A new 3D damage model for concrete under monotonic, cyclic and dynamic loadings
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Abstract
Among the “theories” applied to model concrete behavior, damage mechanics has proven to be efficient. One of the first models for concrete introduced into such a framework is Mazars’ damage model. A new formulation of this model, called the “μ model” and based on a coupling of elasticity and damage within an isotropic formulation, is proposed herein for the purpose of 3D cyclic and dynamic loadings. Unilateral behavior (i.e., crack opening and closure) is introduced by use of two internal variables. A threshold surface is then associated with each of these variables. The shape of such surfaces has been chosen to model as accurately as possible concrete behavior under various loadings, i.e., tension, compression, shear, biaxial and triaxial, in the aim of simulating a large number of loading types (monotonic, cyclic, seismic, blast, impact, etc.). Applications of this model are presented on plain or reinforced concrete elements subjected to a range of loadings (e.g., multiaxial, cyclic, dynamic). A comparison with experimental results serves to demonstrate the effectiveness of these various selected options.
Keywords
Damage models Thermodynamics of irreversible processes Unilateral behavior Concrete Cyclic loading Severe loadings Finite element calculations1 Introduction
Estimating the ultimate capacity of a concrete structure is essential to determining appropriate safety margins. Changes in regulations combined with the objective of building sustainable structures have led engineers to develop efficient modeling tools that remain easy to use. The model presented in this paper has been created for such a purpose.
Concrete is considered as brittle in tension and more ductile under compressive loading. As opposed to uniaxial tension, under which just a single crack propagates, compression caused by the presence of heterogeneities (aggregates surrounded by a cement matrix) produces transverse tensile strains that generate mesoscopic cracks nucleating perpendicularly to the direction of extension. These mesoscopic cracks then coalesce until reaching the point of complete rupture. A pure mode I (extension) is thus considered for the purpose of describing behavior in both tension and compression. This situation can be extended to multiaxial loading at low and moderate confinement that allows for extension in at least one direction. When the material is highly confined, hydrostatic pressure serves to compact the porous cement matrix and shear favors mode II.

the extension allowed (EA) domain: uni, bi or triaxial situations allowing for extension (\(\varepsilon_{i}\) > 0) in at least one direction and, accordingly, for a local cracking (mode I);

the no extension allowed (NEA) domain: a triaxial situation at high confinement generating both the collapse of the cement porous matrix, as a result of the spherical part of the stress tensor, and shear cracking (mode II) due to the deviatoric part.
In the framework of continuum media, to simulate concrete behavior, either plasticity (Ottosen [2], Dragon et al. [3]), fracturebased approaches (Bazant [4], Carpinteri [5]), damage models (Mazars [1], Mazars et al. [6], Simo et al. [7], Jirasek [8]) or a plasticdamage model (Lee and Fenves [9], Jason et al. [10]) is used. Depending on the specific case, one or the other approach may be suitable for certain situations (whether classical or not) found in common structures for conventional loads.
For severe loadings relative to natural or technological hazards (e.g., earthquakes, blasts, impacts), additional aspects must be taken into consideration, namely: the dynamic and cyclic nature of the loading and, for local impacts, the high confinement pressure. Some models provide a description of the cyclic behavior (la Borderie et al. [11], Halm et al. [12], Richard et al. [13]). Very few models however are capable of simulating loading with both confinement and strain rate effects (Pontiroli et al. [14], Gatuingt et al. [15]), though their use often remains complex.
This paper will present a strategy for modeling these types of behavior, in emphasizing efficiency and simplicity. To achieve this objective, a damage model provides good candidate strategies. One of the first models created within such a framework and specifically intended for concrete is Mazars’ damage model [1]. Efficient yet limited to “classical” monotonic loadings, this new proposal, called the “μ model” (μ for Mazars Unilateral), was developed as part of the thermodynamics of irreversible processes (Lemaitre et al. [16]) and has shown itself capable of describing a very broad range of nonlinear behavior (monotonic, cyclic, dynamic, etc.).
2 Theoretical aspects
2.1 Underlying assumptions

Concrete behavior is considered as the combination of elasticity and damage;
 The damage description is assumed to be isotropic and directly affects the stiffness evolution of the material. Let \(\underline{\underline{{\underline{\underline{\Lambda }} }}}\) be the stiffness matrix of the original material, then the matrix for the damaged material is given by:$$\underline{\underline{{\underline{\underline{\Lambda }} }}}_{d} = \underline{\underline{{\underline{\underline{\Lambda }} }}} (1  d)$$(1)

As opposed to classical damage models, d denotes the effective damage. Classically speaking, damage is a variable that describes the microcracking state of the material (Lemaitre and Chaboche [16]). Moreover, d describes the effect of damage on the stiffness activated by loading. In a cracked structure, d must then be able to describe the effects of crack opening and closure (i.e., unilateral effects).

Two principal damage modes are considered cracking (due to tension) and crushing (due to compression), to be subsequently associated with two thermodynamic variables Y _{t} and Y _{c}, which characterize the extreme loading state reached in the tensile part and compressive part, respectively, of the strain space.
2.2 Main concepts
2.2.1 Constitutive equations
2.2.2 Damage evolution

Optimally reproducing the entire set of \(\sigma\)–\(\varepsilon\) curves for the various loading paths in the stress space;

For the specific case of shear, enabling reproduction of the sliding effects related to friction whenever the concrete is cracked, thanks to a residual stress.

Allowing for easy identification of material parameters from uniaxial experiments.
2.3 Model validation
2.3.1 Calculation procedure: model responses for various loading paths
Kupfer et al. [17] conducted a series of tests to investigate the response of plain concrete subjected to a twodimensional loading. During these investigations, concrete plates (200 mm × 200 mm × 50 mm) were loaded until failure at prescribed ratios of \(\sigma_{1}\)/\(\sigma_{2}\), with \(\sigma_{3}\) set equal to zero. \(\sigma_{i}\) denotes the principal stresses.
Concrete characteristics, for the Kupfer tests
Exp. data  Model parameters  

f _{c} (MPa)  f _{t} (MPa)  E (GPa)  ν  ε _{t0}  ε _{c0}  A _{t}  B _{t}  A _{c}  B _{c} 
32.7  3.2  30  0.21  1.1e−4  3e−4  1  1e+4  1.25  517 
In both cases, the μ model yields a very good prediction of maximum strength and quite good results for the strain evolution in various directions, except for extension, in the vicinity of the peak load. This outcome stems from the fact that the Poisson’s ratio remains constant in an isotropic formulation.
2.3.2 Failure surface
3 Applications on beams under cyclic loading
Applications covered by the μ model are mainly severe loadings on concrete structures. Among these applications, earthquakes would be an important issue: earthquakes generate nonlinear cyclic loadings on structural elements. The strain rate is small enough to be eliminated as an issue, unlike with blasts and shocks. The LMT Laboratory at ENS Cachan (France) has conducted an experimental campaign on reinforced concrete (RC) beams in order to study phenomena that play a major role in the response of RC structures during an earthquake (Ragueneau et al. [19], Crambuer et al. [20]). Phenomena such as damage evolution during increased loading, unilateral effects and energy dissipation due to cyclic loads have all been analyzed. These results will serve for the applications that follow.
3.1 Experimental program
Three point RC bend tests, experimental data and material specifications
Experimental data  Concrete model parameters  

Steel  Concrete  
E (GPa)  σ _{rupt} (MPa)  E (GPa)  fc (MPa)  E (GPa)  ν  ε _{t0}  ε _{c0}  A _{t}  B _{t}  A _{c}  B _{c} 
205  530  28  35  28  0.2  1.14e−4  −3.57e−4  0.8  7000  1.25  395 
The RC beams are designed to be tested with a simple threepoint bending setup in the up and down vertical direction. A specific hinge device ensures a freerotation condition at the end of the support beams. The loading path has been reproduced in Fig. 6b; it is displacementcontrolled and includes sets of three cycles with gradually increasing intensity (from ±1 mm to ±8 mm). Figure 6c shows the force–displacement response of beam no. 2 for the full program, indicating: (i) the gradual decrease in stiffness due to concrete damage during the first series of cycles, and (ii) the appearance of rebar plasticity after the ±4 mm cycles and its continued prevalence beyond this stage.
3.2 Numerical modeling approach
The selected model parameter values have been adopted in accordance with the data provided in Table 2.
Regarding the rebar, bar elements are used in compliance with the elastoplastic hardening model developed by Menegotto–Pinto [24]. Figure 7b displays the related \(\sigma\)–\(\varepsilon\) curve.
3.3 Results
3.3.1 Global results

For the total loading path, the calculation curve (Fig. 8a, solid line), performed without any cycle, shows good agreement with the envelope for the entire set of experimental curves.

For the same loading path with cycles up to ±2 mm, the comparison shows very good agreement (Fig. 8b).

For a cyclic loading of ±5 mm, including rebar plasticity (Fig. 8c), the results are also very satisfactory.
It can be concluded from these results that the stiffness recovery, as modeled by the μ model, very accurately reproduces the experimental results and, moreover, the decision not to represent the permanent concrete strain does not penalize these results (some differences are only visible around the zero loading point when rebar plasticity has not been activated– Fig. 8b).
3.3.2 Local results
During a cyclic loading, at a given location, the effective damage d evolves until a maximum value of the load in one direction is reached. When the load is reversed, the local stress is also reversed. As presented Fig. 1, due to changes, first in the triaxial factor r (Eq. 14) and then in the damage driving variable Y (Eq. 13), this maximum value of d vanishes. This specific property may be used as a crack opening indicator.
However, if necessary, it is possible to visualize after a given loading path the total damage area thanks to the thermodynamic variables Y _{t} and Y _{c}. Two images can thus be derived: one is using Y _{t}, while the other is using Y _{c}, both of which are representative of cracking under tension and compression respectively.
These results reveal an efficient model, which has proven to be robust and effective in describing both cyclic behavior and crack opening behavior. The μ model is thus a good candidate for solving seismic problems.
4 Loading with confinement and strain rate effects
Among the range of severe loading situations involving concrete structures, blasts and impacts must be considered. To simulate these effects, the adopted modeling approach must include strain rate effects and confinement effects. This section will propose μ model improvements that allow simulating both phenomena.
4.1 Strain rate effects
An experimental campaign has recently been conducted at the LEM3 Laboratory in Metz (France) (Forquin and Erzar [25], Erzar and Forquin [26]); it has focused on quasistatic tests and dynamic tests performed on both dry and wet concrete specimens. A highspeed hydraulic press was applied for the intermediate strain rate, and an experimental Hopkinson bar device, based on the spalling technique, was used for higher strain rates. From these tests, original identification techniques could be developed in order to deduce the tensile strength of the material.
In the μ model, the dynamic tensile strength is assumed to be the peak of the stress–strain curve: ft = E\(\varepsilon_{{0{\text{t}}}}^{\text{d}}\). From (23), f _{t} = f(\(\dot{\varepsilon }\)) can be obtained. These results have also been plotted in Fig. 11, thus confirming that the calculations provide excellent results. It is a simple step to extend such a calculation to 3D situations by introducing \(\varepsilon_{{0{\text{t}}}}^{\text{d}}\) into Eqs. (12) and (16), in order to define the initial threshold of the 3D driving variable of damage, Y.
4.2 Confinement effects
Confinement corresponds to loadings in the tricompression domain. The triaxial coefficient r, introduced in Eq. (14), lies within the confinement domain and equals 0. As mentioned in Sect. 1, damage modes depend on the presence or absence of extensions. At this point, two distinct domains can be considered: (i) the extension allowed (EA) domain, whereby during loading a positive strain exists in at least one direction; and (ii) a no extension allowed (NEA) domain, when loading prevents any kind of extension.
Positive strains within the EA domain can generate local damage and, as will be seen below, the μ model is a good candidate to describe this situation, which corresponds to a soft impact.
In contrast, within the NEA domain, phenomena are of two orders: (i) strong confinement leading to a gradual collapse of cement matrix porosity; and (ii) the shear caused by stress triaxiality generates local mode II cracking. It has been shown that such phenomena can be advantageously modeled through introducing compaction and plasticity effects. Coupling of the μ model with a plasticity model, as performed in the PRM model [14], will be considered in future developments.
4.2.1 Low and moderate confinement
 1.
radial path: \(\sigma_{2} = \sigma_{3} = \beta \sigma_{1}\) (where β is a constant)
 2.
triaxial path: \(\sigma_{2} = \sigma_{3} = S\) (S is a fixed level), with \(\sigma_{1}\) increasing until failure.
Poisson’s ratio \(\nu\) is responsible for the transverse extension. For a given value of \(\nu\), typically equal to 0.2, it is straightforward to show that C = 0 for β values greater than or equal to 0.25. To avoid any damage during the first step, β = 0.25 has been chosen. Afterwards, during the second step β (= S/ \(\sigma_{1}\)) decreases, allowing d to evolve until failure (Fig. 13).
Figure 13b displays the numerical results derived for the same concrete specimen used earlier and for S values of 0, 20, 35, 50 and 100 MPa. Fig. 13c presents the experimental results obtained by two researchers on a similar concrete and on similar loading paths (Ramtani [27], Gabet [28]). It can be observed that both, the trends and values at failure have been correctly simulated. At a high S value (i.e., 100 MPa) however, it is observed that the phenomena described for the NEA domain have been activated, particularly compaction damage, which along with extension damage serve to accelerate specimen collapse and thus explain the conservative strength value resulting from calculation. This loading situation provides the limitation of the present development of the μ model.
5 Conclusion
The μ model has been based on the principles of isotropic damage mechanics. Two thermodynamic variables have been defined to describe, within a 3D formulation, the unilateral behavior of concrete (crack opening and closure), which is essential for cyclic loadings, in particular the seismic behavior of concrete structures.
Built around a simple formulation that combines elasticity and damage and in respecting the thermodynamic principles of irreversible processes, this model is simply implemented into a finite element code. Furthermore, the material parameters (i.e. eight in all, including elastic parameters) are easily identified solely from individual tensile and compressive tests. A series of applications yields, both at the material level and on reinforced concrete structures, a set of satisfactory experimental results attesting to the model’s effectiveness.
Through variation in the strain rate of the initial threshold of the driving damage variable, it has been proven that the model is able to describe highvelocity loading. The model’s ability to account for low and moderate confinement, i.e. when the load allows for extensions (∑\(\left\langle {\varepsilon_{i} } \right\rangle\) _{+} ≠ 0), was also discussed. Dynamic loadings at either low or high speed, such as a blast or impact, could then be simulated, although an improved description of compaction phenomena and sheardriven local mode II is required to produce a complete model capable of simulating the punching effects that lead to penetration and perforation.
In conclusion, this model can provide a useful tool for engineering applications, as was initially expected, and moreover is able to cover a wide diversity of monotonic or cyclic problems from quasistatic to highvelocity loadings (i.e., earthquakes, blasts, soft impacts). New developments are underway to both: (i) complete the 3D version for high confinement, and (ii) create simplified tools, such as multifiber beams, based on the μ model and enhanced to treat 3D structural problems including torsion, severe loading and cracking indicators [29, 30].
Notes
Acknowledgments
The authors would like to thank EDF for its financial support and both Prof. F. Ragueneau (LMT, ENS Cachan) and Prof. P. Forquin (formally at 3SRGrenoble University) for their valuable assistance in providing the experimental results.
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