A multiscale co-rotational method for geometrically nonlinear shape morphing of 2D fluid actuated cellular structures

Mechanics of Materials 79 (2014) 1–14

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A multiscale co-rotational method for geometrically nonlinear shape morphing of 2D fluid actuated cellular structures J. Lv a,c, H. Liu a,b, H.W. Zhang a,b,⇑ a State Key Laboratory of Structural Analysis for Industrial Equipment, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian, PR China b Department of Engineering Mechanics, Dalian University of Technology, Dalian, PR China c School of Aeronautics and Astronautics, Dalian University of Technology, Dalian, PR China

a r t i c l e

i n f o

Article history: Received 10 January 2014 Received in revised form 6 August 2014 Available online 27 August 2014 Keywords: Fluid actuated Cellular structures Geometrically nonlinear analysis Multiscale finite element method Co-rotational formulation

a b s t r a c t This work investigates geometrically nonlinear shape morphing behaviors of the adaptive bio-inspired fluid actuated cellular structures. An efficient multiscale co-rotational method based on the multiscale finite element framework is proposed for the geometrically nonlinear analysis of the fluidic cellular structures composed of periodical microscopic fluid inclusions. In this method, the multiscale base functions are firstly constructed to establish the relationship between the small-scale fluctuations of the microstructures and the macroscopic deformation on the coarse scale mesh. And then the co-rotational formulation is integrated to the multiscale method to decompose the geometrically nonlinear motion of the coarse-grid element into rigid-body motion and pure deformational displacements. With these formulations, the large displacement-small strain nonlinear problems of the fluid actuated cellular structures can be resolved on the multiscale co-rotational coarse-grid elements with little work. The numerical results indicate that the present multiscale algorithm is simple, accurate and highly efficient and can provide an alternative to model the fluid actuated actuators for morphing wings. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Biologically inspired actuators have receiving growing interest in recent year. Plants such as Dionaea muscipula (Venus flytraps) possess structural and shape adapting functions by actively altering the cell pressures. The leaves of these plants are hydraulic actuators that do not require any complex controls and keep an energy efficiency which is unmatched by natural or artificial muscles (Pagitz et al., 2012; Stahlberg, 2009). Therefore, the bio-inspired fluidic cellular materials that develop pressure and hence strain ⇑ Corresponding author at: State Key Laboratory of Structural Analysis for Industrial Equipment, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian, PR China. Tel./fax: +86 411 84706249. E-mail address: [email protected] (H.W. Zhang). http://dx.doi.org/10.1016/j.mechmat.2014.08.004 0167-6636/Ó 2014 Elsevier Ltd. All rights reserved.

similar to plants, possess some special features, such as light weight, strong energy absorption capability, adaptive ability and large actuation (Luo and Tong, 2013a). These attractive features make the fluidic cellular structures to be useful for smooth shape control. For instance, their potential applications to morphing flexible aircraft wings have been experimentally validated by several authors (e.g. Barbarino et al., 2011; Barrett and Barrett, 2014; Bowman et al., 2007; Gomez and Garcia, 2011; Matthews et al., 2006; Stanewsky, 2001). With the inspiration of the cellular fluid systems in natural plants, the adaptive pressure cellular materials/structures have recently appeared. In the plants, the fluid pressure difference amongst adjacent cells can be generated by the fluid deliver system of the biological membrane. These pressure differences can result in individual

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cell to expand or shrinkage and hence accumulate deformations over all cells in the plants. In other words, the structural morphing of the plants can be obtained by adjusting the differential pressure amongst adjacent cells. Several researchers have been utilized the morphing mechanism of the plants in the design of adaptive structures. Matthews et al. (2006) studied the bioenergetics of a prototype nastic structural system which consists of an array of cylindrical micro-hydraulic actuators embedded in a polymeric plate. Thereafter, Freeman and Weiland (2009) further designed the nastic materials to mimic bulk deformation similar to nastic movements in the plant kingdom. In their works, the controlled transport of charge and fluid across a selectively permeable engineering membrane employing biological process was employed to achieve bulk deformation. And their experiments demonstrated that the bio-inspired actuator can provide large mechanical force from chemical energy and can response quickly. In addition to the above works, some other researchers have also proposed different schemes on this subject. For instance, Shan et al. (2006) and Philen et al. (2007) designed a type of flexible composite material whereby internal fluid pressure was used to cause extension, contraction or twisting motions. Shan et al. (2008) developed a plate consisting of multiple fluidic flexible matrix composites. In the applications of the bio-smart fluidic cellular materials, Vos and Barrett (2011) and Vos et al. (2011) developed adaptive pressurized cellular structures for morphing wings. In their work, the cellular structures were composed of complaint honeycomb cells and could exhibit great strains by relying on a pressure differential to alter the structural stiffness. Pagitz et al. (2012), Pagitz and Bold (2013) and Pagitz and Hühne (2014) proposed a novel pressure actuated cellular structure that was inspired by the nastic movement of plants. The main idea of this structure is to connect prismatic cells with tailored pentagonal and/or hexagonal cross sections such that the resulting cellular structure morphs into given target shapes for certain cell pressures. Vasista and Tong (2012) designed and tested the pressurized cellular planar morphing structures to offer a solution to the challenges faced in designing morphing aircraft structures. Li and Wang (2013) investigated a cellular adaptive structure consisting of a string of fluidconnected fluidic flexible matrix composite cells with different properties. Luo and Tong (2013a,b) designed and validated the adaptive bio-inspired pressure cellular structures for shape morphing optimum designs for cellular structures. Another area worth exploring is the simulation and design methods for the fluid actuated cellular structures with multiple fluid inclusions. In real applications, the fluidic cellular structures commonly consist of a large number of microcapsules, since the size of the components of the motor cells in the cellular structure is generally limited to a small scale level, such as the biological membrane in nastic materials (Sundaresan and Leo, 2006). Due to the difficult to solve such problems by the direct methods (e.g. standard finite element method), some novel multiscale methods have been developed. Zhang and Lv (2011) developed a coupled two-scale model based on the

homogenization theories to simulate nastic structures with periodically arranged fluid-filled cells. Similarly, Ma et al. (2011) proposed an analytical micromechanics model and an asymptotic homogenization-based finite element model to investigate the effective properties of the cellular structure containing pressurized fluid-filled pores. Guiducci et al. (2014) utilized a numerical homogenization and a micromechanical model based on the Born model to simulate cellular materials filled with a variable pressure fluid phase. On the other hand, a new hierarchical multiscale method based on the multiscale finite element framework is developed to simulate linear behaviors of the closed liquid cellular structures with multiscale features (Zhang et al., 2010). The developed multiscale method can efficiently compute the multiscale problems on a relative coarse scale mesh and can easily recover the small-scale information of the microstructures by a downscale technique. Recently, this multiscale method has been successfully extended to perform the shape and topology designs of the fluid actuated morphing structure (Lv et al., 2014). However, these works mainly concentrated on the elastic properties of the fluid actuated cellular structures. In current research, we will focus on the numerical methods to predict the nonlinear characteristics of the fluid actuated cellular structure. In real applications, the fluidic cellular structure often shows large and nonlinear deformations in non-muscular engines, such as snapping motion of the Venus flytrap as illustrated in Fig. 1 (Forterre et al., 2005; Pagitz et al., 2012). The research by Vos et al. (2011) indicated that the adaptive pressurized cellular structures can provide large actuation strains up to 12.5%. As a result, the direct methods for linear problems will show some drawbacks in situations where the problems to be solved contain nonlinear features. Recently, Ma and his coworkers (Ma et al., 2014) successfully developed a micromechanics model and a computational homogenization method to examine the macroscopic elastoplasticity and yield behavior of closed-cell porous materials with varied inner gas pressure. However, this work is limited to the material nonlinearity of the fluid actuated cellular structure with multiple inclusions. On the other hand, due to the geometrical nonlinear behaviors of the bio-inspired smart materials, it will be an interesting and

Fig. 1. Illustration of geometrically nonlinear behaviors in the snap motion of the Venus flytrap (Pagitz et al., 2012).

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challenge works to develop an efficient computational method to predict such nonlinear problems of the fluid actuated cellar structures. The main contribution of this paper is to propose a multiscale co-rotational method for the geometrically nonlinear static analysis of the fluid actuated cellular morphing structure. On the basis of the existing works which mainly focus on the linear behaviors for the cellular materials with multiple fluid inclusions (Lv et al., 2014, 2013; Zhang et al., 2010), the extended multiscale finite element method (EMsFEM) is further developed for the geometrical nonlinear problems of the fluid actuated cellular structures. Firstly, the multiscale base functions for both displacement field and the fluid pressure of the incompressible fluid within the closed cells are constructed to establish the relationship between the macroscopic deformations and microscopic response such as the deformation, stress, strain and fluid pressure. With these multiscale base functions, the complex multiscale problems can be successfully solved only at the macroscale level. Subsequently, EMsFEM is combined with the co-rotational approach to solve the geometrically nonlinear behaviors at the coarse-scale meshes which were generated according to the heterogeneous features in the structures. The co-rotational approach is originally introduced by Wempner (1969) and Belytschko and Hsieh (1973). Some other researchers, such as Argyris (1982), Battini (2008), Crisfield (1990) and Simo (1985) also proposed important works of this approach (see review article of Felippa and Haugen (2005)). The co-rotational formulation is used to decompose the geometrically nonlinear motion of the coarse-grid element into rigid-body motion and pure deformational displacements. And the deformational response of the coarse-grid element is described at the level of local systems, while at least most of the geometric nonlinearity is incorporated in the transformation matrices relating local and global internal force vectors and tangent stiffness matrices. Consequently, the geometrical nonlinear behaviors of the fluid actuated cellular can be efficiently solve at the coarse scale level. Meanwhile, the microscale response of the heterogeneous structures can be easily recovered by a downscaling computation. In the sequel of the paper, Section 2 describes the characteristic of the fluid actuated cellular structures. Section 3 introduces how to incorporate the co-rotational approach into EMsFEM. Three numerical examples are given in Section 4. Conclusions are presented in Section 5.

2. Characteristics of fluid bf actuated cellular structures 2.1. Finite element formation for closed inclusion fully filled with fluid

Fig. 2. 2D Fluidic cellular structure with closed fluid inclusion.

multiphysics problems of such cellular structure, it is important to analyze the coupling between the deformation of the matrix and the pressure exerted by the contained fluid on the boundary of the inclusion. The finite element method (FEM) is introduced for analyzing the elastic deformation of fluidic cellular structure with a fluid-filled cavity (Zhang and Lv, 2011). The virtual work expression of the fluidic cellular structure can be written as:

P¼

1 2

Z

rT edX 

Z

uT  tdS 

S

X

Z

uT  fdX

ð1Þ

X

where r and e represent the stress and strain vectors, respectively. X denotes a volume occupied by a part of the body in the current configuration. S is the surface bounding this volume. The surface traction at any point on S is denoted as t and the body force at any point within the volume is represented as f. As the fluid inside the inclusion is assumed to be incompressible, the volume of the inclusion is satisfied as follows

V f  V 0f ¼ 0

ð2Þ

where V f and V 0f are the current and initial volume of the fluid cavity, respectively. Substituting the condition of incompressibility (i.e. Eq. (2)) into the virtual work expression (1), we immediately get:

P¼

1 2

Z X

rT edX 

Z S

uT  tdS 

Z X

uT  fdX  kðV f  V 0f Þ ð3Þ

In current research, the bio-inspired cellular structures are assumed to be composed of closed fluid inclusion arrays. As illustrated in Fig. 2, the unit cell of the cellular structure consists of elastic matrix and closed inclusion fully filled with incompressible fluid. For simplicity, we assume that no cavitation occurs while the unit cell is under a state of tension. In order to simulate static

where k is the Lagrange multiplier. It can be easily derived out that the hydrostatic pressure of the fluid (PÞ coincides with the Lagrange multiplier. Thus, the pressure and deformation of the fluidic cellular structure under the external forces can be solved by FEM on the fine scale mesh, as shown in Fig. 2.

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2.2. Actuation by volume change of the fluid The fluidic cellular structure can be actuated by the pressure difference inside the closed liquid inclusions. The pressure difference is generally caused by the volume changes of the fluid inside the inclusion, as illustrated in Fig. 3. The fluid lost or diffused into the cells can be controlled by some special driving unit, such as the biological transport systems of membrane of the plant cells. For instance, Sundaresan and Leo (2006) pointed out that the controlled mass transport of charge and fluid across a selectively permeable membrane employing biological processes can achieve bulk deformation of the cells. Denote the volume change of the fluid inside the inclusion by DV f , then the Eq. (2) can be modified as:

V f  V 0f  DV f ¼ 0

θ

ð4Þ

Therefore, the fluidic cellular structure with the actuations of the volume change of the fluid can be modeled by substituting Eq. (4) into Eq. (1) 3. Multiscale co-rotational formations for fluid actuated cellular structures An efficient algorithm is a challenge for the geometrically nonlinear analysis of the fluid actuated cellular structures composed of multiple motor cells, since the multiphysics, multiscale and geometric nonlinearities are simultaneously involved in one problem. This section describes the details of the multiscale co-rotational method for the large displacement-small strain problems about the fluid actuated cellular structures. The implementations of the algorithms are also presented here. 3.1. Overview of the multiscale co-rotational method The proposed multiscale co-rotational method can be divided into three steps, as illustrated in Fig. 4. In the first step, the multiscale base functions in the multiscale finite element framework are calculated on each unit cell to capture the small scale information of the microstructures to a relative coarse scale mesh (coarse-grid element). In this step, the unit cell with a fluid inclusion is equivalent to a

Fig. 4. The procedure for the multiscale co-rotational method for the large displacement-small strain analysis of the fluid actuated cellular structures.

coarse-grid element, which resulting in a coarse scale model of the structure. In the second step, the corotational approach is used to decompose the motion of the coarse-grid element into rigid-body and pure deformational parts and then calculate the equivalent tangent stiffness matrix of the coarse-grid element. And then, the incremental iterations are performed on the coarse scale mesh for the geometrically nonlinear problem until equilibrium is reached. Finally, when the iterations are finished, the downscale computations are carried out to recover the fine scale response from the coarse scale solutions. 3.2. Multiscale finite element methods for fluid actuated cellular structures In current research, the EMsFEM is adopted to modeling the linear multiscale problems of the fluid actuated cellular structures in the local coordinates. EMsFEM can be traced

Fig. 3. The volume and pressure in the cell are controlled by sub-cellular osmotic hydration motors (Pagitz et al., 2012).

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back to the work proposed by Babuska and Osborn (1983), Hou et al. (1999) and Efendiev and Hou (2009). The key idea of EMsFEM is to numerically construct the multiscale base functions on the unit cells with fine scale mesh. These basis functions are used to coarsen (upscale) the heterogeneities of the microstructures to the coarse scale mesh. EMsFEM for fluid actuated cellular structures in the linear systems has been described by Zhang et al. (2010). For the sake of completeness, the main lines are recalled in this paper for ease reading. Therefore, the following subsections are devoted to calculate the multiscale base functions and equivalent stiffness matrix of the coarse-grid element at the local coordinate system. 3.2.1. Multiscale base functions As shown in Fig. 2, the fluidic cellular materials are multiphase materials, which consist of solid matrix and fluid-filled inclusions. Thus, two types of multiscale base functions, i.e., the displacement base functions and the fluid pressure base functions, are introduced to capture the small-scale information (Zhang et al., 2010). For a planar quadrilateral coarse-grid element with four nodes, the multiscale base functions for the displacement and the fluid pressure can be defined as 4   X ~ i þ Niyx V~ i u¼ Nixx U

v¼

~ i þ Niyy V ~i Nixy U

4  X

~ i þ Npiy V ~i Npix U

ð5Þ





ð6Þ

i¼1

where u and v denote the displacement of sub-grid nodes ~i in the unit cell; P is fluid pressure inside the inclusion; U ~ i denote the displacements of the coarse-grid nodes and V i; N i and N pi represent the multiscale base functions of displacement field and fluid pressure of the coarse-grid element, respectively. It should be noted that the multiscale base functions are defined in the local coordinates for geometrically nonlinear problems, as illustrated in Fig. 4 Multiscale base functions can be numerically calculated by solving the equilibrium equations of the unit cells with some specified boundary conditions in the local coordinate system. Still consider the unit cell with closed liquid inclusion as illustrated in Fig. 2, the general expression to construct the multiscale base functions can be given as follows (Zhang et al., 2010)

LNi ¼ 0

3.2.2. Equivalent stiffness matrix of the coarse-grid element The equivalent stiffness matrix (KlE Þ of the coarse-grid element in the local coordinate system can be calculated by m X GTe Kle Ge

ð10Þ

e¼1

i¼1

P¼

ð9Þ

where N0i is the boundary value imposed to construct the multiscale base functions. In the literatures, many different boundary conditions have been developed to solve the multiscale base functions, such as the linear boundary conditions, periodic boundary conditions and oversampling technique (Lv et al., 2013; Zhang et al., 2010). For simplicity, here we assume that the closed liquid inclusions are periodically embedded in the fluidic cellular structure. Thus, periodic boundary conditions are utilized to solve the fluid actuated cellular structures. The detail derivation of periodic boundary conditions can be found in our previous work (Zhang et al., 2010). Fig. 5 illustrates contour of numerical values of the multiscale base functions N 1xx on the sub-grid mesh of the unit cell as well as the fluid pressure base functions N p1x inside the inclusion.

KlE ¼

i¼1 4  X

Ni ¼ N0i on @ X

where Kle denotes element stiffness matrix of the fine-grid element in the local coordinate system. m is the number of the fine-grid elements within the unit cell. Ge is the transition matrix relating the displacements between the nodes of the fine-grid element and the nodes of the coarse-grid element. It can be given as

 Ge ¼ RT1

RT2

. . . RTn

T

ð11Þ

where

Rs ¼ ½ Q 1 ðsÞ Q 2 ðsÞ . . . Q d ðsÞ ;

  Q k ðsÞ ¼ Nijk ðsÞ

ði; j ¼ x; y; k ¼ 1; 2; 3; 4; s ¼ 1; 2; . . . ; nÞ ð12Þ and n is the number of the fine-grid nodes within the unit cell.

in X

Nixx ðxÞ ¼ /ixx ;

Niyx ðxÞ ¼ 0 on @ X

ð7Þ

i ¼ 1; 2; 3; 4 where

L is the elasticity operator and satisfies    Lu ¼ div D : 12 ru þ ðruÞT , D is the fourth-order elasticity tensor. Ni can be expressed as



Ni ¼

Nixx

Nixy

Niyx

N iyy



ð8Þ

Eq. (7) can be solved on the unit cell X with the standard FEM by applying the following boundary conditions

Fig. 5. Illustration of multiscale base functions N 1xx and N p1x .

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3.3. Co-rotational formulation in the macroscale computation

4   X ~2 þ V ~2 U

3.3.1. Element kinematics The geometrically nonlinear behaviors of fluid actuated cellular structure can be performed on the coarse scale mesh (coarse-grid elements). Here the co-rotational approach is introduced to solve the large displacement – small strain problems on the coarse-grid elements. (The presentation is based on the formulations of Battini (2008)). The main idea of the co-rotational approach is that the large deformation of the coarse-grid element is decomposed into a rigid motion in global coordinate system and a small deformation in the convected (local) coordinate system. The pure deformation is captured at the level of local reference frame, whereas the geometric nonlinearity induced by the large rigid-body motion is incorporated in the transformation matrices relating local and global internal force vectors and tangent stiffness matrices. Since the pure deformation is small, the linear theory can be used in the local coordinate system, which means that existing linear coarse-grid elements can be used in local coordinate system (Battini, 2008). There are two reference frames, as illustrated in Fig. 4. The first one is the global coordinates X-Y and the other one is the convected coordinate system xl  yl which is fixed on the coarse-grid element and always moves with it. In this paper, all the variables with subscript l are described in the local coordinate system xl  yl and other variables without subscript l are described in the global coordinate system X-Y. The undeformed configuration is called initial configuration. The element kinematics of the multiscale coarse-grid element is described in Fig. 4. Here we denote the global coordinates and global displacement of the coarse-node i by Ri ¼ ðX i ; Y i Þ and Ui ¼ ðU i ; V i Þ, respectively. Thus, the origin of the local coordinates (o, xl ; yl Þ can be written as follows:

i¼1

i

R o ¼ ðX o ; Y o Þ Xo ¼

4 1X Xi 4 i¼1

ð13Þ Yo ¼

4 1X Yi 4 i¼1

is minimized. Substituting Eq. (17) in Eq. (18), taking the variation with respect to h and setting the resulting expression equal to zero, we can obtain:

P4 ½xi ðY i þ V i  Y o  V o Þ  yi ðX i þ U i  X o  U o Þ tan h ¼ Pi¼1 4 ½ i¼1 xi ðX i þ U i  X o  U o Þ þ yi ðY i þ V i  Y o  V o Þ ð19Þ Two values for the rotation h (i.e., h and h þ pÞ can be calculated from the Eq. (19). Here the one corresponding to the minimization of expression (18) will be chosen as the target rotation angles. 3.3.2. Derivation of the equivalent tangent stiffness matrices In order to deduce the equivalent tangent stiffness matrix, here we denote the local and global displacement vectors of a coarse-grid element by

Ulc ¼



T

~1 U

~1 V

~2 U

~2 V

~3 U

~3 V

~4 U

~4 V

Ugc ¼ f U 1

V1

U2

V2

U3

V3

U4

V4 g

ð20Þ

T

The local and global internal force vectors acting on the l g coarse-grid element are denoted as f c and f c , respectively. Thus, the variation equation of a single coarse-grid element in the global coordinate can be written as: g

df c ¼ Kgc dUgc

ð21Þ

Kgc

where is the equivalent tangent stiffness matrix in the global coordinate system. The virtual work of the coarse-grid element in the global and local systems satisfies:

 T

T g l d Ugc f c ¼ d Ulc f c

ð22Þ

Using the transformation from local to global coordinate system, the internal force in the local coordinate system can be written as: l

ð14Þ

ð18Þ

i

g

df c ¼ TTc df c

ð23Þ

The rigid translation is defined by Uo , the global displacement of the point o, which is calculated by:

where Tc is the transformation matrix relating local and global coordinate system. According to Eqs. (17) and (19), Tc can be expressed as:

Uo ¼ ðU o ; V o Þ

Tc ¼ HGT ;

4 1X Uo ¼ Ui 4 i¼1

ð15Þ

where

4 1X Vo ¼ Vi 4 i¼1

ð16Þ

The displacement at thecoarse-grid nodes in the local  ~i ¼ U ~ i; V ~ i ; i ¼ 1; 2; 3; 4 is calcucoordinate system U lated by

"

~i U V~ i

#

 ¼

cos h

sin h

 sin h cos h



H ¼ I  AB

Xi þ Ui  Xo  Uo Yi þ Vi  Yo  Vo



 

xi

ð24Þ 

G ¼ diag ðN; N; N; NÞ;

N¼

cos h  sin h sin h

 ð25Þ

sin h

A ¼ f yd1 xd1 yd2 xd2 yd3 xd3 yd4 xd4 g;

~ i þ xi xdi ¼ U ~i þ vi y ¼V di

ð26Þ



yi ð17Þ

where h is the rigid rotation. Its value is taken such that the square of the Euclidean norm of the local nodal ~ i (Battini, 2008) displacement U

B ¼ P4

1

i¼1 ðxi xdi þ yi ydi Þ

f y1 x1 y2 x2 y3 x3 y4 x4 g ð27Þ

It is also available for the local and global nodal displacement vectors Ugc and Ulc , i.e.

J. Lv et al. / Mechanics of Materials 79 (2014) 1–14

Ugc ¼ TTc Ulc

ð28Þ Kgo

To get the tangent stiffness matrix in global coordinate system, performing a variation calculation for Eq. (23) yields: g

l

l

df c ¼ TTc df c þ dTTc f c

ð29Þ

The two terms of the above equation can be expressed as: l

ð30Þ

l

ð31Þ

TTc df c ¼ K1c dUgc dTTc f c ¼ K2c dUgc

7

3.5. Incremental iteration algorithm of the macro scale computation The equivalent tangent stiffness matrix of the whole structure on the coarse-grid mesh can be calculated by the expression: c KgC ¼ KNc¼1 Kgc

ð39Þ

Nc c¼1

where K denotes the matrix assembled operator; N c is the number of the coarse-grid elements. The total internal g nodal forces f C and external internal forces FgC applied on the coarse-grid mesh can be assembled in a similar way:

Thus, combining Eqs. (21), (29), (30) and (31), the equivalent tangent stiffness matrix Kgc can be calculated by:

c f C ¼ KNc¼1 fc

ð40Þ

Kgc ¼ K1c þ K2c

c FgC ¼ KNc¼1 Fgc

ð41Þ

The matrix

K1c

¼

ð32Þ

K1c

and

K2c

can be derived as:

ð33Þ

  l @ TTc f c K2c ¼ @Ugc l

ð34Þ

fc

Substituting Eq. (24) into Eq. (34), we can obtain:

h i K2c ¼ G ST B  BT SH GT

ð35Þ

where

n1

n4

n3

n6

n5

n8

n7 g l

ð36Þ l

Here ni is the ith member of the vector HT f c , and HT f c can be written as: l

HT f c ¼ f n1

n2

n3

n4

n5

n6

n7

n8 g

ð37Þ

In the fluid actuated cellular structure, the volume changes of the fluid in each motor cell are regarded as the driving forces. As indicated in Fig. 3, these volume changes are applied within the inclusions at the small scale level. Here an equivalent method is proposed to transfer the microscopic volume expansion of the fluid to the macroscopic equivalent forces on the coarse-grid mesh (Zhang et al., 2010). As illustrated in Fig. 6, the main idea of the equivalent method is that the influence of the volume change of the fluid DV f is equivalent to the superposition of the coarse-grid load FlE (Fig. 6(B)) and the perturbed results (Fig. 6(C)). Some special boundary conditions are applied to the unit cells to calculate the perturbed results and coarse-grid load. More detail of the computation can be found in our previous work (Zhang et al., 2010). It should be noted that the equivalent procedure is performed in the local coordinates. For the geometrically nonlinear problems, the local coarse-grid load FlE is transferred to the global coordinates by the following expansion:

FgV

Fgc ¼ TTc Flc ;

ð38Þ

where denotes the equivalent coarse-grid load of the fluid actuator in the global coordinates.

Flc ¼ NT Fls

ð42Þ

where Flc denotes the external force act on the coarse-grid element, which including the equivalent force of the fluid actuators; Fls is the microscopic force acting on the sub-grid mesh of the unit cells. Finally, the geometrically nonlinear behavior of the fluid actuated structure can be solved on the coarse-scale mesh. The Newton–Raphson algorithm is utilized as the iteration algorithm. In the kth iterative of nth load step,

k the balance force DFgC n of the coarse-grid mesh in the global coordinates can be expressed as:



DFgC

K



g k ¼ FgC n  f C n

n



ð43Þ

g k where is external force and f C n is the internal force. Thus, the incremental displacement vector can be calculated by the following expression:



3.4. Equivalent method for volume changes of the fluid

FgV ¼ TTc FlV

g

and,

TTc Klc Tc

S ¼ f n2

g

KgC

k

n



FgC n

DUgC

k n

k ¼ DFgC n

ð44Þ

Updating the displacement vector by



UgC

kþ1 n

k

k ¼ UgC n þ DUgC n

ð45Þ

The iteration will be performed until the following expression is satisfied:



g k

DFC n

g  < er

F

ð46Þ

C n

where kk denotes the L2 norm of the vectors; er is the control parameter of the relative error. 3.6. Downscale computations The relationship between micro and macro scales is established by multiscale base functions. Downscale computations can be performed to recover the microscale response with these multiscale base functions. Firstly, the macroscale nodal displacement results in the global coordinate system can be calculated by the macroscale computation as illustrated in Section 3.5. Thus, the local nodal displacement Ulc of the each coarse-grid element can be calculated by Eq. (17). On the other hand, the fluid

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Fig. 6. Equivalent method of the volume changes of the fluid in the cells (Zhang et al., 2010).

expansions acting within the inclusion have been transformed into the nodal forces on the coarse-grid element by an equivalent method (see Section 3.4). Therefore, the influence of the fluid expansion at the microscale level should be also taken into account in the downscale computation. Designating the influence of the fluid expansion on the displacement field as Due and the fluid pressure field as DPe . The results of the microstructures on the fine scale can be recovered by:

ue ¼ Ge Ulc þ Due

ð47aÞ

P ¼ Np Ulc þ DPe

ð47bÞ

The microscale strain and stress can be calculated by:

ee ¼ Be ue

ð48Þ

re ¼ De ee ¼ De Be ue

ð49Þ

where ee and re denote the strain and stress of the finegrid element,respectively; Be is the element strain–displacement matrix; And De is the fourth-order elasticity tensor of the fine-grid element. 4. Numerical examples In this section, we proposed three examples to validate the accuracy and efficiency of the multiscale co-rotational method developed. The bio-inspired fluidic cellular structures are idea candidates for such a philosophy as they can role of load carrying members and at the same time act as actuators. Thus, in the first example we will focus on the load carrying ability of the cellular structures by apply large external forces on them. In the second example the actuation ability of the cellular structure is tested by control the amount of the fluid flow into or out of the closed inclusion inside. At the last example, the snap motion of a Venus flytrap-like fluid actuated cellular structure is studied. The plane strain as well as large displacement – small strain state is assumed in all the problems. All the parameters used in the examples are dimensionless. Here we denote the multiscale co-rotational method as ‘MCM’. On the other hand, the comparison of results is made with respect to solutions provided by the commercial nonlinear code ABAQUS as reference. In geometrically nonlinear problems, the ABAQUS solutions are based on

the Total Lagrangian (TL) kinematic description (ABAQUS, 2004). All the calculations of ABAQUS here are carried out on the fine scale mesh of the structures. 4.1. Example 1: Fluidic cellular structure under external forces As illustrated in Fig. 7(A), the fluidic cellular structure is composed of 4  60 periodically arranged unit cells. The structure can be considered as a cantilever beam, of which the left side is fixed. The uniform distributed forces (F) of 5.0  106 act along the right side of the structure. The unit cell contains a square fluid inclusion, and the sub-grid mesh on the cell, which contains 128 sub-grid elements, can be found in Fig. 7(B). The Young’s modulus of the matrix of the cells is 2.0  109, while the Possion’s ratio is 0.3. The multiscale co-rotational analysis is applied on the structure meshed with 240 coarse-grid elements, as illustrated in Fig. 7(C). At the same time, the standard fine scale analysis by ABAQUS is performed on the fine scale mesh of the structure. Both the two analyses are divided into 50 steps. Fig. 8 plots the bottom line of the fluidic cellular beam structure at different load steps in the global coordinates X-Y. It can be observed that the results obtained by MCM and those calculated by ABAQUS exhibit a good agreement with each other. Since the fine-scale response of the microstructure can be easily recovered by MCM with the downscaling technique, we can further compare the stress response of the matrix and the fluid pressure calculated by MCM with those obtained by ABAQUS. The distribution of the microscopic von Mises stress in the deformed structure is shown in Fig. 9. Moreover, the stress results of some local domain of the structure, such as the region where boundary conditions are applied, are magnified. Compared with the reference solution by ABAQUS, the results obtained by MCM have high accuracy. On the other hand, the fluid pressure will be generated in each fluid inclusion when the fluidic structure is subjected to external forces. This pressure is generally important in real applications. For instance, the pressure can greatly affect the driving unit that transports fluid into the inclusions, such as the ion transport system of the bioinspired nastic structure (Freeman and Weiland, 2009; Matthews et al., 2006; Zhang and Lv, 2011). In MCM, the fluid pressure can be calculated by Eq. (47b) in the downscaling computation. To evaluate the accuracy of pressures

J. Lv et al. / Mechanics of Materials 79 (2014) 1–14

9

Fig. 7. (A) Fluidic cellular structure with 4  60 fluid inclusions; (B) Sub-grid mesh of the unit cell; (C) Coarse-grid mesh of the whole structure.

0 -500 -1000

Y

-1500 -2000 -2500 -3000 MCM

ABAQUS

-3500 0

500

1000

1500

2000

2500

3000

3500

X Fig. 8. Comparison of large deformations about the bottom line of the fluidic cellular beam structure under external forces at different load steps in the global coordinate system X-Y.

obtained by the multiscale co-rotational method, the L2 norm is defined as follows

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 12ffi u u X Nc MCM ABAQUS u1 @Pi  P i A LP2 ¼ t N i max P ABAQUS i

ð50Þ

where PMCM and PABAQUS denote the fluid pressure comi i puted by MCM and ABAQUS, respectively. The L2 norm of the results for some increment steps of the analysis are given in Table 1. It can be found that the proposed MCM can accurately calculate the fluid pressure inside the micro-inclusions through the downscaling computation. 4.2. Example 2: Fluid actuated cellular beam structure As mentioned above, the fluidic cellular structure can take the role of actuators to perform shape morphing by controlling the pressure or volumes of the fluid in the closed cells. In this example, we will validate the accuracy

of the multiscale co-rotational method to simulate the geometrically nonlinear behaviors of the cellular structure subjected to volume changes of the fluid. As illustrated in Fig. 10, the cellular structure considered consists of 4  108 periodically arranged unit cells. Each unit cell contains a fluid-filled inclusion (see Fig. 10(A)). The mechanical properties of the matrix in the structure and fluid in the cells are the same as those used in Example 1. The upper left and lower left corners of the structure are mechanically constrained. The volume changes of the fluid in the inclusions vary according to the locations of the inclusions. As illustrated in Fig. 10(B), the volume expansion of the fluid in the each inclusion that distributed at the bottom of the structure is 300, while the volume reduction of the fluid in each inclusion that distributed at the top of the structure is 150. In the other inclusions, the volume changes of the fluid change linearly along the y direction. According to the periodicity of the fluid inclusions, the fluidic structure are meshed with 4  108 coarse-grid element. Each coarse-grid element contains a unit cell that is meshed with 384 fine-grid element (see Fig. 10(A)). Both MCM and ABAQUS are applied to solve the large displacement-small strain behavior of the structure. Both the two calculations are divided into 50step. The transient responses of the cellular structure (bottom line) in the global coordinates X-Y are plot in Fig. 11. The von Mises results of the structure at the 50th step are given in Fig. 12. It can be observed that the displacement results obtained by MCM, especially those calculated in the first few steps, agree well with those calculated by ABAQUS. In these steps, relative small volume increment of the fluid are added in the inclusions of the cellular structures. When the volume increment become larger in the last few steps, there will be some errors occur. Moreover, we can find that these errors are slightly larger than those observed in the Examples 1 (see Fig. 8) where the fluidic cellular structures are subjected to external forces. The main reason that caused these errors is the strong boundary effects, which generally exists in the multiscale strategies. These strong boundary effects will be more obvious when the large volume expansions are applied to those closed liquid cells.

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2.80E+08 2.60E+08 2.40E+08 2.20E+08 2.00E+08 1.80E+08 1.60E+08 1.40E+08 1.20E+08 1.00E+08 8.00E+07 6.00E+07 4.00E+07 2.00E+07

Fig. 9. Von Mises stress distributions of the fluidic cellular beam under external forces.

Table 1 L2 norm of the fluid pressure calculated by the MCM. Time steps

Errors (%)

Step Step Step Step Step

0.1826 0.1978 0.2060 0.2104 0.2131

10 20 30 40 50

For the linear elastic problems, the high order coarse-grid elements of which a node is distributed at the centre of the element are believed to effectively reduce the boundary effects (Zhang et al., 2010). Thus, it will be interesting works to introduce such high order element into the multiscale co-rotational method. 4.3. Example 3: Venus flytrap-like fluid actuated cellular structure In this example, a Venus flytrap-like fluid actuated cellular structure is utilized to validate the methodology

proposed. As depicted in Fig. 13, the L-shaped design is composed of 124  4 periodically arranged unit cells which contain a circular fluid inclusion. The two beams of the structure is designed to imitate leafs of the Venus flytrap. To simulate the shape morphing of the Venus flytrap, different volume changes of the fluid are applied to inclusions inside the structure. According to the mechanism of the Venus flytrap, here we make the water flow into the motor cells that closer to the outer surface of the structure, and cause the water flow out of the motor cells that near the inner surface of the L-shaped structure. This will result in the expansion of the outer surface and shrinkage of inner surface. At last, a closure motion will be observed from the structure. The whole structure are meshed with 124  4 coarsegrid elements, while the sub-grid mesh of the unit cells are the same as those in Example 2, see Fig. 13(A). Here we utilize MCM proposed in current research to simulate the structure. And the results are compared to those reference solution results that computed by ABAQUS to validate the accuracy of the multiscale method. The mechanical properties of the matrix and fluid properties are still the same as those used in Example 1.

Fig. 10. Fluidic cellular beam structure subjected to volume changes (DV f Þ in the structure.

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J. Lv et al. / Mechanics of Materials 79 (2014) 1–14

7000 6000

MCM

of the multiscale method. Furthermore, the von Mises stresses on the fine-scale mesh of the structures are indicated in Fig. 14. To more clearly display the response of the microstructures, the stresses of some local part in the structure are magnified. It can be found that the maximum stress occurs at the inner wall of the fluid inclusion where the fluid expansions are applied. These results indicate that the stresses calculated by the downscale computation of MCM keep good consistence with those calculated by ABAQUS, which further indicates the accuracy of MCM.

ABAQUS

5000

Y

4000 3000 2000 1000 0

0

1000

2000

3000

4000

5000

6000

7000

X

Fig. 11. Comparison of large deformations about the bottom line of the fluid actuated cellular beam structure at different load steps in the global coordinate system X-Y.

The whole solutions of the two methods are separately divided into 50 steps. Some transient deformations (20th, 35th and 50th step) of the Venus flytrap-like cellular structure predicted by the two methods are given in Table 2. It can be observed that the Venus flytrap-like cellular structure can offer a large displacement and small strain closure motions when the fluid is controlled to flow into or out of the inclusions inside the structure. These results imply that morphing structures with large shapes and nonlinear stiffness requirements can be designed by tailor the geometry, internal pressures (fluid) and materials properties of the basic closed liquid cells. Moreover, we can find that the coarse scale deformations of the structure predicted by MCM agree well with those calculated by ABAQUS, which indicates the validity

Fig. 13. The Venus flytrap-like fluid actuated cellular structure: (A) Subgrid mesh of the basic unit cell. (B) Illustration of the fluid changes in the inclusions. (C) Boundary condition applied on the structure.

5.50E+08 5.14E+08 4.79E+08 4.43E+08 4.07E+08 3.71E+08 3.36E+08 3.00E+08 2.64E+08 2.29E+08 1.93E+08 1.57E+08 1.21E+08 8.57E+07 5.00E+07

Fig. 12. Von Mises stress distributions of the fluid actuated cellular beam structure.

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J. Lv et al. / Mechanics of Materials 79 (2014) 1–14

Table 2 Deformed configurations of the fluid actuated cellular structure predicted by the MCM and ABAQUS. Solvers

Step 20

Step 35

Step 50

ABAQUS

MCM

Fig. 14. Von Mises stress results (35th step) on the fine scale meshes of the Venus flytrap-like fluidic cellular structure.

4.4. Cost of the multiscale co-rotational methods

the number of the elements is approximately equivalent to the node number. The direct method, such as the standard FEM, is directly applied on the fine scale mesh. Thus, the memory needed in FEM is

The purpose of MCM proposed is to solve the large nonlinear problems that are hard to handle by direct method even with modern supercomputers. Here, we roughly estimate the cost of the computer memory and CPU time that consumed in MCM and compare them with those used in the direct methods. Let us consider a structure meshed with N c coarse-grid elements and each coarse-grid element consists of m subgrid elements. Then, the total number of the elements at the fine scale mesh is N c  m. For simplicity, we assume that

C FEM ¼ Oða  Nc  mÞ

ð51Þ

where a denotes the number of degrees of freedom of one node; O, the function expression of the operators, is estimated as the power of degrees of freedom if the serial computer is utilized. For the two-dimensional fluidic cellular problems, a is equal to 2. On the other hand, the macroscale calculation in MCM is performed on the coarse grid

Table 3 Comparison of the computational efficiency of the numerical examples. Case Example 1 Example 2 Example 3

Memory cost of MCM 7

3.13  10 2.56  108 2.95  108

Memory cost of FEM 9

7.46  10 1.10  1011 1.45  1011

Time of MCM(s)

Time of ABAQUS(s)

9.5 s 879 s 391 s

300 s 2175 s 2520 s

J. Lv et al. / Mechanics of Materials 79 (2014) 1–14

element and the multiscale base functions only need to be computed once on the sub-grid mesh of the unit cell. Thus, the memory consumed in MCM is

C MCM ¼ Oða  Nc Þ þ Nc  Oða  mÞ

ð52Þ

Utilizing the equations mentioned above, the memory costs of the methods used in the examples are calculated and the results are listed in Table 3. It can be observed that the MCM can save a big computer memory, especially while it is used to solve large scale problems. Furthermore, here we approximately compare the CPU times cost by MCM and the direct methods (e.g., ABAQUS) for the nonlinear problems. In MCM, the multiscale base functions need to be calculated only once, since the fluidic cellular structure are assumed to be composed of periodical arranged unit cells. Thus, the time for this preprocessing computation can be omitted in the nonlinear problems. For the nonlinear iteration computation part, MCM is applied on the coarse-grid element, while the direct methods are carried out on the fine-grid element. On the other hand, the total number of the iterative steps is almost the same for the two methods. Therefore, it can be concluded that MCM will be much more efficient than the direct method. The actual CPU times used in the three examples proposed above are illustrated in Table 3. We can observe that MCM can significantly reduce the computing time. 5. Conclusions The algorithm of the multiscale finite element method for linear problems, which is proposed in the reference, is extended for geometrically nonlinear multiscale analysis of the fluid actuated cellular structures. In the upscale part of the multiscale co-rotational method, the multiscale base functions are numerically constructed in the convected coordinate system to establish the relationship between the macroscopic displacements and the microscopic stress. The fluid changes of the motor cells at the microscale are equivalent to the nodal forces applied on the coarse-grid elements. Furthermore, the co-rotational approach is utilized in the macroscale computation to decompose the geometrically nonlinear motion of the coarse grid element into rigid-body and pure deformation components. The equivalent tangent stiffness matrices of the coarse-grid element are derived for the large displacement-small strain problems. Thus, the Newton–Raphson scheme only needs to be performed on the coarse-grid mesh of the fluid actuated cellular structure. Three numerical examples are studied and the results demonstrate that the multiscale co-rotational method can accurately and efficiently solve the geometrically nonlinear problems of the fluid actuated cellular structure. Acknowledgements This research is supported by the National Natural Science Foundation (11302040, 11232003, 11072051, 11402178), the National Key Basic Research Special Foundation of China

13

(2010CB832704), the 111 Project (No.B08014), Fundamental Research Funds for the Central Universities of China (2013DUT13RC(3)10) and China Postdoctoral Science Foundation (2014T70244, 2013M530908, 2014M552078).

References ABAQUS, 2004. Analysis User’s Manual, vols. II, III, IV, and VI, ABAQUS Inc, Providence, Rhode Island. Argyris, J., 1982. An excursion into large rotations. Comput. Methods Appl. Mech. Eng. 32, 85–155. Babuska, I., Osborn, E., 1983. Generalized finite element methods: their performance and their relation to mixed methods. SIAM J. Numer. Anal. 20, 510–536. Battini, J.M., 2008. A non-linear co rotational 4-node plane element. Mech. Res. Commun. 35, 408–413. Barbarino, S., Bilgen, O., Ajaj, R.M., Friswell, M.I., Inman, D.J., 2011. A review of morphing aircraft. J. Intell. Mater. Syst. Struct. 22, 823–877. Barrett, R., Barrett, C., 2014. Biomimetic FAA-certifiable, artificial muscle structures for commercial aircraft wings. Smart Mater. Struct., 074011. Belytschko, T., Hsieh, B.J., 1973. Nonlinear transient finite element analysis with convected coordinates. Int. J. Numer. Meth. Eng. 7, 255–271. Bowman, J., Sanders, B., Cannon, B., Kudva, J., 2007. Development of next generation morphing aircraft structures, in: 48th AIAA/ASME/ASCE/ AHS/ASC Structures, Structural Dynamics, and Materials Conference, Honolulu, Hawaii. Crisfield, M.A., 1990. A consistent co-rotational formulation for nonlinear, three-dimensional, beam-elements. Comput. Methods Appl. Mech. Eng. 81, 131–150. Efendiev, Y., Hou, T.Y., 2009. Multiscale Finite Element Methods Theory and Applications. Springer, New York. Felippa, C.A., Haugen, B., 2005. A unified formulation of small-strain corotational finite elements: I. Theory. Comput. Methods Appl. Mech. Eng. 194, 2285–2335. Forterre, Y., Skotheim, J., Dumais, J., Mahadevan, L., 2005. How the Venus flytrap snaps. Nature 433, 421–425. Freeman, E., Weiland, L.M., 2009. High energy density nastic materials: parameters for tailoring active response. J. Intell. Mater. Syst. Struct. 20, 1–12. Gomez, J.C., Garcia, E., 2011. Morphing unmanned aerial vehicles. Smart Mater. Struct. 20, 103001. Guiducci, L., Fratzl, P., Bréchet, Y.J.M., Dunlop, J.W.C., 2014. Pressurized honeycombs as soft-actuators: a theoretical study. J.R. Soc. 11, 20140458. Hou, T.Y., Wu, X.H., Cai, Z.Q., 1999. Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comput. 68, 913–943. Li, S., Wang, K., 2013. Synthesizing fluidic flexible matrix composite-based multicellular adaptive structure for prescribed spectral data. J. Intell. Mater. Syst. Struct. http://dx.doi.org/10.1177/1045389X13505254. Luo, Q., Tong, L., 2013a. Adaptive pressure-controlled cellular structures for shape morphing I: design and analysis. Smart Mater. Struct. 22, 055014. Luo, Q., Tong, L., 2013b. Adaptive pressure-controlled cellular structures for shape morphing: II. Numerical and experimental validation. Smart Mater. Struct. 22, 055015. Lv, J., Zhang, H.W., Chen, B.S., 2014. Shape and topology optimization for closed liquid cell materials using extended multiscale finite element method. Struct. Multidiscip. Optim. 49, 367–385. Lv, J., Zhang, H.W., Yang, D.S., 2013. Multiscale method for mechanical analysis of heterogeneous materials with polygonal microstructures. Mech. Mater. 56, 38–52. Ma, L.H., Rolfe, B.F., Yang, Q., Yang, C., 2011. The Configuration evolution and macroscopic elasticity of fluid-filled closed cell composites: micromechanics and multiscale homogenization modelling. CMESComput. Model. Eng. 79, 131–158. Ma, L.H., Yang, Q.S., Yan, X.H., Qin, Q.H., 2014. Elastoplastic mechanics of porous materials with varied inner pressures. Mech. Mater. 73, 58– 75. Matthews, L., Sundaresan, V.B., Giurgiutiu, V., Leo, D.J., 2006. Bioenergetics and mechanical actuation analysis with membrane transport experiments for use in biomimetic nastic structures. J. Mater. Res. 21, 2058–2067. Pagitz, M., Bold, J., 2013. Shape-changing shell-like structures. Bioinspir. Biomim. 8, 016010.

14

J. Lv et al. / Mechanics of Materials 79 (2014) 1–14

Pagitz, M., Hühne, C., 2014. Compliant Pressure Actuated Cellular Structures, pp. 1–30. arXiv1403.2197. Pagitz, M., Lamacchia, E., Hol, J.M.A.M., 2012. Pressure-actuated cellular structures. Bioinspir. Biomim. 7, 016007. Philen, M.K., Shan, Y., Prakash, P., Wang, K.W., Rahn, C.D., Zydney, A.L., Bakis, C.E., 2007. Fibrillar network adaptive structure with iontransport actuation. J. Intell. Mater. Syst. Struct. 18, 323–324. Simo, J.C., 1985. A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Comput. Methods Appl. Mech. Eng. 49, 55– 70. Stanewsky, E., 2001. Adaptive wing and flow control technology. Prog. Aerosp. Sci. 37, 583–667. Shan, Y., Philen, M.P., Bakis, C.E., Wang, K.W., Rahn, C.D., 2006. Nonlinearelastic finite axisymmetric deformation of flexible matrix composite membranes under internal pressure and axial force. Compos. Sci. Technol. 66, 3053–3063. Shan, Y., Philen, M., Lotfi, A., Bakis, C.E., Rahn, C.D., Wang, K.W., 2008. Variable stiffness structures utilizing fluidic flexible matrix composites. J. Intell. Mater. Syst. Struct. 20, 443–456.

Stahlberg, R., 2009. The phytomimetic potential of three types of hydration motors that drive nastic plant movements. Mech. Mater. 41, 1162–1171. Sundaresan, V.B., Leo, D.J., 2006. Chemo-mechanical model for actuation based on biological membranes. J. Intell. Mater. Syst. Struct. 17, 863– 870. Vasista, S., Tong, L., 2012. Design and testing of pressurized cellular planar morphing structures. AIAA J. 50, 1328–1338. Vos, R., Barrett, R., 2011. Mechanics of pressure-adaptive honeycomb and its application to wing morphing. Smart Mater. Struct. 20, 094010. Vos, R., Barrett, R., Romkes, A., 2011. Mechanics of pressure-adaptive honeycomb. J. Intell. Mater. Syst. Struct. 22, 1041–1055. Wempner, G.A., 1969. Finite elements, finite rotations and small strains of flexible shells. Int. J. Solids Struct. 5, 117–153. Zhang, H.W., Lv, J., 2011. Two-scale model for mechanical analysis of nastic materials. J. Intell. Mater. Syst. Struct. 22, 593–609. Zhang, H.W., Lv, J., Zheng, Y., 2010. A new multiscale computational method for mechanical analysis of closed liquid cell materials. CMESComput. Model. Eng. 68, 55–93.