Abstract
A finite element model for cement-stabilized soil block (CSSB) masonry members—including nonlinear stress-strain relationship—has been developed and compared with experimental results. Primarily, this model serves as a simulation tool to study various problems for a large number of stress–strain state and loading conditions of CSSB masonry elements. The model presented is characterized by several parameters experimentally ascertained through triaxial and other testing. Furthermore, these parameters allow the model to capture the elastic, plastic, and softening behavior of CSSB masonry. From a constitutive behavioral standpoint, at small strain levels, the material is approximated as linear elastic. Plastic deformation of the material is captured with a modified version of the Sandia Geomodel, which is specifically designed to replicate geological material behavior. Lastly, at localized softening failure, a damage-like constitutive model which takes into account the normal and shear traction balance on the slip-weakening surface is employed. This model includes cohesion degradation as well as friction under compression. Within the finite element framework, the Strong Discontinuity Approach is used to track localized material failure from element to element. In addition to this, a novel method for modeling interfaces in finite elements is used to replicate the behavior of brick-mortar interfaces. The two featured experiments which are simulated in this study are normal to bedjoint and parallel to bedjoint masonry setups, simplified via a plane strain approximation.
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Abbreviations
- \({\varvec{\sigma }}\) :
-
Stress tensor
- \({\varvec{c}}^e\) :
-
Elastic modulus
- \({\varvec{\epsilon }}\) :
-
Strain tensor
- \({\varvec{\epsilon }}^e\) :
-
Elastic strain
- \({\varvec{\epsilon }}^p\) :
-
Plastic strain
- \({\dot{\gamma }}\) :
-
Consistency parameter
- g :
-
Plastic potential
- f :
-
Elastic modulus
- \({\varvec{\alpha }}\) :
-
Back stress in plasticity model
- \({\varvec{\xi }}\) :
-
Relative stress tensor (i.e \({\varvec{\sigma }} - {\varvec{\alpha }}\))
- \({\varvec{h}}^\alpha\) :
-
Hardening modulus for \({\varvec{\alpha }}\)
- \(\kappa\) :
-
Value of \(I_1\) at which cap function starts in Geomodel
- \(h^\kappa\) :
-
Hardening modulus for \(\kappa\)
- \(I_1\), \(J_2^\xi\), \(J_3^\xi\) :
-
Invariants of the relative stress \({\varvec{\xi }}\)
- \(\beta\) :
-
Lode angle
- \(\Gamma\) :
-
Third-invariant modifying function for yield function (Gudehus type)
- \(\psi\) :
-
Ratio of triaxial extension to compression strength
- \(F_f\) :
-
Shear failure surface for Geomodel yield function
- \(F^g_f\) :
-
Shear potential surface for Geomodel potential function
- A, B, C, \(\theta\) :
-
Material parameters for fitting \(F_f\)
- L, \(\phi\) :
-
Material parameters for fitting \(F_g\)
- H :
-
Heavisde function
- X/3, T :
-
Hydrostatic stress for compression and tensile yielding
- \(I_1^T\) :
-
Vlaue of \(I_1\) for onset of tension cap
- \({\varvec{u}}\) :
-
Displacement field
- \(\bar{{\varvec{u}}}\) :
-
Continuous part of the displacement field (e.g. in a fracture medium)
- \(\llbracket {{\varvec{u}}}\rrbracket = {\varvec{\zeta }}\) :
-
Jump in the displacement field (e.g from an opening fracture)
- \(H_S\) :
-
Heavisde function across a surface S
- \(\zeta _n\), \(\zeta _s\) :
-
Normal and shear jumps across a fracture surface
- \({\varvec{n}}\), \({\varvec{l}}\) :
-
Unit normal and shear directions to a fracture surface
- \({\varvec{\nabla }}^s\) :
-
Symmetric gradient
- \(\delta _S\) :
-
Dirac delta function across a surface
- \(\tilde{{\varvec{c}}}^{ep}\) :
-
Elastic perfectly plastic modulus tensor
- \(\tilde{{\varvec{A}}}\) :
-
Elastic perfectly-plastic acoustic tensor
- \({\varvec{m}}\) :
-
Unit direction of jump in the displacement field
- \(\tau\) :
-
Shear stress
- \(\sigma\) :
-
Normal stress
- f :
-
Friction
- c :
-
Cohesion
- c :
-
Initial cohesion
- \(\alpha _\sigma\) :
-
Ratio of shear to normal strength in localized fracture model
- \(\mu\) :
-
Friction coefficent
- \(<\cdot>\) :
-
McCauley brackets
- \(k_n\) :
-
Normal stiffness of localized fracture surface
- \(k_s\) :
-
Shear stiffness of localized fracture surface
- \(\alpha _\zeta\) :
-
Ratio of impact of shear to normal slip on a fractured surface
- \(\sigma _{eq}\) :
-
Equivalent stress on localized fracture surface
- \(\zeta _{eq}\) :
-
Equivalent slip on localized fracture surface
- \(G_I\), \(G_{II}\) :
-
Mode I, mode II fracture energy
- \(K_I\), \(K_{II}\) :
-
Mode I, mode II stress intensity factors
- \(\zeta ^*\) :
-
Critical shear slip (at which the cohesion becomes 0)
- \({\varvec{u}}^{conf}\) :
-
Conforming (to finite element shape functions) displacement
- \({\varvec{u}}^{enh}\) :
-
Enhanced (in addition to finite element shape functions) displacement
- \(M^h_S\) :
-
Shape function for enhanced displacement
- \(f^{h}\) :
-
Smoothing function; sum of shape functions on the active side of a fracture surface in a finite element
- \(r^e\) :
-
Finite element residual at element level
- \({\varvec{\Phi }}\) :
-
Yield function on fracture surface
References
Reddy BVV, Lal R, Nanjunda Rao KS (2007) Optimum soil grading for the soil-cement blocks. J Mater Civil Eng 19(2):139–148
Reddy BVV, Kumar PP (2011) Cement stabilised rammed earth. Part B: compressive strength and stress-strain characteristics. Mater Struct 44(3):695–707
Burroughs S (2008) Soil property criteria for rammed earth stabilization. J Mater Civil Eng 20(3):264–273
Reddy BVV, Latha M, (2012) Retrieving clay minerals from stabilised soil blocks. In: TERRA 2012 XI international conference on the study and conservation of earthen architecture heritage. Lima, Peru
Williamson LD (2004) Block-ramming machine. \(<\)https://patents.google.com/patent/US7311865B2/en\(>\) ( 12). US Patent 7311865B2
Perrocheau T (2008) Device for manufacturing a compressed brick and brick obtained by such a device. \(<\)https://patents.google.com/patent/FR2937892B1/en\(>\) (6). French Patent 2937892B1
Reddy BVV, Gupta A (2006a) Strength and elastic properties of stabilized mud block masonry using cement-soil mortars. J Mater Civil Eng 18(3):472–476
Rao KVM, Reddy BVV, Jagadish K (1996) Flexural bond strength of masonry using various blocks and mortars. Mater Struct 29(2):119–124
Walker P, Stace T (1997) Properties of some cement stabilised compressed earth blocks and mortars. Mater Struct 30(9):545–551
Walker P (1999) Bond characteristics of earth block masonry. J Mater Civil Eng 11(3):249–256
Reddy BVV, Walker P (2005) Stabilised mud blocks : Problems, prospects. In: 2005 conference on EarthBuild, Australia, UTS, Sydney, pp 63–75
Reddy BVV, Gupta A (2006b) Strength and elastic properties of stabilized mud block masonry using cement-soil mortars. J Mater Civil Eng 18(3):472–476
Reddy BVV, Gupta A (2006c) Tensile bond strength of soil-cement block masonry couplets using cement-soil mortars. J Mater Civil Eng 18(1):36–45
Reddy BVV, Lal R, Rao KSN (2007) Optimum soil grading for the soil-cement blocks. J Mater Civil Eng 19(2):139–148
Reddy BVV, Uday Vyas C (2008) Influence of shear bond strength on compressive strength and stress-strain characteristics of masonry. Mater Struct 41(10):1697–1712
Reddy BVV, Latha MS (2013) Influence of soil grading on the characteristics of cement stabilised soil compacts. Mater Struct 47(10):1633–1645
Rots JG (1988). Computational modeling of concrete fracture. Ph.D. thesis, Technische Hogeschool Delft, Technische Hogeschool Delft
Lourenco PB, Rots JG (1997) Multisurface interface model for analysis of masonry structures. J Eng Mech 123(7):660–668
Lotfi H, Shing P (1991) An appraisal of smeared crack models for masonry shear wall analysis. Comput Struct 41(3):413–425
Giambanco G, Gati LD (1997) A cohesive interface model for the structural mechanics of block masonry. Mech Res Commun 24(5):503–512
Giambanco G, Rizzo S, Spallino R (2001) Numerical analysis of masonry structures via interface models. Comput Methods Appl Mech Eng 190(49–50):6493–6511
Alfano G, Crisfield M (2001) Finite element interface models for the delamination analysis of laminated composites: mechanical and computational issues. Int J Numer Methods Eng 50(7):1701–1736
Massart TJ, Peerlings RHJ, Geers MGD (2007) An enhanced multi-scale approach for masonry wall computations with localization of damage. Int J Numer Methods Eng 69(5):1022–1059
Sacco E (2009) A nonlinear homogenization procedure for periodic masonry. Europ J Mech A/Solids 28(2):209–222
Marfia S, Sacco E (2012) Multiscale damage contact-friction model for periodic masonry walls. Comput Methods Appl Mech Eng 205:189–203
Rots J, Messali F, Esposito R, Jafari S, Mariani V (2016) Computational modelling of masonry with a view to groningen induced seismicity. In: Van Balen K, Verstrynge E, (eds) Structural analysis of historical constructions: anamnesis, diagnosis, therapy, controls—proceedings of the 10th international conference on structural analysis of historical constructions, SAHC 2016, CRC Press/Balkema, pp 227–238
Minga E, Macorini L, Izzuddin BA (2018) A 3D mesoscale damage-plasticity approach for masonry structures under cyclic loading. MECCANICA 53(7):1591–1611
D’Altri AM, Messali F, Rots J, Castellazzi G, de Miranda S (2019) A damaging block-based model for the analysis of the cyclic behaviour of full-scale masonry structures. Eng Fract Mech 209:423–448
D’Altri AM, de Miranda S, Castellazzi G, Sarhosis V (2018) A 3D detailed micro-model for the in-plane and out-of-plane numerical analysis of masonry panels. Comput Struct 206:18–30
Bean Popehn J, Schultz A, Lu M, Stolarski H, Ojard N (2008) Influence of transverse loading on the stability of slender unreinforced masonry walls. Eng Struct 30(10):2830–2839
Minaie E, Moon FL, Hamid AA (2014) Nonlinear finite element modeling of reinforced masonry shear walls for bidirectional loading response. Finite Elements Anal Des 84:44–53
Cundall PA, Strack ODL (1979) A discrete numerical model for granular assemblies. Géotechnique 29(1):47–65
De Lorenzis L, DeJong M, Ochsendorf J (2007) Failure of masonry arches under impulse base motion. Earthquake Eng Struct Dyn 36(14):2119–2136
Tondelli M, Beyer K, DeJong M (2016) Influence of boundary conditions on the out-of-plane response of brick masonry walls in buildings with rc slabs. Earthquake Eng Struct Dyn 45(8):1337–1356
Malomo D, Pinho R, Penna A (2018) Using the applied element method for modelling calcium silicate brick masonry subjected to in-plane cyclic loading. Earthquake Eng Struct Dyn 47(7):1610–1630
Malomo D, Pinho R, Penna A (2019) Applied Element Modelling of the Dynamic Response of a Full-Scale Clay Brick Masonry Building Specimen with Flexible Diaphragms. Int J Architect Heritage
Park K, Paulino G (2011) Cohesive zone models: a critical review of traction-separation relationships across fracture surfaces. ASME Appl Mech Rev 64(6)
Reyes E, Casati M, Gálvez J (2008) Cohesive crack model for mixed mode fracture of brick masonry. Int J Fracture 151(1):29–55
Reyes E, Gálvez J, Casati M, Cendón D, Sancho J, Planas J (2009) An embedded cohesive crack model for finite element analysis of brickwork masonry fracture. Eng Fracture Mech 76(12):1930–1944
Simo JC, Oliver J, Armero F (1993) An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids. Comput Mech 12(5):277–296
Regueiro RA, Borja RI (1999) A finite element model of localized deformation in frictional materials taking a strong discontinuity approach. Finite Elements Anal Des 33(4):283–315
Regueiro RA, Borja RI (2001) Plane strain finite element analysis of pressure sensitive plasticity with strong discontinuity. Int J Solids Struct 38(21):3647–3672
Borja RI, Regueiro RA (2001) Strain localization in frictional materials exhibiting displacement jumps. Comput Methods Appl Mech Eng 190(20–21):2555–2580
Borja RI, Foster CD (2007a) Continuum mathematical modeling of slip weakening in geological systems. J Geophys Res Solid Earth 112(B4)
Foster CD, Borja RI, Regueiro RA (2007) Embedded strong discontinuity finite elements for fractured geomaterials with variable friction. Int J Numer Methods Eng 72(5):549–581
Weed DA, Foster CD, Motamedi MH (2015) A combined opening-sliding formulation for use in modeling geomaterial deformation and fracture patterns. In review. Comput Struct
Motamedi M, Foster CD (2015) An improved implicit numerical integration of a non-associated, three-invariant cap plasticity model with mixed isotropic-kinematic hardening for geomaterials. Int J Numer Anal Methods Geomech 39(17):1853–1883
Allix O, Ladeveze P, Corigliano A (1995) Damage analysis of interlaminar fracture specimens. Compos Struct 31(1):61–74
Mroz Z, Giambanco G (1996) An interface model for analysis of deformation behaviour of discontinuities. Int J Numer Anal Methods Geomech 20(1):1–33
Alfano G, Sacco E (2006) Combining interface damage and friction in a cohesive-zone model. Int J Numer Methods Eng 68(5):542–582
De Lorenzis L (2012) Some recent results and open issues on interface modeling in civil engineering structures. Mater Struct 45(4):477–503
Shieh-Beygi B, Pietruszczak S (2008) Numerical analysis of structural masonry: mesoscale approach. Comput Struct 86(21):1958–1973
Fossum AF, Brannon RM (2004) The Sandia GeoModel: theory and user’s guide. SAND report August. Sandia National Laboratories, Albuquerque
Foster C, Regueiro R, Fossum A, Borja R (2005) Implicit numerical integration of a three-invariant, isotropic/kinematic hardening cap plasticity model for geomaterials. Comput Methods Appl Mech Eng 194(50–52):5109–5138
Sun W, Chen Q, Ostien JT (2014) Modeling the hydro-mechanical responses of strip and circular punch loadings on water-saturated collapsible geomaterials. Acta Geotechnica 9(5):903–934
Bauschinger J (1881) Uber die Vernaderung der Elasticitatsgrenze und elastcitatsmodulverschiedener. Metal Civil NF 27:289–348
Regueiro RA, Foster CD (2011) Bifurcation analysis for a rate-sensitive, non-associative, three-invariant, isotropic/kinematic hardening cap plasticity model for geomaterials: Part i. small strain. Int J Numer Anal Methods Geomech 35(2):201–225
Ortiz M, Leroy Y, Needleman A (1987) A finite element method for localized failure analysis. Comput Methods Appl Mech Eng 61(2):189–214
Foster C, Weed D (2019) A new method for embedding predefined interfaces in finite elements. Finite Elements Anal Des 158:31–42
Gui Y, Bui H, Kodikara J (2015) An application of a cohesive fracture model combining compression, tension and shear in soft rocks. Comput Geotech 66:142–157
Hillerborg A, Modéer M, Petersson P-E (1976) Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement Concrete Res 6(6):773–781
Camacho G, Ortiz M (1996) Computational modelling of impact damage in brittle materials. Int J Solids Struct 33(20–22):2899–2938
de Borst R, Remmers JJ, Needleman A (2006) Mesh-independent discrete numerical representations of cohesive-zone models. Eng Fract Mech 73(2):160–177
Reinhardt H (1984) Fracture mechanics of fictitious crack propagation in concrete. Heron 29(2):3–42
Cornelissen H, Hordijk D, Reinhardt H (1986) Experimental determination of crack softening characteristics of normalweight and lightweight concrete. Heron 31(2):45–56
Guinea G, Planas J, Elices M (1994) A general bilinear fit for the softening curve of concrete. Mater Struct 27(2):99–105
Borja RI, Foster CD (2007b) Continuum mathematical modeling of slip weakening in geological systems. J Geophys Res Solid Earth 112(B4)
Ida Y (1972) Cohesive force across the tip of a longitudinal-shear crack and griffith’s specific surface energy. J Geophys Res 77(20):3796–3805
Rinehart AJ, Bishop JE, Dewers T (2015) Fracture propagation in indiana limestone interpreted via linear softening cohesive fracture model. J Geophys Res Solid Earth 120(4):2292–2308
Simo J, Hughes T (1998) Computational inelasticity, xiv edn. Springer, Berlin
Oliver J, Huespe A, Blanco S, Linero D (2006) Stability and robustness issues in numerical modeling of material failure with the strong discontinuity approach. Comput Methods Appl Mech Eng 195(52):7093–7114
Tennant AG, Foster CD, Reddy BVV (2016) Detailed experimental review of flexural behavior of cement stabilized soil block masonry. J Mater Civil Eng 28(6):06016004
Sarangapani G, Reddy BVV, Jagadish KS (2005) Brick-mortar bond and masonry compressive strength. J Mater Civil Eng 17(2):229–237
Tennant AG, Foster CD, Reddy BVV (2013) Verification of masonry building code to flexural behavior of cement-stabilized soil block. J Mater Civil Eng 25(3):303–307
Uday Vyas CV, Reddy BVV (2010) Prediction of solid block masonry prism compressive strength using fe model. Mater Struct 43(5):719–735
Acknowledgements
The authors wish to acknowledge the Dept. of Civil Engineering at the Indian Institute of Science for facilitating the experimental work, as well as the United States and Indian government for funding through the Fulbright program. The first second, and third authors acknowledge the support of the U.S. National Science Foundation, Grant CMMI-1030398.
Funding
This study was partially funded by U.S. National Science Foundation, Grant CMMI-1030398 and US-India Education Foundation (Fulbright Program).
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Weed, D.A., Tennant, A.G., Motamedi, M.H. et al. Finite element model application to flexural behavior of cement stabilized soil block masonry. Mater Struct 53, 61 (2020). https://doi.org/10.1617/s11527-020-01490-z
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DOI: https://doi.org/10.1617/s11527-020-01490-z