LDPM

Lattice Discrete Particle Model

LDPM is a novel analytical approach for modeling the response of heterogeneous and granular materials,
such as composite laminates, ceramics, concrete, gravels, and soils. The premise of the LDPM formulation
is that most materials are not homogenous when considered at a sufficiently small dimensional scale
(micro- and meso-scale). This heterogeneous character has a paramount relevance for the description of
strain localization, crack initiation, and crack propagation, which, in turn, strongly influence the
ultimate failure mode of a structural system. Continuum-based models, which homogenize material behavior,
are inherently incapable of capturing the mesoscale interactions and as such become complex and inadequate
in the failure range.

LDPM simulates materials as systems of rigid particles connected by deformable links. Particles are
distributed inside a volume according to the characteristics of the mesolevel structure. Links implement
constitutive equations that realize the actual behavior of the material being modeled. The current
implementation of LDPM in MARS has demonstrated the ability to simulate most rheological behaviors of
concrete with relatively simple link models.

A prism specimen of concrete is placed between two rigid platens.
Unconfined compression test is simulated to capture the failure modes in two different situations: high/low friction between specimen and platens.
Different failure modes are expected as follows.

This figure depicts the setup of the mixed-mode fracture test in which a double notched prismatic panel was loaded in tension (T) and shear (S). The shear load was applied first up to a shear force close to the force capacity; then the tension was applied under displacement control keeping the shear load constant. The numerical simulation shows the typical curved cracks propagating from the notch tips towards the opposite side of the specimens.

In this figure the results of biaxial quasi-static tests on plain concrete panels are reported. In the center of the picture one can see the comparison between the numerical (solid curves) and experimental (circles) failure domains normalized with the compressive strength. The top left of the figure shows classical shear band failure characterizing uniaxial unconfined compression tests. The top right shows the failure mode under uniaxial tension. The bottom left is relevant to equi-biaxial tension characterized by a 45 deg fracture. The bottom right reports the failure mode obtained while applying compression in the vertical direction and transverse tension.

This post discusses the numerical simulations of tension testes on fiber reinforced concrete specimens. Figure (a) below shows experimental and numerical stress versus displacement curves for four different fiber volume fractions (Vf ): 0% (plain concrete), 2%, 3%, and 6%. The lattice discrete particle model is able to predict the increased strength and ductility due to the effect of fibers. The behavior gradually transitions from softening for plain concrete and low Vf , to hardening for high Vf . The numerical results are further investigated in Fig. (b), where contours of the mesoscale crack opening at the end of the simulations are reported for three fiber volume fractions. For plain concrete, the crack pattern is characterized by one localized crack that propagates from one side of the specimen towards the other. As fracture propagates, material outsidethe crack unloads as the overall load applied tothe specimen tends to zero. For the 2% Vf , there is still one main crack propagation, but the entire specimen features diffuse cracking and no unloading occurs. Absence of unloading outside the main crack is due to the fact that even though the overall behavior is softening, the stress versus displacement curve shows a non-zero residual stress associated with the fiber crack bridging effect. Finally, for the 6% Vf , the crack pattern is characterized by several branched cracks whose propagation is arrested by the effect of the fibers. No unloading occurs outside the main cracks since the overall behavior is strain-hardening and, up to a displacement of 0.5 mm (average nominal strain of 0.5 mm / 120 mm = 0.42%), no reduction of the load carrying capacity can be observed.

This post deals the results of three-point bending test simulations on notched specimens. Only the central part of the specimen is simulated through the accurate lattice discrete particle model while the two lateral parts are modeled used elastic finite elements. This is reasonable because the presence of the notch ensures that damage localizes ahead of the notch tip. The figures below the meso-scale crack openings (blue=0.0005 mm, red=0.66 mm and above) for plain concrete (top) and fiber reinforced concrete (bottom) with a 0.45% volume fraction. As one can see for the fiber reinforced case fracture are less localized compared to the plain concrete case.

Granular rocks display complex inelastic mechanical properties, such as the transition from brittle to ductile response upon increasing confinement, the tendency to dilate or contract upon shearing, and the formation of a wide range of strain localization mechanisms.
Such rich variety of deformation modes depends on the inelastic properties of rocks, and it is invariably controlled by the confining pressure, which results in pressure-dependent mechanical response in such materials.
Below is a triaxial test of a cylindrical specimen under high confinement using LDPM.
More information can be found in the references with direct download links.^{1} ^{2}

In order to employ LDPM for the simulation of large concrete structures, it is necessary to develop an efficient and accurate multiscale framework.
Therefore, a multiscale homogenization scheme is formulated, in which the Lattice Discrete Particle Model (LDPM) that accounts for the material heterogeneity is employed in the analysis, while the computational cost is considerably reduced.
The basic idea of homogenization is representing each integration point in the macroscopic finite element domain by a representative volume element (RVE).
The RVE is a volume of the material in which related heterogeneity is modeled, while the material is assumed to be homogeneous at the corresponding macroscopic point.
Deformation gradient at each integration point of the macroscopic domain is the input for the solution of the lower scale problem formulated at the RVE level.
Next, using an averaging scheme, the macroscopic response derived from RVE solution is transferred back to the macro-scale.
This flow of information between the two scales of problem continues to the end of the analysis.
In the figure below, results of the analyses of a reinforced concrete column under direct tension performed by both full LDPM model and multiscale homogenization with LDPM RVEs are depicted, which confirms the accuracy of the multiscale analysis ^{1}.

Assigning LDPM RVEs to all macroscopic finite elements from the beginning of the analysis leads to tremendous computational expense necessary to solve the boundary value problems at each integration point at separate scales during the analysis.
In engineering problems dealing with fracture and failure in quasi-brittle materials such as concrete structures, cracking and damage usually localizes in a certain region of the structure, and the rest of the material domain remains elastic.
Therefore, assigning material RVEs to all macroscopic integration points is superfluous.
Therefore, the multiscale homogenization framework into an adaptive scheme, in which no RVE is assigned to macroscopic FEs in advance of the analysis.
A general criterion is formulated to detect the FEs that should be inserted into the multiscale framework.
During the simulation, the adaptive scheme automatically detects the FEs that satisfies the formulated criteria and need to be assigned a material RVE.
In this way, there is a considerable saving of the computational cost especially in the analysis of strain localization problems, in which RVEs are only assigned to FEs that are inside the localization band.
For instance, a four-point bending test on a concrete beams performed by both adaptive homogenization and full LDPM analysis is shown in the figure below.
One can see that the force-displacement curves obtained from the two analyses match very well in terms of elastic stiffness and the strength.

In the adaptive multiscale homogenization analysis, RVEs are assigned to critical macroscopic finite elements during the analysis, which are shown in figure below.
One can see that a small portion of all macroscopic FEs are assigned with RVEs, which tremendously decreases the computational cost ^{2}.