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1. A numerical method for solving equilibrium problems of
no-tension solids subjected to thermal loads
Cristina Padovani *, Giuseppe Pasquinelli, Nicola Zani
Consiglio Nazionale delle Richerche, Istituto CNUCE-CNR, Via Santa Maria 36, 56126 Pisa, Italy
Received 8 September 1998; received in revised form 27 January 1999
Abstract
This paper starts out by recalling a constitutive equation of no-tension materials that accounts for thermal dilatation and the
temperature dependence of the material parameters. Subsequently, a numerical method is presented for solving, via the ®nite element
method, equilibrium problems of no-tension solids subjected to thermal loads. Finally, three examples are solved and discussed: a
spherical container subjected to two uniform radial pressures and a steady temperature distribution, a masonry arch subjected to a
uniform temperature distribution and a converter used in the steel and iron industry. Ó 2000 Elsevier Science S.A. All rights reserved.
Keywords: No-tension materials; Thermal loads; Finite element method
1. Introduction
In many applications it is necessary to model the behaviour of solids not withstanding tension in the
presence of thermal dilatation. For example, molten metal production processes, in particular integrated
steel manufacturing, require refractory linings able to withstand the thermo-mechanical actions produced
by high-temperature ¯uids [1]. Analysis of these coverings is usually carried out by considering the re-
fractory materials to be linear elastic, exhibiting the same behaviour in the presence of tension and com-
pression, though they are actually non-resistant to traction. Results obtained by applying such a
constitutive model are generally characterised by considerable tensile stresses and are thus quite unrealistic.
However, there are many other engineering problems concerning no-tension solids in which thermal
dilatation must be accounted for: consider, for example geological problems connected with the presence of
a volcanic caldera, such as that of Pozzuoli [2], or the in¯uence of thermal variations on stress ®elds in
masonry bridges [3]. In many such cases the thermal variation is so high that the dependence of the material
parameters on temperature cannot be ignored.
In [4] the authors present a constitutive equation for isotropic no-tension materials in the presence of
thermal expansion which accounts for the temperature-dependence of the material's parameters. In par-
ticular, explicit expressions for stress and inelastic strain are given as functions of the strain minus the
thermal dilatation; from these free energy, internal energy and entropy are then obtained, and both coupled
and uncoupled equations of the thermo-mechanical equilibrium of a no-tension solid have been developed.
In this paper we recall the constitutive equation presented in [4] and, by limiting ourselves to thermo-
mechanical uncoupling, we propose a numerical method for solution of the equilibrium problem of solids
not supporting tension that are subjected to thermal loads.
www.elsevier.com/locate/cma
Comput. Methods Appl. Mech. Engrg. 190 (2000) 55±73
*
Corresponding author.
0045-7825/00/$ - see front matter Ó 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 0 4 5 - 7 8 2 5 ( 9 9 ) 0 0 3 4 6 - 1

2. In Section 2 the solution to the constitutive equation proposed in [4] is calculated for the three-di-
mensional case. It is assumed that the thermal dilatation is a temperature-dependent spherical tensor, and
that the total strain minus the thermal dilatation is the sum of two components: an elastic part, on which
the stress, negative semi-de®nite depends isotropically and linearly, and an inelastic part, positive semi-
de®nite and orthogonal to the stress. We thereby obtain a non-linear elastic material conforming to a
masonry-like material [5] when no temperature change occurs. The solution to the equation for plane stress
problems is presented in Appendix A.
By applying a procedure similar to that used in [6], it can be proved that the solution to the equilibrium
problem is unique in terms of stress, and independent of the particular loading process chosen. This latter
result is necessary in order to justify application of incremental numerical techniques that must usually be
used because of the non-linearity of the constitutive equation.
At the end of Section 2 the derivative of the stress with respect to the strain is explicitly calculated for
three-dimensional problems (the derivative for plane stress is reported in Appendix A). This is needed in
order to calculate the tangent sti€ness matrix used in applying the Newton±Raphson method for solution of
the non-linear system obtained by discretising the structure in question into ®nite elements.
The constitutive equation and the numerical techniques for solving the non-linear boundary-value
problem have been implemented in the ®nite element code NOSA [7].
In Section 3, the stress and displacement ®elds are explicitly calculated for a spherical container sub-
jected to two uniform radial pressures and a steady temperature distribution. Finally, two problems are
numerically solved by applying NOSA: a masonry arch subjected to temperature changes and a converter
used in the steel and iron industry.
2. The equilibrium problem
In this section, after recalling the constitutive equation of isotropic no-tension material in the presence of
thermal expansion introduced in [4], we present a numerical procedure for solving the equilibrium problem
via the ®nite element method.
2.1. The constitutive equation
In the following, h 2 ‰
h1;
h2Š will be the current temperature, with h0 2 ‰
h1;
h2Š the reference temperature.
Let V be a three-dimensional linear space, and Lin the space of all linear transformations from V into
V, equipped with the inner product A B ˆ tr AT
B†, A, B 2 Lin, with AT
the transpose of A. Let us in-
dicate as Sym, Sym‡
and Symÿ
, the subsets of Lin constituted by symmetric, symmetric positive semi-
de®nite and symmetric negative semi-de®nite tensors, respectively.
In view of the target applications, no limitations are placed on the range of temperature variation. Let us
now assume that the thermal dilatation due to a temperature change h ÿ h0 is b h†I, where b is a suf®ciently
smooth function, called thermal expansion, such that b h0† ˆ 0, and I is the second-order identity tensor.
When h ÿ h0 is small, the expression of thermal dilatation can be written in the usual way, a h ÿ h0†I, where
a ˆ db h0†=dh is the linear coef®cient of thermal expansion. Denoting E as the symmetric part of the dis-
placement gradient, we assume that tensor E ÿ b h†I is O(d). 1
The strain E ÿ b h†I is assumed to be the
sum of an elastic part Ee
and an inelastic part Ea
positive semi-de®nite
E ÿ b h†I ˆ E e
‡ E a
; 2:1†
E a
2 Sym‡
: 2:2†
Moreover, we suppose that the stress T, orthogonal to Ea
and negative semi-de®nite, depends linearly and
isotropically on Ee
,
1
Given a mapping B from a neighbourhood of 0 in R into a vector space with norm kk, we write B d† ˆ O d† if there exist k 0 and
k0
0 such that kB d†k kjdj whenever jdj k0
.
56 C. Padovani et al. / Comput. Methods Appl. Mech. Engrg. 190 (2000) 55±73

3. T E a
ˆ 0; 2:3†
T 2 Symÿ
; 2:4†
T ˆ
E h†
1 ‡ m h†
E e
‡
m h†
1 ÿ 2m h†
tr E e
†I
; 2:5†
where Young's modulus E(h) and Poisson's ratio m(h) depend on the temperature and satisfy the inequalities
E h† 0; 0 6 m h† 0:5; for each h 2 ‰
h1;
h2Š: 2:6†
Tensor Ea
can be interpreted as fracture strain, in fact, if the inelastic part Ea
of strain is non-null in any
region of the structure, then we can expect fractures to be present in that region. Nevertheless, a simple
analysis of the components of Ea
does not generally yield any information about the direction of eventual
fractures. To this end, we point out that if for any v, v E a
v 0, v is not necessarily a fracture direction; in
other words, v is not necessarily an eigenvector of T corresponding to the zero eigenvalue. Nonetheless,
there must surely exist at least one eigenvector q of Ea
(in view of the coaxiality of Ea
and T, q is also an
eigenvector of T) such that
q Ea
q 0 2:7†
and then, by virtue of the orthogonality of Ea
and T,
q Tq ˆ 0: 2:8†
By such reasoning, we deduce that if F is a fracture surface, then every vector orthogonal to F is an
eigenvector of T corresponding to the eigenvalue 0. This criterion has been used in Section 3 in order to
reveal the regions of the converter where fractures are present and their corresponding directions.
A material having constitutive equation (2.1)±(2.5) is called a no-tension material and in the absence of
thermal variation, conforms to the classical no-tension or masonry-like material described in [5].
By a procedure similar to that used in [8], it is possible to prove that, given E; h† 2 Sym ‰
h1;
h2Š and
the functions b, E and m, the constitutive equation (2.1)±(2.5) has a unique solution (T, Ea
). Moreover,
tensors E, b(h)I, Ea
, Ee
and T are coaxial, and the constitutive equation (2.1)±(2.5) can be written with
respect to the basis fq1; q2; q3g of the eigenvectors of E. Let fe1; e2; e3g; fa1; a2; a3g and ft1; t2; t3g be the
eigenvalues of E, Ea
and T, respectively, with e1; e2 and e3 ordered in such a way that e1 6 e2 6 e3. The
quantities a1; a2; a3 and t1; t2; t3 satisfying (2.1)±(2.5) can be calculated as functions of
e1 ÿ b h†; e2 ÿ b h†; e3 ÿ b h†, their expressions have already been presented in [4] and they are recalled here
only brie¯y. To this end let us de®ne [4] the following subsets of Sym ‰
h1;
h2Š
R1 ˆ f E; h† j 2 e3 ÿ b h†† ‡ c h† tr E ÿ 3b h†† 6 0g; 2:9†
R2 ˆ f E; h† j e1 ÿ b h† P 0g; 2:10†
R3 ˆ f E; h† j e1 ÿ b h† 6 0; c h† e1 ÿ b h†† ‡ 2 1 ‡ c h†† e2 ÿ b h†† P 0g; 2:11†
R4 ˆ f E; h† j c h† e1 ÿ b h†† ‡ 2 1 ‡ c h†† e2 ÿ b h†† 6 0;
2 e3 ÿ b h†† ‡ c h† tr E ÿ 3b h†† P 0g; 2:12†
where we have put
c h† ˆ
2m h†
1 ÿ 2m h†
:
For later use, we observe that from the de®nition of R3 and R4, it clearly follows that in R3 and R4 we have
e1 e2 and e2 e3, respectively.
C. Padovani et al. / Comput. Methods Appl. Mech. Engrg. 190 (2000) 55±73 57

4. Given E ÿ b h†I, with spectral representation
P3
iˆ1 ei ÿ b h††qi qi, by solving the constitutive equa-
tion (2.1)±(2.5), we establish that tensors E a
and T have the following expressions:
if E; h† 2 R1, then
E a
ˆ 0; T ˆ
E h†
1 ‡ m h†
E
ÿ b h†I ‡
m h†
1 ÿ 2m h†
tr E ÿ b h†I†I
; 2:13†
if E; h† 2 R2, then
E a
ˆ E ÿ b h†I; T ˆ 0; 2:14†
if E; h† 2 R3, then
E a
ˆ ‰e2 ÿ b h† ‡ m h† e1 ÿ b h††Šq2 q2 ‡ ‰e3 ÿ b h† ‡ m h† e1 ÿ b h††Šq3 q3;
T ˆ E h† e1 ÿ b h††q1 q;
2:15†
if E; h† 2 R4, then
E a
ˆ
1
1 ÿ m h†
‰e3 ÿ b h† ‡ m h† e1 ‡ e2 ÿ e3 ÿ b h††Šq3 q3;
T ˆ
E h†
1 ÿ m2 h†
f‰e1 ÿ b h† ‡ m h† e2 ÿ b h††Šq1 q1 ‡ ‰e2 ÿ b h† ‡ m h† e1 ÿ b h††Šq2 q2g:
2:16†
We shall denote by ^
T, the function ^
T : Sym ‰
h1;
h2Š ! Sym which associates the stress T ˆ ^
T E; h† given
in (2.13)±(2.16) to every (E, h); ^
T is an isotropic, 2
continuous, non-linear, non-injective function, positively
homogeneous of degree one and di€erentiable with respect to E in the internal part of every region Ri.
2.2. The application of the ®nite element method
The equations governing the thermo-mechanical equilibrium of no-tension solids in the presence of
thermal variations have been set forth in [4]. These equations are: the strain-displacement relation, the
equilibrium equation, the constitutive equation for stress and heat ¯ux, and the equilibrium energy
equation. The system we have obtained is coupled because the temperature coecient and the coecient of
the derivative of temperature with respect to time in the energy equation depend on strain and strain rate. If
we assume, as in this paper, that E; b h†; b0
h†, _
E and _
h are O(d), then the thermoelastic equilibrium
equations are uncoupled and can be integrated separately. In other words, the temperature ®eld is obtained
by integrating the equation of conduction of heat and used as a thermal load in the subsequent resolution of
the equilibrium problem.
In order to solve the equilibrium problems for masonry-like solids by using the ®nite element method, we
must consider loading processes and associated incremental equilibrium problems. As in the isothermal
case [6], it is possible to prove that the numerical solution obtained by using an incremental procedure is
independent of the particular loading process chosen: it depends solely on the ®nal assigned load, provided
that the loading process considered is admissible, in the sense speci®ed as follows.
Let B be a body made of a no-tension material, Su and Sf two subsets of the boundary oB of B such that
their union covers oB and their interiors are disjoint. A loading process l t† ˆ ‰b x; t†; h x; t†; s x; t†Š, with b
and h de®ned on B ‰0;
tŠ, and s de®ned on Sf ‰0;
tŠ, is admissible if for each parameter t 2 ‰0;
tŠ the
corresponding boundary-value problem has a solution, i.e., if there exists a triple [u(t), E(t), T(t)] differ-
entiable with respect to t, constituted by stress ®eld T, strain ®eld E and displacement ®eld u de®ned on the
closure of B such that they satisfy the equations
E ˆ
ru ‡ ruT
2
; 2:17†
2 b
T is an isotropic function in the sense that b
T QEQT
; h† ˆ Qb
T E; h†QT
for each orthogonal tensor Q, for each E 2 Sym and
h 2 ‰h1; h2Š.
58 C. Padovani et al. / Comput. Methods Appl. Mech. Engrg. 190 (2000) 55±73

5. div T ‡ b ˆ 0; 2:18†
T ˆ ^
T E; h† 2:19†
on B, and the boundary conditions
Tn ˆ s on Sf ; 2:20†
u ˆ
u on Su; 2:21†
where n is the unit outward normal to Sf and
u an assigned displacement on Su ‰0;
tŠ.
We now turn to the numerical procedure used for ®nite element analysis of masonry solids. Let w be a
vector ®eld such that w ˆ 0 on Su, From (2.18) and (2.20) it follows that at every t, the following equi-
librium equation must be veri®ed:
Z
B
T rw dV ˆ
Z
Sf
s w dA ‡
Z
B
b w dV : 2:22†
Since T depends non-linearly on E, we must also consider the following incremental equilibrium equation:
Z
B
fDE
^
T E; h†‰ _
EŠg rw dV ˆ
Z
Sf
_
s w dA ‡
Z
B
_
b w dV ÿ
Z
B
fDh
^
T E; h† _
hg rw dV ; 2:23†
where the dot denotes the derivative with respect to t.
The ®nite element method allows us to transform the incremental equation (2.23) into the non-linear
evolution system
‰KŠf_
ug ˆ f _
f g; 2:24†
where f_
ug is the vector of nodal velocities, matrix [K] is obtained from the relation
fcg ‰KŠf_
ug ˆ
Z
B
fDE
^
T E; h†‰ _
EŠg rw dV 2:25†
with fcg the vector of nodal values of ®eld w, and ®nally
fcg f _
f g ˆ
Z
Sf
_
s w dA ‡
Z
B
_
b w dV ÿ
Z
B
fDh
^
T E; h† _
hg rw dV : 2:26†
We assume that Eq. (2.22) holds in correspondence of t, and that the body is therefore in equilibrium;
subsequently, we assign a load increment fDf g de®ned by means of relation
fcg fDf g ˆ
Z
Sf
s t ‡ Dt† ÿ s t†† w dA ‡
Z
B
b t ‡ Dt† ÿ b t†† w dV
ÿ
Z
B
f^
T E; h2† ÿ ^
T E; h1†g rw dV ; 2:27†
where h2 ˆ h t ‡ Dt†; h1 ˆ h t†. It is easy to verify that the following equality:
ÿ
Z
B
f^
T E; h2† ÿ ^
T E; h1†g rw dV ˆ
Z
B
fC h2†‰DE a
Š ‡ 3v h2†DbI ÿ DC‰E ÿ E1
a
Š
‡ 3Dvb h1†Ig rw dV 2:28†
holds, where we have
C h† ˆ
E h†
1 ‡ m h†
I
‡
m h†
1 ÿ 2m h†
I I
;
DC ˆ C h2† ÿ C h1†;
C. Padovani et al. / Comput. Methods Appl. Mech. Engrg. 190 (2000) 55±73 59

6. Db ˆ b h2† ÿ b h1†;
Dv ˆ v h2† ÿ v h1†;
3v ˆ
E
1 ÿ 2m
;
DE a
ˆ E2
a
ÿ E1
a
and E1
a
and E2
a
are the solutions to the constitutive equation corresponding to E; h1† and E; h2†,
respectively.
We then solve the linear system
‰K u†ŠfDug ˆ fDf g 2:29†
and follow the iterative procedure described in [8].
In order to determine the matrix ‰K u†Š of system (2.29) while accounting for (2.25), the derivative of the
stress with respect to the strain must be calculated in the four regions Ri.
For each E 2 Sym with eigenvalues e1; e2; e3 and eigenvectors q1; q2; q3, let us consider the orthonormal
basis of Sym
O1 ˆ q1 q1; O2 ˆ q2 q2; O3 ˆ q3 q3;
O4 ˆ
1

2
p q1 q2 ‡ q2 q1†; O5 ˆ
1

2
p q1 q3 ‡ q3 q1†;
O6 ˆ
1

2
p q2 q3 ‡ q3 q2†:
2:30†
The fourth-order tensor DE
^
T E; h† in the interiors
Qi of the four regions Ri can be calculated by accounting
for (2.13)±(2.16) and recalling that for e1 e2 e3 [6]
DEe1 ˆ O1;
DEe2 ˆ O2;
DEe3 ˆ O3;
DEO1 ˆ
1
e1 ÿ e2
O4 O4 ‡
1
e1 ÿ e3
O5 O5;
DEO2 ˆ
1
e2 ÿ e1
O4 O4 ‡
1
e2 ÿ e3
O6 O6;
DEO3 ˆ
1
e3 ÿ e1
O5 O5 ‡
1
e3 ÿ e2
O6 O6;
where the operator A B, with A; B 2 Lin is the fourth-order tensor de®ned by A B‰HŠ ˆ B H†A, for
every H 2 Lin.
For the derivative of ^
T with respect to E we have
if E; h† 2
R1, then
DE
^
T E; h† ˆ
E h†
1 ‡ m h†
I
‡
m h†
1 ÿ 2m h†
I I
; 2:31†
if E; h† 2
R2, then
DE
^
T E; h† ˆ O; 2:32†
60 C. Padovani et al. / Comput. Methods Appl. Mech. Engrg. 190 (2000) 55±73

7. if E; h† 2
R3, then
DE
^
T E; h† ˆ E h† O1
O1 ÿ
e1 ÿ b h†
e2 ÿ e1
O4 O4 ÿ
e1 ÿ b h†
e3 ÿ e1
O5 O5
; 2:33†
if E; h† 2
R4, then
DE
^
T E; h† ˆ
E h†
2 1 ‡ m h††
O1
ÿ O2† O1 ÿ O2† ‡
1 ‡ m h†
1 ÿ m h†
O1 ‡ O2† O1 ‡ O2† ‡ 2O4 O4
ÿ 2
e1 ‡ m h†e2 ÿ b h† 1 ‡ m h††
1 ÿ m h†† e3 ÿ e1†
O5 O5
ÿ 2
e2 ‡ m h†e1 ÿ b h† 1 ‡ m h††
1 ÿ m h†† e3 ÿ e2†
O6 O6
; 2:34†
where I and O are the fourth-order identity tensor and the fourth-order null tensor, respectively. Since in
R3 and R4 we have e1 e2 and e2 e3, respectively, the derivatives given in (2.33) and (2.34) are well-
de®ned.
3. Numerical examples
Three di€erent examples are dealt with in this section. In the ®rst, the explicit solution of an equilibrium
problem is determined, thus highlighting the di€erence between the thermo-mechanical behaviour of a no-
tension material and a linear elastic one having the same elastic constants and thermal expansion. More-
over, since the solution is determined with the sole aim of validating the numerical method proposed and its
implementation in the ®nite element program, NOSA, a hypothesis on the dependence of YoungÕs modulus
on temperature (cf. relation (3.3)) is made in order to facilitate explicit solution of the problem.
Subsequently, we analyse a masonry arch subjected to its own weight and a temperature distribution
representing the mean seasonal thermal variation, and ®nally, we consider a converter used in the steel and
iron industry, subjected to its own weight and a highly non-uniform temperature distribution. The two
structures, made of no-tension materials having constitutive equation (2.1)±(2.5), have been discretised into
®nite elements and analysed with the NOSA code. For the arch, the stress ®elds and corresponding lines of
thrust have been determined both for when it is subjected to its own weight alone and when subjected to
this weight as well as a temperature distribution. For the converter, on the other hand, in addition to
determining the stress ®eld, we also characterise any fracturing that would occur. The criterion introduced
in Section 2 and based on Eqs. (2.7) and (2.8) has been implemented in a graphic code aimed at revealing
the regions were fractures are present and their corresponding orientations.
3.1. Spherical container subjected to two uniform radial pressures and a steady temperature distribution
A spherical container with inner radius a and outer radius b is subjected to two uniform radial pressures
p1 and p2 acting on the internal and external boundary, respectively, and to a steady temperature distri-
bution h varying with the radius r
h r† ˆ
ab #1 ÿ #2†
b ÿ a
1
r
‡
b#2 ÿ a#1
b ÿ a
‡ h0: 3:1†
In (3.1) #1 and #2, with #1 #2 are de®ned by #1 ˆ h1 ÿ h0 and #2 ˆ h2 ÿ h0, with h1 ˆ h a† and h2 ˆ h b†,
respectively. For the thermal expansion b, we assume that
b h† ˆ a h ÿ h0† 3:2†
with constant a; PoissonÕs ratio is taken equal to zero and ®nally YoungÕs modulus is a decreasing function
of temperature
C. Padovani et al. / Comput. Methods Appl. Mech. Engrg. 190 (2000) 55±73 61

8. E h† ˆ x
ab #1 ÿ #2†
b ÿ a
h
ÿ
b#2 ÿ a#1
b ÿ a
ÿ h0
; 3:3†
with x ˆ E1=a a positive constant. In view of (3.1) we have
^
E r† ˆ E h r†† ˆ x r; 3:4†
in particular
^
E a† ˆ E1; ^
E b† ˆ
b
a
E1 E1:
For the moment, let us suppose that the spherical container is made of a linear elastic material. We denote
by r l†
r and r
l†
t , the radial and tangent stress, respectively. By using a procedure similar to that described in
[9] we get
r l†
r r† ˆ x l1C1rl1
‡ l2C2rl2
ÿ a
ab #1 ÿ #2†
b ÿ a
; 3:5†
r
l†
t r† ˆ x C1rl1
‡ C2rl2
ÿ a
ab #1 ÿ #2†
b ÿ a
; 3:6†
where l1 ˆ ÿ 1 ‡

3
p
†, l2 ˆ ÿ1 ‡

3
p
and C1; C2† is the unique solution of the linear system
l1al1ÿ1
l2al2ÿ1
l1bl1ÿ1
l2bl2ÿ1
C1
C2
ˆ
ÿ
p1
xa
‡
a b #1 ÿ #2†
b ÿ a
ÿ
p2
xb
‡
a a #1 ÿ #2†
b ÿ a
0
B
B
@
1
C
C
A: 3:7†
The radial displacement is
u l†
r† ˆ C1rl1
‡ C2rl2
‡ a
b#2 ÿ a#1
b ÿ a
r: 3:8†
It can be proved that for suitable values of parameters p1; p2; #1 and #2, the stress ®eld with components
(3.5) and (3.6) is purely compressive, while, on the contrary, there exist values of p1; p2; #1 and #2 such that
the radial stress is still negative, whereas the tangent stress is negative for a 6 r 6
r with
r belonging to (a, b),
and then becomes positive. Therefore, the stress ®eld in (3.5) and (3.6) is not the solution to the equilibrium
problem of a spherical container made of a material with constitutive (2.1)±(2.5). In this case, it is possible
to prove that the spherical container is compressed for a 6 r 6 r0, and the region r0 6 r 6 b is characterised
by a null tangent stress and is therefore cracked. The transition radius r0 is the only solution belonging to
the interval ‰a; bŠ to the non-linear equation
C1 r† rl1
‡ C2 r†rl2
ÿ a
ab #1 ÿ #2†
b ÿ a
ˆ 0 3:9†
and C1 r†; C2 r†† is given by
C1 r†
C2 r†
ˆ
l1al1ÿ1
l2al2ÿ1
l1rl1ÿ1
l2rl2ÿ1
ÿ1 ÿ
p1
xa
‡
a b #1 ÿ #2†
b ÿ a
ÿ
p2b2
xr3
‡
aab #1 ÿ #2†
b ÿ a†r
0
B
B
@
1
C
C
A: 3:10†
Through the same procedure used in [10], we arrive at the stress ®eld and radial displacement for the
container made of a no-tension material with constitutive (2.1)±(2.5):
r m†
r r† ˆ
l1C1 r0†rl1 ‡ l2C2 r0†rl2 ÿ a
ab #1 ÿ #2†
b ÿ a
; r 2 ‰a; r0Š;
ÿp2
b2
r2
; r 2 ‰r0; bŠ;
8
:
3:11†
62 C. Padovani et al. / Comput. Methods Appl. Mech. Engrg. 190 (2000) 55±73

9. r
m†
t r† ˆ
C1 r0†rl1 ‡ C2 r0† rl2 ÿ a
ab #1 ÿ #2†
b ÿ a
; r 2 ‰a; r0Š;
0; r 2 ‰r0; bŠ;
8
:
3:12†
u m†
r† ˆ
C1 r0† rl1 ‡ C2 r0† rl2 ‡ a
b#2 ÿ a#1
b ÿ a
r; r 2 ‰a; r0Š;
u0 ‡
p2 b2
2xr2
‡ a
ab #1 ÿ #2†
b ÿ a
lnr ‡
b#2 ÿ a#1
b ÿ a
r
; r 2 ‰r0; bŠ;
8
:
3:13†
where the constant u0 is determined by imposing the continuity of the radial displacement at r0:
u0 ˆ C1 r0†rl1
0 ‡ C2 r0†rl2
0 ‡ a
b#2 ÿ a#1
b ÿ a
r0 ÿ
p2 b2
2xr2
0
ÿ a
ab #1 ÿ #2†
b ÿ a
ln r0
‡
b#2 ÿ a#1
b ÿ a
r0
: 3:14†
The radial inelastic strain is equal to zero, and the tangent one is
e
m†
t r† ˆ
0; r 2 ‰a; r0Š;
u m†
r†
r
ÿ a
ab #1 ÿ #2†
b ÿ a†r
‡
b#2 ÿ a#1
b ÿ a
; r 2 ‰r0; bŠ:
8
:
3:15†
Figs. 1±3 show the behaviour of the radial and tangent stresses and the radial displacement for a spherical
container made of a linear elastic material (dashed line) and a no-tension elastic material (solid line). The
graphs have been obtained from expressions (3.5), (3.6), (3.8) and (3.11)±(3.13), using the following
parameter values:
a ˆ 1 m; b ˆ 2 m;
m ˆ 0; E1 ˆ 2:5 109
Pa;
a ˆ 1 10ÿ5
°C†
ÿ1
;
p1 ˆ 1 106
Pa; p2 ˆ 0:4 106
Pa;
h0 ˆ 30
C; #1 ˆ 400
C; #2 ˆ ÿ10
C:
Fig. 1. r l†
r and r m†
r vs. r.
C. Padovani et al. / Comput. Methods Appl. Mech. Engrg. 190 (2000) 55±73 63

10. With the aim of comparing the explicit solution with the numerical one calculated by means of the ®nite
element program NOSA, we have considered a half spherical container discretised with 800 axisymmetric
eight-node elements. Figs. 4±7 show the values of the stress components, radial displacement and tangent
inelastic strain calculated by using expressions (3.11)±(3.13) and (3.15) (continuous line) and those fur-
nished by NOSA code (markers).
3.2. Masonry arch subjected to its own weight and a uniform temperature distribution
A circular masonry arch having mean radius 110 cm and thickness 20 cm has been discretized into 4800
isoparametric plane stress elements. The arch, whose springings are ®xed, is subjected to its own weight.
The reference temperature is 30°C, and the arch subsequently reaches the temperature of )10°C. The elastic
Fig. 3. u l†
and u m†
vs. r.
Fig. 2. r
l†
t and r
m†
t vs. r.
64 C. Padovani et al. / Comput. Methods Appl. Mech. Engrg. 190 (2000) 55±73

11. Fig. 4. Radial stress vs. r, explicit solution and numerical solution.
Fig. 6. Radial displacement vs. r, explicit solution and numerical solution.
Fig. 5. Tangent stress vs. r, explicit solution and numerical solution.
C. Padovani et al. / Comput. Methods Appl. Mech. Engrg. 190 (2000) 55±73 65

12. constants, assumed to be independent of temperature, are E ˆ 5000 MPa; m ˆ 0:1, while the thermal ex-
pansion is b h† ˆ a h ÿ h0†, with a ˆ 1 10ÿ5
°C†ÿ1
. For the ®rst increment the weight alone has been
assigned, and the temperature variation is divided into the four subsequent increments. Figs. 8 and 9 show
the lines of thrust for the arch subjected to its weight alone, as well as under the action of both weight and a
temperature variation of )40°C. Figs. 10 and 11 present plots of the normal force and bending moment per
unit length vs. the anomaly, for the ®rst (solid line) and ®fth (dashed line) increment.
The lowering of the crown in correspondence to the ®rst and ®fth increment is 4:07 10ÿ3
cm and
1:02 10ÿ1
cm, respectively, while the thrust at the springing is 57.6 N in the ®rst case, and 54.02 N in the
second.
Fig. 7. Inelastic tangent strain vs. r, explicit solution and numerical solution.
Fig. 9. Line of thrust of the arch subjected to its own weight and a temperature variation of )40°C.
Fig. 8. Line of thrust of the arch subjected to its own weight.
66 C. Padovani et al. / Comput. Methods Appl. Mech. Engrg. 190 (2000) 55±73

13. 3.3. The converter
In this subsection we consider a converter used in the iron and steel industry, which is subjected to its
own weight and a high thermal gradient. The structure has been discretized into ®nite elements and
analysed with the NOSA code. In addition to the stress ®eld, the occurrence of fractures has been deter-
mined.
The converter, having axial symmetry, has been discretized into eight-node isoparametric elements
(Fig. 12). It is made up of an outer steel vessel, plus a wear layer and protective lining, both made of
refractory materials, though of di€erent types (Fig. 13). For the refractories, Poisson's ratio is assumed to
be zero, whereas Young's modulus is a function of temperature. The thermal expansion b h† is assumed to
be a linear function of temperature for the refractories and a quadratic function of temperature for the steel.
The converter is subjected to its own weight and the temperature distribution shown in Fig. 14. For the
boundary conditions, we have imposed symmetry conditions and assigned zero axial displacements to
the cylindrical part of the converter where it is attached to the suspension trunnions. Two thermo-me-
chanical analyses have been carried out: one, considering the refractory to be a non-linear elastic material
having constitutive equation (2.1)±(2.5); the other, modelling the refractory as linear elastic. In both the
cases, the steel has been assumed to be a linear elastic material. The results of the analyses have been
compared in order to highlight the di€erences in behaviour.
With regard to the stress ®eld, the numerical analysis shows that the regions that result to be subjected to
tensile stresses in the linear analysis are characterised by zero stress in the non-linear one. Moreover, al-
though the stress ®elds in the other regions are essentially equal in the two cases, the maximum compressive
stresses in the non-linear case are slightly lower. For the sake of comparison, Figs. 15 and 16 show the
Fig. 10. Normal force per unit length vs. the anomaly, at the ®rst (solid line) and ®fth (dashed line) load increment.
Fig. 11. Bending moment per unit length vs. the anomaly, at the ®rst (solid line) and ®fth (dashed line) load increment.
C. Padovani et al. / Comput. Methods Appl. Mech. Engrg. 190 (2000) 55±73 67

14. deformed con®guration superimposed upon the initial one for the two analyses. In the case of non-linear
behaviour, a substantially greater displacement of the lower part is observed. Analysis aimed at revealing
the fracture ®eld was performed according to the criterion based on (2.7) and (2.8) (Figs. 17 and 18).
Fig. 12. The converter discretized into ®nite elements.
Fig. 13. Wear layer, protective lining and steel vessel in the converter.
68 C. Padovani et al. / Comput. Methods Appl. Mech. Engrg. 190 (2000) 55±73

15. Fig. 14. Temperature distribution (°C) within the converter.
Fig. 15. Displacements magni®ed 10 times, non-linear analysis.
C. Padovani et al. / Comput. Methods Appl. Mech. Engrg. 190 (2000) 55±73 69

16. Disregarding the ®lling zones, it can be observed that fractures are concentrated in the protective layer, and
will close during the operational life. Thus, the safety of the converter is con®rmed by this numerical
analysis. Moreover, the presence of fractures near and parallel to the inner surface can explain the so-called
`onion peeling' which is exhibited by a converter as soon as it is set in operation.
Fig. 16. Displacements magni®ed 10 times, linear analysis.
Fig. 17. Fractures belonging to the meridian planes.
70 C. Padovani et al. / Comput. Methods Appl. Mech. Engrg. 190 (2000) 55±73

17. Regarding the behaviour of the steel vessel, the highest stress zone is located near the lower knee, below
the thick reinforcement ring. The stress levels calculated in this zone show that an elastic±plastic analysis
would give better results. Nevertheless, in spite of this simpli®cation, the qualitative predictions have been
clearly con®rmed. In fact, at the end of its operational life, the converter presented a major problem in the
vessel, just near the lower knee zone.
4. Conclusions
The constitutive equation and the numerical procedure proposed in this paper can be used in many
applications. In particular, the equation of no-tension materials in the presence of thermal expansion is
both realistic enough to describe the actual constitutive response of masonry and refractory materials, and
simple enough to be employed in many engineering problems.
The third example provided shows that the proposed numerical method permits determination, not only
of the stress ®eld, but more importantly, the distribution of cracking within the refractory and its critical
zones. The ®nite element code developed for thermo-mechanical analyses can be a useful tool for industries
involved in the production of molten metals and refractories, as well as those concerned with the design of
furnace metalwork.
Acknowledgements
Our thanks to SANAC S.p.A. for the information provided on the converter. The ®nancial support of
Progetto Policentrico `Meccanica Computazionale' of the C.N.R. is gratefully acknowledged.
Appendix A
An analysis of the plane stress is provided here, though for the sake of brevity, plane strain has been
omitted.
Let us suppose t3 ˆ q3 Tq3 ˆ 0. From (2.5) we obtain
e3 ÿ b h† ÿ a3 ˆ
m h†
1 ÿ m h†
a1 ‡ a2 ÿ e1 ÿ e2 ‡ 2b h††;
Fig. 18. Fractures belonging to planes orthogonal to the meridians.
C. Padovani et al. / Comput. Methods Appl. Mech. Engrg. 190 (2000) 55±73 71

18. moreover, since by virtue of (2.3), a3 is arbitrary, it can be assumed to be equal to zero. In order to calculate
the values of a1; a2; t1 and t2 we de®ne the following subsets of Sym ‰
h1;
h2Š
Q1 ˆ f E; h† j c h† e1 ÿ b h†† ‡ 2 1 ‡ c h†† e2 ÿ b h†† 6 0g; A:1†
Q2 ˆ f E; h† j c h† e1 ÿ b h†† ‡ 2 1 ‡ c h†† e2 ÿ b h†† P 0; e1 ÿ b h† 6 0g; A:2†
Q3 ˆ f E; h† j e1 ÿ b h† P 0g: A:3†
In this case Ea
and T have the following expressions:
if E; h† 2 Q1, then
E a
ˆ 0;
T ˆ
E h†
1 ‡ m h†
E
ÿ b h†I ‡
m h†
1 ÿ m h†
tr E ÿ b h†I†I
;
A:4†
if E; h† 2 Q2, then
E a
ˆ ‰m h†e1 ‡ e2 ÿ 1 ‡ m h††b h†Šq2 q2;
T ˆ E h† e1 ÿ b h††q1 q1;
A:5†
if E; h† 2 Q3, then
E a
ˆ e1 ÿ b h††q1 q1 ‡ e2 ÿ b h††q2 q2;
T ˆ 0:
A:6†
Recalling that e1 6 e2 are the eigenvalues of E, and q1, q2 are the corresponding eigenvectors, we de®ne the
tensors
O1 ˆ q1 q1; O2 ˆ q2 q2; O3 ˆ
1

2
p q1 q2 ‡ q2 q1†; A:7†
and we recall that for e1 e2 [6]
DEe1 ˆ O1; DEe2 ˆ O2;
DEO1 ˆ
1
e1 ÿ e2
O3 O3; DEO2 ˆ
1
e2 ÿ e1
O3 O3:
From (A.4)±(A.6) we obtain the derivative of ^
T with respect to E:
if E; h† 2
Q1, then
DE
^
T E; h† ˆ
E h†
1 ‡ m h†
I
‡
m h†
1 ÿ m h†
I I
; A:8†
if E; h† 2
Q2, then
DE
^
T E; h† ˆ E h† O1
O1 ÿ
e1 ÿ b h†
e2 ÿ e1
O3 O3
; A:9†
if E; h† 2
Q3, then
DE
^
T E; h† ˆ O: A:10†
72 C. Padovani et al. / Comput. Methods Appl. Mech. Engrg. 190 (2000) 55±73

19. References
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[4] M. Lucchesi, C. Padovani, G. Pasquinelli, Thermodynamics of no-tension materials, to appear in Internat. J. Solids Struc.
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[6] M. Lucchesi, C. Padovani, N. Zani, Masonry-like materials with bounded compressive strength, Internat. J. Solids Struc. 33 (14)
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