1. University of Houston
Department of Civil & Environmental Engineering
Computational Mechanics Group
Reza Mousavi, Amir Mohammadipour, Masoud DehghaniChampiri
Lattice Approach in Continuum and Fracture Mechanics Mechanical Behavior of Cement-based Materials
Abstract: This study reveals some aspects of lattice formulations to analyze the strain
concentration of a famous classic problem in solid mechanics at two different mechanics
perspectives: continuum and fracture. A 2D plane stress square panel with a single
circular hole is discretized based on Voronoi tessellation. A superelliptic formulation is
used to distribute Lattice computational points over the panel. From the perspective of
linear elasticity under uniaxial and biaxial loading, translational and rotational degrees
of freedom are considered at each computational node of the lattice to obtain strain
concentration factors around the circular perforation. It is observed that the lattice
approach is able to approximate the elastic strain concentration factors of three, four, and
two in uniaxial, biaxial shear, and equibiaxial tension loadings, respectively. To study the
LEFM and Griffith energy balance in uniaxial loading, a brittle lattice erosion technique
is used to compute the energy release rate determined by change in the global stiffness
matrix of the mesh with respect to crack extension. This fracture energy is then used to
determine the mode I stress intensity factor of the crack emanating from the hole which is
validated by the analytical formulation for the same problem. The comparison shows that
both methods give very close results for the mode I stress intensity factor. Being simple in
terms of constitutive formulation and failure criterion for erosion of brittle material, and
also using a propagating crack extension approach, the lattice formulation is used to
determine the fracture properties of cohesive/frictional material.
Keywords: lattice approach, fracture mechanics, continuum mechanics, strain
concentration factor, energy release rate, stress intensity factor
Fig. 1 (a) Degrees-of-freedom and external forces acting on a 2D frame element in local
coordinates and (b) constitutive relation for a single frame element for linear elastic
behavior.
Fracture criterion for the failure of the frame elements under tension and compression was
defined as a function of normal force and bending moments at computational points of
each frame member in the form of a failure condition
𝜎𝜎𝑒𝑒𝑒𝑒𝑒𝑒 =
𝑁𝑁
𝐴𝐴
± 𝛼𝛼′
( 𝑀𝑀𝑖𝑖 , 𝑀𝑀𝑗𝑗 ) 𝑚𝑚𝑚𝑚𝑚𝑚
𝑆𝑆𝑎𝑎
≥ 𝑓𝑓𝑡𝑡 𝑜𝑜𝑜𝑜 𝑓𝑓𝑐𝑐
Fig. 2 Lattice mesh generation of the rectangular panel using the superellipse formulation,
(a) Voronoi particles and their computational points or centroids, (b) Lattice mesh struts
with smooth transition from polar to Cartesian coordinate system. S and ∆θ are parameters
used to define the mesh.
Fig. 3 Determining the mesh parameters 𝑆𝑆𝑚𝑚𝑚𝑚 𝑚𝑚 and 𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 according to the circular hole
radius, 𝑅𝑅, and the panel width, 2𝑏𝑏, for a selected value of ∆𝜃𝜃 = 𝜋𝜋/64.
References:
Mohammadipour A, Willam K., Lattice Simulations for Evaluating Interface Fracture of
Masonry Composites. in Theoretical & Applied Fracture Mechanics, 82, 2016, 152–68.
Mohammadipour A., Willam K., Lattice Approach in Continuum and Fracture Mechanics.
in Journal of Applied Mechanics, 83(7), 2016, 071003-071003-9.
Fig. 4 The distribution of strain concentration factor, SCF, for 𝜀𝜀𝑦𝑦 in a finite thin panel with
a single circular hole, (a) the lattice mesh and corresponding boundary conditions, (b) the
SCF distribution for the panel with uniaxial tension in 𝑦𝑦 direction, (c) the SCF distribution
for the panel with biaxial tension in 𝑥𝑥 and 𝑦𝑦 directions, (d) the SCF distribution for the
panel with biaxial tension-compression in 𝑦𝑦 and 𝑥𝑥 directions, respectively.
Fig. 5 The distribution of strain concentration factor, SCF, for 𝜀𝜀𝑥𝑥 in a finite thin panel with
a single circular hole, (a) the lattice mesh and corresponding boundary conditions, (b) the
SCF distribution for the panel with uniaxial tension in 𝑦𝑦 direction, (c) the SCF distribution
for the panel with biaxial tension in 𝑥𝑥 and 𝑦𝑦 directions, (d) the SCF distribution for the
panel with biaxial tension-compression in 𝑦𝑦 and 𝑥𝑥 directions, respectively.
Fig. 6 Lattice simulation results for Newman problem, (a) Load-displacement curve of the
panel with circular hole in direct tension, (b) Comparing lattice simulations and Newman
analytical solution for the function 𝐹𝐹, c mesh with crack emanating from the circular
hole emulating the Newman problem.
(c)
Materials:
Concrete, Masonry, Rock, Oil-well Cement.
Characteristics of behavior:
Pressure Sensitive, Dilative, Quasi Brittle/Brittle behavior,
Significantly different in tensile and compressive behavior .
Methods of predicting behavior of these materials:
Different material models with different philosophies have been
developed.
plasticity, damage, damaged plasticity, associate vs non-associate,
Different numbers of stress invariants.
Models for different loading scenarios
Uniaxial compression, Confined compression under different levels of
confinement, Oedometer test (Uniaxial Strain)
Use of Digital Image Correlations for Experimental measurement of materials behavior.
Figure 11. Damage based vs. plasticity based philosophy for analysis of quasi-brittle
materials.
Figure7. Representation of boundary
conditions in Oedometer test of concrete
(above)
Figure 8. Typical yield surface of concrete
(left)
Figure 12. Monotonic behavior of concrete (left) vs. Cyclic behavior of concrete
(Right). It has been observed that behavior of concrete is a combination of damaged and
plastic behavior.
Figure 13. Testing of concrete under uniaxial compression, test setup and
instrumentation (strain fields, strain localization in different materials).
-70
-60
-50
-40
-30
-20
-10
0
-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015
AxialStress(MPa)
Strain- Axial (Left), Lateral (Right)
Sigma(con.)=0
Sigma(con.)=0
Sigma(con.)=5MPa
Sigma(con.)=5MPa
Sigma(con.)=15MPa
Sigma(con.)=15MPa
Sigma(con.)=25MPa
Sigma(con.)=25MPa
Sigma(con.)=35MPa
Sigma(con.)=35MPa
Sigma(con.)=45MPa
Sigma(con.)=45MPa
Sigma(con.)=55MPa
-70
-60
-50
-40
-30
-20
-10
0
-0.002 0 0.002 0.004 0.006 0.008 0.01
AxialStress(MPa)
Volumetric Strain
Sigma(con.)=0
Sigma(con.)=5MPa
Sigma(con.)=15MPa
Sigma(con.)=25MPa
Sigma(con.)=35MPa
Sigma(con.)=45MPa
Sigma(con.)=55MPa
Figure 9. Effect of confinement on concrete behavior and dilative post-peak nature or
concrete in compression
Figure 10. Oedometer model results for concrete (same uniaxial compression stress-
strain diagram) in different softwares (ABAQUS, LS-DYNA).
References:
• R. Mousavi, M. D. Champiri, K. J. Willam, "Efficiency of damage-plasticity models in
capturing compaction-expansion transition of concrete under different
compression loading conditions“,[Keynote] ECCOMAS 2016, Crete, Greece.
• R. Mousavi, S. Beizaeei, G. Xotta, K. J. Willam, “Error Analysis of DIC Imaging Data
using Least Square Approximation”, presented at 3D Optical Metrology Workshop,
2016, Houston, TX.
• R.Mousavi, “Ductile vs Brittle Failure: Learning from experimental observation using
digital image correlation”, presented at 3D Optical Metrology Workshop, 2015,
Houston, TX.
• S. Beizaeei, K. J. Willam, G. Xotta, R. Mousavi, “Error Analysis of Displacement
Gradients via Finite Element Approximation of Digital Image Correlation System”,
Framcos-9, 2016, Berkeley, CA, USA.