Two-scale off-and online approaches to geometrically exact elastoplastic rods

This work compares two different computational approaches to geometrically exact elastoplastic rods. The first approach applies an elastoplastic constitutive model in terms of stress resultants, i.e. forces and moments. It requires knowledge of the rod’s elasticity and yield-criterion in terms of stress resultants. Furthermore a resultant-type hardening expression must be formulated. These are obtained by integrating elastoplastic stress and hardening measures from three-dimensional continuum mechanics over the rod’s deformed cross-section, which is performed in an offline stage. The second approach applies an FE2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$FE ^2$$\end{document} approach as established in computational homogenization. Therein, the macro-scale describing the geometrically exact rod is coupled to the micro-scale, i.e., the cross-section of the rod. A novelty of the presented work is the determination of a hardening tensor for use in the stress resultant approach. The mechanical response of both approaches is first compared on the material point level, a single cross-section of a uniformly strained rod. Later, also the mechanical response and the deformation of finitely and non-uniformly strained rods are investigated.


Introduction
Even though the theory of slender bodies such as beams and rods goes back to the renaissance, new applications and new challenges emerge. Representative is the use of rods to model cellular structures as appearing for example in additively manufactured meta-materials [1]. These metamaterials gain in popularity due to their tailorable mechanical properties: we just name spatially varying stiffnesses and density or sophisticated mechanical functionalities such as negative Poisson's ratio [2,3]. Beside applications of cellular meta-materials where the underlying rods may be assumed geometrically linear and purely elastic, there are cases where theses assumptions are not valid. Representative for geometrically linear and purely elastic applications are lightweight structures in aerospace [4] or medical applications, such as B Ludwig Herrnböck ludwig.herrnboeck@fau.de 1 Institute of Applied Mechanics, Friedrich-Alexander University Erlangen-Nuremberg, Egerlandstraße 5, 91058 Erlangen, Germany 2 Department of Applied Mechanics, IIT Delhi, Hauz Khas, New Delhi, India femoral hip stems [5]. Nonlinearities are found on the geometrical as well as on the material level, where they may appear due to hyperelastic or inelastic material behavior. A flexible shoe sole combines both hyperelastic material and large deformation of rods [6]. Representative for inelasticity is an impact absorbing meta-material. This material is used in object and body protection devices and aims at the absorption of kinetic energy at constant stress [7][8][9]. Obviously, in these cases the use of linear rod theory is no more applicable. Rod formulations able to capture large deformations and rotations must be considered. Further, the inelastic behavior of rods must be represented in a satisfactory manner. This contribution is devoted to the modelling and computation of geometrically exact elastoplastic rods. Especially, the derivation of an appropriate resultant-type hardening description is addressed.
The theory of geometrically exact rods (the Cosserat theory of rods [10,11]) employed in this contribution differs from classical beam formulations such as Euler-Bernoulli or Timoshenko beams in that it is not restricted to the geometrically linear case. Rather, large deformations and rotations are captured [11][12][13]. Hence, this theory is widely used to model wires, tubes, cables and biological tissues. There, the theory of geometrically exact rods returns accurate results combined with a tremendeous decrease in simulation time, compared to the same rod modeled by classical three-dimensional continuum mechanics within a finite element framework.
Aiming to model elastoplastic rods, Wackerfuß et al. [14] derived a relation between the rod's strain measures and the continuous Green-Lagrange strain allowing for crosssectional warping and inelastic stress resultants (internal contact forces and internal contact moments) for rectangular cross-sections. This theory has been extended for arbitrary cross-sections in [15]. Elastoplastic rods may also be described in terms of the rod's strain measures and stress resultants. A translation of the rod's strains into continuous strain measures is no more necessary. The thermodynamically consistent extension from the elastic to the inelastic case is given in [16] and further elaborated in [17,18]. The challenge in modelling geometrically exact elastoplastic rods is the lack of knowledge of an appropriate yield-function in terms of the rod's stress resultants and an appropriate resultant-type hardening formulation. Applying upper and lower bound techniques of limit analysis, yield-surfaces in terms of stress resultants were derived in [19][20][21][22]. There, a complete plastification of the cross-section is assumed. In [23], the authors present yield-surfaces obtained in terms of dissipated work. The framework proposed there is general in that it is not restricted by the rod's cross-sectional shape and its constitutive behavior. Furthermore, the yield-criterion chosen is more flexible -full plastification of the crosssection need not be assumed as in prior works. However, the yield-surface defines just the onset of yield. Resultant-type hardening, which allows for the evolution of the yield-surface [24] has not yet been considered for the case of geometrically exact rods. Hardening is commonly captured with a diagonal hardening tensor [17,18].
The aim of this work is to further drive the use of geometrically exact elastoplastic rods within the modeling of lattice structures and slender load bearing structures such as rotors or lattice cranes. The definition of limit loads is of crucial importance in engineering applications and requires a profound knowledge on the inelastic behavior of slender structures, which, in the following, shall be modeled by rod models. Compared to structures modeled by threedimensional volume elements, the use of rods enables to solve large problems within moderate time. To this end, this contribution encourages the investigations on plastic behavior, not only in terms of an FE 2 homogenization framework but especially in terms of stress resultant dependent plasticity. In particular, the focus is on capturing hardening in geometrically exact elastoplastic rods by two different approaches. The frameworks presented here may be further used to investigate on limit loads of slender structures or periodically repeating lattice structures [17]. Both presented approaches make use of two-scale homogenization techniques and distinguish in the following way. The first approach expresses the elastoplastic behavior of the rod with an elastoplastic constitutive model formulated in terms of stress resultants and is denominated the stress resultants approach. This approach implies the knowledge of the yieldsurface and the resultant-type hardening behavior in terms of stress resultants. The determination of both the yieldsurface and the hardening behavior take place a priori in an offline stage. The yield-surface is addressed in our previous research [23]. In this contribution, a quadratic resultanttype hardening potential depending on an internal variable vector and a hardening tensor with yet unknown entries is introduced. The entries of the hardening tensor are fitted using an approach, which integrates the elastoplastic continuum stress over the rod's cross-section based on classical three-dimensional continuum mechanics [25,26], the cross-sectional warping problem. The same is employed in the second discussed approach to capture hardening: the FE 2 approach. The FE 2 approach is motivated by computational homogenization, where the mechanical properties of the macro-scale result from online integration over a representative volume element [27][28][29]. In this contribution, the geometrically exact rod represents the macro-scale, whereas the rod's cross-section represents the micro-scale. The description of the cross-sectional warping problem in terms of three-dimensional continuum mechanics enables the use of well known elastoplastic constitutive models which then return elastoplastic stress resultants and the corresponding tangent stiffness to the macro-scale. This implies that the hardening used at the micro-scale naturally reflects in the results of the macro-scale. The transfer of the required quantities and history variables from the macro-to the microscale and vice versa follows a FE 2 -inspired framework. Both approaches are presented and their results are comprehensively discussed and compared. This work has the following structure. Sect. 2 reiterates the theory of geometrically exact rods with rigid crosssections. First, the kinematics are introduced for the elastic and elastoplastic case, which allows to solve for elastoplastic rods within the stress resultants approach. Next, stress measures are introduced. Finally, the constitutive relation and the balance equations are addressed. Sect. 3 describes the cross-sectional warping problem. After resuming different attempts in the literature to incorporate cross-sectional warping an approach is presented that allows to model the non-linear warping of a cross-section of strained rod. Furthermore, it allows to determine resulting forces and moments and the rod's tangent stiffness. The approach is general in that it is not restricted by the cross-sectional shape and constitutive model. Here, multiplicative elastoplasticity will be used. Sect. 4 first introduces the FE 2 approach. Then the continuum elastoplastic models used in the FE 2 and stress resultants approach are recaptured. In the latter, a yield- Table 1 Strain measures and their energetic conjugates are denoted differently in the context of geometrically exact rods and three-dimensional continuum mechanics. This terminology is consistent with [17,23] Geometrically exact rods Three-dimensional continuum mechanics strain measures strain prescriptors strain energetic conjugates stress resultants stress surface in terms of stress resultants is used and a convenient resultant-type hardening potential is introduced. Finally in sect. 5 the numerical results on both approaches are compared. The comparison is made first on the material point level and eventually also for an entire rod. Throughout this contribution, we restrict ourselves to rods with circular crosssections. However, the framework presented here is general and could also be applied for rods with non-circular crosssections. Before setting the stage by recalling the theory of geometrically exact rods, the terminology used throughout this contribution is introduced. As in Herrnböck et al. [17,23] strain measures and their energetic conjugates are denoted differently in the context of three-dimensional continuum mechanics and geometrically exact rods. The different terminology is summarized in table 1.
All simulations carried out to generate the results shown in this contribution are based on the open source finite element library deal.II [30].

Geometrically exact rods with rigid cross-sections
In this section the theory of geometrically exact rods with rigid cross-sections is recalled. We begin with the presentation of the kinematics followed by the stress resultants and the balance equations for linear and angular momentum. This section follows [12,13,23]. Figure 1 shows a rod deforming from its undeformed straight material configuration B 0 into its deformed spatial configuration B t . In this contribution, the rod is assumed to be straight in its undeformed configuration, with coordinates,

Kinematics
The undeformed cross-section is denoted by X CS = X α E α . The deformed centerline is defined by r(s), with s := X 3 the rod's undeformed arc-length. Orthogonal directors d α (s) span the deformed cross-section Ω t which is assumed planar. Together with d 3 (s) = d 1 (s) × d 2 (s) Fig. 1 A geometrically exact rod deforming from its straight material configuration B 0 into its deformed, spatial configuration B t . Any point in the material configuration is denoted by X. Material points in the spatial configuration are denoted by x an orthogonal basis is constructed. Note that d 3 (s) need not be tangential to r(s). The orientation of the deformed cross-section is described by d i and the orientation of the undeformed cross-section is described by D i . Here, D i = E i . Usually, the principal directions of the undeformed crosssection are aligned with D α .
The directors in the spatial configuration are related to the directors in the material configuration via with R ∈ SO(3), the special orthogonal group. The quantities r(s) and R(s) are the kinematic unknowns of this theory. Note that the rotation described by the tensor R(s) can also be parametrised by a vector ϑ(s), where ϑ(s) equals the angle of the rotation, and ϑ(s)/ ϑ(s) describes the direction of rotation [12,18]. Assuming a rigid cross-section, the position x(s) of any material point in the spatial configuration is described by For the sake of readability, the dependencies on s and X CS are omitted from now on. Strain prescriptors are introduced as the translational strain v and the rotational strain k (also called curvature and twist): where the derivative with respect to the arc-length is denoted as The rotational pull backs v 0 and k 0 of v and k, respectively, are defined as: Here K and K 0 are skew-symmetric tensors whose axial vectors are k and k 0 respectively 2 . The deformation gradient F of the deformation map (2) is defined as Inserting equation (2) into equation (7) yields Recalling the dependencies r = r(s), R = R(s) and By factoring out R in equation (9) the deformation gradient can be written in terms of strain prescriptors as follows:

Stress resultants and balance equations
The internal (contact) force n and internal (contact) moment m, collectively called stress resultants, are defined by: Their rotational pull backs are 2 The operator ax(A) transforms an arbitrary skew-symmetric tensor A into its axial vector a such that: For the elastic case and assuming the existence of a scalarvalued strain energy density function ψ rod (v 0 , k 0 ) the stress resultants follow as The second derivative of ψ rod (v 0 , k 0 ) with respect to the strain prescriptors yields the 6 × 6 tangent stiffness C 0 which relates the change in the stress resultants to the change in the strain prescriptors as Assuming ψ rod to be quadratic in the strain prescriptors, a linear relation between stress resultants and strain prescriptors results In the elastoplastic case the definition of the stress resultants (13) is no longer valid. The key components of modeling elastoplastic rods are briefly presented here. The strain prescriptors are assumed to decompose into an elastic and a plastic part as follows: [16,23,31].
A free energy density function Ψ rod is introduced as where ψ rod (v e 0 , k e 0 ) denotes the strain energy density and H rod (ξ ) describes the resultant-type hardening depending on the internal variables ξ . The dissipation power D is the difference between the stress resultants power and the material time derivative of the free energy density and must be positive: where• := d• dt X . Computing the time derivative by applying chain rule and requiring that the result has to hold for any admissible process, one obtains the constitutive equations: Furthermore, the resultant-type hardening stress β is defined as Inserting equations (20) and (21) into the dissipation inequality yields its reduced form Expression (22) is maximized under the constraint Φ(n 0 , m 0 , β) ≤ 0. Here Φ describes a yield-function in terms of stress resultants. Finally, the evolution equations for the plastic quantities are obtained: Together with Karush-Kuhn-Tucker (KKT) conditions, the Lagrange multiplierγ ensures admissibility of the resulting stress resultantṡ The changes in stress resultants are related to the changes in strain prescriptors via the 6×6 elastoplastic tangent stiffness C ep 0 Due to the additive nature of the strain prescriptors, methods from small-strain plasticity may be applied to derive an update algorithm and the algorithmic elastoplastic tangent [18]. Using a plasticity formulation in terms of stress resultants is further called the stress resultants approach to solve elastoplastic rods.
To conclude this section the balance equations of linear and angular momentum are presented. In absence of distributed external forces and moments they take the following localized forms The kinematic unknowns (r, ϑ) can be determined by solving (26) using the finite element method. The computational framework used in this work is presented in [13]. Two-noded finite elements with three translational degrees of freedom (DOF) and three rotational DOFs assigned to each node are used. The kinematic unknowns within one element are approximated by linear shape functions. The discretized balance equations are evaluated using a one-point (uniformly reduced) Gauss integration, sufficient to exactly integrate linear shape functions within beam elements and commonly applied in literature [13,18,32].

Cross-sectional warping-problem
Attempts to incorporate warping to obtain accurate stiffnesses and stress resultants of rods have been tackled in different approaches. Simo et al. [33] allowed the crosssection to display out-of-plane warping as result of twist but neglect in-plane warping. Missing out warping due to shear, the resulting shear stiffness neglects the shear correction factor leading to non accurate stiffness values. Mora et al. [34] used Γ -convergence to obtain linear constitutive relations of arbitrariliy shaped rods. However, their approach is restricted to Kirchhoff rods. Yu et al. [35] enabled linear constitutive relations using the variational asymptotic beam selection analysis. An alternative method to obtain the rod's stress resultants and stiffnesses is given by Klarmann et al. [36] using a first-order homogenization framework fulfilling the Hill-Mandel condition. There, a rod segment of finite thickness serves as a micro-problem to the whole rod model, which is then defined as macro-problem. By solving the micro-problem and allowing warping the stiffness of the rod may be evaluated. In this section, however, we reiterate the cross-sectional warping problem as presented in [25] and elaborated for the elastic and inelastic case in [26] and [23], respectively. There, the cross-sectional warping problem allows to compute the cross-sectional deformation of the rod depending on strain prescriptors only. Similar to the above cited contributions, it further enables to compute the stress resultants and tangent stiffness of a rod, depending on strain prescriptors. The model is not restricted in the choice of the constitutive model and thus allows the use of inelastic continuum constitutive relations. In accordance with multiscale homogenization techniques, the cross-sectional warping problem may be treated as a micro-problem, since each cross-section describes a distinct material point in the one-dimensional rod theory.
In general a geometrically exact rod deforms such that the strain prescriptors vary along the rod's arc-length s. Thus, the deformation of the cross-section at s depends not only on the strain prescriptors at s but also on the strain prescriptors in the neighborhood of s. If, however, the strain prescriptors vary slowly along s, the deformation can be assumed to depend only on the local strain prescriptors. For the special case that v 0 (s) = v 0 and k 0 (s) = k 0 the rod takes the shape of an helix [25]. Let us rewrite the deformation map as [26]. Consequently, the deformation gradient based on equation (27) is Equation (28) reveals that the deformation in the crosssection is merely dependent on the strain prescriptors v 0 and k 0 and the warped cross-sectionX CS .
The warped cross-section may be evaluated for the purely elastic case by minimizing the stored energy density ψ (F) in the rod's cross-section under appropriate constraints [23,25,26]. These are given by where The first constraint in (30) ensures that the center of gravity of the cross-section remains fixed, whereas the second constraint ensures that the average rotation about E 3 vanishes and that the principal axes of the cross-section align with E 1 and E 2 .
To solve the constrained minimization problem the following constrained energy functional is defined: Here, the Lagrange multipliers λ and μ enforce the constraints (30). The minimization of (32) leads to the equi-librium equation of linear momentum, which may also be solved for the inelastic case [23].
Once the deformed configuration of the rod's cross-section is calculated, the micro-problem enables the computation of stress resultants by integrating the stresses (P, the Piola stress) in the rod's cross-section As introduced in equation (15) the tangent stiffness C 0 relates the changes in stress resultants to the changes in strain prescriptors. Thus C 0 is obtained by the derivative of the stress resultants (33) with respect to the strain prescriptors The entries of (34) are computed via where The derivatives ∂X CS ∂ p and ∂F ∂ p are presented in detail in [26].
Recalling from this chapter's introduction the crosssectional warping problem may be treated as a microproblem within the one-dimensional rod theory. It is obvious that the micro-problem may also be used to solve geometrical exact rods in an FE 2 approach. However, if the rod undergoes only moderate strain prescriptors, the mechanical response is nearly linear with elastic material behavior 3 Note that for the elastoplastic case A = A ep and subsequently C 0 = C ep 0 . and a FE 2 approach does not come with notable gains. But the FE 2 approach allows the use of inelastic constitutive relations without any knowledge about macro-scale plasticity in terms of stress resultants. We make use of this feature in subsequent sections.

Elastoplastic rods
In this section, the ingredients needed for the two different approaches to computationally model elastoplastic rods are introduced. One approach we pursue is the description of the inelastic behavior by a yield-surface in terms of stress resultants. Since this approach is solved directly in the geometrically exact rod setting, it has been named the stress resultants approach. In the second approach we make use of methods from multiscale homogenization, in particular the FE 2 method, where the mechanical properties at the macro-scale (the geometrically exact rod) are incorporated -while solving -by integrating the solution at the representative micro-scale (the cross-sectional warping problem). We refer to this approach as the FE 2 approach. A further possibility to incorporate inelastic material behavior within the theory of geometrically exact rods is by translating the rod's strain prescriptors into three-dimensional strain measures within the rod's cross-section. This approach allows cross-sectional warping and is pursued by Wackerfuß et al. [14,15]. In [14] the strain prescriptors are transformed into the Green-Lagrange strain on the crosssection. Both the transformation and the calculation of the warping function, enabling an exact calcultion of the rod's stiffnesses, make use of the description of the rod's crosssection by global shape functions. Thus the framework is first restricted to squared cross-sections. By translating the strain prescriptors into three-dimensional strain measures, the use of any constitutive relation, well known from threedimensional continuum mechanics is enabled. In [15] the warping is no more described by global shape functions but by local shape functions, which removes the constraint of squared cross-section. In contrast to these approaches, the FE 2 approach here shows a separation of the rod and its cross-sections. The strains of the rod are transformed into a deformation gradient, describing the actual warped configuration of the cross-section. Thus, a problem described by three-dimensional continuum mechanics emerges on the cross-sectional level. Further, the geometry of the crosssection undergoes an FE-discretization. Thus no restrictions to the cross-sectional shape are given. As mentioned earlier the stress resultants approach is fully described by measures from the rod theory. A transformation of strain prescriptors into three-dimensional strain measures is not required. This section first introduces the FE 2 approach. Then the continuum elastoplastic constitutive models of both approaches are presented.

FE 2 approach
The FE 2 approach is motivated by techniques well known from multiscale homogenization, in particular the FE 2 method [28]. Each material point on the macro-scale (a point on the centerline of the geometrically exact rod) is resolved by a micro-problem (the cross-sectional warping problem). The micro-problem is solved at every support point of the Gauss integration in the macro-scale and a transfer from mechanical quantities and history variables takes place. In detail, strain prescriptors, which serve as boundary conditions for the micro-problem are transferred to the microproblem. Once the warping of the cross-section is solved depending on the strain prescriptors and internal history variables, the stress resultants and the rod's tangent stiffness are obtained from integration of stress-measures over the microproblem (compare equation (33) and (35)). With these, the balance equations of the macro-scale can be solved. In other words, the micro-problem takes the role of the constitutive relation in terms of strain prescriptors and stress resultants. The framework is sketched in fig. 2. Let us note that history variables such as the plastic strain and the internal hardening variables are stored for each material point in the micro-scale. No history variables appear on the macro-scale, since plasticity is only considered on the micro-scale. In practice, at each support point of the macro-problem the deformed configuration of the micro-problem and a set of history variables is stored. Fig. 2 The FE 2 approach. The micro-scale (cross-sectional warping problem) is solved for strain prescriptors resulting from the macroscale simulation. Stress resultants and tangent stiffness are integrated over the cross-section and transferred to the macro-scale. The transfer is performed when evaluating the discretized balance equations of the macro-scale, thus at every quadrature point QP. Dirichlet boundary conditions are introduced by ϑ p and r p For the case that an elastoplastic constitutive model is used on the micro-scale, elastoplastic stress resultants and tangent stiffness are returned. Thus, it is possible to model geometrically exact elastoplastic rods without any knowledge of an elastoplastic rod formulation as presented in sect. 2. This includes of course that a yield-surface in terms of stress resultants and resultant-type hardening is not required. A further benefit of the FE 2 approach is that any inelastic constitutive model from three-dimensional continuum mechanics may be used. This implies well known finite strain J 2-plasticity [23], as well as finite strain rate-dependent and rate-independent crystal plasticity or any other conceivable model [37,38]. An FE 2 approach is also used in Klarmann et al. [36] to model inelastic rods. The difference to present approach is that the micro-problem is decribed by a segment of the rod (RVE) with finite thickness and not by its cross-section incorporating cross-sectional warping. Since warping must be considered, the approach in [36] shows dependencies of the results on the thickness of the RVE. Additional constraints may be introduced to the RVE leading to result independence towards the RVE's thickness. In the same way as the FE 2 approach presented here, it allows the use of ineasltic constitutive behavior described by classical three-dimensional continuum mechancis, and thus, the computation of inelastic deflection curves. Later in this contribution we will compare the FE 2 approach with the stress resultants approach. Thus we restrict ourselves to J 2-plasticty, since in [23] a yield-surface in terms of stress resultants for a circular rod applying J 2-plasticity has already been derived. Varying the plasticity model would require the determination of a new yield-surface.
In contrast to the stress resultants approach, the FE 2 approach captures the exact plastification of the cross-section and its hardening behavior. It is shown in [23] that, depending on the strain state, different areas of the cross-section plastify at different magnitudes of macroscopic strain. Let us think of a rod subjected to axial strain. The stress is distributed uniformly in the cross-section and the plastification is likewise homogeneously distributed in the cross-section. This is different for twist, where the stress shows a gradient in the cross-section. There, the outer region of the cross-section plastifies earlier. This effect is not captured by the stress resultants approach, where the whole cross-section is assumed to plastify simultaneously. Finally, the FE 2 approach is able to consider geometrical effects, e.g., warping and necking of the cross-section. This effect may influence the results and must be considered, when comparing results from the FE 2 and the stress resultants approach.

Elastoplastic continuum models
This section presents the employed elastoplastic models. The FE 2 approach requires an elastoplastic constitutive model in terms of three-dimensional continuum mechanics. A J 2plasticity model for finite strains as introduced in [39] and applied to the cross-sectional warping problem in [23] is used. The model is not discussed in detail here. However, for completeness, let us state the free energy density as with the strain energy density ψ (b e ) depending on the elastic left Cauchy-Green strain b e = F e F e T and the hardening potential H (ξ ) depending on the scalar internal variable ξ . The strain energy density ψ (b e ) is defined in terms of the logarithmic principle stretches of b e . For details the reader is referred to [23,39]. The hardening potential is defined as where H is a scalar hardening parameter. The yield-criterion is given by the von-Mises yield-surface where τ defines the Kirchhoff stress and q = − ∂H ∂ξ (ξ ). The material parameters for the following simulations are given by compression modulus κ = 164210 GPa, shear modulus μ = 80193 GPa and yield limit σ y = 450 GPa. The value of the hardening parameter H will be specified later. In the sequel, forces take the unit [N] and moments [Nmm]. Further, the scalar hardening parameter H is of unit [GPa].
To properly model the inelastic rod with the stress resultants approach the knowledge of a constant elastic stiffness C 0 and yield-surface in terms of stress resultants is of crucial importance. Together, they allow the computation of elastoplastic stress resultants and tangent stiffnesses. The constant elastic stiffness C 0 is obtained by solving the cross-sectional warping problem at zero strain with the above material constants (see equation (35)). For circular cross-sections the stiffness takes a diagonal shape. In this contribution the yieldsurface derived in [23] is extended by the resultant-type hardening stress β as with β = − ∂H rod ∂ξ (ξ ). Note that equation (39) is only valid for a rod with circular cross-section and radius r = 1 mm. However, as shown in [23] the yield-surface may be scaled by the cross-sectional area without considerable loss in accuracy. The structure and the parameters of the yield-surface (39) results from the fit of a generic continuous yield-surface to a discrete yield-surface obtained by solving the elastoplastic cross-sectional warping problem [23]. Thus, Φ rod implies the continuous elastoplastic model (here J 2-plasticity) in terms of stress resultants. The parameters in (39) are unique for the used continuous plasticity model and cross-sectional shape. A systematic relation between the continuous constitutive model and the plasticity in terms of stress resultants is yet not known and is not addressed in this contribution. Isotropic hardening on the micro-scale reflects in a shift of the yield-limit in each stress resultants entry, which reveals kinematic hardening in (39). The resultant type hardening potential introduced in equation (18) is postulated as where H rod is an invertible symmetric 6 × 6 tensor The symmetry of the hardening tensor results from the quadratic dependency on ξ 4 . Further, invertibility is required, since the hardening tensor is inverted within the algorithm to compute the plastic strain prescriptors and stress resultants [18]. Invertibility is ensured while performing the fit of H rod as presented in sect. 5.1. The computation of the entries of H rod remains an unsolved task, which we tackle in the next section. For a general symmetric invertible hardening tensor H rod , the resultant-type hardening stress equals In subsequent sections we omit units for the sake of simplicity.

Remark 1
Let us briefly make a note on two effects that may impact the entries of the hardening matrix H rod . First, we note that the hardening matrix H rod must be symmetric but may be non positive definite. We demonstrate this on a numerical example. Let us consider the case of a longitudinal strained rod with v 0 = [0 0 1+λ] T and k 0 = [0 0 0] T . The cross-sectional warping problem is solved with the above introduced material models (36)- (38). Hardening is set to H = 0. The course of the stress-resultant n 0 3 is plotted in fig. 3 (40) could no more be applied. For these reasons we restrict ourselves to small values in v 0 and k 0 and to hardening values far away from 0. Thus, the influence of geometrical effects in the cross-section is still there, but contrary behavior as shown in fig. 3 is reduced (see fig. 4).

Numerical investigations
In this section the mechanical response of rods using the FE 2 and the stress resultants approach is investigated. First, on the material point level, later both approaches are compared for finitely strained rods.

Determining hardening tensor H rod
The yield-surface determines the onset of yield in terms of stresses or stress resultants. The evolution of the yield-  fig. 3 but with H = 20000. Here, in both cases hardening is observed. The geometric effects in the cross-section do not reflect in contrary hardening or softening, respectively surface is described by hardening. In the following, the hardening parameter for the finite strain plasticity model (37) is set to H = 20000, H = 10000 and H = 5000, respectively. Since the yield-surface in terms of stress resultants is defined with respect to the dissipated work within the rod's cross-section [23], the hardening parameter slightly impacts the resulting onset of yield in terms of stress resultants. However, the influence is marginal and not further considered. To model elastoplastic rods with the stress resultants approach in a satisfactory way, the entries of the resultant-type hardening tensor H rod must be evaluated. Note that in this contribution H rod is assumed to be constant throughout the deformation. A requirement to H rod is that it must be invertible.
To obtain the entries of H rod we optimize the stress resultants of uni-and biaxial loading at one single material point towards the stress resultants of the cross-sectional warping problem, i.e. the micro-problem. Uniaxial strain states are obtained when every entry of the strain prescriptors = [v 0 k 0 ] T equals zero except entry i 5 . In biaxial states, the entries i and j are non equal to zero. Here i, j = 1, ..., 6. The nonzero value is set to a constant value λ c with the load parameter λ taking values from 0 to 1 in 100 equidistant steps of width Δλ = 0.01 and c being a constant terminating value. The design variables of the optimization are the entries H rod i j . Since H rod is symmetric, H rod i j = H rod ji . The following minimization problem is formulated 5 In order to obtain a zero vector at zero strains we redefine v 0 → and σ = [n 0 m 0 ] T . The indices • SR and • CS indicate the approach to obtain the stress resultants. It distinguishes between the stress resultants approach (• SR ) and the microor cross-sectional warping problem (• CS ) used in the FE 2 approach. This notation is used further on. In the objective function, the difference in slope at every equidistant load step is normalized and squared. The evaluation of H rod i j takes place with respect to the slope, since hardening influences the elastoplastic tangent stiffness, thus the slope of the stress resultantsstrain prescriptors curves. The minimization procedure is repeated until every entry of H rod is computed. In this contribution, first the diagonal entries of H rod are evaluated (i = j), while fixing the remaining entries. Then, using the diagonal entries, the off-diagonal entries are computed (i = j). The detailed procedure is presented in Algorithm 1. The minimization problems presented here are solved by an appropriate optimizer, such as the particle swarm optimizer implemented in the commercial software Matlab. Within the optimization it is ensured that H rod is invertible. The inverse of H rod is needed in the return algorithm to solve the evolution equations (23). If H rod is not invertible the computation of σ SR fails and a new guess for H rod is taken within the particle swarm optimizer. It has been shown in remark 1 that H rod may be non positive definite. If however positive definiteness is a requirement, instead of fitting the entries of H rod , the entries of its Cholesky decomposition may be fitted. This approach is not pursued in this contribution. The One may observe that the tensor has not only considerable entries on its diagonal but also on its off-diagonals. The fit of σ SR H rod , and σ CS ( ) is presented in appendix 1. There, fig. 19 displays the results for H = 20000, fig. 20 for H = 10000 and fig. 21 for H = 5000. The stress resultants emerging from the micro-problem are displayed in solid lines, whereas the stress resultants resulting from the stress resultants approach are dashed. We refer to the solution from the micro-or cross-sectional warping problem as the reference solution. The figure reads as an i × j matrix. The subplot at position i, j results from a strain state where i = j = λ c and the remaining terms equal zero. The stress resultants approach fits the reference solution in a rather satisfactory way. The following observations are valid for all three considered values of H . Slight differences between the stress resultants approach and the reference solution are visible at the onset of yield. This however is not astonishing. Onset of yield is sudden in the stress resultants approach, whereas the cross-sectional warping problem enables a gradual plastification of the rod's cross-section. This effect is already known and discussed in [23]. Further, notable discrepancies are obvious for combined longitudinal strain and bending. There, the stress resultants approach overestimates the resulting bending moment. This effect increases with decreasing value of H . When comparing the stress resultants approach with the reference solution the following aspect must be considered and may not be neglected. The stress resultants approach shows a constant relation between stress resultants and strain prescriptors in the elastic region (see equation 16). The crosssectional shape does not vary in both the elastic and inelastic region. Nonlinear geometric effects on the cross-sectional level, such as warping and necking, are not captured. But these effects influence the stress resultants. As an example consider the case of shear, where 1 = 2 = λ c. Beside resulting shear forces, geometric effects on the cross-section induce a coupling term between shear and tensile forces in the tangent stiffness C 0 , such that a tensile force appears. The direct approach is not able to capture this effect, if coupling is nonlinear (compare fig. 21, row 1, column 1).

Dependence of hardening tensor H rod on the scalar hardening parameter H
In sect. 5.3 hardening matrices for three different values of the scalar hardening parameter H were presented. In appendix 2, fig. 22 where f (H ) is a linear function relating H rod i j to H . Slight deviations from the linear relation are observed for small values of H . Compared to the elastic material parameters, these are already so small that they refer to ideal plasticity. This section is just an observation. Reasons for the linear dependency are not further discussed. Further one may not expect a linear relation for any arbitrary rod's cross-section.

Elastoplastic response of rods at material points
For an arbitrarily strained material point (one cross-section of the rod), the stress resultants are compared, once obtained by the cross-sectional warping problem used in the FE 2 approach and once by the stress resultants approach. Besides the uni-and biaxial strain cases, this gives an additional possibility to check if on the material point level the hardening tensor replicates the solution from the cross-sectional warping problem sufficiently good. Again, the cross-sectional warping problem serves as reference solution. The imposed strain prescriptors are The strain prescriptors are chosen such that all stress resultants are nonzero. Figure 5 compares the stress resultants from the reference solution (solid lines) and stress resultants approach (dashed lines). In the cross-sectional warping problem the scalar hardening parameter is set to H = 20000. The hardening tensor used in the stress resultants approach is H rod = H rod 20000 , which is fitted to the micro-problem. The course of the stress resultants reveals a small elastic region. The strain state leads almost to immediate plastification which is indicated by a kink in the curves. After plastification a more or less linear relation between λ and the stress resultants is discernible. Concentrating on the forces, the agreement between the stress resultants approach and the reference solution is remarkable. A slightly different picture is depicted for the moments. In general the stress resultants approach fits the reference solution. But one can detect an offset in the final values. Further, when plastifying, the moments of the reference solution decrease suddenly. The stress resultants approach is not able to capture this effect in detail. To demonstrate that the fitted hardening tensor is a valid approximation to the reference solution, the stress resultants obtained by the stress resultants approach with a naively introduced diagonal hardening tensor H rod n are displayed by dotted lines. The hardening tensor H rod n is motivation by the fact that H = 20000 is approximately 1/10 of the Young's modulus E and 1/4 of the shear modulus μ used in the crosssectional warping problem 6 . In the elastic stiffness C 0 of the rod (which is diagonal for circular cross-sections) the Young's modulus and the shear modulus appear in combination with the areas and moments of areas [17]. We apply the relation mentioned above to obtain a diagonal hardening tensor with This naively introduced hardening tensor leads to nonsatisfactory results (dotted lines). It strongly underestimates the stress resultants. Figure 6 is structured the same way as fig. 5, except that here the hardening parameter used in the cross-sectional warping problem is set to H = 10000. The hardening tensor used in the stress resultants approach is H rod = H rod 10000 . Now, the naively defined diagonal hardening tensor H rod n is calculated as which considers the relation of H to E and μ. The results show similar behavior as discussed above. Focusing on the forces, the stress resultants approach (dashed lines) fits the reference solution (solid lines) in a satisfactory way. A discrepancy in the values is observed for the moments. But still, the stress resultants approach using H rod 10000 shows a considerable improvement to the solution obtained with H rod n (dotted lines), which are not satisfying.
Finally, fig. 7 shows the stress resultants for H = 5000, H rod = H rod 5000 and Fig. 6 Stress resultants obtained by the cross-sectional warping problem (reference solution, solid lines) and the stress resultants approach (dashed lines) of a strained material point. Additionally, results of the stress resultants approach with a diagonal hardening tensor are displayed (dotted lines). Despite an offset, the stress resultants approach using the fitted hardening tensor matches the reference solution better than the direct approach with diagonal hardening poorly fits the reference. The scalar hardening variable is set to H = 10000 respectively. Compared to fig. 5 the agreement between the stress resultants approach obtained by using the fitted hard- Fig. 7 Stress resultants obtained by the cross-sectional warping problem (reference solution, solid lines) and the stress resultants approach (dashed lines) of a strained material point. Additionally, results of the stress resultants approach with a diagonal hardening tensor are displayed (dotted lines). Compared to fig. 6 and 5 the fit is less exact. But still the fitted hardening tensor matches the reference solution better than the naively derived diagonal hardening tensor. The scalar hardening variable is set to H = 5000 ening and the reference solution is less accurate. Especially, the course of the moments shows deviations. The second and third component of the moments show different levels after plastification. But still, the naively introduced hardening tensor shows much poorer results Concluding, one may state that the stress resultants approach using the fitted hardening tensor does not match the reference solution used in the FE 2 approach perfectly, but by ways better than commonly used naive definition of resultant-type hardening.

Boundary value problems involving non-uniformly strained elastoplastic rods
Having shown on the material point level that the mechanical response of the stress resultants approach follows the mechanical response of the cross-sectional warping problem in a satisfactory manner, the next step is to compare the results for strained elastoplastic rods. In the following, we investigate the reaction forces n and moments m of an elastoplastic rod subjected to prescribed displacements and rotations. Here, the mechanical response is obtained by using once the FE 2 approach and once the stress resultants approach. The rod of circular cross-section with r = 1 has length l = 100.

Bent rod
In a first example, the boundary conditions are given as follows r (s = 0) = 0, r 1 (s = l) = 0, r 2 (s = l) = λ 40, where 0 ≤ λ ≤ 1 denotes the load factor. The constant a takes the values 0, 0.5 and 1 and prescribes the axial twist of the rod. The resulting strain state of the rod is not uniform along s. A s-bent rod will form. Let us in advance consider the spatial configuration resulting from the prescribed boundary conditions for λ = 1 in fig. 8. Here, the results are obtained by applying the FE 2 approach for a rod discretized with 32 elements. The twist angle at s = 0 is set to ϑ 3 = π/2 and the hardening parameter on the micro-scale is set H = 20000. In addition to the obvious deformation, the twist is visualized by displaying the directors d 1 and d 2 in blue and red, respectively. Further, twist induces an out-of-plane deflection as visible in the right figure. The actual rod is colored in gray for the sake of better visualization.
Implicit to the above decision to discretize the rod into 32 elements is a study of the convergence behavior of the FE 2 approach with respect to the number of elements within the rod. Therein, the rod is subdivided into 8, 16 and 32 elements. The stress resultants at s = l are plotted as a function of λ in fig. 9. The constant a is set to a = 1 and the scalar hardening parameter H = 20000.
The first, second and third component of n and m with respect to the director basis are characterized by blue, red and green color. One can nicely observe that all stress resultants except for n 3 take values which are nonzero. The reason therefore is that all degrees of freedom on the boundary are constrained except r 3 . A distinct kink in the course of n 2 and m 1 is visible. The kink visualizes the onset of plastic yield. Further, it is obvious that increasing the number of elements does not affect the results in a major way. The results for 8 (dash-dotted lines), 16 (dashed lines) and 32 (dotted lines) elements coincide. Figure 10 and 11 show the same setting as fig. 9. However there, the scalar hardening variable is set to H = 10000 and H = 5000, respectively. Again, the results do not show a significant sensitivity towards the discretization. Compared with the results displayed in fig. 9 lower  In the sequel, to solve the elastoplastic rods with the stress resultants or the FE 2 approach, 32 elements will be used. Figure 12 compares the reaction stress resultants of the strained rod when simulating with the FE 2 approach (solid lines), the stress resultants approach using the fitted hardening tensor (dashed lines) and the stress resultants approach using the naively derived diagonal hardening tensor (dotted lines). In analogy to the above examples, the FE 2 approach serves as reference solution. The first row shows the reaction Fig. 12 Comparison of reaction stress resultants coming from the FE 2 (reference solution, solid lines) and the stress resultants approach using the fitted hardening tensor (dashed lines) and the naively derived diagonal hardening tensor (dotted lines) for H = 20000. The stress resultants approach is in good agreement with the reference solution. The onset of plastification is visible by a kink in the curves stress resultants for the case of no twist (a = 0 in equation (52)). In the second and third row the twist parameter is set to a = 0.5 and a = 1, respectively. For the case of no twist the reaction stress resultants of the stress resultants approach coincide with the FE 2 approach. A slight deviation in the beginning of yield is discernible. This is due to the fact that the FE 2 approach is able to capture gradual plastification of the cross-section, whereas the stress resultants approach models instantaneous plastification. Both cases with included twist show good agreement between the FE 2 approach and the stress resultants approach using the fitted hardening tensor. There, slight deviations are visible in n 1 and m 2 . However, they are considered to be acceptable. Figure 13 shows the same setting as fig. 12 but for H = 10000. Again the stress resultants from the stress resultants approach fit the stress resultants from the FE 2 approach, serving as reference solution. Especially the fit for the untwisted case (a = 0) is of very good accuracy. Fig. 13 Comparison of reaction stress resultants coming from the FE 2 (reference solution, solid lines) and the stress resultants approach using the fitted hardening tensor (dashed lines) and the naively derived diagonal hardening tensor (dotted lines) for H = 10000. In analogy to fig. 12 the stress resultants approach is in good agreement with the reference solution. The onset of plastification is visible by a kink in the curves Eventually, fig. 14 shows the reaction stress resultants for H = 5000. The stress resultants from the stress resultants approach match the stress resultants from the FE 2 approach. In contrast to the previous figures note the reduced stress resultants after plastification. In fig. 12, 13 and 14 the naively derived diagonal hardening tensor yields results which are not in good agreement with the reference solution. Especially for high values of H the deviation of n 2 and m 1 is not negligible. The other stress resultants match the reference solution better.
Concluding, let us note that the spatial configuration resulting from the stress resultants and the FE 2 approach do not differ in a significant manner and a not further discussed.

Curled rod
A second example of a finitely strained rod is given by boundary conditions Fig. 14 Comparison of reaction stress resultants coming from the FE 2 (reference solution, solid lines) and the stress resultants approach using the fitted hardening tensor (dashed lines) and the naively derived diagonal hardening tensor (dotted lines) for H = 5000. In analogy to fig. 12 the stress resultants approach is in good agreement with the reference solution. The onset of plastification is visible by a kink in the curves r (s = 0) = 0, r 2 (s = l) = λ 20, The spatial configuration of the strained rod for λ = 1 is visualized in fig. 15. There, the FE 2 approach is used. The rod is subdivided into 32 elements and the hardening parameter is set to H = 20000. The boundary conditions lead to a curled rod.
Again, the reaction stress resultants at s = l are plotted over the load factor λ. The results are compared when applying the FE 2 approach (solid lines), the stress resultants approach using the fitted hardening tensor (dashed lines) and the stress resultants approach using a naively derived hardening tensor (dotted lines). In fig. 16 the hardening parameter H is set H = 20000. Focusing on the forces, we distinct a difference in the values for λ → 1 when applying differ-   16 Reaction stress resultants coming from the FE 2 approach (solid lines) and stress resultants approach using the fitted hardening tensor (dashed lines) and the naively derived hardening tensor (dotted lines) for H = 20000. Discrepancies are visible in n 2 . Further, in m 1 and m 2 oscillations appears due to the curl of the rod ent approaches. Especially the discrepancy between the FE 2 approach and the stress resultants approach using the naively derived hardening tensor is remarkable. It behaves slightly better when considering the moments. There, the curling of the rod induces an oscillation in the moments. Figure 17 shows the same setting but for H = 10000. Again discrepancies in the values are visible for λ → 1. The stress resultants approach with the naively derived hardening tensor shows poor results.
Eventually, fig. 18 shows the reaction stress resultants for H = 5000. For all three values of hardening considered in figs. 16-18 the stress resultants approach does not fit the FE 2 approach considered as the reference solution in a satisfactory way.
A reason for the mismatch between the different approaches is most probably the following. The stress resultants approach is not able to capture gradual plastification of the cross-section. A possible gradual loading and unloading of the cross-section can not be captured. Clearly, this is a limitation of the stress resultants approach that may not be neglected. The differences in the reaction stress resultants also lead to subtle differences in the spatial configurations. However, these are not further treated here. Fig. 17 Reaction stress resultants coming from the FE 2 approach (solid lines) and stress resultants approach using the fitted hardening tensor (dashed lines) and the naively derived hardening tensor (dotted lines) for H = 10000 Fig. 18 Reaction stress resultants coming from the FE 2 approach (solid lines) and stress resultants approach using the fitted hardening tensor (dashed lines) and the naively derived hardening tensor (dotted lines) for H = 5000 Remark 2 A detailed runtime comparison between the FE 2 and the stress resultants approach is omitted in this contribution for obvious reasons. The FE 2 approach requires solving a micro-problem, the cross-sectional warping problem, at every integration point of the macro-problem, the one-dimensional rod. This leads to large computational costs which are not present in the stress resultants approach. Thus, the stress resultants approach outperforms the FE 2 approach in terms of runtime by magnitudes. Let us just state that the rod subjected to the boundary condition (52) solved with 32 elements took ∼ 1000 s in the FE 2 approach compared to ∼ 2.2 s in the stress resultants approach. Note that here the micro-problems are solved in parallel on six cores on a conventional workstation. The obvious difference in runtime makes a detailed comparison superfluous.

Summary
This contribution compares two different possibilities to computationally model geometrically exact elastoplastic rods. Both apply a two-scale approach, where the macroscale is described by the one-dimensional rod and the microscale by the rod's cross-section. Both approaches distinguish as follows. In the stress resultants approach the elastoplastic behavior is described in terms of rod specific quantities. An appropriate yield-surface and the hardening behavior are obtained by offline computations of a model enabling cross-sectional warping. A second approach is motivated by multiscale homogenization. The FE 2 approach combines the geometrically exact rod theory on the macro-scale with the cross-sectional warping problem on the micro-scale. The transfer of rod specific quantities between the scales takes place while solving the rod problem, thus online. Using an elastoplastic constitutive continuum model on the microscale opens the opportunity to model elastoplastic rods. In this contribution the FE 2 approach serves as reference solution. To replicate the mechanical response on the material point level, the knowledge of the yet unknown hardening tensor for the stress resultants approach is necessary. A hardening tensor is evaluated by fitting its entries in such way that the mechanical response of the stress resultants approach fits the mechanical response of the FE 2 approach on the material point level. The invertible hardening tensor shows not only entries on its diagonal but also on the off-diagonals. Compared with a naively computed diagonal hardening tensor, the fitted tensor shows improved results on the material point level for arbitrary strains. Finally the mechanical response of two differently strained elastoplastic rod is compared using both introduced approaches. Therein, rods undergoing nonuniform strain states are considered. In the first example the results of both approaches are in good agreement. The second example shows discrepancies which are mainly explained by the missing ability to show gradual plastification of the cross-section, when using the stress resultants approach. Considering that the computational cost of the FE 2 approach is by magnitudes larger than the cost of the stress resultants approach, the use of the stress resultants approach may be a good compromise.
In further steps the authors aim to apply both the online and the offline approach to rods with arbitrary cross-sections and constitutive behavior. Especially inelasticity modeled in terms of crystal plasticity is of big interest when considering additively manufactured rod lattices. Eventually it is to be examined whether a parametrization of the hardening tensor and yield-surface in terms of stress resultants with respect to the cross-sectional geometry and constitutive model is feasible.
Acknowledgements The authors acknowledge support from the German Science Foundation (DFG) within the Collaborative Research Center 814 "Additive Manufacturing" and from the Indo-German DST-DAAD PPP exchange program within the project "Micro-resolved finite element modeling and simulation of nonwovens" (grant number 57519760).
Funding Open Access funding enabled and organized by Projekt DEAL.
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