Proceedings of the International rilem conference Materials, Systems and Structures in Civil Engineering 2016

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). 

 

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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, 

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