120
International RILEM Conference on Materials, Systems and Structures in Civil Engineering
Conference segment on Service Life of Cement-Based Materials and Structures
22-24 August 2016, Technical University of Denmark, Lyngby, Denmark
shrinkage in the outer layer and the almost zero shrinkage strains in the core, tensile stresses
in the outer layer build up and the specimen is no longer stress free. This effect is already
present for a macroscopic model, but additional restraining stresses build up when
considering the mesoscale structure. As a consequence, a calibration of a strain based
shrinkage model can only be obtained by a direct modelling of the experimental setup (and
not a smeared diagram where weight loss and thus water volume fraction is plotted over
macroscopic strains).
Another option to model shrinkage strains is based on the Biot-theory of porous media with
an additional stress component resulting from the moisture distribution.
( )
where the capillary pressure
can be calculated from the local relative humidity using the
Kelvin equation as discussed in [6]. The parameter
comprises the water volume fraction,
and is the identity tensor to link the scalar variables to the hydro-static stress tensor.
Shrinkage in this model is then interpreted as a hydrostatic pressure on the solid skeleton.
According to [7], an additional contribution is due to surface adsorbed water as well as
interlayer water. The latter is neglected in our model, since it does not contribute for relative
Figure 4 : Minimum principal stress after drying of
28 days (strain based approach).
Figure 5 : Maximum principal stress after drying of
28 days (strain based approach).
121
International RILEM Conference on Materials, Systems and Structures in Civil Engineering
Conference segment on Service Life of Cement-Based Materials and Structures
22-24 August 2016, Technical University of Denmark, Lyngby, Denmark
humidities above 20%. The results of the simulation are shown in Figure . As discussed
before, this only corresponds to the capillary pore water, the effects for surface adsorbed
water for 40% RH are in the same order of magnitude, resulting in roughly twice as much
global shrinkage strains. For the calculation, a Young’s modulus of the matrix material of
30GPa has been used, the particles have a Young’s modulus of 60GPa. According to the
figure, the strain distribution is strongly heterogeneous along the horizontal direction with a
maximum at the two ends and in the vertical direction at the top and bottom. This is due to the
core of the sample that is still at a RH of 95% and thus constrains the shrinkage of the outer
layer in the central part. Furthermore, the strong influence of the mesostructured is recognized
that is assumed to be inert regarding shrinkage.
The corresponding principal stresses are plotted in Figure 2 and Figure 3 for the stress based
approach, where a large hydrostatic stress state is obtained. The consideration of mesoscale
structure directly creates tensile stresses in some parts of the specimen that are in the order of
the material strength. For the strain based approach shown in Figure 4 and Figure 5, the
tensile stresses in the matrix are significantly larger.
Both approaches (strain or stress based) result in a very similar distribution of shrinkage
deformations. The main difference is that a coupling with a mechanical load does not induce
any influence on the strength for a strain based coupling (at least not for specimens that are
not restrained while drying), whereas the stress based approach shifts the failure surface in the
principal stress space of the mechanical failure surface along the hydrostatic axis. For the
additional hydrostatic pressure, shrinkage would be accompanied with an increase of the
uniaxial compressive strength, whereas the strain based model predicts a strength independent
of the moisture content.
4. Mesoscale modelling of fatigue with a continuum model
Evaluation of fatigue life is usually derived from a linear elastic simulation of the very first
cycle in order to comprise the stress level, more precisely the mean stress and the oscilation
amplitude. In combination with the experimentally determined Wöhler lines, the safety of the
structure is verified. For different loading amplitudes and mean values, a damage
accumulation theory such as the Palmgren Miner rule is used. This assumes a linear
accumulation of damage for different stress levels. The approach seems to be natural from an
engineering point of view, but it inherently has many weak points including both
experimental limitations
the experimental determination of Wöhlerlines is very time consuming, especially for
relatively low stresses with a large number of cycles up to final failure. Furthermore, a
high number of samples per setup is required to obtain statistically reliable results,
since the scatter for fatigue tests is in the order of a factor of 10;
standard Wöhler lines do not include a mean value of the stress. The additional
consideration of the mean stress in the test program is often not feasible. In addition,
the real stress in a structure is rarely a uniaxial stress, but a full 3D problem. The
extrapolation from 1D to 3D requires additional assumptions,
and model assumption. These include
the assumption of the linear damage accumulation which is only a rough
approximation. It has been shown [8] that first cycling with low amplitudes and a
subsequent large amplitude leads to a longer life time compared to linear theory,