# Two- and three-dimensional elastic networks with rigid junctions: modeling within the theory of micropolar shells and solids

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## Abstract

For two- and three-dimensional elastic structures made of families of flexible elastic fibers undergoing finite deformations, we propose homogenized models within the micropolar elasticity. Here we restrict ourselves to networks with rigid connections between fibers. In other words, we assume that the fibers keep their orthogonality during deformation. Starting from a fiber as the basic structured element modeled by the Cosserat curve beam model, we get 2D and 3D semi-discrete models. These models consist of systems of ordinary differential equations describing the statics of a collection of fibers with certain geometrical constraints. Using a specific homogenization technique, we introduce two- and three-dimensional equivalent continuum models which correspond to the six-parameter shell model and the micropolar continuum, respectively. We call two models equivalent if their approximations coincide with each other up to certain accuracy. The two- and three-dimensional constitutive equations of the networks are derived and discussed within the micropolar continua theory.

## 1 Introduction

As the basic structural element of considered structures is a beam, in Sect. 2 we briefly consider the nonlinear beam theory. Let us note that mechanics of one-dimensional structures has a long history of development after such names as Euler and Bernoulli. Mechanics of rods and beams is summarized in many books; see, for example, [1, 22, 41, 43, 51, 67, 77, 81]. Nowadays, the rod and beam structures are widely used in engineering as relatively simple robust models; see, e.g., [5, 20, 83]. Here we use the model originally introduced by the Cosserat brothers and nowadays known as a Cosserat curve. Within this direct approach, the mechanics of rods was developed in seminal papers by Ericksen and Truesdell [29], Green et al. [38, 39], DeSilva and Whitman [15], and Kafadar [44]. Let us note that despite their one-dimensional nature for the solution of the nonlinear beam equations we require a proper numerical tool; see, e.g., [10, 11, 14, 37, 54, 79, 82] and the references therein. Here we consider the Cosserat model with certain constraints. In particular, we neglect shear deformations. The equilibrium equations consist of a system of ordinary differential equations.

For this net, we introduce an averaged continuum model that is a deformable material surface with particular material properties described within the framework of the six-parameter shell theory [13, 49, 50]. Within the six-parameter shell theory, the kinematics of a shell is described by three translations and three rotations as in the rigid body dynamics or in the Cosserat (micropolar) continuum [30]. So, the model is also called the micropolar shell theory [24, 26]. In the theory, only forces and moments including drilling ones are considered as stress characteristics which are also used in the static boundary conditions. Within the theory, solutions of many problems were given numerically and/or analytically; see, e.g., [8, 9, 13, 60, 85]. We briefly recall the basic equations of the micropolar shell statics paying the most attention to the constitutive relations and possible constraints for deformations. Let us note that usually shell equations are derived for a solid shell-like body; see, for example, the through-the-thickness integration technique presented in [13, 49, 50]. As here we consider deformations of completely different structures similar to a fishnet, one needs another approach; see, e.g., [45, 64, 86] for lattice shells.

Comparing discrete models for a 2D network and shell in Sect. 4, we present the constitutive equations for the shell which is equivalent in certain sense to the considered network. In fact, the homogenized constitutive equation for an elastic network has a particular form of strain energy of a six-parameter shell. The obtained results can be relatively easy generalized for networks made of extensible, shear deformable, initially curved and naturally twisted fibers. The only crucial point for the derivation is the orthogonality of fibers during deformations. This assumption results in the possibility to describe rotations of all fibers using one rotation tensor field.

Finally, in Sect. 5 we consider 3D structures made of three orthogonal families of flexible elastic fibers. In a similar way, comparing the discretization for the 3D network and a micropolar continuum we derive the continuous nonlinear micropolar model. As was mentioned in [26], homogenization is one of the main sources for derivation of the constitutive equations of micropolar solids; see, e.g., [6, 18, 19, 31, 35, 69] and the references therein.

## 2 Cosserat curve as a fiber model

*s*is the referential arc-length parameter. In order to describe the cross-sectional orientation of the beam, we introduce the unit orthogonal vectors \({\mathbf {D}}_k={\mathbf {D}}_k(s)\) called directors, \({\mathbf {D}}_k\cdot {\mathbf {D}}_m=\delta _{km}\), where \(\delta _{km}\) is the Kronecker symbol, the centered dot stands for the scalar product and Latin indices take values 1, 2, 3. In what follows, we use the direct tensor calculus as introduced in [25, 47, 52, 71]. Without loss of generality, we assume that \({\mathbf {D}}_1\) is tangent to \(\mathcal {C}_0\).

*s*as a coordinate.

*s*, \((\ldots )'=\frac{\partial (\ldots )}{\partial s}\).

## 3 Discrete beam lattice

*n*horizontal and

*m*vertical beams with distance

*h*between intersection points. Using the orthogonality, we can chose directors \({\mathbf {D}}_1\) and \({\mathbf {D}}_2\) as tangent vectors to the first and second family of beams, respectively. In what follows, we denote the quantities related to these two families by indices 1 and 2. Introducing the Cartesian coordinates \(s_1\) and \(s_2\) and numbering nodes as in Fig. 4, we get the formulae for the position of (

*i*,

*j*)-node,

*m*and

*n*are numbers of beams in these families, and we introduced the line strain energy densities

*equivalent*. In order to characterize the equivalent continuous beam lattice shell, we consider the nonlinear resultant shell theory.

## 4 Continuous beam lattice shell: micropolar shell

Following [24, 26], let us briefly introduce the governing equations used within the six-parameter shell theory. Here the kinematics of a shell is described by six scalar degrees of freedom that are three translations and three rotations as in the case of the Cosserat (micropolar) continuum [26, 30]. So the model is also called the micropolar shell theory. The basic equations of the six-parameter shell theory can be derived using through-the-thickness integration of the 3D equations of motion [13, 49, 50, 58] or within the so-called direct approach [24, 26]. In a current placement, the base surface of the shell has the position vector \({\mathbf {x}}={\mathbf {x}}(q^1,q^2)\), whereas its orientation is determined by the rotation tensor \({\mathbf {Q}}(q^1,q^2)\). Here \(q^1\) and \(q^2\) are Lagrangian surface convective coordinates.

In [27], the detailed analysis of constitutive relations for shells was provided considering the material symmetries and the invariance properties of \(\mathcal {W}\). Here the material symmetry group contains rotations about \({\mathbf {D}}_3={\mathbf {N}}\) of angles \(\pm \frac{\pi }{2}\) and mirror reflections \({\mathbf {I}}-{\mathbf {D}}_1\otimes {\mathbf {D}}_1\) and \({\mathbf {I}}-{\mathbf {D}}_2\otimes {\mathbf {D}}_2\). So (33) belongs to the class of orthotropic shells.

## 5 3D elastic network and its continuous counterpart

*x*,

*y*, and

*z*are Lagrangian Cartesian coordinates,

*h*is the cell size, and

*m*,

*n*, and

*l*are the numbers of beams in

*x*-,

*y*-, and

*z*-directions, respectively.

It is worth noting that (48) has the form proposed for the nonlinear elasticity by Valanis and Landel [80]. So (33) and (48) can be treated as a generalization of the Valanis–Landel hypothesis for micropolar solids.

## 6 Conclusions

In the paper, we discussed the governing equations for an elastic network which consists of flexible fibers undergoing large deformations. Here we restrict ourselves to networks with orthogonal fibers with rigid connections such that the fibers keep their orthogonality during deformations. Let us note that this assumption plays a key role in the analysis, as it gives the possibility to describe the rotations of the fibers using one rotation tensor. As similar assumption was used for the derivation of the micropolar beam model considering the homogenization of beamlike lattice structures by Noor and Nemeth [55, 56]. As a result, we came to a special case of micropolar materials with the strain energy density which inherits all properties of the fibers. Let us note that in the case of micropolar materials as for any generalized medium the derivation of the constitutive equations is a rather complex quest. In addition to rather rare direct experimental data, see, e.g., [46, 68], the homogenization of highly inhomogeneous materials brings us the constitutive equations of micropolar materials; see [6, 18, 19, 31, 35, 69] and the references therein. Here we also consider a certain type of homogenization for the finite deformations as we replaced the semi-discrete network by a homogeneous medium. The presented constitutive equations can also be treated as a generalization of the Valanis–Landel hypothesis [80] for the case of micropolar shells and solids. Relaxing the assumption of orthogonality, we can obtain more general models of the equivalent medium such as strain gradient and micromorphic ones; see, e.g., [16, 42, 65].

## Notes

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