1. Introduction
Several studies have been conducted using probabilistic numerical models that deal directly with the heterogeneity of concrete [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13]. With the statistical distribution of mechanical properties and possible use of fracture mechanics concepts, the failure process of structures can be analyzed in a more realistic and comprehensive way. In such models, manifestations of the scale effect, which involve the variation in global mechanical property evaluations with the change in the structure size, and softening behavior (gradual loss of load-bearing capacity) are intrinsic consequences, and the statistical treatment of global results is enabled by the Monte Carlo method.
One of the probabilistic approaches directed to the modeling of concrete behavior is the explicit cracking approach originally proposed by Rossi and Richer [
1]. This approach is based on the finite element method (FEM), being mainly characterized by the use of zero-thickness interface elements to which the following purposes are assigned: to receive mechanical properties distributed randomly and to represent the material discontinuities (failure planes) explicitly. The models developed from this idea have advantages in situations of crack opening assessment, which can be performed directly, in addition to favoring the implementation of strategies to deal directly with phenomena that occur during the fracture process, such as the contact between crack surfaces.
The development of 3D probabilistic explicit cracking models remains at the initial stage, with considerable limitations imposed by their high computational cost. The inclusion of interface elements in the mesh significantly increases the number of nodes and, consequently, the execution time of the analyses, especially when quadratic elements are employed. It is important to think of strategies to deal with this issue, since three-dimensional analyses are indispensable in some cases, with the cracking process tending to be naturally three-dimensional. In such cases, for more realistic and comprehensive modeling, the fracture mechanisms cannot be neglected in any dimension. For safety and durability reasons, an important potential application of 3D models concerns, for example, the study of thermal cracking of massive concrete structures, such as dams and nuclear power plants. The computational cost issue can be handled by using a finite element code designed for high performance, proper data structures [
14,
15], and mesh adaptivity and computational parallelization strategies.
Mota et al. [
11] developed a 3D probabilistic explicit cracking model capable of reproducing different levels of softening behavior and predicting the scale effect in tension with an intensity similar to that experimentally observed. This is of great importance, since the scale effect and softening behavior occur naturally for quasi-brittle materials such as concrete and strongly influence the fracture process of these materials. The mentioned model was implemented following the classical procedure of using interface elements between all the solid elements since the beginning of the analysis. Although the code was written on a high-performance FEM platform and parallelization strategies were employed, the computational cost can still be considerably reduced by adopting a mesh adaptivity strategy for the controlled use of interface elements.
As the use of interface elements is a strong time-consuming factor, an adaptivity strategy for inserting interface elements only in the regions of the mesh where they are indispensable is an important alternative to speed up the analyses. There are studies performed with adaptive mesh procedures in accordance with this idea [
16,
17,
18,
19,
20,
21,
22,
23], but these works do not encompass probabilistic cracking models and almost all of them [
16,
17,
18,
19,
20,
21,
22] do not take into account the influence of the proposed adaptivity strategy on the redistribution of stresses after cracking. Furthermore, the adaptivity strategies proposed by Zhan and Meschke [
22] and Rodrigues et al. [
23] consider a different type of interface element (interface solid element with a high aspect ratio). Thus, the adaptive mesh strategies developed in all the mentioned studies [
16,
17,
18,
19,
20,
21,
22,
23] cannot be applied to the probabilistic explicit cracking approach, so a new strategy is necessary. An adaptivity procedure appropriate for probabilistic explicit cracking models can prove to be very efficient, for example, in simulations in which the structural failure occurs due to concentrated cracking in delimited regions of the mesh.
In view of the foregoing considerations, this work reports the development of a 3D adaptive probabilistic cracking model based on the explicit cracking approach proposed by Rossi and Richer [
1]. The model in question represents an improvement of the one implemented by Mota et al. [
11]. The contribution offered herein consists in an adaptive mesh strategy designed to optimize the use of interface elements in probabilistic explicit cracking models, taking into account possible influences on the redistribution of stresses after cracking. This strategy is markedly different from the strategies found in the literature [
16,
17,
18,
19,
20,
21,
22,
23] and can also be applied to purely deterministic cracking models that employ interface elements.
2. Empirical Relations for Scale Effect
Rossi et al. [
24], by means of an extensive experimental investigation, obtained analytical expressions that define correlations between concrete heterogeneity and scale effects, contributing to the validation of probabilistic models. These expressions depend on the
ratio, which is a parameter of heterogeneity, and on the compressive strength (
). The parameter
is the total volume of the specimen, and
is the volume of the coarsest aggregate, modeled by a sphere. The expressions are applicable to any concrete, except fiber-reinforced and lightweight concretes, with an
ranging from 35 MPa to 130 MPa and with the
ratio varying from 10 to 10,000 in order of magnitude. For the mean value of tensile strength or mean tensile strength (
) and the coefficient of variation (
), which is a measure of dispersion defined as the ratio between the standard deviation (
) and the mean tensile strength, Rossi et al. [
24] obtained the following relations:
in which
MPa,
, and the values of
and
are defined by
with
MPa.
Figure 1 portrays the diagrams of
and
versus
. With the aid of these diagrams, it is possible to note that, with
and
being always positive, the higher the
ratio in Equations (1) and (2), the lower the mean tensile strength and the coefficient of variation.
Figure 1 also shows that, with the compressive strength being a quality indicator of the cement paste, when
increases,
decreases, leading to higher values of mean tensile strength. On the other hand, higher values of
result in higher values for
and, consequently, in lower values for
.
According to Rossi et al. [
24], the tensile strength dispersion is occasioned by the material heterogeneity, being directly associated with the scale effect. Small values of
, obtained either by increasing the
ratio or by increasing the value of
, represent a low influence of the heterogeneity on the results and, consequently, a low-intensity scale effect.
3. 3D Adaptive Probabilistic Explicit Cracking Model
Figure 2 outlines the 3D probabilistic explicit cracking model without mesh adaptivity developed by Mota et al. [
11], which is the basis of the present work, including for comparison purposes. The main characteristics of this model are described as follows:
The three-dimensional treatment is basically given by the use of tetrahedral solid elements of 10 nodes;
Zero-thickness interface elements of 6 + 6 nodes are previously inserted between all the solid elements in accordance with the classical method of mesh formation;
Quadratic interpolation functions are used in the elements to obtain results with a higher precision;
The heterogeneous and probabilistic character is established by the random assignment of tensile strength values to the interface elements according to the Weibull distribution. Thus, concrete is being defined as a heterogeneous material, since there is no tensile strength variation for a homogenous material. Weibull’s statistical law was chosen for two reasons: (1) it was developed for brittle materials, which can be associated with the fact that a brittle law is considered for the material locally; (2) negative values of random properties are not possible;
Solid elements have a linear elastic behavior;
Interface elements follow a sufficiently rigid and brittle behavior under tension, which is naturally associated with the Rankine failure criterion, resulting in local geometric and material discontinuity when the tensile strength of a given interface element is reached, causing the occurrence of a cracked interface element. Thus, during the computational analysis, there are local nonlinearities that generate nonlinearity in the global response;
Structural softening is naturally obtained according to the distribution of tensile strength among the interface elements;
When there is contact between the faces of cracked interface elements (contact problem locally), there is a reactivation of stiffness in the interface elements to take into account compression and friction effects;
The acquisition of structural responses occurs by means of the Monte Carlo method.
The constitutive relation for the interface element can be basically written in the form of the following damage-type model [
11]:
in which
is the damage parameter,
and
are penalty parameters associated with the tangential stiffness,
is related to the normal stiffness,
and
are tangential relative displacements, and
is the normal relative displacement. In order to obtain a brittle behavior, it is assumed that
for the intact interface element, and
when the tensile stress reaches the tensile strength (occurrence of a cracked interface element).
The friction between crack surfaces, which are represented by faces of cracked interface elements, is taken into account by means of the Mohr–Coulomb criterion [
25] without cohesion:
where
is the shear stress or, more specifically, the acting friction stress in the cracked interface element;
is the absolute value of the normal compressive stress; and
is the angle of friction. A more detailed description of the friction model is provided by Mota et al. [
12].
The next subsections describe the implementation of mesh adaptivity in the model portrayed in
Figure 2. For convenience, the regions where interface elements can be inserted are referred to as interfacial regions of solid elements or simply interfacial regions.
3.1. Generation and Distribution of Random Tensile Strength
For the Weibull distribution [
26], the cumulative distribution function and the inverse cumulative distribution function related to a random variable
can be written respectively as
in which
, with this variable corresponding to the tensile strength in the present paper;
is the shape parameter, associated with the coefficient of variation; and
is the scale parameter, directly related to the mean value of
.
The generation of random values of tensile strength is based on the inverse transformation method or uniform probability transformation technique [
27]. These values are generated and distributed to interfacial regions of solid elements according to Algorithm A1 (
Appendix A). If the values of
and
are fixed, a different Monte Carlo sample for the same material will be obtained each time Algorithm A1 is used.
3.2. Adaptive Mesh Strategies for Efficient Use of Interface Elements
Regarding finite element meshes for mechanical problems, a relevant challenge is how to represent the crack explicitly. For a mesh composed only of solid elements, a crack at an interfacial region of the elements, as indicated in
Figure 3a, can be detected relatively simply through the calculation of nodal stresses in the region. However, the incorporation of the geometric discontinuity generated by the crack requires strategies of greater complexity.
Figure 3b illustrates the classical strategy adopted in probabilistic explicit cracking models to deal with the appearance of cracks, while
Figure 3c–e present the three adaptive mesh strategies developed and evaluated in the context of this work.
Figure 4 provides more details about the adaptivity options, with the solid elements being displayed in a reduced size in the crack zone so that the interface elements are highlighted. Two dimensional representations are used for simplification purposes.
In the here-called Classical Strategy of Mesh Formation (CSMF), shown in
Figure 3b, the mesh is initially prepared with interface elements inserted between all the solid elements. Thus, the crack observed in
Figure 3a is detected directly in the interface element, which produces a geometric discontinuity when its rigidity is nullified, instantly generating stress redistribution around the crack. However, this strategy causes a significant increase in the number of nodes from the beginning of the analysis, with interface elements being used even in regions where these elements may be unnecessary.
The three previously mentioned mesh adaptivity strategies were developed so that interface elements could be employed only in regions where they are indispensable. In these strategies, the mesh is initially composed only of solid elements, with interface elements being used during the analysis as cracking occurs. Consequently, after the beginning of the analysis, there may or may not be interface elements in the interfacial regions of the solid elements. When a crack is detected at an interfacial region with no interface element, a cracked interface element is inserted in this region. However, for mesh conformity, it is also necessary to place intact interface elements in the vicinity of the cracked interface element.
In the first mesh adaptivity strategy, referred to as Adaptive Mesh Strategy 1 (AMS1) and portrayed in
Figure 3c and
Figure 4a, intact interface elements are inserted around the solid element that provides the top face of the cracked interface element. As the choice for the solid element related to the top face was purely arbitrary, one could opt to use intact interface elements around the solid element associated with the base face of the cracked interface element instead. In a 2D mesh composed of triangular elements, this strategy would entail the inclusion of at most two intact interface elements for each detected crack. In a 3D configuration with tetrahedral solid elements, at most three intact interface elements are included for each new crack.
In Adaptive Mesh Strategy 2 (AMS2), depicted in
Figure 3d and
Figure 4b, intact interface elements are placed around the solid elements that provide the base and top faces of the cracked interface element. In this way, for each new crack, a maximum of four intact interface elements would be inserted in a 2D case with triangular elements; on the other hand, in 3D meshes of tetrahedrons, a maximum of six intact interface elements can be added.
In Adaptive Mesh Strategy 3 (AMS3), which is represented in
Figure 3e and
Figure 4c, intact interface elements are placed around all the solid elements in the vicinity of the cracked interface element. In this case, for each new crack, the maximum number of intact interface elements that can be inserted depends on the level of mesh discretization in the crack zone.
Details of the computational implementation of AMS1, AMS2 and AMS3, such as utilized arrays, explained synthesized algorithms, and further information about the insertion of cracked and intact interface elements in the mesh, are described in
Appendix B.
3.3. Computational Program for Finite Element Analyses
The finite element code, written in Fortran language, was implemented in a high-performance computational platform developed by Ribeiro and Ferreira [
15]. This platform allows for the use of algorithms and data structures designed for high performance [
14,
15], having already been used as a basis for several works [
28,
29,
30,
31]. Details related to the developed computational program are described in
Appendix C, with an overview of the implemented code being presented in Algorithm A5. This appendix includes the description of a method adopted to favor proper stress redistributions during the cracking process and to avoid any dependence of results on the load increment size, called the most dangerous element method.
The application of the Monte Carlo method requires Algorithm A5 to be executed a number of times, corresponding to a number of Monte Carlo samples whose load–displacement responses may naturally vary considerably from one sample to another. In order to deal automatically with this variation, two simulation stopping criteria were implemented, taking into account some load–displacement diagram information: maximum load (
), ultimate load (
), and ultimate displacement (
), as indicated in
Figure 5.
The simulation stopping criteria are named as follows: SSC1, whose evaluation parameter is the ratio between and , with being necessarily less than ; and SSC2, whose evaluation parameter is simply . It is also possible to opt for both criteria simultaneously. More specifically, these criteria are defined as follows:
SSC1 [] the criterion is met, and the simulation is terminated, when ;
SSC2 [] the criterion is met, and the simulation is terminated, when .
with the limit parameters and being set according to the particularities of the considered analysis, so that and .
It is important to point out that values of close to 1 should be avoided, since such values do not guarantee the effective knowledge of the peak load. As load fluctuations can naturally occur in the load–displacement diagram, inappropriate values of can easily cause false peak-load results. For each type of mechanical problem, it is necessary to carry out previous analyses to determine suitable values of .
4. Analyzed Cases
The numerical simulations were divided into four stages: (1) mechanical evaluation of the adaptive mesh strategies; (2) inverse analysis; (3) validation; (4) assessment of time savings.
Figure 6,
Figure 7 and
Figure 8 show the geometries and meshes of the analyzed cases, with the details presented in
Table 1. In this table, the columns related to interface elements and nodes inform the maximum possible number of these entities in the mesh, remembering that, in case of adaptive mesh use, the informed numbers are not necessarily reached at the end of the analysis; the parameter
denotes the average value of the sum of the volumes of the two solid elements that provide the base and the top faces of the interface elements.
In stage 1, direct tensile test simulations were performed by using the square-based prisms (P1, P2, and P3) shown in
Figure 6, with P3 corresponding to a cube. Boundary and loading conditions in agreement with the simulated test have been considered, with displacement restrictions being imposed on the base nodes, and displacement increments being applied on the top nodes in the longitudinal direction. Mechanical responses obtained by using the adaptive mesh strategies were compared with the responses provided by the classical strategy of mesh formation. For P1 and P2, the comparisons refer to values of the average, standard deviation, and coefficient of variation of tensile strength, and to the global load–displacement responses. For P3, the comparisons concern the local results of stress redistribution after crack appearance. The purpose of these comparisons was to identify the most suitable adaptive mesh strategy to be adopted in stages 2, 3, and 4, using the classical strategy of mesh formation (CSMF) to provide reference results. The mean tensile strength of each set of Monte Carlo samples (
) and the corresponding standard deviation (
) are calculated by the following expressions [
27]:
in which
is the number of Monte Carlo samples, and
, where
is the maximum load of the
-th Monte Carlo sample, and
is the cross-sectional area.
In stages 2 and 3, the simulations with P1 and P2 were based on the experimental investigation conducted by Rossi et al. [
24], which was chosen for the following reasons:
The study refers to concrete behavior in a direct tensile test, in which the transition from diffuse to localized cracking is the most important phenomenon and the most difficult to capture;
Concretes with different mechanical properties were produced, enabling the evaluation of different scenarios;
A large number of experimental samples were tested, implying statistical relevance;
Experimental results are expressed using Equations (1) and (2) with great accuracy, in such a way that it is completely possible to use these equations to obtain expected values according to the scale effect.
In stage 2, inverse analysis procedures described by Mota et al. [
11] were carried out with the purpose of determining the parameters
and
(statistical parameters related to Equations (7) and (8)) for P1 and B1. It is important to highlight that these parameters naturally depend on the volume of the mesh elements, in a manner similar to that in which the values of
and
given by Equations (1) and (2) depend on the volume of the considered specimens. The parameters
and
are scale effect parameters at the local level, while the values of
and
provided by Equations (1) and (2) are global-level scale effect results (specimen responses).
Basically, each inverse analysis procedure corresponds to analyzing the same case with different pairs of
and
to define the most appropriate pair of such parameters, taking into account empirical results. A considerable number of Monte Carlo processes is necessary. One Monte Carlo process provides, for example, the mean and the standard deviation of a certain global mechanical response for a given pair of
and
. Repeating this procedure for different pairs of
and
, two surfaces are obtained: one for the mean and another for the standard deviation. These surfaces are properly fitted using polynomial expressions. The expected results for the mean and standard deviation, which can be provided using Equations (1) and (2) for simulations with the prism P1, can be considered as horizontal planes on the
-
domain. In the next step, the intersection curves between these planes and the fitted surfaces must be determined.
Figure 9 shows these curves, considering that the tensile strength of the samples, which is given by the maximum load divided by the cross-sectional area, is the mechanical response to be observed. In this figure, the intersection point of the curves represents the most suitable pair of
and
. More details related to the inverse analysis procedures are presented by Mota et al. [
11].
For beam B1, detailed in
Figure 7 and experimentally tested by Hordijk [
32], the boundary and loading conditions were consistent with the simulated four-point bending test, with points A and B indicating the locations of displacement restrictions and points C and D signaling the places of application of displacement increments. Point E marks the deflection measurement location. The acting load was measured as the sum of the forces applied to the supports that contain points C and D. The possibility of using interface elements was restricted to the region highlighted in red in
Figure 8. In the simulated test, due to its particularities, the failure is due to tension.
In stage 3, the ability to predict the scale effect was investigated. The values of and obtained from an inverse analysis with P1 were used in simulations with P2, which has larger dimensions, for the same concretes. In addition to depending on the material, the parameters and , as previously mentioned, depend on the volume of the mesh solid elements. Therefore, the meshes for P1 and P2 were prepared keeping the same average size of elements. In this way, it was possible to evaluate only the effect of increasing the size of the simulated specimen.
Also due to the dependency between the parameters
and
and the size of the elements, the mesh for B1 was generated with the elements of the central region having a very low level of volumetric variation. In relation to P3, there was no need to guarantee low volumetric variation among the elements, since there was no inverse analysis for this case. For this reason, the value of
is not important information for P3, as indicated in
Table 1.
In stage 4, the efficiency of the adaptive mesh strategy selected in stage 1 was investigated. In order to do this, the simulation times for the analyses performed with the adaptive mesh strategy and the CSMF were compared. As shown in
Table 1, the results for P1, P2, and B1 were taken into account in this stage.
Table 2 displays the compressive strength, the modulus of elasticity (
), and the volume of the coarsest aggregate of the considered concretes, as well as the heterogeneity parameters
and
. Concretes C1, C2, and C3, experimentally studied by Rossi et al. [
24], were applied to P1 and P2. Concrete C4, related to the experimental investigation conducted by Hordijk [
32], was applied to B1, for which the value of
corresponds to the region in red in
Figure 8.
Table 3 contains the values of complementary modeling parameters: the angle of friction, which corresponds to an average estimate based on studies found in the literature [
33,
34]; the Poisson’s ratio of concretes C1, C2, C3, and C4; and the Poisson’s ratios and moduli of elasticity for P3 and for the steel support of B1. The adopted value for the penalty parameters
,
, and
related to the constitutive relation of intact interface elements was
. This value was chosen from a calibration for which the results of a numerical linear analysis were similar to the analytical ones, without causing problems in the solution.
The nomenclature applied to the Monte Carlo samples involves the geometry, mesh, and concrete in question, according to the terminology used in
Table 1 and
Table 2. Altogether, there are seven classes of Monte Carlo samples: P1-C1, P1-C2, P1-C3, P2-C1, P2-C2, P2-C3, and B1-C4.
It is worth mentioning that, for the considered probabilistic cracking model, there are some restrictions for performing a mesh sensitivity analysis in which a convergence of results is evaluated as the degree of mesh refinement increases. The variation in the degree of mesh refinement naturally implies a change in the volume of the solid elements. However, the values of and depend on the volume of the solid elements (as mentioned before), which means that different values of and need to be used when the volume of the solid elements changes, modifying the characteristics of the problem and compromising the mesh sensitivity analysis. A mesh sensitivity analysis can be performed considering the elastic response (in order to ensure the correct elastic response, which is guaranteed in the cases analyzed in this paper) or considering the problem without tensile strength variation (with the same value of tensile strength for all interface elements in the model). In the present paper, the degree of mesh refinement is not a problem, since the adopted meshes provide a level of cracking information that enables the investigation described herein to be carried out. If results with a very high level of cracking information were needed, the meshes could be more refined (which would naturally generate a higher computational cost). But this is not the case in this work, whose purpose is to present a new adaptive mesh strategy and its viability, which does not require results with a very high level of cracking information.
6. Conclusions
This paper communicates the development of an adaptive mesh strategy to optimize the use of interface elements in a 3D probabilistic explicit cracking model for concrete. The investigation involved the evaluation of three adaptive mesh strategies: AMS1, AMS2, and AMS3. Based on comparisons between mechanical behaviors, only AMS3 proved to be able to replace the classical strategy of mesh formation (CSMF) generally adopted in probabilistic explicit cracking models.
AMS1 and AMS2 are not suitable to replace the CSMF due to divergences in the redistribution of stresses during fracturing, which lead to remarkably different mechanical responses. On the other hand, the mechanical responses produced by using AMS3 tend to present a very high degree of equivalence with those obtained through the CSMF, as there is similarity in the redistribution of stresses. Thus, an important conclusion is that different adaptive mesh strategies can lead to distinct stress redistribution processes and, consequently, to very different mechanical responses.
Possible differences between the mechanical responses related to AMS3 and the CSMF are basically concentrated in the post-peak behavior. Such differences are generally of little importance and stem from the fact that the calculation of stresses in interfacial regions of solid elements does not happen in the exact same way for the two strategies. Nevertheless, with the application of the Monte Carlo method, the divergent behaviors tend to compensate for each other, so that there is convergence on very similar average results.
The adaptive probabilistic model based on AMS3, as well as the classical alternative, is capable of predicting the scale effect at a level similar to that experimentally observed, besides covering the occurrence of different levels of softening behavior.
Regarding the efficiency of AMS3, in terms of the computational execution time, it can be concluded that this adaptive procedure is advantageous. This strategy reduced the average simulation time for all the analyzed cases. Understandably, the level of efficiency depends on the characteristics of the considered problem. The smallest average time savings, in percentage terms, were observed for cases in which the number of interface elements quickly reached values very close to the maximum possible number. In cases where this did not occur, there were significant time savings, corresponding to reductions in the average simulation time that ranged from 32% to 77%.
It is also important to highlight that, as the probabilistic nature of the cracking model used here did not affect the development of the adaptivity strategies, AMS3 can also be applied to purely deterministic cracking models that employ interface elements.