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

122

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 

 

whereas first loading with large amplitudes with subsequent lower amplitudes results 

in a reduced life time. As a consequence, this assumption may or may not be 

conservative; 

 

ignorance of the stress redistribution which is attributed to accumulated damage 

within the fatigue cycles as well as creep deformations that reduce stress 

concentrations; 

 

size effects that are well captured to influence the static behaviour of concrete are not 

taken into account;  

 

the evaluation of the structural performance with the reduced stiffness and permanent 

deformations is not possible at intermediate stages; 

 

a combination with other influences is only possible to macroscopic 

phenomenological coupling terms that are difficult to calibrate and often lack a clear 

physical interpretation. 

For this purpose, a continuum model for fatigue deterioration of concrete has been developed 

[9]. This model is based on a viscoplastic formulation in the effective stress space, where 

isotropic damage is related to the irreversible strains. For details, the reader is referred to the 

original paper.  As can be seen in Figure 6 in [9], the model is able to approximate the general 

trend of the Wöhler line.  A better lifetime prediction in the low cycle regime can be achieved 

once the reversible strain is assumed to contribute to damage as well. Furthermore, the 

application to structural problems requires additional regularization procedures such as the 

gradient enhanced damage model [10]. An generalization of the model is required to extend 

the model towards applications in fatigue modelling.  

The stress   in the new model is given as a function of the elasticity tensor C and the local 

elastic strain   

 

( ) 

with an isotropic damage variable  . Generally, the regularization can be formulated by either 

expressing the isotropic damage using an integral type nonlocal model, or as a gradient 

enhanced model using additional nonlocal quantities as unknowns and an additional 

Helmholtz equation to describe its evolution. The latter approach is proposed here, since the 

bandwidth of the corresponding stiffness matrices is constant. The evolution equation for the 

equivalent nonlocal strain  is given by 

 

( ) 

with additional Neumann boundary condition for the nonlocal quantities. For a discussion on 

the model see e.g. [11]. The local equivalent strain is  given  by a modified von Mises 

criterion according to [12]. In the original formulation by Peerlings, damage occurs once the 

maximum equivalent strain   achieves the threshold strain     

 if  

 

( ) 

 if  

 

( ) 

An exponential softening is assumed in the present model. The damage variable is now driven 

by the accumulated strain instead of the accumulated irreversible strain as in the previous 

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123

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 

 

model. In order to allow for the cyclic damage accumulation, the evolution equation was 

reformulated by the initial value problem 

 

( ) 

where the history variable   is no longer the maximum equivalent strain in the material point. 

Using this continuum model to simulate fatigue requires a time discretization that decomposes 

each cycle into subincrements, e.g. 12 subincrements. For high cycle fatigue with a total 

number of cycles until failure in the order of 10

6

, this exceeds most computational resources, 

especially when dealing with complex heterogeneous mesoscale models that itself require a 

fine resolution of the spatial discretization. 

For this purpose, a cycle jump method has been implemented. The idea is based on the 

assumption that the stress within consecutive cycles is quasi-periodic. The increment of the 

history variables that characterize the evolution of damage is calculated within a single cycle 

and then extrapolated to the next 10, 100 or 1000 cycles. After the extrapolation step, an 

equilibrium solution has to be calculated and another cycle is fully integrated to obtain the 

next jump for the extrapolation. A result of the calculation is shown in Figure 6, where the 

evolution of the equivalent nonlocal strain is plotted for different stages of the lifetime. A 

displacement controlled tensile analysis is performed in vertical direction with a mean 

displacement amplitude corresponding to 10% of the mean displacement.  A stress amplitude 

resulting in the first cycle  achieves 45% of the tensile strength (f

t

 = 3MPa, f

c

 = 20 MPa, E = 

40 GPa, c=18mm², n=1, 

, 

). In this test case, only voids are 

considered to validate the cycle jump method and the extension of the model to fatigue for 

heterogeneous models. 

The example should only demonstrate that a continuum model for fatigue has to include the 

influence of the heterogeneous microstructure. It allows to evaluate the damage state and the 

crack distribution within the specimen/structure at intermediate stages of the lifetime. 

Furthermore, interactions with other effects such as thermal loads, creep/shrinkage or the 

influence of the mean stress as well as the full 3D-stress state can be taken into account. It is 

further to be highlighted that, in this model, the static strength is assumed to be the limiting 

case of  fatigue failure with a failure after a single cycle. As a consequence, standard material 

test can be used to calibrate the fatigue model with only a few additional tests required to 

a)

 

After the first cycle 

b)

 

After 20% of the 

lifetime 

c)

 

Close to failure 

    3·10

-4 

 

 

 

 

 

     0  

Figure 6 : Evolution of the nonlocal equivalent strain 

 at different stages of the lifetime.  

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