Derivation of a homogenized bending–torsion theory for rods with micro-heterogeneous prestrain
In this paper we investigate rods made of nonlinearly elastic, composite–materials that feature a micro-heterogeneous prestrain that oscillates (locally periodic) on a scale that is small compared to the length of the rod. As a main result we derive a homogenized bending–torsion theory for rods as -limit from 3D nonlinear elasticity by simultaneous homogenization and dimension reduction under the assumption that the prestrain is of the order of the diameter of the rod. The limit model features a spontaneous curvature–torsion tensor that captures the macroscopic effect of the micro-heterogeneous prestrain. We device a formula that allows to compute the spontaneous curvature–torsion tensor by means of a weighted average of the given prestrain. The weight in the average depends on the geometry of the composite and invokes correctors that are defined with help of boundary value problems for the system of linear elasticity. The definition of the correctors depends on a relative scaling parameter , which monitors the ratio between the diameter of the rod and the period of the composite’s microstructure. We observe an interesting size-effect: For the same prestrain a transition from flat minimizers to curved minimizers occurs by just changing the value of . Moreover, in the paper we analytically investigate the microstructure-properties relation in the case of isotropic, layered composites, and consider applications to nematic liquid–crystal–elastomer rods and shape programming.
MSC2010: 74B20, 74K10, 35B27, 74Q05.
Keywords: homogenization, dimension reduction, elastic rods, prestrain, residual stress
- 1 Introduction
- 2 General framework and statement of main results
- 3 Evaluation of the homogenization formulas via BVPs
- 4 Examples and explicit formulas for isotropic materials
- 5 Proofs
- 6 Acknowledgments
- A Appendix
Residual stress can have a tremendous effect on the mechanical behavior of slender elastic structures: Equilibrium states of elastic thin films and rods with residual stresses often have a complex shape in equilibrium, and may feature wrinkling and symmetry breaking. Many natural and synthetic materials feature residual stresses due to different physical principles, e.g., growth of soft tissues [6, 11], swelling and de-swelling in polymer gels , thermo-mechanical coupling in nematic liquid crystal elastomers , and thermal expansion in production processes. These mechanisms may be triggered by different stimuli (such as temperature, light, and humidity), and are exploited in the design of active thin structures—elastic structures that are capable to change from an initially flat state into a 3D “programmed” configuration in response to external stimuli, see  and  for a recent review on shape shifting flat soft matter. Modeling of such structures, requires (next to a description of the stimuli process) a good understanding of the highly nonlinear relation between residual stresses and the geometry of the equilibrium shape. Although intensively studied, no satisfying understanding of this relation has been obtained so far. This is especially the case for composite materials, where material properties and residual stresses feature microstructure—a situation that is relevant for future applications, since “Shape-changing materials offer a powerful tool for the incorporation of sophisticated planar micro- and nano-fabrication techniques in 3D constructs” as pointed out in .
Overview of results.
In this paper we investigate rods made of nonlinearly elastic, composite–materials that feature a micro-heterogeneous prestrain (or residual stress) that oscillates (locally periodic) on a scale that is small compared to the length of the rod. Our starting point is the energy functional of -nonlinear elasticity with a cylindrical reference domain :
It depends on two small parameters and (describing the thickness of the rod and the period of the composite), and describes prestrain with help of a tensor field , see Section 2.1 for the continuum–mechanical interpretation. We suppose to describe a non-degenerate, nonlinear material with stress–free reference state. Moreover, we assume that the amplitude of the prestrain is comparable to the diameter of the rod, i.e., so that .
As a main result (see Theorem 2.7) we derive the -limit as of (1) in the bending regime. In this simultaneous homogenization and dimension reduction limit, we obtain a homogenized bending–torsion theory for rods that features a spontaneous curvature–torsion tensor . It captures the macroscopic effect of the micro-heterogeneous prestrain:
where bending and torsion of the rod is described by the isometry and an attached orthonormal frame , . The elastic moduli of the rod are described by the quadratic form . It is positive definite on skew symmetric matrices and can computed by a linear relaxation and homogenization formula from —the fourth order elasticity tensor obtained by linearizing at identity. While it is difficult to study energy minimizers of (1) directly, energy minimizers of (2) can easily be obtained by integrating the spontaneous curvature–torsion field . It turns out that depends on (nonlinearly) and on (linearly). In addition, we observe that both and depend on the relative–scaling parameter .
Next to the -convergence result, we introduce an effective scheme to evaluate and , which invokes the definition of suitable correctors that are characterized by corrector equations which essentially come in form of boundary value problems for the system of linear elasticity, see Proposition 3.1. The spontaneous curvature–torsion tensor is than obtained as weighted average of the prescribed prestrain tensor with weights given by the correctors. For isotropic composites with a laterally layered microstructure, we can solve the corrector equations by hand and we obtain explicit formulas for the and , see Lemma 4.1. We observe a significant qualitative and quantitative dependence of on the relative–scaling parameter . In particular, we device an example of a prestrain that yields a transition from a straight minimizer (i.e., ) to a curved minimizers (i.e., ) by only changing the value of , see Section 4.2. Moreover, we briefly discuss applications to nematic liquid crystal elastomers in Section 4.3, shape programming in Section 4.4, and comment on potential applications to rods with varying cross-section in Remark 3.
Survey of the literature.
The derivation of mechanical models for rods has a long history. For modeling based on equilibria of forces or conservation of momentum, and derivations via formal asymptotic expansions or based on the assumption of a kinematic ansatz we refer the reader to [4, 5, 8]. In contrast to these works, we take the perspective of energy minimization, and our result is an ansatz-free derivation that is based on the -convergence methods developed by Friesecke, James & Müller in , in particular the geometric rigidity estimate. If we replace in (1) the prestrain tensor by the identity matrix, then we recover a standard 3D nonlinear elasticity model without prestrain, i.e., with a stress–free reference configuration. In that case the limit with fixed, corresponds to a dimension reduction problem (without homogenization) studied by Mora & Müller in  where for the first time a bending–torsion theory for inextensible rods has been derived via -convergence. On the other hand, the limit corresponds to simultaneous homogenization and dimension reduction and is studied by the second author in [32, 33], see also [34, 35, 18, 45] where the same problem for plates is considered. First results that combine dimension reduction in the presence of a prestrain are due to Schmidt: In [40, 41] prestrained bending plates are obtained from 3D nonlinear elasticity; see also  on the derivation of a model for prestrained von Kármán plates, see also  where applications to models for nematic liquid crystal elastomers are studied. Our result can be viewed as a combination of Schmidt’s work with [32, 33]. We note that a simplified version of our main result is announced in the second author’s thesis  (together with a rough sketch of the proof). Recently, the derivation of prestrained bilayer rods has been investigated by Kohn & O’Brien  and Cicalese, Ruf & Solombrino . In these interesting works not only energy minimizers are studied, but also the convergence of critical points is established and comparision with experiments  are discussed. Another interesting direction of active research on related topics are the derivation and analysis of ribbons, e.g., [2, 14, 13].
In the results discussed so far the prestrain (if present) is assumed to be infinitesimally small. In the last decade, dimension reduction for finite prestrain (yet smoothly varying on a macroscopic scale) has been studied in the framework of non-Euclidean elasticity theory , e.g., [24, 27, 28, 7, 26] for the derivation of non-Euclidean theories for rods and plates. Rods and shells with nontrivially curved reference configuration lead to similar models when being pulled back to a flat reference configuration (cf. Remark 3 below), e.g., see [39, 44, 19, 20] for shells.
Structure of the paper.
We introduce the general framework in Section 2. In particular, we explain the modeling of prestrained composites (based on a multiplicative decomposition of the strain) in Section 2.1. The 3D model and the limiting are described in Sections 2.2 and 2.3. In Section 2.4 we present an abstract definition of the homogenization and averaging formulas that determine and . In Proposition 3.1 in Section 3 we describe the effective evaluation scheme for these formulas. It is based on the notion of suitable correctors. Eventually, in Section 4 we discuss various applications of the theory to isotropic material for which the correctors, homogenization-, and averaging formulas can be evaluated by hand. All proofs are contained in Section 5.
denotes the standard basis of .
Given we write to denote the unique matrix in given by for all .
We write , , and for the space of symmetric, skew-symmetric, and rotation matrices in . We denote the identity matrix by .
We decompose into the in-plane component and the out-of-plane components .
For all we set . We tacitly drop the argument and simply write (instead of ).
2 General framework and statement of main results
In this section we state the general framework and our main result.
2.1 A model for prestrain in nonlinear elasticity.
We start by presenting a model for prestrained composites in nonlinear elasticity. We first introduce a class of stored energy functions:
Definition 2.1 (Nonlinear and linearized material law).
Let , , and let denote a monotone function satisfying .
The class consists of all measurable functions such that,
is frame indifferent: for all , ;
is non degenerate:
is minimal at : .
admits a quadratic expansion at :
where is a quadratic form.
The class consists of all quadratic forms on such that
We associate with the fourth order tensor defined by the polarization identity .
Stored energy functions of class describe materials that have a stress-free reference state (cf. ), and that can be linearized at that state (e.g., in the sense of -convergence, see [10, 31, 16, 32]). The elastic moduli of the linearized model are given by the quadratic form in condition , and we have:
Lemma 2.2 (see Lemma 2.7 in ).
Let and denote by the quadratic form in . Then .
We describe prestrained composites with help of a multiplicative decomposition of the strain. To motivate this decomposition, we consider for a moment a composite consisting of two materials. We suppose that each of the materials can be described w.r.t. their individual stress-free reference configurations by stored energy functions , respectively. Let denote a common reference configuration of the composite and suppose that material–one (resp. –two) occupies the subdomain (resp. ). We suppose that material–one is stress-free in the reference configuration , and thus the elastic energy coming from material–one is captured by . On the other hand, we suppose that material–two is prestrained in the following sense: If we separate an (infinitesimally small) test-volume from the rest of the body, then it relaxes to a stress-free (energy minimizing) state described by an affine deformation where is positive definite and independent of . Thus, defines an alternative, stress-free reference state for material–two, and the elastic energy of a deformation defined relative to is given by . Since the original deformation and are related by (for ), we deduce that the energy functional on the level of associated with material–two is given by
Hence, the energy functional for the whole composite takes the form
This is corresponds to a multiplicative decomposition of the strain. Similar decomposition are used in models for finite strain elasto-plasticity  (where is called the plastic strain tensor and is given by a flow rule), or in biomechanical models for growth and remodeling of tissues and plants, e.g., see [38, 17].
If the prestrain is small, then we can simplify decomposition: Suppose that with , , and . Then for , can be inverted by the Neumann Series . Moreover, . Hence, we arrive at an energy functional of the form with and a tensor . The functional describes (up to an error of order smaller than ) a composite material with heterogeneous prestrain .
2.2 The three-dimensional model.
Let be a Lipschitz domain (open, bounded and connected)—the cross-section of the rod. We may assume without loss of generality that
(otherwise we apply a rigid motion). Set . We denote by the reference configuration of the rod with thickness . For our purpose it is convenient to describe the deformation w.r.t. the rescaled reference domain , and thus consider for the scaled deformation gradient,
Rescaling (1) and assuming that the prestrain takes the form yields an energy functional of the form ,
This parametrized energy functional is the starting point of our derivation. We make the following assumption on the material law:
Assumption 2.3 (Material law).
Let be fixed (as in Definition 2.1). Let be a sequence of Borel-functions such that,
for almost every and for every .
We suppose that there exists such that
is a quadratic form that is piecewise continuous in and periodic in . More precisely,
for a.e. ,
is -measurable for all ,
The fourth order tensor associated with (cf. Definition 2.1) satisfies
is periodic for a.e. and .
The quadratic expansion at identity of (cf. ) satisfies
Regarding the prestrain, we suppose that is locally periodic. Our precise assumption on involves the notion of two-scale convergence in a variant for slender domains [32, 33] (see [36, 3] for the original definition of two-scale convergence). Since this variant of two-scale convergence is sensitive to the relative scaling between and , we introduce a parameter describing the relative scaling of and .
Assumption 2.4 (Relative scaling of and ).
We suppose that there exists and a monotone function such that and .
Definition 2.5 (Two-scale convergence).
Let and denote by the one-dimensional torus. We say a sequence , , weakly two-scale converges in to a function as , if is bounded in and
where is as in Assumption 2.4. We say strongly two-scale converges to if additionally . We write in (resp. ) for weak (resp. strong) two-scale convergence in .
Note that this notion of two-scale convergence changes if we change the parameter . A prototypical example of a strongly two-scale convergent sequence is as follows: Let , then strongly two-scale converges in to .
Assumption 2.6 (Prestrain).
We suppose that there exists such that
2.3 Limiting model and -convergence.
Under the assumptions above, we can pass to the -limit of as . We obtain as a limit a functional defined on the the set of all deformations of the rod that describe (length-preserving) bending- and twisting-deformations, and an infinitesimal stretch:
The -limit is given by ,
where (the homogenized elastic moduli), (the spontaneous curvature–torsion tensor), (the spontaneous infinitesimal stretch), and (the incompatibility of the prestrain) are quantities that only depend on the linearized material law , the prestrain , the geometry of the cross-section , and the scale ratio ; in particular,
is a constant given in Definition 2.11 below,
is a positive–definite quadratic form given by the homogenization formula of Definition 2.8 below,
and are given by the averaging formula of Definition 2.11 below.
Our main result establishes -convergence of to :
Theorem 2.7 (-convergence).
the (scaled) nonlinear strain tensor. Then:
(Compactness). Let be a sequences with equibounded energy, i.e.
Then there exists and a subsequence (not relabeled) such that
(For the proof see Section 5.1).
Theorem 2.7 also yields a compactness and -convergence result towards a (more conventional) pure bending–torsion model. Indeed, by part (a) of Theorem 2.7 every sequence with equibounded energy satisfies (10) for some rod-deformation satisfying (6). Furthermore, by minimizing over the statements of the parts (b) and (c) in Theorem 2.7 hold with and replaced by and (see Remark 5 below for a more explicit characterization of ).
Energies of the type (4) naturally emerge in models of rods with varying cross-section. We describe this in the simple set-up of an homogeneous material law that occupies a cylindrical domain with a rapidly oscillating cross-section : For set , where is a Lipschitz domain satisfying (3) and is sufficiently smooth, and consider the reference domain
where , and
2.4 Homogenization- and averaging formulas.
The definitions of , , and rely on the two-scale structure of limiting strains. To motivate the upcoming formulas, we recall a two-scale compactness statement for the nonlinear strain, see [33, Theorem 3.5] (and also Proposition 5.1 below): Suppose is a sequence with equibounded energy (cf. (9)) with limit (cf. (10), (11)), then (up to a subsequence) the associated scaled nonlinear strain tensors weakly two-scale converge in to a limiting strain of the form
Above, where is defined as follows:
Note that on the right-hand side in (13) the first and second term are determined by the limiting deformation . Only the third term —the only term that involves the fast variable —depends on the chosen subsequence. We call it the strain corrector. For the following discussion it is convenient to define for the affine map
and to introduce the two-scale strain space
Since is skew-symmetric (almost surely) for ), the limiting strain of (13) satisfies .
Formula for .
As in  the homogenized quadratic form is defined by minimizing out the energy contribution coming from :
Definition 2.8 (homogenization formula for ).
We define by
We emphasize that the definition of depends on the small-scale coupling via the relaxation space . For almost every , defines a positive definite quadratic form on and the map is piecewise continuous for every (see Proposition 3.1 below).
Formulas for and .
We first present a “geometric” definition—an alternative “algorithmic” definition that is more practical for numerical investigations is presented in Section 3 below. The geometric definition invokes the following Hilbert-space structure on : Let denote the symmetric fourth-order tensor obtained from the quadratic form by polarization, and consider for ,
Since is positive-definite and bounded on symmetric matrices, defines a scalar product on . We write for the associated norm and note that
and (see (14) and (16)) are closed, linear subspaces of . We denote by (resp. ) the -orthogonal complement of in (resp. in ), and by and the associated -orthogonal projections. We thus have the orthogonal decomposition,
A direct consequence is the following observation:
Lemma 2.9 (Pythagoras).
For all and ,
In particular, we obtain the following characterization of :
It turns out that any admits a representation via a unique pair :
Lemma 2.10 (Representation).
For all the map
defines a linear isomorphism and there exists a constant such that
We denote by the unique bounded operator on defined by the identity
and define and analogously. We are now in position to define and :
Definition 2.11 (averaging formula for and ).
and define as the unique field such that
3 Evaluation of the homogenization formulas via BVPs
The definitions of , and (see Definitions 2.8 and 2.11) are rather abstract. In this section we present a characterization that replaces the “abstract” operator in these definitions by boundary value problems for the system of linear elasticity on the domain . To benefit from the linearity of the map , we set
and note that this defines an orthonormal basis of . Moreover, we introduce the maps ,
see (15) for the definition of . Note that spans the macroscopic strain space. In particular, corresponds to an infinitesimal stretch (in tangential direction); () corresponds to bending in direction , and corresponds to a twist.
We have the following scheme to evaluate the homogenized quantities:
For we define the following objects:
The strain correctors () as the unique solution in to
The averaging matrix as the unique matrix with entries
The vector representation of the strain