Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

41, 3, pp. 643-673, Warsaw 2003

SHAPE SENSITIVITY ANALYSIS OF ELASTIC SHELLS WITH

CRACKS

Edgardo Taroco

Raúl A. Feijóo

National Laboratory of Scientific Computing (LNCC/MCT), Petrópolis, Brasil

e-mail: etam@lncc.br; feij@lncc.br

This study concerns the application of shape sensitivity analysis as a sys-
tematic methodology to determine the energy release rate of cracked shells,
within the framework of a linear elastic approach that takes into account
the effect of transverse shear deformation. This methodology and the direct
method of shape sensitivity analysis is applied to shells with an arbitra-
ry middle surface and leads to an explicit general expression for the shape
sensitivity of the total potential strain energy. In elastic shells with cracks,
crack initiation is simulated by a change of shape characterizedby a suitable
tangential velocity distribution over the middle surface of the shell. In this
case, a useful expression of energy release rate is expressed in terms of the
strain-stress state and the adopted shape change velocity field. Finally, sha-
pe sensitivity analysis is applied to the circular cylindrical shell and thus the
condition of null divergence of the corresponding Eshelby tensor is verified.

Key words: shape sensitivity analysis, elastic shells, linear elastic fracture
mechanics

1. Introduction

It is well known that a curved sheet containing a through crack has a
reduced resistance to fracture initiation. Moreover, it has been verified that
cracks in shells can severely decrease their strength, their load-carrying capa-
city, and in limit situations can also cause sudden failure. For this reason, the
interaction of flaws with shells curvature is a subject that has received careful
attention. To ensure structural integrity failure criterions have been developed
by the simultaneously application of shell theory and fracturemechanical con-
cepts. To establish a proper failure criterion, the knowledge of the shell stress



644 E.Taroco, R.A.Feijóo

distribution due to the presence of cracks and the fracture initiation law are
necessary, as it was pointed out by Folias (1974).

Concerning the shell stress distribution it is also well known in fracture
mechanics that in the vicinity of the crack tip or on the crack surface, the
transverse shear deformation can not be ignored. On the other hand, the ap-
proximateKirchhoff boundary conditions used in the classical theory of plates
and shells are inadequate to determine the stress field in the neighborhood of
the crack tip or the displacement of the crack surface. Nevertheless, this diffi-
culty can be overcome by selecting a shell theory that takes into account the
effect of transverse deformation. Such a hypothesis leads to a more accurate
solution for the stress analysis of cracked shells, as it was reported by Sih and
Hagendorf (1974).

In reference to the fracture initiation laws, the earliest work of Griffith
introduced the first statement of the energy balance criterion for crack ini-
tiation. Therefore, it has been accepted that the energy release rate provides
the work required to create new fracture surface in elastic materials. Thus,
the energy release rate has become a very useful parameter in linear elastic
fracture mechanics.

Since the pioneer work of Griffith, the continued interest in developping
the procedures to determine the energy release rate has been maintained up
to the present time.

Regarding the conservation laws and path-independent integrals that have
been widely used in the analysis of cracked bodies, we must go back to the
classic papers of Eshelby (1956, 1975) in which the notion of a force on a lat-
tice defect and the concept of the energy momentum tensor, were addressed.
Particularly, the J-integral proposed by Rice (1968) exemplifies how success-
fully these invariant integrals can be applied in fracture mechanics and also
indicates the usefulness of the J-integral in determining the energy release rate
of cracked bodies.

In a subsequent paper, Bergez and Radenkovic (1973) extended the con-
cept of path-independent integrals to the shell theory, even though they did
not place any restrictions on the geometry of themiddle surface.However, it is
accepted today, that such integrals are not path-independent in general. Fur-
ther, Lo (1980) has shown that J and related integrals, are path-independent
for circular cylindrical shell in the context of the Koiter linear elastic theory.
Later on, Kienzler andGolebiewska-Herrmann (1985) discussed the conserva-
tion laws in higher order shell theory. Recently, Li and Shyy (1997) derived
several new invariant integrals for shallow shells withinMarguerre’s approach.
More recently, Kienzler andHerrmann (2000) turned on to circular cylindrical



Shape sensitivity analysis of elastic shells with cracks 645

shell theory that accounted transverse shear deformation and the associate
Eshelby’s tensor in the framework of material space.

Since the energy release rate is generally interpreted as the rate of the
energy dissipated in the fracture process per unit crack propagation length,
and due to the difficulty of obtaining expressions for the potential energy as
explicit functions of the crack length, which enable us to obtain derivatives in
a direct form, several procedures, both numerical and experimental, have been
developed in fracturemechanics. Among them, the Shape SensitivityAnalysis
(proposed originally by Cèa (1981) developed mainly by Zolésio (1981), Ma-
smoudi (1987) andwidely discussed byHaug et al. (1986), can be successfully
applied. As shown in Feijóo et al. (2000), the crack growth is simulated as a
shape change of a 3D cracked body. Then, using the well known results from
the shape sensitivity analysis, the general expression for the energy release ra-
te was obtained in that paper.Moreover, this general expression is a function
of a velocity field describing the change of shape.

On the other hand, shape sensitivity analysis for curved elements was first
applied by Chenais andRousselet (1984) and later byRousselet (1987) in the
shape optimization of arches submitted to static loads. In addition, theoretical
aspects of axisymmetric shells and numerical results were reported by Mota
Soares et al. (1987).

In the case of arches, the analysis was performed along themid line and in
axisymmetric shells-along the meridian curve. In both cases the local system
of coordinates is orthogonal and the vector base is given by the unit normal
and the unit tangent to the curve (themid line in the case of the arch and the
meridian in the case of the axisymmetric shell).

However, in shells with arbitrary shape the coordinate curves over the
middle surface generally are not orthogonal. Therefore the corresponding equ-
ations have been established by using curvilinear coordinates over the middle
surface of the shell. Covariant and contravariant components of vectors and
tensors have to be introduced and partial derivatives are to be replaced by
covariant derivatives with the help of Christoffel’s symbols.

In the casewhen themiddle surface of a general shell is definedbya smooth
mapping of a two-dimensional domain, shape derivatives may be performed
by differentiation with respect to the mapping. In other words, the change of
the shape can be seen as the change of the mapping.

This approach and the LagrangianMethod, which allows to carry out the
derivative of any functional,wereappliedbyBernadou et al. (1991).Their aims
were in shape optimization of a thin shell within the framework of Koiter’s
theory formulated in an arbitrary curvilinear coordinates and subjected to



646 E.Taroco, R.A.Feijóo

different kinds of loads over the middle surface of the shell. A comprehensive
general analysis of shape sensitivity was included.

A rather simple and direct derivation in an orthogonal coordinate system,
shell theories and shape sensitivity analysis becomemore involved in arbitrary
curvilinear coordinates,making itdifficult to follow thephysicalmeaningof the
model.Toovercome this difficulty and towork independently of the coordinate
system, we adopted in this paper the intrinsic base defined at each point of
the middle surface by its unit normal vector and its tangent plane, cf. Valid
(1981).

Among a number of different possible approaches to the theory of small-
strain linear elastic shells, we have selected the one developed by Reissner
(1941), that takes into account the effect of transverse shear deformation. Re-
issner’s approach appears to us to bepreferable to other since it uses relatively
simple formulation and requires in the definition of strains only the first order
gradient of displacements. Moreover, it leads to results of considerable gene-
rality and is also suitable for the applications that are in focus of attention.

Considering the application to fracture mechanics to be conducted later
and to demonstrate the simplicity of the approach adopted in this work, we
have limited ourselves to carry out exclusively the shape sensitivity analysis
of the total potential energy of the shell submitted to static loads along its
boundary.

2. Shell shape change

In the present section, we introduce the concept of shape change of the
middle surface of the shell thatwill allow us to study the behavior of functions
and functionals when the shape of a shell is modified. Proposed originally by
Cèa (1981) andwidelydiscussedbyHaug et al. (1986), this approach simulates
the change in shape by a motion from an initial configuration to a known
deformed configuration characterized by the adopted velocity field defining
the shape change.
On the other hand, the basic idea behind almost all theories of shells is to

reduce theanalysis over themiddle surfacebymeansof simplifiedassumptions.
According to this, wemay characterize the shape of the shell by the geometry
of its middle surface.

Then, let us consider an elastic shell characterized by a smooth middle
surface Ωo, boundedbya curve thatwe also assume tobe smooth anddenoted
by ∂Ωo. The shape change of the shell will be defined by a known smooth



Shape sensitivity analysis of elastic shells with cracks 647

vector field V (X),X ∈ Ωo.Using this approach, the shapechangeof the shell,
and more precisely the shape change of its middle surface, can be described
by the transformation χτ, given by

χτ : Ωo 7→ Ωτ
(2.1)

xτ = χτ(X)=X+ τV (X) τ ∈ R
+

for τ ∈ R+ sufficiently small.

Thus, the shape change is a smooth one-parameter family of transforma-
tions where V (X) is the direction of the domain variation. This means that,
for a given direction V (X), the shape change of Ωo is uniquely determined
by the parameter τ ∈ R+.

The transformed domain Ωτ might be considered as a deformed confi-
guration of the initial domain Ωo under the transformation from Ωo to Ωτ
defined by (2.1). Furthermore, introducing the continuummechanics termino-
logy, Gurtin (1981), an analogy can be drawn between change of shape and
motion of a body. From this point of view, V (X) can be seen as the shape
change velocity field.

From now on and to simplify the notation, we will omit the subscript τ
identifying Ωτ (∂Ωτ) with Ω (∂Ω). Moreover, the surface Ω can be seen as
the actual description of the middle surface at each value of τ. Therefore,
the surface Ω might be considered as a perturbation of the initial surface Ωo
and the transformation from Ωo to Ω, as a function of the point X and the
parameter τ.

Since at each τ, the shape change is a one-to-one transformation from Ωo
to Ω, there is a unique inverse transformation χ−1τ of Ω to Ωo.

Hence, any scalar, vector, or tensor field associated with the shape change
can be expressed as a function over the initial surface Ωo, or a function over
the actual surface Ω.Within the continuummechanics analogy, we call them
material and spatial descriptions, respectively. For instance, in the particular
case of the shape change velocity field, wemay write for both descriptions

V =V (X) v=v(τ;x) (2.2)

In this paper we shall carry out the analysis over the middle surface of
the shell in the actual configuration. In other words we will adopt the spatial
description, taking advantage of thewell-known expressions of the material or
total (time) derivatives of spatial fields developed in Continuum Mechanics,
Gurtin (1981). From this analogy, the shape sensitivity of any regular functio-



648 E.Taroco, R.A.Feijóo

nal characterized by its spatial description Ψ(τ;x), can formally be defined as

dΨ

dτ
=
{∂Ψ(τ;x)

∂τ

∣∣∣
x=χτ(X)

}∣∣∣
X=χ

−1

τ (x)
(2.3)

Furthermore, in the shape sensitivity analysis of shells it is convenient to
describe vector and tensor fields using the intrinsic shell frame defined at
each point x of the middle surface Ω by its unit normal n, and its tangent
plane Tp.
In addition, we introduce the projection tensor operator over the tangent

plane Tp and the projection tensor operator over the normal vector n, respec-
tively denoted by Π and n⊗n. Hence, the unit tensor Imay be described as

I=Π+(n⊗n) (2.4)

where ⊗ denotes the tensorial product of vectors.
Considering the application to cracked shells, we assume that the shape

change velocity at each point of themiddle surface lies over the corresponding
tangent plane, thus the spatial description of this velocity is denoted by vt.
Moreover, we assume that both the unit normal vector n and the tangent
vector vt, are smooth fields on Ω.
We also define the surface gradient of spatial fields (Gurtin, 2000). This

surface gradient can be seen as the restriction to Tp of the usual gradient

grads(·)= grad(·)|Tp (2.5)

Fore instance, the surface gradient of the spatial description of the velocity
field, admits a unique decomposition into tangential and normal components

gradsvt =Πgradsvt+n⊗ (gradsvt)
⊤
n (2.6)

In particular, we denote by gradsn, the surface gradient of the unit nor-
mal n. This gradient, known in the literature as the curvature tensor, has a
central place in the theory of surfaces and is concomitant with the formulation
of theory of shells that reduces the analysis to the middle surface.
Since n is a unit vector, the surface gradient of the scalar product n·n=1

leads to
(gradsn)

⊤n=0 (2.7)

From (2.7), we conclude that gradsn lies on the tangent plane Tp at the
point xunder consideration. In addition, it can be easily verified that gradsn
is a symmetric tensor (Gurtin, 2000), thus

gradsn=(gradsn)
⊤ (2.8)



Shape sensitivity analysis of elastic shells with cracks 649

On the other hand, as a consequence of the orthogonality between the
vectors vt and n, the surface gradient of the scalar product vt ·n = 0,
leads to

(gradsvt)
⊤n=−(gradsn)vt (2.9)

Subsequently, inserting (2.9) into (2.6), we may write

gradsvt =Πgradsvt−n⊗ (gradsn)vt (2.10)

Here, thefirst termon the right-hand side represents the tangential component
and the last term – the normal component of the surface gradient of the
velocity.
It will be evident in the following sections of this paper that the surface

gradient of the velocity plays an outstanding role in the shape sensitivity
analysis of shells.

3. The potential energy of the shell

The purpose of this section is to introduce the mechanical model and the
expression of the cost function, using the well-known terminology of optimi-
zation. As a first step we assume the selection of the mechanical model and
the cost function. In spite of the fact that classical shell theories are quite ap-
propriate for problems without stress singularities, when fracture mechanics
must be included, more accurate theories are necessary to adequately model
thebehavior of the shell near the crack tip region.Amonganumberof different
possible approaches to the analysis of elastic shells, we have selected onewhich
appears to us to be preferable to others since it leads to results of considerable
generality using only first order gradient in the strain-displacement relations.
For simplicity we shall be concerned with a shell within the framework of a
linear elastic small-strain approach that takes into account the effect of trans-
verse shear deformation, known in the shell literature as Reissner’s theory.
Furthermore, considering the application to shells containing cracks to be ac-
complished later, we adopt as cost function the total potential energy of the
shell under analysis, given by

ψ(ut,un,ϑ)= U −W =
∫

Ω

φ(εs,γ,κs) dΩ −W (3.1)

Here, φ denotes the specific strain energy of the shell, ut, un,ϑ the kinema-
tically admissible displacement fields, εs, γ, κs the strains associated to the



650 E.Taroco, R.A.Feijóo

displacements by kinematical relations and the super index (•)s the symme-
tric part of the tensor (•). The domain integral on the right-hand side of the
above expression represents the total strain energy stored in the shell and W
the external work.

Within the framework of Reissner’s approach, the strain-displacement re-
lations take the form

εs =(Πgradsut)
s+ungradsn

γ=−ϑ+ gradsun− (gradsn)ut (3.2)

κ
s =(Πgradsϑ)

s

Here, the vector field ut denotes the tangent displacement, the scalar field
un is the normal displacement, the tangential vector field ϑ is the rotational
angle of the normal at any point of the middle surface.

Hence, from the above definition of the strains, stretching of the middle
surface εs leads to a tangent second order tensor field, the transversal shearing
γ to a tangent vector field, and the flexural strain κs to a tangent secondorder
tensor field.

As it was noted, the strain-displacement relations of this approach, given
by (3.2), involve all three displacements and require in its definition only the
first order gradient.

Then, to throw additional light onReissner’s kinematical assumptions, the
meaning of the surface gradient of scalar, tangential vector and unit normal
fields are essential.

3.1. Total strain energy

According to Reissner’s assumptions, the total elastic strain energy stored
in the shell U may be expressed as the sum of the stretching Uε, shearing Uγ
and flexural strain energy Uκ

U = Uε+Uγ +Uκ (3.3)

given, respectively, by

Uε =

∫

Ω

φε dΩ Uγ =

∫

Ω

φγ dΩ Uκ =

∫

Ω

φκ dΩ (3.4)

where the scalars φε, φγ and φκ denote, respectively, the specific elastic stret-
ching, shearing and flexural shell energy.



Shape sensitivity analysis of elastic shells with cracks 651

3.2. External work

For simplicitywewill not consider thebody forces. In this case the external
work is performed by a system of loads applied along the boundary, hence

W =

∫

∂Ωt

(t ·ut+qun+m ·ϑ) d∂Ω (3.5)

Here, t, q and mdenote the loads prescribedon theboundary ∂Ωt, the vector
field t denotes the tangent force, the scalar field q – the shearing force and
the tangential vector field m the moment. These loads are compatible with
the shell model under analysis.

4. Variational form of the equilibrium

Now, making use of the Principle of Virtual Power (which is equivalent to
the Principle of Minimum Total Potential Energy due to the assumption ad-
opted in thiswork), the equilibriumof the shell can bewritten in the following
variational form:

• Find ut, un and ϑ∈ Kin such that

∫

Ω

N · (Πgradsût+ ûngradsn) dΩ +

+

∫

Ω

Q ·
[
− ϑ̂+ gradsûn− (gradsn)ût

]
dΩ +

∫

Ω

M ·Πgradsϑ̂ dΩ −(4.1)

−
∫

∂Ωt

(t · ût+qûn+m · ϑ̂) d∂Ω =0

for all ût, ûn and ϑ̂ ∈ Kin, and where Kin is the space of admissible
kinematical displacements.

We also assume that the fields ut, un and ϑ are prescribed (null for sim-
plicity) along the boundary ∂Ωu (∂Ω = ∂Ωu∪∂Ωt ; ∂Ωu∩∂Ωt = ∅).
In theabovevariational form, the tangential symmetric secondorder tensor

N denotes the membrane force of the shell, the tangential vector Q – the
transverse shearing force and the tangential symmetric second order tensor M



652 E.Taroco, R.A.Feijóo

– the bending moment in the middle surface of the shell. They are given by
the formulae

N=
∂φε
∂ε

Q=
∂φγ
∂γ

M=
∂φκ
∂κ

(4.2)

Next, we insert the following surface tensor relations

N ·Πgradsût = divs(N
⊤
ût)− ût ·ΠdivsN

Q · gradsûn = divs(ûnQ)− ûndivsQ (4.3)

M ·Πgradsϑ̂= divs(M
⊤ϑ̂)− ϑ̂ ·ΠdivsM

Further, by the use of the surface divergence theorem

∫

Ω

divs(N
⊤
ût) dΩ =

∫

∂Ωt

Nm · ût d∂Ω

∫

Ω

divs(ûnQ) dΩ =

∫

∂Ωt

(Q ·m)ûn d∂Ω (4.4)

∫

Ω

divs(M
⊤ϑ̂) dΩ =

∫

∂Ωt

Mm · ϑ̂ d∂Ω

the Principle of Virtual Power (4.1) can be rewritten as

−
∫

Ω

[
ΠdivsN+(gradsn)Q

]
· ût dΩ −

∫

Ω

[
divsQ−N · gradsn

]
ûn dΩ −

−
∫

Ω

[
ΠdivsM+Q

]
· ϑ̂ dΩ + (4.5)

+

∫

∂Ωt

[
(Nm− t) · ût+(Q ·m−q)ûn+(Mm−m) · ϑ̂

]
d∂Ω =0

where divs denotes the spatial surface divergence of vector or tensor fields
and m is the outward unit normal vector to the boundary curve ∂Ω. This
normal lies on the intersection of the tangent plane to the middle surface of
the shell and the normal plane orthogonal to the unit tangent vector of the
curve ∂Ω at the point under consideration. In the theory of surface curves,
the unit tangent vector to the curve ∂Ω together with the normal vectors n
and m, mutually orthogonal, compose the intrinsic frame of ∂Ω.



Shape sensitivity analysis of elastic shells with cracks 653

The above expression furnishes the Euler surface equations of the shell
(strong form of the equilibrium) associated to the Principle of Virtual Power
in Ω

ΠdivsN+(gradsn)Q=0

divsQ−N · gradsn=0 (4.6)

ΠdivsM+Q=0

as well as the natural boundary conditions on ∂Ωt

Nm= t Q ·m= q Mm=m (4.7)

The coupled nature of the equilibrium equations is a direct consequence of
the strain-displacement relations adopted.

5. Shape derivative of vectors

In this section we start applying the analogy between the material (total
time) derivative and shape derivative to obtain the shape derivatives of the
tangential vector ut and the unit normal vector n. When the direct method
is used, these derivatives are useful to perform the shape derivative of the
corresponding surface gradients.
In its general form, thematerial (total time) derivative of superficial fields

defined by (2.3), may be rewritten as

d

dτ
Ψ(τ;x)=

{ ∂
∂τ
{Ψ(τ;x)}m

}

sp
(5.1)

Here, Ψ denotes a scalar, vector or tensor field, the subscript m – thematerial
description and sp – the spatial description.
To this end, we focus our attention on shape changes characterized by

the spatial description of a tangential velocity field given by vt(x). Then, the
shape change gradient F defined at each material point X (dx = FdX) is
such that its partial time (τ) derivative is given by

∂

∂τ
F= {gradsvt}mF (5.2)

Moreover, from FF−1 = I we have

∂

∂τ
F
−1 =−F−1{gradsvt}m (5.3)



654 E.Taroco, R.A.Feijóo

Derivative of the tangent vector field ut

The spatial description of the tangent vector ut and its material descrip-
tion denoted by ũt are related as follows

{ut}m =Fũt (5.4)

Next we apply the partial derivative with respect to τ on both sides of
(5.4) and, inserting (5.2), we have

∂

∂τ
{ut}m =

∂

∂τ
(Fũt)=

∂F

∂τ
ũt = {gradsvt}mFũt (5.5)

Further, inserting (5.5) into (5.2), we obtain

dut
dτ
=
{
{gradsvt}mFũt

}

sp
=(gradsvt)ut (5.6)

thus, combining (2.10) and (5.6), the vector dut/dτ can be written in the
convenient form

dut
dτ
=Π

dut
dτ
− [(gradsn)vt ·ut]n (5.7)

Here, thefirst termon the right-hand side represents the tangential component
and the second one – the normal component of the total derivative of ut.

Derivative of the normal vector field n

Since n is a unit vector field, the first information about dn/dτ may be
found by differentiating the scalar product n ·n=1with respect to τ

dn

dτ
·n=0 (5.8)

Thus, the total time derivative dn/dτ results in a tangential vector.
Moreover, as the vectors ut and n remain orthogonal, the differentiation

of the scalar product n ·ut =0with respect to τ, leads to

dn

dτ
·ut =−n ·

dut
dτ

(5.9)

From (5.7) and (5.9) it is finally obtained

dn

dτ
=(gradsn)vt (5.10)

As itwill be seen later, the total derivative of nwill beused in theapproach
of shape sensitivity analysis of shells presented in this work.



Shape sensitivity analysis of elastic shells with cracks 655

6. Shape derivative of surface gradients

While in the formulation of Reissner’s shell model, the knowledge of first
order surface gradients were necessary, in the shape sensitivity analysis the
shape derivative of these gradients must be carried out.
For a superficial vector field, we may write

{gradsu}m =(∇sũ)F
−1 (6.1)

where u and ũ denote respectively the spatial and material (referential)
descriptions of the displacement, ∇s is the material surface gradient and
F – the shape change gradient.
From (6.1) and (5.4), we have

∂

∂τ
[(∇sũ)F

−1] =
(
∇s

∂ũ

∂τ

)
F
−1+(∇sũ)

∂

∂τ
F
−1 =

(6.2)

=
{
grads

du

dτ

}

m
−{gradsu}m{gradsvt}m

Thus, combining (5.1) and (6.2), we obtain

d

dτ
gradsu= grads

du

dτ
− gradsugradsvt (6.3)

On the other hand, in terms of intrinsic components of the displacement
(ut,un), we may write

u=ut+unn

du

dτ
=
dut
dτ
+
dun
dτ
n+un

dn

dτ
gradsu= gradsut+ungradsn+n⊗ gradsun (6.4)

grads
du

dτ
= grads

dut
dτ
+
dun
dτ
gradsn+n⊗ grads

dun
dτ
+

+ungrads
dn

dτ
+
dn

dτ
⊗ gradsun

If we insert (6.4)3,4 into (6.3), we obtain

d

dτ
gradsun = grads

dun
dτ
− (gradsvt)

⊤gradsun

d

dτ
gradsn= grads

dn

dτ
− gradsngradsvt (6.5)

d

dτ
gradsut = grads

dut
dτ
− gradsutgradsvt



656 E.Taroco, R.A.Feijóo

Derivative of the gradient of scalar field un

In the case of the gradient of a scalar-valued field, we recall (6.5)1. Again,
since gradsun is a tangent vector field, introduction (2.6) into (6.5)1 yields

d

dτ
gradsun = grads

dun
dτ
− (Πgradsvt)

⊤gradsun (6.6)

Therefore, the total derivative of gradsun leads also to a tangent vector.

Derivative of the gradient of the normal vector field n

Now, we return to (6.5)2. Upon inserting (5.10) into (6.5)2 and making
further use of the following tensorial relation

grads
[
(gradsn)

⊤vt
]
=(gradsgradsn)

⊤vt+(gradsn)
⊤gradsvt (6.7)

the total derivative of gradsn becomes

d

dτ
gradsn=(gradsgradsn)

⊤
vt (6.8)

Derivative of the gradient of the tangential vector field ut

For any superficial vector field, in particular for the tangential vector field
ut, similarly to (2.10), the following relation holds

Πgradsut = gradsut+n⊗ (gradsn)ut (6.9)

Next, by differentiating with respect to τ, we obtain

d

dτ
(Πgradsut)=

d

dτ
gradsut+

dn

dτ
⊗ (gradsn)ut+n⊗

d

dτ
[(gradsn)ut]

(6.10)
If we apply the projector tensor Π on both sides, and introducing (5.10), the
foregoing expression reduces to

Π
d

dτ
(Πgradsut)=Π

d

dτ
gradsut+(gradsn)vt⊗ (gradsn)ut (6.11)

From (6.5)3 and (6.11), we may write

Π
d

dτ
(Πgradsut)=Πgrads

dut
dτ
−Πgradsutgradsvt+

(6.12)

+(gradsn)vt⊗ (gradsn)ut



Shape sensitivity analysis of elastic shells with cracks 657

Finally, substitution of (5.7) into (6.12) yields

Π
d

dτ

(
Πgradsut

)
=ΠgradsΠ

dut
dτ
− [(gradsn)vt ·ut]gradsn−

(6.13)

−Πgradsutgradsvt+(gradsn)vt⊗ (gradsn)ut

With the preceding results, we can now perform the shape derivative of
the strains in Reissner’s model.

7. Shape derivative of the strains

Inorder to obtain the total derivatives of the stretching strain ε, transverse
shearing strain γ, and flexural strain κ, with respect to the parameter τ, we
return to the strain-displacement relations given by (3.2). An inspection of
these equations shows us that we have to perform the total derivatives of the
unit normal vector field n, the tangent vector fields ut and ϑ. Further we
shall perform the total derivatives of the surface gradient of un,n,ut and ϑ.
Within the continuum mechanics analogy and using the general expressions
of thematerial (total time) derivative of superficial fields, we carry out in this
section the shape derivatives following the definition given in (2.3).

Shape derivative of the stretching strain ε

Fromthedefinition of stretching (3.2)1, the total timederivative of εyields

dε

dτ
=

d

dτ
(Πgradsut)+

dun
dτ
gradsn+un

d

dτ
gradsn (7.1)

As it will be seen later, in the present approach the tangential component
of dε/dτ will be relevant, thus

Π
dε

dτ
=Π

d

dτ
(Πgradsut)+

dun
dτ
gradsn+unΠ

d

dτ
gradsn (7.2)

Upon introducing (6.13) into (7.2), the tangential component of the total de-
rivative of the stretching strain can be written as

Π
dε

dτ
=−Πgradsutgradsvt+unΠ

d

dτ
gradsn− [(gradsn)vt ·ut]gradsn+

(7.3)

+(gradsn)vt⊗ (gradsn)ut+ΠgradsΠ
dut
dτ
+
dun
dτ
gradsn



658 E.Taroco, R.A.Feijóo

Shape derivative of the shearing strain γ

Likewise, from (3.2)2, the total derivative of γ, takes the form

dγ

dτ
=−

dϑ

dτ
+

d

dτ
gradsun−

( d
dτ
gradsn

)
ut− (gradsn)

dut
dτ

(7.4)

By inserting the derivative of gradsun from (6.6) into the above expression,
the total derivative of the shearing strain leads to

dγ

dτ
=−(Πgradsvt)

⊤gradsun−
( d
dτ
gradsn

)
ut−

dϑ

dτ
+

(7.5)

+grads
dun
dτ
− (gradsn)

dut
dτ

Shape derivative of the flexural strain κ

From (3.2)3, the tangential component of the total derivative of κmay be
written as

Π
dκ

dτ
=Π

d

dτ
Πgradsϑ (7.6)

Since the rotation ϑ is a tangent vector, the evaluation of its total time deri-
vative will be entirely similar to the evaluation of Πd(Πgradsut)/dτ shown
in (6.13), thus (7.6) may be rewritten as

Π
dκ

dτ
=−Πgradsϑgradsvt− [(gradsn)vt ·ϑ]gradsn+

(7.7)

+(gradsn)vt⊗ (gradsn)ϑ+ΠgradsΠ
dϑ

dτ

8. Shell shape sensitivity

Let us begin the present section with differentiating the cost functionwith
respect to the parameter τ. Due to the approach adopted in this work, in
which the potential energy is chosen as the cost function, combining (3.1) and
(3.3), we may rewrite dψ/dτ as

dψ

dτ
=

d

dτ
Uε+

d

dτ
Uγ +

d

dτ
Uκ−

d

dτ
W (8.1)

Thus, to perform the total derivative of the potential energy of the shell,
it will be required to calculate the total derivative of each one of its terms.



Shape sensitivity analysis of elastic shells with cracks 659

Shape derivative of the stretching strain energy Uε

Let us first consider the elastic stretching strain energy Uε, given by (3.4)1.
Upon the application of Reynolds’ transport theorem, the total derivative of
Uε with respect to τ yields

d

dτ
Uε =

∫

Ω

(dφε
dτ
+φεdivsvt

)
dΩ =

∫

Ω

(∂φε
∂εs
·
dεs

dτ
+φεdivsvt

)
dΩ (8.2)

where divsvt represents the surface divergence of the shape change velocity,
defined by

divsvt = Ip ·Πgradsvt (8.3)

where Ip denotes the identity tensor over the tangent plane.

Expression (8.2) can be rewritten as

d

dτ
Uε =

∫

Ω

(
N ·Π

dε

dτ
+φεdivsvt

)
dΩ (8.4)

Furthermore, by substituting (7.3) and (8.3) into (8.4), the total derivative of
the stretching energy may be rewritten as

d

dτ
Uε =

∫

Ω

[φεIp− (Πgradsut)
⊤
N] · gradsvtdΩ +

∫

Ω

unN ·Π
( d
dτ
gradsn

)
dΩ −

−
∫

Ω

[(N · gradsn)ut−N(gradsn)ut] · (gradsn)vt dΩ + (8.5)

+

∫

Ω

N ·
[
ΠgradsΠ

dut
dτ
+
dun
dτ
gradsn

]
dΩ

Shape derivative of the shearing strain energy Uγ

In the same manner as before, the total derivative of Uγ given in (3.4)2
can be written as

d

dτ
Uγ =

∫

Ω

(dφγ
dτ
+φγdivsvt

)
dΩ =

∫

Ω

(
Q ·

dγ

dτ
+φγdivsvt

)
dΩ (8.6)



660 E.Taroco, R.A.Feijóo

Thus, from (7.5), (8.3) and (8.6), the total derivative of the shearing energy
becomes

d

dτ
Uγ =

∫

Ω

[φγIp− (gradsun⊗Q)] · gradsvt dΩ −

−
∫

Ω

(Q⊗ut) ·
( d
dτ
gradsn

)
dΩ + (8.7)

+

∫

Ω

Q ·
[
−
dϑ

dτ
+ grads

dun
dτ
− (gradsn)

dut
dτ

]
dΩ

Shape derivative of the flexural strain energy Uκ

Similarly, from (3.4)3, we obtain

d

dτ
Uκ =

∫

Ω

(dφκ
dτ
+φκdivsvt

)
dΩ =

∫

Ω

(
M ·Π

dκ

dτ
+φκdivsvt

)
dΩ (8.8)

Finally, from (7.7) and (8.3), the total derivative of the flexural energy
equals

d

dτ
Uκ =

∫

Ω

[φκIp− (Πgradsϑ)
⊤
M] · gradsvt dΩ −

−
∫

Ω

[(M · gradsn)ϑ−M(gradsn)ϑ] · (gradsn)vt dΩ + (8.9)

+

∫

Ω

M ·ΠgradsΠ
dϑ

dτ
dΩ

Shape derivative of the external work W

Proceeding as before and assuming that the prescribed load at the boun-
dary remains unchanged, the material derivative of W may be expressed as

d

dτ
W =

∫

∂Ωt

[
t ·

d

dτ
ut+q

d

dτ
un+m ·

d

dτ
ϑ+(t ·ut+qun+m ·ϑ)divsvt

]
d∂Ω

(8.10)

In what follows we also assume that the shape change of the shell, charac-
terized by the velocity field vt, is such that divsvt =0 on ∂Ω and vt =0 on



Shape sensitivity analysis of elastic shells with cracks 661

∂Ωu. Hence, the preceding expression takes the more simple form

d

dτ
W =

∫

∂Ωt

[
t ·

d

dτ
ut+ q

d

dτ
un+m ·

d

dτ
ϑ
]
d∂Ω =

(8.11)

=

∫

∂Ωt

[
t ·Π

d

dτ
ut+q

d

dτ
un+m ·Π

d

dτ
ϑ
]
d∂Ω

Furthermore, by combination of (8.1), (8.5), (8.7), (8.9) and (8.11), the shape
derivative of the potential energy of the shell can be written as

dψ

dτ
=

∫

Ω

[
(φε+φγ +φκ)Ip− (Πgradsut)

⊤
N− gradsun⊗Q−

−(Πgradsϑ)
⊤
M

]
· gradsvt dΩ +

∫

Ω

(unN−Q⊗ut) ·
d

dτ
gradsndΩ −

−
∫

Ω

[
(N · gradsn)ut−N(gradsn)ut+(M · gradsn)ϑ− (8.12)

−M(gradsn)ϑ
]
· (gradsn)vt dΩ +

∫

Ω

N ·
(
ΠgradsΠ

dut
dτ
+
dun
dτ
gradsn

)
dΩ +

+

∫

Ω

Q ·
[
−
dϑ

dτ
+ grads

dun
dτ
− (gradsn

)dut
dτ

]
dΩ +

∫

Ω

M ·ΠgradsΠ
dϑ

dτ
dΩ −

−
∫

∂Ωt

(
t ·Π

dut
dτ
+q

dun
dτ
+m ·Π

dϑ

dτ

)
d∂Ω

In addition, since the shell is in equilibriumwith theprescribed loads along
the boundary and taking into account that Πdut/dτ, dun/dτ and Πdϑ/dτ
belong to Kin, the Principle of Virtual Power (4.1), leads to

∫

Ω

N ·
(
ΠgradsΠ

dut
dτ
+
dun
dτ
gradsn

)
dΩ +

+

∫

Ω

Q ·
[
−
dϑ

dτ
+ grads

dun
dτ
− (gradsn

)dut
dτ

]
dΩ + (8.13)

+

∫

Ω

M ·ΠgradsΠ
dϑ

dτ
dΩ −

∫

∂Ωt

(
t ·Π

dut
dτ
+q

dun
dτ
+m ·Π

dϑ

dτ

)
d∂Ω =0



662 E.Taroco, R.A.Feijóo

Further, by combination of (8.12) and (8.13), the shape derivative of the po-
tential energy of the shell reduces to

dψ

dτ
=

∫

Ω

[
(φε+φγ +φκ)Ip− (Πgradsut)

⊤
N− gradsun⊗Q−

−(Πgradsϑ)
⊤
M

]
· gradsvt dΩ +

∫

Ω

(unN−Q⊗ut) ·
d

dτ
gradsndΩ −

(8.14)

−
∫

Ω

[
(N · gradsn)ut−N(gradsn)ut+(M · gradsn)ϑ−

−M(gradsn)ϑ
]
· (gradsn)vt dΩ

The expression in brackets in the first integral on the right-hand side of
the above equation we denote as

Σ=(φε+φγ+φκ)Ip−(Πgradsut)
⊤
N−gradsun⊗Q−(Πgradsϑ)

⊤
M (8.15)

An inspection of this expression enables us to recognize the similarity be-
tween Σ and the energy momentum tensor, introduced by Eshelby (1975) in
the analysis of defects in three-dimensional elasticity in the context of infini-
tesimal deformation. Thus Σ could be viewed as an extension of Eshelby’s
tensor used for the analysis of elastic shells within Reissner’s approach. The
energy momentum tensor Σ yields a tangential tensor and to point out the
effects of the stretching, transversal shearing and flexural strain, it can be
expressed as the sum of three terms

Σ=Σε+Σγ +Σκ (8.16)

where

Σε = φεIp− (Πgradsut)
⊤
N Σγ = φγIp− gradsun⊗Q

(8.17)

Σκ =φκIp− (Πgradsϑ)
⊤
M

Next, we insert (6.8) and (8.15) into (8.14), to obtain

dψ

dτ
=

∫

Ω

Σ · gradsvt dΩ +
∫

Ω

(unN−Q⊗ut) · [(gradsgradsn)
⊤vt] dΩ −

−
∫

Ω

[
(N · gradsn)ut−N(gradsn)ut+(M · gradsn)ϑ− (8.18)

−M(gradsn)ϑ
]
· (gradsn)

⊤
vt dΩ



Shape sensitivity analysis of elastic shells with cracks 663

Finally, taking into account the definition of the transpose of the second and
third order tensors, this expression may be written in a more suitable form

dψ

dτ
=

∫

Ω

Σ · gradsvt dΩ +
∫

Ω

(gradsgradsn)(unN−Q⊗ut) ·vt dΩ −

−
∫

Ω

(gradsn)
[
(N · gradsn)ut−N(gradsn)ut+(M · gradsn)ϑ− (8.19)

−M(gradsn)ϑ
]
·vt dΩ

It should be emphasized that the shape derivative of the potential strain
energy of the shell, given by the foregoing expression, is exclusively a function
of the strain-stress state and the adopted shape change velocity field vt.

9. Boundary integral

Let us review in this section the shape derivative of the total potential
energy of the shell in the light of the expression of theReynolds’ theorem that
allows us to rewrite the mentioned derivative as a boundary integral.

To do so, we assume that the shape change of the shell is given by the
tangential velocity field vt defined along its boundary ∂Ωt (vt = 0 along
∂Ω −∂Ωt). Then, we recall the definition of the potential energy of the shell
that, in the present analysis, takes the following form

ψ(ut,un,ϑ)= U−W =
∫

Ω

φ(εs,γ,κs) dΩ−
∫

∂Ωt

(t·ut+qun+m·ϑ) d∂Ω (9.1)

Next, consider the first termon the right-hand side of the above expression
and by using Reynolds’ theorem, the shape derivative of the strain energy of
the shell yields

d

dτ
U =

∫

Ω

∂φ

∂τ
dΩ +

∫

∂Ω

φvt ·m d∂Ω =
∫

Ω

∂φ

∂τ
dΩ +

∫

∂Ωt

φvt ·m d∂Ω (9.2)

Further consider the second term on the right-hand side of (9.1). In the same
manner as before and with the assumption that the prescribed loads at the



664 E.Taroco, R.A.Feijóo

boundary remainunchanged (td∂Ω, qd∂Ω and md∂Ω are independent of the
parameter τ), the shape derivative of the external work may be expressed as

d

dτ
W =

∫

∂Ωt

t ·Π
[ ∂
∂τ
ut+(gradsut)vt

]
d∂Ω +

(9.3)

+

∫

∂Ωt

q
[ ∂
∂τ
un+(gradsun) ·vt

]
d∂Ω +

∫

∂Ωt

m ·Π
[ ∂
∂τ
ϑ+(gradsϑ)vt

]
d∂Ω

Upon combining (9.1), (9.2), (9.3), the total derivative of the potential energy
becomes

dψ

dτ
=

∫

Ω

∂φ

∂τ
dΩ −

∫

∂Ωt

[
t ·Π

∂

∂τ
ut+q

∂

∂τ
un+m ·Π

∂

∂τ
ϑ
]
d∂Ω +

(9.4)

+

∫

∂Ωt

[
φvt ·m− t · (Πgradsut)vt−q(gradsun) ·vt−m · (Πgradsϑ)vt

]
d∂Ω

Moreover, as the shell is in equilibriumwith the applied loads along its boun-
dary, from the Principle of MinimumTotal Potential Energy, we have

∫

Ω

∂φ

∂τ
dΩ −

∫

∂Ωt

[
t ·Π

∂

∂τ
ut+q

∂

∂τ
un+m ·Π

∂

∂τ
ϑ
]
d∂Ω =0 (9.5)

Therefore, from (9.4), (9.5) and the natural boundary conditions (4.7), the
shape derivative of the total potential energy of the shell, becomes

dψ

dτ
=

∫

∂Ωt

Σm ·vt d∂Ω =
∫

∂Ω

Σm ·vt d∂Ω (9.6)

This expression points out that when the change of the shape of the shell is
performed by a tangential velocity defined along the boundary, according to
our assumption, the shape derivative of the potential energy leads to a path
integral that represents thefluxof theEshelby energymomentumtensor along
the boundary of the shell.

The foregoing result allows us to know somethingmoreabout theEshelby’s
tensor for elastic shells.



Shape sensitivity analysis of elastic shells with cracks 665

10. Eshelby’s momentum energy tensor

This section is devoted to perform the surface divergence of Eshelby’s
tensor within the framework ofReissner’s shell theory. It should be noted that
the surface divergence of a tangential tensor results in a vector with tangential
and normal components. Nevertheless we will now focus our attention on the
tangential component; the application to cracked shells will be conducted in
the next section.
To do so, let us compare the shape derivative of the total potential energy

expressed as a domain integral (8.19) with the same shape derivative carried
out in (9.6). Thus, wemay write

dψ

dτ
=

∫

∂Ω

Σm ·vt d∂Ω =
∫

Ω

Σ ·Πgradsvt dΩ +

+

∫

Ω

(gradsgradsn)(unN−Q⊗ut) ·vt dΩ −
∫

Ω

(gradsn)
[
(N · gradsn)ut−

(10.1)

−N(gradsn)ut+(M · gradsn)ϑ−M(gradsn)ϑ
]
·vt dΩ

Subsequently, we introduce the following tensor relation

divs(Σ
⊤vt)=ΠdivsΣ ·vt+Σ ·Πgradsvt (10.2)

Next, we integrate the preceding expression over the domain Ω and further
we make use of the divergence theorem to obtain

∫

∂Ω

Σm ·vt d∂Ω =
∫

Ω

Σ ·Πgradsvt dΩ +
∫

Ω

ΠdivsΣ ·vt dΩ (10.3)

Then, from (10.1) and (10.3), we have
∫

Ω

{
ΠdivsΣ− (gradsgradsn)(unN−Q⊗ut)+(gradsn)

[
(N · gradsn)ut−

(10.4)

−N(gradsn)ut+(M · gradsn)ϑ−M(gradsn)ϑ
]}
·vt dΩ =0

Since the above equation should hold for arbitrary tangential velocity vt,
the quantity in the parentheses must vanish, then

ΠdivsΣ=Π(gradsgradsn)(unN−Q⊗ut)− (gradsn)
[
(N · gradsn)ut−

(10.5)

−N(gradsn)ut+(M · gradsn)ϑ−M(gradsn)ϑ
]



666 E.Taroco, R.A.Feijóo

Consequently, we have obtained the expression of the projection over the
tangent plane of the surface divergence of Eshelby’s tensor of elastic shells
within the framework of Reissner’s approach. On the other hand, using (10.5)
together with the shape derivative of the potential energy expressed as a do-
main integral (8.19), we arrive to the equivalent expression of shape derivative
as a path integral (9.6).

10.1. Circular cylindrical shell

Now, the shape derivative of the potential energy will be applied to a
circular cylindrical shell. At the begining we carry out the first and second
order surface gradient of n, that characterize the geometry of the middle
surface of the shell.
For circular cylindrical shell, one of the principal radii of curvature is infi-

nite and the other is constant. Consequently, the second order surface gradient
of n vanishes

gradsgradsn=0 (10.6)

Moreover, if we assume over the tangent plane orthogonal base vectors eφ and
ex, respectively following the circumferential and the longitudinal directions,
the only non-vanishing component of gradsn is given by

(gradsn)φφ = eφ · (gradsn)eφ = r
−1 (10.7)

where r denotes the radius of curvature.
If we assume that (10.7) and (10.6) hold, it can be easily verified that the

right-hand side of (10.5) vanishes, thus

Π(gradsgradsn)(unN−Q⊗ut)− (gradsn)
[
(N · gradsn)ut−

(10.8)

−N(gradsn)ut+(M · gradsn)ϑ−M(gradsn)ϑ
]
=0

Then, combining (10.5) and (10.8) we obtain

ΠdivsΣ=0 (10.9)

Therefore, for displacement fields ut, un and ϑ in equilibrium with the
applied loads on the boundary of a circular cylindrical shell, the surface diver-
gence of Eshelby’s tensor projected over the tangent plane vanishes.
Finally, combining (10.1) and (10.8), the following relations holds

dψ

dτ
=

∫

∂Ω

Σm ·vt d∂Ω =
∫

Ω

Σ ·Πgradsvt dΩ (10.10)



Shape sensitivity analysis of elastic shells with cracks 667

11. Cracked elastic shells

In the present section we apply the analysis to the case of an elastic shell
containing cracks.

SinceGriffith provided the primary criterion for crack extension in linearly
elastic bodies, the energy release rate has played an essential role in fracture
mechanics. Therefore, our aim in this section is to evaluate the energy release
rate of an elastic shellwith arbitrary smoothmiddle surface containing a crack.

Todo this, let us consider a plane P cutting themiddle surface of the shell
along the (smooth) curve C, see Fig.1. The crack is a part of this curve and
its faces are denoted by C+c and C

−
c , respectively. We assume that the shell

is in equilibrium with a given traction at the boundary. For simplicity, body
forces will not be considered and null traction along the faces of the crack will
be assumed.

Fig. 1. Cracked shell

Let us also assume that the crack advances in a such form that the crack
tip remains over the curve C. Then, the crack initiation can be simulated as a
shape change of the shell by choosing a suitable tangential velocity field, vt,
over the cracked domain Ω (see Fig.1). This tangential velocity functionmust
be smooth, takes unitary value at the crack tip and remains tangent to the
faces of the crack, vt ·m

± = 0, and also vanishes along the boundary of the
uncracked shell domain (∂Ω).



668 E.Taroco, R.A.Feijóo

With the aid of the previously developed shape sensitivity analysis, we can
easily obtain the expression for the derivative of the potential strain energy
with respect to crack advance.This derivativewithnegative sign, traditionally
denoted by G, is known in fracturemechanics as the energy release rate. From
(10.1) follows

G =−
∫

Ω

Σ ·Πgradsvt dΩ −
∫

Ω

(gradsgradsn)(unN−Q⊗ut) ·vt dΩ +

+

∫

Ω

(gradsn)
[
(N · gradsn)ut−N(gradsn)ut+(M · gradsn)ϑ− (11.1)

−M(gradsn)ϑ
]
·vt dΩ

11.1. Circular cylindrical shell with a crack

In this section, the above expression for G will be written for the case of
a circular cylindrical shell containing a crack through its thickness, Yahsi and
Erdogan (1983). The cutting plane P is inclined by an arbitrary angle θ to
the circumferential direction (see Fig.2).

Fig. 2. Circular cylindrical shell with a crack



Shape sensitivity analysis of elastic shells with cracks 669

For this shell, the tangent plane at any point x of the middle surface Ω
(x ∈ Ω), can be described by the intrinsic base system {ex,eφ}. Then, the
tangential unit vector t, always tangent to the curve C, can be defined as

t=cosθeφ+sinθex (11.2)

From the above definition, the shape change velocity field vt takes the
form

vt = αt (11.3)

where α(x) ∈ [0,1] is a smooth scalar (realvalued) field. At the tip of the
crack α(x) = 1, on both crack surfaces, 0 < α < 1, and along the boundary
of the domain (∂Ω) α(x)= 0.
From the definition of vt follows

gradsvt = t⊗ gradsα+αgradst (11.4)

Then, projection of this expression on the tangent plane is

Πgradsvt = t⊗ gradsα+αΠgradst (11.5)

From (10.10) the energy release rate as a domain integral can be expressed
by

G =−
dψ

dτ
=−
∫

Ω

Σ · gradsvt dΩ =−
∫

Ω

Σ · (t⊗ gradsα+αΠgradst) dΩ =

(11.6)

=−
∫

Ω

(t ·Σgradsα+αΣ ·Πgradst) dΩ

As a path integral, the energy release rate becomes

G =−
dψ

dτ
= J =

∫

Γ

αΣm · t dΓ (11.7)

Here Γ is any contour around the tip of the crack over the middle surface of
the shell and whose two end points lie on the crack faces C+c and C

−
c .

12. Final remarks

The present paper shows a straightforward use of the (continuous) varia-
tional formulation linked to the directmethod of sensitivity analysis to obtain



670 E.Taroco, R.A.Feijóo

the shape derivative of the total potential energy stored in a shell within the
framework of Reissner’s theory.

To perform the shape derivative, the analogy with material (total time)
derivative of Continuum Mechanics is widely explored. In fact, the spatial
description of this derivative and the use of some well-known expressions of
mechanics vastly simplify this task.

The intrinsic surface frame composed of both the unit normal vector and
the tangent plane in each point of themiddle surface of the shell is employed.
Moreover, the procedure and the results are presented in a compact notation
(independent of the coordinate system) to point out the advantage of this
formulation. By doing so the physical meaning of the model and the shape
derivatives are preserved and the resulting expressions are not obscured by an
excess of notations.

In dealing with general elastic shells containing through cracks, if crack
advance is simulated by a suitable change of shape, the shape sensitivity ana-
lysis can be used as a systematic methodology to obtain the energy release
rate.

Moreover, the energy release rate expression obtained in the present work
requires the evaluation of the displacement (ut,un,ϑ) solution of the state
equation (equilibrium equation in our case) and the definition of the shape
change velocity field vt. In practical evaluation of the energy release rate, as
we are free to select the velocity, we can take advantage of choosing themore
convenient distribution over the middle surface of the shell. Thus, the energy
release rate expression for cracked shells conducted in the present study is
meaningful in both the theoretical and practical aspects.

In shells with arbitrary middle surface, this procedure led to a surface
integral in which the Eshelby’s tensor naturally appears. It was also verified
that, in spite of the considered null body force in the analysis of the shell, the
divergence of Eshelby’s tensor did not vanish. However, in the particular case
of circular cylindrical shell, the divergence of Eshelby’s tensor vanishes. In this
case it is simple to show the equivalence between the surface integral and the
integral along a contour around the crack tip lying over the middle surface of
the shell. This integral, well known in fracturemechanics as theRice-Eshelby-
Cherepanov J-integral, remains path independent and also represents a useful
alternative to evaluate the energy release rate of circular cylindrical shells
containing through cracks.

Acknowledgement

This research was supported by FINEP/CNPq-LNCC PRONEX Project,

CTPETRO andCNPq, Brazil.



Shape sensitivity analysis of elastic shells with cracks 671

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Analiza wrażliwości kształtu podatnych powłok z pęknięciami

Streszczenie

Badania opisane w pracy dotyczą zastosowania analizy wrażliwości kształtu jako
systematycznej metodologii wyznaczania tempa uwalnianej energii powłok z pęknię-
ciami w ramach liniowego podejścia uwzględniającego efekt deformacji od ścinania
poprzecznego.Tametodologia i bezpośrednia analiza wrażliwości kształtu została za-
stosowana do powłok o dowolnej powierzchni środkowej, pozwalając na znalezienie



Shape sensitivity analysis of elastic shells with cracks 673

jawnego i ogólnego wyrażenia na pochodną całkowitej energii odkształcenia. W po-
datnychpowłokachzpęknięciami symulację inicjacji pęknięciadokonanonapodstawie
zmiany kształtu określonej odpowiednim rozkłademprędkości powierzchni środkowej
powłoki.W takim przypadku użyteczną formułę określającą tempo uwalnianej ener-
gii wyznaczono w funkcji stanu naprężenia i odkształcenia oraz zmian rozkładu pola
prędkości powierzchni środkowej.Na koniec, analizęwrażliwości kształtu zastosowano
do szczególnegoprzypadkupowłoki cylindrycznej, gdziewarunek zerowej dywergencji
odpowiadającego tensora Eshelby’ego został potwierdzony.

Manuscript received December 11, 2003; accepted for print March 5, 2003