Keywords: hysteresis, hysteretic, Preisach, damage, plasticity, cohesive crack, partition of unity (PUM).

Cyclic quasi-static test results of quasi-brittle materials reveal hysteretic loops during unloading and reloading in the post-peak region. Physically, this region corresponds with a cracked state of the material and the hysteretic loops are formed due to the frictional sliding of the crack faces. Such a hysteretic loop can be considered as a thermodynamic cycle and thus can be considered as an irreversible thermodynamic process. According to the second law of thermodynamics, this is accompanied with energy dissipation.

It is generally known that an accurate constitutive model is required to assure an accurate FEM-prediction of the structure. This means that the constitutive model in the post peak-region should incorporate the hysteretic phenomena. The constitutive model used is in fact an extension of a combined damage-plasticity model proposed by De Proft [1]. This model is capable of capturing the observed permanent deformations and the observed reduction of Young's modulus. The hysteretic loops are included by the coupling of the Preisach hysteresis model with the combined damage-plasticity model. The Preisach hysteresis model is capable of simulating a single hysteretic loop. It originates from electromagnetism and corresponds with the opening and closing of small relays simulating the orientation effects of small magnetic particles. This model was extended to a general mathematical hysteresis model as shown in [2]. The coupling of the Preisach hysteresis model and the combined damage-plasticity model is realized by the physical origin of the hysteretic loops. As mentioned before, the hysteretic phenomenon is caused by the sliding of the crack faces, the hysteretic phenomena can therefore be coupled with the number and size of the cracks. In its turn, the number and size of the cracks are coupled with the amount of damage. Consequently, it can be concluded that the combined damage-plasticity model and the Preisach hysteresis model are coupled by the damage parameter.

Subsequently, the model generated is been implemented in the PUM-FEM algorithm. This algorithm is chosen because of the fact that it is capable of simulating discrete cracks in a mesh-independent fashion (the kind of cracks that occur in quasi-brittle materials). However, before the implementation has been done, the continuum constitutive model has been discretised, to allow its implementation in the PUM-FEM model.

Furthermore, the numerical test results obtained with the PUM-FEM algorithm of a double notched test specimen under cyclic tensile loading are compared with the published experimental test results by Gopalaratnam, et al. [3]. It can be concluded that the observed phenomena can be simulated. However, a more advanced parameter study of the numerical material model is required to obtain a good matching of the numerical results with the experimental results.

1

K. De Proft, "Combined experimental-computational study to discrete fracture of brittle materials", PhD-thesis, Vrije Universiteit Brussel, 2003.

2

I.D. Mayergoyz, "Mathematical Models of Hysteresis and Their Applications", Academic Press, 2003

3

V.S. Gopalaratnam & S.P. Shah, "Softening response of plain concrete in direct tension", ACI Journal 82, p. 310-323, 1985.

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