Theoretical and Applied Fracture Mechanics 96 (2018) 308–325
Contents lists available at ScienceDirect
Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec
Numerical simulation of fracture of concrete at different loading rates by using the cohesive crack model
T
⁎
Gustavo Morales-Alonsoa, , Víctor Rey-de-Pedrazab, Francisco Gálvezb, David A. Cendónb a b
E.T.S.I. Industriales, Universidad Politécnica de Madrid, c/ José Gutiérrez Abascal, 2, 28006 Madrid, Spain E.T.S.I. Caminos, Canales y Puertos, Universidad Politécnica de Madrid, c/ Profesor Aranguren, s/n, 28040 Madrid, Spain
A R T I C LE I N FO
A B S T R A C T
Keywords: Numerical implementation Material user subroutine Cohesive fracture Embedded crack High strain rate Blast loading
This paper presents an implementation of the Cohesive Crack Model using the Strong Discontinuity Approach technique to simulate the fracture of the concrete and other quasi-brittle materials at different strain rates. Since one of the principal applications of this model is to simulate the fracture at high strain rates, it has been programmed as a user subroutine in the commercial explicit finite element solver LS-DYNA. For the validation of the model, in addition to studying the sensitivity to element size, it has been compared with a collection of experimental results, both in quasi-static and dynamic regimes. The results prove the ability of the model to accurately reproduce different parameters, such as load-displacement curves, crack trajectories or fracture mechanisms.
1. Introduction Important efforts have been devoted in recent years to developing robust models capable of modeling the fracture of concrete and other quasi-brittle materials when subjected to loads of different strain rates. Among the different approaches for modeling the fracture of concrete, the Cohesive Zone Model or Cohesive Crack Model (CCM), initially proposed by the seminal works of Dugdale [1] and Baremblatt [2] and successfully introduced for concrete by Hillerborg and co-workers [3], is considered nowadays the most reliable methodology for addressing the simulation of fracture of concrete pieces [4]. Besides the simulation of fracture of plain concrete, its use has been also extended to several applications related with concrete, such as steel corrosion [5,6], splitting tests [7], and different materials, like fiber reinforced concrete [8,9], aluminum [10], rocks [11], gray cast iron [12] and even explosive propellants [13]. Up to the early 1990s, in spite of its capability for providing accurate predictions of the structural behavior of concrete specimens, some of the main drawbacks of the CCM were the need (i) to know the crack trajectory in advance for its implementation [14,15], (ii) to implement complex remeshing techniques in those cases where such crack trajectory was not known in advance [16,17] and (iii) to insert multiple interface elements in the element boundaries in order to allow the crack to propagate as a result of the calculation [18]. During the 1990s, different finite element techniques emerged to overcome this drawback, such as the generalized finite element method (GFEM) [19], the
⁎
Corresponding author. E-mail address:
[email protected] (G. Morales-Alonso).
https://doi.org/10.1016/j.tafmec.2018.05.003 Received 31 August 2017; Received in revised form 9 February 2018; Accepted 3 May 2018 0167-8442/ © 2018 Elsevier Ltd. All rights reserved.
extended finite element method (XFEM) [20] and the strong discontinuity approach (SDA) [21]. These techniques, also known as embedded discontinuities, allow us to insert the crack through the finite elements, being the technique thoroughly described in [22–24]. The crack trajectory is obtained as a result of the simulation, thus making the use of the CCM possible in cases of nontrivial crack trajectories, even with relatively simple meshes. But crack path prediction is not the only benefit of these techniques. Since the crack is inserted inside the elements, all parameters accessible at the element level are available for the crack. This fact allows for making the CCM influenced by many different variables if required, such as the stress triaxiality [12] or the strain rate, as in the case of the simulations presented in this work. Moreover, when the SDA is implemented by means of single integration points, the equilibrium equations can be solved at the material level [25,26], enhancing the possibilities of this technique even more since most commercial finite element codes allow the user to implement his own created material models. This makes it possible to combine the CCM with some of the features offered by commercial finite element codes, such as impact modelling, blast modelling, heat transfer, etc. Given that the formulation of the CCM is based on the so-called softening curve, which relates stresses across the crack with crack opening displacements, the most consistent way to approach dynamic loading effects seems to be making the softening curve dependent on the crack opening velocity [27]. However, most experimental data about the effect of dynamic loadings on the different parameters of the
Theoretical and Applied Fracture Mechanics 96 (2018) 308–325
G. Morales-Alonso et al.
Nomenclature
ftd fts GF I N Nα(x) T u+ u− uα W ∼ W
Acronyms CCM DIF FOI GFEM SDA XFEM
Cohesive Crack Model Dynamic Increase Factor Swedish Defence Agency generalized finite element method strong discontinuity approach extended finite element method
Latin symbols 1 A+ A− b+ D f( ) fct
unity second order tensor part of the element that contains the solitary nodes part of the element that does not contain the solitary nodes vector that defines the contribution of the crack opening vector to the strain field cauchy elastic moduli fourth order tensor softening curve function concrete tensile strength
concrete tensile strength under high strain rates concrete tensile strength under static strain specific fracture energy fourth order identity tensor normal vector to the fracture surface shape function associated with node α traction vector between crack borders displacement field vector for A+ region displacement field vector for A− region displacements vector of node α crack opening vector normalized crack opening parameter
Greek symbols
ε̇ ε0̇ ɛa ɛc λ μ
strain rate threshold of static strain rate apparent strain tensor continuum strain tensor Lamé elastic constants Lamé elastic constants
The constitutive model presented is based on the combination of the CCM with the SDA, so it is possible to take into account the strain rate effect on the softening curve of concrete in an easy manner. The model has been programmed in a commercial finite element code [29] as a material user subroutine. The remainder of the paper is organized as follows. The second section is devoted to summarizing the basis of fracture mechanics of concrete from the perspective of the CCM, as well as the effect of the strain rates in the mechanical properties of concrete. In the third section, the constitutive model is presented, along with the main details about its implementation. In the fourth section, the model is validated, first by discarding possible mesh dependency and then, by confronting the numerical predictions achieved by the model with a series of quasistatic and dynamic tests taken from the literature. Finally, the fifth section presents a summary and conclusions from this research.
softening curve, such as the tensile strength or the specific fracture energy, characterize the loading velocity through the strain rate in the continuum [28] and not through the crack opening velocity (since this latter parameter is extremely difficult to measure). For this reason and as already discussed in the previous paragraph, CCM implementations based on the SDA like the one used in this paper are especially well suited for implementing dynamic effects via the strain rate calculated in the continuum. Focusing on the strain rate effect on the mechanical properties of concrete, most studies that can be found in the literature are based on the measurement of the Dynamic Increase Factor (DIF) as a function of the strain rate. The DIF is the ratio between a given material property under high strain rate and under static conditions. Several studies can be found in the literature about the DIF at different strain rates for the compressive strength and for the tensile strength. An extent compilation of DIFs for concrete can be found in [28]. However, the number of references that address the characterization of the DIF for the specific fracture energy is considerably lower. This paper presents a constitutive model for modeling the fracture of concrete and other quasi-brittle materials under both static and dynamic loadings. When compared with other publications with similar approach, the main contributions of this paper are: (i) 3D simulations with hexahedral single-integration-point elements are used (in contrast with Sancho et al. [25,26]), (ii) simulation of existing quasi-static experimental results and (iii) an increased number of highly dynamic experimental results simulated, finding a good match in both terms of cracking pattern and failure mode.
2. Fracture behavior of concrete under tensile stresses 2.1. The cohesive crack model The response of concrete under tension can be roughly considered as linear elastic under pure tension loading (mode I) until it reaches its tensile strength. When this occurs, damage appears in the material and the stresses that it can withstand are progressively reduced, as depicted in Fig. 1a. This phenomenon can be simulated through the CCM [3], according to which damage is assumed to concentrate in a discontinuity surface (crack) that is governed by the so-called softening curve
Fig. 1. (a) Fracture of concrete under tensile stresses. (b) Softening curve. 309
Theoretical and Applied Fracture Mechanics 96 (2018) 308–325
G. Morales-Alonso et al.
vector which defines the width and direction of the displacements jump (discontinuity). This crack opening vector is assumed to have a constant value along the discontinuity. As shown in Fig. 2, the discontinuity “cuts” the element into two and as a result, some nodes are moved away from the element. Such nodes are termed “solitary nodes”. From now on, we shall denote as A+ the element part separated away by the crack, which contains the solitary nodes, while A− region is the remaining part of the element. While the element is not fractured, the displacement jump vector w equals 0. When fracture takes place, a crack is inserted and the displacement field is decoupled into a continuous and a discontinuous part (see Fig. 2). In this decoupling, the discontinuous part is taken into account through the displacement jump vector, w, which allows for normal opening of the crack and a relative sliding between both parts A+ and A− of the element. The expression of w can be calculated as the subtraction between the displacement fields at both sides of the crack, A− and A+, which can be obtained as:
(Fig. 1b). This curve relates the mode I crack opening (w) and the stress transmitted across the crack sides (f(w)). Decohesion strength and fracture energy are the two most important properties for the definition of the softening curve [30]. Decohesion strength is defined as the stress at which the crack is created and starts to open, and in the case of concrete and other quasi-brittle materials, it is usually assumed to be equal to the tensile strength. Second, cohesive fracture energy is the external energy supply required to create and fully break a unit surface area of a cohesive crack, and coincides with the area under the softening function. Both properties can be experimentally measured through standard tests [31] for static loading conditions. 2.2. Strain rate effects However, when it comes to highly dynamic loadings, mechanical properties change. This effect is considerably important in the case of concrete and needs to be accounted for [32,33]. Both compressive and tensile strength of concrete have been reported to be enhanced under highly dynamic loading, when compared with static loading. This is attributed to the high strain rates imposed on a solid by high impulsive loads. Particularly, the blast waves generated in a detonation may induce strain rates between the order of 10 s−1 and 1000 s−1 [28,34]. The Dynamic Increase Factor (DIF) accounts for this effect, which is defined as the ratio between the dynamic and the static strengths. The DIF in the case of concrete may vary between 1.5 and 4 (or even higher) for both compression and tension [28,35,36]. A formulation for the DIF of concrete for tension was proposed by the CEB (1) [35]. f
ε ̇ 1.016δ
DIF= ftd = ⎡ ε ̇ ⎤ ts ⎣ 0⎦ f
ε̇
1 3
DIF= ftd = η ·⎡ ε ̇ ⎤ ts ⎣ 0⎦ where δ =
1 10 + 6·fcs
For A−: u− (x) = For A+: u+ (x) =
and η =
107.11·δ − 2.33 ,
Nα (x)·u α +
∑
∑
Nα (x)·(u α−w)
(2)
α ∈ A+
Nα (x)·u α +
α ∈ A+
∑
Nα (x)·(u α + w)
(3)
α ∈ A−
⎡ w = ‖u (x s)‖ = u+ (x)−u− (x) = ⎢ ∑ Nα (x s) + ⎣ α ∈ A−
∑ α ∈ A+
⎤ Nα (x s)⎥·w ⎦
⎡ ⎤ = ⎢ ∑ Nα (x s)⎥·w = w ∈ α A ⎣ ⎦
for ε ̇ ⩽ 30s−1 for ε ̇ > 30 s−1
∑ α ∈ A−
−
(4) −
where u and u define the displacement fields for the A and A+ regions, respectively, Nα(x) is the shape function associated with node α, uα is the displacements vector of node α and w is the displacement jump vector. Note that, in (4), the property of the shape functions has been applied according to which ∑α ∈ A Nα (x s) = 1. Eqs. (2) and (3) can be combined into a single equation depicting the displacement field of a strong discontinuity on finite elements as in [40]:
(1) ftd and fts are the dynamic and
static tensile strengths of concrete, respectively, ε ̇ is the actual strain rate and ε0̇ is the static strain rate, which is taken as 3 · 10−6 s−1. As can be seen in (1), the DIF depends on compressive and tensile strengths of concrete. But what about the fracture energy and the softening curve? How do they vary under high strain rates when compared with static loading? Regarding the effect of strain rate in the specific fracture energy of concrete, the literature shows a less developed state of the art, with scarce and sometimes contradictory conclusions. Some researchers have attempted to obtain the DIF for the fracture energy of concrete [37–39]. In spite of the limited experimental data available, it is agreed that the strain rates modify both fracture energy and tensile strength in a similar way.
u (x) =
∑
+
Nα (x)·u α + [H (x)−N+ (x)]·w
(5)
α∈A
N+ (x)
= ∑α ∈ A+ Nα (x) and H(x) is the Heaviside In expression (5), function that complies with H(x) = 0 if x ∈ A−; H(x) = 1 if x ∈ A+. As previously mentioned, α is the node index, Nα(x) is the shape function associated with node α, uα is the corresponding nodal displacement vector and w is the displacement jump vector. The continuum strain tensor can be obtained by taking the symmetric gradient to this equation, following a small strains approach. Hence,
3. Constitutive model
εc (x) = εa (x)−[b+ (x) ⊗ w]S
(6)
where ɛ is the continuum strain tensor and ɛ is the apparent strain tensor of the element, obtained from the total nodal displacements, whose form is: c
The model presented here combines the cohesive crack concept with the technique of embedded discontinuities. Therefore, cracks are simulated within the finite element mesh as discontinuities in the displacement field of the elements. Crack opening is ruled through a softening curve, in accordance with the cohesive crack model. For this reason crack propagation depends on the release of fracture energy, assumed to be a material property, which is especially convenient in terms of mesh objectivity. This approach allows calculating not only the response of the material, but also the cracking patterns, even when these are multiple.
a
3.1. Kinematics of strong discontinuity The kinematics of a strong discontinuity is described in Fig. 2, in which a strong discontinuity, defined by a plane given by its normal vector n, divides the element. In this figure, w is the crack opening
Fig. 2. Kinematics of strong discontinuity, where node 2 would be termed as “solitary node”. 310
Theoretical and Applied Fracture Mechanics 96 (2018) 308–325
G. Morales-Alonso et al.
εa (x) =
∑
[bα (x) ⊗ u α ]S
α∈A
b+ (x) =
∑
bα (x)
α ∈ A+
bα (x) = grad Nα (x)
(7) (8) (9)
The used S index in Eqs. (6) and (7) denotes the symmetric part of the tensor. It is worthy to highlight that the apparent strain tensor of the element (ɛa) represents the strain field that would have the element if no displacement discontinuity was present on it (see Fig. 2), given that for this term the strain field is obtained from taking the derivative to the shape functions directly applied to the nodal displacements. The b+ vector plays a paramount role, since it defines the contribution of the crack opening vector to the strain field. Its calculation is described in next section. 3.2. Obtaining n and b+ vectors
Fig. 4. Central forces model and softening curves with and without dynamic factor increase.
According to the classical approach of Rankine’s criterion, the orientation of the crack (n) is given by the direction of the maximum principal stress. For this reason, the crack orientation n is computed as the unit eigenvector associated with the maximum principal stress when the latter exceeds the tensile strength for the first time. On the other hand, the b+ vector enriches the apparent strain ɛa with the strains associated with the discontinuity, while ɛc accounts for its continuum components. As stated in Eqs. (8) and (9), this vector is calculated through the addition of gradients of the linear shape functions corresponding to the solitary nodes. These nodes are defined as those that are separated on one side of a new crack or discontinuity on an element, which is defined by its crack orientation vector n (see Fig. 2). However, given a crack orientation (defined by its normal vector n), there are multiple solitary nodes combinations, and therefore, several b+ possible vectors, as shown in Fig. 3, for a 2D case. Similarly to [25,26,41], in this research, the solitary nodes are chosen to be those in which the angle between vectors n and b+ is the smallest possible, which can be expressed as:
n ·b+ = max |b+|
the numerical simulation of mixed mode crack propagation in concrete under static loading. This behavior is formulated in expression (11) and graphically depicted in Fig. 3.
t=
∼) f (w ∼ ·w w
(11)
∼) stands for an equivalent crack opening value In this expression, (w ∼) is the softening defined as the maximum historical crack opening. f (w function that relates the stress across the crack with crack opening. It must be noted that expression (11) defines not only the traction vector, but also the material behavior when the cohesive crack is unloaded. Since the actual crack opening vector w is divided by the ∼) under crack unloads (w ∼) remaximum historical crack opening, (w mains constant while the crack opening vector diminishes. Therefore, ∼) according to Eq. (11), the norm of the traction vector t is equal to f (w but decreased by a factor |w|/|w|max. Fig. 5 shows the unloading behavior according to Eq. (11) for a mode I crack opening case.
(10)
3.4. Definition and solution of the constitutive equations Concrete can be roughly considered to behave as linear elastic when subjected to tensile loading. The stress state under these conditions is described by (12), where D is the Cauchy elastic moduli fourth order tensor, given by (13), εc is the continuum strain tensor of the element, 1 is the unity second order tensor, I is the fourth order identity tensor and λ and μ are the Lamé elastic constants:
3.3. Crack behavior and orientation The CCM as defined by Hillerborg and his coworkers [3] was initially formulated for mode I crack opening. In the model described here, the crack may open in any combination of modes (I, II or III) and therefore the constitutive behavior of the crack must account for it. To this end, the traction vector t between crack lips is assumed to be parallel to the direction of the crack opening vector, w. This formulation was originally proposed by Costanzo [42] as a necessary and sufficient condition in order to guarantee the frame independency of a cohesive crack model. It was later adopted by Sancho et al. [25,26] for
σ = D·εc = D·εa
(12)
D = λ·1 ⊗ 1 + 2·μ·I
(13)
As shown in Eq. (12), before a crack is generated inside the element, the apparent strain and the continuum strain are equal. Once the
Fig. 3. Different solitary nodes (highlighted in red) combinations for the same crack orientation n. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 311
Theoretical and Applied Fracture Mechanics 96 (2018) 308–325
G. Morales-Alonso et al.
Fig. 5. Examples of softening functions.
Fig. 6. Benchmark test used for the mesh sensitivity analysis.
complex formulation that would require for further equations to reach a solution. For the 3D calculations presented in this paper, one integration point brick elements have been used (see [29]). When compared to previous implementations in which tetrahedrons and triangular elements were used, the use of hexahedral finite elements offers two major advantages: (i) the existence of more possible combinations of solitary nodes, which allows for a better description of the crack kinematics, and (ii) a savings in computation expenses when a structured mesh is used, since in this case nodal coordinates are not required for the b+ vector computation. The only unknown in Eq. (15) is the crack displacement vector w, which needs to be obtained numerically. The strategy for solving this equation is different depending on whether the element is being loaded or unloaded. For both loading and unloading cases, the right-hand side of Eq. (15) can be re-written as
maximum principal stress exceeds the tensile strength of the material, a crack is inserted. The stress tensor in such an element is given by Eq. (14), where the elastic moduli tensor is applied to the continuum strain, which now is obtained by subtracting the crack contribution to the apparent strain, according to (6).
σ = D·[εa −[b+ ⊗ w]S ]
(14)
In (14), the superscript S denotes the symmetric part of the tensor. However, stresses in the crack are governed by the cohesive behavior and therefore, tractions in the discontinuity are given by (11). Since the stress field in the continuum must satisfy equilibrium with tractions at the crack lips, Eq. (15) turns out:
∼) f (w a + S ∼ ·w = D·[ε −[b ⊗ w] ]·n w
(15)
Following previous approaches [25,26,43,44], in this implementation, constant stress (single integration point) elements have been used. Therefore, expression (15) is valid for the whole element. Moreover, the crack opening vector w is also constant along the element. Otherwise the apparent strain tensor ɛa in expression (15) would be no longer constant and consequently the crack opening vector w should be also variable along the element. This would result in a considerably more
[D·εa −D·[b+ ⊗ w]S ]·n = D·εa ·n−D ·[b+ ⊗ w]S ·n = t a−M·w
(16)
where ta is the apparent (elastic) traction vector and M is a matrix that relates the jump in displacements to the traction that must be substracted from the continuum one. Expression (15) calls
312
Theoretical and Applied Fracture Mechanics 96 (2018) 308–325
G. Morales-Alonso et al.
Fig. 7. Meshes for the sensitivity analysis.
Fig. 8. Mesh sensitivity analysis results.
∼) f (w a ∼ ·w = t −M·w w
solver are summarized in Appendix A. On the other hand, when ẇ < 0 , the governing equation is also (17), ∼ is known, and therefore the vector w can but in this case, the value of w be isolated, as in Eq. (18).
(17)
For the case in which ẇ > 0 , the w vector cannot be isolated, since it is implicitly in. For this reason, the Newton-Raphson method for nonlinear solving of equations is used. The equations developed for this 313
Theoretical and Applied Fracture Mechanics 96 (2018) 308–325
G. Morales-Alonso et al.
recalculated at each simulation step while the norm of the crack opening vector w does not exceed a user-defined value (wadm), that is to say, while a certain fraction of the total fracture energy of the element has not been consumed. Therefore while the crack is adapting (w < wadm) in a given element, the material is treated as linear elastic, ignoring that it was cracked at the previous increment. The n and the b+ vectors are freshly calculated from the apparent total strain, and a new crack opening w is calculated. This is possible since Eq. (16) is not formulated in an incremental way, providing the total crack opening vector w as a function of the total apparent strain ɛa, without depending on the crack opening values calculated in previous increments. Once the crack opening exceeds the predefined value (w > wadm), the n and its corresponding b+ vectors are frozen for the rest of the simulation. In summary, the crack is allowed to rotate while a certain userdefined amount of fracture energy has not been released. It must be noted that in the classical approach of the smeared crack the rotating crack concept implies recalculation of the n vector, while the adapting crack concept in the embedded crack approach implies recalculation of both the n and the b+ vectors.
Fig. 9. Test setting by Gálvez et al. [50,51], measurements in mm. −1 ∼) f (w w = t a ·⎛ ∼ ·I + M⎞ ⎝ w ⎠ ⎜
⎟
(18)
3.5. Crack adaptation 3.6. Implementation of strain rate effects The selection of the solitary nodes takes place locally at every cracked finite element, with no crack continuity being enforced between adjacent elements. That is to say, the proposed algorithm is strictly local, without a previously defined crack path or enforcing global crack continuity between adjacent elements. The consequence of this approach is that the tip of the crack must find its way through the mesh. However, the spurious stress oscillations caused by stress waves in explicit calculations and poor directional resolution given the limited combinations of b+ may lead to significantly different maximum principal stress directions between adjacent elements on the onset of cracking. This may lead to crack locking due to a crack path topology which is kinematically non-compatible. To solve this problem, the approach of the adapting crack, first introduced by Sancho et al. [25,26], is adopted here. The rationale behind this adaptation algorithm is that vectors n and subsequently b+ are
As described in Section 2.2, the softening behavior of concrete under high dynamic loading is different from that under static loading. Hence, some considerations should be made to simulate it properly. The most usual way of taking into account the effects of strain rates is by affecting material properties by the so-called Dynamic Increase Factor (DIF). In this research in particular, in view of the lack of concluding experimental data about the DIF for the fracture energy (see Section 2.2.), both the tensile strength and the fracture energy are affected by the same DIF, in line with the experimental results from [38]. The approach followed consists of multiplying the whole softening curve by the DIF, as depicted in Fig. 4. The value of ε ̇ is evaluated as the average maximum principal strain rate in the element before the crack is created.
Fig. 10. Results of test 1 by Gálvez et al. [50,51]. 314
Theoretical and Applied Fracture Mechanics 96 (2018) 308–325
G. Morales-Alonso et al.
Fig. 11. Results of test 2 by Gálvez et al. [50,51]. Fig. 12. Crack pattern on the simulation developed for type 1 test.
Fig. 13. Crack pattern on the simulation developed for type 2 test.
3.7. Explicit vs. implicit finite element solvers
view of the results shown in the following sections, the use of explicit calculations for some static problems can be a sound approach in some cases. It must be recalled that explicit codes do not suffer from convergence problems, which are a classical challenge in crack propagation models for implicit codes.
The implementation described in this paper is primarily intended for the numerical simulation of concrete and other quasi-brittle materials when subjected to impulsive loads such as blast or impacts. These are events in which the material may fail before structural equilibrium is reached. In this situation, finite element solvers with explicit time integration are preferred. For this reason, the explicit finite element solver LS-DYNA has been used in this research. On the other hand, for the simulation of quasi-static problems, in which structural equilibrium is assured through the loading process, implicitly integrated finite elements codes are commonly used. Nevertheless, explicit codes may also be used if the loading rate is carefully selected, avoiding inertial effects. This is the strategy used in the quasi-static simulations presented in this paper, in which the DIF is set equal to 1. It is possible to calculate the stiffness tangent matrix and to implement the model in an implicit code. However, the main interest of this research is the response of the material under impulsive loadings and, as mentioned before, only the explicit solver has been chosen. In
4. Validation 4.1. Choice of basic model parameters As mentioned, in the Cohesive Crack Model, the cohesive crack introduces a softening function that simulates damage process of the material in the fracture process zone. The relation between the stress transferred and crack opening is given by the softening function, which is considered to be a material property independent of geometry and size. Several formulations have been proposed for the softening curve, such as the rectangular, the linear [3], the bilinear [45], the exponential [46], the General Bilinear Fit (GBF) [47] and the Extra Long Tail (ELT) [48]. Five different softening curves [49] are depicted in 315
Theoretical and Applied Fracture Mechanics 96 (2018) 308–325
G. Morales-Alonso et al.
Fig. 14. Prediction of crack trajectories for type tests 1 and 2 by Gálvez et al. [50,51].
Fig. 5 for the same value of tensile strength and fracture energy. As can be seen, one of the biggest differences is how fast the fracture energy is released as the crack opens. The fastest energy release occurs with the rectangular curve, while the slowest takes place in the ELT curve, according to the steepness of the first part of the curve. Although several shapes for the softening curve have been proposed, in this work the exponential softening curve will be used because of its simplicity and the continuity of its derivative. In addition, its definition under quasi-static conditions only needs two material parameters that can be easily estimated with laboratory tests [31], namely, the tensile strength (fct) and the fracture energy (GF) of concrete. The exponential softening curve is defined through expression [49] Fig. 15. Test setting by Schlangen and van Mier [52], measurements in mm.
f
∼) = f ·e− GctF ·w∼ f (w ct
Fig. 16. Results obtained for the test by Schlangen and van Mier [52]. 316
(19)
Theoretical and Applied Fracture Mechanics 96 (2018) 308–325
G. Morales-Alonso et al.
Fig. 17. Crack pattern on the simulation developed.
material properties, corresponding to a conventional concrete, has been set for the analysis: Young's modulus E = 20GPa, Poisson's ratio ν = 0.15, specific fracture energy GF = 100 N/m and tensile strength fct = 2.5 MPa. An exponential softening curve was been chosen. Being a quasi-static problem, the DIF was set equal to 1. The abovementioned material properties lead to a characteristic length of lch = EGF/fct2 = 0.32 m. Three different mesh sizes have been used. The number of elements are 1250, 5000 and 20,000, which correspond to element lengths of 28.17, 14.18 and 7.02 mm, respectively (see Fig. 7). It can be observed how these three element sizes are far below the characteristic length and hence, any of them should adequately reproduce the fracture process zone. The simulations were run assuming plane strain conditions; hence, only one row of elements through the thickness has been modeled, constraining the displacements out of the specimen plane. The comparison of results in terms of load-displacement is depicted in Fig. 8. The oscillation exhibited by the curves is due to the explicit nature of the finite element solver. Besides obtaining similar results with the three element sizes, the areas contained below the P-δ plot for each of the mesh sizes are 91.1%, 92.5% and 106% respectively of the theoretical value corresponding to the specific fracture energy multiplied by the specimen ligament. These results reveal the low mesh sensitivity exhibited by the model.
Fig. 18. Prediction of crack trajectories for the test by Schlangen and van Mier [52].
4.2. Mesh sensitivity
4.3. Quasi-static tests
Material models based on the release of an specific fracture energy, such as the Cohesive Crack Model, have the capability of providing objective solutions regarding the change of the mesh size [40] and therefore, a low mesh sensitivity can be beforehand expected from this model. However, numerical simulation of materials exhibiting strength softening are prone to exhibit mesh dependency to some extent. For this reason, in order to assess the reliability of the model proposed here, a mesh sensitivity study has been conducted, using a three-point bending test benchmark under quasi-static conditions (see Fig. 6). The following
4.3.1. Tests of Gálvez et al. [50,51] An experimental campaign on concrete beams of different sizes subjected to mixed mode fracture was presented by Gálvez and his coworkers in [50,51]. Two different test setups were presented: threepoint bending tests (type 1) and four-point bending tests (type 2) (see Fig. 9). Both test setups were applied to three different specimen sizes, with depths of 75 mm, 150 mm and 300 mm. The mechanical properties, according to the authors, were fct = 3.0 MPa, GF = 69 N/m and E = 38 GPa.
Fig. 19. Reinforced slab testing device as reported by Thiagarajan et al. [59]. 317
Theoretical and Applied Fracture Mechanics 96 (2018) 308–325
G. Morales-Alonso et al.
Fig. 20. Reflected pressure histories measured on the experiments as reported by Thiagarajan et al. [59].
Table 1 Mechanical properties of concrete used in the simulations. Mechanical property 3
Density [kg/m ] Young’s modulus [GPa] Poisson’s ratio [–] Tensile strength [MPa] Fracture energy [N·m] Softening curve Yield stress [MPa] Tangent Young’s modulus [GPa]
NSC
HSC
2350 33.33 0.20* 4.00* 100* Exponential – –
2368 40.78 0.26 6.00* 120* Exponential – –
Table 2 Mechanical properties for steel used in the simulations. Mechanical property
NSS
HSS
Density [kg/m3] Young’s modulus [GPa] Poisson’s ratio [–] Tangent Young’s modulus [GPa] Input constant A for strain hardening [GPa] Input constant B for strain hardening [GPa] Input constant n for strain hardening Input constant C for strain rate hardening Input constant m for thermal softening
7850 200 0.30* 4.25 0.482 0.00 0.01 0.025 0.00
7850 200 0.30* 3.20 0.565 0.00 0.01 0.025 0.00
From this experimental campaign, we have chosen the intermediate size specimen (depth 150 mm, as shown in Fig. 9) to check the model. The simulations have been conducted under plain strain conditions with a 5000 elements mesh of 4.47 mm of element size. The load was applied by prescribing a displacement in the load point at a speed low enough to prevent the generation of inertial effects in the material. The same mechanical properties measured by the authors of the test were used, plus an exponential softening curve. Both test types 1 and 2 have been simulated; the results obtained are presented in Figs. 10–14. In these figures, the results of the numerical predictions are shown together with the experimental scatter and the numerical predictions originally obtained by Gálvez et al. [50,51]. A good level of prediction is obtained with the model presented here. The availability of a detailed record of the crack trajectories
Fig. 21. View of the finite element model.
obtained in the experimental tests of Gálvez et al. [50,51] allows for a quantitative comparison of the crack trajectories obtained in the simulation (Fig. 14), where a good concordance is found. 4.3.2. Test of Schlangen and van Mier [52] The second quasi-static test modeled here corresponds to the singleedge notched beam reported by Schlangen and van Mier [52], subjected to four point shear, as shown in Fig. 15. Material properties of the concrete reported by the authors of the test were fct = 2.8 MPa, 318
Theoretical and Applied Fracture Mechanics 96 (2018) 308–325
G. Morales-Alonso et al.
Fig. 22. Deflection histories of the central point of the slab for sets 1a and 1b.
Fig. 23. Deflection histories of the central point of the slab for sets 2a and 2b.
4.4. Dynamic tests
GF = 100 N/m and E = 35 GPa. The simulation was conducted under plain strain condition with a 20,000 elements mesh of 1.41 mm of element size and a load application speed low enough to prevent the generation of inertial effects in the material. As in the previously presented simulations, the load application was simulated with an imposed displacement in the point where the load was applied, and the DIF was set equal to 1. Again, the same mechanical properties measured by the authors of the test were used, plus an exponential softening curve. Results obtained are depicted in Figs. 16–18. While the result obtained in terms of crack trajectory is average, probably due to the impossibility for reproducing a crack initiating in the corner of the notch, a remarkable level of agreement has been achieved in terms of Load-CMOD.
Although the material model presented only accounts for failure under tension, it is suitable for modeling highly dynamic problems, since it has been proved that in concrete structures subjected to such events, most structural members exhibit tensile and/or shear failures [53–58]. 4.4.1. Tests of Thiagarajan et al. [59] The experimental program conducted on reinforced concrete slabs tested under an explosion load in a shock tube by Thiagarajan et al. [59] has been used to test the ability of the developed model to predict deflection and damage of concrete elements subjected to blast loading (see Fig. 19). 319
Theoretical and Applied Fracture Mechanics 96 (2018) 308–325
G. Morales-Alonso et al.
Fig. 24. Comparison of cracking pattern and simulation result of the slab for set 1a.
Fig. 26. Schematic depiction of the test as reported by Magnusson et al. [53,54]. Table 3 Mechanical properties and reinforcement amounts.
Reinforcement amount Concrete uniaxial compressive strength [MPa] Concrete tensile strength [MPa] Concrete elastic modulus [MPa] Steel yield stress [MPa] Steel elastic modulus [MPa]
Fig. 25. Comparison of cracking pattern and simulation result of the slab for set 2a.
B100-D3(12)
B100-D3(16)
B200-D1
4ϕ12 101
5ϕ16 101
5ϕ16 204
5.80 41,700 580 203,000
5.80 41,700 580 203,000
9.90 74,500 580 203,000
material parameters for the Johnson-Cook model were the ones summarized in Table 2, taken from the report provided by the authors of the tests. The results obtained, expressed as the deflection in the center of the slab with respect to time, can be seen in Fig. 22 for set 1 and Fig. 23 for set 2. As shown in Figs. 22–25, both the global response of the slabs and the crack patterns predicted by the model fit the experimental observations, with a spread cracking on their rear faces and rebound of the slabs due to the elastic energy stored by the reinforcing bars.
Out of the tests conducted, four of them (termed 1a, 1b, 2a and 2b by the authors) have been used in this research. In this nomenclature, 1 stands for normal concrete with steel rebars and 2 for high strength concrete with steel rebars, while a and b stand for different pressure actions on the slab. The recorded reflected pressure histories for all the sets are shown in Fig. 20. The dimensions of the slabs were 1625 × 857 × 101 mm (64 × 33.75 × 4 in.) and the reinforcement was #3 bars (9.5 mm in diameter). Material properties of concrete and steel are summarized in Tables 1 and 2, respectively, where values marked with ∗ have been estimated with the use of Model Code 2010 from the data given by Thiagarajan et al. [59]. The model used in the simulation (see Fig. 21) included both the slab and the steel frame that acted as a supporting frame. Concrete was simulated with solid 3D elements, amounting to 548,864 elements, whereas truss elements were used for steel bars (2890 beam elements). The average element size was 6.35 mm (0.25 in). Interaction between concrete and steel reinforcement elements was set as a perfect bond by sharing nodes between the truss elements of the rebars and the solid elements of concrete. The blast load was modeled through a distributed pressure applied on the side of the slabs facing the pressure wave front, with the pressure histories given in Fig. 20. Regarding constitutive models, the constitutive model presented here was used for concrete, while the Johnson-Cook model [60] was used for the steel rebars. The
4.4.2. Tests of Magnusson et al. (2010) Lastly, the shock tube tests conducted at the FOI (Swedish Defence Agency) by Magnusson and coworkers between 1997 and 2000 have been simulated [53,54]. Concrete beams were placed vertically and simply supported at one end of the tube, while the explosive was detonated at a stand-off distance of 10 m (see Fig. 26). Tested beams were 1.70 m long, with cross-sectional dimensions of 0.30 × 0.16 m (width × height). Concrete types and reinforcement ratios were varied, but for the present simulations, just three beam configurations were selected (see Table 3). Similar to the simulations presented in Section 4.4.1, solid elements were used for concrete (mesh size 5 mm, total number of elements = 350,000), while truss elements were used for reinforcement, in a total number of 900. The blast load was applied again as a distributed pressure history over the side of the beam facing the blast wave, where the overpressures have been taken from the original source [53,54], see 320
Theoretical and Applied Fracture Mechanics 96 (2018) 308–325
G. Morales-Alonso et al.
Fig. 27. Reflected pressure histories measured on the experiments as reported by Magnusson et al. [53,54].
Fig. 28. Bonding law provided by the Model Code 2010 [62].
of the beam supports. For this reason, the results have been filtered in order to have them smoothed. By doing so, it can be seen that the average numerical result fits well with the experimental records, in spite of not having an experimental envelope to compare, but rather a single-result experimental test.
Fig. 27. In this case, attempting to take into account all possible failure modes, bonding between concrete and steel was modeled by setting non-linear springs that link the nodes of reinforcing steel and concrete, similar to [61]. The response of the springs follows the tangential bonding stress law provided by Model Code 2010 [62] (see Fig. 28). The three beams chosen for validating the model were selected because they presented different failure modes when subjected to the shock tube test, that is, B100-D3(12) failed in a bending mode, whereas B100-D3(16) and B200-D1 failed in a shear mode. Results obtained are presented in Figs. 29–31, in which the mid-span deflection is plotted against the reactions on the support of the beams (see Figs. 32–34). The results obtained show good agreement with the failure modes, the cracking patterns and the support reaction – deflection curves. In the reaction forces-deflection curves, the raw numerical results numerical results exhibit large load oscillations due to the elastic behavior
5. Summary, conclusions and limitations A constitutive model for simulating fracture of concrete and other quasi-brittle materials has been presented in this paper. It is based on the Cohesive Crack Model and follows the Strong Discontinuity Approach. It has been implemented in LS-DYNA as a user subroutine, and allows for predicting the progress of fracture processes under different strain rates. A brief overview of the simulation of fracture with the CCM has been provided and the numerical implementation of the 321
Theoretical and Applied Fracture Mechanics 96 (2018) 308–325
G. Morales-Alonso et al.
Fig. 29. Plot of support reactions versus mid-span deflection for the beam B100-D3(12).
Fig. 30. Plot of support reactions versus mid-span deflection for the beam B100-D3(16).
simulated. On the other hand, two blast tests on reinforced concrete elements have been referred to and simulated. Good accuracy is found in load displacement curves, crack trajectories and progress and fracture typologies. Therefore, the constitutive model presented has proved its ability to simulate the behavior of concrete in both quasi-static and dynamic regimes. However, this study is not without limitations. The first and most relevant, the constitutive model implemented does not account for failure under compression. Hence, it is not valid for simulating ballistic impact nor other situations in which failure of concrete occurs due to crushing (e.g. explosions in the very close range or very rigid structural members). The second limitation is related to the importance of size effect. That is, when structural elements of much greater size than finite
model has been thoroughly presented. Special attention has been paid to the formation and progress of the cracking process, and the introduction of the Dynamic Increase Factor in order to account for different material responses under different strain rates. In relation to the former, the crack adaptation that allows the progression of the crack without its locking has been discussed. Regarding the latter, a novel strategy for introducing the DIF in the softening curve has been presented. In this sense, the existing experimental results of how fracture energy is affected by different strain rates have been accounted for. Lastly, validation examples for both quasi-static and dynamic tests have been presented. On one hand, two different tests of concrete specimens subjected to mixed mode under static conditions have been 322
Theoretical and Applied Fracture Mechanics 96 (2018) 308–325
G. Morales-Alonso et al.
Fig. 31. Plot of support reactions versus mid-span deflection for the beam B200-D1.
Fig. 32. Crack pattern predicted by the numerical simulation for the beam B100-D3(12) and comparison with experimental result.
Fig. 33. Crack pattern predicted by the numerical simulation for the beam B100-D3(16) and comparison with experimental result.
element mesh size in use, the cohesive crack approach is cost ineffective when compared to a more simple failure criterion, such as erosion under maximum principal stress. The third and last limitation is of a
technological type: one-single point integration elements have been used. In this element type, hourglassing effects are prone to appear, which can be controlled with specific algorithms. In the presented 323
Theoretical and Applied Fracture Mechanics 96 (2018) 308–325
G. Morales-Alonso et al.
Fig. 34. Crack pattern predicted by the numerical simulation for the beam B200-D1 and comparison with experimental result.
simulations, hourglassing has not been detected, and for this reason no hourglassing control has been introduced.
and the Spanish Ministry of Economy and Innovation [grant number BIA2014-54916-R].
Acknowledgements This work was supported by the Fundación Agustín de Betancourt Appendix A. Newton-Raphson method for solving the loading case Starting from (17): ∼) f (w g (w ) = ∼ ·w−t a + M ·w = 0 w
(A.1)
In step n + 1:
g (wn + 1) = g (wn ) +
∂g δw = 0 ∂w
(A.2)
And then
∂g δw = g (wn + 1)−g (wn ) ∂w
(A.3)
The first term in (A.3) can be calculated, introducing (A.1) as: ∂g ∂w
=
∂ ∂w
(
∼) f (w a ∼ · w−t w
=M + w ⊗ =M + ⎡ ⎣
∂ ⎡ ∂w ⎣
)
+ M ·w =
∼) f (w ∼ w
( ) ⎤⎦ +
∼)·w ∼ − f (w ∼) f ̇ (w w ⎤· w∼ ∼2 w
⎦
∂ ∂w
(
∼) ∂w f (w ∼ · ∂w w
∼) f (w ∼ ·w w
) + M= ∂
= M + w ⊗ ⎡ ∂w∼ ⎣
∼) f (w ∼ w
( )·
∼ ∂w ⎤ ∂w ⎦
+
∼) f (w ∼ ·I = w
∼) f (w ∼ ·I w
⊗ w+
(A.4)
Introducing (A.4) in (A.3): ̇ ∼ ∼
∼
⎡M + ⎡ f (w )·w∼2− f (w ) ⎤· w ∼ ⊗ w + w ⎣ ⎦ w ⎣ J ·δw =
∼) f (w 0−⎡ w∼ n ·wn−t a
⎣
n
where J = M + Therefore:
∼) f (w ∼ · I⎤ · δw w
= g (wn + 1)−g (wn )
+ M ·wn⎤ ⎦
∼)·w ∼ − f (w ∼) f ̇ (w w ⎡ ⎤· w∼ ∼2 w
⎣
⎦
⎦
⊗ w+
(A.5) ∼) f (w ∼ ·I w
∼) f (w n δw = J −1·⎡t a− ∼ ·wn−M ·wn⎤ = J −1·Res ⎥ ⎢ w n ⎦ ⎣
(A.6)
where Res is the residuum vector, whose module is evaluated on every iteration of the Newton – Raphson method and used as a tolerance measure to determine when the convergence is reached.
70121-2. [3] A. Hillerborg, M. Modéer, P. Petersson, Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements, Cem. Concr. Res. 6 (1976) 773–782. [4] M. Elices, G.V. Guinea, F.J. Gómez, J. Planas, The cohesive zone model: advantages, limitations and challenges, Eng. Fract. Mech. 69 (2) (2002) 137–163. [5] S. Guzmán, J.C. Gálvez, J.M. Sancho, Cover cracking of reinforced concrete due to
References [1] D.S. Dugdale, Yielding of steel sheets containing slits, J. Mech. Phys. Solids 8 (2) (1960) 100–104, http://dx.doi.org/10.1016/0022-5096(60)90013-2. [2] G.I. Barenblatt, The mathematical theory of equilibrium cracks in brittle fracture, Adv. Appl. Mech. 7 (1962) 55–129, http://dx.doi.org/10.1016/S0065-2156(08)
324
Theoretical and Applied Fracture Mechanics 96 (2018) 308–325
G. Morales-Alonso et al.
[6] [7]
[8]
[9] [10]
[11]
[12]
[13]
[14] [15]
[16] [17] [18] [19] [20] [21]
[22]
[23] [24]
[25]
[26]
[27]
[28] [29] [30]
[31]
[32] [33] [34]
[35] Comité Euro-International du Béton, Concrete structures under impact and impulsive loading, Bulletin d' Information 187, 1988. [36] C.A. Ross, J.W. Tedesco, S.T. Kuennen, Effects of strain rate on concrete strength, Mater. J. 92 (1) (1995) 37–47. [37] H. Schuler, C. Mayrhofer, K. Thoma, Spall experiments for the measurement of the tensile strength and fracture energy of concrete at high strain rates, Int. J. Impact Eng. 32 (10) (2006) 1635–1650. [38] J. Weerheijm, J.C.A.M. Van Doormaal, Tensile failure of concrete at high loading rates: new test data on strength and fracture energy from instrumented spalling tests, Int. J. Impact Eng. 34 (3) (2007) 609–626. [39] X.X. Zhang, G. Ruiz, R.C. Yu, M. Tarifa, Fracture behaviour of high-strength concrete at a wide range of loading rates, Int. J. Impact Eng. 36 (10) (2009) 1204–1209. [40] J. Oliver, Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. Part 1: Fundamentals, Int. J. Numer. Meth. Eng. 39 (21) (1996) 3575–3600, http://dx.doi.org/10.1002/(SICI)1097. [41] R.I. Borja, A finite element model for strain localization analysis of strongly discontinuous fields based on standard galerkin approximation, Comput. Meth. Appl. Mech. Eng. 190 (11) (2000) 1529–1549. [42] F. Costanzo, A continuum theory of cohesive zone models: deformation and constitutive equations, Int. J. Eng. Sci. 36 (15) (1998) 1763–1792. [43] C. Feist, G. Hofstetter, An embedded strong discontinuity model for cracking of plain concrete, Comput. Meth. Appl. Mech. Eng. 195 (52) (2006) 7115–7138. [44] J. Alfaiate, G.N. Wells, L.J. Sluys, On the use of embedded discontinuity elements with crack path continuity for mode-I and mixed-mode fracture, Eng. Fracture Mech. 69 (6) (2002) 661–686. [45] P.E. Petersson, Crack growth and development of fracture zones in plain concrete and similar materials (Ph.D. thesis), Lund Institute of Technology, 1981. [46] J. Planas, M. Elices, Towards a measure of Gf: An analysis of experimental results, in: F.H. Wittmann (Ed.), Fracture Toughness and Fracture Energy of Concrete, Elsevier, Amsterdam, 1986. [47] G.V. Guinea, J. Planas, M. Elices, A general bilinear fit for the softening curve of concrete, Mater. Struct. 27 (2) (1994) 99–105. [48] J. Planas, M. Elices, Shrinkage eigenstress and structural size-effect, in: Z.P. Bazant (Ed.), Fracture of Concrete Structures, Elsevier, London, 1992, pp. 939–951. [49] Z.P. Bazant, J. Planas, Fracture and size effect in concrete and other quasibrittle materials, CRC Press, 1998. [50] J.C. Gálvez, M. Elices, G.V. Guinea, J. Planas, Mixed mode fracture of concrete under proportional and nonproportional loading, Int. J. Fract. 94 (1998) 267–284. [51] J.C. Gálvez, D.A. Cendón, J. Planas, G.V. Guinea, M. Elices, Fracture of Concrete under Mixed Loading. Experimental Results and Numerical Prediction, in: H. Mihashi, K. Rokugo (Eds.), Fracture Mechanics of Concrete Structures, AEDIFICATIO Publ., 1998, pp. 729–738. [52] E. Schlangen, J.G. van Mier, Mixed-mode fracture propagation: a combined numerical and experimental study. In: Rossmanith, H.P. (Ed.), Fracture and Damage of Concrete and Rock - FDCR-2, 166–75, 1993. [53] J. Magnusson, M. Hallgren, A. Ansell, Air-blast-loaded, high-strength concrete beams Part I: Experimental investigation, Mag. Concr. Res. 62 (2) (2010) 127–136. [54] J. Magnusson, A. Ansell, H. Hansson, Air-blast-loaded, high-strength concrete beams. Part II: Numerical non-linear analysis, Mag. Concr. Res. 62 (4) (2010) 235–242. [55] G. Morales-Alonso, D.A. Cendón, F. Gálvez, B. Erice, V. Sánchez-Gálvez, Blast response analysis of reinforced concrete slabs: experimental procedure and numerical simulation, J. Appl. Mech. 78 (5) (2011) 051010. [56] M. Oña, G. Morales-Alonso, F. Gálvez, V. Sánchez-Gálvez, D.A. Cendón, Analysis of concrete targets with different kinds of reinforcements subjected to blast loading, Eur. Phys. J. Special Topics 225 (2) (2016) 265–282. [57] G. Morales-Alonso, D.A. Cendón, F. Gálvez, V. Sánchez-Gálvez, Influence of the softening curve in the fracture patterns of concrete slabs subjected to blast, Eng. Fract. Mech. 140 (2015) 1–16. [58] G. Morales-Alonso, Experimental and numerical analysis of reinforced concrete elements subjected to blast loading (Ph.D. thesis). Universidad Politécnica de Madrid, 2013. [59] G. Thiagarajan, A.V. Kadambi, S. Robert, C.F. Johnson, Experimental and finite element analysis of doubly reinforced concrete slabs subjected to blast loads, Int. J. Impact Eng. 75 (2015) 162–173. [60] G.R. Johnson, W.H. Cook, A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures, in: Proceedings of the 7th International Symposium on Ballistics, vol. 21(1), 1983, pp. 541–547. [61] G. Ruiz, J.R. Carmona, D.A. Cendón, Propagation of a cohesive crack through adherent reinforcement layers, Comput. Meth. Appl. Mech. Eng. 195 (52) (2006) 7237–7248. [62] Comité Euro-International du Béton, Model Code 2010, 2012.
rebar corrosion induced by chloride penetration, Cem. Concr. Res. 41 (8) (2011) 893–902. B. Sanz, J. Planas, J.M. Sancho, Influence of corrosion rate on the mechanical interaction of reinforcing steel, oxide and concrete, Mater. Struct. 50 (4) (2017) 195. C. Rocco, G.V. Guinea, J. Planas, M. Elices, Review of the splitting-test standards from a fracture mechanics point of view, Cem. Concr. Res. 31 (1) (2001) 73–82, http://dx.doi.org/10.1016/S0008-8846(00)00425-7. A. Enfedaque, M.G. Alberti, J.C. Gálvez, J. Domingo, Numerical simulation of the fracture behaviour of glass fibre reinforced cement, Constr. Build. Mater. 136 (2017) 108–117. K. Park, G.H. Paulino, J. Roesler, Cohesive fracture model for functionally graded fiber reinforced concrete, Cem. Concr. Res. 40 (6) (2010) 956–965. A. García-Collado, J.M. Vasco-Olmo, F.A. Díaz, Numerical analysis of plasticity induced crack closure based on an irreversible cohesive zone model, Theor. Appl. Fract. Mech. 89 (2017) 52–62. T. Saksala, D. Brancherie, A. Ibrahimbegovic, Numerical modeling of dynamic rock fracture with a combined 3D continuum viscodamage-embedded discontinuity model, Int. J. Numer. Anal. Meth. Geomech. 40 (9) (2016) 1339–1357. D.A. Cendón, N. Jin, Y. Liu, F. Berto, M. Elices, Numerical assessment of gray cast iron notched specimens by using a triaxiality-dependent cohesive zone model, Theor. Appl. Fract. Mech. 90 (2017) 259–267. Y.L. Liu, D.A. Cendón, P.W. Chen, K.D. Dai, Fracture of PBX notched specimens: Experimental research and numerical prediction, Theor. Appl. Fract. Mech. 90 (2017) 268–275. W.H. Gerstle, M. Xie, FEM modeling of fictitious crack propagation in concrete, ASCE J. Eng. Mech. 118 (1992) 416–434. D.A. Cendón, J.C. Gálvez, M. Elices, J. Planas, Modelling the fracture of concrete under mixed loading, Int. J. Fract. 103 (3) (2000) 293–310, http://dx.doi.org/10. 1023/A:1007687025575. C. Koenke, R. Harte, W.B. Krätzig, O. Rosenstein, On adaptive remeshing techniques for crack simulation problems, Eng. Comput. 15 (1) (1998) 74–88. D. Colombo, M. Giglio, A methodology for automatic crack propagation modelling in planar and shell FE models, Eng. Fract. Mech. 73 (4) (2006) 490–504. G.T. Camacho, M. Ortiz, Computational modelling of impact damage in brittle materials, Int. J. Solids Struct. 33 (20–22) (1996) 2899–2938. M. Melenk, On generalized Finite element method (Ph.D. thesis). University of Maryland, 1995. T. Belytschko, T. Black, Elastic crack growth in finite elements with minimal remeshing, Int. J. Numer. Meth. Eng. 45 (1999) 601–620. J.C. Simo, J. Oliver, F. Armero, An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids, Comput. Mech. 12 (5) (1993) 277–296. J. Oliver, Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. Part 1: Fundamentals, Int. J. Numer. Meth. Eng. 39 (21) (1996) 3575–3600. M. Jirásek, Comparative study on finite elements with embedded discontinuities, Comput. Meth. Appl. Mech. Eng. 188 (1) (2000) 307–330. B. Sanz, J. Planas, J.M. Sancho, An experimental and numerical study of the pattern of cracking of concrete due to steel reinforcement corrosion, Eng. Fract. Mech. 114 (2013) 26–41. J.M. Sancho, J. Planas, D.A. Cendón, E. Reyes, J.C. Gálvez, An embedded crack model for finite element analysis of concrete fracture, Eng. Fract. Mech. 74 (1) (2007) 75–86. J.M. Sancho, J. Planas, A.M. Fathy, J.C. Gálvez, D.A. Cendón, Three-dimensional simulation of concrete fracture using embedded crack elements without enforcing crack path continuity, Int. J. Numer. Anal. Meth. Geomech. 31 (2) (2007) 173–187. M. Tarifa, E. Poveda, R.C. Yu, X. Zhang, G. Ruiz, Effect of loading rate on highstrength concrete: numerical simulations, in: VIII International Conference on Fracture Mechanics of Concrete and Concrete Structures, FraMCoS-8, 2013. L.J. Malvar, J.E. Crawford, Dynamic increase factors for concrete, Naval Facilities Engineering Service Center, Port Hueneme, CA, 1998. Livermore Software Technology Corporation (LSTC) LS-DYNA version 971, 2007. X.H. Guo, F. Tin-Loi, H. Li, Determination of quasibrittle fracture law for cohesive crack models, Cem. Concr. Res. 29 (7) (1999) 1055–1059, http://dx.doi.org/10. 1016/S0008-8846(99)00089-7. J. Planas, G.V. Guinea, J.C. Gálvez, B. Sanz, A.M. Fathy, Chapter 3. Indirect tests for stress-crack opening curve. Experimental determination of the stress-crack opening curve for concrete in tension. RILEM TC 187-SOC Final report, 2003. Y.S. Tai, T.L. Chu, H.T. Hu, J.Y. Wu, Dynamic response of a reinforced concrete slab subjected to air blast load, Theor. Appl. Fract. Mech. 56 (3) (2011) 140–147. O. Yilmaz, J.F. Molinari, A mesoscale fracture model for concrete, Cem. Concr. Res. 97 (2017) 84–94. P.H. Bischoff, S.H. Perry, Compressive behaviour of concrete at high strain rates, Mater. Struct. 24 (6) (1991) 425–450.
325
