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Unsteady free surface flow in porous media: One-dimensional model equations including vertical effects and seepage face

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Michael Brun

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Xu Wang

Peter Schiavone

C. R. Mecanique 346 (2018) 477–481Contents lists available at ScienceDirect
Comptes Rendus Mecanique
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A coated rigid elliptical inclusion loaded by a couple in 
the presence of uniform interfacial and hoop stresses
Xu Wang a, Peter Schiavone b,∗
a School of Mechanical and Power Engineering, East China University of Science and Technology, 130 Meilong Road, Shanghai 200237, China
b Department of Mechanical Engineering, University of Alberta, 10-203 Donadeo Innovation Centre for Engineering, Edmonton, Alberta 
T6G 1H9, Canada
a r t i c l e i n f o a b s t r a c t
Article history:
Received 14 February 2018
Accepted 22 March 2018
Available online 28 March 2018
Keywords:
Rigid inclusion
Confocal coating
Uniform interfacial and hoop stresses
Couple
Rigid body rotation
Harmonic inclusion
We consider a confocally coated rigid elliptical inclusion, loaded by a couple and 
introduced into a remote uniform stress field. We show that uniform interfacial and 
hoop stresses along the inclusion–coating interface can be achieved when the two remote 
normal stresses and the remote shear stress each satisfy certain conditions. Our analysis 
indicates that: (i) the uniform interfacial tangential stress depends only on the area of the 
inclusion and the moment of the couple; (ii) the rigid-body rotation of the rigid inclusion 
depends only on the area of the inclusion, the coating thickness, the shear moduli of 
the composite and the moment of the couple; (iii) for given remote normal stresses 
and material parameters, the coating thickness and the aspect ratio of the inclusion are 
required to satisfy a particular relationship; (iv) for prescribed remote shear stress, moment 
and given material parameters, the coating thickness, the size and aspect ratio of the 
inclusion are also related. Finally, a harmonic rigid inclusion emerges as a special case if 
the coating and the matrix have identical elastic properties.
© 2018 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
1. Introduction
It is well known that a uniform distribution of interfacial normal and tangential stresses is optimal in that it will elim-
inate any stress peaks at the interface between an inclusion and the surrounding material [1]. Furthermore, a uniform 
distribution of hoop stress along the edge of a hole or inclusion is also known to be ideal in the design of what Cherepanov 
refers to as “equally strong outlines of holes” [2]. The design of inclusions with constant interfacial and hoop stresses has 
attracted considerable attention in the literature (see, for example, [1,3–7]). In particular, confocal elliptical interfaces can 
be used to achieve the design objective of uniform interfacial and hoop stresses for a three-phase elliptical inclusion [1]. In 
all these previous investigations, the inclusion itself is free of any external loading. It is of great practical and theoretical 
interest to ask whether, if a rigid inclusion were loaded, for example, by a couple, it would still be possible to achieve 
uniform interfacial and hoop stresses along the inclusion boundary. This question forms the basis of the present study.
In this paper, we consider a rigid elliptical inclusion, loaded by a couple moment and bonded to an infinite elastic matrix 
through a coating consisting of two confocal elliptical interfaces. We develop one condition on remote normal stresses and 
another on remote shear stress that ensure that the interfacial and hoop stresses along the inclusion–coating interface are 
* Corresponding author.
E-mail addresses: xuwang @ecust .edu .cn (X. Wang), p .schiavone @ualberta .ca (P. Schiavone).https://doi.org/10.1016/j.crme.2018.03.005
1631-0721/© 2018 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
478 X. Wang, P. Schiavone / C. R. Mecanique 346 (2018) 477–481Fig. 1. A confocally coated rigid elliptical inclusion loaded by a couple.
uniformly distributed. For prescribed remote normal stresses and material parameters, a relationship is established between 
the coating thickness and the aspect ratio of the inclusion. In the case of prescribed remote shear stress, couple moment and 
material parameters, we establish a relationship among the coating thickness, the size, and the aspect ratio of the inclusion.
2. Complex variable formulation
In a Cartesian coordinate system O x1x2x3, in the case of plane elastostatics, the stresses (σ11, σ22, σ12), displacement 
components (u1, u2) and the stress functions (φ1, φ2) can be expressed in terms of two analytic functions ϕ(z) and ψ(z) of 
the complex variable z = x1 + ix2 as [8]
σ11 + σ22 = 2
[
ϕ′(z) + ϕ′(z)],
σ22 − σ11 + 2iσ12 = 2
[
z̄ϕ′′(z) + ψ ′(z)], (1)
2μ(u1 + iu2) = κϕ(z) − zϕ′(z) − ψ(z)
φ1 + iφ2 = i
[
ϕ(z) + zϕ′(z) + ψ(z) ] (2)
where κ = 3 − 4ν for plane strain, κ = (3 − ν)/(1 + ν) for plane stress, μ and ν (0 ≤ ν ≤ 1/2) are the shear modulus and 
Poisson’s ratio, respectively. In addition, the stresses are related to the stress functions through [9]
σ11 = −φ1,2, σ12 = φ1,1
σ21 = −φ2,2, σ22 = φ2,1 (3)
3. General solution
Consider a rigid elliptical inclusion bonded to an infinite elastic matrix through a confocal coating. Let S0, S1 and S2
denote the rigid inclusion, the coating and the matrix, respectively, all of which are perfectly bonded across two confocal 
elliptical interfaces L1 and L2, the common foci of which are located at z = ±2R (R > 0) on the real axis, as shown in Fig. 1. 
The rigid inclusion is loaded by a couple of moment M and the matrix is subjected to remote uniform in-plane stresses 
(σ∞11 , σ∞22 , σ∞12 ). In what follows, the subscripts 1 and 2 will be used to identify the respective quantities in S1 and S2.
We first introduce the following conformal mapping function [8]
z = ω(ξ) = R
(
ξ + 1
ξ
)
, ξ = ω−1(z) = z
2R
+
√
z2
4R2
− 1 (4)
which maps the segment [−2R, 2R] onto the unit circle in the ξ -plane and the two interfaces L1 and L2 onto two coaxial 
circles with radii R1 and R2. Thus S1 and S2 are mapped onto R1 ≤ |ξ | ≤ R2 and |ξ | ≥ R2, respectively. For the sake of 
convenience and without loss of generality, we write ϕi(ξ) = ϕi(ω(ξ)), ψi(ξ) = ψi(ω(ξ)), i = 1, 2.
X. Wang, P. Schiavone / C. R. Mecanique 346 (2018) 477–481 479In the image ξ -plane, the boundary value problem has the form
κ1ϕ1(ξ) − ω(ξ)
ω′(ξ)
ϕ′1(ξ) − ψ1(ξ) = 2iμ1
ω(ξ), |ξ | = R1 (5a)
ϕ2(ξ) + ω(ξ)
ω′(ξ)
ϕ′2(ξ) + ψ2(ξ) = ϕ1(ξ) +
ω(ξ)
ω′(ξ)
ϕ′1(ξ) + ψ1(ξ)
κ2ϕ2(ξ) − ω(ξ)
ω′(ξ)
ϕ′2(ξ) − ψ2(ξ) = κ1ϕ1(ξ) −
ω(ξ)
ω′(ξ)
ϕ′1(ξ) − ψ1(ξ), |ξ | = R2
(5b)
ϕ2(ξ) = R(σ
∞
11 + σ∞22 )
4
ξ + O (1),ψ2(ξ) = R(σ
∞
22 − σ∞11 + 2iσ∞12 )
2
ξ + O (1), |ξ | → ∞ (5c)
where 
 is the (to be determined) rigid-body rotation of the inclusion and  = μ2/μ1.
In order to ensure that the interfacial and hoop stresses along the inclusion–coating interface L1 are uniform, the analytic 
function ϕ1(ξ) should take the following form
ϕ1(ξ) = X
(
ξ + 1
ξ
)
, R1 ≤ |ξ | ≤ R2 (6)
where X is a complex number to be determined. We can thus write X = X ′ + iX ′′ identifying X ′ and X ′′ as the real and 
imaginary parts of X .
Using the expression for ϕ1(ξ) from Eq. (6) and imposing the interface condition on the inner interface |ξ | = R1, from 
Eq. (5a) we obtain
ψ1(ξ) = (κ1 X̄ − X + 2iRμ1
)
(
R21
ξ
+ ξ
R21
)
, R1 ≤ |ξ | ≤ R2 (7)
Similarly, using Eqs. (6) and (7) and imposing the two interface conditions on the outer interface |ξ | = R2 from Eq. (5b), we 
arrive at
ϕ2(ξ) = X(κ1 + 1) + X̄(1 − )
κ2 + 1
(
ξ + 1
ξ
)
+ (1 − )(κ1 X − X̄ − 2iRμ1
)
κ2 + 1
(
ρξ + 1
ρξ
)
ψ2(ξ) = X(κ2 + ) + X̄(κ2 − κ1)
κ2 + 1
(
R22
ξ
+ ξ
R22
)
− ( + κ2)(X − κ1 X̄ − 2iRμ1
)
κ2 + 1
(
R21
ξ
+ ξ
R21
)
−
R22
ξ
+ ξ
R22
1 − 1
ξ2
ϕ′2(ξ), |ξ | ≥ R2
(8)
where ρ = R21/R22 (0 ≤ ρ ≤ 1) is a measure of the relative thickness of the coating.
Using Eq. (8) and imposing the asymptotic conditions from Eq. (5c) as well as the moment balance for the circular disk 
|z| = R0 → ∞, we finally obtain

 = M R
2
1[ + ρ(1 − )]
4πμ2 R2(R41 − 1)
= M[ + ρ(1 − )]
4μ2 A
, X ′′ = 2ρRμ1
(1 − )
(κ1 + 1)[ + ρ(1 − )] =
MρR21(1 − )
2π(κ1 + 1)R(R41 − 1)
(9)
σ∞12 =
M[ + κ2 + ρ2(1 − )]
2πR2(R41 − 1)(κ2 + 1)
(10)
σ∞11 + σ∞22 =
4X ′[(κ1 − 1) + 2 + ρ(1 − )(κ1 − 1)]
R(κ2 + 1)
σ∞22 − σ∞11 =
2X ′{2ρ[κ2 − 1 − (κ1 − 1)] + ( + κ2)(κ1 − 1) + ρ2( − 1)(κ1 − 1)}
R R21(κ2 + 1)
(11)
where A is the area of the elliptical inclusion S0.
The necessary and sufficient condition for the existence of the real coefficient X ′ simultaneously satisfying the two 
conditions in Eq. (11) is
σ∞22 − σ∞11
σ∞11 + σ∞22
R21 =
2ρ[κ2 − 1 − (κ1 − 1)] + ( + κ2)(κ1 − 1) + ρ2( − 1)(κ1 − 1)
2[(κ1 − 1) + 2 + ρ(1 − )(κ1 − 1)] (12)
which is found to be in agreement with Eq. (3.7) or Eq. (4.3) given in Ru [1]. Furthermore, the interfacial normal stress σnn
and interfacial tangential stress σnt are uniformly distributed along the inclusion–coating interface L1 as follows:
480 X. Wang, P. Schiavone / C. R. Mecanique 346 (2018) 477–481σnn = (κ1 + 1)X
′
R
= (κ1 + 1)(κ2 + 1)(σ
∞
11 + σ∞22 )
4[(κ1 − 1) + 2 + ρ(1 − )(κ1 − 1)]
σnt = − M R
2
1
2πR2(R41 − 1)
= − M
2A
= − R
2
1(κ2 + 1)σ∞12
 + κ2 + ρ2(1 − ) , z ∈ L1
(13)
It is deduced from Eqs. (6) and (9) that the constant mean stress and uniform rigid body rotation in the coating are given 
by
σ11 + σ22 = 4X
′
R
= (κ2 + 1)(σ
∞
11 + σ∞22 )
(κ1 − 1) + 2 + ρ(1 − )(κ1 − 1)

0 = 1
2
(u2,1 − u1,2) = (κ1 + 1)X
′′
2μ1 R
= MρR
2
1(1 − )
4πμ2 R2(R41 − 1)
= 
ρ(1 − )
 + ρ(1 − ) , z ∈ S1
(14)
Consequently, from Eqs. (6), (7) and (13), it follows that the hoop stress σtt is uniformly distributed along the inclusion–
coating interface L1 on the coating side as follows:
σtt = (3 − κ1)X
′
R
= (3 − κ1)(κ2 + 1)(σ
∞
11 + σ∞22 )
4[(κ1 − 1) + 2 + ρ(1 − )(κ1 − 1)] , z ∈ L1 (15)
In view of the fact that σ∞12 	= 0 (see Eq. (10)), the remote principal stresses are inclined at an angle to the principal axes of 
the elliptical inclusion. It is seen from the above results that: (i) the uniform interfacial tangential stress depends only on 
the area of the inclusion and the moment of the couple; (ii) the rigid-body rotation of the rigid inclusion is dependent only 
on the area of the inclusion, the relative coating thickness ρ , the shear moduli of the composite and the moment; (iii) 

and 
0 have the same sign when the coating is stiffer than the matrix ( < 1) and opposite signs when the coating is more 
compliant than the matrix ( > 1); (iv) the magnitude of the interfacial tangential stress is always greater than that of the 
remote shear stress if the coating is stiffer than the matrix, although it can be greater than, equal to, or smaller than that of 
the remote shear stress if the coating is more compliant than the matrix; (v) for given remote normal stresses and material 
parameters, the relative coating thickness and the aspect ratio of the inclusion here characterized by R1 should satisfy the 
relationship given in Eq. (12); (vi) for prescribed remote shear stress, couple moment and given material parameters, the 
relative coating thickness, the size of the inclusion (characterized by R R1) and the aspect ratio of the inclusion should 
satisfy the relationship given in Eq. (10).
In the case when the elliptical inclusion contracts to a line inclusion (R1 → 1), the denominator of the right-hand side 
of Eq. (10) will become zero. This implies that it is impossible to constrain a rigid line inclusion to continue to maintain 
uniform interfacial and hoop stresses along its boundary L1 when both the remote shear stress and the moment take 
finite values. At the other extreme, when the elliptical inclusion becomes circular (as R1 → ∞) (in which case the coating 
becomes an annulus), we deduce from Eqs. (10) and (12) that σ∞11 = σ∞22 , σ∞12 = 0, which indicates that the remote stress 
field should now be hydrostatic in order to ensure that the interfacial and hoop stresses along the circular interface L1
remain uniform.
4. A harmonic rigid inclusion
When the materials comprising the coating and the matrix are identical ( = 1 and κ1 = κ2), we have from Eqs. (9)–(15)
that

 = M
4μ1 A
, 
0 = 0 (16)
σ∞12 =
M
2πR2(R41 − 1)
,
σ∞22 − σ∞11
σ∞11 + σ∞22
= κ1 − 1
2R21
(17)
σnn = (κ1 + 1)(σ
∞
11 + σ∞22 )
4
, σtt = (3 − κ1)(σ
∞
11 + σ∞22 )
4
, σnt = − M
2A
, z ∈ L1 (18)
In this case,
ϕ1(z) = ϕ2(z) = σ
∞
11 + σ∞22
4
z, z ∈ S1 ∪ S2 (19)
which implies that the trace of the original stress field in the matrix is unaltered following the introduction of the loaded 
(by a couple) rigid inclusion. Thus the loaded rigid inclusion now satisfies the harmonic condition of Bjorkman and Richards 
[3,10–12].
X. Wang, P. Schiavone / C. R. Mecanique 346 (2018) 477–481 4815. Conclusions
We investigate the possibility of uniform interfacial and hoop stresses for a loaded (by a couple) rigid elliptical inclusion 
with confocal coating inserted into an infinite matrix subjected to a uniform stress field at infinity. Using complex variable 
methods, we have shown that constant interfacial and hoop stresses along the inclusion–coating interface can indeed be 
realized when the two conditions specified in Eqs. (10) and (12) are satisfied. The presence of the couple loading the 
inclusion means that the interfacial tangential stress at the inclusion–coating interface and the remote shear stress are no 
longer zero. When the coating and the matrix have identical elastic properties, our analysis identifies a harmonic rigid 
inclusion.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant No. 11272121) and through a Discov-
ery Grant from the Natural Sciences and Engineering Research Council of Canada (Grant No. RGPIN – 2017-03716115112).
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A coated rigid elliptical inclusion loaded by a couple in the presence of uniform interfacial and hoop stresses

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Sergio Zlotnik

Alberto García-González

Antonio Huerta

Algebraic PGD for tensor separation and compression: An algorithmic approach

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C. R. Mecanique 346 (2018) 997–998Contents lists available at ScienceDirect
Comptes Rendus Mecanique
www.sciencedirect.com
Computational methods in welding and additive manufacturing/Simulation numérique des procédés de soudage et de fabrication additive
Foreword
Numerical simulation is a tool widely used nowadays in the field of materials manufacturing processes, both in academic 
laboratories and in industry. Indeed, more than 40 years of development have led to efficient numerical methods imple-
mented in robust codes. The power of the current computational resources makes it possible to enrich the physical models 
used and to carry out multi-physics and multi-scale couplings that are increasingly relevant. Similarly, the relationship be-
tween simulation and experimental approach is becoming ever closer and more dynamic. Obviously, the objectives are to 
predict the properties of the material, to anticipate possible defects, and ultimately to optimize the process, taking into 
account the different thermo-mechanical treatments applied.
Welding, which has been the subject of much work since the 1970s, is a perfect illustration of this evolution. Simulation 
of the various welding processes has always required (and still requires) the use of the most advanced numerical methods. 
Consider that the full simulation of a welding assembly process requires, in absolute terms, taking account of the passage 
from a solid to a molten state under the action of a heat source, the flow in this molten pool with surface tension and 
Marangoni effect, its interaction with the heat source, the cooling and the metallurgical transformations that go with, in 
order to predict the residual stresses and distortions or the initiation of cracks.
During this decade, additive manufacturing processes have matured technologically. Announced as a revolution, they 
have already changed, at least in some areas, our perception of materials manufacturing. These processes allow, on the one 
hand, the production of parts to be relocated as close as possible to the site of use; on the other hand, they allow the 
production of custom-made parts, with complex geometries that cannot be achieved by traditional manufacturing methods. 
However, beyond the technological aspect, the classic questions in materials manufacturing are still there: what is the qual-
ity of the manufactured part, how to optimize the process, how to optimize the geometry of the part? Strong similarities 
exist between metal additive manufacturing processes and welding processes. For example, in Powder Bed Fusion technolo-
gies, which are commonly used printing techniques, powder particles from particular regions of a powder bed are fused 
together by thermal energy. In this context, the strategies developed for modeling and simulating the welding processes 
are of great help in understanding the thermo-mechanical-metallurgical cycles that materials undergo during an additive 
manufacturing process.
The technological and scientific importance of ‘Computational methods in welding and additive manufacturing’ justifies 
that the journal Comptes rendus Mecanique devotes a thematic dossier to this topic. The present issue compiles eight peer-
reviewed contributions giving an overview of the work performed in this area. We hope that this thematic issue brings to 
the readers’ knowledge new points of view and advances in the field and leads to further contributions to the journal on 
this topic.
As guest editors, we would like to thank the editorial staff and Jean-Baptiste Leblond, editor-in-chief, as well as all the 
authors and reviewers who helped make this dossier.
Julien Bruchon
École des mines de Saint-Étienne, Université de Lyon, CNRS, UMR 5307 LGF, Centre SMS, Departement MPE,
42023 Saint-Étienne, France
E-mail address: bruchon@emse.fr
Jean-Michel Bergheau
Université de Lyon, École nationale d’ingénieurs de Saint-Étienne, LTDS, UMR CNRS 5513, 58, rue Jean-Parot,
42023 Saint-Etienne cedex 2, France
E-mail address: jean-michel.bergheau@enise.frhttps://doi.org/10.1016/j.crme.2018.08.009
1631-0721/© 2018 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license 
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
998 Computational methods in welding and additive manufacturing/Simulation numérique des procédés de soudage et de fabrication additiveRomain Boman
University of Liège, Department of Aerospace and Mechanical Engineering, 9, allée de la Découverte, B-4000 Liège, Belgium
E-mail address: r.boman@ulg.ac.be
Available online 24 August 2018


Foreword

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Application of multi-phase viscoplastic material modelling to computational welding mechanics of grade-s960ql steel

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