Recent Researches in Applied Mathematics and Informatics
Numerical method for handling the interface
conditions in equations of elasticity
M. Michaeli, F. Assous.
physical material, but describe the cohesive forces which
occur when material elements are being pulled apart. Other
approaches about the oscillating stress singularities for the
interface crack problems are considered from the numerical
point of view in [16], [17] for the isotropic case, whereas
anisotropic case is considered in [24].
Abstract—This work deals with the crack problem simulation
in dissimilar media. It proposes a new numerical method derived
from a Nitsche approach for handling interface conditions in the
Elasticity equations. The Nitsche method, introduced to impose
weakly essential boundary conditions in the scalar Laplace operator, has been then worked out more generally and transferred
to continuity conditions. We propose here an extension of this
method to the Navier-Lame equations. We derive a variational
formulation that provides the solution in terms of displacements
field in the case of a crack existence in a plate domain Ω, made
of several different layers characterized by different material
properties. We formulate the method for both the homogeneous
and the dissimilar material domains and report some numerical
experiments.
This work deals with the crack problem simulation in
dissimilar material, and proposes a new numerical approach
which is based on Nitsche’s variational formulation of
Navier-Lame equations in two dimensional domain. We
present the variational formulation which provides the
solution in terms of displacements field in the case of a crack
existence, first in homogeneous material domain, then for a
two dimensional plate domain Ω made of several different
layers characterized by different material properties. The
classical Nitsche formulation [25] was introduced several
years ago to impose weakly essential boundary conditions
in the scalar Laplace operator. Then, it has been worked out
more generally and transferred to continuity conditions by
Stenberg et al. [27], [8], [19] to many physical fields and
particulary to the Maxwell equations [6], [7]. The Nitsche
formulation has several advantages. First, it is well adapted to
conforming finite element,that can lead to efficient numerical
scheme in the time dependent cases. Note also that following
[27], a nice property of Nitsche method is the optimal
order of convergence. Moreover, the Nitsche approach is an
efficient way to reuse available codes, built on conforming
finite element methods. In addition, the Nitsche formulation
leads to a symmetric, definite, positive discrete formulation
(and then to symmetric, definite, positive matrices), in
agreement with symmetry and ellipticity of the boundary
value problem formulation. The Nitsche’s method proposed
here is concerned with the handle of interface transmission
conditions in the Navier-Lame equations between two
or more subdomains characterized by different material
constants. This gives a numerical method easy to implement
by simply adding integral terms on the boundary, that is
able to solve problems in singular domains, for instance
in domains with cracks, where the stress fields tend to infinity.
I. I NTRODUCTION
Fracture mechanics deals with the study of the formation of
cracks in materials, using methods of analytical mechanics to
calculate the force on a crack and the material’s resistance
to fracture. It applies the physics of stress and strain, the
theories of elasticity and plasticity, to the microscopic defects
found in real materials in order to predict the mechanical
failure of bodies, and a crack propagation.
The early works in this area refer to the linear elastic material
problems in simple geometries [18], [30], which were solved
by Irwin and Williams using the energy release rate concept,
where the stresses and displacements near the crack tip
were described by a single parameter, which is known as
the Stress Intensity Factor (SIF). Stress intensity factor is
directly proportional to the applied load on the material and
its magnitude depends on the size and location of the crack.
Another approach was introduced by Westergaard [29] and
Williams [31]. They presented a specialized complex variable
method to determine the stresses around the crack in the
case of homogeneous materials and in the case of interface
of two different homogeneous isotropic materials. Later,
more complicated theoretical approaches were developed
for elastic-plastic cracks, cracks under thermal-mechanical
loading conditions and the interface cracks for bimaterial
media, see [11], [12], [13] and references therein. In last few
decades there was a major progress in development of new
methods for crack problems, where most of them are based on
the numerical approaches derived from finite element method.
One of this approaches is Cohesive zone element (CE) [23],
which is a phenomenological model for crack propagation
analysis that was introduced in recent few years and deals
with the stresses calculations at the crack tip. According to
this method, cohesive zone elements do not represent any
ISBN: 978-1-61804-059-6
The paper is organized as follows: In section 2, we recall
the equations of the boundary value problem (B.V.P.). These
equations are formulated in terms of displacements in cartesian
coordinate systems and the stress-strain relation is applied
according to the elastic isotropic form of Hooks law. In
section 3, we introduce a first variational formulation of the
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Recent Researches in Applied Mathematics and Informatics
elasticity equations, which provides the solution of the B.V.P.
in the case of homogeneous elastic material. The section 4
is devoted to the dissimilar elastic material, there we deal
with the transimission conditions. The appropriate B.V.P is
presented together with the boundary and interface conditions
and the Nitsche variational formulation is derived. Numerical
results are also shown in section 5 and concluding remarks
follow.
T
L1
Σ
R
Ω
L2
II. E QUATIONS OF E LASTICITY
B
According to the classical theory of elasticity (see [26],
[28]) the two-dimensional formulation of equilibrium equations in Cartesian coordinate system can be formulated in
tensor form using the divergence operator:
−div(S) = F
Fig. 1.
Cracked Domain - Homogeneous elastic plate
(1)
forces at B and T in the vertical directions, and fixed
on R (see Fig. 1). We denote the domain boundaries in
the following manner: ΓC = R, ΓF = Σ ∪ L1 ∪ L2 and
ΓL = T ∪ B, where the letters C, F and L refer to clamped,
free and loaded boundaries respectively. In that case our
problem is formulated as follows:

−divS = F in Ω



u = 0 on ΓC
(6)
S = 0 on ΓF



S · n = G on ΓL
where S is a stress tensor and F is a acting force vector:
σx τxy
Fx
S=
, F=
(2)
τxy σy
Fy
σx , σy are the normal stresses and τxy is the shearing stress.
From the isotropic form of Hook’s law, the allowable stresses
reduce to
σx = λ(x +y )+2µx , σy = λ(x +y )+2µy , τxy = 2µxy
(3)
where x , y and xy are the strain components and λ, µ
are defined below. The strain components are defined via
the displacement field variable vector u = [ux , uy ] which
contains the displacements in x and y directions respectively.
The strain-displacement relation has the following form:
∂uy
1 ∂ux
∂uy
∂ux
, y =
, xy =
+
(4)
x =
∂x
∂y
2 ∂y
∂x
where n is the outward pointing unit normal of surface
element, and the load F ∈ L2 (Ω). To derive the variational
formulation, we classically (see [10]) introduce the Sobolev
space
H1C (Ω) = {v ∈ L2 (Ω); ∇v ∈ L2 (Ω); v = 0 on ΓC }
and consider V a linear closed subspace of H1C (Ω). Then,
multiplying the first equation of (6) by v ∈ V, we apply
the fundamental Green’s identity for the tensor field. For any
symmetric tensor field S and vector field v ∈ V, the Green
formula
Z
Z
Z
−
divS · vdΩ =
S : ∇vdΩ − (S · n) · vdΓ
(8)
where the coefficients λ and µ characterize the elastic material:
λ is the elastic coefficient, called Lame’s constant and µ is
referred to so called shear modulus. This pair of characterization coefficients can be determined from other two well
known characterization coefficients E (Young’s modulus) and
ν (Poisson’s ratio), where the relation between them is as
follows:
E
Eν
µ=
, λ=
(5)
2(ν + 1)
(1 + ν)(1 − 2ν)
Ω
Ω
Γ
holds (see [10], p. 288), where the symbol : stands for the
contracted product of two tensors and Γ = ΓC ∪ ΓF ∪ ΓL .
Because the load is applied only on ΓL , S = 0 on ΓF and (8)
becomes as follows:
Z
Z
Z
−
divS · vdΩ =
S : ∇vdΩ −
(S · n) · vdΓL (9)
In the text, names of functional spaces of scalar fields usually
begin by an italic letter, whereas they begin by a bold letter
for spaces of vector fields.
Ω
III. VARIATIONAL FORMULATION OF E LASTICITY
Ω
ΓL
Taking in account the problem configuration , i.e. ∇v = (v)
[26], where (v) is a strain tensor, and denoting S · n = G we
obtain that
Z
Z
Z
Z
−
divS·vdΩ =
S : (v)dΩ−
G·vdΓL =
F·vdΩ
PROBLEM
A. Homogeneous elastic material
In order to obtain the variational formulation for the
elasticity problem, we consider the implementation of
deformation in domain Ω which interpreters the elastic
homogeneous plate, with the crack whose undeformed shape
is a curve Σ, and which is loaded by the opposed surface
ISBN: 978-1-61804-059-6
(7)
Ω
Ω
ΓL
Ω
(10)
The last expression leads us to the variational form of the
B.V.P, formulated for the displacement vector u (see Theorem
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Recent Researches in Applied Mathematics and Informatics
6.3-1 in [10]):
Z
{λ tr ((u)) tr ((v)) + 2µ(u) : (v)} dΩ =
Ω
Z
Z
=
F · vdΩ +
G · vdΓL , ∀v ∈ V . (11)
Ω
T2
ΓL
L1
C2
B2
C1
T1
Ω1
R1
E1 , ν1
B1
Fig. 2.
i = 1, 2:
The mathematical meaning of σy is that it represented by
some continuous function σy = f (x) which defines the load
distribution along the boundary and is chosen according to
applied force configuration. In this case we have the following
variational formulation based on displacement field:
Cracked Domain - Dissimilar elastic plate

−divSi = Fi in Ωi



ui = 0 on ΓCi
S

i = 0 on ΓFi


Si · n = Gi on ΓLi
(13)
where the load (F1 , F2 ) ∈ L2 (Ω). In this case, λ and µ are
the Lamé parameters, assumed constant in each domain Ωi ,
i = 1, 2. Denoting the interface between the two sub domains
Ω1 and Ω2 by Υ = {(x, y) : 0 ≤ x ≤ 1, y = 0} = T1 = B2 ,
we have to define the appropriate boundary interface conditions (see [15], [14]). The interface conditions refer to problem
configuration, where the both of the half plates are perfectly
jointed on the interface. In this case the interface transmission
conditions are defined as follows:
 1
2
= 0 on Υ
 τxy = τxy
(14)
[u] = 0 on Υ

[σy ] = 0 on Υ
Find u = [ux , uy ] ∈ V, such that
∂uy ∂vy
∂ux ∂vx
+
dΩ+
2µ
∂x ∂x
∂y ∂y
Ω
Z
∂ux
∂uy
∂vx
∂vy
+
µ
+
+
dΩ+
∂y
∂x
∂y
∂x
Ω
Z ∂ux
∂vx
∂uy
∂vy
+λ
+
+
dΩ =
∂x
∂y
∂x
∂y
Ω
Z
Z
=
f (x)vy dT −
f (x)vy dB, ∀v = [vx , vy ] ∈ V
T
R2
E2 , ν2
In the problem we consider here, we assume that there are no
applied body forces acting inside Ω, namely F = 0, only forces
acting on the upper and lower faces ΓL (ΓL = T ∪ B, see
Fig.1). The applied forces define the stresses which obtained
on those boundaries from stress tensor S. Taking into account
the directions of the outward pointing unit normals on the
boundaries B ant T , and the fact that there are no shear stresses
on that boundaries, we obtain:
0
0
G=
on T, G =
on B
(12)
σy
−σy
Z
Ω2
L2
where the brackets [·] denotes the jump across the interface Υ.
The physical meaning of the first condition is that there is no
shear stress on the interface. The second condition expresses
the continuity of the displacement fields u the across the
interface between the sub domains Ω1 and Ω2 , whereas the
third one asserts the continuity of the stress field in the y
direction.
B
IV. D ISSIMILAR ELASTIC MATERIAL
For the simplicity of the presentation, we consider the case
of a two layer elastic material. The extension to multilayer
material is straightforward. Numerical examples will be given
in the section 5.2.
B. The classical Nitsche method
A. Setting of the problem
As it was mentioned before, Nitsche method [25] was
introduced several years ago for imposing essential boundary
conditions weakly in the finite element method approximation
of Poisson equation with Dirichlet boundary conditions. Basically, Nitsche approach consists in penalizing the difference
between the approximate solution and the Dirichlet boundary
data rather than trying to interpolate that data directly. It leads
to symmetric positive definite linear systems that can be solved
very quickly for instance using gradient or multigrid numerical
methods. The main advantage of Nitsche method is that it
keeps the convergence rate of the finite elements method [27],
as opposed to the standard penalty method. We introduce
the classical Nitsche’s method as it was derived in [25] for
imposing a Dirichlet boundary condition. Let us consider the
In order to obtain the variational formulation for the problem with dissimilar elastic material, we consider the deformation of the domain Ω1 ∪ Ω2 which interpreters the elastic
dissimilar plate, with the crack whose undeformed shape is a
curve C1 ∪ C2 and which is perfectly jointed on the interface
B2 ∪ T1 . As in the first problem, the plate is loaded by the
opposed surface forces at B1 and T2 in the vertical directions,
and fixed on R1 ∪ R2 (see Fig.2). We denote the domain
boundaries in the following manner: ΓCi = Ri , for i = 1, 2,
ΓF1 = C1 ∪ L1 , ΓF2 = C2 ∪ L2 , ΓL1 = B1 and ΓL2 = T2 ,
where the notations Ci , Fi and Li stand for clamped, free
and loaded boundaries respectively for each subdomain Ωi ,
i = 1, 2. In this case, the problem is defined as follows, for
ISBN: 978-1-61804-059-6
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presentation, we denote Sui , i = 1, 2 (resp: Svi , i = 1, 2) the
stress tensor associated to the displacement ui , i = 1, 2 (resp:
vi , i = 1, 2). Hence, we rewrite our problem in the following
manner


−divSu1 = F1 in Ω1
−divSu2 = F2 in Ω2








u
=
0
on
Γ
u2 = 0
on ΓC2


1
C
1






S
=
0
on
Γ
S
=
0
on ΓF2


u
F
u
1
1
2




Su1 · n = G1 on ΓL1
Su2 · n = G2 on ΓL2
,
1
2
τxy
=0
on Υ
τxy
=0
on Υ






1
2
2
1


u
=
u
on
Υ
u
=
u
on Υ


x
x
x
x




1
2
2
1


u
=
u
on
Υ
u
=
u
on
Υ


y
y
y
y




σy1 = σy2
on Υ
σy2 = σy1
on Υ
(19)
Again, we assume that there are no applied body forces acting
inside Ω1 , Ω2 , that is F1 = F2 = 0, and that the only forces
are acting on the upper and lower faces ΓL1 , ΓL2 (ΓL1 =
B1 , ΓL2 = T2 , see Fig.2). Applying the fundamental Green’s
identity for the tensor fields, as it was done in (10), we obtain
the natural variational formulation for each sub domain Ω1 ,
Ω2 :
Find ui ∈ H1C (Ωi ) such that
following problem:
For given functions f ∈ L2 (ω) and g ∈ H 1/2 (γ), find u
solution to
−∆u = f in ω ,
(15)
u
= g on γ .
where ω is a bounded, open subset in R2 with the boundary
γ, and H 1/2 (γ) is the trace on γ of elements of H 1 (ω).
To formulate the Nitsche method, we first introduce a shape
regular finite element partition Th = ∪K of the domain ω. For
any element K of the mesh Th , let Pk (K) be the set of all
polynomials on K of degree ≤ k. We denote by E an edge
of the element of Th and by Ch the trace mesh induced by Th
on the boundary γ, that is Ch = {E; E = K ∩ γ, K ∈ Th }.
Moreover, we assume that the elements of Ch verify the
regularity condition, i.e. for the diameter hE of an element
E ∈ Ch and the diameter ρh of the largest inscribed circle
of E, we have hE ≤ Cρh , where C is independent constant
of E and h. Finally we introduce the finite element space:
Vh = v ∈ H 1 (ω) ; v|K ∈ Pk (K) Denote by uh the finite
element approximation of u in Vh , the Nitsche formulation for
the problem (15) is written:
Find uh ∈ Vh such that
Bh (uh , v) = Fh (v) ∀v ∈ Vh
Z
Z
Sui : (vi )dΩi −
Sui · n · vi dΥ =
Ωi
Υ
Z
=
Gi · vi dΓLi , ∀vi ∈ H1C (Ωi ), for i = 1, 2 (20)
(16)
where the bilinear form Bh (·, ·) is defined on Vh × Vh by
Z
∂uh
, v >Γ −
Bh (uh , v) =
grad uh · grad v dx− <
∂n
Ω
X 1
∂v
−<
, uh >Γ +β
< uh , v >E , (17)
∂n
hE
ΓLi
Up to now, we have not dealt with the transmission condition
[u] = 0. For this purpose, we formulate the Nitsche method
for this elasticity problem, following the same approach
and the same notations as for the Laplace operator. Note
that the boundary conditions on ∂Ω, which are the same
here as for the homogeneous problem (6), are treated in the
same traditional way. Nitsche’s method will only be used for
handling the interface conditions.
E∈Ch
and the linear form Fh (·) on Vh is equal to
Z
X 1
∂v
Fh (v) =
f v dx− <
, g >Γ +β
< g, v >E .
∂n
hE
Ω
E∈Ch
(18)
Above, β is some positive sufficiently large constant (to be
specified subsequently), and the bracket < ·, · >E denotes the
L2 (E) scalar product. Essentially, Nitsche’s method imposes
the boundary conditions via three boundary terms. Two of
them containthe weak form of the normal derivatives of the
solution and the test functions. These two terms cause the
method to be symmetric and consistent. The third term (with
β) depends on the domain triangulation, and causes the method
to be stable. As expected, the solution u of the Eq.(15) satisfies
the variational Eq.(16), or in other words, u is consistent with
the Nitsche approach (16). A nice property of Nitsches method
is the optimal order of convergence. Indeed, Nitsche proved
[25] that if β is a sufficiently large constant, then the discrete
solution converges to the exact one with optimal order in
H 1 (ω) and L2 (ω).
Assuming that we have a regular finite element mesh Th
of the domain Ω, we introduce the finite element approximation space Vh of vectorized functions as Vhi = {v ∈
H1C (Ωi ); v|K ∈ Pk (K)}, where Pk (K) denotes the set of
all vector fields which are polynomials componentwise on K
with degree ≤ k . In this case, Ch denotes the trace mesh
induced by Th on the interface Υ. As for the Laplace operator,
we also assume a regularity condition for the elements of
Ch . Denoting also by ui the approximate solution of ui in
Vhi , we readily get the discrete formulation associated to (20).
From the transmission condition [u] =R 0 on Υ, we derive the
variational expression, for i = 1, 2, − Υ Svi · n · [u]dΥ that we
add to the discrete approximation of formulation (20). This
causes the method to be symmetric and consistent. Finally, to
ensureP
the method to be stable, for a sufficient large β, we also
add β E∈Ch h1E < [u], vi >E , Hence the Nitsche variational
formulation of the problem (13) together with the second
interface condition of (14) is written, for each subdomain Ωi ,
i = 1, 2 as:
C. Nitsche formulation for the elasticity problem in dissimilar
domain with interface
Let us consider now the problem (13) for a two-layer media,
together with the interface conditions (14). For the easiness of
ISBN: 978-1-61804-059-6
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Recent Researches in Applied Mathematics and Informatics
Find ui ∈ Vhi such that
Z
Z
Z
Sui : (vi )dΩi −
Sui · n · vi dΥ −
Svi · n · ui dΥ+
Ωi
Υ
Υ
Z
X 1
+β
< ui , vi >E +
Svi · n · ui+1 dΥ−
hE
Υ
E∈Ch
Z
X 1
−β
< ui+1 , vi >E =
Gi · vi dΓLi (21)
hE
ΓLi
ΓL1 = B1 and ΓL2 = T2 . Taking into account the directions
of the outward pointing unit normals on those boundaries
(see Fig.2), we obtain the expression for load distribution
components along those loaded boundaries:
0
0
G1 =
on B1 , G2 =
on T2
(28)
−σy1
σy2
As it was explained in homogeneous material problem, the
stress obtained on the loaded boundaries, is represented by
some continuous function. In our case we chose it as follows:
E∈Ch
where u3 stands for u1 . We have now to handle the first
1
2
interface condition of (14), that is τxy
= τxy
= 0. This
is performed in a straightforward way by substituting the
vanishing shear stress components into the stress tensor Sui
for i = 1, 2, for each integral over Υ in (21). Hence, the
stress tensor becomes diagonal and we get
i
σx 0
Sui =
on Υ
(22)
0 σyi
σy1 = f1 (x), σy2 = f2 (x)
If the tensile test is done with uniform load, we obtain that
f2 (x) = −f1 (x), i.e. f2 (x) = f (x) and f1 (x) = −f (x).
V. N UMERICAL R ESULTS
A. The case of a bimaterial plate
We consider the elastic bi material plate depicted in Fig.2
with given material properties E1 , E2 , ν1 , ν2 , which is fixed on
R1 and R2 . As in the first problem, the plate is loaded only by
the opposed surface forces at B1 and T2 in the vertical directions. The numerical results are shown in Fig. 3. Using still
Finally, It remains to take into account the last interface
condition of (14), namely σy1 = σy2 , that refers to the continuity
of the stress field in y direction. In order to impose it, we
couple the second term of each bilinear form (21) for i = 1, 2,
in the following manner: For Ω1 , we have:
σy1 =σy2
def
(23)
def
(24)
Su1 · n · v1 = σx1 nx vx1 + σy2 ny vy1 = Sx1 ,y2 · n · v1
(29)
Similarly for Ω2 , we get:
σy1 =σy2
Su2 · n · v2 = σx2 nx vx2 + σy1 ny vy2 = Sx2 ,y1 · n · v2
Note that in our case, since the interface between the two
sub domains is part of the axis y = 0, we have that Sx1 ,y2
coincides with Su2 and Sx2 ,y1 coincides with Su1 . Summing
up, our Nitsche type variational formulation of problem (19)
can be written as (still with the y3 stands for y1 in Sx2 ,y3 )
Find uh = (u1 , u2 ) ∈ Vh = Vh1 × Vh2 such that
∀v ∈ Vh ,
ah (uh , v) = Lh (v)
(a) Displacement in direction x
(25)
where
ah (uh , v) =
Z
X Z
=
Sui : (vi )dΩi −
Sxi ,yi+1 · n · vi dΥ −
Ωi
i=1,2
−
X 1
< ui , vi >E
Svi · n · ui dΥ + β
hE
Υ
X
i=1,2
+
X
i=1,2
Υ
Z
!
+
E∈Ch
Z
X 1
Svi · n · ui+1 dΥ − β
< ui+1 , vi >E
hE
Υ
!
E∈Ch
(b) Displacement in direction y
(26)
and
Fig. 3.
Lh (v)
=
X Z
i=1,2
Gi · vi dΓLi
(27)
the stress-strain relations in (3) and the strain-displacement
relations in (4) we obtain the stress fields represented in Fig. 4.
In that case, up to our knowledge, analytical (even estimated)
solutions do not exist. The discontinuity in σx appears on
ΓLi
Concerning the load vector, we still consider the case where
the only forces are acting on the upper and the lower faces
ISBN: 978-1-61804-059-6
Displacement fields in dissimilar material - numerical results
65
Recent Researches in Applied Mathematics and Informatics
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(a) σx
(b) σy
Fig. 4.
Stress fields in dissimilar material - numerical results
Fig.4(a) as a consequence of the dissimilarity of the material.
At the opposite, the σy is continuous through the interface,
that shows that the third interface condition in (14) is well
satisfied.
VI. C ONCLUSION
In this work we considered the Navier Lame equations
in two dimensional cracked plate in the case of similar
and dissimilar materials. We presented a new concept of
handling the interface conditions in the case of interface crack
existence in dissimilar materials. The numerical formulation
for multilayered dissimilar materials has been shown. Our
approach is derived from a Nitsche type method, which
was introduced several years ago to impose weakly essential
boundary conditions in the scalar Laplace operator and has
been transferred to many physical fields. The formulation was
developed in two dimensional case, and numerical examples
were given as a first attempt to show the efficiency of the
proposed formulation. This method seems promising even to
compute the stress fields in the case of different materials, like
elasto-plastic, viscoelastic and hyper elastic materials.
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ISBN: 978-1-61804-059-6
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