Inverse Lagrangian Formulation for the Deformation of Hyperelastic Solids
Wei Hong*
Department of Aerospace Engineering, Iowa State University, Ames, IA 50010
Abstract
Although the field equations of solid mechanics are commonly written in the undeformed
configuration, neither fundamental theories nor mathematics prohibits the analysis from being
carried out in a deformed configuration. In this letter, following recent developments in inverse
deformation problems, the governing equations for static hyperelasticity problems are
formulated in the current configuration after deformation. An inverse mapping from the
deformed geometry to the original one is solved as the unknown field. Such an approach,
herein referred to as the inverse Lagrangian formulation, is exemplified with several applications
in addition to the inverse problem of known deformed geometry by design and determination
of the original geometry. Applications are found in steady-state fluid-structure interaction
problems, such as the design of microfluidic devices. More importantly, the method is also
found to be useful in analyzing mechanical instability and bifurcation problems, as the inverse
mapping from a buckled state to the original remains unique.
Keywords
inverse Lagrangian formulation; inverse design problem; inverse elastostatics; instability
*Email: whong@iastate.edu
1. Introduction
In continuum mechanics, the Lagrangian formulation, which traces the motion of material
particles from a specific reference state, is often used for the deformation of solid structures. On the
other hand, the Eulerian formulation which directly solves for the fields in the current state, is more
often used for fluid dynamics problems. Although tremendous effort has been put on integrating the
different algorithms in the two formulations, such as the Arbitrary-Lagrangian-Eulerian (ALE) method
which bridges the two by introducing a deforming mesh (Hirt et al, 1974; Liu et al, 1988), solving soliddeformation problems directly in a deformed state is not as common. Fundamental theories never
prevent the Eulerian formulation of elasticity problems. In fact, the Navier-Stokes equations are equally
1
applicable to deformable solids. The technical difficulties hindering the direct implementation of
elasticity problems in Eulerian formulation are two folds: the constitutive laws of solids, especially
elastic solids, usually require the knowledge of the undeformed configurations; the deformed geometry
of a solid structure is often unknown.
In the last two decades, various numerical methods have been developed to tackle the inverse
elasto-static problem (e.g. Govindjee and Mihalic, 1996, 1998; Yamada, 1998; Koishi and Govindjee,
2001; Lu et al., 2007, 2008; Rajagopal et al., 2007; Fachinotti et al., 2008; Gee et al., 2009; Albanesi et al.,
2010; Sellier, 2011): Given the deformed geometry of a body subject to prescribed loads, what is the
original undeformed geometry? While differ in mathematical details, most studies formulate the
problem in the deformed configuration. To obtain the geometry of the undeformed state, the mapping
from the original to the current spatial location of the same material particle is often calculated as the
main dependent variable, just as in the common Lagrangian formulation of hyperelasticity. However,
instead of a function the original coordinates of the particle, the mapping is written as a field in the
deformed frame. One direct application of these methods is the inverse design problem, in which the
deformed shape of a part under the actual working condition is of practical interest and is prescribed by
design. With the geometry designed in the working condition under the desired loads, the original
undeformed geometry, together with the stress and strain fields, is obtained as a solution to the
mathematical problem. Although the resulting undeformed geometry could be more complex and
tends to have non-straight edges or non-flat surfaces, the fabrication would not be overly complicated
due to the widely available additive manufacturing technologies. Practical examples of this kind include
the design of high-precision forming (Sellier, 2006), turbine blades (Fachinotti et al, 2008), compliant
mechanisms (Albanesi et al, 2013), and structures with elastomeric parts (Govindjee and Mihalic, 1996,
1998; Koishi and Govindjee, 2001). A related but special application is the stress analysis of biological
tissues, of which the geometry of the undeformed reference state is unknown (e.g. Rajagopal et al.,
2007; Lu et al., 2007, 2008; Gee et al., 2009, 2010).
With the field equations for the elastic deformation written in the deformed configuration, such
methods are naturally compatible with the governing equations in fluid dynamics which are usually in
Eulerian formulation. Fluid-structure interaction problems, especially those in steady-state, can be
solved entirely over the deformed frame without invoking ALE or moving boundaries (Sellier, 2011;
Vavourakis et al., 2011). As will be shown in the current letter, such an approach is very useful in
designing soft structures for the purpose of directing fluid flow, e.g. in a microfluidic device.
2
The mathematical formulation in this letter follows closely to Fachinotti et al (2008), with the
extension of handling the volumetric incompressibility by using a mixed displacement-pressure
formulation. This formulation differs from Govindjee and Mihalic (1996, 1997) mainly in the material
laws: the same material laws in terms of the Lagrangian strain measures are used just as regular
hyperelasticity. In the literature, similar approaches do not carry a common name, and are instead
often named after specific applications, such as the inverse design problem (Fachinotti et al, 2008) or
the inverse elastostatics problem (Govindjee and Mihalic, 1996; Lu et al, 2007). However, as will be
demonstrated in this letter, the method has wider applications than just the inverse design problems.
To differentiate from common Eulerian approaches which solves the history-independent velocity or
acceleration field, it is referred to as the inverse Lagrangian formulation in the current letter.
Besides applications in inverse design problems and fluid-structure interaction, the main focus
of the current letter is the application of the inverse Lagrangian approach to the analyses of instability
problems in solid structures. Not only because of the importance in engineering applications, but also
due to the intriguing mechanics, instability phenomena have always been a topic of interest. As an
example, the rich and controllable surface instabilities on a soft material (or in a stiff-film-soft-substrate
system) have been studied extensively (Biot, 1963; Huang et al., 2005; Genzer and Groenewold, 2006;
Hong et al., 2009; Hohlfeld and Mahadevan, 2011; Li et al, 2012; Wang and Zhao, 2014) and found
promising applications in material characterization (Stafford et al, 2004), fabrication of flexible
electronics (Choi et al., 2007; Kim and Rogers, 2008; Baca et al., 2008), surface functionalization (Chan
and Crosby, 2006; Zang et al., 2013). It is non-trivial, however, to numerically predict the onset of
instability or carry out post-buckling analysis. Each instability instance mathematically corresponds to a
bifurcation point, beyond which the solution is non-unique, i.e. there exists more than one mapping
from the undeformed to the deformed state satisfying the equilibrium and boundary conditions.
Besides the singularity of the linearized coefficient matrix, numerical calculation tends to stay on a
uniform (trivial) solution instead of branching out to the physically stable one with lower energy. To
overcome the critical point and perform post-buckling analysis, numerical schemes usually introduce
imperfections of small amplitude as well as artificial damping to ensure convergence. In some
circumstances, the result is found to be sensitive to the imperfection (Cao and Hutchinson, 2012).
Nevertheless, if the mathematical problem is reformulated and instead the inverse mapping from the
deformed state is sought for, no bifurcation may be involved and the solution is usually unique. As will
be demonstrated in this letter, without the need for artificial imperfection or damping, the inverse
3
Lagrangian approach is a robust numerical method for determining the onset of instability as well as the
post-buckling behavior in soft elastic solids.
2. The Inverse Lagrangian Formulation
Although the method can be applied to general elastic or inelastic solids, it is illustrated here
through the deformation of hyperelastic solids as a timely example. Let us first review the mathematical
description via the commonly used Lagrangian approach. Consider a material particle located at X in
the undeformed configuration, and denote its position after deformation by x . The mapping X  x
uniquely specifies the motion of all particles in the continuum, and the field of deformation is commonly
described by the deformation gradient tensor
FiK 
xi
,
X K
(1)
which is the spatial gradient of xX  with respect to the undeformed geometry. The behavior of a
hyperelastic material is often written in terms of the relation between the deformation gradient and its
work conjugate, the nominal stress
siK  siK F  
W
,
FiK
(2)
where W is the elastic free energy per unit undeformed volume. The material behavior may also be
written in terms of various other stress and strain measures.
In equilibrium, the field of nominal stress is in balance with body force b in the bulk (e.g. Bower,
2009)
siK
 bi  0 ,
X K
(3)
and surface traction t on any surface with prescribed load
siK N K  t i .
(4)
Here, the unit normal vector N is defined on the surface before deformation. Equations (1)-(3) form a
closed differential system to determine the mapping xX  when proper boundary conditions such as Eq.
4
(4) or a given displacement boundary condition are prescribed. Such a framework is commonly adopted
in the analyses of hyperelasticity problems, and is often referred to as the Lagrangian formulation.
The Lagrangian formulation is mathematically equivalent to the following framework which is
based on the inverse mapping x  X , if it exists. For (locally) differentiable Xx  , the inverse
deformation gradient
H Ki 
X K
xi
(5)
captures the field of deformation. H is the inverse tensor of the deformation gradient, H  F 1 . In
the deformed configuration, the equilibrium condition is more conveniently written in terms of the true
stress tensor σ as (e.g. Bower, 2009)
 ij
x j
 bi  0 ,
(6)
and the surface traction is balanced by
 ij n j  t i .
(7)
Here, the unit normal vector n is defined on the deformed surface. By using the geometric relation
between different stress measures, the material constitutive relation (2) can be written in terms of the
true stress
 ij 
siK F jK
det F
 det H
W
F jK .
FiK
(8)
Just like the Lagrangian formulation, Eqs. (5), (6), and (8) also consist a differential system,
through which the inverse mapping Xx  can be identified when proper boundary conditions are
prescribed. This formulation described above follows closely to Fachinotti et al (2008), in which it was
mainly applied to inverse design problems. In the limiting case of small deformation, when the
difference between the deformed and undeformed geometries is negligible, the inverse Lagrangian
formulation simply reduces to the Lagrangian formulation. Only for problems involving finite
deformation, the two formulations will lead to different equations.
5
The inverse Lagrangian formulation is mathematically well posed. It is less often applied,
perhaps because the deformed geometry of solid objects is seldom known. Nevertheless, some unique
features of the inverse Lagrangian formulation make it suitable for the special circumstances as will be
demonstrated in the following sections.
Due to the mathematical similarity between the two formulations, almost all numerical methods
for elasticity problems in the Lagrangian formulation will equally be applicable to those in the inverse
Lagrangian formulation. Here as a demonstration, a finite-element method will be adopted. Through
simple mathematical operations such as integration by parts and the divergence theorem, the
differential equation (6) and the traction boundary condition (7) can be written into a weak form
(Fachinotti et al, 2008)
 ui

  ij
dv  tiui da , ui x  .

b

u
i
i
  x j


 t


(9)
Here, the integrations are carried out over the volume  of the domain under consideration, and the
surfaces  t on which the traction boundary condition is prescribed. Both  and  t are defined in
the deformed configuration. The displacement vector ux  x  Xx . Unlike its counterpart in the
Lagrangian formulation, the weak form (9) may not have a clear physical meaning or correspond to a
virtual work principal. However, the lack of physical interpretation does not prohibit the numerical
implementation. In fact, the procedure is almost identical due to the mathematical similarity. The only
difference is that all fields, known or unknown, are expressed as functions of the coordinates x in the
deformed configuration.
In the Lagrangian formulation, the thermodynamic stability dictates the stiffness tensor to be
positive definite. Here, it is easy to show that the tensor derivative of the true stress σ with respect to
the Cauchy deformation tensor c  H T  H is negative definite.
Although the inverse Lagrangian formulation can take any constitutive models, here the neoHookean model will be used for the purpose of demonstration. With the free-energy function given by
W

2
F : F  3 , a neo-Hookean solid has true stress σ   det H F  F T , where 
modulus of the material.
6
is the initial shear
Hyperelastic materials are often assumed to be incompressible. Inspired by the displacementpressure mixed formulation (e.g. Zienkiewicz et al., 2005; Govindjee and Mihalic, 1998), we enforce the
volume incompressibility by adding the pressure field px  as a Lagrange multiplier to the weak form:

 F


iK
F jK det H   p    ij 

ui
 biui  dv   det H  1pdv   tiui da , ui x, px . (10)
x j


 t
Here, the specific weak form (10) is taken such that the combination
 ij   FiK FjK det H   ij   p ij
(11)
represents the true stress.
3. Inverse Problems
a
A
Fig. 1. Axisymmetric deformation of a quarter-circular wedge. Constrained by frictionless rollers on the
flat surfaces, the quarter cylinder of radius
A is deformed into a half cylinder of radius a  A
2 . The
thickness direction is constrained and the mode of deformation is plane strain.
To verify the formulation and the numerical implementation, we first test the method on an
example with known analytical solution. As sketched in Fig. 1, a quarter cylinder of radius A is
constrained with rollers on the two flat surfaces. Subject to a displacement boundary condition, one of
the two surfaces is rotated by  2 , and the sample deforms into a half cylinder. The mode of
deformation is plane strain, and the material is assumed to be incompressible neo-Hookean. The
problem is kinematically determinate, and the field of deformation can be readily calculated from
volume conservation. In polar coordinates, the principal stretches are r  1
the stress-stretch relation of a neo-Hookean material, the true stresses are
7
2 and   2 . Using


  r   pr 
,

2
   2  pr 
(12)
where pr  is the field of hydrostatic pressure, which could be determined from the equilibrium
equation in the radial direction, d r dr   r     r  0 . Using the boundary condition  r a   0 ,
the pressure field is calculated as pr  

r
1  3 ln  , where a  A
2
a
2 is the radius in the
deformed configuration. The corresponding stress field is then given by
3 r

  r  2 ln a
.

3 
r
  
1  ln 

2 
a
(13)
Numerically, we build the 2D model in COMSOL Multiphysics under the inverse Lagrangian
frame, by implementing the weak form (11) in the deformed semi-circular domain. Given displacement
boundary conditions are prescribed on the two straight edges, and a traction-free boundary condition is
prescribed on the curved edge. To aid the convergence, gradually changing rotation angles on one
displacement boundary is applied. A triangular mesh is created on the semi-circular domain, denser
near the center ( ~ 0.03a ) and coarser near the perimeter ( ~ 0.1a ). The resulting true stress
distribution along the radius of the cylinder is plotted in Fig. 2, together with the analytical solution (13).
The good agreement between the numerical and analytical results validates the method.
r

Fig. 2. Radial distribution of dimensionless true stresses,
r 
and
  .
The circles and squares are
numerical results by using the finite element method via the inverse Lagrangian formulation, and the
continuous curves are plotted from the anylitical solution, Eq. (13).
8
As introduced in Section 1, the inverse Lagrangian formulation is naturally suitable for solving
inverse problems, namely the computation of the original undeformed geometry for manufacturing,
when the geometry in the deformed state under working load is known (or specified by design). Limited
by conventional manufacturing techniques, structural designs commonly chose relatively simple
geometric features (e.g. flat surfaces and straight edges) in the undeformed configuration (when the
part is machined). Emerging techniques of additive manufacturing no longer require the original shape
to be simple. One can simply design the geometry in the working state, use the inverse Lagranian
method to calculate the shape before deformation, and then directly print it.
m 
Deformed
Undeformed
(calculated)
Fig. 3. Deformation and stress distribution of an inflated tubular structure. By symmetry, only a quarter
of the structure is modeled. The computation is carried out over the deformed geometry with the inverse
Lagrangian formulation. The mesh grids in the deformed and undeformed configurations represent the
distortion, instead of the actual elements for calculation. The shades indicate the distribution of the
normalized mean stress
 m    kk  .
As an illustrative example, let us consider the following inverse problem. A hollow tube made of
rubbery elastomer is designed to work in an inflated state of internal pressure 0.1 , with  being the
material’s initial modulus. The functional design requires the external shape of the cross section to be
circular and the internal to be a rounded square, as shown by Fig. 3. With conventional finite element
method, such an inverse problem will require multiple rounds of trial and error (or parametric
optimization). Here by taking the inverse Lagrangian approach, we formulate and solve the problem in
9
the deformed geometry within one step. The resulting undeformed shape and stress distribution are
shown in Fig. 3. The resulting undeformed state has smooth but non-flat surfaces, which should not
pose difficulty to existing 3D printing technologies (e.g. Shepherd et al., 2011). With a calibrated
material constitutive relation, such an approach could lead to very accurate geometries for soft
structures in their working conditions.
4. Fluid-Structure Interaction
Since the inverse Lagrangian method solves the unknown fields in the deformed configuration, it
can be seamlessly integrated with common fluid solvers, which are formulate in Eulerian description in
terms of the current coordinates. If the deformed geometry of the immersed solid structure is known,
the coupling between the flow and the solid structure will be unidirectional and easy to handle: the fluid
flow provides pressure and viscous stress to the solid surface as boundary conditions, with which the
fields of deformation and stress in the solid can be determined. Because the deformed shape is already
prescribed, the fluid solver requires no additional information from the structural solution, and no
further iteration is needed.
p
  (kPa)
Fig. 4. Steady flow around an inflated soft membrane. The mesh shows the undeformed position of the
membrane, calculated via the inverse Lagrangian approach. The shades in the fluid domain shows the
pressure distribution, and that in the membrane shows the hoop stress, both in the unit of kPa. The
curves in the fluid domain represent the streamlines of the flow.
The only technical difficulty remains in the determination of the deformed geometry. However,
to most engineering applications, the deformed geometry is actually of more importance. Take
microfluidic devices as an example, it has long been noticed that the deformation of soft components
caused by fluid pressure may be significant and should be taken into consideration in design (Gervais et
10
al., 2006). Such complexity can be totally avoided if one directly designs the deformed geometry. To
illustrate this application, we calculate the following microfluid device as an example. A soft rubbery
membrane (initial shear modulus 0.2 MPa) is inflated in a fluid channel to perturb the flow, as shown by
Fig. 4. The fluid channel has a height of 2mm and length of 5mm, and is filled with water (viscosity 1
mPa·s). It is assumed that a semi-circular shape of radius 1mm is preferred by design. The device
operates at an inlet velocity of 1 m/s and the rubber membrane is inflated with an internal pressure of 5
kPa. For simplicity, the deformed thickness of the membrane is set to be 0.1 mm uniformly. For the
fluid flow, non-slip boundary conditions are applied to both the top and bottom surfaces, and to the
deformed membrane surface. For the hyperelastic membrane, the bottom edges are fixed, a uniform
pressure is applied to the internal surface, and the calculated fluid pressure and viscous stress are
applied to the outer surface. The flow problem is solved with the built-in steady laminar fluid solver in
COMSOL Multiphysics, while the deformation of the elastic membrane is solved with the inverse
Lagrangian formulation, also implemented in COMSOL Multiphysics. The resulting deformation and
stress fields in the membrane are plotted in Fig. 4, together with the flow field in the fluid channel. For
this soft membrane, the deformed geometry differs significantly from the original shape. Clearly, to
work against the flow-induced pressure gradient in the fluid, the undeformed geometry of the
membrane needs to be asymmetric and leaning towards upstream. If the as-manufactured membrane
is symmetric and circular instead, the flow would be very different. To achieve the same design task
with conventional approaches, even with the arbitrary Lagrangian-Eulerian formulation, much more
effort is needed to determine the geometry for manufacture.
Just as all nonlinear problems, the solution may be non-unique and the deformed configuration
may be dependent on the actual deformation history. For problems with loads given in terms of
displacements, it is possible to add intermediate steps to prescribe the loading history. In general,
however, it is not easy to include such information in the current framework, as the intermediate
geometries are often unknown. It is even more difficult for fluid-solid interaction problems in which the
steady-state flow fields will take time to establish. To overcome these difficulties, one will need to
extend the inverse Lagrangian formulation to include dynamic terms, so that the entire deformation
process can be predicted incrementally, and the effect of loading history can be accounted for. Such a
development, however, is beyond the scope of the current letter.
11
5. Instability and Bifurcation Problems
For decades, instability phenomena have been a focal point for researches in the field of soft
matter mechanics. The inevitable geometric nonlinearity brings varieties to the form of instability, and
at the same time complicates the analysis. For example, conventional finite-element method
formulated under the Lagrangian description may encounter singularity in the coefficient matrix at the
critical point. In addition, due to the non-uniqueness in solution, the numerically converged solution
may not be the physical one. In the absence of perturbation, numerical solvers tend to converge to the
homogeneous solution even beyond the critical point, even though the homogeneous solution may be
physically unstable. To obtain an inhomogeneous solution, artificial defects or numerical perturbations
will need to be added. Such an approach lowers the accuracy of computation, especially in systems that
are sensitive to the mode or amplitude of perturbation.
Response
S1
S2
f
fc
O
Generalized load
Fig. 5. Schematic illustration of a bifurcation problem. At the critical point
into two branches. Under a higher load,
undeformed state O:
f c , the solution bifurcates
f  f c , multiple solutions exist for mapping from the original
O  S1 and O  S2 . On the other hand, the inverse mapping from one of the
deformed states, e.g.
S2  O , could be unique.
By using the inverse Lagrangian formulation, some of these numerical difficulties could be
avoided. As illustrated by Fig. 5, beyond the critical point f c , the mathematical system with the
conventional Lagrangian formulation loses the solution uniqueness, and has multiple solutions in terms
of the mapping from the undeformed to the deformed state: O  S1 and O  S2 . Nevertheless, if
the same problem is formulated in the deformed configuration via the inverse Lagrangian approach, the
uniqueness of the solution could be maintained even beyond the critical point. From a deformed state
12
S2 , there is only one inverse mapping S2  O which satisfies the equilibrium and boundary conditions.
The other solution S1  O , formulated over a different deformed geometry, now becomes an entirely
different problem.
y
Deflection wx 
Deformed geometry
P
P
x
Undeformed geometry
(to be solved)
Fig. 6. Sketch of a classic Euler buckling problem considered under the inverse Lagrangian frame. The
deformed geometry is taken to be known, while the undeformed geometry (or the deflection) is an
unknown field to be solved.
To show the mathematical difference between the two approaches, let us temporarily depart
from the finite-element method and consider the classic Euler buckling problem. A column of length L
and bending stiffness EI , is pinned at both ends as sketched in Fig. 6. Under the assumptions of the
linear beam theory, the amplitude of the post-buckling deflection is indeterministic, and the stiffness is
singular at the critical point. Now consider the equilibrium in the deformed state. With a deformed
geometry given by y  A sin x L , the moment balance requires that M   Py . Substituting in the
relation between the lateral deflection wx  and the moment, we arrive at
EI
d 2w
  Py .
dx
(14)
Equation (14) can be directly integrated to obtain the solution
wx  
L2 PA
x
sin .
2
 EI
L
(15)
Unlike its counterpart via the conventional Lagrangian approach, solution (15) exists and is unique for
any axial load P , although the general case would represent the bending and compression from an
initially curved beam, as sketched in Fig. 6. The critical state of buckling corresponds to the case when
the column is originally straight, or w  y , which gives the classic result of the critical axial force
13
Pc   2 EI L2 . However, it does not cause the ill conditioning of the governing equation (14) or affect
the uniqueness of the solution.
a
b
y 
Fig. 7. (a) Sketch of a Euler buckling problem. (b) Deformation and stress distribution in the buckled
hyperelastic column, in two modes of buckling. The shades indicate the normalized true axial stress
 y  , and the mesh shows the calculated geometry of the undeformed configuration.
To further demonstrate the application of the inverse Lagrangian formulation to instability
analyses, we will now use it to solve an Euler buckling problem numerically. As sketched in Fig. 7a, a
column of aspect ratio 10:1 is subject to axial compression. The rotation in both ends are constrained,
but the top end moves freely in the lateral direction. The deformation is plane strain. For simplicity, the
material is also assumed to be neo-Hookean and incompressible, so that the formulation in Section 2
can be directly applied. As shown by Fig. 7b, the computation is carried out in the domains
corresponding to the deformed states. The deformed geometries are obtained by superimposing a
sinusoidal deflection onto the neutral axis of the column. As examples, the first two modes of buckling
are calculated, and the representative fields of deformation and stress are plotted in Fig. 7b. To identify
the critical points and illustrate the bifurcation, we further plot the amplitudes of lateral deflection as a
function of the axial compression for both modes in Fig. 8. The critical strain of each mode can be easily
identified from the intersection between the computed curve corresponding to the buckled states and
that of the homogeneously deformed states. The resulting critical strains are very close to those given
by the Euler buckling theory:  c  0.0082 for the first mode, and  c  0.033 for the second mode.
14
Although the uniformly compressed states can also be calculate via the inverse Lagrangian approach,
such calculation is omitted as the solutions all fall on the horizontal axis in Fig. 8.
Deflection amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.02
0.04
Nominal compressive strain
0.06
Fig. 8. Deflection amplitudes in the first two buckling modes of an axially compressed column, plotted as
functions of the nominal compressive strain. As the homogeneous solution is just the horizontal axis, the
plot indicates bifurcation at the critical points.
Now let us turn to the wrinkling instability of a stiff thin film bonded to a soft thick substrate,
which has been extensively studied (Cerda and Mahadevan, 2003; Chen and Hutchinson, 2004; Huang et
al., 2005; Genzer and Groenewold, 2006) due to various applications (e.g. Choi et al., 2007; Stafford et
al., 2004). Here as an example, we consider the film-substrate system in which the initial shear modulus
of the film is 100 times that of the substrate, and the substrate thickness is 30 times film thickness h
before deformation. In contrast to the conventional approach, we solve the problem in the wrinkled
geometry via the inverse Lagrangian formulation. As shown by Fig. 9, a rectangular region with
sinusoidal top edge is used as the computation domain. Only half of the wrinkling wavelength is
computed, and symmetry boundary conditions are assumed on left and bottom edges. For simplicity,
incompressible neo-Hookean material model is adopted for both the film and the substrate, and the
mode of deformation is taken to be plane strain. A displacement boundary condition is applied
gradually over the right boundary. When the computed top surface flattens, the undeformed state is
thought to be reached. It should be noted that due to the numerical error in the prescription of the
deformed geometry, the top surface may not reach a mathematically flat state. In practice, the state in
which the peak and trough points have the same vertical coordinate is regarded as the flat state. A
representative solution is shown in Fig. 8, in which the mesh indicates the calculated original geometry,
15
and the shades represent the horizontal component of the Eulerian-Almansi strain e 
1
2
I  H
T

H .
As shown by Fig. 8, for wrinkles of relatively small amplitude, the resulting undeformed top surface is
relatively flat, indicating that the prescribed sinusoidal surface is quite close to the actual geometry of
the wrinkled state.
e xx
Fig. 9. Calculated field of deformation of a wrinkled film-substrate system by using the inverse
Lagrangian formulation. The shades indicate the horizontal component
exx of the Eulerian-Almansi strain,
and the mesh represents the undeformed geometry from computation. The mesh is also that used for
finite element calculation. For the case shown, the wrinkle wavelength is taken to be 20 times the film
thickness, and the substrate thickness is 30 times film thickness before deformation. Only a half period
is included for computation and symmetry conditions are assumed on corresponding boundaries.
To identify the critical point of the wrinkling instability, we plot the wrinkle amplitude as a
function of the applied compression in Fig. 10a. Unlike the Euler column, here the wavelength of the
wrinkles is a continuous variable: wrinkles of any wavelength could be a possible deformed state. As
shown by Fig. 10a, each wavelength corresponds to a critical compressive strain, shown as the
intersection between the corresponding curve and the horizontal axis. In the absence of external
constraints or perturbation, the system will branch into the wrinkling state of the lowest critical strain.
To determine this state, we plot the critical strains as a function of the dimensionless wavelength  h
in Fig. 10b. The dependence of the critical strain on the wrinkle wavelength is non-monotonic, and
16
shows a clear minimum point. For this example, the wavelength of the lowest critical strain is c  20h ,
and the corresponding nominal compressive strain for the onset of instability is  c  0.0237 . Both
values are very close to the analytical solution for the case of linear elastic materials and infinite
substrate, c  20.2h and  c  0.0241 (Chen and Hutchinson, 2004; Huang et al., 2005). As the strain
is relatively low, the small discrepancy is mainly attributed to the finite thickness effect of the substrate.
b 0.02405
0.024
0.02395
0.0239
0.02385
0.0238
0.02375
0.0237
0.02365
λ = 16h
0.5
Critical nominal strain εc
Wrinkle amplitude A/h
a 0.6
λ = 20h
0.4
λ = 28h
0.3
0.2
0.1
0
0.022
0.024 0.026 0.028 0.03
Nominal compressive strain ε
0.032
18
19
20
21
Wavelength (λ/h)
22
Fig. 10. (a) Normalized wrinkle amplitudes plotted as functions of the applied compressive strain, for
various wrinkle wavelengths. The intersection of each curve with the horizontal axis corresponds to the
critical point of instability. (b) Critical compressive strain for wrinkling instability as a function of the
dimensionless wavelength
 h.
The minimum point is the strain and wavelength of the wrinkles that
will spontaneously develop in a large system with no additional constraints or perturbation.
As the last example, we will study the instability of a periodical porous structure under equal biaxial compression. A 2D structure contains a periodical array of circular holes, as shown by Fig. 11a, is
made of incompressible neo-Hookean solid. When compressed (either uniaxially or biaxially) beyond a
critical strain, the four-fold symmetry of the structure will be broken, and the circular holes will turn into
oval shapes of alternating orientations. This instability phenomenon has been demonstrated
experimentally and analyzed numerically (Mullin et al., 2007; Bertoldi and Boyce, 2008; Bertoldi et al.,
2008). Very recently, it is also proposed that such structures can be regarded as a meta material
exhibiting phase transition (Yang et al., 2016). Here, it is used as an example to show case the
application of the inverse Lagrangian formulation. The computation is carried out over a representative
unit cell of the structure (quarter of that shown on Fig. 11a), and symmetric boundary conditions are
applied on all edges except the free surface of the holes. The ratio between the hole diameter and the
17
center-to-center distance is taken to be 9:10 in the undeformed state. To prescribe the deformed
geometry, we first utilize the incompressibility and evaluate the area of each hole under a given nominal
compressive strain. For simplicity, we assume the deformed shape of each hole to be an ellipse. The
aspect ratio of the ellipse is then determined through a line search to achieve a symmetric undeformed
structure. Using such a strategy, we compute the deformation and stress distribution in both branches
of the solution, and plot the results in Fig. 11.
b
Nominal compressive stress σ/μ
a
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0
0.01
0.02
0.03
Nominal compressive strain
0.04
Fig. 11. (a) Stable equilibrium state of a periodic porous structure under biaxial compression of linear
strain
  0.035 .
The shades show the mean stress normalized by the initial modulus,
 m  , and the
mesh represents the computed undeformed geometry. The mesh is also that used for the finite-element
calculation. (b) Nominal compressive stress as a function of the applied compressive strain, in both the
symmetrically compressed state (unstable, circles) and the state with broken symmetry (squares). The
intersection at
 c  0.007
is the bifurcation point or the onset of instability.
A representative result from computation is shown by Fig. 11a, in which the mesh indicates the
calculated undeformed geometry, and the shades represent the normalized mean stress  m  . By
breaking the four-fold symmetry of the structure and rotating the region in the midst of four holes, the
mode of deformation transitions from a hydrostatic-compression-dominated state to one mainly
manifested by bending of the thin ligaments, and the overall stress is greatly relieved. It can also be
noticed on Fig. 11a that the computed undeformed shape of the holes is not exactly circular. Such a
discrepancy is again due to the inaccuracy in prescribing the deformed geometry. In other words, the
result suggests that the buckled shape of the holes is not exactly an ellipse. While the numerical results
18
could be further improved, e.g. by describing the deformed geometry with more parameters and
carrying out a parametric optimization, such a process is not the main focus of the current letter. In Fig.
11b, by integrating the true stresses along the boundaries, we computed the nominal stress of the
structure, and plotted it against the applied strain for both non-buckled and buckled states. The
bifurcation point  c  0.007 indicates the onset of the instability.
Finally, it should be noted that although the inverse Lagrangian formulation changes the
mathematics to avoid singularities or non-uniqueness of solutions near the critical points, it by no means
alters the intrinsic physical behaviors of the structures. If the structure is sensitive to geometric
imperfections, a small change in the original geometry may lead to a large difference in the deformed
configuration. In these cases, the inverse Lagrangian formulation may still be used to analyze the
instability and post-buckling behaviors, but is not suitable for the inverse design of structures. In fact,
for better control of the target geometry, critical points of mechanical instabilities should be avoided in
structural design if at all possible.
6. Conclusion
In this letter, following Fachinotti et al (2008), the field equations of static hyperelastic problems
are formulated in the deformed configuration, via the inverse Lagrangian approach. Different from the
conventional Lagrangian formulation which solves for the undeformed-to-deformed mapping, or the
Eulerian formulation of fluid dynamics which solves for the current-time velocity, the inverse Lagrangian
formulation solves for the backward mapping from the deformed state to the original undeformed one.
For linear elasticity problems with indistinguishable undeformed and deformed geometries, the inverse
Lagrangian and Lagrangian formulations lead to identical equations.
Through numerical examples, the inverse Lagrangian formulation has found some unique and
interesting applications. Similar as those demonstrated in the literature, a direct application is in the
inverse problems of identifying undeformed geometries from the prescribed geometries after
deformation, especially in the design of soft structures which focuses more on the geometric accuracy
under working conditions. With the aid of 3D printing technologies, the elevated complexity in the
undeformed geometry should not become a challenge to manufacturing processes. Similarly, the
method can be applied to the design of soft microfluid devices by solving the steady-state fluid-structure
interaction as a decoupled problem. More importantly, the application of the inverse Lagrangian
formulation to mechanical instability problems is demonstrated. By solving for the inverse mapping
19
from the deformed to the undeformed state, such an approach avoids the solution non-uniqueness in
the conventional Lagrangian formulation. No artificial imperfection or numerical damping is needed for
critical or post-buckling analysis. It should be admitted that the unknown geometry of the deformed
configuration remains a major drawback for the inverse Lagrangian approach, especially for the postbuckling analysis at relatively large amplitudes. In future works, this method could be improved by using
parametric optimization to more accurately determine the deformed geometries.
References
Albanesi, A.E., Fachinotti, V.D. and Cardona, A., 2010. Inverse finite element method for
large‐displacement beams. International Journal for Numerical Methods in Engineering, 84(10), pp.11661182.
Albanesi, A.E., Pucheta, M.A. and Fachinotti, V.D., 2013. A new method to design compliant mechanisms
based on the inverse beam finite element model. Mechanism and Machine Theory, 65, pp.14-28.
Baca, A.J., Ahn, J.H., Sun, Y., Meitl, M.A., Menard, E., Kim, H.S., Choi, W.M., Kim, D.H., Huang, Y. and
Rogers, J.A., 2008. Semiconductor Wires and Ribbons for High‐Performance Flexible Electronics.
Angewandte Chemie International Edition, 47(30), pp.5524-5542.
Bertoldi, K. and Boyce, M.C., 2008. Mechanically triggered transformations of phononic band gaps in
periodic elastomeric structures. Physical Review B, 77(5), p.052105.
Bertoldi, K., Boyce, M.C., Deschanel, S., Prange, S.M. and Mullin, T., 2008. Mechanics of deformationtriggered pattern transformations and superelastic behavior in periodic elastomeric structures. Journal
of the Mechanics and Physics of Solids, 56(8), pp.2642-2668.
Biot, M.A., 1963. Surface instability of rubber in compression. Applied Scientific Research, Section A,
12(2), pp.168-182.
Bower, A.F., 2009. Applied mechanics of solids. CRC press, New York.
Cao, Y. and Hutchinson, J.W., 2012, January. From wrinkles to creases in elastomers: the instability and
imperfection-sensitivity of wrinkling. In Proceedings of the Royal Society of London A: Mathematical,
Physical and Engineering Sciences (Vol. 468, No. 2137, pp. 94-115). The Royal Society.
Chan, E.P. and Crosby, A.J., 2006. Fabricating microlens arrays by surface wrinkling. ADVANCED
MATERIALS-DEERFIELD BEACH THEN WEINHEIM-, 18(24), p.3238.
20
Chen, X. and Hutchinson, J.W., 2004. Herringbone buckling patterns of compressed thin films on
compliant substrates. Journal of applied mechanics, 71(5), pp.597-603.
Choi, W.M., Song, J., Khang, D.Y., Jiang, H., Huang, Y.Y. and Rogers, J.A., 2007. Biaxially stretchable
“wavy” silicon nanomembranes. Nano Letters, 7(6), pp.1655-1663.
Cerda, E. and Mahadevan, L., 2003. Geometry and physics of wrinkling. Physical review letters, 90(7),
p.074302.
Fachinotti, V.D., Cardona, A. and Jetteur, P., 2008. Finite element modelling of inverse design problems
in large deformations anisotropic hyperelasticity. International Journal for Numerical Methods in
Engineering, 74(6), pp.894-910.
Gee, M.W., Förster, C. and Wall, W.A., 2010. A computational strategy for prestressing patient‐specific
biomechanical problems under finite deformation. International Journal for Numerical Methods in
Biomedical Engineering, 26(1), pp.52-72.
Gee, M.W., Reeps, C., Eckstein, H.H. and Wall, W.A., 2009. Prestressing in finite deformation abdominal
aortic aneurysm simulation. Journal of biomechanics, 42(11), pp.1732-1739.
Genzer, J. and Groenewold, J., 2006. Soft matter with hard skin: From skin wrinkles to templating and
material characterization. Soft Matter, 2(4), pp.310-323.
Gervais, T., El-Ali, J., Günther, A. and Jensen, K.F., 2006. Flow-induced deformation of shallow
microfluidic channels. Lab on a Chip, 6(4), pp.500-507.
Govindjee, S. and Mihalic, P.A., 1996. Computational methods for inverse finite elastostatics. Computer
Methods in Applied Mechanics and Engineering, 136(1), pp.47-57.
Govindjee, S. and Mihalic, P.A., 1998. Computational methods for inverse deformations in
quasi‐incompressible finite elasticity. International Journal for Numerical Methods in Engineering, 43(5),
pp.821-838.
Hirt, C.W., Amsden, A.A. and Cook, J.L., 1974. An arbitrary Lagrangian-Eulerian computing method for all
flow speeds. Journal of Computational Physics, 14(3), pp.227-253.
Hohlfeld, E. and Mahadevan, L., 2011. Unfolding the sulcus. Physical review letters, 106(10), p.105702.
Hong, W., Zhao, X. and Suo, Z., 2009. Formation of creases on the surfaces of elastomers and gels.
Applied Physics Letters, 95(11), p.111901.
21
Huang, Z.Y., Hong, W. and Suo, Z., 2005. Nonlinear analyses of wrinkles in a film bonded to a compliant
substrate. Journal of the Mechanics and Physics of Solids, 53(9), pp.2101-2118.
Koishi, M. and Govindjee, S., 2001. Inverse design methodology of a tire. Tire Science and Technology,
29(3), pp.155-170.
Li, B., Cao, Y.P., Feng, X.Q. and Gao, H., 2012. Mechanics of morphological instabilities and surface
wrinkling in soft materials: a review. Soft Matter, 8(21), pp.5728-5745.
Liu, W.K., Herman, C., Jiun-Shyan, C. and Ted, B., 1988. Arbitrary Lagrangian-Eulerian Petrov-Galerkin
finite elements for nonlinear continua. Computer methods in applied mechanics and engineering, 68(3),
pp.259-310.
Lu, J., Zhou, X. and Raghavan, M.L., 2007. Computational method of inverse elastostatics for anisotropic
hyperelastic solids. International journal for numerical methods in engineering, 69(6), pp.1239-1261.
Lu, J., Zhou, X. and Raghavan, M.L., 2008. Inverse method of stress analysis for cerebral aneurysms.
Biomechanics and modeling in mechanobiology, 7(6), pp.477-486.
Mullin, T., Deschanel, S., Bertoldi, K. and Boyce, M.C., 2007. Pattern transformation triggered by
deformation. Physical review letters, 99(8), p.084301.
Rajagopal, V., Chung, J.H., Bullivant, D., Nielsen, P.M. and Nash, M.P., 2007. Determining the finite
elasticity reference state from a loaded configuration. International Journal for Numerical Methods in
Engineering, 72(12), pp.1434-1451.
Sellier, M., 2006. Optimal process design in high-precision glass forming. International Journal of
Forming Processes, 9(1), p.61.
Sellier, M., 2011. An iterative method for the inverse elasto-static problem. Journal of Fluids and
Structures, 27(8), pp.1461-1470.
Shepherd, R.F., Ilievski, F., Choi, W., Morin, S.A., Stokes, A.A., Mazzeo, A.D., Chen, X., Wang, M. and
Whitesides, G.M., 2011. Multigait soft robot. Proceedings of the National Academy of Sciences, 108(51),
pp.20400-20403.
Stafford, C.M., Harrison, C., Beers, K.L., Karim, A., Amis, E.J., VanLandingham, M.R., Kim, H.C., Volksen,
W., Miller, R.D. and Simonyi, E.E., 2004. A buckling-based metrology for measuring the elastic moduli of
polymeric thin films. Nature materials, 3(8), pp.545-550.
22
Vavourakis, V., Papaharilaou, Y. and Ekaterinaris, J.A., 2011. Coupled fluid–structure interaction
hemodynamics in a zero-pressure state corrected arterial geometry. Journal of biomechanics, 44(13),
pp.2453-2460.
Wang, Q. and Zhao, X., 2014. Phase diagrams of instabilities in compressed film-substrate systems.
Journal of applied mechanics, 81(5), p.051004.
Yamada, T., 1995. Finite element procedure of initial shape determination for hyperelasticity. In
Computational Mechanics’ 95 (pp. 2456-2461). Springer Berlin Heidelberg.
Yang, D., Jin, L., Martinez, R.V., Bertoldi, K., Whitesides, G.M. and Suo, Z., 2016. Phase-transforming and
switchable metamaterials. Extreme Mechanics Letters, 6, pp.1-9.
Zang, J., Ryu, S., Pugno, N., Wang, Q., Tu, Q., Buehler, M.J. and Zhao, X., 2013. Multifunctionality and
control of the crumpling and unfolding of large-area graphene. Nature materials, 12(4), pp.321-325.
Zienkiewicz, O.C., Taylor, R.L. and Zhu, J.Z., 2005. The finite element method: its basis and fundamentals.
6th ed., Elsevier, Oxford.
23