The aim of this paper is to model the effective viscoelastic properties of micro-cracked masonry based on a homogenization approach. The masonry is modeled as a mixture of a micro-cracked viscoelastic mortar matrix and isotropic elastic brick inclusions. Firstly, the effect of micro-cracks on the effective viscoelastic properties of mortar is considered at micro-scale. Secondly, brick inclusions are added in the micro-cracked mortar matrix at mesoscale to simulate effective viscoelastic properties of masonry. The modified Maxwell viscoelastic model is employed for noncracked, micro-cracked mortar matrix and micro-cracked masonry. The use of the same modified Maxwell model of the noncracked mortar to model the viscoelastic properties of the micro-cracked mortar is an approximation. This approximation is carried out in the short- and the long-term behaviors in the Laplace–Carson space. It is validated in transient situation with exact solutions obtained from the inverse Laplace–Carson transform for simple loading conditions. This technique allows avoiding the complexity of the inverse Laplace–Carson transform. The isotropic case with random orientation distribution of open micro-cracks in mortar and with spherical brick inclusions is considered.
BudianskyBO’connellRJ (1976) Elastic moduli of a cracked solid. International Journal of Solids and Structures12(2): 81–97.
2.
CecchiASabK (2002) Out of plane model for heterogeneous periodic materials: The case of masonry. European Journal of Mechanics-A/Solids21(5): 715–746.
3.
CecchiATralliA (2012) A homogenized viscoelastic model for masonry structures. International Journal of Solids and Structures49(13): 1485–1496.
4.
ChoiKKLisselSLReda TahaMM (2007) Rheological modelling of masonry creep. Canadian Journal of Civil Engineering34(11): 1506–1517.
5.
Cohen AM (2007) Numerical Methods for Laplace Transform Inversion (vol. 5). New York: Springer Science & Business Media.
6.
DonolatoC (2002) Analytical and numerical inversion of the Laplace–Carson transform by a differential method. Computer Physics Communications145(2): 298–309.
7.
DormieuxLKondoD (2009) Stress-based estimates and bounds of effective elastic properties: The case of cracked media with unilateral effects. Computational Materials Science46(1): 173–179.
8.
Dormieux L, Kondo D and Ulm FJ (2006) Microporomechanics. Chichester: John Wiley & Sons.
9.
EvansRHMaratheMS (1968) Microcracking and stress-strain curves for concrete in tension. Materials and Structures1(1): 61–64.
10.
Haddad YM (1995) Viscoelasticity of Engineering Materials. Kluwer, Dordrecht, The Netherlands: Springer.
11.
HillerborgAModéerMPeterssonPE (1976) Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research6(6): 773–781.
12.
Hoang-DucHBonnetGMeftahF (2013) Generalized self-consistent scheme for the effective behavior of viscoelastic heterogeneous media: A simple approximate solution. European Journal of Mechanics-A/Solids39: 35–49.
13.
HoriiHNemat-NasserS (1985) Compression-induced microcrack growth in brittle solids: Axial splitting and shear failure. Journal of Geophysical Research: Solid Earth (1978–2012)90(B4): 3105–3125.
14.
IgnoulSSchueremansLTackJ, (2006) Creep behavior of masonry structures–failure prediction based on a rheological model and laboratory tests. Structural Analysis of Historical Constructions2: 913–920.
15.
LahellecNSuquetP (2007) Effective behavior of linear viscoelastic composites: A time-integration approach. International Journal of Solids and Structures44(2): 507–529.
16.
LeQVMeftahFHeQC, (2007) Creep and relaxation functions of a heterogeneous viscoelastic porous medium using the Mori-Tanaka homogenization scheme and a discrete microscopic retardation spectrum. Mechanics of Time-Dependent Materials11(3–4): 309–331.
17.
LencznerD (1986) Creep and prestress losses in brick masonry. The Structural Engineer64(3): 57–62.
18.
LévesqueMGilchristMDBouleauN, (2007) Numerical inversion of the Laplace–Carson transform applied to homogenization of randomly reinforced linear viscoelastic media. Computational Mechanics40(4): 771–789.
19.
Monagan G, Heal L and Vorkoetter M (2007) Maple 12 introductory programming guide. Maplesoft, Waterloo.
20.
MoriTTanakaK (1973) Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica21(5): 571–574.
21.
NedjarBLe RoyR (2013) An approach to the modeling of viscoelastic damage. Application to the long-term creep of gypsum rock materials. International Journal for Numerical and Analytical Methods in Geomechanics37: 1066–1078.
22.
Nguyen ST (2014) Generalized Kelvin model for micro-cracked viscoelastic materials. Engineering Fracture Mechanics 127: 226–234.
23.
Nguyen ST (2015a) Generalized Maxwell model for micro-cracked viscoelastic materials. International Journal of Damage Mechanics 26(5): 697–710.
24.
Nguyen ST (2015b) Approches multi-échelles pour des maçonneries viscoélastiques. PhD Thesis, Université d'Orléans, France, ftp://ftp.univ.orleans.fr/theses/thithunga-nguyen_3812.pdf
25.
Nguyen ST and Dormieux L (2015) Viscoelastic properties of transversely isotropic micro-cracked materials. International Journal of Damage Mechanics 1056789515575798.
26.
NguyenSTDormieuxLLe PapeY, (2011) A Burger model for the effective behavior of a microcracked viscoelastic solid. International Journal of Damage Mechanics20(8): 1116–1129.
27.
Nugeyre JB (1960) Inversion numérique de la transformation de Laplace-Carson. Grenoble: Doctoral Dissertation, Université Joseph-Fourier-Grenoble I.
28.
PapaETaliercioA (2005) A visco-damage model for brittle materials under monotonic and sustained stresses. International Journal for Numerical and Analytical Methods in Geomechanics29(3): 287–310.
29.
Piessens R (1971, October) Numerical inversion of the Laplace transform using generalised Laguerre polynomials. In Proceedings of the Institution of Electrical Engineers. IET Digital Library, vol. 118(10), pp. 1517–1522.
30.
PouyaAZaouiA (1999) Linéarisation et homogénéisation en viscoélasticité. Comptes Rendus de l'Académie des Sciences-Series IIB-Mechanics-Physics-Astronomy327(4): 365–370.
31.
RekikABrennerR (2011) Optimization of the collocation inversion method for the linear viscoelastic homogenization. Mechanics Research Communications38: 305–308.
32.
Salençon J (1983) Viscosité. Paris: Presse des Ponts et Chaussées.
33.
SayersCMKachanovM (1995) Microcrack-induced elastic wave anisotropy of brittle rocks. Journal of Geophysical Research: Solid Earth (1978–2012)100(B3): 4149–4156.
34.
SuquetP (2012) Four exact relations for the effective relaxation function of linear viscoelastic composites. Comptes Rendus Mécanique340(4): 387–399.
35.
VerstryngeESchueremansLVan GemertD (2011) Time-dependent mechanical behavior of lime-mortar masonry. Materials and Structures44(1): 29–42.
36.
ZhuQZKondoDShaoJF (2009) Homogenization-based analysis of anisotropic damage in brittle materials with unilateral effect and interactions between microcracks. International Journal for Numerical and Analytical Methods in Geomechanics33(6): 749–772.
37.
ZobackMDByerleeJD (1975) The effect of microcrack dilatancy on the permeability of Westerly granite. Journal of Geophysical Research80(5): 752–755.
