Finite element model application to flexural behavior of cement stabilized soil block masonry


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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\({\varvec{\sigma }}\) :

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 :


c :


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


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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.


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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Correspondence to Craig D. Foster.

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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).

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  • Soil-cement
  • Masonry
  • Stabilized soil block
  • Compressed earth block
  • Embedded discontinuity
  • Damage