AGENERAL THEORETICAL SCHEME FOR SHAPE -PROGRAMMING OF INCOMPRESSIBLE HYPERELASTIC SHELLS THROUGH DIFFERENTIAL GROWTH

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AGENERAL THEORETICAL SCHEME FOR SHAPE-PROGRAMMING

OF INCOMPRESSIBLE HYPERELASTIC SHELLS THROUGH

DIFFERENTIAL GROWTH

A PREPRINT

Zhanfeng Li1,3, Jiong Wang∗,1,2, Mokarram Hossain3, Chennakesava Kadapa4

1School of Civil Engineering and Transportation, South China University of Technology, Guangzhou, China

2State Key Laboratory of Subtropical Building Science, South China University of Technology, Guangzhou, China

3Zienkiewicz Centre for Computational Engineering (ZCCE), Swansea University, Swansea, United Kingdom

4School of Engineering and the Built Environment, Edinburgh Napier University, Edinburgh, United Kingdom

October 13, 2022

ABSTRACT

In this paper, we study the problem of shape-programming of incompressible hyperelastic shells

through differential growth. The aim of the current work is to determine the growth tensor (or

growth functions) that can produce the deformation of a shell to the desired shape. First, a consistent

ﬁnite-strain shell theory is introduced. The shell equation system is established from the 3D governing

system through a series expansion and truncation approach. Based on the shell theory, the problem

of shape-programming is studied under the stress-free assumption. For a special case in which the

parametric coordinate curves generate a net of curvature lines on the target surface, the sufﬁcient

condition to ensure the vanishing of the stress components is analyzed, from which the explicit

expression of the growth tensor can be derived. In the general case, we conduct the variable

changes and derive the total growth tensor by considering a two-step deformation of the shell. With

these obtained results, a general theoretical scheme for shape-programming of thin hyperelastic

shells through differential growth is proposed. To demonstrate the feasibility and efﬁciency of the

proposed scheme, several nature-inspired examples are studied. The derived growth tensors in these

examples have also been implemented in the numerical simulations to verify their correctness and

accuracy. The simulation results show that the target shapes of the shell samples can be recovered

completely. The scheme for shape-programming proposed in the current work is helpful in designing

and manufacturing intelligent soft devices.

Keywords

Hyperelastic shell

Differential growth

Shape-programming

Theoretical scheme

Numerical simulations

1 Introduction

Growth of soft biological tissues and swelling (or expansion) of soft polymeric gels are commonly observed in nature

[Ambrosi et al., 2011, Liu et al., 2015]. Due to the inhomogeneity or incompatibility of the growth ﬁelds, soft material

samples usually exhibit diverse morphological changes and surface pattern evolutions during the growing processes,

which is referred to as the ‘differential growth’ and has attracted extensive research interest in recent years [Goriely

and Ben Amar, 2005, Li et al., 2012, Kempaiah and Nie, 2014, Huang et al., 2018]. To fulﬁll the requirements of

engineering applications, it is usually desired that the conﬁgurations of soft material samples are controllable during the

growing processes, such that certain kinds of functions are realized. This goal can be achieved through sophisticated

composition or architectural design in the soft material samples. The technique is known as ‘shape-programming’ [Liu

et al., 2016, van Manen et al., 2018], which has been utilized for manufacturing a variety of intelligent soft devices, e.g.,

∗Corresponding author. Email: ctjwang@scut.edu.cn Tel.: +86 13926459861, Fax: +86 21-87114460.

arXiv:2210.06202v1 [math.NA] 9 Aug 2022

A general theoretical scheme for shape-programming of shells A PREPRINT

biomimetic 4D printing of ﬂowers [Gladman et al., 2016], pressure-actuated deforming plate [Siéfert et al., 2019], pasta

with transient morphing effect [Tao et al., 2021], and polymorphic metal-elastomer composite [Hwang et al., 2022].

From the viewpoint of solid mechanics, soft materials can be treated as certain kinds of hyperelastic materials. The

growth ﬁeld in a soft material sample is usually modeled by incorporating a growth tensor. Due to the residual stresses

triggered by the incompatibility of the growth ﬁeld, as well as the external loads and boundary restrictions, the sample

also undergoes elastic deformations. Thus, the total deformation gradient tensor should be decomposed into an elastic

strain tensor and a growth tensor [Kondaurov and Nikitin, 1987, Rodriguez et al., 1994, Ben Amar and Goriely, 2005].

The elastic incompressible constraint should also be adopted since the elastic deformations of soft materials are typically

isochoric [Wex et al., 2015, Kadapa et al., 2021]. Based on these constitutive and kinematic assumptions, the growth

behaviors of soft material samples can be studied by solving the system of mechanical ﬁeld equations. Because of the

inherent nonlinearities in the large growth-induced deformations, mechanical instabilities can also be triggered in the

soft material samples [Ben Amar and Goriely, 2005, Li et al., 2011, Goriely, 2017, Pezzulla et al., 2018, Xu et al.,

2020].

Despite the numerous studies on the growth behaviors of soft material samples, the majority of the modeling works pay

attention to the direct problem, i.e., determining the deformations of soft material samples when the growth ﬁelds are

speciﬁed. However, in order to utilize the shape-programming technique for engineering applications, one needs to

study the inverse problem. That is, how to determine the growth ﬁelds in the samples such that the current conﬁgurations

induced by differential growth can achieve any target shapes? This inverse problem has also been studied in some

previous works [cf. Dias et al., 2011, Jones and Mahadevan, 2015, Acharya, 2019, Wang et al., 2019a, Nojoomi et al.,

2021, Li et al., 2022, Wang et al., 2022]. In these works, the initial conﬁgurations of soft material samples usually have

the thin plate form. Although the shell form is more common in nature and engineering ﬁelds, it is seldom chosen as

the initial conﬁguration of the soft material samples due to the difﬁculties associated with modelling shell structures.

To achieve the goal of shape-programming, a prerequisite is to predict the relations between the growth ﬁelds and the

morphologies of soft material samples. It is thus of signiﬁcance to establish an efﬁcient and accurate mathematical

model by taking conﬁgurations of samples, material properties, boundary conditions and other factors into account. In

terms of shell theories for growth deformations, the Kirchhoff shell theory has been adopted to describe mechanical

behavior in growing soft membranes [Vetter et al., 2013, Rausch and Kuhl, 2014], which relies on ad hoc assumptions

of the stress components and deformation gradient. Another shell theory is proposed based on the non-Euclidean

geometry, where the deformation of samples is determined by the intrinsic geometric properties attached to surfaces,

such as the ﬁrst and second fundamental forms, and the applied growth ﬁelds [Souhayl Sadik et al., 2016, Pezzulla

et al., 2018]. In Song and Dai [2016], a consistent ﬁnite-strain shell theory has been proposed within the framework

of nonlinear elasticity, where the shell equation is derived from the 3D formulation through a series-expansion and

truncation approach. To apply this theory for growth-induced deformations, Yu et al. [2022] incorporated the growth

effect through the decomposition of the deformation gradient and derived the shell equation system for soft shell

samples.

In this paper, we aim to propose a general theoretical scheme for shape-programming of incompressible hyperelastic

shells through differential growth. Following the shell theory proposed in Yu et al. [2022], the shell equation system

is established from the 3D governing system, where a series expansion and truncation approach is adopted. To fulﬁll

the purpose of shape-programming, the shell equation system is tackled by assuming that all the stress components

vanish. Under this stress-free assumption, we ﬁrst consider a special case in which the parametric coordinate curves

generate a net of curvature lines on the target surface. By analyzing the sufﬁcient condition to ensure the vanishing of

the stress components, the explicit expression of the growth tensor is derived (i.e., the inverse problem is solved), which

depends on the intrinsic geometric properties of the target surface. In the general case that the parametric coordinate

curves cannot generate a net of curvature lines on the target surface, we conduct the variable changes and derive the

total growth tensor by considering a two-step deformation of the shell sample. Based on these results, a theoretical

scheme for shape-programming of hyperelastic shells is formulated. The feasibility and efﬁciency of this scheme are

demonstrated by studying several typical examples.

This paper is organized as follows. In section 2, the ﬁnite-strain shell theory for modeling the growth behaviors of thin

hyperelastic shells is introduced. In section 3, the problem of shape programming is solved and the theoretical scheme

is proposed. In section 4, some typical examples are studied to show the efﬁciency of the theoretical scheme. Finally,

some conclusions are drawn. In the following notations, the Greek letters

(α, β, γ...)

run from 1 to 2, and the Latin

letters

(i, j, k...)

run from 1 to 3. The repeated summation convention is employed and a comma preceding indices

(·),

represents the differentiation.

A general theoretical scheme for shape-programming of shells A PREPRINT

2 The ﬁnite-strain shell theory

In this section, we ﬁrst formulate the 3D governing system for modeling the growth behavior of a thin hyperelastic

shell. Then, through a series-expansion and truncation approach, the ﬁnite-strain shell equation system of growth will

be established.

2.1 Kinematics and the 3D governing system

We consider a thin homogeneous hyperelastic shell locating in the three-dimensional (3D) Euclidean space

. Within

an orthonormal frame

{O;e1,e2,e3}

, the reference conﬁguration of the shell occupies the region

Kr=Sr×[0,2h]

where the thickness parameter

is much smaller than the dimensions of the base (bottom) surface

and its local

radius of curvature. The position vector of a material point in the reference conﬁguration

is denoted by

X=Xiei

(cf. Fig. 1(a)). The geometric description of a shell has been systematically reported in the literature [cf. Ciarlet, 2005,

Steigmann, 2012, Song and Dai, 2016], which is simply introduced below.

First, a curvilinear coordinate system

{θα}α=1,2

is utilized to parametrize the base surface

of the shell in the

reference conﬁguration, which yields the parametric equation as

s(θα) = X1(θα), X2(θα), X3(θα),(θα)α=1,2∈Ωr.(1)

This parametric equation represents a continuous map from the region

Ωr⊂R2

to the surface

Sr⊂R3

. At a generic

point on

, the tangent vectors along the coordinate curves are given by

gα=s,α =∂s/∂θα

, which span the tangent

plane to the surface

at that point. The two vectors

{gα}α=1,2

are also referred to as the covariant basis of the tangent

plane. Another two vectors

{gα}α=1,2

on the tangent plane can be determined unambiguously through the relations

gα·gβ=δβ

, which form the contravariant basis of the tangent plane. Then, the unit normal vector of the surface

should be deﬁned by

n= (g1∧g2)/|g1∧g2|

(cf. Fig. 1(b)). By denoting

g3=g3=n

{gi}i=1,2,3

and

{gi}i=1,2,3

constitute two sets of right-handed orthogonal bases on the base surface

. The ﬁrst and second fundamental forms of

the surface Srcan be written into

Ir=gαβdθαdθβ,IIr=bαβdθαdθβ,(2)

where

gαβ =gα·gβ

and

bαβ =s,αβ ·n

are the fundamental quantities. Conventionally, the fundamental quantities are

also denoted by

Er=g11, Fr=g12 =g21, Gr=g22,

Lr=b11, Mr=b12 =b21, Nr=b22.(3)

(a)

base surface 

, 

 

, 

 

(b)













Figure 1: Position vector in the reference conﬁguration

: (a) reference conﬁguration of the shell and decomposition

of the position vector

; (b) the curvilinear coordinate system and the local covariant basis on the base surface

the shell.

As shown in Fig.1(a), the position vector

of a material point in the reference conﬁguration

of the shell can be

decomposed into

X=s(θα) + Zn(θα),0≤Z≤2h, (4)

where Zis the coordinate of the point along the normal direction n. Accordingly, the differential of Xyields that

dX=ds+Zdn+ndZ =gαdθα+Zn,αdθα+ndZ. (5)

A general theoretical scheme for shape-programming of shells A PREPRINT

From the Weingarten equation[Chen, 2017], we have

dn=n,αdθα= (n,α ⊗gα)gβdθβ=−Kds,(6)

where K=−n,α ⊗gαis the curvature tensor. The mean and Gaussian curvatures of the surface Srare given by

H=1

2tr (K), K = Det (K).(7)

By substituting (6) into (5), we obtain

dX=Uds+ndZ =ˆ

gαdθα+ndZ, (8)

where

U=gα⊗gα−ZK

and

gα=Ugα

. We further denote

gα=U−Tgα

, then

{ˆ

gα}α=1,2

and

{ˆ

gα}α=1,2

form

the covariant and contravariant base vectors at an arbitrary point in the shell, which are also orthogonal to

. Notice

that the thickness of the shell is much smaller than the radius of curvature of

; thus,

should be an invertible tensor.

From (8), the area element on the base surface and the volume element in the shell can be written into

dA =|g1∧g2|dθ1dθ2=qg11g22 −g2

12 dθ1dθ2,

dV = Det(U)dAdZ =1−2HZ +KZ2dAdZ.

(9)

Regarding area element on the lateral surface da, the local differential follows (8) that

Nda = (Uτ)×ndsdZ, (10)

where

is the outward normal unit vector of the lateral surface, and

is the unit tangent vector along the edge curve

∂Sr

of the base surface, and

is the arc-length variable on the edge curve

∂Sr

of the base surface. The norm of vector

(Uτ)×nis denoted by √gτsuch that da =√gτdsdZ.

Due to the growth effect and the external loads, the conﬁguration of the shell will deform from

to the current

conﬁguration Ktin R3. Within the orthonormal frame {O;e1,e2,e3}, the position vector of a material point in Ktis

denoted by x(θα, Z) = xi(θα, Z)ei. The deformation gradient tensor Fcan then be calculated through

F=x,α ⊗ˆ

gα+∂x

∂Z ⊗n= (∇x)U−1+∂x

∂Z ⊗n,(11)

where ∇is the in-plane 2-D gradient on the base surface Sr(∇x=x,α ⊗gα).

Following the basic assumption of growth mechanics [Kondaurov and Nikitin, 1987, Rodriguez et al., 1994, Ben Amar

and Goriely, 2005, Groh, 2022, Dortdivanlioglu et al., 2017, Mehta et al., 2021], the deformation gradient tensor

decomposed into

F=AG,(12)

where

is the elastic strain tensor and

is the growth tensor. It is known that the rate of growth is relatively slow

compared with the elastic response of the material, thus the distribution of the growth tensor

in the shell is assumed

to be given and does not change.

As the elastic deformations of soft materials (e.g., soft biological tissues, polymeric gels) are generally isochoric, the

following constraint equation should be adopted

R(F,G) = JGR0(A) = JG(Det(A)−1) = 0,(13)

where JG= Det(G). Furthermore, we suppose the material has an elastic strain-energy function

φ(F,G) = JGφ0(A) = JGφ0(FG−1).(14)

Then, the nominal stress tensor Scan be calculated through the constitutive equation

S=∂φ

∂F−p∂R

∂F=JGG−1∂φ0(A)

∂A−p∂R0(A)

∂A,(15)

where p(θα, Z)is the Lagrange multiplier associated with the constraint (13).

During the growing process, the hyperelastic shell satisﬁes the following mechanical equilibrium equation

Div S= (S,α)Tˆ

gα+∂S

∂Z T

n=0,in Sr×[0,2h].(16)

A general theoretical scheme for shape-programming of shells A PREPRINT

We suppose that the bottom and top surfaces of the shell are subjected to the applied traction

q±

, which yields the

boundary conditions

STn|Z=0 =−q−,STn|Z=2h=q+,on Sr.(17)

On the lateral surface

∂Sr×[0,2h]

of the shell, we suppose the applied traction is

q(s, Z)

, where

is the arc-length

variable of boundary curve ∂Sr. So, we also have the boundary condition

STN=q(s, Z) on ∂Sr×[0,2h].(18)

Eqs.

(13)

and

(16)

together with the boundary conditions

(17)

and

(18)

constitute the 3D governing system of the shell

model, which contains the unknowns {x, p}.

2.2 Shell equation system

Starting from the 3D governing system of the shell model, the shell equation system can be derived through a series-

expansion and a truncation approach. This approach has been proposed in Dai and Song [2014], Song and Dai [2016],

Wang et al. [2016] for developing the consistent ﬁnite-strain plate and shell theories without the growth effect. In Wang

et al. [2018], Yu et al. [2022], the ﬁnite-strain plate and shell theories of growth have also been established through this

approach. For the sake of completeness of the current paper, the key steps of this approach to derive the shell equation

system are introduced below (see Yu et al. [2022] for a comprehensive introduction). It should be noted that the derived

shell equation system can attain the accuracy of

O(h2)

. However, to fulﬁll the requirements of shape-programming in

the following sections, we only need to present the shell equation to the asymptotic order of O(h).

To eliminate the thickness variable

from the 3D governing system, we ﬁrst conduct the series expansions of the

unknowns as follows

x(θα, Z) =

n=0

x(n)

n!Zn+OZ3, p(θα, Z) =

n=0

p(n)

n!Zn+OZ3,(19)

where

(·)(n)=∂n(·)/ ∂Zn|Z=0

. According to

(19)

, the deformation gradient tensor

, the elastic strain tensor

and

the nominal stress tensor Scan also be expanded as

F=F(0) +ZF(1) +O(Z2),

A=A(0) +ZA(1) +O(Z2),

S=S(0) +ZS(1) +O(Z2).

(20)

Furthermore, we denote

G=G(0) +ZG(1) +O(Z2),

G−1=¯

G(0) +Z¯

G(1) +O(Z2),

JGG−1=ˆ

G(0) +Zˆ

G(1) +O(Z2).

(21)

Once the growth tensor Gis given, ¯

G(n)and ˆ

G(n)(n= 0,1) can be calculated directly.

By using the kinematic relations

(11)

and

(12)

, the concrete expressions of

F(n)

and

A(n)(n= 0,1)

in terms of

x(n)

(n= 0,1,2) can be derived. Further from the constitutive equation (15), we obtain

S(0) =ˆ

G(0) A(0) −p(0)R(0),

S(1) =ˆ

G(0) A(1) :A(1) −p(0)R(1) :A(1) −p(1)R(0)+ˆ

G(1) A(0) −p(0)R(0),

(22)

where A(n)=∂n+1φ0/∂An+1|A=A(0) and R(n)=∂n+1R0/∂An+1|A=A(0) (n= 0,1).

We substitute

(19)

and

(20)

into the constraint equation

(13)

and the mechanical equilibrium equation

(16)

. The

coefﬁcients of Zn(n= 0,1) in these equations should be zero, which yield that

Det A(0)−1 = 0,R(0) :A(1) = 0,(23)

and

∇ · S(0) +S(1)T

n=0,

∇ · S(1) +S(2)T

n+KTgα·S(0)

,α =0.

(24)

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AGENERALTHEORETICALSCHEMEFORSHAPE-PROGRAMMINGOFINCOMPRESSIBLEHYPERELASTICSHELLSTHROUGHDIFFERENTIALGROWTHAPREPRINTZhanfengLi1;3,JiongWang;1;2,MokarramHossain3,ChennakesavaKadapa41SchoolofCivilEngineeringandTransportation,SouthChinaUniversityofTechnology,Guangzhou,China2StateKeyLaboratoryofSubtropica...

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