Different strategies applied to model interface transition zone of concrete using a computational homogenization approach

ABSTRACT

This work models numerically the concrete mechanical behaviour using a two-dimensional model at mesoscopic level and using the concept of Representative Volume Element (RVE). Concrete is considered as three phases material: mortar/aggregate interface, mortar matrix and aggregate zones, where each constituent is modelled properly. The aggregates are considered to behave elastically, while the Mohr-Coulomb criterion defines the mechanical behaviour in the mortar matrix. Different strategies are used to model the fracture process at the interface transition zone: i) defining rectangular finite elements along interfaces where a fracture/contact model is incorporated; ii) adopting triangular finite elements where the Mohr-Coulomb model is used with lower strength characteristics compared to the mortar matrix. In the numerical examples, we study which of these two strategies is more efficient for modelling the transition zone. Besides, in the RVEs we consider different shapes for the aggregates, which are randomly arranged, with different volume fractions. The results evidence the potentialities of the proposed modelling, but they also show the high sensibility of parameters related to fracture and contact models what can restrict their use for interface zone modelling.

Keywords:
Homogenization; Interface transition zone; Concrete; Cohesive fracture; Multiscale modelling

1. INTRODUCTION

Modelling properly the mechanical behaviour of composite materials is very important as this kind of material is used in different types of structures. Concrete can be considered as a composite material as it is composed of mortar matrix, aggregates and interface transition zone (ITZ). Due to the heterogeneity and complexity of concrete microstructure, it is difficult to propose a constitutive model capable of reproducing its mechanical behaviour properly [¹, ²]. In this context, phenomenological constitutive models based on Damage and Fracture Mechanics as well as Plasticity [¹, ³,⁴,⁵,⁶,⁷], have been considered to model the mechanical behaviour of materials, especially concrete, where the material is assumed as a homogeneous medium and the modelling is based on the dissipative processes that occur at microscale. Usually, this kind of modelling leads to complex constitutive models which have several parameters to be identified.

Nevertheless, the macroscopic properties can be modified according to the aggregate shapes and the typology at microstructure level. On the other hand, the ITZ plays an important role in the mechanical behaviour of the concrete. This zone, defined between the aggregates and the mortar matrix, is directly related to important characteristics of the concrete as the brittle behaviour in tension and a ductile behaviour in compression resulting in a compression strength much bigger than the one in tension. Although the ITZ region is composed of the same constituents of the mortar, their properties are quite different as they are related to a more brittle behaviour. So that, it must be considered as a different phase of the concrete microstructure.

The first works on the mechanical behaviour of the ITZ of the concrete microstructure are from the fifties, but since then the subject has not been well explored [⁸]. MEHTA and MONTEIRO [⁹] say that the volume as well as the dimensions of the voids are bigger at the ITZ when compared to the mortar zone. Besides, the dimensions and concentration of crystalline composites as the calcium hydroxide and the ettringite are also bigger at the ITZ leading to the microcracking propagation. These characteristics lead to smaller strength at ITZ when compared to the mortar zone, showing the importance of studying the ITZ behaviour in order to better simulate the mechanical behaviour of the concrete.

In this context, the using of multiscale modelling is gaining importance because the dissipative phenomena are considered directly at the microscale of the material, adopting simple constitutive models for the different phases of the material. In this case, the values related to the microstructure problem are transferred to the macro-continuum by using the Hill- Mandell principle as well as by applying homogenization techniques [¹⁰,¹¹,¹²,¹³]. The disadvantage of the full coupled multiscale analysis is the high computational cost. Despite the high computational cost, the numerical models based on a multiscale framework are very attractive because they allow defining the different constituents of the microstructure and their respective elastic properties and constitutive models. Criteria based on Damage Mechanics, Fracture Mechanics or Plasticity can be used to govern the mechanical behaviour of a particular phase of the microstructure. Note that in the case of the concrete, the plastic processes in the matrix as well as the microcracking propagation in the ITZ are important phenomena to be modelled as pointed out in the works [¹⁴,¹⁵,¹⁶,¹⁷,¹⁸,¹⁹,²⁰]. Besides, more recently, some more realistic mesoscopic models can be found in the literature, as instance [²¹]. In that work, the proposed modelling deals with 3D cohesive cracks using zero-thickness cohesive interface elements to model ITZ and computed tomography images to obtain RVEs for concrete. Despite the massive computational effort, the proposed model is able to explain complex mechanisms of microcracking and its propagation. On the other hand, it is not trivial to model the ITZ, as its mechanical properties are not easy to obtain experimentally. But recent studies [²¹,²²,²³] have contributed to better understand the chemical composition, the dimension and the strength of the ITZ as well as to develop different numerical models to simulate the mechanical behaviour of the concrete. In these works, important characteristics of concrete microstructure have been discussed, such as: i) the shape, dimensions and distributions of aggregates; ii) the ITZ thickness; iii) mechanical properties of the concrete. As example, the mechanical behaviour of the concrete microstructure has been studied in [¹⁴] by using a FEM model where different shapes have been considered for the aggregates as well as different ITZ thickness have been adopted.

In the present work, a multiscale framework is considered to model the mechanical behaviour of the concrete where two different ITZ modelling have been adopted: i) additional cohesive and contact rectangular finite elements are defined on the interfaces [¹, ²⁴, ²⁵]; ii) triangular finite elements with different material properties are adopted in the ITZ, i.e., the elements are adopted continuous at the interface. We intend to discuss the advantages and disadvantages of each modelling developed in this work applied to concrete submitted to tension predominant regime considering different shapes and volume fractions for the aggregates. Thus, this work contributes to understanding the influence of each proposed modelling on the concrete mechanical behaviour and which one is more efficient for using.

The present work is divided into four sessions: a brief description of the proposed modelling for the concrete is presented in section 2, where the homogenization techniques, the Mohr-Coulomb model adopted to represent the matrix and ITZ mechanical behaviours, as well as a cohesive contact finite element used to model the interface zone are discussed. In section 3, numerical examples are analysed and discussed to show the potentialities and limitations of the proposed models. Finally, in section 4, final considerations are presented.

2. MATERIALS AND METHODS

2.1. RVE modelling

We simulate the mechanical behaviour of the material by using a modelling at mesoscale level where the mechanical behaviour of each constituent is treated properly as well as the iterations between them. The microstructure at mesoscale level is defined by the RVE whose domain is discretized by the FEM, using triangular finite elements. Two different modelling are considered for the interface transition zone (ITZ): i) additional rectangular finite elements are defined on the interfaces (superposed to the triangular elements mesh) where a fracture and contact model is considered; ii) triangular elements are adopted in the ITZ region where the Mohr-Coulomb model is defined, and which have strength lower than the mortar. On the other hand, we consider an elastic behaviour for the aggregates while the material of the matrix (mortar) is assumed to have elastoplastic behaviour governed by the Mohr-Coulomb model. Figure 1 defines a RVE with dimensions 100 mm × 100 mm. For all RVEs considered in the numerical examples we adopt, respectively, the following radius for the coarse aggregates: 10 mm, 5 mm and 2.5 mm.

Figure 1
Representative Volume Element (RVE).

2.1.1. The RVE mesoscale formulation

The RVE formulation is detailed in the works [¹¹, ²⁶] where the interfaces between the matrix and the aggregates are adopted perfectly bonded, i.e., the phase debonding at ITZ is not taken into account [²⁷]. On the other hand, the RVE formulation where the rectangular finite elements are defined on the interfaces to model the phase debonding is developed in [²⁴]. Observe that to obtain the model presented in [²⁴], the rectangular finite elements contributions have been added to the formulation developed in [¹¹, ²⁶]. Therefore, in the present work we present briefly the RVE formulation, but for more details, please consider the works cited previously.

One important aspect of the present formulation is that the macro-strain ε(x,t) and the macro-stress σ(x,t) at a particular point x of the macro-continuum are defined by the volume average of their respective fields in the RVE (ε_μ=ε_μ(y,t) and σ_μ= σ_μ(y,t)) at an instant t of the load incremental process, i.e.:

(1) $$\epsilon (x,t)=\frac{1}{V_{\mu }}{\displaystyle \underset{{\Omega }_{\mu }}{\int }{\epsilon }_{\mu }(y,t)dV}$$

(2) $$\sigma (x,t)=\frac{1}{V_{\mu }}{\displaystyle \underset{{\Omega }_{\mu }}{\int }{\sigma }_{\mu }(y,t)dV}$$

where V_μ is the RVE volume, y denotes a point in the RVE domain Ω_μ and the subscript μ refers to the RVE microstructure.

In Eq. (2) the stress at a point y is defined as σ_μ(y,t) = fy(ε_μ(y,t)), where fy represents the constitutive model used to govern the material behaviour. In the present paper, if y is a point inside the mortar matrix, fy is defined by the Mohr-coulomb model while inside the aggregate fy is given by Hooke’s law because the material behaviour is assumed elastic.

Therefore, by using the average values defined by Eqs. (1)-(2) we represent macroscopic quantities. On the other hand, the microscopic displacement field u_μ is assumed to have a linear part (ε(x,t)y) plus the displacement fluctuation field ũ_μ(y, t),i.e.:

(3) $$u_{\mu }(y,t)=\epsilon (x,t)y+{\tilde{u}}_{\mu }(y,t)$$

where the quantity ε(x,t) is constant and it refers to the macro-continuum strain imposed on the RVE and y represents the coordinate vector of a particular RVE point.

Similarly, the RVE strain field can be decomposed into the following form:

(4) $${\epsilon }_{\mu }(y,t)=\epsilon (x,t)+{\tilde{\epsilon }}_{\mu }(y,t)$$

where $\tilde{\epsilon }$_μ(y, t) denotes the RVE strain fluctuation field.

As the RVE equilibrium problem is solved in terms of displacements fluctuations, it is important to write the RVE strain fields defined in Eq. (4) in terms of their respective displacement fields as follows: ε_μ(y, t) = ∇^su_μ (y, t) and $\tilde{\epsilon }$_μ= ∇^sũ_μ(y, t).

The RVE solution, i.e., the fields of displacements, strain and stress as well as the constitutive tensor related to all finite elements are found when the convergence of the RVE equilibrium problem is achieved. For that, we must impose boundary conditions in terms of displacement fluctuations on the RVE and find the displacement fluctuation field that satisfies the equilibrium equation. Each boundary condition leads to different numerical results, resulting in different multiscale models. In the present paper we adopt periodic fluctuations as the boundary conditions to be imposed to the RVE. In fact, this type of boundary condition is attractive for modelling the behaviour of materials with periodic structure, see [¹⁰], although it can be shown that any material behaviour presents a periodic response when fine meshes are considered [¹¹]. Thus, there are situations of random distributions of heterogeneities that allow adopting a periodic approximation for composite materials as long as the RVE is progressively repeated in a fine mesh in the macrostructure. This is the case of the present work that is developed to be applied to concrete structures. In this context, the displacement fluctuations of a point y+ on the boundary Γ_i⁺ (whose normal direction is n_i⁺) are equal to the displacement fluctuations of its opposite point y− on the boundary Γ_i^– (side opposite to Γ_i⁺ and whose normal direction is n_i^–), being n_i⁺ = −n_i^–. Therefore, we impose that:

(5) $${\tilde{u}}_{\mu }(y^{+},t)={\tilde{u}}_{\mu }(y^{-},t)\begin{array}{cc}& \end{array}\forall \left\{y^{+},y^{-}\right\}\in \partial {\mathrm{\Omega }}_{\mu }$$

To define the RVE equilibrium equation, the Hill-Mandel principle must be considered which establishes that the power of the macroscopic stress must be equal to the volume average of the microscopic stress power, see [²⁸]. Besides, considering a load increment of time defined as Δt = t_n+1 – t_n, the displacement fluctuations field to be found is defined as ũ_μ(n+1) = ũ_μ(n) + Δũ_μ(n) and the RVE equilibrium equation, for a discretization h, is given by:

(6) $$G_h^{n+1}={\displaystyle \underset{{\Omega }_{\mu }^h}{\int }{B}^T{f}_y({\epsilon }_{n+1}+B{\tilde{u}}_{\mu (n+1)})dV}=0$$

where B is the global strain-displacement matrix and Ω^h_u denotes the discretised RVE domain.

When Eq. (6) is not automatically satisfied in a load increment n, it is solved by the application of the Newton-Raphson Method which consists of computing the correction δũ_μⁱ⁺¹ at iteration i+1 by the following expression:

(7) $$F^i+K^i\delta {\tilde{u}}_{\mu }^{i+1}=0$$

where F is the force vector and K the rigidity matrix.

Considering a discretization with Ne elements and denoting De the constitutive tangent tensor at element e given by df_y ⁄ dε_μ, we can define at an iteration i:

(8) $$F^i={\displaystyle {\sum }_{e=1}^{Ne}{B}_e^T}{f}_y({\epsilon }_{n+1}^i{B}_e{\tilde{u}}_{n+1}^i)V_e={\displaystyle {\sum }_{e=1}^{Ne}(B^T{D}_e^i}{\epsilon }_{\mu (n+1)}^{i}V{)}_e$$

(9) $$K^i={\displaystyle {\sum }_{e=1}^{Ne}{B}_e^T}{D}_e^i{B}_e{V}_e$$

Observe that Eqs. (8)-(9) are valid when only triangular finite elements are used to discretize the RVE, i.e., for the case where the ITZ is modelled by triangular finite elements with lower strength. For the case where the fracture process at ITZ is simulated by additional rectangular finite elements defined on the interfaces, we have to add to the Eqs. (8)-(9) the terms related to these elements. Thus, in this case, Eqs. (8)-(9) become:

(10) $$F^i={\displaystyle \int {}_{{\Omega }_{\mu }^h}}{B}^T{f}_y(\epsilon +B{\tilde{u}}_u^i)dV+{\displaystyle \int {}_{{\Omega }_{\mu }^h}}{F}_f^{int(i)}dV={\displaystyle {\sum }_{e=1}^{Ne}{B}_e^T{\sigma }_{\mu }^{e(i)}{V}_e+}{\displaystyle {\sum }_{ef=1}^{Nf}{F}_{ef}^{int(i)}}$$

(11) $$K^i={\displaystyle {\sum }_{e=1}^{Ne}{B}_e^T{D}_e^i{B}_e{V}_e+{\displaystyle {\sum }_{ef=1}^{Nf}{K}_{ef}^{(i)}}}$$

where K_ef is the rigidity related to the contact cohesive finite element ef, Nf is the number of contact cohesive fracture elements, $F_{ef}^{int}$ is the internal forces vector related to the contact cohesive fracture elements [²⁴].

After computing the corrections δũ_μ,ⁱ⁺¹ the displacement fluctuations field ${\tilde{u}}_{\mu }^{i+1}$ to be applied at iteration i+1 can be computed by: ${\tilde{u}}_{\mu }^{i+1}={\tilde{u}}_{\mu }^i+\delta {\tilde{u}}_{\mu }^{i+1}$.

It is important to stress that the present RVE formulation is solved using the consistent tangent operator leading to a quadratic rate of the iterative procedure. This is important when a full coupled multiscale analysis is performed because this kind of modelling is very expensive computationally.

In order to obtain the homogenized stress, let us consider that the RVE is composed of a solid part defined by the matrix, the ITZ and the aggregates. After discretizing the RVE domain in Ne elements, we can obtain from Eq. (2) the following expression to compute the homogenized stress σ [¹¹, ²⁶]:

(12) $$\sigma =\frac{1}{2V_{\mu }}(\overline{\sigma }+{\overline{\sigma }}^T)$$

where $\overline{\sigma }={\displaystyle \sum _{N=1}^{Nb}{\left\{F\right\}}_N{\left\{y\right\}}_N^T}$, being Nb the number of external boundary nodes of the RVE.

On the other hand, the homogenized constitutive tensor is computed from the following expression:

(13) $$D=D^{(Taylor)}+\tilde{D}$$

where D^Taylor is denoted the Taylor model tangent tensor (obtained by assuming ∇^s ũ_μ=0 ) and computed by the volume average of the microscopic constitutive tangent tensor; D represents the influence of the displacement fluctuations on the homogenized tangent tensor; they are given by:

(14) $$D^{(Taylor)}={\displaystyle {\sum }_{p=1}^{Np}\frac{V_p}{V_{\mu }}}{D}_{\mu }^{(p)}$$

(15) $$\tilde{D}=-\frac{1}{V_{\mu }}{G}_R{K}_R^{-1}{G}_R^T$$

where $G={\displaystyle {\sum }_{e=1}^{Ne}{D}_u^e{B}_e{V}_e}$, Np is the number of phases p defined in the RVE, D_u is the microscopic constitutive tangent tensor and Vp is the volume of the phase p. The matrices K_R and G_R are, respectively, reduced forms of matrix K (see Eq. (11)) and G for particular boundary conditions imposed to EVR, i. e., these matrices are defined according to the adopted multiscale model [¹¹, ²⁶].

2.1.2. The cohesive fracture model considering plasticity

To model the dissipative process that occurs in the mortar when the concrete structure is subjected to loads, we use the Mohr-Coulomb model. This model will be also used to simulate the ITZ fracture process in the case where the ITZ region is modelled by triangular finite elements with lower strength. The proposed modelling is an extension of the one developed in [²], where an elastic behaviour has been assumed for the matrix and cohesive and contact finite elements have been defined on the ITZ. In this previous work, even considering an elastic behaviour for the matrix, macroscopic homogenized plastic strains have been obtained in unloading cases as well as for loading inversion, evidencing the unilateral effect of the material.

Alternatively, the ITZ fracture process is also modelled by additional cohesive contact finite elements which are defined on the interfaces superposed to the triangular finite element nodes. In this case, cohesive fracture and contact models developed in [²⁴] govern the ITZ behaviour. Observe that the fracture model presented in [²⁴] has been obtained from the model proposed by CIRAK et al. [²⁹] which has been modified to simulate the fracture process in ductile materials until their rupture. The model describes the cohesive law of irreversible finite strains, being the energy released Φ given by:

(16) $$\mathrm{\Phi }=\mathrm{\Phi }({\delta }_n,{\delta }_S,q)$$

where, δn is the opening in mode I (normal); δs is the opening in mode II (sliding) and q is a variable which describes the inelastic processes of cohesion.

We assume that the strain due to the opening in mode II is a scalar value independent of the crack direction. Then we assume δ_s = |δ_s|, which defines an isotropic behaviour. For the case of mixed mode, it is introduced an effective opening displacement given by:

(17) $$\delta =\sqrt{{\beta }^2{\delta }_S^2+{\delta }_n^2}$$

The parameter β assumes different values for the openings, which can vary from 0 to 1. On the other hand, assuming that the cohesive energy released Φ depends on the value δ, the cohesive law is written as:

(18) $$t=\frac{t}{\delta }({\beta }^2{\delta }_S+{\delta }_{n}n)$$

where n is the normal direction to the crack; δs is the sliding opening vector on the crack surface; t is the cohesive stress vector along the crack. The relations to obtain t are given by:

(19) $$t={\sigma }_c{e}^{-\delta /{\delta }_c}if\delta ={\delta }_{\max }and\dot{\delta }\ge 0$$

(20) $$t=\frac{t_{\max }}{{\delta }_{\max }}\delta if\delta <{\delta }_{\max }or\dot{\delta }<0$$

where e is the exponential (e≅2,71828), σ_c is the maximum cohesive normal traction, δ˙ is the opening velocity and δc is the characteristic opening displacement. Before the crack initiation, we define a scalar parameter denoted as penalty factor λ_P which represents a stiffness between the future crack lips in order to not allow the penetration of the crack surfaces. In general, we adopt high values for the penalty factor to increase the approximation. This procedure ensures that the crack remains closed until the debonding criterion is achieved and at the same time it ensures the physical admissibility of the process.

To detect the cohesive contact phenomenon, we consider the differences between the displacements of the gauss points which lead to the relative displacement between the crack lips (λp). Therefore, in the case of having the contact phenomenon Eq. (21) is adopted.

(21) $$t={\lambda }_p\delta if{\lambda }_p\delta \le {\sigma }_c$$

The cohesive and contact finite element is rectangular, with four nodes, being its surfaces coincident in the non-deformed configuration. Besides, in the non-deformed configuration, its nodes are also coincident to the nodes of the triangular finite elements used to discretize the inclusions and the matrix [², ²⁴].

3. RESULTS, DISCUSSION AND ANALYSIS

In the numerical analyses we used the FX+ for Diana to generate 62 different RVE microstructures as well as their meshes. We have adopted the dimensions 100 mm × 100 mm for all RVEs as well as the following radius for the coarse aggregates: 10 mm, 5 mm and 2.5 mm. TERADA et al. [³⁰] evaluated the convergence of the solution sought in computational homogenization approaches. In general, as the dimension of the EVR increases, the characteristics of the material heterogeneities are incorporated more realistically. Consequently, the computationally homogenized response becomes more accurate and approaches the real or exact value. In this context, following [¹⁴], the adopted size of the RVE is representative of the statistical distribution of aggregates in this kind of concrete as well as to capture its heterogeneities. Therefore, we have adopted the same RVE dimensions considered in [¹⁴]. On the other hand, in [³¹] is reported the structural size effect issue presented in concrete-like materials and the necessity to understand the correlation between this effect to the heterogeneity of the material. Following [³¹], this issue can be solved by repeated numerical simulations to find the critical size of the RVE. Nevertheless, the present work aims to study the influence of two different strategies of modelling of ITZ on the mechanical behaviour of the material. The size effect would be one more variable in the study requiring much more numerical analyses. However, the structural size effect is not studied here. But we intend to investigate this issue in future works using the RVEs used in the present work.

We assume an elastic behaviour for the aggregates while the mortar material is governed by the Mohr coulomb criterion. In all numerical analyses we have imposed periodic fluctuations on the RVE boundary using a MATLAB computational code where the multiscale formulation described in item 2 has been implemented. The results are compared to the ones presented in the work [¹⁴], where a similar study has been considered.

In the case where triangular finite elements with lower strength are adopted to represent the mechanical behaviour of the ITZ (the green band represented in Figure 1) we define 41 different RVEs. Within these RVEs we have defined 5 RVEs with different shapes for the aggregates, 4 RVEs with different distributions for the aggregates, 12 RVEs with different volume fractions of aggregates and 20 RVEs where 5 different ITZ thicknesses have been combined with 4 different volume fractions of aggregates. Table 1 shows the elastic properties used for the 3 RVEs phases: mortar, ITZ and aggregates while Table 2 presents the material parameters related to the Mohr-coulomb model. Observe that in the present paper we have only used the parameters in tension, because this was the case considered in [¹⁴].

Table 1
Elastic properties for the three phases: coarse aggregates, mortar and ITZ [¹⁴].

	E (MPa)	ʋ
Mortar	26000	0.20
ITZ	25000	0.20
Aggregates	55500	0.20

Table 2
Mohr-Coulomb parameters for mortar and ITZ.

			HARDENING POINTS (εP × σP)
	FRICTION (º)	DILATANCY (º)	εP	σP (MPa)
Mortar	2	2	0.0	1.65
			0.017	0.4
ITZ	2	2	0.0	1.05
			0.019	0.001

When additional rectangular finite elements are defined on interfaces to model the ITZ fracture process, we have defined 21 RVEs, where 5 RVEs have different shapes of aggregates, 4 RVEs have different distributions of aggregates and 12 RVEs have different volume fractions of aggregates. In this case, the RVE has only 2 phases, being the material parameters related to the mortar matrix and to the aggregates also defined by Tables 1, 2. Besides, the following parameters are adopted for the rectangular finite elements: λp = 200000, β = 0.707, σc = 0.09 MPa and a limit opening displacement δc = 0,0568 mm. These values have also been adopted in the works [², ¹⁵, ¹⁸, ³²].

To obtain the parameters shown in Tables 1, 2 we have considered a uniform RVE, i.e., composed of only one phase. To obtain the mortar parameters we define the mortar as the RVE phase while the RVE phase is defined by the ITZ to obtain the ITZ parameters. For both cases we have considered tension regimes, and we have calibrated the parameter values to find the constitutive responses (stress × strain relation) presented by KIM and AL-RUB [¹⁴] used as reference response. Observe that [¹⁴] used a damage model to govern the material behavior. Note that Table 2 presents the pair of hardening points of the constitutive model to describe the relation ε^P × σ^P. We adopted only two pairs of points because the idea is to use constitutive models on the RVE as simple as possible. Besides, the plasticity model presents characteristics of a hardening curve and, therefore, the obtained curvature, even if the number of pair of points has been increased, explains the difference between the softening branches of the reference and adopted constitutive model, see Figures 2 and 3. Finally, Figure 2 shows the results for the mortar while Figure 3 presents the results for the ITZ.

Figure 2
Stress × strain relation for the mortar.

Figure 3
Stress × strain relation for the ITZ.

In Figures 2 and 3, we observe that, for the softening branch, the parametric identification for the mortar presents more similar results to the work [¹⁴] than the identification related to the ITZ. To improve the approximations, several points should be used to define the branches of hardening or softening related to the Mohr-Coulomb model, but in the present paper we have considered only two points. Our intention is to verify if adopting simple constitutive models with few parameters for the RVE phases produce good results. Observe that in the present work we model the mechanical behaviour of the concrete using a formulation developed within a multiscale framework which allows to define for each constituent its material properties. Therefore, this kind of modelling allows to model detailed the material microstructure, resulting in a more reliable modelling and usually there is no need of using sophisticated constitutive models for the RVE phases in order to have good results. Moreover, in Figures 2 and 3 we can observe that the ITZ phase presents a strength smaller than the mortar strength, due to the heterogeneity of its microscopic composition.

3.1. Influence of the aggregate shapes

Different shapes have been adopted for the aggregates to verify its influence on the material strength in predominant tension regimes. In agreement to the work [¹⁴], we have considered the following shapes: Circular, Hexagonal, Pentagonal, Tetragonal and an arbitrary polygonal shape (see Figure 4). In Table 3 is shown the quantity of each kind of coarse aggregate while in Table 4 the number of nodes and elements used in the meshes related to the two different modelling are defined. As defined previously, the two modelling are: i) ITZ modelled by triangular finite elements in the region of thickness equal to 0.2 mm, ii) ITZ modelled by additional rectangular finite elements defined on the interfaces. Observe that different meshes have been tested in order to obtain the results convergence. Therefore, in Table 4 we present the meshes whose results had already converged. Besides, in Table 3 for the case of arbitrary polygonal shape we have considered the aggregates with 5 mm and 2.5 mm as a single group because their separation is not very clear in Figure 4e.

Figure 4
Different shapes of aggregates: (a) Circular, (b) Hexagonal, (c) Pentagonal, (d) Tetragonal and (e) Polygonal [¹⁴].

Table 3
Number of inclusions for the RVEs with different shapes of aggregates.

RVE	COARSE AGGREGATES NUMBER
	10 mm	5 mm	2,5 mm	TOTAL
Circular	8	19	51	78
Hexagonal	8	19	51	78
Pentagonal	8	19	51	78
Tetragonal	8	19	51	78
Polygonal	7	73		80

Table 4
Number of finite elements for the RVEs with different aggregate shapes.

RVE	ITZ REGION MODELLED WITH TRIANGULAR FINITE ELEMENTS			ITZ MODELLED WITH ADDITIONAL RECTANGULAR FRACTURE ELEMENTS (FE) ON THE INTERFACES
	THICKNESS (mm)	Nº OF NODES	Nº OF ELEMENTS	Nº OF FE	Nº OF NODES	Nº OF TRIANGULAR ELEMENTS
Circular	0.2	2172	4306	780	2299	2996
Hexagonal	0.2	6038	12062	1404	6521	10220
Pentagonal	0.2	4446	8878	1170	4802	7250
Tetragonal	0.2	2598	5182	936	2759	3632
Polygonal	0.2	4955	9896	1467	5305	7662

In the numerical analyses, each RVE has been subjected to an arbitrary macrostrain vector related to a predominant tension regime and whose bigger component was in direction x. Besides, the imposed macrostrain is such that the RVE reaches its limit strength in tension, which leads to an instability in the numerical responses and difficult convergence of the iterative procedure. After the RVE equilibrium problem is solved, the homogenized stress vector can be computed, being the component in direction × presented in Figure 5 for the case where the ITZ is modelled by triangular finite elements while Figure 6 shows the results when additional rectangular finite elements are defined on the interfaces. Observe that in Figures 5 and 6 we present the normalized values for stress and strain, being the results compared to the ones obtained by Kim and Al-Rub [¹⁴]. In these figures, we assume that the beginning of the dissipative process in the material (plastic process as well as crack initiation) occurs when the limit strength in tension is achieved [³³, ³⁴] which is related to the strain on damage onset.

Figure 5
Normalized homogenized stress and macro strain in direction x for RVEs with different aggregates shapes and modelling the ITZ by triangular finite elements.

Figure 6
Normalized homogenized stress and macro strain in direction x for RVEs with different aggregates shapes and modelling the ITZ by additional rectangular finite elements.

We can observe in Figure 5 that when the ITZ is modelled by triangular finite elements, the smaller strength refers to the polygonal shape, which corresponds to 97% of the bigger strength related to the pentagonal shape. The polygonal shape also presents the smaller value for the strain at the beginning of the dissipative process, which is 89% of the bigger value which is related to the pentagonal shape. Besides, the strengths and strains at the beginning of the dissipative process obtained with the proposed model compare well to the results obtained by [¹⁴]. The maximum difference observed between the two models was 8%, being the hexagonal and polygonal shapes the ones with more similar results. In this case, we can conclude that the aggregate shape has not a significant influence on the material strength in tension but can reduce up to 11% the strain at the beginning of the dissipative process. Note that a similar conclusion has been achieved by KIM and AL-RUB [¹⁴]. However, in the proposed model the tetragonal shape presents the bigger value for the maximum stress while the pentagonal shape presents the bigger value for the strain at the beginning of the damage. In the model developed in [¹⁴], the circular shape leads to the bigger values in both cases.

In Figure 6 we can observe that when cohesive and contact finite elements are considered to model the phase debonding, the strength and the strain at the beginning of the dissipative process related to the pentagonal shape are very similar to the results presented by KIM and AL-RUB [¹⁴]. But the other shapes lead to a strong reduction in these values. Adopting the results of pentagonal shape as reference, the strength of the circular, hexagonal, tetragonal and polygonal is, respectively, equal to 75%, 60%, 62% and 25% of the reference value. Regarding the strain at the beginning of the dissipative process, the strain of the circular, hexagonal, tetragonal and polygonal shapes is, respectively, equal to 58%, 0.2% and 12% of the reference value. Therefore, the cohesive and contact finite elements lead to an important reduction of the RVE strength as well as of the strain value at the beginning of the dissipative process. In the numerical analyses we have evidenced difficulty of using fracture models at the mesoscale of the material, mainly due to numerical instabilities and the lacky of reliable values for the parameters. However, we can have interesting conclusions when contact and cohesive elements are used. First, we can observe that RVE deformability is smaller when this kind of element is considered. Besides, we can conclude that the identification of the parameter σc is crucial, because it governs if the element is in a fracture process or if the contact phenomenon is acting, meaning that the crack remains closed, which increases the RVE strength. In this context, in [²⁵] the authors have verified that small variations in the parameter values can produce significant differences in the macroscopic response of the material.

In Figure 7 we compare only the results shown in Figures 5 and 6 related to the proposed model. The analysis of this figure is very similar to Figure 6, i.e., when cohesive-contact finite elements are used in the proposed model, we have great reduction in the strength and deformability of the RVE. The exception is for the pentagonal shape which produces similar results. We can note that both the tetragonal and polygonal shapes lead to a massive loss of strength and deformability what have not been observed in the values related to Figure 5, where the ITZ has been considered as a continuum medium without cracks. Note that in the work [¹⁴] the authors have adopted a damage and plasticity model to govern the material behaviour of the mortar and ITZ. Therefore, some differences are expected between the present formulation and the one developed by KIM and AL-RUB [¹⁴] which can be explained by the different constitutive models adopted as well as by the parametric identification. In summary, the results obtained by using the cohesive-contact finite elements present a limitation of this kind of the finite element to model the ITZ when the number of corners of the aggregates is increased due to stress concentration at the sharp edges of aggregates. As example, when tetragonal and polygonal shapes are modelled which lead to poor qualitative and quantitative responses. For the numerical examples analysed in this work, the application of this strategy for modelling ITZ presented qualitative responses similar to [¹⁴] for circular, hexagonal and pentagonal shapes, i. e., when the shapes approximate to the circular ones.

Figure 7
Normalized homogenized stress and macro strain in direction x for RVEs with different aggregates shapes considering two modelling: ITZ modelled by fracture elements and ITZ modelled by triangular finite elements.

3.2. Influence of the aggregate distribution

Four different distributions for the aggregates are shown in Figure 8, where all RVEs have the same volume fraction of aggregates equal to 50%. All aggregates are randomly distributed with the following proportion: 5:3:2. Despite the using of RVEs randomly generated, there are other methods, as example the Computed Tomography based methods [²¹, ³⁵], where the phases of the microstructure of the materials are more realistic represented, but for the objectives of the present work, the generation using random algorithm is quite satisfactory for different aggregate shapes used in the present work. The meshes used in the modelling where the fracture process in ITZ is simulated using triangular finite elements with lower strength are detailed in Table 5. It was seen in item 3.1 that the results using the cohesive fracture elements were unstable and strongly dependent on parametric identification, this also was confirmed in the analyses performed in the present section and the results were not representative. One solution would be to drastic increasing σc, but this would lead to a linear elastic behaviour response with strong strength that would not match what was expected in a predominant tension regime. The material properties are defined in Tables 1, 2.

Figure 8
RVEs with different distributions for the aggregates (a) Model 1. (b) Model 2. (c) Model 3. (d) Model 4 [¹⁴].

Table 5
Number of finite elements for the RVEs with different distributions of aggregates.

RVE	ITZ MODELLED WITH TRIANGULAR FINITE ELEMENTS
	THICKNESS (mm)	Nº OF NODES	Nº OF ELEMENTS
Model 1	0.2	2264	4490
Model 2	0.2	2216	4394
Model 3	0.2	2166	4294
Model 4	0.2	2184	4330

A macrostrain vector ε = [εx; −0.2 εx; 0] in predominant tension regime, with the biggest strain defined in direction x (εx = 0.52 x 10−6) has been imposed to the RVEs defined in Figure 8. Figure 9 shows the normalized homogenized stress, taking the highest strength as reference, versus the imposed macro-strain, in the direction x, being the homogenized stress obtained after the convergence of the RVE iterative procedure is achieved. We can observe that the curves are identical, which indicates that the distribution of the aggregates does not influence the homogenized response of the RVE. This conclusion is in agreement to the work [¹⁴], where after performing a full coupled multiscale analysis, the authors have concluded that the macroscopic response defined by the load versus displacement of concretes, until the ultimate load, does not depend on the distribution of aggregates in the microstructure.

Figure 9
Normalized homogenized stress versus imposed macrostrain x for different distributions of aggregates: (a) Model 1. (b) Model 2. (c) Model 3. (d) Model 4 [¹⁴].

It is important to stress that when contact and cohesive fracture elements are considered on the interfaces, we have observed numerical instabilities when the fracture process is accentuated, because in this case there is no stress transmissibility between the coarse aggregates and the mortar matrix, what increases the stress field of the mortar matrix. This issue has been observed in other works [¹⁵, ³⁶] when dealing with tension regimes and this is the case of the present work. To overcome this problem, some different fracture models have been proposed as the following works [¹⁶, ³⁷,³⁸,³⁹]. In those works, different models have been developed which take into account the localization problem in the microstructure and its influence on the macrostructure response where a crack is nucleated. Besides, the proposed modelling is capable of presenting the microcracking process that initiates in the ITZ and propagates to the mortar matrix as shown in [⁴⁰], where it was used a damage model combined to contact and cohesive fracture elements.

3.3. Different volume fractions of aggregates combined to different distributions of aggregates

The influence of the volume fraction of aggregates is studied in this section for predominant regimes in tension. In Figure 10 are shown 12 different RVEs which have been separated into two groups, according to the distribution of the aggregates. Each group has 6 RVEs, being the RVEs of the first group denoted as a, b, c, d, e and f while the RVEs of the other group are denoted as g, h, i, j, k and l. For the RVEs a to f the random distribution number 1 has been assumed while for RVEs g to l we adopt the random distribution number 2. Both groups present a gradual increase of the volume fraction of aggregates from 10% to 60%.

Figure 10
RVEs with different combinations of volume fractions of aggregates with their random distributions [¹⁴].

The number of aggregates as well as the volume fraction (vf) of aggregates are detailed in Table 6 for each RVE while in Table 7 the mesh used in each analysis has been defined.

Table 6
Number of aggregates and the volume fraction (vf) of aggregates used for RVEs a to l.

RVE	NUMBER OF AGGREGATES				AREA OF AGGREGATES (MM2)	VOLUME FRACTION VF (%)
	COARSE (10 mm)	AVERAGE (5 mm)	FINE (2,5 mm)	TOTAL
a (vf = 10%)	1	5	15	21	1001.38	10.01
b (vf =20%)	3	8	21	32	1983.13	19.83
c (vf =30%)	5	11	29	45	3004.15	30.04
d (vf =40%)	6	17	40	63	4005.53	40.06
e (vf =50%)	8	19	51	78	5006.91	50.07
f (vf =60%)	10	22	60	92	6047.57	60.48
g (vf =10%)	1	5	15	21	1001.38	10.01
h (vf =20%)	3	8	21	32	1983.13	19.83
i (vf =30%)	5	11	29	45	3004.15	30.04
j (vf =40%)	6	17	40	63	4005.53	40.06
k (vf =50%)	8	19	51	78	5006.91	50.07
l (vf =60%)	10	22	60	92	6047.57	60.48

Table 7
Meshes for the analyses of the RVEs a to l.

RVE	ITZ REGION MODELLED WITH TRIANGULAR FINITE ELEMENTS			ITZ MODELLED WITH ADDITIONAL RECTANGULAR FRACTURE ELEMENTS (FE) ON THE INTERFACES
	THICKNESS (mm)	Nº OF NODES	Nº OF ELEMENTS	Nº OF FE	Nº OF NODES	Nº OF TRIANGULAR ELEMENTS
a	0.2	1695	3352	210	1842	3222
b	0.2	1587	3136	320	1734	2786
c	0.2	1788	3538	450	1899	2856
d	0.2	1992	3946	630	2117	2932
e	0.2	2153	4268	780	2266	2930
f	0.2	2377	4716	920	2461	3040
g	0.2	1705	3372	210	1893	3324
h	0.2	1667	3296	320	1784	2886
i	0.2	1806	3574	450	1916	2890
j	0.2	2002	3966	630	2133	2964
k	0.2	2222	4406	780	2346	3090
l	0.2	2396	4754	920	2477	3072

A macrostrain vector ε = [εx; −0.2 εx; 0] in predominant tension regime, with the biggest strain defined in direction x has been imposed to the RVEs defined in Figure 10. The value for εx is dependent of the RVE rigidity varying from 5.0 x 10−5 up to 6.3 x 10−5. A similar study has been performed by KIM and AL-RUB [¹⁴]. But in that work the authors did not analyse only the microstructure mechanical behaviour as we consider in the present work. Their study has been made in the context of studying the macro-mechanical response of a structure composed of the material defined with the different volume fractions shown in Figure 10. KIM and AL-RUB [¹⁴] verified that the decreasing of the volume fraction of aggregates leads to an increasing of the displacement related to the ultimate load, evidencing a bigger deformability of the structure. However, the model proposed by KIM and AL-RUB [¹⁴] did not present a significant increase of the structure strength when the volume fraction of aggregates in the microstructure was increased.

Figure 11 shows the normalized homogenized stress, taking the highest strength among the models (Figure 10) as reference, versus the imposed macrostrain in the direction x, for the case where the ITZ is modelled by triangular finite elements with lower strength. And Figure 12 shows the results for the modelling with contact and cohesive finite elements.

Figure 11
Normalized homogenized stress versus imposed macrostrain with different volume fractions of aggregates where the ITZ is modelled by triangular finite elements with lower strength.

Figure 12
Normalized homogenized stress versus imposed macrostrain with different volume fractions of aggregates where the ITZ is modelled by contact and cohesive finite elements on the interfaces.

We can observe in Figure 11 that when the ITZ is modelled by triangular finite elements, the results of the proposed model are in agreement to the conclusions presented by KIM and AL-RUB [¹⁴] and that the bigger the volume fraction of aggregates the bigger the material rigidity. Besides, we can observe a small increase of the RVE strength when the volume fraction of aggregates is increased. Therefore, increasing the volume fraction of inclusions does not significantly change the RVE strength. This can be explained by the high values for the ITZ parameters obtained from the numerical responses presented by [¹⁴], see Table 1, Figures 2 and 3. These values are similar to the mortar matrix values, therefore, the increase of number of aggregates and consequently, the number of ITZ zones, does not lead to the increase of material strength in this material. Moreover, we can note from Figure 11, that the distribution of the aggregates does not change the RVE homogenized response for the same volume fraction of aggregates. This conclusion is in agreement to the results discussed in section 3.2.

In Figure 12 we can observe that increasing the volume fraction of aggregates also increases the RVE rigidity, but in this case the impact on the RVE rigidity is bigger when compared to the case where triangular finite elements are used to model the ITZ. Besides, in this case the distribution of aggregates also does not affect the RVE rigidity for the following volume fractions: 10%, 20%, 30%, 40% and 50%, but the strength of these RVEs (except for the RVE with 10%) is decreased when the number 2 distribution is adopted. On the other hand, for the case where the volume fraction is adopted equal to 60%, both the rigidity and the strength are decreased when the number 2 distribution is assumed. This can be explained by the fact that for high volume fractions of aggregates, they can be stacked in different directions according to the adopted distribution. Once more time, the application of the strategy using cohesive contact fracture elements has limitations.

3.4. Influence of the ITZ thickness

Using the modelling where the ITZ fracture process is simulated by triangular finite elements, in this section we investigate the influence on the RVE tension strength when the ITZ thickness is changed. The results are compared to the work [¹⁴] where a similar study has been made. Five thickness values have been adopted: 0.1 mm, 0.2 mm, 0.4 mm, 0.6 mm and 0.8 mm. Besides, we have adopted 4 different volume fractions of aggregates, resulting into 20 different EVRs, after the combination of these 2 factors, as depicted in Figure 13.

Figure 13
RVEs with different ITZ thickness and volume fraction of aggregates [¹⁴].

Table 8 details the number of aggregates as well as the number of finite elements considered for the 20 RVEs defined in Figure 13. As the same way of others numerical examples described in this work, we have imposed to the RVEs defined in Figure 13, a macrostrain vector ε = [εx; −0.2 εx; 0] in predominant tension regime, with the biggest strain defined in direction x. In Figure 14 we show the relation between the damage strain and the ITZ thickness for different volume fractions of aggregates. The damage strain is adopted as macrostrain in x direction when dissipative processes are initiated. We can observe in this figure that the damage strain is bigger when the ITZ thickness is increased, which is coherent because the bigger is the ITZ thickness the bigger must be the RVE deformability and smaller must be its rigidity. But this has not been observed with the Kim and Al-Rub´s model, because according to the work [¹⁴], this strain does not change when different ITZ thickness are adopted. Besides, we also note that increasing the volume fraction of inclusions leads to smaller damage strain, as expected, because the bigger is the volume fraction of aggregates, the bigger is the RVE rigidity, which is in agreement to [¹⁴].

Table 8
Number of aggregates and finite elements for the RVEs where different thickness combined to different volume fraction of aggregates are assumed.

RVE		NUMBER OF AGGREGATES				NUMBER OF FINITE ELEMENTS
VOLUME FRACTION (VF)	ITZ THICKNESS (mm)	COARSE (10 mm)	AVERAGE (5 mm)	FINE (2,5 mm)	TOTAL	Nº OF NODES	Nº OF ELEMENTS
20%	0.1	3	8	21	32	1666	3294
	0.2	3	8	21	32	1587	3136
	0.4	3	8	21	32	1497	2956
	0.6	3	8	21	32	1408	2778
	0.8	3	8	21	32	1335	2632
30%	0.1	5	11	29	45	1831	3624
	0.2	5	11	29	45	1788	3538
	0.4	5	11	29	45	1682	3309
	0.6	5	11	29	45	1585	3128
	0.8	5	11	29	45	1518	2994
40%	0.1	6	17	40	63	2057	4072
	0.2	6	17	40	63	1992	3946
	0.4	6	17	40	63	1905	3768
	0.6	6	17	40	63	1816	3590
	0.8	6	17	40	63	1757	3472
50%	0.1	8	19	51	78	2213	4384
	0.2	8	19	51	78	2153	4268
	0.4	8	19	51	78	2063	4084
	0.6	8	19	51	78	2017	3992
	0.8	8	19	51	78	1978	3914

Figure 14
Damage strain for different ITZ thickness and volume fraction of aggregates.

It is interesting to note that despite the work [¹⁴] models the mortar matrix behaviour by a damage model, their conclusions are similar to the ones discussed in the present paper, where the mortar matrix behaviour is modelled by an elastoplastic model. This indicates that it is not mandatory modelling the mortar matrix by a damage model to represent satisfactorily the concrete mechanical behaviour.

4. CONCLUSIONS

The present work studied the potentialities as well as the limitations of two different modelling to simulate the ITZ fracture process, in order to capture some phenomena of the concrete mechanical behaviour at mesoscale. For that, we have used a FEM formulation written within a multiscale framework and based on homogenization techniques, which allows to define the material constitutive response as the homogenized response of its microstructure. This kind of technique has shown to be promising and a good alternative to the complex phenomenological constitutive models to simulate the mechanical behaviour of quasi-brittle materials.

The results presented in the work [¹⁴] for different microstructures have been assumed as reference ones in order to compare to the proposed modelling. Note that [¹⁴] did not present experimental results of the concrete they simulated numerically. So that, the present work only made comparisons to the numerical results presented in [¹⁴]. Besides, we investigate the potentialities and limitations of the proposed modelling. We have observed that, for the case where the ITZ is modelled by triangular finite elements, the proposed modelling captures a small increasing of the RVE strength for bigger volume fractions of aggregates. Besides, we have also verified that the distribution of aggregates does not change the homogenized response of the microstructure.

The strategy of using triangular finite elements to model the ITZ fracture process leaded to stable numerical results, while the modelling where contact and cohesive finite elements, which are defined on the interfaces, leaded to convergence issues in the numerical analyses. Moreover, with the increasing of number of corners in the aggregates lead to poor qualitative and quantitative responses, as example when tetragonal and polygonal shapes are modelled. Besides, the results are very dependent on the value of parameter σc, [²⁵]. This can be explained by the fact that in the present work we used the cohesive fracture contact elements to model the ITZ which have a finite thickness. However, the adoption of a null thickness should be more adequate [³⁹]. On the other hand, the contact and cohesive finite elements have been successfully used in [³⁶] to model the ITZ of concrete, producing good numerical results when compared to the experimental results. In that work, a step-by-step procedure to perform a parametric identification for a real tested concrete was presented.

This modelling is capable of representing the ascend branch of the strass-strain relationship and capturing the ultimate stress. However, when damage localization occurs, the numerical responses become instable and hard to converge. These numerical problems could be overcame by adopting different kinds of modelling as example: i) if constitutive models presenting softening regime is used to model the matrix behaviour, such as damage models, combined to the adoption of triangular finite elements with high aspect ratio [³⁷], ii) by using formulations with strong discontinuities incorporated [¹⁶, ⁴¹], iii) by using a RVE formulation that simulates the localization problem at microstructure which leads to a crack nucleation at the macro-continuum in order to perform a full coupled multiscale analysis of a structure composed of quasi brittle materials, such as the concrete [³⁹]. However, the choice of a properly and robust numerical technique to solve the equilibrium problem presenting softening regimes is also important. As example, the Arch-Length Method [⁴²] is a potential numerical tool to solve this kind of problem.

Finally, we conclude that the formulation developed in this work to analyse the mechanical behaviour of the concrete is capable to capture the dissipative phenomena that occur in its microstructure. Despite the use of the contact and cohesive fracture elements presented limitations, if a reliable parametric identification is performed, this strategy can be used. Besides, we have shown that the modelling where the fracture process of the ITZ is simulated by triangular finite elements can represent properly the mechanical behaviour of the concrete.

5. BIBLIOGRAPHY

• ^[1] PITUBA, J.J.C., “A damage model formulation: unilateral effect and RC structures analysis”, Computers and Concrete, v. 15, n. 5, pp. 709–733, 2015. doi: http://doi.org/10.12989/cac.2015.15.5.709.
  » https://doi.org/10.12989/cac.2015.15.5.709
• ^[2] PITUBA, J.J.C., SOUZA NETO, E.A., “Modeling of unilateral effect in brittle materials by a mesoscopic scale approach”, Computers and Concrete, v. 15, n. 5, pp. 735–758, 2015. doi: http://doi.org/10.12989/cac.2015.15.5.735.
  » https://doi.org/10.12989/cac.2015.15.5.735
• ^[3] NGHIA NGUYEN, T., LE, T.C., KATHIR, S., et al, “A novel approach to the complete stress strain curve for plastically damaged concrete under monotonic and cyclic loads”, Computers and Concrete, v. 28, pp. 39–53, 2021. doi: http://doi.org/10.12989/CAC.2021.28.1.039.
  » https://doi.org/10.12989/CAC.2021.28.1.039
• ^[4] ARSLAN, A., Gümüş, M., “A simple and robust approach for 2D-simulation of reinforced concrete member”, Structures, v. 32, pp. 1701–1716, 2021. doi: http://doi.org/10.1016/j.istruc.2021.03.050.
  » https://doi.org/10.1016/j.istruc.2021.03.050
• ^[5] BRÜNIG, M., MICHALSKI, A., “Numerical analysis of damage and failure behavior of concrete”, International Journal of Damage Mechanics, v. 29, n. 4, pp. 570–590, 2020. doi: http://doi.org/10.1177/1056789519866005.
  » https://doi.org/10.1177/1056789519866005
• ^[6] JIAO, Y., WANG, B., SHEN, Z.A., “New 3D empirical plastic and damage model for simulating the failure of concrete structure”, International Journal of Concrete Structures and Materials, v. 13, n. 1, pp. 57, 2019. doi: http://doi.org/10.1186/s40069-019-0362-z.
  » https://doi.org/10.1186/s40069-019-0362-z
• ^[7] ZHANG, X., WU, H., LI, J., et al, “A constitutive model of concrete based on Ottosen yield criterion”, International Journal of Solids and Structures, v. 193–194, pp. 79–89, 2020. doi: http://doi.org/10.1016/j.ijsolstr.2020.02.013.
  » https://doi.org/10.1016/j.ijsolstr.2020.02.013
• ^[8] MASO, J.C., Interfacial Transition Zone in Concrete, 5th ed, Toulouse, France, LMDC Editors, 1996. doi: doi: http://doi.org/10.1201/9781482271560.
  » https://doi.org/10.1201/9781482271560
• ^[9] MEHTA, P.K., MONTEIRO, P.J.M., Concreto: Microestrutura, Propriedades e Materiais, 2. ed, São Paulo, Brazil, Pini Editors, 2014.
• ^[10] PERIC, D., DE SOUZA NETO, E.A., FEIJÓO, R., et al, “On micro-to-macro transitions for multiscale analysis of heterogeneous materials: unified variational basis and finite element implementation”, International Journal for Numerical Methods in Engineering, v. 87, n. 1–5, pp. 149–170, 2011. http://doi.org/10.1002/nme.3014.
  » https://doi.org/10.1002/nme.3014
• ^[11] FERNANDES, G.R., PITUBA, J.J.C., DE SOUZA NETO, E.A., “FEM/BEM formulation for multi-scale analysis of stretched plates”, Engineering Analysis with Boundary Elements, v. 54, pp. 47–59, 2015. doi: http://doi.org/10.1016/j.enganabound.2015.01.005.
  » https://doi.org/10.1016/j.enganabound.2015.01.005
• ^[12] FERNANDES, G.R., MARQUES, M.J., VIEIRA, J.F., et al, “A RVE formulation by the boundary element method considering phase debonding”, Engineering Analysis with Boundary Elements, v. 104, pp. 259–276, 2019. doi: http://doi.org/10.1016/j.enganabound.2019.03.018.
  » https://doi.org/10.1016/j.enganabound.2019.03.018
• ^[13] MENTGES, J., CARNIEL, T.A., “A note on the multiscale volume changes of auxetic metamaterials assessed via digital image correlation”, Matéria (Rio de Janeiro), v. 27, n. 2, pp. e13206, 2022. doi: http://doi.org/10.1590/s1517-707620220002.1306.
  » https://doi.org/10.1590/s1517-707620220002.1306
• ^[14] KIM, S.M., AL-RUB, R.K.A., “Meso-scale computational modeling of the plastic-damage response of cementitious composites”, Cement and Concrete Research, v. 41, n. 3, pp. 339–358, 2011. doi: http://doi.org/10.1016/j.cemconres.2010.12.002.
  » https://doi.org/10.1016/j.cemconres.2010.12.002
• ^[15] BORGES, D.C., PITUBA, J.J.C., “Analysis of quasi-brittle materials at mesoscopic level using homogenization model”, Advances in Concrete Construction, v. 5, n. 3, pp. 221–240, 2017. doi: http://doi.org/10.12989/acc.2017.5.3.221.
  » https://doi.org/10.12989/acc.2017.5.3.221
• ^[16] OLIVER, J., CAICEDO, M., ROUBIN, E., et al, “Continuum approach to computational multiscale modeling of propagating fracture”, Computer Methods in Applied Mechanics and Engineering, v. 294, pp. 384–427, 2015. doi: http://doi.org/10.1016/j.cma.2015.05.012.
  » https://doi.org/10.1016/j.cma.2015.05.012
• ^[17] BLANCO, P.J., SÁNCHEZ, P.J., SOUZA NETO, E.A., et al, “Variational foundations and generalized unified theory of RVE-based multiscale models”, Archives of Computational Methods in Engineering, v. 23, n. 2, pp. 191–253, 2016. doi: http://doi.org/10.1007/s11831-014-9137-5.
  » https://doi.org/10.1007/s11831-014-9137-5
• ^[18] OLIVER, J., CAICEDO, M., ROUBIN, E., et al, “Multi-scale (FE2) analysis of material failure in cement/aggregate-type composite structures”, In: Proceedings of Computational Modelling of Concrete Structures, St. Anton am Arlberg, Austria, March 2014.
• ^[19] NGUYEN, V.P., LLOBERAS-VALLS, O., STROEVEN, M., et al, “Homogenization-based multiscale crack modeling: from micro-diffusive damage to macro-cracks”, Computer Methods in Applied Mechanics and Engineering, v. 200, n. 9–12, pp. 1220–1236, 2011. doi: http://doi.org/10.1016/j.cma.2010.10.013.
  » https://doi.org/10.1016/j.cma.2010.10.013
• ^[20] NGUYEN, V.P., LLOBERAS VALLS, O., STROEVEN, M., et al, “On the existence of representative volumes for softening quasi-brittle materials - a failure zone averaging scheme”, Computer Methods in Applied Mechanics and Engineering, v. 199, n. 45–48, pp. 3028–3038, 2010. doi: http://doi.org/10.1016/j.cma.2010.06.018.
  » https://doi.org/10.1016/j.cma.2010.06.018
• ^[21] HUANG, Y., NATARAJAN, S., ZHANG, H., et al, “A CT image-driven computational framework for investigating complex 3D fracture in mesoscale concrete”, Cement and Concrete Composites, v. 143, pp. 105270, 2023. doi: http://doi.org/10.1016/j.cemconcomp.2023.105270.
  » https://doi.org/10.1016/j.cemconcomp.2023.105270
• ^[22] TORRENCE, C.E., TRAGESER, J.E., JONES, R.E., et al, “Sensitivity of the strength and toughness of concrete to the properties of the interfacial transition zone”, Construction & Building Materials, v. 336, pp. 126875, 2022. doi: http://doi.org/10.1016/j.conbuildmat.2022.126875.
  » https://doi.org/10.1016/j.conbuildmat.2022.126875
• ^[23] ZHAO, H., WU, Z., LIU, A., et al, “Numerical insights into the effect of ITZ and aggregate strength on concrete properties”, Theoretical and Applied Fracture Mechanics, v. 120, pp. 103415, 2022. doi: http://doi.org/10.1016/j.tafmec.2022.103415.
  » https://doi.org/10.1016/j.tafmec.2022.103415
• ^[24] PITUBA, J.J.C., FERNANDES, G.R., DE SOUZA NETO, E.A., “Modeling of cohesive fracture and plasticity in composite microstructures”, Journal of Engineering Mechanics, v. 142, n. 10, pp. 04016069, 2016. doi: http://doi.org/10.1061/(ASCE)EM.1943-7889.0001123.
  » https://doi.org/10.1061/(ASCE)EM.1943-7889.0001123
• ^[25] SANTOS, W.F.D., FERNANDES, G.R., PITUBA, J.J.D.C., “Analysis of the influence of plasticity and fracture processes on the mechanical behavior of metal matrix composites microstructures”, Matéria (Rio de Janeiro), v. 21, pp. 577–598, 2016. doi: http://doi.org/10.1590/S1517-707620160003.0056.
  » https://doi.org/10.1590/S1517-707620160003.0056
• ^[26] FERNANDES, G.R., PITUBA, J.J.C., SOUZA NETO, E.A., “Multi-scale modelling for bending analysis of heteregeneous plates by coupling BEM and FEM”, Engineering Analysis with Boundary Elements, v. 51, pp. 1–13, 2015. doi: http://doi.org/10.1016/j.enganabound.2014.10.005.
  » https://doi.org/10.1016/j.enganabound.2014.10.005
• ^[27] SANTOS, W.F.D., PITUBA, J.J.D.C., “Yield surfaces of material composed of porous and heterogeneous microstructures considering phase debonding”, Latin American Journal of Solids and Structures, v. 14, n. 8, pp. 1387–1415, 2017. doi: http://doi.org/10.1590/1679-78253776.
  » https://doi.org/10.1590/1679-78253776
• ^[28] GIUSTI, S.M., BLANCO, P.J., DE SOUZA NETO, E.A., et al, “An assessment of the Gurson yield criterion by a computational multi-scale approach”, Engineering Computations, v. 26, n. 3, pp. 281–301, 2009. doi: http://doi.org/10.1108/02644400910943626.
  » https://doi.org/10.1108/02644400910943626
• ^[29] CIRAK, F., ORTIZ, M., PANDOLFI, A., “A cohesive approach to thin-shell fracture and fragmentation”, Computer Methods in Applied Mechanics and Engineering, v. 194, n. 21–24, pp. 2604–2618, 2005. doi: http://doi.org/10.1016/j.cma.2004.07.048.
  » https://doi.org/10.1016/j.cma.2004.07.048
• ^[30] TERADA, K., HORI, M., KYOYA, T., et al, “Simulation of the multi-scale convergence in computational homogenization approaches”, International Journal of Solids and Structures, v. 37, n. 16, pp. 2285–2311, 2000. doi: http://doi.org/10.1016/S0020-7683(98)00341-2.
  » https://doi.org/10.1016/S0020-7683(98)00341-2
• ^[31] ZHANG, H., HUANG, Y., GUO, F., et al, “A meso-scale size effect study of concrete tensile strength considering parameters of random fields”, Engineering Fracture Mechanics, v. 269, pp. 108519, 2022. doi: http://doi.org/10.1016/j.engfracmech.2022.108519.
  » https://doi.org/10.1016/j.engfracmech.2022.108519
• ^[32] ORTIZ, M., PANDOLFI, A., “Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis”, International Journal for Numerical Methods in Engineering, v. 44, n. 9, pp. 1267–1282, 1999. doi: http://doi.org/10.1002/(SICI)1097-0207(19990330)44:9<1267::AID-NME486>3.0.CO,2-7.
  » https://doi.org/10.1002/(SICI)1097-0207(19990330)44:9<1267::AID-NME486>3.0.CO,2-7
• ^[33] GOPALARATNAM, V., SHAH, S.P., “Softening response of plain concrete in direct tension”, Journal of the American Concrete Institute, v. 82, n. 3, 1985. doi: http://doi.org/10.14359/10338.
  » https://doi.org/10.14359/10338
• ^[34] TERRIEN, M., “Emission acoustique et comportement mécanique postcritique d’un béton sollicité en traction”, In: Bulletin de Liaison Ponts et Chaussées, n. 105, France, 1980.
• ^[35] HUANG, Y., YANG, Z., REN, W., et al, “3D meso-scale fracture modelling and validation of concrete based on in-situ X-ray Computed Tomography images using damage plasticity model”, International Journal of Solids and Structures, v. 67–68, pp. 340–352, 2015. doi: http://doi.org/10.1016/j.ijsolstr.2015.05.002.
  » https://doi.org/10.1016/j.ijsolstr.2015.05.002
• ^[36] SILVA, M.J.M., PITALUGA, C.G., FERNANDES, G.R., et al, “Meso-scale modeling of the compressive mechanical behavior of concrete by a RVE-based BEM formulation”, Mechanics of Advanced Materials and Structures, v. 31, n. 9, pp. 1877–1896, 2022. doi: http://doi.org/10.1080/15376494.2022.2144974.
  » https://doi.org/10.1080/15376494.2022.2144974
• ^[37] RODRIGUES, E.A., MANZOLI, O.L., BITENCOURT JUNIOR, L.A.G., et al, “2D mesoscale model for concrete based on the use of interface element with a high aspect ratio”, International Journal of Solids and Structures, v. 94–95, pp. 112–124, 2016. doi: http://doi.org/10.1016/j.ijsolstr.2016.05.004.
  » https://doi.org/10.1016/j.ijsolstr.2016.05.004
• ^[38] MANZOLI, O.L., GAMINO, A.L., RODRIGUES, E.A., et al, “Modeling of interfaces in two-dimensional problems using solid finite elements with high aspect ratio”, Computers & Structures, v. 94–95, pp. 70–82, 2012. doi: http://doi.org/10.1016/j.compstruc.2011.12.001.
  » https://doi.org/10.1016/j.compstruc.2011.12.001
• ^[39] TORO, S., SÁNCHEZ, P.J., BLANCO, P.J., et al, “Multiscale formulation for material failure accounting for cohesive cracks at the macro and micro scales”, International Journal of Plasticity, v. 76, pp. 75–110, 2016. doi: http://doi.org/10.1016/j.ijplas.2015.07.001.
  » https://doi.org/10.1016/j.ijplas.2015.07.001
• ^[40] BORGES, D.C., PITUBA, J.J.C., “Homogenized damage model for brittle materials”, Mechanics of Advanced Materials and Structures, v. 31, n. 26, pp. 8356–8368, 2023. doi: http://doi.org/10.1080/15376494.2023.2258365.
  » https://doi.org/10.1080/15376494.2023.2258365
• ^[41] OLIVER, J., CERVERA, M., MANZOLI, O.L., “Strong discontinuities and continuum plasticity models: the strong discontinuity approach”, International Journal of Plasticity, v. 15, n. 3, pp. 319–351, 1999. doi: http://doi.org/10.1016/S0749-6419(98)00073-4.
  » https://doi.org/10.1016/S0749-6419(98)00073-4
• ^[42] SOUZA NETO, E.A., PERIC, D., OWEN, D.R.J., Computational Methods for Plasticity: Theory and Applications, Swansea: Wiley, 2008. doi: http://doi.org/10.1002/9780470694626.
  » https://doi.org/10.1002/9780470694626

Publication Dates

• Publication in this collection
  07 Feb 2025
• Date of issue
  2025

History

• Received
  23 Oct 2024
• Accepted
  30 Dec 2024