Keywords: quarry masonry, fracture energy, wedge splitting test, homogenization, periodic unit cell.

Estimates of the ultimate load bearing capacity of historical structures often require complex nonlinear full scale analysis. The problem becomes particularly difficult when addressing the behavior of masonry structures with either a regular or irregular arrangement of stone blocks bond to a mortar phase. Clearly, introducing all geometrical details of the meso-structure within a macroscopic computational model would be prohibitively expensive. A crucial step thus appears in the derivation of the estimates of the macroscopic or homogenized effective properties. As suggested in [1,2] the elements of first order homogenization presented in the framework of a certain statistically equivalent periodic unit cell (SEPUC) may serve as a reasonable tool to accomplish this task. A single numerical experiment then readily provides the desired fracture energy as the area under macroscopic stress-strain curve multiplied by the area of the unit cell and divided by the total crack length (the extent of traction free surfaces).

No objections with reference to periodic fields are expected when estimating the elastic effective properties only. However, when it comes to material parameters describing failure one may argue that the concept of homogenization based on periodicity assumptions and existence of uniform fields is objectionable especially if dealing with quasi-brittle materials prone to localized rather than distributed damage. In the present approach this remark can be safely overruled providing the analysis on two relevant scales is totally uncoupled so that the bulk properties derived from homogenization are introduced directly into the macroscopic constitutive law with no back reference to the actual heterogeneous meso-structure thus allowing for evolution of a highly localized failure zone due to strain softening feature of quasi-brittle materials.

With reference to the strain softening character of the failure of masonry structures the most appealing material characteristic is the homogenized macroscopic fracture energy. Derivation of this quantity, keeping in mind the above comment, was discussed in [1] with applications to both regular or irregular stone masonry walls. In the present contribution this approach will be extended to cover also quarry masonry that often serves as the filling of pillars and interior part of historical stone bridges. An improved material model for such a meso-structure is proposed in [3]. This model fundamentally differs from standard or simplified models, prevailingly describing the mortar joints between the stone blocks by contact elements of zero thickness. In the improved model the mortar is discretized by finite elements (similarly to the dicretization of stone blocks) but moreover the contact elements cover the stone boundaries to comprise the impaired material properties of the interfacial transition zone (ITZ) between the mortar and stones. The Mohr-Coulomb material model is applied in conjunction with the contact elements and the reduced values of cohesion and tensile strength represent the special properties of the ITZ. As evident from [3], the improved model exhibits a better mesoscopic response when compared with the simplified model and that is why it is applied also in the present paper to predict the macroscopic (effective) fracture energy of quarry masonry.

To advocate the applicability of homogenization technique the paper offers another (rather different) way of the determination of macroscopic or homogenized fracture energy of quarry masonry. This approach complies with the RILEM recommendations and draws on the series of numerical representations of the macroscopic wedge splitting test assuming specimens of variable ligament lengths. While the estimated fracture energies vary with the depth of the wedge their plot gives, in the limit for the wedge depth approaching zero, the size independent effective fracture energy.

Comparison of the results suggests a good agreement between individual approaches and therefore their applicability for the solution of the present problem. Owing to its relative simplicity over more tedious wedge splitting test we recommend the latter approach as the more effective one particularly in case of numerical experiments.

1

J. Novák, M. Šejnoha and J. Zeman, "On Representative Volume Element Size for Analysis of Masonry Structures", Proceedings of The Tenth International Conference on Civil, Structural and Environmental Engineering Computing, Edited by B.H.V. Topping, Rome-Italy, 30.8.-2.9. 2005. doi:10.4203/ccp.81.188

2

M. Šejnoha, J. Zeman and J. Novák, "Homogenization of random masonry structures - comparison of numerical methods", In: EM 2004 - 17th ASCE Engineering Mechanics Division Conference, Newark, University of Delaware, 1-8, 2004.

3

J. Šejnoha, M. Šejnoha, J. Sýkora and J. Vorel, "An improved material model for quarry masonry", Proceedings of the Eighth International Conference on Computational Structures Technology. Civil-Comp. Press, Stirling, UK, 2006. doi:10.4203/ccp.83.88

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