The document discusses the thermoelastic behavior of masonry-like solids where the Young's modulus is temperature-dependent. It presents a nonlinear elastic constitutive model for masonry-like materials under non-isothermal conditions. The model equations are solved using finite element analysis to study the stress fields in a spherical masonry container subjected to thermal loads. The results show that accounting for temperature-dependent material properties can influence stress distributions compared to assuming temperature-independent properties. Further experiments are needed to better understand these effects.

1. CC2011, Chania, Crete, Greece,6-9 September 2011
Thermoelastic behaviour of masonry-like solids
with temperature-dependent Young’s modulus
Maria Girardi, Cristina Padovani,
Andrea Pagni, Giuseppe Pasquinelli
Laboratory of Mechanics of Materials and Structures
Institute of Information Science and Technologies “A. Faedo”
Italian National Research Council
Pisa, Italy

2. CC2011, Chania, Crete, Greece,6-9 September 2011
Masonry-like solids in the presence of thermal variations
Masonry-like (no-tension) materials are nonlinear elastic materials whose constitutive equation is adopted
to model the mechanical behaviour of solids that do not withstand tensile stresses, such as masonry and
stones.
There are many engineering problems in which the presence of thermal variations (and then thermal
dilatation) must be taken into account
Masonry constructions subjected to
Refractory linings of converters and ladles employed
seasonal thermal variations
in the iron and steel industry
Δθ ≤ 1600 ºC
Δθ = − 40 ºC
The thermal variation is so
high that the dependence
of material constants on
temperature cannot be
ignored

3. CC2011, Chania, Crete, Greece,6-9 September 2011
Masonry-like materials under non-isothermal conditions
Sym the space of symmetric tensors
Sym+ the set of positive semidefinite symmetric tensors
Sym- the set of negative semidefinite symmetric tensors
the absolute temperature, the reference temperature
E Sym, the symmetric part of the displacement gradient, I the identity tensor
T Sym, the Cauchy stress tensor
the thermal expansion
I, the thermal dilatation due to the the thermal variation
the Young’s modulus, the Poisson’s ratio
No limitations on = (is small)

4. CC2011, Chania, Crete, Greece,6-9 September 2011
For Sym × there exists a unique triple of elements of Sym such that
(1)
The nonlinear elastic material with stress function = T is called masonry-like material
In the absence of thermal
variations we get the
constitutive equation of
masonry-like materials
introduced by Heyman and Di
Pasquale and Del Piero in the
80s

5. CC2011, Chania, Crete, Greece,6-9 September 2011
The equations of the thermoelasticity of masonry-like materials are coupled
If we assume that
then the equations of thermoelasticity are uncoupled and can be integrated separately.
The uncoupled equilibrium problem of masonry-like solids with temperature dependent
material properties subjected to thermal loads is solved via the finite element code NOSA
developed by the Mechanics of Materials and Structures Laboratory for nonlinear structural
analysis

6. CC2011, Chania, Crete, Greece,6-9 September 2011
Spherical container made of a masonry-like material
subjected to thermal loads
Let us consider a spherical container S with inner radius a and outer radius b, made of a masonry-
like material in the absence of body forces, subjected to surface tractions and temperatures such
that the problem has spherical symmetry
The displacement vector and the infinitesimal strain tensor are
with
The stress tensor and the fracture strain are

7. CC2011, Chania, Crete, Greece,6-9 September 2011
The solution to the constitutive equation (1) in the case of spherical symmetry is given by
If belongs to
If belongs to
If belongs to
If belongs to

8. CC2011, Chania, Crete, Greece,6-9 September 2011
The equilibrium equation is
with the boundary conditions
For with we consider the steady temperature distribution
we assume that the thermal expansion is with α positive constant
We want to compare the solutions to the equilibrium problem corresponding to different choices
of the Young’s modulus (with ν=0)

9. CC2011, Chania, Crete, Greece,6-9 September 2011
If the Young’s modulus does not depend on temperature, then the solution can be calculated
explicitly
If the Young’s modulus depends on temperature, then the solution can be calculated numerically
via the finite element code NOSA

10. CC2011, Chania, Crete, Greece,6-9 September 2011
is the transition radius
which separates the
entirely compressed
region from the cracked
region Cracked region =0
<0
=0 >0
compressed region
<0
<0
=0
=0

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Masonry-like material

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Masonry-like material

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Masonry-like material

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Masonry-like material

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Conclusions
In order to model numerically the behaviour of masonry-like solids in the presence of thermal variations
it is fundamental
1. To have realistic constitutive equations for the material
2. To know the dependence of the material constants on temperature (experimental data)
Further investigation is necessary to assess how temperature-dependent material properties influence the
stress field and the crack distribution in the case of thermomechanical coupling