nach oben

Computational Mechanics

Erschienen in:

Open Access 02.07.2022 | Original Paper

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

verfasst von: Ludwig Herrnböck, Ajeet Kumar, Paul Steinmann

Erschienen in: Computational Mechanics | Ausgabe 1/2023

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Patentsuche

Aus

Abstract

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 $FE ^2$ 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.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

1 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 meta-materials 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 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 [7‐9]. 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 [11‐13]. 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 cross-sectional 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 [19‐22]. 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 cross-section 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].

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

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 three-dimensional 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 yield-surface and the resultant-type hardening behavior in terms of stress resultants. The determination of both the yield-surface 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 resultant-type 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 [27‐29]. 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 micro-scale 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 cross-sections. 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-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 cross-sections. However, the framework presented here is general and could also be applied for rods with non-circular cross-sections.

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].

2 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].

2.1 Kinematics

Figure 1 shows a rod deforming from its undeformed straight material configuration $\mathcal {B}_0$ into its deformed spatial configuration $\mathcal {B}_t$. In this contribution, the rod is assumed to be straight in its undeformed configuration, with coordinates, $\mathbf {X}=X _i\mathbf {E}_{i}$¹. The undeformed cross-section is denoted by $\mathbf {X}_{CS }=X _{\alpha }\mathbf {E}_{\alpha }$. The deformed centerline is defined by $\mathbf {r}{(s)}$, with $s:=X _3$ the rod’s undeformed arc-length. Orthogonal directors $\mathbf {d}_{\alpha }{(s)}$ span the deformed cross-section $\varOmega _t$ which is assumed planar. Together with $\mathbf {d}_{3}{(s)}=\mathbf {d}_{1}{(s)}\times \mathbf {d}_{2}{(s)}$ an orthogonal basis is constructed. Note that $\mathbf {d}_{3}{(s)}$ need not be tangential to $\mathbf {r}{(s)}$. The orientation of the deformed cross-section is described by $\mathbf {d}_{i}{}$ and the orientation of the undeformed cross-section is described by $\mathbf {D}_{i}{}$. Here, $\mathbf {D}_{i}{}=\mathbf {E}_{i}$. Usually, the principal directions of the undeformed cross-section are aligned with $\mathbf {D}_{\alpha }{}$.

The directors in the spatial configuration are related to the directors in the material configuration via

$$\begin{aligned} \mathbf {d}_{i}{(s)}=\mathbf {R}{(s)}\mathbf {D}_{i}, \end{aligned}$$

(1)

with $\mathbf {R}\in SO (3)$, the special orthogonal group. The quantities $\mathbf {r}$(s) and $\mathbf {R}$(s) are the kinematic unknowns of this theory. Note that the rotation described by the tensor $\mathbf {R}$(s) can also be parametrised by a vector $\varvec{\vartheta }(s)$, where $\Vert \varvec{\vartheta }(s)\Vert $ equals the angle of the rotation, and $\displaystyle {\varvec{\vartheta }(s)/\Vert \varvec{\vartheta }(s)\Vert }$ describes the direction of rotation [12, 18].

Assuming a rigid cross-section, the position $\mathbf {x}(s)$ of any material point in the spatial configuration is described by

$$\begin{aligned} \mathbf {x}(s,\mathbf {X}_{CS })=\mathbf {r}{(s)}+\mathbf {R}{(s)}\mathbf {X}_{CS }. \end{aligned}$$

(2)

For the sake of readability, the dependencies on s and $\mathbf {X}_{CS }$ are omitted from now on.

Strain prescriptors are introduced as the translational strain $\mathbf {v}_{}^{}{}$ and the rotational strain $\mathbf {k}_{}^{}{}$ (also called curvature and twist):

$$\begin{aligned} \begin{aligned}&\mathbf {v}_{}^{}{}=\mathbf {r}{}'=v _i\mathbf {d}_{i}{},\\&{\mathbf {K}}=\mathbf {R}{}'\mathbf {R}{}^{T},\quad \mathbf {k}_{}^{}{}=\text {ax}\left( \mathbf {K}\right) =k _i\mathbf {d}_{i}{}, \end{aligned} \end{aligned}$$

(3)

where the derivative with respect to the arc-length is denoted as

$$\begin{aligned} \bullet ':=\frac{\partial \bullet }{\partial s}. \end{aligned}$$

(4)

The rotational pull backs $\mathbf {v}_{0}^{}{}$ and $\mathbf {k}_{0}^{}{}$ of $\mathbf {v}_{}^{}{}$ and $\mathbf {k}_{}^{}{}$, respectively, are defined as:

$$\begin{aligned} \begin{aligned}&\mathbf {v}_{0}^{}{}=\mathbf {R}{}^{T}\mathbf {r}{}'=v _i\mathbf {D}_{i},\\&{\mathbf {K}}_0=\mathbf {R}{}^{T}{\mathbf {K}}\mathbf {R}{},\quad \mathbf {k}_{0}^{}{}=\text {ax}\left( \mathbf {K}_0\right) =\mathbf {R}{}^{T}\mathbf {k}_{}^{}{}=k _i\mathbf {D}_{i}{}. \end{aligned} \end{aligned}$$

(5)

Here ${\mathbf {K}}$ and ${\mathbf {K}}_0$ are skew-symmetric tensors whose axial vectors are $\mathbf {k}_{}^{}{}$ and $\mathbf {k}_{0}^{}{}$ respectively². The deformation gradient $\mathbf {F}$ of the deformation map (2) is defined as

$$\begin{aligned} \mathbf {F}=\frac{\partial \mathbf {x}}{\partial \mathbf {X}}. \end{aligned}$$

(8)

Inserting equation (2) into equation (7) yields

$$\begin{aligned} \begin{aligned} \mathbf {F}=&\frac{\partial \mathbf {r}}{\partial \mathbf {X}}+\frac{\partial \mathbf {R}}{\partial \mathbf {X}}\mathbf {X}_{CS }+ \mathbf {R}\frac{\partial \mathbf {X}_{CS }}{\partial \mathbf {X}}. \end{aligned} \end{aligned}$$

(9)

Recalling the dependencies $\mathbf {r}=\mathbf {r}(s)$, $\mathbf {R}=\mathbf {R}(s)$ and $\mathbf {X}=\mathbf {X}(s,\mathbf {X}_{CS })=s\,\mathbf {E}_3+\mathbf {X}_{CS }=s\,\mathbf {E}_3+X _{\alpha }\mathbf {E}_{\alpha }$ respectively, equation (8) simplifies

$$\begin{aligned} \begin{aligned} \mathbf {F}=&\mathbf {r}'\otimes \mathbf {E}_{3}+\left[ \mathbf {R}'\mathbf {X}_{CS }\right] \otimes \mathbf {E}_{3}+\mathbf {R}\left[ \mathbf {E}_{\alpha }\otimes \mathbf {E}_{\alpha }\right] . \end{aligned} \end{aligned}$$

(10)

By factoring out $\mathbf {R}$ in equation (9) the deformation gradient can be written in terms of strain prescriptors as follows:

$$\begin{aligned} \mathbf {F}=\mathbf {R}\bigg [\mathbf {v}_{0}^{}{}\otimes \mathbf {E}_{3}+\left[ \mathbf {k}_{0}^{}{}\times \mathbf {X}_{CS }\right] \otimes \mathbf {E}_{3}+\mathbf {E}_{\alpha }\otimes \mathbf {E}_{\alpha }\bigg ]. \end{aligned}$$

(11)

2.2 Stress resultants and balance equations

The internal (contact) force $\mathbf {n}_{}{}$ and internal (contact) moment $\mathbf {m}_{}{}$, collectively called stress resultants, are defined by:

$$\begin{aligned} \mathbf {n}_{}{}=n _i\mathbf {d}_{i}{},\quad \mathbf {m}_{}{}=m _i\mathbf {d}_{i}{}. \end{aligned}$$

(12)

Their rotational pull backs are

$$\begin{aligned} \begin{aligned} \mathbf {n}_{0}{}=n _i\mathbf {D}_{i},\quad \mathbf {m}_{0}{}=m _i\mathbf {D}_{i}. \end{aligned} \end{aligned}$$

(6)

For the elastic case and assuming the existence of a scalar-valued strain energy density function $\psi ^{rod }(\mathbf {v}_{0}^{}{}, \mathbf {k}_{0}^{}{})$ the stress resultants follow as

$$\begin{aligned} \begin{aligned} \mathbf {n}_{0}{}=\frac{\partial \psi ^{rod }}{\partial \mathbf {v}_{0}^{}{}}(\mathbf {v}_{0}^{}{}, \mathbf {k}_{0}^{}{}),\quad \mathbf {m}_{0}{}=\frac{\partial \psi ^{rod }}{\partial \mathbf {k}_{0}^{}{}}(\mathbf {v}_{0}^{}{}, \mathbf {k}_{0}^{}{}). \end{aligned} \end{aligned}$$

(13)

The second derivative of $\psi ^{rod }(\mathbf {v}_{0}^{}{},\mathbf {k}_{0}^{}{})$ with respect to the strain prescriptors yields the $6\times 6$ tangent stiffness $\mathbf {C}_0$

$$\begin{aligned} \begin{aligned} \mathbf {C}_0&= \begin{bmatrix} \mathbf {C}_{0_{vv }} &{} \mathbf {C}_{0_{vk }}\\ \mathbf {C}_{0_{kv }} &{} \mathbf {C}_{0_{kk }} \end{bmatrix}\\&= \begin{bmatrix} \displaystyle { \frac{\partial ^2\psi ^{rod }}{\partial \mathbf {v}_{0}^{}{}\partial \mathbf {v}_{0}^{}{}}\left( \mathbf {v}_{0}^{}{},\mathbf {k}_{0}^{}{}\right) } &{} \displaystyle {\frac{\partial ^2\psi ^{rod }}{\partial \mathbf {v}_{0}^{}{}\partial \mathbf {k}_{0}^{}{}}\left( \mathbf {v}_{0}^{}{},\mathbf {k}_{0}^{}{}\right) } \\ \displaystyle {\frac{\partial ^2\psi ^{rod }}{\partial \mathbf {k}_{0}^{}{}\partial \mathbf {v}_{0}^{}{}}\left( \mathbf {v}_{0}^{}{},\mathbf {k}_{0}^{}{}\right) } &{} \displaystyle {\frac{\partial ^2\psi ^{rod }}{\partial \mathbf {k}_{0}^{}{}\partial \mathbf {k}_{0}^{}{}}\left( \mathbf {v}_{0}^{}{},\mathbf {k}_{0}^{}{}\right) } \end{bmatrix}, \end{aligned} \end{aligned}$$

(14)

which relates the change in the stress resultants to the change in the strain prescriptors as

$$\begin{aligned} \begin{bmatrix} d {{\textbf {n}}}_0\\ d {{\textbf {m}}}_0 \end{bmatrix} =\mathbf {C}_0 \begin{bmatrix} d {{\textbf {v}}}_0 \\ d {{\textbf {k}}}_0 \end{bmatrix}. \end{aligned}$$

(15)

Assuming $\psi ^{rod }$ to be quadratic in the strain prescriptors, a linear relation between stress resultants and strain prescriptors results

$$\begin{aligned} \begin{bmatrix} {{\textbf {n}}}_0\\ {{\textbf {m}}}_0 \end{bmatrix} =\mathbf {C}_0 \begin{bmatrix} {{\textbf {v}}}_0-\mathbf {E}_3 \\ {{\textbf {k}}}_0 \end{bmatrix}. \end{aligned}$$

(16)

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:

https://static-content.springer.com/image/art%3A10.1007%2Fs00466-022-02204-8/MediaObjects/466_2022_2204_Equ17_HTML.png

(17)

A free energy density function ${\varPsi }^{rod } $ is introduced as

$$\begin{aligned} {\varPsi }^{rod } (\mathbf {v}_{0}^{e}{},\mathbf {k}_{0}^{e}{}, \varvec{\xi })=\psi ^{rod }(\mathbf {v}_{0}^{e}{},\mathbf {k}_{0}^{e}{}) +\mathcal {H}^{rod }(\varvec{\xi }), \end{aligned}$$

(18)

where $\psi ^{rod }(\mathbf {v}_{0}^{e}{},\mathbf {k}_{0}^{e}{})$ denotes the strain energy density and $\mathcal {H}^{rod }(\varvec{\xi })$ describes the resultant-type hardening depending on the internal variables $\varvec{\xi }$. The dissipation power $\mathcal {D}$ is the difference between the stress resultants power and the material time derivative of the free energy density and must be positive:

$$\begin{aligned} \mathcal {D} =\mathbf {n}_0\cdot \dot{\mathbf {v}}_0+\mathbf {m}_0\cdot \dot{\mathbf {k}}_0-\frac{d {\varPsi }^{rod }}{d t}(\mathbf {v}_{0}^{e}{},\mathbf {k}_{0}^{e}{},\varvec{\xi })\ge 0, \end{aligned}$$

(19)

where $\displaystyle {\dot{\bullet }:=\left. \frac{d \bullet }{d t}\right| _{\mathbf {X}}}$. 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:

$$\begin{aligned} \mathbf {n}_0=\frac{\partial \psi ^{rod }}{\partial \mathbf {v}_{0}^{e}{}}(\mathbf {v}_{0}^{e}{},\mathbf {k}_{0}^{e}{}),\quad \mathbf {m}_0=\frac{\partial \psi ^{rod }}{\partial \mathbf {k}_{0}^{e}{}}(\mathbf {v}_{0}^{e}{},\mathbf {k}_{0}^{e}{}). \end{aligned}$$

(20)

Furthermore, the resultant-type hardening stress $\varvec{\beta }$ is defined as

$$\begin{aligned} \varvec{\beta }=-\frac{\partial \mathcal {H}^{rod }}{\partial \varvec{\xi }}(\varvec{\xi }). \end{aligned}$$

(21)

Inserting equations (20) and (21) into the dissipation inequality yields its reduced form

$$\begin{aligned} \mathcal {D}^{red}=\mathbf {n}_0\cdot \dot{\mathbf {v}}_0^p+\mathbf {m}_0\cdot \dot{\mathbf {k}}_0^p+\varvec{\beta }\cdot \dot{\varvec{\xi }}\ge 0. \end{aligned}$$

(22)

Expression (22) is maximized under the constraint $\displaystyle {\varPhi }(\mathbf {n}_0,\mathbf {m}_0,\varvec{\beta })\le 0$. Here $\varPhi $ describes a yield-function in terms of stress resultants. Finally, the evolution equations for the plastic quantities are obtained:

$$\begin{aligned} \begin{aligned}&\dot{\mathbf {v}}_0^p=\dot{\gamma }\frac{\partial \varPhi }{\partial \mathbf {n}_0}(\mathbf {n}_0,\mathbf {m}_0,\varvec{\beta }), \quad \dot{\mathbf {k}}_0^p=\dot{\gamma }\frac{\partial \varPhi }{\partial \mathbf {m}_0}(\mathbf {n}_0,\mathbf {m}_0,\varvec{\beta }), \quad \\&\dot{\varvec{\xi }}=\dot{\gamma }\frac{\partial \varPhi }{\partial \varvec{\beta }}(\mathbf {n}_0,\mathbf {m}_0,\varvec{\beta }). \end{aligned} \end{aligned}$$

(23)

Together with Karush-Kuhn-Tucker (KKT) conditions, the Lagrange multiplier $\dot{\gamma }$ ensures admissibility of the resulting stress resultants

$$\begin{aligned} \dot{\gamma }\ge 0,\quad \dot{\gamma }\,\varPhi (\mathbf {n}_0,\mathbf {m}_0,\varvec{\beta })=0,\quad \varPhi (\mathbf {n}_0,\mathbf {m}_0,\varvec{\beta })\le 0. \end{aligned}$$

(24)

The changes in stress resultants are related to the changes in strain prescriptors via the $6\times 6$ elastoplastic tangent stiffness $\mathbf {C}_0^{ep }$

$$\begin{aligned} \begin{bmatrix} d {{\textbf {n}}}_0\\ d {{\textbf {m}}}_0 \end{bmatrix} =\mathbf {C}_0^{ep } \begin{bmatrix} d {{\textbf {v}}}_0 \\ d {{\textbf {k}}}_0 \end{bmatrix}. \end{aligned}$$

(25)

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

$$\begin{aligned} \begin{aligned}&\mathbf {n}'=\mathbf {0},\\&\mathbf {m}'+\mathbf {r}'\times \mathbf {n}=\mathbf {0}. \end{aligned} \end{aligned}$$

(26)

The kinematic unknowns ($\mathbf {r}$, $\varvec{\vartheta }$) 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].

3 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 cross-section 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 $\varGamma $-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 $\mathbf {v}_{0}^{}{}(s)=\mathbf {v}_{0}^{}{}$ and $\mathbf {k}_{0}^{}{}(s)=\mathbf {k}_{0}^{}{}$ the rod takes the shape of an helix [25]. Let us rewrite the deformation map as

$$\begin{aligned} \mathbf {x}(s;\mathbf {X}_{CS })=\mathbf {r}{(s)}+\mathbf {R}{(s)}\hat{\mathbf {X}}_{CS }\left( \mathbf {X}_{CS }\right) , \end{aligned}$$

(27)

where $\hat{\mathbf {X}}_{CS }\left( \mathbf {X}_{CS }\right) =\mathbf {X}_{CS }+\mathbf {U}\left( \mathbf {X}_{CS }\right) $. Here $U _{\alpha }\left( \mathbf {X}_{CS }\right) $ denotes in-plane, whereas $U _{3}\left( \mathbf {X}_{CS }\right) $ denotes out-of-plane cross-sectional displacement [26]. Consequently, the deformation gradient based on equation (27) is

$$\begin{aligned} \mathbf {F}=\mathbf {R}\bigg [\mathbf {v}_{0}^{}{}\otimes \mathbf {E}_{3}+\left[ \mathbf {k}_{0}^{}{}\times \hat{\mathbf {X}}_{CS }\right] \otimes \mathbf {E}_{3}+\nabla _{\alpha }\hat{\mathbf {X}}_{CS }\bigg ]. \end{aligned}$$

(28)

Equation (28) reveals that the deformation in the cross-section is merely dependent on the strain prescriptors $\mathbf {v}_0$ and $\mathbf {k}_0$ and the warped cross-section $\hat{\mathbf {X}}_{CS }$.

The warped cross-section may be evaluated for the purely elastic case by minimizing the stored energy density $\psi \left( \mathbf {F}\right) $ in the rod’s cross-section

$$\begin{aligned} \underset{\hat{\mathbf {X}}_{CS }}{\min }\int _{\varOmega _0}\psi \left( \mathbf {F}\right) ~d A \end{aligned}$$

(29)

under appropriate constraints [23, 25, 26]. These are given by

$$\begin{aligned} \int _{\varOmega _0}\,\hat{\mathbf {X}}_{CS }~d A=\mathbf {0}, and~ \int _{\varOmega _0}\,\varvec{\mathfrak {M}}~d A=\mathbf {0}, \end{aligned}$$

(30)

where

$$\begin{aligned} \begin{aligned} \varvec{\mathfrak {M}}&=\,\hat{X }_{CS _2}\hat{X }_{CS _3}\mathbf {E}_1+\hat{X }_{CS _1}\hat{X }_{CS _3}\mathbf {E}_2\\&\quad +\left[ atan \left( \frac{\hat{X }_{CS _1}}{\hat{X }_{CS _2}}\right) -atan \left( \frac{X _{CS _1}}{X _{CS _2}}\right) \right] \mathbf {E}_3. \end{aligned} \end{aligned}$$

(31)

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 $\mathbf {E}_3$ vanishes and that the principal axes of the cross-section align with $\mathbf {E}_1$ and $\mathbf {E}_2$.

To solve the constrained minimization problem the following constrained energy functional is defined:

$$\begin{aligned} \begin{aligned}&\mathcal {F}\left( \hat{\mathbf {X}}_{CS },\varvec{\lambda },\varvec{\mu },\mathbf {v}_0,\mathbf {k}_0\right) \\&\quad =\int _{\varOmega _0}\psi \left( \mathbf {F}\right) +\left[ \varvec{\lambda }\cdot \hat{\mathbf {X}}_{CS }+\varvec{\mu }\cdot \varvec{\mathfrak {M}}\right] ~d A. \end{aligned} \end{aligned}$$

(32)

Here, the Lagrange multipliers $\varvec{\lambda }$ and $\varvec{\mu }$ enforce the constraints (30). The minimization of (32) leads to the equilibrium 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 ($\mathbf {P}$, the Piola stress) in the rod’s cross-section

$$\begin{aligned} \begin{aligned} \mathbf {n}_0=&\int _{\varOmega _0}\mathbf {P}\mathbf {E}_{3}~d A, \\ \mathbf {m}_0=&\int _{\varOmega _0}\hat{\mathbf {X}}_{CS }\times \left[ \mathbf {P}\mathbf {E}_3\right] ~d A. \end{aligned} \end{aligned}$$

(33)

As introduced in equation (15) the tangent stiffness $\mathbf {C}_0$ relates the changes in stress resultants to the changes in strain prescriptors. Thus $\mathbf {C}_0$ is obtained by the derivative of the stress resultants (33) with respect to the strain prescriptors

$$\begin{aligned} \mathbf {C}_0= \begin{bmatrix} \displaystyle \frac{\partial n _{0_1}}{\partial v _{0_1}} &{}\displaystyle \frac{\partial n _{0_1}}{\partial v _{0_2}} &{}\displaystyle \frac{\partial n _{0_1}}{\partial v _{0_3}} &{} \displaystyle \frac{\partial n _{0_1}}{\partial k _{0_1}} &{}\displaystyle \frac{\partial n _{0_1}}{\partial k _{0_2}} &{} \displaystyle \frac{\partial n _{0_1}}{\partial k _{0_3}} \\ \displaystyle \frac{\partial n _{0_2}}{\partial v _{0_1}} &{}\displaystyle \frac{\partial n _{0_2}}{\partial v _{0_2}} &{} \displaystyle \frac{\partial n _{0_2}}{\partial v _{0_3}} &{} \displaystyle \frac{\partial n _{0_2}}{\partial k _{0_1}} &{}\displaystyle \frac{\partial n _{0_2}}{\partial k _{0_2}} &{}\displaystyle \frac{\partial n _{0_2}}{\partial k _{0_3}} \\ \displaystyle \frac{\partial n _{0_3}}{\partial v _{0_1}} &{}\displaystyle \frac{\partial n _{0_3}}{\partial v _{0_2}} &{} \displaystyle \frac{\partial n _{0_3}}{\partial v _{0_3}} &{} \displaystyle \frac{\partial n _{0_3}}{\partial k _{0_1}} &{}\displaystyle \frac{\partial n _{0_3}}{\partial k _{0_2}} &{} \displaystyle \frac{\partial n _{0_3}}{\partial k _{0_3}} \\ \displaystyle \frac{\partial m _{0_1}}{\partial v _{0_1}} &{}\displaystyle \frac{\partial m _{0_1}}{\partial v _{0_2}} &{} \displaystyle \frac{\partial m _{0_1}}{\partial v _{0_3}} &{}\displaystyle \frac{\partial m _{0_1}}{\partial k _{0_1}} &{} \displaystyle \frac{\partial m _{0_1}}{\partial k _{0_2}} &{} \displaystyle \frac{\partial m _{0_1}}{\partial k _{0_3}} \\ \displaystyle \frac{\partial m _{0_2}}{\partial v _{0_1}} &{}\displaystyle \frac{\partial m _{0_2}}{\partial v _{0_2}} &{}\displaystyle \frac{\partial m _{0_2}}{\partial v _{0_3}} &{}\displaystyle \frac{\partial m _{0_2}}{\partial k _{0_1}} &{}\displaystyle \frac{\partial m _{0_2}}{\partial k _{0_2}} &{}\displaystyle \frac{\partial m _{0_2}}{\partial k _{0_3}} \\ \displaystyle \frac{\partial m _{0_3}}{\partial v _{0_1}} &{}\displaystyle \frac{\partial m _{0_3}}{\partial v _{0_2}} &{} \displaystyle \frac{\partial m _{0_3}}{\partial v _{0_3}} &{} \displaystyle \frac{\partial m _{0_3}}{\partial k _{0_1}} &{} \displaystyle \frac{\partial m _{0_3}}{\partial k _{0_2}} &{}\displaystyle \frac{\partial m _{0_3}}{\partial k _{0_3}}\\ \end{bmatrix}. \end{aligned}$$

(34)

The entries of (34) are computed via

$$\begin{aligned} \begin{aligned} \frac{\partial \mathbf {n}_0}{\partial p}&=\int _{\varOmega _0}\left[ \mathbb {A}:\frac{\partial \mathbf {F}}{\partial p}\right] \mathbf {E}_3~d A, \\ \frac{\partial \mathbf {m}_0}{\partial p}&=\int _{\varOmega _0}\frac{\partial \hat{\mathbf {X}}_{CS }}{\partial p}\times \left[ \mathbf {P}\mathbf {E}_3\right] \\&\quad +\hat{\mathbf {X}}_{CS }\times \left[ \left[ \mathbb {A}:\frac{\partial \mathbf {F}}{\partial p}\right] \mathbf {E}_3\right] ~d A, \end{aligned} \end{aligned}$$

(35)

where $p\in \left\{ v _{0_1},v _{0_2},v _{0_3},k _{0_1},k _{0_2},k _{0_3}\right\} $ and $\displaystyle {\mathbb {A}=\frac{\partial \mathbf {P}}{\partial \mathbf {F}}}\left( \mathbf {F}\right) $³. The derivatives $\displaystyle {\frac{\partial \hat{\mathbf {X}}_{CS }}{\partial p}}$ and $\displaystyle {\frac{\partial \mathbf {F}}{\partial p}}$ are presented in detail in [26].

Recalling from this chapter’s introduction the cross-sectional warping problem may be treated as a micro-problem 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 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.

4 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 cross-section. 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 cross-section 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 three-dimensional 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 cross-section 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.

4.1 $FE ^2$ 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 micro-problem. 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 micro-problem (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.

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 J2-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 J2-plasticty, since in [23] a yield-surface in terms of stress resultants for a circular rod applying J2-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.

4.2 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 J2-plasticity 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

$$\begin{aligned} \varPsi \left( \mathbf {b}^e,\xi \right) =\psi \left( \mathbf {b}^e\right) +\mathcal {H}\left( \xi \right) , \end{aligned}$$

(36)

with the strain energy density $\psi \left( \mathbf {b}^e\right) $ depending on the elastic left Cauchy-Green strain $\mathbf {b}^e=\mathbf {F}^e\mathbf {F}^{e^T}$ and the hardening potential $\mathcal {H}\left( \xi \right) $ depending on the scalar internal variable $\xi $. The strain energy density $\psi \left( \mathbf {b}^e\right) $ is defined in terms of the logarithmic principle stretches of $\mathbf {b}^e$. For details the reader is referred to [23, 39]. The hardening potential is defined as

$$\begin{aligned} \mathcal {H}\left( \xi \right) =\frac{1}{2}H\xi ^2, \end{aligned}$$

(37)

where H is a scalar hardening parameter. The yield-criterion is given by the von-Mises yield-surface

$$\begin{aligned} \varPhi \left( \varvec{\tau },q\right) =\Vert \text {dev}\varvec{\tau }\Vert -\sqrt{\frac{2}{3}}\left[ \sigma _y-q\right] \le 0, \end{aligned}$$

(38)

where $\varvec{\tau }$ defines the Kirchhoff stress and $\displaystyle {q=-\frac{\partial \mathcal {H}}{\partial \xi }\left( \xi \right) }$. The material parameters for the following simulations are given by compression modulus $\kappa = 164210~\text {GPa}$, shear modulus $\mu = 80193~\text {GPa} $ and yield limit $\sigma _y = 450~\text {GPa}$. 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 $\mathbf {C}_0$ 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 $\mathbf {C}_0$ 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 yield-surface derived in [23] is extended by the resultant-type hardening stress $\varvec{\beta }$ as

$$\begin{aligned} \varPhi ^{rod }&=\left| \frac{n _{0_1}}{700~N -\beta _1}\right| ^{2.04}+\left| \frac{n _{0_2}}{700~N -\beta _2}\right| ^{2.04}\nonumber \\&\quad +\left| \frac{n _{0_3}}{1470~N -\beta _3}\right| ^{1.76}+\left| \frac{m _{0_1}}{620~Nmm -\beta _4}\right| ^{2.09}\nonumber \\&\quad +\left| \frac{m _{0_2}}{620~Nmm -\beta _5}\right| ^{2.09}{+}\left| \frac{m _{0_3}}{560~Nmm {-}\beta _6}\right| ^{1.73}{-}1\nonumber \\&\le 0. \end{aligned}$$

(39)

with $\displaystyle {\varvec{\beta }=-\frac{\partial \mathcal {H}^{rod }}{\partial \varvec{\xi }}\left( \varvec{\xi }\right) }$. 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, $\varPhi ^{rod }$ implies the continuous elastoplastic model (here J2-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

$$\begin{aligned} \mathcal {H}^{rod }\left( \varvec{\xi }\right) =\frac{1}{2}\varvec{\xi }\cdot \left[ \mathbf {H}^{rod }\varvec{\xi }\right] , \end{aligned}$$

(40)

where $\mathbf {H}^{rod }$ is an invertible symmetric $6\times 6$ tensor The symmetry of the hardening tensor results from the quadratic dependency on $\varvec{\xi }$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 $\mathbf {H}^{rod }$ as presented in sect. 5.1. The computation of the entries of $\mathbf {H}^{rod }$ remains an unsolved task, which we tackle in the next section. For a general symmetric invertible hardening tensor $\mathbf {H}^{rod }$, the resultant-type hardening stress equals

$$\begin{aligned} \varvec{\beta }=-\mathbf {H}^{rod }\varvec{\xi }. \end{aligned}$$

(41)

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 $\mathbf {H}^{rod}$. First, we note that the hardening matrix $\mathbf {H}^{rod}$ 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 $\mathbf {v}_0=\left[ 0~ 0~ 1\!+\!\lambda \right] ^{T }$ and $\mathbf {k}_0=\left[ 0~ 0~ 0\right] ^{T }$. 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 (red curve). One may observe that after a small elastic region, $n_{0_{3}}$ does not remain constant, as expected for zero hardening, but decreases - softening reveals. The reason therefore is the necking of the cross-section due to large longitudinal strains. For small strain prescriptors ($\lambda <0.05$) the relation may be considered linear in the inelastic region. To capture the softening effect in the stress resultants approach, the corresponding entry in $\mathbf {H}^{rod }$ may be negative and thus for the simple assumption of a diagonal structured $\mathbf {H}^{rod }$ the hardening tensor may be non positive definite. Further, the displayed example also demonstrates that for the cases of $H\rightarrow 0$ the stress resultants approach behaves non symmetric for the longitudinal case. Geometric effects reveal softening under tension, which is not expected in the compression case $\mathbf {v}_0=\left[ 0~ 0~ 1\!-\!\lambda \right] ^{T }$ (fig. 3 blue curve), since different geometrical effects on the cross-sectional level appear.Whereas the cross-sectional area reduces under tension, it increases under compression. Consequently, the entries in the hardening matrix would depend on the direction of loading. A simple but handy model as shown in (40) could no more be applied. For these reasons we restrict ourselves to small values in $\mathbf {v}_{0}$ and $\mathbf {k}_0$ 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).

5 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.

5.1 Determining hardening tensor $\mathbf {H}^{rod }$

The yield-surface determines the onset of yield in terms of stresses or stress resultants. The evolution of the yield-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 $\mathbf {H}^{rod }$ must be evaluated. Note that in this contribution $\mathbf {H}^{rod }$ is assumed to be constant throughout the deformation. A requirement to $\mathbf {H}^{rod }$ is that it must be invertible.

To obtain the entries of $\mathbf {H}^{rod }$ 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 $\varvec{\epsilon }=\left[ \mathbf {v}_0~ \mathbf {k}_0\right] ^T$ equals zero except entry $\epsilon _i$⁵. In biaxial states, the entries $\epsilon _i$ and $\epsilon _j$ are non equal to zero. Here $i,j=1,...,6$. The nonzero value is set to a constant value $\lambda \,c$ with the load parameter $\lambda $ taking values from 0 to 1 in 100 equidistant steps of width $\varDelta \lambda =0.01$ and c being a constant terminating value. The design variables of the optimization are the entries $H ^{rod }_{ij}$. Since $\mathbf {H}^{rod }$ is symmetric, $H ^{rod }_{ij}=H ^{rod }_{ji}$. The following minimization problem is formulated

$$\begin{aligned} \underset{\left( \text {H}_{ij}\right) }{\min }~y\left( H ^{rod }_{ij}\right) , \end{aligned}$$

(42)

with the objective function

$$\begin{aligned} \begin{aligned}&y\left( H ^{rod }_{ij}\right) \\&\quad =\left[ \sum _{n}^{100}\left\Vert \left[ \frac{\varDelta \varvec{\sigma }_{SR }\left( H ^{rod }_{ij},\varvec{\epsilon }\right) }{\varDelta \lambda }-\frac{\varDelta \varvec{\sigma }_{CS }\left( \varvec{\epsilon }\right) }{\varDelta \lambda }\right] _{n}\right\Vert \right] ^2 \end{aligned} \end{aligned}$$

(43)

and $\varvec{\sigma }=\left[ \mathbf {n}_0~ \mathbf {m}_0\right] ^T$. The indices $\bullet _{SR }$ and $\bullet _{CS }$ indicate the approach to obtain the stress resultants. It distinguishes between the stress resultants approach ($\bullet _{SR }$) and the micro- or cross-sectional warping problem ($\bullet _{CS }$) 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 $H ^{rod }_{ij}$ takes place with respect to the slope, since hardening influences the elastoplastic tangent stiffness, thus the slope of the stress resultants - strain prescriptors curves. The minimization procedure is repeated until every entry of $\mathbf {H}^{rod }$ is computed. In this contribution, first the diagonal entries of $\mathbf {H}^{rod }$ are evaluated ($i=j$), while fixing the remaining entries. Then, using the diagonal entries, the off-diagonal entries are computed ($i\ne j$). 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 $\mathbf {H}^{rod }$ is invertible. The inverse of $\mathbf {H}^{rod }$ is needed in the return algorithm to solve the evolution equations (23). If $\mathbf {H}^{rod }$ is not invertible the computation of $\varvec{\sigma }_{SR }$ fails and a new guess for $\mathbf {H}^{rod }$ is taken within the particle swarm optimizer. It has been shown in remark 1 that $\mathbf {H}^{rod }$ may be non positive definite. If however positive definiteness is a requirement, instead of fitting the entries of $\mathbf {H}^{rod }$, the entries of its Cholesky decomposition may be fitted. This approach is not pursued in this contribution.

The resultant-type hardening tensors for $H=20000$, $H=10000$ and $H=5000$ take the following values

https://static-content.springer.com/image/art%3A10.1007%2Fs00466-022-02204-8/MediaObjects/466_2022_2204_Equ44_HTML.png

(44)

https://static-content.springer.com/image/art%3A10.1007%2Fs00466-022-02204-8/MediaObjects/466_2022_2204_Equ45_HTML.png

(45)

https://static-content.springer.com/image/art%3A10.1007%2Fs00466-022-02204-8/MediaObjects/466_2022_2204_Equ46_HTML.png

(46)

One may observe that the tensor has not only considerable entries on its diagonal but also on its off-diagonals. The fit of $\varvec{\sigma }_{SR }\left( \mathbf {H}^{rod },\varvec{\epsilon }\right) $ and $\varvec{\sigma }_{CS }\left( \varvec{\epsilon }\right) $ 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 $i\times j$ matrix. The subplot at position i, j results from a strain state where $\epsilon _i=\epsilon _j=\lambda \,c$ 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 cross-sectional 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 $\epsilon _1=\epsilon _2=\lambda \,c$. Beside resulting shear forces, geometric effects on the cross-section induce a coupling term between shear and tensile forces in the tangent stiffness $\mathbf {C}_0$, 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).

5.2 Dependence of hardening tensor $\mathbf {H}^{rod }$ 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 shows the relation of $H ^{rod }_{ij}$ to the scalar hardening parameter H. There, H takes the values 200, 3000, 5000, 7500, 10000, 15000, 20000 and 30000. A nearly linear relation between $\text {H}_{ij}^{rod }$ and H is discernible. Given the hardening parameter on the micro-problem, one may interpolate the component of the hardening tensor for the stress resultants approach by

$$\begin{aligned} H ^{rod }_{ij}\left( H\right) =f\left( H\right) , \end{aligned}$$

(47)

where $f\left( H\right) $ is a linear function relating $H ^{rod }_{ij}$ 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.

5.3 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

$$\begin{aligned} \mathbf {v}_0=\begin{bmatrix} 0.05 \\ -0.02 \\ 1.06 \end{bmatrix} ,~ \text {and}~ \mathbf {k}_0= \begin{bmatrix} 0.04 \\ 0.07 \\ 0.05 \end{bmatrix}. \end{aligned}$$

(48)

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 $\mathbf {H}^{rod }=\mathbf {H}^{rod }_{20000}$, 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 $\lambda $ 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 $\mathbf {H}^{rod }_{n }$ are displayed by dotted lines. The hardening tensor $\mathbf {H}^{rod }_{n }$ 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 $\mu $ used in the cross-sectional warping problem ⁶. In the elastic stiffness $\mathbf {C}_0$ 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

$$\begin{aligned} H ^{rod }_{n _{ii}}={\left\{ \begin{array}{ll} \frac{1}{10}C _{0_{ii}},~\text {if}\quad i=3,4,5,\\ \frac{1}{4}C _{0_{ii}},~\text {if}\quad i=1,2,6. \end{array}\right. } \end{aligned}$$

(49)

This naively introduced hardening tensor leads to non-satisfactory 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 $\mathbf {H}^{rod }=\mathbf {H}^{rod }_{10000}$. Now, the naively defined diagonal hardening tensor $\mathbf {H}^{rod }_{n }$ is calculated as

$$\begin{aligned} H ^{rod }_{n _{ii}}={\left\{ \begin{array}{ll} \frac{1}{20}C _{0_{ii}},~\text {if}\quad i=3,4,5,\\ \frac{1}{8}C _{0_{ii}},~\text {if}\quad i=1,2,6. \end{array}\right. } \end{aligned}$$

(50)

which considers the relation of H to E and $\mu $. 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 $\mathbf {H}^{rod }_{10000}$ shows a considerable improvement to the solution obtained with $\mathbf {H}^{rod }_{n }$ (dotted lines), which are not satisfying.

Finally, fig. 7 shows the stress resultants for $H=5000$, $\mathbf {H}^{rod }=\mathbf {H}^{rod }_{5000}$ and

$$\begin{aligned} H ^{rod }_{n _{ii}}={\left\{ \begin{array}{ll} \frac{1}{40}C _{0_{ii}},~\text {if}\quad i=3,4,5,\\ \frac{1}{16}C _{0_{ii}},~\text {if}\quad i=1,2,6, \end{array}\right. } \end{aligned}$$

(51)

respectively. Compared to fig. 5 the agreement between the stress resultants approach obtained by using the fitted hardening 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.

5.4 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 $\mathbf {n}$ and moments $\mathbf {m}$ 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 $l=100$.

5.4.1 Bent rod

In a first example, the boundary conditions are given as follows

$$\begin{aligned} \begin{aligned}&\mathbf {r}\left( s=0\right) =\mathbf {0},\quad r _1\left( s=l\right) =0,\quad r _2\left( s=l\right) =\lambda \,40, \\&\varvec{\vartheta }\left( s=0\right) =\begin{bmatrix} 0 \\ 0 \\ \lambda \,a\frac{\pi }{2} \end{bmatrix} ,\quad \varvec{\vartheta }\left( s=l\right) =\mathbf {0}, \end{aligned} \end{aligned}$$

(52)

where $0\le \lambda \le 1$ 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 $\lambda =1$ 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 $\vartheta _3=\pi /2$ 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 $\mathbf {d}_1$ and $\mathbf {d}_2$ 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 $\lambda $ 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 $\mathbf {n}$ and $\mathbf {m}$ 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 hardening leads to lower values of stress resultants. Even a reduction in stress resultants just after onset of yield is observed.

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 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.

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.

5.4.2 Curled rod

A second example of a finitely strained rod is given by boundary conditions

$$\begin{aligned} \begin{aligned}&\mathbf {r}\left( s=0\right) =\mathbf {0},\quad r _2\left( s=l\right) =\lambda \,20, \\&\varvec{\vartheta }\left( s=0\right) =\mathbf {0} ,\quad \varvec{\vartheta }\left( s=l\right) =\begin{bmatrix} 0 \\ \lambda \,2\pi \\ 0 \end{bmatrix}, \end{aligned} \end{aligned}$$

(53)

The spatial configuration of the strained rod for $\lambda =1$ 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 $\lambda $. 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 $\lambda \rightarrow 1$ when applying different 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 $\lambda \rightarrow 1$. 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.

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 $\sim 1000~s $ in the $FE ^2$ approach compared to $\sim 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.

6 Summary

This contribution compares two different possibilities to computationally model geometrically exact elastoplastic rods. Both apply a two-scale approach, where the macro-scale is described by the one-dimensional rod and the micro-scale 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 micro-scale 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).

Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Nächster Artikel Numerical treatment of a static thermo-electro-elastic contact problem with friction

1. Fit of stress resultants approach to $FE^{2}$ approach

Figures 19-21 show the fit of the stress resultants at one material point resulting from the stress resultants approach and the $FE ^2$ approach. Uni- and biaxial strain states are considered. The figures are discussed in detail in sect. 5.3.

2. Evolution of the entries of the hardening tensor $H^\mathrm{rod}$ with the hardening parameter H

Figure 22 shows the evolution of the individual entries of $\mathbf {H}^{rod }$ with their corresponding scalar hardening parameter H. Here H takes values from 200 to 30000. The curves are discussed in detail in sect. 5.2.

In the sequel, Latin indices run from 1 to 3, whereas Greek indices run from 1 to 2.

The operator $ax (\mathbf {A})$ transforms an arbitrary skew-symmetric tensor $\mathbf {A}$ into its axial vector $\mathbf {a}$ such that:

$$\begin{aligned} \mathbf {a}=\text {ax}(\mathbf {A})\overset{\wedge }{=}\begin{bmatrix} A _{32} \\ A _{13} \\ A _{21} \end{bmatrix}= -\begin{bmatrix} A _{23} \\ A _{31} \\ A _{12} \end{bmatrix}. \end{aligned}$$

(7)

Note that for the elastoplastic case $\displaystyle {\mathbb {A}=\mathbb {A}^{ep }}$ and subsequently $\mathbf {C}_0=\mathbf {C}_0^{ep }$.

since $\varvec{\beta }=-\frac{\partial \mathcal {H}^{rod }}{\partial {\varvec{\xi }}}=-\frac{1}{2}\left[ \mathbf {H}^{rod ^{T }}+\mathbf {H}^{rod }\right] $, we further define $\mathbf {H}^{rod }=sym \left( \mathbf {H}^{rod }\right) $

In order to obtain a zero vector $\varvec{\epsilon }$ at zero strains we redefine $\mathbf {v}_0\rightarrow \left[ \mathbf {v}_0-\mathbf {E}_3\right] $. The redefinition of $\mathbf {v}_0$ is only valid in this special context.

The Young’s modulus is calculated as $\displaystyle {E=\frac{9\kappa \mu }{3\kappa +\mu }}$.

Riva L, Ginestra P, Ceretti E (2021) Mechanical characterization and properties of laser-based powder bed-fused lattice structures: a review. The International Journal of Advanced Manufacturing Technology 113:1. https://doi.org/10.1007/s00170-021-06631-4CrossRef

Jamshidian M, Boddeti N, Rosen D, Weeger O (2020) Multiscale modelling of soft lattice metamaterials: Micromechanical nonlinear buckling analysis, experimental verification, and macroscale constitutive behaviour. International Journal of Mechanical Sciences 188. https://doi.org/10.1016/j.ijmecsci.2020.105956

Lee JH, Singer JP, Thomas EL (2012) Micro-/nanostructured mechanical metamaterials. Advanced materials (Deerfield Beach, Fla.) 24(36):4782–4810. https://doi.org/10.1002/adma.201201644CrossRef

Blakey-Milner B, Gradl P, Snedden G, Brooks M, Pitot J, Lopez E, Leary M, Berto F, du Plessis A (2021) Metal additive manufacturing in aerospace: A review. Materials and Design 209:110008. https://doi.org/10.1016/j.matdes.2021.110008. https://www.sciencedirect.com/science/article/pii/S0264127521005633

Jetté B, Brailovski V, Dumas M, Simoneau C, Terriault P (2018) Femoral stem incorporating a diamond cubic lattice structure: Design, manufacture and testing. Journal of the Mechanical Behavior of Biomedical Materials 77:58. https://doi.org/10.1016/j.jmbbm.2017.08.034. https://www.sciencedirect.com/science/article/pii/S175161611730382X

Dong G, Tessier D, Zhao Y (2019) Design of Shoe Soles Using Lattice Structures Fabricated by Additive Manufacturing. Proceedings of the Design Society: International Conference on Engineering Design 1:719. https://doi.org/10.1017/dsi.2019.76CrossRef

Wang See C, Kim T, Zhu D (2020) Hernia Mesh and Hernia Repair: A Review. Engineered Regeneration 1: 19. https://doi.org/10.1016/j.engreg.2020.05.002. https://www.sciencedirect.com/science/article/pii/S2666138120300025

Ozdemir Z, Hernandez-Nava E, Tyas A, Warren JA, Fay SD, Goodall R, Todd I, Askes H (2016) Energy absorption in lattice structures in dynamics: Experiments. Int J Impact Eng 89:49. https://doi.org/10.1016/j.ijimpeng.2015.10.007CrossRef

Jin N, Wang F, Wang Y, Zhang B, Cheng H, Zhang H (2019) Failure and energy absorption characteristics of four lattice structures under dynamic loading. Materials and Design 169. https://doi.org/10.1016/j.matdes.2019.107655

10.

Cosserat E, Cosserat F (1968) Theory of deformable bodies

11.

Antman S (2005) Nonlinear Problems of Elasticity. Springer-Verlag, New YorkMATH

12.

Simo J (1985) A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Comput Methods Appl Mech Eng 49(1):55. https://doi.org/10.1016/0045-7825(85)90050-7CrossRefMATH

13.

Simo J, Vu-Quoc L (1986) A three-dimensional finite-strain rod model. part II: Computational aspects. Comput Methods Appl Mech Eng 58(1):79. https://doi.org/10.1016/0045-7825(86)90079-4CrossRefMATH

14.

Wackerfuß J, Gruttmann F (2009) A mixed hybrid finite beam element with an interface to arbitrary three-dimensional material models. Comput Methods Appl Mech Eng 198:2053. https://doi.org/10.1016/j.cma.2009.01.020CrossRefMATH

15.

Wackerfuß J, Gruttmann F (2011) A nonlinear Hu-Washizu variational formulation and related finite-element implementation for spatial beams with arbitrary moderate thick cross-sections. Comput Methods Appl Mech Eng 200:1671. https://doi.org/10.1016/j.cma.2011.01.006CrossRefMATH

16.

Smriti A, Kumar A, Großmann P (2019) Steinmann, A thermoelastoplastic theory for special Cosserat rods. Math Mech Solids 24(3):686. https://doi.org/10.1177/1081286517754132CrossRefMATH

17.

Herrnböck L, Steinmann P (2021) Homogenization of fully nonlinear rod lattice structures: on the size of the RVE and micro structural instabilities. Comput Mech. https://doi.org/10.1007/s00466-021-02123-0CrossRefMATH

18.

Smriti A, Kumar P (2020) Steinmann, A finite element formulation for a direct approach to elastoplasticity in special Cosserat rods. Int J Numer Meth Eng. https://doi.org/10.1002/nme.6566CrossRef

19.

Drucker D (1956) The Effect of Shear on the Plastic Bending of Beams. Journal of Applied Mechanics 23(4). https://doi.org/10.1115/1.4011392

20.

Neal BG (1961) The Effect of Shear and Normal Forces on the Fully Plastic Moment of a Beam of Rectangular Cross Section. J Appl Mech 28(2):269. https://doi.org/10.1115/1.3641666CrossRefMATH

21.

Hajjar J (2003) Evolution of stress-resultant loading and ultimate strength surfaces in cyclic plasticity of steel wide-flange cross-sections. J Constr Steel Res 59:713. https://doi.org/10.1016/S0143-974X(02)00063-9CrossRef

22.

Duan L, Chen WF (1990) A yield surface equation for doubly symmetrical sections. Eng Struct 12(2):114. https://doi.org/10.1016/0141-0296(90)90016-LCrossRef

23.

Herrnböck L, Kumar A, Steinmann P (2021) Geometrically exact elastoplastic rods - determination of yield surface in terms of stress resultants. Comput Mech 67:723. https://doi.org/10.1007/s00466-020-01957-4CrossRefMATH

24.

Simo JC, Hughes TJR (1998) Computational Inelasticity

25.

Kumar A, Kumar S, Gupta P (2016) A Helical Cauchy-Born Rule for Special Cosserat Rod Modeling of Nano and Continuum Rods. J Elast 124:81. https://doi.org/10.1007/s10659-015-9562-1CrossRefMATH

26.

Arora A, Kumar A, Steinmann P (2019) A computational approach to obtain nonlinearly elastic constitutive relations of special Cosserat rods. Computer Methods in Applied Mechanics and Engineering 350:295. https://doi.org/10.1016/j.cma.2019.02.032. https://www.sciencedirect.com/science/article/pii/S0045782519301033

27.

Miehe C, Schröder J, Becker M (2002) Computational homogenization analysis in finite elasticity: material and structural instabilities on the micro- and macro-scales of periodic composites and their interaction. Comput Methods Appl Mech Eng 191(44):4971. https://doi.org/10.1016/S0045-7825(02)00391-2CrossRefMATH

28.

Geers M, Kouznetsova V, Brekelmans W (2010) Multi-scale computational homogenization: Trends and challenges. Journal of Computational and Applied Mathematics 234(7):2175. https://doi.org/10.1016/j.cam.2009.08.077. https://www.sciencedirect.com/science/article/pii/S0377042709005536. Fourth International Conference on Advanced COmputational Methods in ENgineering (ACOMEN 2008)

29.

Geers M, Kouznetsova V, Matous K, Yvonnet J (2017) Homogenization Methods and Multiscale Modeling: Nonlinear Problems. https://doi.org/10.1002/9781119176817.ecm107

30.

Bangerth W, Hartmann R, Kanschat G (2007) Deal.II-A General-Purpose Object-Oriented Finite Element Library. ACM Trans. Math. Softw. 33(4): 24-es. https://doi.org/10.1145/1268776.1268779

31.

Dörlich V, Linn J, Scheffer T, Diebels S (2016) Towards viscoplastic constitutive models for Cosserat rods. Archive of Mechanical Engineering 63(2):215CrossRef

32.

Crisfield M, Jelenić G (1999) Objectivity of strain measures in geometrically exact 3D beam theory and its finite element implementation. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 455. https://doi.org/10.1098/rspa.1999.0352

33.

Simo J, Vu-Quoc L (1991) A Geometrically-exact rod model incorporating shear and torsion-warping deformation. Int J Solids Struct 27(3):371. https://doi.org/10.1016/0020-7683(91)90089-XCrossRefMATH

34.

Mora MG, Müller S (2003) Derivation of the nonlinear bending-torsion theory for inextensible rods by $\Gamma $-convergence. Calculus of Variations and Partial Differential Equations 18. https://doi.org/10.1007/s00526-003-0204-2

35.

Yu W, Hodges DH, Ho JC (2012) Variational asymptotic beam sectional analysis - An updated version. International Journal of Engineering Science 59:40. https://doi.org/10.1016/j.ijengsci.2012.03.006. https://www.sciencedirect.com/science/article/pii/S0020722512000493. The Special Issue in honor of VICTOR L. BERDICHEVSKY

36.

Klarmann S, Gruttmann F, Klinkel S (2020) Homogenization assumptions for coupled multiscale analysis of structural elements: beam kinematics. Computational Mechanics 65. https://doi.org/10.1007/s00466-019-01787-z

37.

Steinmann P, Stein E (1996) On the numerical treatment and analysis of finite deformation ductile single crystal plasticity. Comput Methods Appl Mech Eng 129(3):235CrossRefMATH

38.

Miehe C, Schröder J (2001) A comparative study of stress update algorithms for rate-independent and rate-dependent crystal plasticity. International Journal for Numerical Methods in Engineering 50(2):273. https://doi.org/10.1002/1097-0207(20010120)50:2<273::AID-NME17>3.0.CO;2-Q

39.

Simo J (1992) Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory. Comput Methods Appl Mech Eng 99(1):61. https://doi.org/10.1016/0045-7825(92)90123-2CrossRefMATH

Titel: Two-scale off-and online approaches to geometrically exact elastoplastic rods
verfasst von: Ludwig Herrnböck
Ajeet Kumar
Paul Steinmann
Publikationsdatum: 02.07.2022
Verlag: Springer Berlin Heidelberg
Erschienen in: Computational Mechanics / Ausgabe 1/2023
Print ISSN: 0178-7675
Elektronische ISSN: 1432-0924
DOI: https://doi.org/10.1007/s00466-022-02204-8

Neuer Inhalt

Bildnachweise

VDI-Icon, Profil Icon, inhalt2, Springer Professional Modul/© Springer Fachmedien Wiesbaden GmbH, Nachhaltigkeitsaward Key Visual/© Cometis AG/Global ESG Monitor | Daniel Rupp | Generiert mit KI, Search Icon, Banner Hanser, Buchstaben, die aus einem Megaphon kommen/© MicroStockHub/Getty Images/iStock, Digitale Lieferkette/© zapp2photo / stock.adobe.com, Arbeitszeit/© granata68 / Fotolia, Zeitschrift Wissensmanagement Cover, PatentFit-Logo/© Springer Fachmedien Wiesbaden GmbH, Sustainibility Finance/© Robert Kneschke / stock.adobe.com / Springer Fachmedien Wiesbaden GmbH, Zukunftswerkstatt Sales Excellence 2024/© AndreyPopov / Getty Images / iStock, 2023_Antrieb/© supervisuell

Springer Professional

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

Abstract

Publisher's Note

1 Introduction

2 Geometrically exact rods with rigid cross-sections

2.1 Kinematics

2.2 Stress resultants and balance equations

3 Cross-sectional warping-problem

4 Elastoplastic rods

4.1 \(FE ^2\) approach

4.2 Elastoplastic continuum models

5 Numerical investigations

5.1 Determining hardening tensor \(\mathbf {H}^{rod }\)

5.2 Dependence of hardening tensor \(\mathbf {H}^{rod }\) on the scalar hardening parameter H

5.3 Elastoplastic response of rods at material points

5.4 Boundary value problems involving non-uniformly strained elastoplastic rods

5.4.1 Bent rod

5.4.2 Curled rod

6 Summary

Acknowledgements

Publisher's Note

1. Fit of stress resultants approach to \(FE^{2}\) approach

2. Evolution of the entries of the hardening tensor \(H^\mathrm{rod}\) with the hardening parameter H

Neuer Inhalt

Springer Professional

Abstract

Publisher's Note

1 Introduction

2 Geometrically exact rods with rigid cross-sections

2.1 Kinematics

2.2 Stress resultants and balance equations

3 Cross-sectional warping-problem

4 Elastoplastic rods

4.1 \(FE ^2\) approach

4.2 Elastoplastic continuum models

5 Numerical investigations

5.1 Determining hardening tensor \(\mathbf {H}^{rod }\)

5.2 Dependence of hardening tensor \(\mathbf {H}^{rod }\) on the scalar hardening parameter H

5.3 Elastoplastic response of rods at material points

5.4 Boundary value problems involving non-uniformly strained elastoplastic rods

5.4.1 Bent rod

5.4.2 Curled rod

6 Summary

Acknowledgements

Publisher's Note

1. Fit of stress resultants approach to \(FE^{2}\) approach

2. Evolution of the entries of the hardening tensor \(H^\mathrm{rod}\) with the hardening parameter H

Weitere Artikel der Ausgabe 1/2023

An equilibrium finite element method for contact problem with application to strict error estimation

Computational multi-phase convective conjugate heat transfer on overlapping meshes: a quasi-direct coupling approach via Schwarz alternating method

Proper generalized decomposition solutions for composite laminates parametrized with fibre orientations

An efficient algorithm for rigid/deformable contact interaction based on the dual mortar method

Hybrid of monolithic and staggered solution techniques for the computational analysis of fracture, assessed on fibrous network mechanics

FFT-based homogenization at finite strains using composite boxels (ComBo)

Neuer Inhalt

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.