In order to preserve the objectivity and path-independence in the rotation interpolation, several methods have been developed; for examples, orthogonal interpolation of relative rotation vectors [30], [34], geodesic interpolation [35], interpolation of quaternion parameters [36]. Romero [37] compared several rotation interpolation schemes in perspective of computational accuracy and efficiency. A more comprehensive review on geometrically exact finite element beam formulations can be found in [38]. In the isoparametric approximation of directors, employed in our beam formulation, the director vectors belong to R3, that is, no orthonormality condition is imposed. This means that the cross-section can undergo in-plane deformations like transverse normal stretch and in-plane shear deformations. Further, it enables us to avoid the rotation group, which is a nonlinear manifold, in the configuration space of the beam, and consequently complicates the configuration and strain update process [13]. Coda [14] and Coda and Paccola [39], who employed an isoparametric interpolation of directors without orthonormality condition, presented several numerical examples showing the objectivity and path-independence of the finite element formulation.

Classical beam theories introduce the zero transverse stress condition based on the assumption that the transverse stresses are much smaller than the axial and transverse shear stresses. Thus, six stress components in the three-dimensional theory reduce to three components including the transverse shear components in the Timoshenko beam theory. However, this often complicates the application of three-dimensional nonlinear material laws, and requires a computationally expensive iteration process. Global and local iteration algorithms to enforce the zero stress condition at Gauss integration points were developed in [40], [41], respectively. One can also refer to several recent works on Kirchhoff–Love shell formulations with general three-dimensional constitutive laws, where the transverse normal strain component can be condensed out by applying the plane stress condition in an analytical or iterative manner, for example, for hyperelasticity by Kiendl et al. [42] and Duong et al. [43], and elasto-plasticity by Ambati et al. [44]. There are several other finite element formulations to dimensionally reduce slender three-dimensional bodies and incorporate general three-dimensional constitutive laws. The so-called solid beam formulation uses a single brick element3 in thickness direction. To avoid severe stiffening effects typically observed in low-order elements, a brick element was developed based on the EAS method in geometrically nonlinear problems [45]. A brick element combined with EAS, ANS, and reduced integration methods in order to alleviate locking was presented in [46]. The absolute nodal coordinate (ANC) formulation uses slope vectors as nodal variables to describe the orientation of the cross-section. The fully parameterized ANC element enables straightforward implementation of general nonlinear constitutive laws. A comprehensive review on the ANC element can be found in [47], and one can also refer to a comparison with the geometrically exact beam formulation in [48]. Wackerfuß and Gruttmann [22], [23] presented a mixed variational formulation, which allows a straightforward interface to arbitrary three-dimensional constitutive laws, where each node has the common three translational and three rotational degrees-of-freedom, as the additional degrees-of-freedom are eliminated on element level via static condensation.

Isogeometric analysis (IGA) was introduced in [49] to bridge the gap between computer-aided design (CAD) and computer-aided engineering (CAE) like finite element analysis (FEA) by employing non-uniform rational B-splines (NURBS) basis functions to approximate the solution field as well as the geometry. IGA enables exact geometrical representation of initial configuration in CAD to be directly utilized in the analysis without any approximation even in coarse level of spatial discretization. Further, the high-order continuity in NURBS basis function is advantageous in describing the beam and shell kinematics under the Kirchhoff–Love constraint, which requires at least C1-continuity in the displacement field. IGA was utilized for example in [42], [43], [44] for Kirchhoff–Love shells, and in [50] for Euler–Bernoulli beams. For geometrically exact Timoshenko beams, an isogeometric collocation method was presented by Marino [51], and it was extended to a mixed formulation in [52]. An isogeometric finite element formulation and configuration design sensitivity analysis were presented in [53]. Recently, Vo et al. [54] used the Green–Lagrange strain measure with the St. Venant-Kirchhoff material model under the zero stress condition. There have been several works to develop optimal quadrature rules for higher order NURBS basis functions to alleviate shear and membrane locking, for examples, a selective reduced integration in [20], and Greville quadrature in [21]. Since our beam formulation allows for additional cross-sectional deformations from which another type of locking due to the coupling between bending and cross-section deformations appears, it requires further investigation to apply those quadrature rules to our beam formulation, which remains future work.

There are many applications where one may find deformable cross-sections of rods or rod-like bodies with low or moderate slenderness ratios. Although one can find many beam structures with open and thin-walled cross-sections in industrial applications, which requires to consider torsion-warping deformations, we focus on convex cross-sections in this paper, and the incorporation of out-of-plane deformations in the cross-section remains future work. Our beam formulation is useful for the analysis of beams with low to moderate slenderness ratios, where the deformation of cross-section shape is significant, for examples, due to local contact or the Poisson effect. For example, our beam formulation can be applied to the analysis of lattice or textile structures where individual ligaments or fibers have moderate slenderness ratio, and coarse-grained modeling of carbon nanotubes and DNA. Those applications are often characterized by circular or elliptical cross-section shapes. For highly slender beams, it has been shown that the assumption of undeformable cross-sections and shear-free deformations, i.e., Kirchhoff–Love theory, can be effectively and efficiently utilized [38], since it enables to further reduce the number of degrees-of-freedom and avoid numerical instability due to the coupling of shear and cross-sectional deformations with bending deformation. This formulation was successfully applied to contact problems, for example, contact interactions in complex system of fibers [55]. As the slenderness ratio decreases, the analysis of local contact with cross-sectional deformations becomes significant. One example is the coupling between normal extension of the cross-section and bending deformation that can be found in the works of Naghdi and Rubin [56] and Nordenholz and O’Reilly [57]. Especially, Naghdi and Rubin [56] illustrated that the difference in the transverse normal forces on the upper and lower lateral surfaces leads to flexural deformation via the Poisson effect. They also showed that the consideration of transverse normal strains plays a significant role to accurately predict a continuous surface force distribution. Another example that can lead to significant deformation of the beam cross section is local contact and adhesion of soft beams. For example, in [58], the adhesion mechanism of geckos was described by beam-to-rigid surface contact, where no deformation through the beam thickness was assumed, even though local contact can be expected to have a significant influence on beam deformation. Olga et al. [59] applied the Hertz theory to incorporate the effect of cross-section deformation in beam-to-beam contact, where the penalty parameter in the contact constraint was obtained as a function of the amount of penetration. Another interesting application can be found in the development of continuum models for atomistic structures like carbon nanotubes. Kumar et al. [60] developed a beam model for single-walled carbon nanotubes that allows for deformation of the nanotube’s lateral surface in a one-dimensional framework, which can be an efficient substitute to two-dimensional shell models.

The remainder of this paper is organized as follows. In Section 2, we present the beam kinematics based on extensible directors. In Section 3, we derive the momentum balance equations from the balance laws of a three-dimensional continuum, and define stress resultants and director stress couples. In Section 4.1, we derive the beam strain measures that are work conjugate to the stress resultants and director stress couples. Further, the expression of external stress resultants and director stress couples are obtained from the surface loads. In Section 4.2 we detail the process of reducing three-dimensional hyperelastic constitutive laws to one-dimensional ones. In Section 5, we present the enhanced assumed strain method to alleviate Poisson locking. In Section 6, we verify the developed beam formulation in various numerical examples by comparing the results with those of IGA brick elements. For completeness, appendices to the beam formulation and further numerical examples are given in Appendix A Appendix to the beam formulation, Appendix B Appendix to numerical examples, respectively.

Section snippets

Beam kinematics

The configuration of a beam is described by a family of cross-sections whose centroids4 are connected by a spatial curve referred to as the central axis. An initial (undeformed) configuration of the central axis C0 is given by a spatial curve parameterized by a parametric coordinate ξR1, i.e., C0:ξφ0(ξ)R3. The initial configuration of the central axis is

Three-dimensional elasticity

We recall the equilibrium equations and boundary conditions of a three-dimensional deformable body, which occupies an open domain Bt bounded by the boundary surface StBt in the current configuration. The boundary is composed of a prescribed displacement boundary StD and a prescribed traction boundary StN, which are mutually disjoint, i.e.6 St=StDStN,andStDStN=.The

Weak form of the governing equation

We define a variational space by VδyδφT,δd1T,δd2TTH1(0,L)dδφ=δd1=δd2=0onΓD,where H1(0,L) defines the Sobolev space of order one which is the collection of all continuous functions whose first order derivatives are square integrable in the open domain (0,L)s. Here the components of δy in the global Cartesian frame are considered as independent solution functions, so that the dimension becomes d=9. In the following, we restrict our attention to the static case. By multiplying the linear and

Alleviation of Poisson locking by the EAS method

In order to alleviate Poisson locking, the in-plane strain field in the cross-section should be at least linear. We employ the EAS method, and we modify the Green–Lagrange strain tensor as E=Eccompatible+Ẽenhanced,where the compatible strain part is the same as in Eq. (77), and the additional strain part Ẽ, which is incompatible, is intended to enhance the in-plane strain components of the cross-section, expressed by Ẽ=ẼαβGαGβ.The enhanced strain components are assumed as the linear and

Numerical examples

We verify the presented beam formulation by comparison with reference solutions from the isogeometric analysis of three-dimensional hyperelasticity using brick elements. The brick elements use different degrees of basis functions in each parametric coordinate direction. We denote this by ‘deg.=(pL,pW,pH)’, where pL, pW, and pH denote the degrees of basis functions along the length (L), width (W), and height (H), respectively. Further, we indicate the number of elements in each of those

Conclusions

In this paper, we present an isogeometric finite element formulation of geometrically exact Timoshenko beams with extensible directors. The presented beam formulation has the following advantages.

  • The extensible director vectors allow for the accurate and efficient description of in-plane cross-sectional deformations.

  • They belong to the space R3, so that the configuration can be additively updated.

  • In order to alleviate Poisson locking, the complete in-plane strain field has been added in the form

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

M.-J Choi would like to gratefully acknowledge the financial support of a postdoctoral research fellowship from the Alexander von Humboldt Foundation in Germany .

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