Abstract
This paper presents an isogeometric finite element formulation for nonlinear beams with impenetrability constraints, based on the kinematics of Cosserat rods with unconstrained directors. The beam crosssectional deformation is represented by director vectors of an arbitrary order. For the frictionless lateral beamtobeam contact, a surfacetosurface contact algorithm combined with an active set strategy and a penalty method is employed. The lateral boundary surface of the beam is parameterized by its axis and crosssectional boundary curves with NURBS basis functions having at least \(C^2\)continuity, which yields a continuous surface metric and curvature for the closest point projection. Threedimensional constitutive laws of hyperelastic materials are considered. Several numerical examples verify the accuracy and efficiency of the proposed beam contact formulation in comparison to brick element solutions. The lateral contact pressure distribution of the beam formulation is in excellent agreement with the contact pressure of the brick element formulation while requiring much less degreesoffreedom.
Introduction
The simulation of interacting rods or rodlike bodies has been investigated across many applications, including wire strands [26], cables [2], biopolymer networks [8], woven fabrics [9, 12], entangled fibrous materials [36], DNA supercoiling [18], and deformations in adhesive microstructures [38]. In such examples the large number of bodies and their contact interactions typically cause significant computational costs, which calls for the development of efficient and accurate beam and beamtobeam contact formulations.
A beam in solid mechanics refers to a dimensionally reduced model of a threedimensional slender body based on a suitable kinematic assumption. In Cosserat (or directed) rod theory, a material point position in the current configuration, in threedimensional space, is given by [29]
where \(\varvec{\varphi }(s)\) denotes the position vector of the current axis,^{Footnote 1} and \(\varvec{d}_\gamma (s)\,(\gamma \in \left\{ 1,2\right\} )\) represent two director vectors spanning the planar crosssection. s denotes the arclength coordinate along the initial axis, and \(\zeta ^1\) and \(\zeta ^2\) denote two transverse coordinates of the crosssection. This kinematic expression is sufficiently general to encompass constrained theories like Euler–Bernoulli, and Timoshenko beams [32]. One can refer to Meier et al. [24] and the references therein for various nonlinear beam formulations with orthonormality constraints on the two directors, which are typically satisfied by a parameterization using an orthogonal tensor with three rotational degreesoffreedom (DOFs) in space, or constraints of shearfree deformations. There are several previous works employing Eq. (1) in its unconstrained form, which allows to use nine DOFs per crosssection, for example, Rhim and Lee [35], Durville [10] and Choi et al. [5]. The same DOFs appear in the brick element formulation with six nodes in Schweizerhof et al. [41] and Konyukhov et al. [15] that use polar coordinates for the elliptical crosssection, combined with a linear approximation along the longitudinal direction.
The extensibility of the two directors in Eq. (1) gives several advantages including an additive configuration update procedure, inplane crosssectional deformations, and a straightforward implementation of threedimensional constitutive laws. Such formulations have been considered in several works. For example, Frischkorn and Reese [11] presented a brick element formulation combined with an enhanced assumed strain (EAS) method, an assumed natural strain (ANS) method, and a reduced integration method in order to alleviate locking. Wackerfuß and Gruttmann [45] developed a mixed hybrid finite beam element formulation where an arbitrary threedimensional constitutive laws can be easily implemented. The first order expression of Eq. (1) in terms of the transverse coordinates leads to constant inplane crosssectional strains, which suffers from artificial increase of bending stiffness for nonzero Poisson’s ratio. In order to circumvent this Poisson locking, two quadratic order terms are additionally introduced in Coda [6]; however, this formulation still suffers from locking due to the missing bilinear terms. An EAS method of enriching all linear inplane Green–Lagrange strain components was verified to effectively alleviate Poisson locking in Choi et al. [5].
It was shown in Naghdi and Rubin [30] that the consideration of transverse normal strains of the crosssection significantly contributes to correctly predict the contact force distribution. In Kawa et al. [14], a penalty parameter depending on the amount of penetration due to the Hertz theory of elastic contact is utilized within a small strain range in order to consider crosssectional strains. The beam kinematics of Eq. (1) combined with a simplified constitutive equation to alleviate Poisson locking was employed in Durville [10] for frictional beamtobeam contact.
In order to more accurately capture crosssectional strains in a consistent way from threedimensional elasticity, Eq. (1) can be generalized to have an arbitrary order of approximation in the transverse directions by using the series expansion [1]
with the directors
where m and n are nonnegative integers, and N is a positive integer representing the order of approximation in transverse direction. It is noted that Eq. (1) is a special case of Eq. (2) with \(N=1\), \({\varvec{\varphi }}\equiv {{\varvec{d}}^{(0,0)}}\), \({\varvec{d}}_1\equiv {\varvec{d}}_{}^{(1,0)}\), and \({\varvec{d}}_2\equiv {\varvec{d}}_{}^{(0,1)}\). Here and hereafter, we often omit the argument s in directors for brevity. The higher order kinematics of Eq. (2) with \(N\ge 2\) was employed in Moustacas et al. [28] to obtain a homogenized model of fiber bundles in contact with a rigid surface. Compared to existing works, our beam and contact formulations introduce the following novelties:

1.
Unconstrained directors enable an efficient and accurate description of crosssectional strains. Further we verify the beam solutions by comparison with those using brick elements.

2.
We present a frictionless beam contact formulation considering crosssectional strains, based on a Gauss pointtosurface contact algorithm. The impenetrability constraint is enforced using a penalty method combined with an active set method. We verify the contact pressure distribution obtained from the beam contact formulation by comparison with the results of the brick element formulation.

3.
A parameterization of the lateral boundary surface with NURBS basis functions with at least \(C^2\)continuity in both longitudinal and circumferential directions enables to have continuous surface metric and curvature within the local Newton–Raphson iteration for the closest point projection.
For very slender beams, a contact force on the lateral surface is often assumed to act directly on the center axis. Such curvetocurve contact formulation can be divided into the following two categories [23]. First, a pointtopoint contact formulation considers a discrete contact force between two interacting bodies. It is typically based on a bilateral closest point projection, e.g., see Wriggers and Zavarise [49], and provides an efficient formulation, since the evaluation of the contact integral is not required. However, it suffers from the nonuniqueness of solution in the closet point projection if the intersection angle between the curves is small, e.g., in the case of two nearly parallel beams. Further discussions on the uniqueness of solution in the closest point projection can be found in Konyukhov and Schweizerhof [16] and Meier et al. [23]. One can also refer to Weeger et al. [46] for a relevant discussion in the framework of isogeometric collocation method. Second, a linetoline (or Gauss pointtoaxis) contact formulation considers a distributed contact force. It is also applicable to small intersection angles between curves, although it requires larger computational cost, compared to the pointtopoint contact formulation, as the angle decreases. In order to consider an arbitrary intersection angle in a more efficient way, those two formulations are combined in Meier et al. [25] with a regularized transition. The curvetocurve contact formulation is efficient due to its onedimensional contact search; however, for beams having low to moderate slenderness ratios, considering a surface load formulation rather than applying an equivalent load to the axis becomes crucial due to the additional effects like external moments and crosssectional stretching [5]. Further the closest point projection to the center axis might lead to an error in the calculation of contact forces since the crosssection is not always orthogonal to the axis due to transverse shear deformations [40]. Therefore, in this paper, we present a surfacetosurface contact formulation with a closest point projection to the lateral beam surface, based on a parameterization of the lateral surface with at least \(C^2\)continuity. Neto et al. [31] presented a finite element formulation of pointtopoint frictionless contact on the lateral surface based on the bilateral closest point projection, which still suffers from the nonuniqueness of solution for small intersection angles.
A seamless integration of geometry and analysis is developed, in Hughes et al. [13], by employing nonuniform rational Bsplines (NURBS) basis functions for the spatial discretization of the solution field as well as the geometry, which is known as isogeometric analysis (IGA). The higherorder interelement continuity in IGA yields smooth contact pressure distributions [20, 44], and enables to reduce numerical instabilities and oscillations in classical nodetosegment (NTS) and Gausspointtosegment contact algorithms associated with kinks between elements [22, 39]. Thus, surface smoothing [33, 43, 48] and surface enrichment [7, 39], developed for classical finite element discretizations based on Lagrange polynomials are intrinsically captured by IGA. In this paper, we parameterize the initial lateral surface of the beam by the center axis curve and the crosssection’s boundary curve defined in the plane spanned by two orthonormal directors. Especially, we employ an unclamped knot vector in the crosssection’s boundary curve in order to have higher order continuity, which gives us continuous metric and curvature components for the surface that are necessary for robustness of the local Newton–Raphson iteration in the closest point projection.
The remainder of this paper is organized as follows. In Sect. 2, we present the beam kinematics based on Cosserat rod theory, and the parameterization of the lateral boundary surface of the rod. In Sect. 3, the frictionless beamtobeam contact formulation is presented. In Sect. 4, an isogeometric finite element discretization of the beam and contact formulations is presented. In Sect. 5, we present several numerical examples of beamtorigid body and beamtobeam contact. In three appendices, we present detailed algorithms for the beam and its contact formulation, and supplementary information of the numerical examples.
Cosserat rods with deformable crosssection
In this paper, in order to capture the crosssectional strains more accurately, the isogeometric finite element formulation of Choi et al. [5] is extended to incorporate an arbitrary order of approximation in the transverse directions, i.e., N, based on the kinematics of Eq. (2). For \(N\ge {2}\), since the crosssection can represent at least linear inplane strains properly, one does not require any special treatment to alleviate Poisson locking. For \(N=1\), we use the EAS method in Choi et al. [5].
Beam kinematics: Cosserat rod theory
The initial configuration of a beam is typically described by a family of crosssections whose (mass) centroids^{Footnote 2} are connected by a spatial curve called the line of centroids or the initial axis. For an initial (undeformed) axis, we consider a spatial curve \(\mathcal {C}_0\) parameterized by a coordinate \({\xi }\in {\mathbb {R}^1}\), i.e., \({\mathcal {C}_0}:\,{\xi } \rightarrow {{\varvec{\varphi }}_0}({\xi }) \in {{\mathbb {R}}^3}\). We reparameterize the curve as \({\mathcal {C}_0}:\,s \rightarrow {{\varvec{\varphi }}_0}(s) \in {{\mathbb {R}}^3}\) by an arclength parameter \(s \in \left[ {0,L} \right] \subset {{\mathbb {R}}^1}\), where L represents the length of the initial axis. The arclength coordinate is defined by the mapping
and the Jacobian of the mapping is defined as \({\tilde{j}}\,{:}{=}\,{\textrm{d}s/\textrm{d}{\xi }}={\left\ {{{\varvec{\varphi }}_{0,{\xi }}}({\xi })} \right\ }\). The arclength coordinate enables to simplify the subsequent expressions by \(\left\ {{{\varvec{\varphi }}_{0,s}}} \right\ = 1\). Here and hereafter, we often use \(s\equiv s(\xi )\) for brevity, and \((\bullet )_{,s}\) denotes the partial differentiation with respect to the arclength parameter s. The initial crosssection domain \(\mathcal {A}_0\subset {\mathbb {R}^2}\) is spanned by two initial directors \({{\varvec{D}}_{\gamma }}(s) \in {\mathbb {R}}^3\) \(\left( \gamma \in \left\{ 1,2\right\} \right) \), which are orthonormal, and aligned with the principal directions of the second moment of inertia of the crosssection. Further, it is assumed that, in the initial configuration, the crosssection is orthogonal to the initial axis; thus, we simply obtain \({\varvec{D}}_3(s)\,{:}{=}\,{\varvec{\varphi }}_{0,s}(s)\). The current (deformed) configuration of the axis is defined by the spatial curve \({{\mathcal {C}}_t}:\,s \rightarrow {{\varvec{\varphi }}}(s,t) \in {{\mathbb {R}}^3}\), where \(t\in {{\mathbb {R}}}^{+}\) denotes time. It should be noted that the current axis does not always pass through the geometrical centroid of the crosssection if the order of approximation in the transverse direction, i.e., N in Eq. (2) is greater than 1, see Remark 2. We define \(\left\{ {{{\varvec{e}}_1},{{\varvec{e}}_2},{{\varvec{e}}_3}} \right\} \) as a standard Cartesian basis in \({{{\mathbb {R}}}^3}\), and \({\varvec{e}^i}\equiv {\varvec{e}_i}\), \(i\in \left\{ 1,2,3\right\} \). Figure 1 schematically illustrates the above kinematic description of the initial and current beam configurations.
We define a reference domain \({\mathcal {B}}\,{:}{=}\,(0,L) \times {\mathcal {A}}\), where \({\mathcal {A}}\) denotes the open domain of coordinates \(\zeta ^1\) and \(\zeta ^2\). For example, for a circular crosssection with radius R we have \(\mathcal {A} \,{:}{=}\,\left\{ { {\left( {\zeta ^1},{\zeta ^2}\right) } \big {{\left( {{\zeta ^1}} \right) }^2} + {{\left( {{\zeta ^2}} \right) }^2} < R^2} \right\} \), see Fig. 2 for an illustration. The location of each point in the reference domain is expressed in terms of the coordinates \({\zeta ^1}\), \({\zeta ^2}\), and \({\zeta ^3}\) in the standard Cartesian basis in \({{{\mathbb {R}}}^3}\) denoted by \({{\varvec{E}}_1}\), \({{\varvec{E}}_2}\), and \({{\varvec{E}}_3}\), where we use \(\zeta ^3\equiv s\). We then define two mappings from the reference domain to the initial configuration \(\mathcal {B}_0\) and to the current configuration \(\mathcal {B}_t\), respectively, by \({{\varvec{X}}}:{\mathcal {B}} \rightarrow {{\mathcal {B}}_0}\) and \({{\varvec{x}}}:{\mathcal {B}} \rightarrow {{\mathcal {B}}_t}\). The deformation from the initial to the current configuration is then expressed by the mapping
We assume a smooth crosssection boundary such that the initial boundary surface \({\mathcal {S}}_0\equiv \partial {\mathcal {B}_0}\) consists of the lateral surface \({\mathcal {S}}^{\textrm{L}}_0\), the crosssections \({\mathcal {A}}_0\) at ends, and their interface edges \({\mathcal {E}_0}\equiv {\partial {{\mathcal {S}}^{\textrm{L}}_0}} \equiv {\partial {{{\mathcal {A}}_0}\big }_{s \in \{0,L\} }}\), i.e., \({{\mathcal {S}}_0} = {{\mathcal {S}}}_0^{\textrm{L}} \cup { {{{\mathcal {A}}_0}} \big _{s \in \{ 0,L\} }} \cup {\mathcal {E}_0}\). Further, in the current configuration, the boundary surface \(\mathcal {S}_t\equiv \partial {\mathcal {B}_t}\) consists of the lateral surface \({\mathcal {S}}^{\textrm{L}}_t\), the crosssections \({\mathcal {A}}_t\) at ends, and their interface edges \({\mathcal {E}_t}\equiv {\partial {{\mathcal {S}}^{\textrm{L}}_t}} \equiv {\partial {{{\mathcal {A}}_t}\big }_{s \in \{0,L\} }}\), i.e., \({{\mathcal {S}}_t} = {{\mathcal {S}}}_t^{\textrm{L}} \cup {{{{\mathcal {A}}_t}} \big _{s \in \{ 0,L\} }} \cup {\mathcal {E}_t}\), see Fig. 1 for an illustration. The initial position vector of any point of the beam is given as
Here and hereafter, unless otherwise stated, repeated Latin indices like i and j imply summation over 1–3, and repeated Greek indices like \(\alpha \), \(\beta \) and \(\gamma \) imply summation over 1–2. It should be noted that the coordinates \(\zeta ^\gamma \,(\gamma \in \left\{ 1,2\right\} )\) are chosen to have dimensions of length, such that the initial directors \(\varvec{D}_\gamma \) are dimensionless. Initial covariant base vectors are obtained by \({{\varvec{G}}_{i}} \,{:}{=}\,\partial {{\varvec{X}}}/\partial {\zeta ^i}\) (\(i\in \left\{ 1,2,3\right\} \)), so that we have
Further we define \(j_0\) as the Jacobian of the mapping \({\varvec{X}}\big (\zeta ^1,\zeta ^2,\zeta ^3\big )\) such that the corresponding infinitesimal volume in the domain \(\mathcal {B}_0\) can be expressed by [5]
with
Remark 1
We consider a pair of nonnegative integers (p, q) in Eq. (2), which belong to
where \({\mathbb {Z}^+}\) denotes the set of positive integers, and \({Z^*}\,{:}{=}\,\{0\}\cup {\mathbb {Z}^+}\), and the number of elements in \(C_n\) is \(\left{{C_n}} \right= n + 1\). In this paper, n of Eq. (10) is called the order of directors, for examples, \(\varvec{\varphi }\equiv {\varvec{d}^{(0,0)}}\) is called the 0th order director, and \(\varvec{d}_1\equiv {\varvec{d}^{(1,0)}}\) and \(\varvec{d}_2\equiv {\varvec{d}^{(0,1)}}\) are called the 1st order directors. Then, for an Nth order of approximation in the transverse directions, the number of directors in each crosssection is
and the number of DOFs in each crosssection is simply obtained by \(n_\textrm{cs}=3{n_\textrm{dir}}\). For a single brick element of order N, the number of nodes in each crosssection is typically \(n_\textrm{node}=(N+1)^2\), and then the number of DOFs is \(n^\textrm{brick}_\textrm{cs}=3n_\textrm{node}\). Figure 3 compares the number of DOFs per crosssection in the beam and brick elements. It is shown that for the same order of approximation in the crosssection, the beam formulation uses less DOFs due to the symmetry with respect to the axis \(\zeta ^1=0\) or \(\zeta ^2= 0\) in the kinematic assumption of Eq. (2).
Equation (2) can be rewritten in the compact form [27]
with
and
where we define the operator
\(n\in \left\{ 0,1,\ldots ,N\right\} \), with
Here \({{\varvec{q}}_n}(\zeta ^3)\) denotes a column array of nth order directors, and \({\varvec{q}}(\zeta ^3)\) is a column array of \({{\varvec{q}}_n}(\zeta ^3 )\,(n\in \left\{ 1,\ldots ,N\right\} )\), which is called a generalized director vector. \({{\varvec{1}}_{m}}\) denotes the identity matrix of dimension m. Note that \({\varvec{\pi }}_n\) solely depends on the transverse coordinates, which is independent from the deformations. Thus, taking the first variation of Eq. (12) yields
In the current configuration, the covariant base vectors are defined by \(\varvec{g}_i\,{:}{=}\,\partial {\varvec{x}}/\partial {\zeta ^i}\,(i=1,2,3)\), so that we have
where \((\bullet )_{,{\zeta ^i}}\) denotes the partial derivative with respect to \(\zeta ^i\). The deformation gradient can be expressed by [47, p. 478]
such that the Green–Lagrange strain tensor is obtained by
where \(\varvec{1}\) represents the identity tensor in \(\mathbb {R}^3\), and \(g_{ij}\,{:}{=}\,\varvec{g}_i\cdot \varvec{g}_j\), and \(G_{ij}\,{:}{=}\,\varvec{G}_i\cdot \varvec{G}_j\).
Remark 2
Geometrical centroid of the current crosssection. The position of the geometrical centroid in the current crosssection is defined by
with the current crosssectional area
It can be easily verified that, for \(N=1\) in Eq. (2), we have \(\varvec{C}=\varvec{\varphi }\). That is, for \(N=1\), the current axis of the beam always coincides with the geometrical centroid. For \(N>1\), however, this is not always the case.
Parameterization of a lateral boundary surface using NURBS
The initial geometry of the lateral boundary surface of the beam is decomposed into two parts: First, the directed axis curve, described by a NURBS curve with two attached orthonormal directors, \(\varvec{D}_1\) and \(\varvec{D}_2\). Second, the crosssection defined by the NURBS curve in the plane spanned by the two initial directors
where \({\big ({C^1_{I},C^2_{I}}\big )}\in \mathbb {R}\times \mathbb {R}\) represents the position of control point in the plane, and \({{m}}_{\textrm{cp}}\) denotes the total number of control points along the curve, and \(M^q_I(\xi ^2)\) denotes the Ith NURBS basis function of order q. The definition and basic properties of NURBS basis function can be found in Piegl and Tiller [34]. \(\xi ^1\) and \(\xi ^2\) denote the convective coordinates along the axis and the crosssection’s boundary curve, respectively. In this paper, we define those two convective coordinates by parametric coordinates of NURBS. Here and hereafter, we often use \(\varvec{\xi }\,{:}{=}\,\left[ {\xi ^1},{\xi ^2}\right] ^\textrm{T}\) for brevity. It is noted that \({C}^\gamma _{I}\) depends on the coordinate \(\xi ^1\) only if a dimension of the initial crosssection is varying along the center axis. However, in this paper, we restrict our discussion to nonvarying crosssections along the axis, that is, we consider only those cases where \({C}^\gamma _{I}\) does not depend on the coordinate \(\xi ^1\). In the subsequent formulation of the closest point projection based on a local Newton–Raphson iteration, we need at least a \(C^2\)continuous surface parameterization. However, if we use a clamped knot vector for the closed crosssection boundary curves, the displacement continuity is typically \(C^0\) at the interface between the two end points where the end control points are matched, which may lead to difficult convergence of the local Newton–Raphson iteration. Further, contact tractions are singular at \(C^0\) surface points, which can lead to spurious contact deformations. Thus, so as to have at least \(C^2\)continuity of the whole lateral boundary surface, we employ the following approaches:

1.
In the representation of the crosssection’s boundary curve, an unclamped knot vector is utilized, which allows higher order continuity at the interface between two end points of the closed boundary curves.

2.
We also reduce the multiplicity of internal knots to one, so that we have \(C^{q1}\) continuity in the entire curve of the crosssection’s boundary, where q denotes the order of basis functions.

3.
In the spatial discretization of kinematic variables as well as the initial geometry of the axis, we use NURBS basis functions, in the framework of isogeometric analysis, in order to have higher order continuity in the axial direction of the beam as well.
The removal of knots may change the initial geometry of the crosssection. However, increasing the DOFs in the geometry, e.g., the number of control points, and the order of basis functions, before the knot removal enables to reduce the loss of geometry. For example, we consider a circular crosssection of radius \(R=1\,\textrm{m}\). NURBS is capable of exactly representing conic sections like circle. Figure 4a shows quadratic NURBS basis functions utilized to represent the circle in Fig. 5a, which have \(C^0\)continuity at the interface between nonzero knot spans. To remove those nonsmooth points we apply the operations, unclamping and knot removal for the end and internally repeated knots, respectively, and then we finally obtain the basis functions, shown in Fig. 4b. Further details on these unclamping, and knot removal procedures can be found in Piegl and Tiller [34] and Rogers [37]. We obtain those periodic NURBS curves by using the function MakePeriodic in the commercial program Rhinoceros 3D.^{Footnote 3} It is clearly shown in Fig. 5b that the initial geometry of the crosssection is not an exact circle anymore. However, as Fig. 5c shows, by increasing the number of control points before those operations, the loss of geometry approaches zero. Further, it should be noted that the DOFs in the geometry of initial crosssection is associated with the parameterization only, and does not have any influence on the computational costs, since the deformation of the current crosssection is solely described by director vectors.
Convective base vectors and metrics
Using the parameterization of the coordinates \(s=s(\xi ^1)\) in Eq. (4) and \(\zeta ^{\gamma }=\zeta ^{\gamma }(\varvec{\xi })\) of Eq. (22), the initial position vector of the beam in Eq. (6) can be reparameterized on the initial lateral boundary surface \(\mathcal {S}_0^\textrm{L}\), as
Similarly, we can also reparameterize the current position vector of the beam in Eq. (12), on the current lateral boundary surface \(\mathcal {S}_t^\textrm{L}\), as
For the given parameterization of Eq. (23), we define the covariant base vectors on the initial lateral boundary surface \(\mathcal {S}_0^\textrm{L}\), as
and let \(A_{\alpha \beta }\,{:}{=}\,{\varvec{A}_\alpha }\cdot {\varvec{A}_\beta }\) denote the covariant components of the surface metric, and here and hereafter, we define \({(\bullet )_{,\alpha }} \,{:}{=}\,\partial (\bullet )/\partial {\xi ^\alpha }\), and \({(\bullet )_{,\alpha \beta }} \,{:}{=}\,\partial ^2 (\bullet )/\partial {\xi ^\alpha }\partial {\xi ^\beta }\) \((\alpha ,\beta \in \left\{ 1,2\right\} )\). An initial convective surface basis \(\left\{ {{{\varvec{A}}_1},{{\varvec{A}}_2},{\varvec{\nu }}_0} \right\} \) on \(\mathcal {S}_0^\textrm{L}\) can be obtained by defining an outward unit normal vector on the surface as \({{\varvec{\nu }}_0}\,{:}{=}\,{{\varvec{A}}_1} \times {{\varvec{A}}_2}/\left\ {{{\varvec{A}}_1} \times {{\varvec{A}}_2}} \right\ \), see Fig. 6 for an illustration.
Remark 3
The first order derivatives of \(\zeta ^{\alpha }\) with respect to \(\xi ^\beta \), i.e., \(\zeta ^\alpha _{,\beta }\) \((\alpha ,\beta \in \left\{ 1,2\right\} )\) can be simply obtained, from Eq. (22), as
and the second order derivatives are
where the dependence of \(C^\alpha _I\) on \(\xi ^1\) vanishes due to the assumption of nonvarying initial crosssections along the axis.
Further we define the covariant base vectors on the current lateral surface \(\mathcal {S}_t^\textrm{L}\), as
and let \({{a}_{\alpha \beta }} \,{:}{=}\,{{\varvec{a}}_\alpha } \cdot {{\varvec{a}}_\beta }\) denote the covariant components of the surface metric. The contravariant base vectors \({{\varvec{a}}^\alpha }\) are defined by \({{\varvec{a}}_\alpha }\cdot {{\varvec{a}}^\beta }=\delta _{\alpha }^{\beta }\) \(\left( \alpha ,\beta \in \left\{ 1,2\right\} \right) \), where \(\delta _{\alpha }^{\beta }\) denotes the Kroneckerdelta symbol, which leads to \({{\varvec{a}}^\alpha } = {a^{\alpha \beta }}{{\varvec{a}}_\beta }\), where \(a^{\alpha \beta }\,{:}{=}\,{{\varvec{a}}^\alpha }\cdot {{\varvec{a}}^\beta }\) denotes the contravariant components of the surface metric, calculated from
A current convective surface basis \(\left\{ {{{\varvec{a}}_1},{{\varvec{a}}_2},{\varvec{\nu }}_t} \right\} \) on \(\mathcal {S}_t^\textrm{L}\) can be obtained by defining an outward unit normal vector on the surface as \({{\varvec{\nu }}_t}\,{:}{=}\,{{\varvec{a}}_1} \times {{\varvec{a}}_2}/\left\ {{{\varvec{a}}_1} \times {{\varvec{a}}_2}} \right\ \). The covariant components of curvature are defined by \(b_{\alpha \beta }\,{:}{=}\,{{\varvec{a}}_{\alpha ,\beta }} \cdot {{\varvec{\nu }}_t}\), where \(\varvec{a}_{\alpha ,\beta }=\varvec{x}_{,\alpha \beta }\), and
Initial infinitesimal area element
In the initial configuration, from the parametrization of the lateral boundary surface in Eq. (23), the infinitesimal area element of the initial lateral boundary surface can be expressed, using \(\textrm{d}s={{\tilde{j}}}\,\textrm{d}\xi ^1\) and the surface Jacobian \({{\tilde{J}}}\,{:}{=}\,\left\ {{{\varvec{A}}_{1}}} \times {{\varvec{A}}_{2}} \right\ \), by
See Fig. 6 for an illustration.
Variational formulation
Boundary value problem in strong form
The (static) local linear momentum balance equations in combination with displacement and traction boundary conditions is stated in the initial configuration by the boundary value problem [4]
where \({\mathcal {S}^\textrm{D}_0},\,{\mathcal {S}^\textrm{N}_0}\subset \mathcal {S}_0\) denote the boundary surfaces where the displacement \(\varvec{u}\,{:}{=}\,{\varvec{x}}{\varvec{X}}\) and traction are prescribed, respectively, and \(\varvec{S}\) denotes the second Piola–Kirchhoff stress tensor. Further, \(\varvec{\nu }_0\) denotes the unit outward normal vector on the surface of the undeformed configuration, \(\varvec{b}_0\) denotes the body force per unit undeformed volume, \(\textrm{Div}(\bullet )\) represents the divergence operator with respect to the initial configuration, and \({{\bar{\varvec{u}}}}\in {{\mathbb {R}}^3}\) denotes the prescribed displacement vector. Here we consider hyperelastic materials, where the strain energy density, defined by the strain energy per unit undeformed volume, is given in terms of the Green–Lagrange strain tensor \(\varvec{E}\), as \(\Psi = \Psi ({\varvec{E}})\). Then, the constitutive equation is expressed by
Weak form
Exploiting the symmetries, \({\varvec{E}}\) and \({\varvec{S}}\) can be expressed in array form (Voigt notation), as \({\varvec{\underline{S}}} \,{:}{=}\,{\left[ {{S^{11}},{S^{22}},{S^{33}},{S^{12}},{S^{13}},{S^{23}}} \right] ^{\textrm{T}}}\) and \({\varvec{\underline{E}}} \,{:}{=}\,{\left[ {{E_{11}},{E_{22}},{E_{33}},2{E_{12}},2{E_{13}},2{E_{23}}} \right] ^{\textrm{T}}}\). From Eq. (20), the first variation of the covariant components of the Green–Lagrange strain tensor can then be expressed, in Voigt notation, by
where
Then, the internal virtual work can be written as
where the resultant of the stress is obtained by
with \(\textrm{d}\mathcal {A}\,{:}{=}\,\textrm{d}\zeta ^1\textrm{d}\zeta ^2\). Further the external virtual work due to the body force, and surface tranctions on the lateral surface and the crosssection at the ends of rod, i.e., \({{{\mathcal {A}_0}} \big _{s \in \Gamma _\textrm{N} }}\) is given by
where the resultant of the body force and traction on the lateral surface is expressed by
Further, the resultant of the traction on the crosssection at the ends is expressed, using \(\textrm{d}\mathcal {A}_0=\left\ {{{\varvec{G}}_1} \times {{\varvec{G}}_2}} \right\ \textrm{d}\mathcal {A}\), as
Finally, the variational equation can be stated as: Find the generalized director vector \(\varvec{q}\in {{\mathcal {V}}}\) such that
where represents the internal virtual work due to normal contact whose detailed expression is given in Sect. 3, and denotes the external virtual work due to nonconservative loads, e.g., the distributed follower load in Remark 4. We also define
and
The director vectors are prescribed at the boundary \({\Gamma _\textrm{D}}\ni {s}\). It is noted that \({\Gamma _\textrm{D}} \cap {\Gamma _\textrm{N}} = \emptyset \), and \({\Gamma _\textrm{D}} \cup {\Gamma _\textrm{N}} = \left\{ {0,L} \right\} \).
Linearization
For hyperelastic materials, in general, the constitutive relation between \(\varvec{S}\) and \(\varvec{E}\) is nonlinear. Taking the directional derivative of \(\varvec{S}\) gives
where \(D(\bullet )\cdot (*)\) denotes the directional derivative of \((\bullet )\) in the direction of \((*)\), and \(\Delta \varvec{x}\) represents the increment of the current position of a material point. The material (Lagrangian) elasticity tensor \(\varvec{\mathcal {C}}\) is expressed by
with
Equation (44) can be rewritten, using Eq. (34), as
where
The internal virtual work of Eq. (36) is nonlinear in terms of the generalized director \(\varvec{q}\). Thus, in order to solve the variational equation using the Newton–Raphson iteration, we need to linearize Eq. (36). The directional derivative of \(\varvec{R}\) can be obtained by using Eq. (47), as
Thus, the directional derivative of the internal virtual work of Eq. (36) with the first variation of the Green–Lagrange strain tensor held constant, i.e., the material part of the tangent stiffness, is simply obtained by
Further, the geometric part of the tangent stiffness is derived by taking the directional derivative of Eq. (36) with the second Piola–Kirchhoff stress part held constant, as
with
where
with
and
Combining Eqs. (50) and (51), we finally obtain the following increment of the internal virtual work
Remark 4
Application of a moment load by a distributed follower load. An end moment can be applied to a beam by employing the following linear distribution of the first Piola–Kirchhoff stress over the height, i.e., \(h/2\le {\zeta ^1}\le {h/2}\) of the rectangular crosssection with dimension \(h\times w\) on \({{{{{\mathcal {A}_0}}\big }_{s \in {\Gamma _{\textrm{N}}}}}}\) [3, 5]
and the outward unit normal vectors on the crosssections at the ends of beam’s axis in the initial and current configurations are obtained by
and
respectively, where the values of the signum functions can be simply determined by
Then, the prescribed surface traction vector can be expressed by \({\bar{\varvec{T}}}_0=\varvec{P}\,\varvec{\nu }_0={\bar{p}}\,{\varvec{\nu }}_t\), and the external virtual work due to the follower load is obtained from Eq. (40), as
with
Further, the increment of Eq. (59a) is derived as
where
with
Here, \(\widehat{{{\varvec{g}}}_\alpha }\) denotes the skewsymmetric tensor associated with the dual vector \({{\varvec{g}}}_\alpha \).
A frictionless beamtobeam contact formulation
Impenetrability condition
We present a continuum formulation for frictionless lateral contact between two slender bodies^{Footnote 4} based on the beam formulation presented in Sects. 2.1–2.2. We employ a Gauss pointtosurface contact formulation. The interacting bodies are denoted by \({{\mathcal {B}}}_0^{(\alpha )}\) (\(\alpha \in \left\{ 1,2\right\} \)) in their initial configurations. All the geometrical or physical quantities for the two bodies are indicated by the super or subscript \((\alpha )\). Let \(\mathcal {R}_t\,{:}{=}\,{\mathcal {R}_t^{(1)}}={\mathcal {R}_t^{(2)}}\) be the current contact interface between the two contacting bodies, which are subsets of \(\mathcal {S}_t^\textrm{L}\). The contact interface is pulled back to the boundary surface in the initial configuration of each body \({\mathcal {R}_0^{(\alpha )}}\), and the contact conditions are evaluated on \(\mathcal {R}_0\,{:}{=}\,\mathcal {R}_0^{(1)}\ne \mathcal {R}_0^{(2)}\) [17]. We designate the bodies \({\mathcal {B}}_0^{(1)}\) and \({\mathcal {B}}_0^{(2)}\) as slave and master, which implies that all current points \({\varvec{x}}\in {\mathcal {R}_t^{(1)}}\) are supposed to not penetrate the surface \({\mathcal {R}_t^{(2)}}\). The boundary value problem of Eq. (32) can be rewritten for each of the interacting bodies, as
\(\alpha \in \left\{ 1,2\right\} \). Let \({\varvec{\bar{x}}}\in \mathcal {S}_t^{\textrm{L}(2)}\) be the solution of the following minimal distance problem
for a given point \({\varvec{x}^{(1)}\in {{\mathcal {S}}_t^{\textrm{L}(1)}}}\), and \({\bar{\varvec{\nu }}}_t\) denote the outward unit normal vector at \({\varvec{{\bar{x}}}}\in \mathcal {S}_t^{\textrm{L}(2)}\). The impenetrability condition can then be evaluated, on the initial lateral surface \(\mathcal {S}_0^{\textrm{L}{(1)}}\ni {\varvec{X}}^{(1)}\), as [42]
where \({\varvec{\nu }}^{(1)}_0\) denotes the outward unit normal vector on the initial lateral surface of the slave body. Equation (64a) denotes the impenetrability condition, and Eq. (64b) represents that contact should be compressive, where \(p_\textrm{N}\) denotes the contact pressure. Further, Eq. (64c) means the contact pressure applies only if the impenetrability condition is active, and Eq. (64d) implies no friction is considered, where \(\varvec{t}_\textrm{T}\) denotes the tangential component of surface traction. The constraints of Eqs. (64a)–(64c) can be replaced by the penalty regularization of the contact pressure [42]
with
where the constant \(\epsilon _\textrm{N}>0\) is the penalty parameter, and \({g_\textrm{N}}\rightarrow 0\) and \(p_\textrm{N}\) converges as \({\epsilon _\textrm{N}}\rightarrow \infty \). That is, in order to reduce the error in the contact constraints or to avoid unphysical penetration, it is required to use a sufficiently large penalty parameter. However, it should be noted that, as we discuss in Sect. 5.3, a larger penalty parameter typically requires more surface Gauss integration points for the contact integral, and a smaller load increment, which makes the computation less efficient. An adaptive adjustment of penalty parameter remains future work. One can adjust the penalty parameter locally in order to control the penetration globally, such that the penetration does not exceed the maximum allowed value, see, e.g., the iterative adjustment approach of Durville [10].
Active set iteration
If the contact surface \(\mathcal {R}_0\) is known, the impenetrability condition becomes an equality constraint, and Eqs. (64a)–(64c) can be rewritten as
and the penalty regularization in Eq. (65) can be rewritten as
However, since the contact surface is also unknown in general, we employ an iterative procedure, called the active set method, in order to find the current contact area. The underlying idea of this method is to divide the inequality constraints into two groups—the active set and the inactive set of constraints [21]—and use two nested loops—an outer loop to find the correct set of active constraints, and an inner loop to solve the nonlinear boundary value problem of Eq. (62) using the Newton–Raphson iteration with a fixed set of active constraints. Let \(\mathcal {R}_0^k\) denote the contact area at the kth iteration (\(k=1,2,\ldots \)). For a given \(\mathcal {R}_0^{k1}\) within the inner loop, we solve the variational equation of Eq. (41) using the Newton–Raphson iteration, and then we update the contact area by a contact search process, which is explained in Sects. 3.2 and 3.3. The outer loop continues until the active set converges. The search procedure to find the active set has to be performed at every iteration step of the active set loop. It contains two cases:

Case 1: update the contact state of those contact pairs in the existing active set, and remove any inactive contact pairs,

Case 2: add new contact pairs to the active set.
In the first case, we keep only the set elements (contact pairs) with positive contact pressure, i.e., \(p_\textrm{N}={\epsilon _\textrm{N}}{g_\textrm{N}}>0\), and remove the others from the active set. In the second case, new set elements are added to the active set if \(g_\textrm{N}<0\). For a more efficient contact search, we divide the search procedure into two steps: global and local searches.
Remark 5
One can employ a regularized penalty law, which updates the active set simultaneously during the Newton–Raphson iteration, instead of employing an additional outer loop. For example, quadratically regularized penalty laws are used by Durville [10] and Meier et al. [23]. Although in our formulation the active set method requires an additional outer loop, it has the following advantages:

It fixes the active set within the inner Newton–Raphson iteration, so that it gives an improved convergence behavior for the same size of load increment.

It does not require an additional userdefined parameter like a regularization threshold.

The quadratic regularization leads to zero contact stiffness initially, leading to inaccuracies. If a positive regularization threshold is used, unphysical contact force may initially occur.
Global contact search
The global search finds possible contact pairs using the normal distance between the axes of interacting beams. We first find the convective axial coordinate \({\widetilde{\xi }}^1\) of the closest point \({\widetilde{\varvec{\varphi }}}\,{:}{=}\,{\varvec{\varphi }}^{(2)}\big (\widetilde{\xi }^{1}\big )\), for a given slave point \(\varvec{\varphi }^{(1)}\), where we use several initial guesses \({}^{(0)}{\widetilde{\varvec{\varphi }}}\) along the axis of the master body, see “Appendix A.1” for the detailed procedure, and Fig. 7 for an illustration. Then we exclude those surface Gauss points corresponding to \(\varvec{\varphi }^{(1)}\), if \({\widetilde{\varvec{\varphi }}}\) lies outside of the range
where \({{\varvec{\varphi }}_{\textrm{d}}} \,{:}{=}\,{{\varvec{\varphi }}^{(1)}}  {\widetilde{\varvec{\varphi }}}\) defines a relative position vector between the axis in slave body and the axis in master body, and \(r_\textrm{c}>0\) is a chosen cutoff radius. Selecting \(r_\textrm{c}\) too small may result in some contact areas being undetected, leading to unphysical penetration. As \(r_\textrm{c}\) increases, the contact search procedure becomes less efficient. It may also depend on crosssectional dimensions in the current configurations as well, so that it might need to be adjusted, if crosssectional deformations are very large. In this paper, we use a fixed parameter \(r_\textrm{c} = 3R\) for an initially circular crosssection of radius R, and the development of an efficient algorithm to adjust the parameter \(r_\textrm{c}\) remains future work. We further exclude those surface Gauss points that lie outside of the angular range
with
and \(0 < {\varepsilon _\theta } \le \pi \,\,[\textrm{rad}]\). As \(\varepsilon _\theta \) increases, the more Gauss points in the crosssections’s boundary of the slave body are considered as contact candidates, and the contact search becomes less efficient. On the other hand, too small \(\varepsilon _\theta \) may lead to undetected contact areas or oscillation in the active set iteration, see the relevant discussion in Sect. 5.3.1. The subsequent local contact search considers only those surface Gauss points of the slave body which satisfy the criteria of Eqs. (68) and (69a).
Local contact search
The local contact search is an iterative procedure to solve a Gauss pointtosurface minimal distance problem using a local Newton–Raphson iteration.
Unilateral minimal distance problem
We determine the distance between two surface points from
Here and hereafter, we often use \({\varvec{x}}^{(2)}\equiv {\varvec{x}}^{(2)}({\varvec{\xi }})\equiv {{\varvec{x}}^{(2)}({\varvec{\xi }}^{(2)})}\) for brevity. The parametric coordinates of the closest master point \({\bar{\varvec{\xi }}}\equiv {\bar{\varvec{\xi }}}(\varvec{\xi }^{(1)})\) to a given surface point \({{\varvec{x}}^{(1)}} \equiv {{\varvec{x}}^{(1)}}({\varvec{\xi }}^{(1)})\) in the slave body is determined as the solution of the following unilateral minimal distance problem:
where
and, in this paper, we normalize both convective coordinates such that \(\xi ^1_\textrm{min}=\xi ^2_\textrm{min}=0\) and \(\xi ^1_\textrm{max}=\xi ^2_\textrm{max}=1\). As we consider a closed curve for the boundary of the crosssection, the coordinate \(\xi ^2\) should have a periodic property. Therefore, for \(\alpha =2\), we employ
with
if \({\xi ^\alpha } > \xi _{\max }^\alpha \), and
if \({\xi ^\alpha } < \xi _{\min }^\alpha \), where \(\xi ^\alpha _\textrm{r}\,{:}{=}\,{\xi ^\alpha _\textrm{max}}{\xi ^\alpha _\textrm{min}}\), and \(\left\lfloor {{\xi ^\alpha }} \right\rfloor \) denotes the largest integer smaller than \(\xi ^\alpha \). If the axis curve is closed, Eq. (72) can also be applied for \(\alpha =1\). Otherwise, we divide the constraint of Eq. (71b) for \(\alpha =1\) into two parts: domain (i.e., \({\xi ^1}\in (\xi ^1_\textrm{min},\xi ^1_\textrm{max})\)) and end edges (i.e., \(\xi ^1\in \left\{ \xi ^1_\textrm{min},\xi ^1_\textrm{max}\right\} \)) and treat those two cases separately. In this paper, we focus on the former as the latter is not relevant to the envisioned examples. But in general both domaintoedge or the edgetoedge contacts might contribute significantly in some applications including an arbitrary arrangement or configuration of beams. For example, in the simulation of biopolymer networks of Meier et al. [23], it turns out that the end pointtocurve and end pointtoend point contacts play a significant role. The solution of Eq. (71a) satisfies the first order necessary condition
with
\(\alpha \in \left\{ 1,2\right\} \), where
Hereafter, for brevity, we often use \(\varvec{a}_\alpha ^{(2)}\equiv {\varvec{a}_\alpha ^{(2)}}(\varvec{\xi }^{(2)})\), \(\alpha \in \left\{ 1,2\right\} \). The first order necessary condition of Eq. (73) in the minimal distance problem finds a local extremum solution. In order to find the closest point, it is required to select an initial guess sufficiently close to the closest point. We present a geometrical approach to efficiently determine an initial guess in Sect. 3.3.2. Within the Newton–Raphson iteration, for given coordinates \(\varvec{\xi }^{(2)}_{(i1)}\) at the \((i1)\)th iteration, we find the increment \(\Delta \varvec{\xi }^{(2)}\) such that
and the convective coordinates are updated by
until the condition \(\left\ {{\varvec{f}}\big ( {{{\varvec{\xi }}^{(2)}_{(i)}}} \big )} \right\ < {\varepsilon _{{\textrm{cpp}}}}\) is satisfied, where \(\varepsilon _\textrm{cpp}>0\) denotes a chosen tolerance, and \(\varvec{\xi }^{(2)}_{(0)}\) is an initial guess. Assuming \({\varvec{f}}^*({{\varvec{\xi }}^{(2)}_{{(i  1)}}})\) is invertible, we obtain
where
with
\(\alpha ,\beta \in \left\{ 1,2\right\} \). Hereafter, the notation \(\overline{(\bullet )}\) denotes the variable at \(\varvec{\xi }^{(2)}=\varvec{{\bar{\xi }}}\) in the master body, unless otherwise stated. For example, \({\varvec{{\bar{x}}}} \,{:}{=}\,{{\varvec{x}}^{(2)}}({\varvec{{\bar{\xi }} }})\). The overall procedure of the closest point projection is given in Algorithm 3. In the following we explain the determination of an initial guess \(\varvec{\xi }^{(2)}_{(0)}\equiv {{\bar{\varvec{\xi }}}}_{{(0)}}=\left[ {\bar{\xi }}^{1}_{(0)},{\bar{\xi }}^{2}_{(0)}\right] ^\textrm{T}\).
Determination of an initial guess
We select the intersection point between the relative position vector \({{\varvec{\varphi }}_{\textrm{d}}}\) and the crosssection at \(\widetilde{\varvec{\varphi }}\) as an initial guess for the local contact search. However, the intersection point, may not always exist on the boundary of the current crosssection at \(\widetilde{\varvec{\varphi }}\) due to the following two aspects:

The relative position vector is always normal to the axis of the master body; however, it may not be on the crosssection at \(\widetilde{\varvec{\varphi }}\) due to the transverse shear deformation.

For \(N>1\), the crosssection in the current configuration may not be planar.
Thus, we first consider a projection of the domain of the current crosssection \({\mathcal {A}^{(2)}_t}\) onto its tangent plane. This projected domain \(\widetilde{\mathcal {A}_t}\) can be expressed by
Note that for \(N=1\), \(\mathcal {A}^{(2)}_t\equiv {\widetilde{\mathcal {A}_t}}\). For convenience, we define \(\varvec{d}_3\,{:}{=}\,{{\varvec{d}}_1} \times {{\varvec{d}}_2}/\left\ {{{\varvec{d}}_1} \times {{\varvec{d}}_2}} \right\ \), and reciprocal base vectors \(\varvec{d}^i\) on the tangent plane such that \({{\varvec{d}}_i} \cdot {{\varvec{d}}^j} = \delta _i^j\) \((i,j\in \left\{ 1,2,3\right\} )\), as
We also project the relative position vector \(\varvec{\varphi }_\textrm{d}\) onto the tangent plane, as^{Footnote 5}
Then, \(\widetilde{\varvec{\varphi }_\textrm{d}}\) always intersects the boundary of \(\widetilde{\mathcal {A}_t}\), denoted as \(\partial {\widetilde{\mathcal {A}_t}}\), and we find the intersection between \(\partial \widetilde{{\mathcal {A}}_t}\) and the projected vector \(\alpha _\textrm{ig}{\widetilde{\varvec{\varphi }_\textrm{d}}}\) with the length adjusted by a parameter \(\alpha _\textrm{ig}>0\), and the intersection point is selected as an initial guess for the closest point in the local contact search, see Fig. 8. That is, for a given \(\varvec{d}^\beta _{(2)}\) and \(\widetilde{\varvec{\varphi }_\textrm{d}}\), we find the convective circumferential coordinate \({\xi ^{2}_{(2)}}={\bar{\xi }^{2}_{(0)}}\in \left[ \xi ^2_\textrm{min},\xi ^2_\textrm{max}\right] \), and \(\alpha _\textrm{ig}>0\) such that
Since Eq. (81) is nonlinear with respect to \({{\bar{\xi }}^{2}_{(0)}}\), we need an iterative solution process. It should be noted that we use \({{\bar{\xi }}^{1}_{(0)}}={\widetilde{\xi }^1}\). Further details can be found in Appendix B.2.1.
Remark 6
For typical shapes of the initial crosssection, \(\zeta ^\gamma \) satisfying Eq. (81) can be analytically found. For example, for an initially circular crosssection of radius R such that
we have
\(\gamma \in \left\{ 1,2\right\} \). The process to find \({\bar{\xi }_{(0)}^{2}}\) satisfying Eq. (83) requires an iterative process due to the parameterization of the coordinates \(\big (\zeta ^1,\zeta ^2\big )\) in Eq. (22). Instead, in this paper, we directly solve Eq. (81), where both \({\bar{\xi }_{(0)}^{2}}\) and \(\alpha _\textrm{ig}\) are determined simultaneously.
Variational formulation
The first variation of the position vector of the closest point on the master surface, i.e., \({\bar{\varvec{x}}} \equiv {\bar{\varvec{x}}}(\bar{\varvec{\xi }})\) is obtained by [47]
where \(\delta {\bar{\varvec{u}}}\) defines the directional derivative of \(\bar{\varvec{x}}\) with the dependence of the parametric coordinate \(\bar{\varvec{\xi }}\) on \(\varepsilon \) suppressed, i.e.,
For beams, evaluating Eq. (12) at \({\varvec{\xi }}={\bar{\varvec{\xi }}}\) gives the position vector of the closest point on the surface \(\mathcal {R}^{(2)}_t\)
where \(\bar{\varvec{x}}\,{:}{=}\,{\varvec{x}}^{(2)}(\bar{\varvec{\xi }})\), \({\bar{\varvec{\Pi }}}\,{:}{=}\,{{\varvec{\Pi }}}^{(2)}(\zeta ^1(\bar{\varvec{\xi }}),\zeta ^2(\bar{\varvec{\xi }}))\), and \({\bar{\varvec{q}}}\,{:}{=}\,{\varvec{q}^{(2)}}{(s({\bar{\xi }^1}))}\). Eq. (84) can thus be rewritten as
where we define
For a slave point, we simply obtain
At \({\varvec{\xi }}^{(2)}={\bar{\varvec{\xi }}}\) for a given \(\varvec{x}^{(1)}\), by the definition of \(g_\textrm{N}\) in Eq. (64a), we have
Taking the first variation of Eq. (90), and substituting Eqs. (87) and (89), we obtain
Then, by taking the inner product of Eq. (91) and \({\varvec{{\bar{\nu }}}}_t\), and using \({{\varvec{\bar{\nu }}}}_t \cdot {{\varvec{{\bar{\nu }} }}}_t = 1\) and \(\delta {{\varvec{{\bar{\nu }}}}}_t \cdot {{\varvec{{\bar{\nu }} }}}_t = {{\varvec{{\bar{\nu }}}}}_t \cdot {{\varvec{{\bar{a}}}}_\alpha } = 0\), we have
The internal virtual work due to the contact pressure can be written as
Using Eqs. (31) and (92), Eq. (93) can be rewritten into
with the normal contact stress resultant vector
where \(\varXi ^2\,{:}{=}\,[\xi ^2_\textrm{min},\xi ^2_\textrm{max})\) denotes the domain of the parametric coordinate \(\xi ^2\) in the crosssection’s boundary curve.
Remark 7
Units of the contact stress resultant. Each component of the stress resultant \(\varvec{r}_\textrm{N}\), which is energy conjugate to the nth order director, has units of \({F_0}{L_0}^{n1}\), where \(F_0\) and \(L_0\) denote a unit force and unit length, respectively. For example, those components of \(\varvec{r}_\textrm{N}\) corresponding to \(n=0\) represent a resultant force per unit undeformed arclength, and those corresponding to \(n=1\) represent a resultant director moment per unit undeformed arclength.
Linearization of the contact variational form
In order to solve the nonlinear variational equation, we linearize the contact variational form of Eq. (93). Taking the directional derivative of Eq. (93) gives
where we employ the Heaviside function \(\omega \equiv \omega (\varvec{X})\), defined as
Further, using Eq. (92), we have
where
We obtain the increment of \(\delta {g_\textrm{N}}\) by taking the directional derivative of Eq. (91)^{Footnote 6} and applying Eq. (87), as (see “Appendix A.2” for details)
where the symmetric matrix \(\varvec{k}^\textrm{G}_\textrm{N}\) is given by Eq. (A19). Finally, substituting Eqs. (97) and (99) into Eq. (95), we have
with
which is symmetric.
Isogeometric finite element discretization
Beam formulation
We discretize the contact variational form and its increment using NURBS basis functions. A discussion on the crucial properties of NURBS in isogeometric analysis can be found in Hughes et al. [13]. The geometry of the beam’s initial axis is described by a NURBS curve, as
where \({n_{{\textrm{cp}}}}\) denotes the total number of basis functions (or control points) of the axis, and \({{\varvec{P}}_I}\in \mathbb {R}^3\) denotes the position vector of the Ith control point. Using the NURBS basis functions \(N_I=N_I(\xi ^1)\), the finite element approximation \(\delta {{\varvec{q}}^h}=\delta {{\varvec{q}}^h}(s(\xi ^1))\) is expressed by
where \({\delta {\textbf{q}}_I} \in {{\mathbb {R}}^{n_{{\textrm{cs}}}}}\) denotes the coefficient vector of the Ith control point (\(I\in \left\{ 1,\ldots ,n_e\right\} \)), and \({n_e}\) denotes the number of basis functions having local support in the knot span \({\varXi _e}\) with \(e \in \left\{ {1,\ldots ,{n_{{\textrm{el}}}}} \right\} \), and \(n_\textrm{el}\) denotes the total number of nonzero knot spans. Substituting Eq. (103) into Eq. (36), and using the standard finite element assembly operator \(\textbf{A}\), we have
with , and \(\delta \textbf{q}\) denotes the global coefficient vector of the generalized directors. The element internal load vector is obtained by
where we define
with \({\varvec{\Xi }}_e^h \,{:}{=}\,\left[ {\varvec{\Xi }}_1^h, \ldots ,{\varvec{\Xi }}_{{n_e}}^h\right] _{6 \times {n_{{\textrm{cs}}}{n_e}}}\), and
It is noted that, for brevity, we often use [5]
The external virtual work of Eq. (38) is discretized as
with \(\mathbf{{F}}_{\mathrm{{ext}}} := \mathbf{{A}}_{e = 1}^{{n_{\mathrm{{el}}}}} \mathbf{{F}}_{\mathrm{{ext}}}^e + \mathbf{{A}}{\left[ {{{{\varvec{\bar{R}}}}_0}} \right] _{s \in {\Gamma _\mathrm{{N}}}}}\), where
Similarly, the increment of the internal virtual work is discretized as
with . The element tangent stiffness matrix is
where
and \({{\mathbb {Y}}_e} \,{:}{=}\,{\left[ {\begin{array}{*{20}{c}} {{{\textbf{Y}}_1}}, \ldots , {{{\textbf{Y}}_{{n_e}}}} \end{array}} \right] _{2n_{{\textrm{cs}}} \times n_{{\textrm{cs}}}{n_e}}}\) with \({{\textbf{Y}}_I} \,{:}{=}\,{\left[ {\begin{array}{*{20}{c}} {N_I}{{{\textbf{1}}_{{n_{{\textrm{cs}}}}}}}, {N_{I,s}}{{{\textbf{1}}_{{n_{{\textrm{cs}}}}}}} \end{array}} \right] ^{\textrm{T}}}\). It is noted that the global tangent stiffness matrix is symmetric, since the matrix \({{\mathbb {C}}_e}\) and \({\varvec{k}}_\textrm{G}\) are symmetric.
Beam contact formulation
Let \(\mathcal {W}_k\) denote the active (working) set of indices of the surface Gauss point at the kth iteration of the outer loop, defined by
Hereafter, we often use \({{\varvec{\xi }}_i}\,{:}{=}\,[\xi _i^{1},\xi _i^{2}]^\textrm{T}\), which denotes the parametric coordinates of the ith surface Gauss integration point of the slave body, and \({\bar{\varvec{\xi }}}_i\,{:}{=}\,{\bar{\varvec{\xi }}}({{\varvec{\xi }}_i})\) with \({\bar{\varvec{\xi }}}_i\,{:}{=}\,[{\bar{\xi }^1_i},{\bar{\xi }^2_i}]^\textrm{T}\). We also use \(g^i_{\textrm{N}}\,{:}{=}\,{g_{\textrm{N}}}({\varvec{x}}^{(1)}({{\varvec{\xi }}_i}),{{\bar{\varvec{x}}}}({\bar{\varvec{\xi }}}_i))\), and let \(n^\textrm{L}_\textrm{G}\) denote the total number of Gauss integration points on the lateral surface of the slave body. \(w^\alpha _i\) denotes the weight of the Gauss integration point in the domain \({\varXi _{(1)}^\alpha }\ni {\xi ^\alpha }\), \(\alpha \in \left\{ 1,2\right\} \). Then, from Eq. (94b), we define
\(i\in {\mathcal {W}_k}\), with \({{\tilde{J}}^{(1)}_i}\,{:}{=}\,{\tilde{J}^{(1)}}{({{\varvec{\xi }}_i})}\), \({{{\tilde{j}_i^{(1)}}}}\,{:}{=}\,{{{{\tilde{j}}^{(1)}}(\xi _i^1)}}\), \({{\zeta }}^\alpha _i\,{:}{=}\,\zeta ^\alpha ({\varvec{\xi }}_i)\), \({{\bar{\zeta }}^\alpha _i}\,{:}{=}\,\zeta ^\alpha ({\bar{\varvec{\xi }}}_i)\), \(\alpha \in \left\{ 1,2\right\} \), and \({\bar{\varvec{\nu }}_t^{i}}\,{:}{=}\,{\varvec{\nu }_t^{(2)}}\big ({\bar{\varvec{\xi }}_i}\big )\). Substituting Eq. (103) into the contact variational form of Eq. (94) gives
with
where
with \({\mathbb {N}_{e}^{(1)}}\equiv {{\mathbb {N}_{e}^{(1)}}}(\xi ^1_i)\), and \({{\bar{\mathbb {N}}}_{{\bar{e}}}}\,{:}{=}\,{{{\mathbb {N}}}^{(2)}_{\bar{e}}}({{\bar{\xi }}}^1_i)\). e and \({\bar{e}}\) represent the indices of the nonzero knot spans including the coordinates \({\xi }_i^{1}\) and \({{\bar{\xi }}}_i^1\), respectively, i.e., \(\varXi ^{1(1)}_{e}\ni {{\xi }_i^1}\) and \(\varXi ^{1(2)}_{\bar{e}}\ni {{{\bar{\xi }}}_i^1}\). Similarly, the increment of the contact variational form of Eq. (100) can be discretized as
where
From Eq. (101), we have
where \(i\in {\mathcal {W}_k}\), and \({\mathbb {k}}^{\textrm{M}}_\textrm{N}\) and \(\mathbb {k}^{\textrm{G}}_\textrm{N}\) are defined by Eqs. (A21) and (A22), respectively.
Numerical examples
We verify the presented isogeometric finite element formulation for beams and frictionless contact by comparison with reference solutions based on isogeometric brick elements. In the latter, we employ an active set strategy for Gauss pointtosurface contact combined with a penalty regularization. For the brick element formulation, we denote the degrees of basis functions in each parametric coordinate direction by \(\mathrm {deg.}=(p_\textrm{L},p_\textrm{W},p_\textrm{H})\), where \(p_\textrm{L}\), \(p_\textrm{W}\), and \(p_\textrm{H}\) denote the degrees of basis functions along the length (L), width (W), and height (H), respectively. Further, the number of elements in each of those directions are indicated by \({n_\textrm{el}}={n_\textrm{el}^\textrm{L}}\times {{n_\textrm{el}^\textrm{W}}}\times {{n_\textrm{el}^\textrm{H}}}\). In both beam and brick element formulations, for the contact integral, we employ a subdivision of elements in order to increase the accuracy of numerical integration. The number of subelements are given in each numerical example, and we always use a single Gauss integration point per subelement, so that the number of subelements is the same as the number of Gauss integration points for the contact integral. In the beam formulation, for the evaluation of the contact integral along the circumferential direction, we subdivide the whole domain \(\left[ \xi ^2_\textrm{min},\xi ^2_\textrm{max}\right] \ni {\xi ^2_{(1)}}\) of each curve patch into \(m_\textrm{el}^\textrm{sub}\) subelements with uniform intervals. In the brick formulation, \(m^\textrm{sub}_\textrm{el}\) denotes the number of subelements within each nonzero knot span (element) along the circumferential direction. In both beam and brick element formulations, the number of subelements in each element (nonzero knot span) along the axial direction is denoted by \(n^\textrm{sub}_\textrm{el}\). Thus, in the beam formulation, the total numbers of subelements for the evaluation of contact integral is \({n_\textrm{el}}\times {n_\textrm{el}^\textrm{sub}}\) and \(m_\textrm{el}^\textrm{sub}\) in the axial and circumferential directions, respectively. In the brick element formulation, the total numbers of subelements in two transverse directions are \({n^\textrm{W}_\textrm{el}}\times {m^\textrm{sub}_\textrm{el}}\), and \({n^\textrm{H}_\textrm{el}}\times {m^\textrm{sub}_\textrm{el}}\), respectively. In the numerical examples, we consider a compressible NeoHookean material, which is the simplest type of hyperelastic material for arbitrarily large deformations. The St. Venant–Kirchhoff material, which provides a simpler formulation, is not valid for arbitrarily large deformations. Especially in the example of Sect. 5.4, we apply a large prestrain in the axial direction, in which the St. Venant–Kirchhoff material might show unphysical decrease of volume for positive Poisson’s ratio, as discussed in Choi et al. [5]. In Sects. 5.3 and 5.4, the boundary curves of initial (circular) crosssections are represented by a single patch of periodic NURBS with \(p=3\), and \(m_\textrm{cp}=67\).
Lateral contact of a straight beam under end moments
This example investigates the alleviation of Poisson locking in the beam formulation, and the verification of the contact pressure distribution in lateral contact to a rigid flat surface. We consider a straight beam with length \(L=10\,\textrm{m}\) and a rectangular crosssection of height \(h=0.2\,\textrm{m}\), and width \(w=1\,\textrm{m}\), and choose a NeoHookean material with Young’s modulus \(E=1.2\times {10^7}\,\textrm{Pa}\), and the Poisson’s ratio \(\nu =0.25\). As shown in Fig. 9a, the straight beam is aligned in Xdirection and it has an initial lateral distance \({\epsilon }L\) from the rigid flat surface, with chosen \(\epsilon =2\times 10^{2}\). An end moment \(M=\alpha \lambda _\textrm{load}EI/L\) is applied, where \(\lambda _\textrm{load}\) denotes the load parameter, satisfying \(0\le \lambda _\textrm{load}\le 1\), and I denotes the second moment of inertia of the rectangular crosssection, obtained by \(I=w{h^3}/12\). It is noted that for \(\alpha =2\pi \) the two end points of the beam’s axis meet, leading to an indeterminacy for rotation around the Yaxis, which is why we choose \(\alpha =1.8\pi \). In this example, we use uniform load increments, and the load parameter is obtained by \(\lambda _\textrm{load}=n/{n_{\textrm{load}}}\), where \(n_\textrm{load}\) denotes the total number of load step, and \(n=0,1,\ldots ,n_\textrm{load}\) denotes the load step number. We use \(n_\textrm{load}=20\) for both beam and brick elements. In both beam and brick element solutions, we use \(n_\textrm{el}^\textrm{sub}=10\) and \(m_\textrm{el}^\textrm{sub}=20\). For the beam formulation, a hinge condition is imposed at the left end (\(s=0\)), as \(\Delta {\varphi }_{1} = \Delta {\varphi }_{2} = \Delta {\varphi }_{3} = 0\), and \(\Delta {d}_{22}=0\) is further imposed in order to avoid rigid body rotation around the Xaxis, where \(\Delta \varphi _i\,{:}{=}\,{\Delta \varvec{\varphi }\cdot {\varvec{e}}_i}\) and \({\Delta }d_{{\alpha }i}\,{:}{=}\,\Delta {{\varvec{d}}_{\alpha }}\cdot \varvec{e}_i\) (\(i\in \left\{ 1,2,3\right\} \), \(\alpha \in \left\{ 1,2\right\} \)). In the brick element formulation, at the left end, we also impose the condition \({{\bar{v}}}=0\) along the vertical line \(Y=0\) in the initial configuration (see the vertical blue line in Fig. 9b) in order to avoid the rigid body rotation around Xaxis. The detailed formulation of the end moment condition can be found in Remark 4. In the brick element model, we apply the same traction boundary condition of Remark 4 (see Eq. (55)) with \({{\bar{p}}}={MZ/I}\) (\( h/2\le Z\le h/2\)) at both end faces at \(X=0\) and \(X=L\). At the right end of the axis (\(s=L\)), roller conditions are applied by \({{\bar{v}}}={{\bar{w}}}=0\), and \(\Delta {\varphi _2}=\Delta {\varphi _3}=0\) in the brick and beam element formulations, respectively, see Fig. 9c. Since the surface basis functions do not have an interpolatory property in the domain of the end faces, these displacement boundary conditions along the selected points or lines in the initial configuration of the brick element model are imposed by using a penalty method with penalty parameter \(\epsilon _\textrm{D}=10^{7}E/{L_0}\), where \(L_0\) denotes the unit length. In the beam formulation, however, those displacement boundary conditions can be imposed exactly since the basis functions using the clamped knot vectors in the axis satisfies the Kroneckerdelta property at the ends.
Deformation without contact
We first consider a problem without the impenetrability condition. Figure 10 compares the Xdisplacement at the point A (marked in Figure. 9a) between beam element solutions and reference solutions for two different values of Poisson’s ratios: \(\nu =0.25\) and \(\nu =0\). We have the following two reference solutions:

Analytical solution: Under the assumption of pure bending, the applied moment M at both ends deforms the beam axis into a circle with radius \(R=EI/M\), where the Xdisplacement at point A can be obtained as
$$\begin{aligned} u_\textrm{A} = L\left\{ \frac{\sin \left( {{\alpha {\lambda }_\textrm{load}/2}}\right) }{\alpha {\lambda }_\textrm{load}/2}  1\right\} , \end{aligned}$$(122)for \(0<\lambda _\textrm{load}\le 1\), and \(u_\textrm{A}=0\) if \(\lambda _\textrm{load}=0\),

The brick element solution using Bspline basis functions of \(\mathrm {deg.}=(3,3,3)\) and \(n_\textrm{el}=80\times 10\times 10\).
The results using \(N=1\) suffers from Poisson locking such that the bending stiffness is artificially increased due to the inability to represent linear inplane strains of the crosssection. By employing the EAS method in Choi et al. [5], it is shown that the beam solution agrees very well with the analytical solution. However, the beam solution for \(N=1\) (EAS) does not consider crosssectional warping properly, so that it is shown to slightly deviate from the brick element solution. By increasing the order of approximation in the transverse direction to \(N=2\), which enables to represent crosssectional warping properly, the beam solution comes very close to the brick element solution. In case of \(\nu =0\), there is no Poisson effect, so that the beam solution for \(N=1\) already agrees very well with the analytical solution, but still slightly deviates from the brick element solution. Similar to the results in the case of \(\nu =0.25\), the beam solution of \(N=2\) agrees very well with the brick element solution. Figure 11 compares the deformed crosssection shapes for the two different Poisson’s ratios: \(\nu =0.25\) and \(\nu =0.25\). It is shown that the beam solution for \(N=1\) (EAS) always maintains straight boundaries, while the beam solution for \(N=2\) properly represents the curved crosssection boundary due to the Poisson effect in the cases of \(\nu =0.25\), and \(\nu =0.25\). This is essential for accurately capturing contact, as is shown next.
Contact pressure distribution on the bottom surface
Next, we verify the contact formulation by comparing the pressure distribution between the presented beam contact formulation and the brick element solutions. Figure 12 shows the comparison for the three different values of Poisson’s ratios: \(\nu =0.25\), \(\nu =0\), and \(\nu =0.25\). For a positive Poisson’s ratio, the bottom surface deforms into a concave shape, so that the contact pressure is concentrated around the two lateral edges, see Fig. 12a–c. On the other hand, for a negative Poisson’s ratio, the bottom surface deforms into a convex shape, so that the contact pressure is concentrated around the center of the bottom surface, see Fig. 12d–f. For zero Poisson’s ratio, the contact pressure is higher in the center region than that around the lateral edge in the results of the brick and beam (\(N=2\)) element formulations, see Fig. 12g–i. Further, we define the total contact force by the integral of the contact pressure in the lateral surface, as
where the contact force per unit initial arclength is obtained by
Figure 13 compares the distribution of \(r_\textrm{N}\) along the axis in the final deformed configuration from the beam solution with that of the brick element solution. In the beam solution for \(N=1\), the crosssection boundary always remains straight (hence the constant contact pressure in transverse direction in Fig. 12b, e and h), which leads to an overestimation of the contact force. However, it is seen that the agreement improves, as N increases. Figure 14 also shows that the total contact force of Eq. (123) converges to the brick element solution, as N increases.
Lateral contact of a circular ring
In this example, we consider contact between an elastic circular ring and a rigid flat surface. The circular ring has a square crosssection with dimension \(h=w=5\,\textrm{m}\), and the inner and outer radii of the ring are \(R_\textrm{i}=20\,\textrm{m}\) and \(R_\textrm{o}=25\,\textrm{m}\), respectively, see Fig. 15a. We choose the NeoHookean material model with Young’s modulus \(210\,\textrm{MPa}\) and Poisson’s ratio \(\nu =0.3\). Impenetrability is enforced by the penalty method using the penalty parameter \({\epsilon _{\textrm{N}}}={10^2}E/{L_0}\). The displacement is prescribed at the inner surface \(\mathcal {S}^\textrm{D}_0\) (see Fig. 15a), as
where we choose the prescribed displacement vector \(\bar{\varvec{u}}=[0,2\,{\textrm{m}},0]^\textrm{T}\), which means that the inner surface of the circular ring is rigidly translated in the negative Ydirection. For the beam formulation, the prescribed displacement boundary condition on the lateral surface \(\mathcal {S}^\textrm{D}_0\) can be enforced using a penalty method, and the detailed formulation can be found in Appendix A.3. The corresponding penalty parameter is chosen as \({\epsilon _{\textrm{D}}} = {10^7}E/{L_0}\). For computational efficiency, we locally refine the mesh along the longitudinal direction in the domain quarter at the bottom (see Fig. 15). In the following, for brevity, we specify only the number of elements in the bottom parts for both beam and brick simulations. In the other parts, we use \(n^\textrm{L}_\textrm{el}=10\) and \(n_\textrm{el}=10\) for brick and beam element solutions, respectively. In this example, we use uniform load increments with a total number of load steps \(n_\textrm{load}=5\), and \(n_\textrm{load}=40\) for beam and brick simulations, respectively. In both beam and brick simulations, we use \(n_\textrm{el}^\textrm{sub}=10\) and \(m_\textrm{el}^\textrm{sub}=20\). Figure 16 compares the displacement of the line \(\overline{AB}\), defined in Fig. 15b. This displacement contains a Zcomponent due to the Poisson effect. In case of \(N=1\) (EAS), we have only one director along \(\zeta ^2\), see Fig. 15b, and the homogeneous boundary condition on the Zdisplacement at \(\mathcal {S}^\textrm{D}_0\) constrains the magnitude of the director \(\varvec{d}_2\). Therefore, the lateral displacement component \(u_Z\) vanishes. As the order of approximation N is increased, the lateral displacement of the beam formulation approaches that of the brick element solution, as Fig. 16 shows. Further we verify the contact pressure distribution at the bottom surface, i.e., the greencolored surface in Fig. 15b. Due to the symmetry with respect to the \(YZ\) plane, we plot the pressure only within the range \({0^ \circ } \le \theta \le {30^ \circ }\). Figure 17 compares the contact pressure distributions of the beam and brick element models. Figure 18 compares the distribution of the contact force per unit initial arclength, i.e., \(r_\textrm{N}\) of Eq. (124). In case of \(N=1\), the contact pressure is constant in transverse direction due to the first order approximation of the displacement field, which leads to an overestimation of the contact force. As the order of approximation N increases, the agreement between the contact pressure distribution and the brick element solution improves significantly, as Figs. 17 and 18 show. Further, Fig. 19 shows that the total contact force of Eq. (123) converges to the corresponding brick element solution with increasing order N.
Sliding contact between two initially straight beams
We consider sliding contact between two initially straight beams. The two beams have the same length \(L=6\,\textrm{m}\) and a circular crosssection of radius \(R=0.1\,\textrm{m}\), and they are initially perpendicular to each other with vertical distance \(d=10^{3}\,\textrm{m}\). We select the NeoHookean material model with Young’s modulus \(E=210\times 10^9\,\textrm{Pa}\) and Poisson’s ratio \(\nu =0.3\). In the contact formulation, we choose the lower beam as the master body, and the upper beam as the slave body. The lower beam is fixed at both ends, and the upper beam is under nonhomogeneous displacement boundary conditions at both ends of the beam, see Fig. 20. We investigate the following two cases of deformability.

Case 1: the slave body (upper beam) is rigid, and the master body (lower beam) is deformable,

Case 2: both beams are deformable. It should be noted that the prescribed displacement boundary conditions apply to the whole end faces. Thus, the crosssections at both ends are not deformable.
We use the beam formulation with \(N=1\) combined with the EAS method, and choose the cutoff radius \(r_\textrm{c}=3R=0.3\,\textrm{m}\), and \(\varepsilon _\theta =0.4\pi \) in the global contact search.
Case 1: Rigid slave beam
The first case considers a rigid slave body, and we implement the rigidity by constraining all DOFs in the finite element discretization of the beam, and the closest point projection employs the numerical scheme presented in Sects. 3.2 and 3.3. Figure 21 shows the deformed configuration. During the sliding contact, as we consider frictionless contact, no Ydirectional contact force should act on the lower beam. However, if the number of Gauss integration points along the axis of the slave body is not sufficient like the case \(n^\textrm{sub}_\textrm{el}=4\), it is seen in Fig. 22 that an unphysical Ydisplacement occurs. For both cases \(n^\textrm{sub}_\textrm{el}=10\) and 20, it is shown that the Ydisplacement vanishes to machine precision. However, it is observed that, if \(\varepsilon _\theta \) is too small (e.g., \(\varepsilon _\theta =0.25\pi \)), the activeset iteration does not converge but oscillates between two different contact states in several load steps, e.g., the last one, see Fig. 23. With \(\varepsilon _\theta =0.4\pi \), the active set iteration converges in all load steps. We calculate the average normal gap over the whole contact area by
Figure 24a shows that the average normal gap decreases, and the total contact force converges, with increasing penalty parameter. Table 1 shows the selected number of subelements in axial and circumferential directions for the contact integral, and Table 2 shows the selected load increment sizes for each case of the penalty parameter. It is typically required to increase the number of surface Gauss integration points for the contact integral and reduce the load increment size in order to achieve convergence in the solution process using a larger penalty parameter. Too many Gauss integration points or high penalty parameters may lead to an overconstrained system, a case also called contact locking, especially for low order finite elements. In this paper, we basically use higher order basis functions, so that we do not observe such locking. Further steps to alleviate this locking, e.g., using a mortartype discretization, remains future work. One can also develop a scheme to deactivate superfluous Gauss integration points in the active set (outer) loop, or adaptively control the penalty parameter.
Case 2: Two deformable beams
Next, we consider both beams deformable. Figure 25 shows the final deformed configuration. In Fig. 26, we compare the lateral (Ydirectional) displacement at the center of the lower beam’s axis during the deformation for two different numbers of DOFs in the upper beam (slave body): \(n^{(1)}_\textrm{el}=160\) and \(n^{(1)}_\textrm{el}=320\) with the number of subelements \(n^\textrm{sub}_\textrm{el}=20\), and \(n^\textrm{sub}_\textrm{el}=10\) for the evaluation of the contact integral, respectively, such that both cases have the same total number of Gauss integration points. In the former case with less DOFs in the slave body, the lateral displacement exhibits spurious oscillations. This is associated with an oscillatory change of crosssectional area along the axis in the deformed configuration of the slave body due to curvature thickness locking, which means an artificial coupling between the crosssectional stretching and the bending deformation. In Fig. 27, we compare the distribution of the crosssectional area along the axis at the final deformed configuration for the two cases of the number of DOFs in the slave body. In Fig. 27, the reference solution of the crosssectional area (\(A_\textrm{ref}\)) is obtained by using Bspline basis functions with \(p=4\), \(n^{(1)}_\textrm{el}=n^{(2)}_\textrm{el}=320\), \(n^\textrm{sub}_\textrm{el}=20\), and \(m^\textrm{sub}_\textrm{el}=100\). Figure 27 shows that the crosssectional area decreases in the whole domain due to the Poisson effect caused by the axial stretching. It leads to large curvature in the lateral surface around the loaded area as well as the fixed boundary, which eventually leads to the oscillatory lateral displacement during the sliding contact. As shown in Choi et al. [5], this locking can be alleviated by mesh refinement, see Fig. 27b. Thus, in Fig. 26, it is seen that as we increase the number of DOFs in the slave body while maintaining the same total number of Gauss integration points for the evaluation of the contact integral, the amplitude of oscillation significantly decreases. The selected load increment sizes for each case of the results in Figs. 26 and 27 can be found in Table 3.
Twisting of a wire strand
A wire rope usually consists of twisted strands, where each strand is also composed of several twisted wires. Here we show a simulation of the twisting process of strands made of initially straight wires, which has been commonly employed to verify the applicability of the developed beamtobeam contact formulations to cases with significantly small intersection angles, for example, see the relevant examples in Meier et al. [23], Konyukhov et al. [15], and Durville [9, 10]. We consider two different cases of strands with either two or seven initially straight wires aligned with the Xaxis. In both cases, we consider initially straight beams with \(L=10\,\textrm{m}\), and initially circular crosssections of radius R. Two initial directors of the crosssection are chosen as \({{\varvec{D}}_1} = {\varvec{e}_2}\) and \({{\varvec{D}}_2} = {\varvec{e}_3}\). A NeoHookean material model with Young’s modulus \(E=210\,\textrm{GPa}\), and Poisson’s ratio \(\nu =0.3\) is considered. We use the beam formulation with \(N=1\) combined with the EAS method, and consider the following boundary conditions.

First, we constrain the axis displacement \(\Delta \varvec{\varphi }\) at the end \(s=0\), i.e,
$$\begin{aligned} \Delta \varvec{\varphi }=\varvec{0}\,\,\,\,\textrm{at}\,\,s=0. \end{aligned}$$(127) 
Second, an axial tension is imposed as a predeformation by prescribing the Xdisplacement of the axis, as
$$\begin{aligned} \Delta \varvec{\varphi } = {\bar{u}_1}{\varvec{e}_1}\,\,\,\,\textrm{at}\,\,s=L, \end{aligned}$$(128)where we choose \({\bar{u}}_1=1\,\textrm{m}\). This leads to a loss of contact due to the Poisson effect in the early phase of the twisting motion.

Third, the end position of the axes at \(s=L\) are prescribed such that they follow a circular path (see Fig. 28), i.e.,
$$\begin{aligned} \left\{ {\begin{array}{*{20}{c}} {{\varvec{\varphi }\cdot \varvec{e}_2}{c_2}}\\ {{\varvec{\varphi }\cdot \varvec{e}_3}{c_3}} \end{array}} \right\} = {\bar{\varvec{\Lambda }}}\left\{ {\begin{array}{*{20}{c}} {{{\varvec{\varphi }_0\cdot \varvec{e}_2}}  {c_2}}\\ {{{\varvec{\varphi }_0\cdot \varvec{e}_3}}  {c_3}} \end{array}} \right\} , \end{aligned}$$(129a)with
$$\begin{aligned} {\bar{\varvec{\Lambda }}}\,{:}{=}\,\left[ {\begin{array}{*{20}{c}} {\cos {{\bar{\theta }}}}&{}{  \sin {{\bar{\theta }}}}\\ {\sin {{\bar{\theta }}}}&{}{\cos {{\bar{\theta }}}} \end{array}} \right] ,\,\,0 \le {{\bar{\theta }}} \le 2\pi , \end{aligned}$$(129b)where \(c_2\) and \(c_3\) denote the fixed Y and Zcoordinates of the selected rotation center, and \({\bar{\theta }}=2\pi \lambda _\textrm{load}\) denotes the prescribed rotation angle with the load parameter \(0\le \lambda _\textrm{load}\le 1\). Note that this prescribed circular path is imposed after applying the axial tension of Eq. (128).

Fourth, in order to prevent the rigid body rotation around the axis, we constrain the displacement components of directors \(\varvec{d}_1\) and \(\varvec{d}_2\) at the end \(s=0\), as
$$\begin{aligned} \Delta {\varvec{d}_{1}}\cdot {\varvec{e}_3} = \Delta {\varvec{d}_{2}}\cdot {\varvec{e}_2} = 0\,\,\,\,\textrm{at}\,\,s=0. \end{aligned}$$(130)
We choose the cutoff radius \(r_\textrm{c}=3R\), and \(\varepsilon _\theta =0.25\pi \) in the global contact search.
A strand of two wires
We first consider a strand of two wires initially parallel to the Xaxis, and a circular path is prescribed by the angle \({\bar{\theta }}=2\pi \lambda _\textrm{load}\,[\textrm{rad}]\) at \(s=L\) using Eq. (129), see Fig. 28. Figure 29 shows the deformed configurations for two different values of the initial crosssection radius R. In Fig. 30, we compare the Zcoordinate in the axis of the final deformed configuration of the wires with that of an analytical circular helix of radius R, given by
where \(\ell \) denotes the deformed length of the beam’s axis after the prestretch, and \({\varphi ^h_i}\,{:}{=}\,{\varvec{\varphi }^h}\cdot \varvec{e}_i\), \(i\in \left\{ 1,2,3\right\} \). Further we define the relative \(L^2\)norm of the difference by
where the \(L^2\)norm for \(u=u(s)\) in the domain \((0,L)\ni {s}\) is defined as
It is shown in Fig. 30 that the deformed axis of the beam slightly deviates from the analytical helix curve, and the difference decreases as the initial crosssectional radius decreases. The difference from the analytical solution is mainly attributed to the following reasons, which are not considered in the analytical solution:

Crosssectional contraction due to the prestretch,

Crosssectional deformations due to the contact interactions,

A slight penetration allowed in the beam contact formulation using the penalty method.
Figure 31 shows that the difference decreases as the amount of penetration decreases due to increasing the penalty parameter. It is also seen in Fig. 32a that the difference linearly decreases, as the slenderness ratio increases, since the Zdisplacement is linearly proportional to the crosssectional radius. However, the relative difference does not completely vanish but converges to a value around \(e_{\varphi _3}=0.03\), which is attributed to the nominal transverse normal strain in the crosssection. Table 4 shows the selected number of subelements for the contact integral. Tables 5, 6 and 7 show the selected load increment sizes used in the results of Fig. 31 for initial crosssectional radii \(R=0.25\,\textrm{m}\), \(0.125\,\textrm{m}\), and \(0.0625\,\textrm{m}\), respectively.
A strand of seven wires
We consider twisting of a strand with seven wires. The beams are initially aligned with the Xaxis, and have the same length \(L=10\,\textrm{m}\), and a circular crosssection of radius \(R=0.25\,\textrm{m}\). Figure 33 shows the initial arrangement of the wires and the prescribed rotation \({\bar{\theta }}=2\pi \lambda _{\textrm{load}}\,[\textrm{rad}]\) at the end \(s=L\) of the outer wires. We consider the following two cases of selecting contact pairs:

M1S6: the inner beam is selected as master body, and the other six outer beams are selected as slave bodies. Thus, we have a total of six contact pairs.

M6S1A: the inner beam is selected as slave body, and the other six outer beams are selected as master bodies. Additionally, the interaction between outer wires is considered. Thus, we have six inner wireouter wire contact pairs, and also six outer wireouter wire contact pairs, see Table 8 for the chosen contact pair information.
Figure 34 shows the final deformed configuration in case 1. In Fig. 35, the average of the total contact forces between the contact pairs in each case is plotted. In the result for M6S1A, the contact force between inner and outer wires, and outer wires are plotted separately. Interestingly, the contact force between inner and outer wires in those two cases are significantly different, if we additionally consider the interaction between outer wires. Table 9 shows the selected load increment sizes in each case of M1S6 and M6S1A. In both cases, we use \(n_\textrm{el}^\textrm{sub}=20\), and \(m_\textrm{el}^\textrm{sub}=300\) for the contact integral.
Conclusions
In this paper, we present an isogeometric finite element formulation for beams and beamtobeam frictionless contact, based on the kinematics of Cosserat rod with unconstrained directors. The beam crosssectional deformation is efficiently and accurately described by unconstrained directors of an arbitrary order. The beam contact formulation is based on a Gauss pointtosurface contact algorithm, where the impenetrability constraint is enforced by a penalty method combined with an active set method. Further we present a geometrical approach to efficiently determine an initial guess in the local Newton–Raphson iteration of the closest point projection. In numerical examples, we show that the beam contact formulation can provide contact pressure distributions which agree well with brick element solutions while requiring much less DOFs. The following areas could be interesting future research directions.

Alleviation of locking: In Sect. 5.3.2, it is shown that the curvaturethickness locking may lead to an unphysical oscillation of displacements in sliding contact. An extension of the assumed natural strain method in Betsch and Stein [3] to a higher order basis functions would be interesting. Further, an alleviation of contact locking (overconstrained system) due to the contact constraints will be investigated further in the future, e.g., using a mortartype discretization method, a deactivation of superfluous Gauss integration points in the active set (outer) loop, or an adaptive control of the penalty parameter.

A mixedvariational formulation could alleviate locking and improve robustness in larger load steps, see Wackerfuß and Gruttmann [45]. The static condensation of the strains associated with higher order directors could also improve the efficiency of the beam formulation.

It is observed that an increase of the penalty parameter typically requires larger number of surface Gauss integration points for the contact integral and larger number of load steps, which eventually makes the solution process less efficient. One can consider an augmented Lagrangian method or Nitsche’s method in order to exactly satisfy the impenetrability constraint using low or moderate penalty parameters.

In the surfacetosurface beam contact formulation, we consider contact between smooth lateral surface domains only. If a nonsmooth edge exists in the lateral surface, it is required to additionally consider a contact integral along the edge, for example, see Litewka and Wriggers [19]. Further, the crosssections at the ends of axis may also contribute to contact interactions. For example, their boundary edges may undergo edgetosurface or edgetoedge contact.

The present beam contact formulation can be extended to incorporate tangential contact conditions including sticking and sliding friction. In sliding friction, an additional history variable is required for the amount of irreversible tangential slip, and it typically requires much smaller load increment sizes for accuracy, compared with frictionless or sticking contact formulations.
Change history
08 November 2022
A Correction to this paper has been published: https://doi.org/10.1007/s00466022022360
Notes
The axis is typically defined by a spatial curve connecting (mass) centroids of the crosssections, which is often called line of centroids.
In this paper, we assume a constant mass density, so that the mass centroid coincides with the geometrical centroid.
Version 7, Robert McNeel and Associates.
This formulation can be easily extended to problems with more bodies by applying it to each pair of bodies.
The projection of the relative position vector was also used for the contact search in Durville [10].
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Acknowledgements
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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Appendices
Appendix A
A.1 Gauss pointtoaxis closest point projection
We first define the distance between the material points on the axes of slave and master bodies, as
The convective axial coordinate \(\widetilde{\xi ^1}\) of the closest point in the master body to a given point \(\varvec{\varphi }^{(1)}\) along the axis of the slave body is determined as a solution of the following unilateral minimal distance problem:
where
Hereafter, for brevity, we often omit the argument \(\xi ^1_{(2)}\). The solution of Eq. (A2) satisfies the first order necessary condition
This is a single nonlinear equation with respect to \({{\xi ^1_{(2)}}}\), which can be iteratively solved using a Newton–Raphson iteration. The linearization of Eq. (A4) leads to
such that the convective axial coordinate is updated by
until a convergence criterion \(\left{{f_\varphi }} \right< \varepsilon _{{\textrm{cpp}}}^\varphi \) is satisfied, where \(\varepsilon _{{\textrm{cpp}}}^\varphi >0\) is a chosen tolerance, and
and is a chosen initial guess. Assuming \({f_\varphi ^ *}\ne 0\), from Eq. (A5), we obtain
A.2 Contact variational form
A.2.1 Linearization
Taking the directional derivative of Eq. (91) leads to [47]
with the following matrix operators
and
\(\delta {\bar{\varvec{\xi }}}\) in Eq. (A9) can be expressed in terms of \(\delta {\varvec{{\bar{u}}}}\). Taking the first order variation of Eq. (73), we have
where we obtain [47]
by evaluating Eq. (77) at \({{\varvec{\xi }}^{(2)}} = {\varvec{{\bar{\xi }} }}\), and we define the operator
where
Here we assume the matrix \(\bar{{\varvec{f}}^{*}}\) is invertible, then we have
We further define a matrix operator \({\varvec{\Xi }_\textrm{u}}\) such that
with
Then, substituting Eqs. (A16) and (A17a) into Eq. (A9), we finally obtain
where
with
A.2.2 Spatial discretization
In the material and geometric part of the tangent stiffness matrices, we define
with
and
where
Those matrices \({\varvec{\Xi }}_{\textrm{u}}^e\) and \({{\varvec{\Xi }}_{\textrm{f}}^e}\) are defined for the eth element, as
where
and
with
A.3 Weak enforcement of displacement boundary conditions
We present a weak enforcement of the displacement boundary condition on the lateral boundary surface, which is utilized in the numerical example of Sect. 5.2. Let \(\mathcal {S}^\textrm{D}_0\subset \mathcal {S}^\textrm{L}_0\) be a region of the initial boundary surface where the displacement \(\varvec{u}\) is prescribed, i.e., \(\varvec{u}={\bar{\varvec{u}}}\) on \(\mathcal {S}^\textrm{D}_0\), \({\bar{\varvec{u}}}\in \mathbb {R}^3\).
A.3.1 Variational formulation
We employ a penalty method, and the penalty functional can be expressed as
where \(\epsilon _\textrm{D}>0\) denotes the chosen penalty parameter, and \({{\bar{\omega }}}={{\bar{\omega }}}(\varvec{X})\) denotes a Heaviside function, defined by
Taking the first variation of Eq. (A27), and using Eq. (31), we have
where
with \(\varvec{f}_\textrm{D} \,{:}{=}\,{\epsilon _{\textrm{D}}}\,{\bar{\omega }}\,(\varvec{u}  {\bar{\varvec{u}}})\). Taking the directional derivative of Eq. (A29), we obtain
where
A.3.2 Spatial discretization
Substituting Eq. (103) into Eq. (A29) leads to
where the element load vector is defined as
Similarly, using (103) into Eq. (A31) gives
with \({{\textbf{K}}_{\textrm{D}}} \,{:}{=}\,\mathop {\textbf{A}}_{e = 1}^{{n_{{\textrm{el}}}}} {\textbf{K}}_{\textrm{D}}^e\), where the element tangent stiffness matrix is defined as
Appendix B
B.1 Global contact search
The global contact search scheme in Sect. 3.2 finds contact candidate Gauss integration points on the lateral surface of the slave body. Algorithm 1 shows the overall procedure of the global contact search, and Algorithm 2 presents the procedure of the Gauss pointtoaxis closest point projection, presented in Sect. A.1. Algorithm 3 shows the overall process of the local contact search. Since the NURBS basis functions are evaluated in the range of parametric coordinate \(\left[ \xi ^1_\textrm{min},\xi ^1_\textrm{max}\right] \ni \xi ^1\) only, we move to the next initial guess if the coordinate \(\xi ^1\) goes outside of this range, see lines 20–22 of Algorithm 2, and lines 21–23 of Algorithm 3. We still need to check if the solution of Eq. (73) found by the iterative method is a local maximum solution, based on the condition
with
Thus, if the condition of Eq. (B.1a) is violated, we could try a new initial guess or simply omit the given Gauss integration point. In this paper, we utilize the latter approach (see lines 11–16 in Algorithm 3), since it does not significantly affect the accuracy of the solution, if sufficient number of Gauss integration points are used.
B.2 Local contact search
B.2.1 Determination of an initial guess
We determine the initial guess \({{\bar{\xi }}^2_{(0)}}\) as the position of the intersection point, shown in Fig. 8, by solving Eq. (81) using a Newton–Raphson iteration. For a given \(\big ({}^{(i1)}{\bar{\xi }^2_{(0)}},{}^{(i1)}\alpha _\textrm{ig}\big )\), we first calculate
where \({}^{(i  1)}{e^\gamma } \,{:}{=}\,{e^\gamma }\big ({}^{(i1)}{{\bar{\xi }}^{2} _{(0)}},{}^{(i  1)}\alpha _\textrm{ig} \big )\), \(\gamma \in \left\{ 1,2\right\} \), and with
and then update
until a convergence criterion \(e < {\varepsilon _{{\textrm{ig}}}}\) is satisfied, where
and \(\varepsilon _\textrm{ig}>0\) is a chosen tolerance. We choose , and several values of with a uniform interval, see Algorithm 4 for the details.
Appendix C
In this appendix, we provide supplementary information in the numerical examples.
C.1 Sliding contact between two initially straight beams
C.1.1 Case 1
Table 1 shows the selected number of subelements for the contact integral. Table 2 shows the selected load increment size in each interval of the load parameter for each case of the penalty parameters.
C.1.2 Case 2
Figure 36 shows the original graph of the magnified one in Fig. 27b. Table 3 shows the selected load increment size in each interval of the load parameter.
C.2 Twisting of wire strands
C.2.1 A strand of two wires
Table 4 shows the selected number of subelements for the contact integral. Tables 5, 6 and 7 show the chosen load increment sizes in each case of the initial radii \(R=0.25\,\textrm{m}\), \(0.125\,\textrm{m}\), and \(0.0625\,\textrm{m}\), respectively.
C.2.2 A strand of seven wires
Table 8 shows the information of the chosen slavemaster contact pairs for the case of M6S1A in theexample of seven wires. Table 9 shows the selected load increment sizes in each case of M1S6 and M6S1A.
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Choi, MJ., Klinkel, S. & Sauer, R.A. An isogeometric finite element formulation for frictionless contact of Cosserat rods with unconstrained directors. Comput Mech 70, 1107–1144 (2022). https://doi.org/10.1007/s00466022022235
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DOI: https://doi.org/10.1007/s00466022022235
Keywords
 Cosserat rod
 Crosssectional deformation
 Frictionless contact
 Surfacetosurface contact
 Beamtobeam contact
 Isogeometric analysis