CNTs and CNFs have been introduced as a nanoscale reinforcing material to cementitious composites, for stiffening and strengthening the microstructure. This technology is motivated by the need to control crack initiation in the cementitious gel before it propagates into visible crack formations. Experimental evidence supports this concept; however, testing at the nanoscale may only be conducted through nanoindentation, which has a limited range only providing localized results that cannot be extrapolated to general stress states. To evaluate the restraining action of nanomaterials in the gel microstructure, a computational mechanistic model has been developed where the material phases (gel, nanotubes, and pores) are modeled explicitly allowing for natural randomness in their distribution and orientation. Repeated analysis with identical input data reproduces the statistical scatter observed in laboratory tests on identical material samples. The formulation uses a discrete element approach; the gel structure is represented by a random network of hydrates and successfully reproduces the known trends in mechanical behavior of cementitious materials (pressure and restraint sensitive material behavior) and the small ratio of tensile to compressive strength. Simulations illustrate that it is possible to computationally reproduce the measured properties and behavior of fiberreinforced cement composites using information from simple laboratory tests.
New generations of functionally graded cementitious materials have emerged in recent years designed to meet specific performance objectives. These developments in the materials technology open up the prospects of a new era in construction, where the traditional concrete will be displaced in favor of more durable cementitious materials that possess higher fracture energy and are resilient to tensile deformation. An important milestone towards this objective is the new generation of fiberreinforced matrices that contain a hybrid mix of fibers as a means to enhance the performance through the synergistic effects imparted by the various types of fibers functioning at the various scales of the FRC microstructure [
Although the synergistic effects of fibers with reinforcement, as well as synthetic and steel fiber cocktails, have been studied, only recently have pioneering efforts to address the flaws at the nanoscale been made through the addition of nanomaterials to the cementitious mix [
Studying the nanoreinforced material response to general states of stress requires a calibrated constitutive model that reflects the parametric sensitivities as well as the uncertainties necessarily embedded in the macroscopic properties of the composite. Recall that cementitious materials are amorphous, whereas past works that have published pictures of the nanostructure obtained through electronscanning microscopy (ESM) reveal that CNTs often are inadequately distributed unless properly treated with surfactants [
After hydration, the amount of water that was originally mixed with cement occurs in two different states: chemically bound in the hydrates (nonevaporable) or occupying the volume of pores; water in larger pores (capillary) may flow out easily, whereas water in the smaller pores is held by surface forces in the internal surface area of the hydrate and may be only lost upon heating. Therefore, the entire volume of the porous phase represents space that was originally occupied by water molecules which were subsequently chemically bound in the solid phase of the cement hydrates. The origin of strength and stiffness is due to the cement hydrates. As the chemical reaction proceeds, the solid phase grows radially from the reacting cement grain. During this process, growths from adjacent grains meet and merge in space, developing a solid network of amorphous mass; in the context of the present study, the axis of a branch is modeled by a linear element although it is implicit that it is endowed by hydrate volume that provides it with solid mass; the organization and scale dimension of the hydrates depend very much on the water/binder ratio,
Discrete square space
A characteristic measure of internal geometry in cementitious solids is the socalled mean hydraulic distance which is an idealized estimate of the mean separation distance between solid particles in the undisturbed (unstressed) condition. Another way to measure this distance is the diameter of the mean pore,
The problem is idealized in a twodimensional plane; that is, planestress conditions are considered in the model. In principle, the model may be extended to the three dimensions, albeit at significant computational expense. The essential attributes of the solid structure of cementitious gel are emulated by abstracting the solid phase so as to represent it as linear segments spanning along the axes of the dendrites which emanate from a mesh of nodes which correspond to the initial position of the cement grains.
Therefore, the first step in generating the computational mesh is the random placement of cement grains in a rectangular solid section that extends in
The mesh considered for response analysis under various boundary conditions extends over a square zone with a size that ranges between
Trimming of the 15 × 15 MHD “box.” (a) New boundary nodes (green) are created where branches cross the perimeter of the box. (b) Trimming of elements to the “box.” Dashed lines represent trimmed dendrites (red) and CNTs (black). Space not occupied by the solid structure represents the voids (pores) of the material.
The dendrite material is considered linear elastic in tension and compression. Basic properties refer to the tensile stress and strain values at rupture,
A similar generation procedure was used in order to define CNTs in the mesh, so as to enable a random definition of their layout in the sample specimen. Considering the usual CNTs that are currently available on the market, their length is in general about two orders of magnitude longer than dendrites and one order of magnitude longer than the sample region. So, CNTs are defined in the greater generation area and then only those that cross the boundary of the “box” are considered in studying the mechanics of the state of stress of the composite (Figures
In the present section, the mechanical behavior of the “box” model is studied under various controlled boundary conditions, with the objective of identifying and interpreting the contribution of the CNTs in the apparent properties of the reinforced cementitious matrix. Analysis in 2D is conducted using constant strain increments, Δ
Loading on the model may be conducted under either load or displacement control using the master nodes which control the kinematics of the boundary nodes as described in Section
Individual response curves of unreinforced discrete specimens (
Individual response curves, along with sample mean response curves (in solid black) and 2
Each time the program is run with the same input values, a different random truss is created, owing to the random generation process. Hence, two otherwise identical execution instances of the algorithm will not lead to identical results. Variability occurs not only in the net magnitude of the values but also in the state of convergence of the algorithm at various levels of applied displacement. This is reflected in the occasional digressions from the calculated stressstrain curves, most of which are “local” and swiftly return to the backbone curve. Nevertheless, all calculated stressstrain envelopes (Figures
The parametric performance of the discrete material model is illustrated in the graphs of Figures
Each example material and stress state is studied with several analytical specimens, as discussed earlier. Therefore, the bundle of curves depicted in any given type of plot has been obtained by randomly generating a study “box” each time and repeating the solution using identical input values for each run. So, samples 1, 2, 3, and so forth represent cases that are in principle identical but are practically different, since they have been randomly generated. Note that the observed variability between curves is expected and represents the analytical analogue of the statistical variability witnessed in real lab work, whenever a group of identical specimens is physically tested. Each specimen is codified in the legend of the figure by a composite ID, which includes the specimen number, its volume density, and the corresponding CNT volume ratio (
An important open issue of the investigation is to determine whether the computational results are susceptible to mesh sensitivity. Towards this objective, several groups of specimens of plain paste (nominally differing only in box dimensions) were examined in uniaxial compression, uniaxial tension, and pure shear. Results are summarized in Table
Summary of mesh sensitivity study: normalized stresses and strains at limit points.
Mesh size in MHD multiples 







6 




−2.12 (±13%)  −0.35 (±16%) 
9 




−1.99 (±10%)  −0.31 (±10%) 
15 




−1.90 (±6%)  −0.30 (±3%) 
21 




−1.84 (±5%)  −0.25 (±8%) 
30 




−1.72 (±4%)  −0.23 (±4%) 
It was observed that, in cases dominated by
Consistent with the apparent uniaxial stressstrain response recorded during tests, analyses of computational samples where compressive stresses dominate failure are marked by pronounced nonlinearity before the limit point. For
Analysis under
To assess the contribution of CNT reinforcement on the mechanical properties of the material, different CNT contents were considered in the study; nominal CNT volumetric ratios, calculated with reference to the final material volume, were targeted to values of 1%, 0.5%, and 0.1%. These are usual contents that have been tried in tests [
Calculated effect of CNT reinforcement in the cementitious matrix.
Target CNT per vol.  Actual CNT per vol. 





1%  0.3%–0.8%  25%  40%  30% 
0.5%  0.28%–0.41%  10%  10%  10% 
0.1%  0.03–0.1%  —  —  1% 
Comparison of results at these levels of reinforcement with results from plain paste samples indicates the following effects (Figures
This paper presents results from computational simulations of the mechanical behavior of cementitious matrices reinforced by the addition of carbon nanotubes. These additives operate at the nanoscale of the matter and are intended to control the creation of flaws (cracks) within the gel structure, well before these may propagate to higher geometric scales (micro and mesoscales), where larger, visible steel or synthetic fibers may operate. Simulations are carried out using a novel discrete representation of the nanostructure of the gel, which has been developed in the present paper from first principles. In deriving the model, the intent was to reproduce and highlight the interactions occurring between cementitious material particles and nanoreinforcement such as CNTs and CNFs. In this regard, a novel approach is used to model the hydrates as randomly generated dendrites emanating from partially hydrated cement grains and intertwining to create load and resistance paths randomly oriented in the solid. Dendrite geometric properties and cement grain positioning are randomly generated, so that responses obtained are quantified by the mean values and dispersion as occurs in laboratory testing of several identical specimens. The model provides a testbed for the study of the material behavior under stress and strain. For this purpose, the material model is subjected to constraints that mimic various controlled laboratory experiments that are conducted for material characterization (uniaxial tension/compression, uniaxial stretching/contraction, and pure kinematic shear). Calibrated to fit the macroscopic physical characteristics of the matrix (density and mean pore size), the model reproduces successfully several known macroscopic behavioral traits of cementitious materials, including the characteristics of the yield and failure surfaces under states of plane stress/strain. The discrete model is combined with a random network of CNTs of known geometric and mechanical properties to computationally reproduce the parametric sensitivities and measured properties of nanoreinforced cementbased composites successfully. It is shown that CNTs effectively confine the material passively, leading to increased strength and stiffness in compression with a much slower rate of degradation and lateral dilation (phenomenological Poisson’s ratio), consistently with experimental evidence of a significant confining effect occurring in the microstructure of the matrix and effectively increasing the apparent material stiffness [
The authors declare that there are no conflicts of interest regarding the publication of this article.
This research was cofinanced by the EU (European Social Fund, ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF), Research Funding Program: “ThalesDemocritus University of ThraceCenter for Multifunctional Nanocomposite Construction Materials” (MIS 379496).