An enhanced damage plasticity model for predicting the cyclic behavior of plain concrete under multiaxial loading conditions

Some of the current concrete damage plasticity models in the literature employ a single damage variable for both the tension and compression regimes, while a few more advanced models employ two damage variables. Models with a single variable have an inherent difficulty in accounting for the damage accrued due to tensile and compressive actions in appropriately different manners, and their mutual dependencies. In the current models that adopt two damage variables, the independence of these damage variables during cyclic loading results in the failure to capture the effects of tensile damage on the compressive behavior of concrete and vice-versa. This study presents a cyclic model established by extending an existing monotonic constitutive model. The model describes the cyclic behavior of concrete under multiaxial loading conditions and considers the influence of tensile/compressive damage on the compressive/tensile response. The proposed model, dubbed the enhanced concrete damage plasticity model (ECDPM), is an extension of an existing model that combines the theories of classical plasticity and continuum damage mechanics. Unlike most prior studies on models in the same category, the performance of the proposed ECDPM is evaluated using experimental data on concrete specimens at the material level obtained under cyclic multiaxial loading conditions including uniaxial tension and confined compression. The performance of the model is observed to be satisfactory. Furthermore, the superiority of ECDPM over three previously proposed constitutive models is demonstrated through comparisons with the results of a uniaxial tension-compression test and a virtual test.

The confinement is typically provided by transverse reinforcement, which improves the ultimate strength and ductility of the concrete.
Constitutive models for simulating the behavior of concrete under cyclic and dynamic loads have also been proposed. While various capabilities of these models have been demonstrated, uncertainties on their ability to predict the cyclic behavior of concrete still remain. Some of these models are briefly reviewed next, and a particular key shortcoming of the two most advanced models is discussed to provide context for the model that will be proposed herein.
Bićanić and Zienkiewicz [40] proposed a constitutive model for analyzing plain concrete structures under dynamic loading conditions. The model follows Perzyna's theory of elasto-viscoplasticity [41]. Suaris and Shah [42] presented a constitutive model based on continuum damage mechanics to predict the nonlinear biaxial tension-compression behavior of concrete under dynamic loading. Rabczuk and Eibl [43] proposed a numerical model to capture the effects of large dynamic loading conditions and large deformations on concrete failure. Sima et al. [44] developed a constitutive model for concrete that describes the strength and stiffness degradation for both compression and tension using two independent damage variables for the said behavior regimes. The model also describes the opening and closing of cracks under uniaxial cyclic loading. He et al. [45] proposed a constitutive model based on the smeared rotating crack approach to capture the cyclic behavior of concrete, including inelastic strains, crack opening and closing, and the degradation of its strength and stiffness. Tasnimi and Lavasani [46] developed a nonlinear constitutive model for the uniaxial cyclic behavior of plain and reinforced concrete elements based on the elastoplastic-fracture-damage (EPFD) theory using one parameter for the damage due to compression and another parameter for the fracture under tension. Breccolotti et al. [47] enhanced the constitutive model of Sima et al. [44], which has not considered the damage accumulation during unloading. Moharrami and Koutromanos [48] formulated a constitutive model to simulate the cyclic behavior of concrete by combining classical plasticity with the smeared-crack model. Huguet et al. [49] proposed a constitutive model for reinforced concrete members. The model explicitly considers crushing under compression, cracking under tension, bond slip between concrete and rebars, and steel yielding at crack-crossing.
A number of numerical studies have investigated cracking in concrete at both the material and structural levels and presented 2D/3D meshfree approaches for modeling arbitrary crack initiation, crack growth, and discrete cracks [50][51][52][53]. Some recent studies include the assessment of fractures in various materials using a dualhorizon peridynamics method for fractures in granular and rock-like materials [54,55]; a phase field model for fractures in 2D/3D continua in homogeneous and heterogeneous materials [56]; self-propagating strong-discontinuity embedded and cracking-element methods for quasibrittle [57] and dynamic fractures [58], respectively; and a robust discontinuity-embedded approach for fractures in quasi-brittle materials such as glass and concrete [59].
Among the proposed models for concrete, one particular model has arguably found the most practical use in simulating the cyclic behavior of concrete at both the material and structural member scales. This model, which was proposed by Lee and Fenves [60], combines the classical plasticity and continuum damage mechanics approaches, and is typically referred to as the concrete damage plasticity model (CDPM). CDPM is able to account for the stiffness degradation and strength recovery due to the opening and closing of cracks, as well as the effects of low levels of lateral confinement on the strength and ductility. A more recent multi-axial model (henceforth CDPM2) was proposed by Grassl et al. [61] as an improvement over the CDPM1 proposed by Grassl and Jirásek [29] for monotonic loading. CDMP2 aims to capture the increase in the load-carrying capacity with increasing confinement, as well as the transition between the tension and compression regimes through the use of two damage variables [61]. Despite their advanced features, neither CDPM, nor the more advanced CDPM2, can describe the influence of prior compressive/tensile damage on the tensile/compressive behavior. This particular shortcoming is addressed in the present study through the development of a new model, namely the enhanced concrete damage plasticity model (henceforth, the ECDP model or ECDPM), that aims to capture the behavior of plain concrete under cyclic multiaxial loading conditions. The latter is one of the advantages of ECDPM over the two prior cyclic damage plasticity models [60,61] whose abilities to predict the cyclic response of plain concrete under biaxial and triaxial cyclic compression conditions have neither been demonstrated nor examined in the open literature.

An enhanced damage plasticity model for concrete
The proposed new ECDPM is an enhanced version of the damage plasticity model proposed by Grassl et al. [29,61]. Its features include isotropic hardening and softening, the generation and accumulation of inelastic strains, as well as stiffness degradation due to damage in both the compressive and tensile regimes. The nominal stress-strain relationship in ECDPM takes the form where σ denotes the nominal stress tensor, D is the scalar isotropic damage variable ranging from 0 (undamaged concrete) to 1 (fully damaged concrete), E 0 is the initial elastic stiffness tensor, and ε and ε p are the total and plastic strain tensors, respectively. The plastic part of the model is formulated in terms of the effective stress, which is defined as The model takes the softening in the post-peak region caused by crushing under compression or cracking under tension into account by splitting the damage variable (D) into two damage variables in the two regimes as follows: where D c and D t denote the scalar isotropic damage variables under compression and tension, respectively. The split weight factor r w ranges from 0 (purely 3D compressive strain state) to 1 (total tension) and is defined as [62] r w ¼ where hε I i is the positive part of the principal strain component ε I and I = 1,2,3 [62]. Other damage plasticity models employ a stress-based split weight factor [60,61]. The effects of various split weight factors on the cyclic response of plain concrete in ECDPM and the other two damage plasticity models considered herein (i.e., CDPM by Lee and Fenves [60] and CDPM2 by Grassl et al. [61]) are shown in Fig. 1. It can be observed from Fig. 1(a) that the material experiences tensile damage (D t > 0), which results in the reduction of tensile stiffness. However, because the loading direction varies when the stress enters into the compressive regime (D c = 0), the value of the stiffness suddenly changes to its intact value because of the stress-based split weight factor. It is worth noting that Grassl et al. [61] proposed a bilinear model for tensile softening. The present study uses the exponential tensile softening response proposed in Ref. [29]. Figure 1(b) demonstrates the influence of the strain-based split weight factor on the stress-strain response and the compressive stiffness. It can be observed in Fig. 1(b) that the tensile damage causes a reduction of the compressive stiffness. The employment of a strain-based split weight factor is one of the main differences between ECDPM and the other damage plasticity models [60,61].
The elastic domain is delimited by employing a single yield surface f p formulated in the Haigh-Westergaard coordinates of the effective stress tensor [29]: The effective mean stress m , the effective deviatoric radius , and the Lode angle q are three invariants of the effective stress tensor; f cu is the uniaxial compressive strength of concrete; m 0 is a frictional model parameter; r(q) is a function describing the shape of the yield surface in the deviatoric planes [23]; and q h denotes a stress-like internal hardening variable. The evolution of the yield surface in the principal stress space during hardening is illustrated in Fig. 2.
The evolution of the plastic strain is given by where _ l is the consistency parameter, and g p is the nonassociated plastic potential function, which is defined as [29] where m g m ð Þ controls the ratio of the volumetric and deviatoric parts of the flow direction. It is a function of the effective mean stress and takes the form where A g and B g are model parameters (more information can be found in Ref. [29]) and f tu is the tensile strength of concrete. The hardening law is defined through the function q h in Eq. (7). q h is a function of the strain-like internal hardening variable α p , and is defined as where f cy is the uniaxial yield stress under compression. The direction of the plastic flow, m ¼ ∂g p =∂σ, can be defined using the volumetric and deviatoric parts as In Eq. (10), the substitutions ∂ m =∂σ ¼ δ=3 and ∂=∂σ ¼ s= can be made, where δ is the Kronecker delta (i.e., identity) tensor and s is the deviatoric effective stress tensor. The other two terms ∂g=∂ m and ∂g=∂ take the forms and The evolution of the strain-like internal hardening variable, α p , can be defined by employing the modified hardening law by Unteregger et al. [63]: x h m ð Þ denotes a hardening ductility measure and can be written as where A h , B h , C h , and D h are model parameters that can be calibrated from the experimentally measured values of the strain at the ultimate tensile stress. The variable R h m ð Þ takes the form The isotropic damage variables, which represent the softening behavior of concrete, can be defined in the compressive and tensile regimes as respectively. It is worth noting that Eqs. (16) and (17) are defined by splitting the damage parameter proposed in Ref. [29] into two damage variables for the tensile and compressive regimes. According to Grassl et al. [61], the softening modulus under compression is given by wherein the compressive fracture energy can be defined as [64] G The softening modulus under tension ϵ ft can be computed using the specific mode-I fracture energy G FI , the characteristic size of a finite element l char , and the Young's modulus E as [65] The evolution of the damage driving the internal variables under compression α dc and tension α dt is given by respectively. α c is a variable ranging from 0 (uniaxial tension) to 1 (uniaxial compression) that distinguishes between tensile and compressive loading and is given by [61] where PCI and PTI denote the components of the compressive and tensile part of the principal effective stresses, respectively, and x s m ð Þ denotes a softening ductility measure The model parameter A s can be calibrated from the softening response in uniaxial compression tests, and the function R s m ð Þ is defined as  Fig. 3(a). The comparisons are presented in Fig. 3(b), from which it can be observed that there is very good agreement between the ECDPM predictions and the experimental data. Next, the performance of ECDPM in modeling the cyclic behavior of plain concrete under uniaxial compression is evaluated. For this, the ECDPM predictions are compared against data from two experiments performed by Karsan and Jirsa [67] and Van Mier [68].

Validation of ECDPM
Karsan and Jirsa's experiment is a uniaxial compression test for which the material parameters are E = 30 GPa, n = 0.20, f cu = 28 MPa, and f tu = 2.8 MPa, and the model parameter was identified as A s = 5. The specific Mode-I fracture energy G FI was estimated using the FIB (International Federation for Structural Concrete) model code for concrete structures [69], and the Poisson's ratio n and the uniaxial compression yield stress f cy were assumed based on similar grades of concrete. The model predictions and experimental data are shown in Fig. 4, from which it can be seen that ECDPM predicts the ultimate compressive strength and softening responses very well.
The material parameters for the uniaxial compression test performed by Van Mier [68] are E = 33 GPa, n = 0.20, f cu = 42.3 MPa, f tu = 2.8 MPa, and G FI = 0.15 N/mm, and the model parameter was identified as A s = 10. It is noted here that the Young's modulus E was computed from the experimental data [68]. The details of the single loading frame employed by Van Mier [68] are shown in Fig. 5(a). The predicted and observed results for the tested specimen are shown in Fig. 5(b). It can be seen in the left half of Fig. 5(b) that the softening response of the plain concrete specimen along the loading direction is captured very well by ECDPM. On the other hand, it can be seen in the right half of Fig. 5(b) that ECDPM considerably underestimates the lateral strains measured at the free surfaces. Van Mier [68] stated that the measurements of the lateral strains were probably disturbed by the splitting phenomena. As such, it is not clear whether the discrepancy observed between ECDPM and Van Mier's reported lateral strains is indicative of a problem with ECDPM, or due to an artifact of the specimen-sensor configuration.
It should be noted that the stress-strain diagrams in the softening region in Figs. 2, 4, and 5 were computed using the characteristic length l char of 100 mm.

Biaxial cyclic compression tests
Data from two biaxial (monotonic and cyclic loading) experiments performed by Kupfer et al. [70] and Van Mier [68] were also employed to evaluate ECDPM.
The first of these data sets is from the uniaxial and biaxial monotonic tests by Kupfer et al. [70]. The material parameters for concrete in these tests are E = 32 GPa, n = 0.18, f cu = 32.8 MPa, and f tu = 3.3 MPa, and the Mode-I fracture energy G FI was estimated as 0.145 N/mm [69]. The model constant A s was reported as 1.5 in Ref. [61]. Lateral confining pressures (σ 2 ) of 50% and 100% of the uniaxial compressive strength were applied to the specimens. Figure 6 displays the responses predicted by ECDPM and the measurements from the tests. As can be seen, ECDPM captures the concrete behavior in these uniaxial and biaxial tests very well.
In the biaxial cyclic loading tests by Van Mier [68], the model parameters are E = 25 GPa, n = 0.20, f cu = 47.9 MPa, f tu = 3.1 MPa, and G FI = 0.15 N/mm, and the model constant was identified as A s = 7.5. The lateral confining stress (σ 2 ) was considered to be 5% of the uniaxial compressive strength and σ 3 = 0 [68]. Figure 7 displays the predicted and measured responses. It can be seen in Fig. 7(a) that the ECDP model predicts the stress-strain response (σ 1 -ε 1 ) of the tested specimen very well, especially the influence of the lateral confining pressure on both the strength and ductility, as well as the softening Fig. 3 (a) Scheme of wedge-type frictional grips adopted from Ref. [66]; (b) comparison of predicted and observed concrete responses in a uniaxial cyclic tension test by Gopalaratnam and Shah [66]. Fig. 4 Comparison of predicted and observed concrete behavior in the uniaxial cyclic compression test by Karsan and Jirsa [67]. response in the post-peak region. Again, the issue discussed in the preceding section regarding lateral strains at the free-surface of the specimen is observed in this test as well in Fig. 7(b).
The characteristic length for computing the softening region of the stress-strain diagrams employed is 100 mm for all the biaxial compression tests, which follows from the specimen dimensions provided in the test reports.

Triaxial cyclic compression tests
Data from tests on confined concrete specimens subjected to triaxial monotonic and cyclic loads by Imran and Pantazopoulou [71] and Van Mier [68] were also considered.
The first of these data sets is from the tests carried out by  MPa, and f tu = 4.74 MPa. The specific Mode-I fracture energy G FI was computed as 0.15 N/mm [69] and the model constant A s was taken as 15, as reported in Ref. [61]. Figure 8 illustrates the experimental setup of the confined compression tests, which was adopted from Ref. [71]. The computed ECDPM results are shown along with the test data in Fig. 8, wherein very good agreement is observed at all levels of lateral confinement pressure. The performance of ECDPM in capturing the cyclic response of plain concrete under triaxial loading conditions is evaluated by considering two triaxial cyclic tests conducted by Van Mier [68]. In the first test, the reported concrete material parameters are E = 35 GPa, n = 0.20, f cu = 47.9 MPa, f tu = 3.1 MPa, and G FI = 0.15 N/mm, and the model constant was identified as A s = 2.5. In this test, lateral confining pressures σ 2 and σ 3 of 10% and 5% of the uniaxial compressive strength were respectively applied to the specimen [68]. The predicted results from ECDPM are illustrated in Fig. 9. It can be seen in Fig. 9 that the concrete response is well predicted by ECDPM along the loading direction (σ 1 -ε 1 ). Nevertheless, ECDPM overestimates the lateral strain (ε 2 ) in the direction with higher lateral pressure and considerably underestimates the lateral strain (ε 3 ) in the direction with lower lateral pressure.
Next, the ECDP model is also examined with data from Fig. 7 Comparison of predicted and observed concrete behavior in biaxial cyclic compression tests conducted by Van Mier [68].
(a) Stress along the loading direction (σ 1 ) versus the strain along the loading direction (ε 1 ) and the strain along the lateral pressure directions (ε 2 ); (b) stress along the loading direction (σ 1 ) versus the strain at the free surface (ε 3 ). another triaxial cyclic test performed by Van Mier [68]. In this test, the model parameters are E = 27 GPa, n = 0.20, f cu = 45.3 MPa, f tu = 2.8 MPa, and G FI = 0.15 N/mm, and the model constant was identified as A s = 1.5. In this test, equal lateral confining stresses σ 2 = σ 3 were applied to the specimen at 10% of the uniaxial compressive strength [68]. The responses predicted by ECDPM are shown together with the measurements in Fig. 10 and very good agreement is observed. It is useful to note here that all of the three strains are well reproduced by ECDPM for this triaxial test wherein there were no free surfaces or surfaces with low levels of lateral pressure, unlike the case for Van Mier's uniaxial, biaxial, and triaxial tests which were examined above. The softening response in Figs. 8, 9, and 10 was computed by employing the characteristic length of 100 mm based on the specimen dimensions provided in the test reports.

Uniaxial cyclic tension-compression test
Herein, the ECDP model is assessed with data from a cyclic uniaxial test with alternating tension-compression by Reinhardt et al. [72], and it is also compared to other damage plasticity models for concrete, namely CDPM by Lee and Fenves [60], CDPM1 by Grassl and Jirásek [29], and CDPM2 by Grassl et al. [61]. It is important to note that the uniaxial cyclic test by Reinhardt et al. [72] is, to the best of the authors' knowledge, the only experimental data set with alternating tension-compression available in the open literature that can be used to examine the effects of compressive damage on the tensile behavior of plain concrete and vice versa.
In this experiment, the concrete attained its ultimate tensile strength while its behavior under compression was probed within the elastic regime. It is also useful to note that both CDPM and CDPM2 have been previously examined and have demonstrated their capabilities in capturing the tensile response of plain concrete well (the same/similar demonstrations are omitted here for brevity). However, unlike ECDPM, neither CDPM nor CDPM2 has been devised to capture the influence of tensile damage on the (effective) compressive stiffness. As such, the Fig. 9 Comparison of predicted and observed concrete behavior in triaxial cyclic compression test carried out by Van Mier [68].
measurements from the test by Reinhardt et al. [72] form an especially important and unique data set for model validation.
The material parameters for concrete are E = 24 GPa, f cu = 47.1 MPa, f tu = 3.20 MPa, and G FI = 0.06 N/mm. The Poisson's ratio was assumed to be n = 0.20, and l char was taken as 35 mm. The responses predicted by ECDPM and the other two damage plasticity models are presented along with the test data in Fig. 11.
The test data are compared to the predictions from CDPM2 and CDPM in Figs. 11(b) and 11(c), respectively, from which it can be seen that neither of the models can capture the transitions from tension to compression and vice vera observed in the test. The compressive stiffnesses from both models are equal to the intact/initial elastic modulus, and do not evolve with the cyclic transitions between the tensile and compressive loading regimes. In short, it can be concluded that tensile damage does not influence the compressive stiffness in these models, which is expected from their formulation. It can also be seen in Figs. 11(b) and 11(c) that the residual tensile strains are inaccurately predicted by both CDPM and CDPM2 when σ = -3.4 MPa. In contrast, ECDPM predicts the near-lack of residual tensile strains at σ = -3.4 MPa very well because the compressive stiffness is reduced by tensile damage, as shown in Fig. 11(a). It is worth noting that the numerical test on CDPM [60] here was performed using the preexisting implementation in the finite element software package ABAQUS [73].
In Fig. 11(d), the predicted cyclic response of another concrete damage plasticity model dubbed CDPM1 by Grassl and Jirásek [29] is also compared to the test data. Unlike ECDPM, CDPM, and CDPM2, this model (CDPM1) features a single damage variable and for the comparison at hand, appears to produce a reasonable prediction which is nearly identical to that of the ECDP model. Moreover, the capabilities of CDPM1 in capturing the responses of plain concrete and reinforced concrete structures under monotonic loading scenarios have been demonstrated in prior studies [74,75]. However, it is still unclear whether the single-damage formulation of CDPM1 is capable of capturing the cyclic behavior under the full range of compressive stresses, especially up to the compressive strength, and then within the post-compressive failure regimes.
To clarify the potential shortcomings of CDPM1 (and incidentally, those of CDPM and CDPM2), the cyclic behavior envelope curves of plain concrete in a straindriven virtual test (ε = 0, 0.25Â10 -3 , -2Â10 -3 , unloading) predicted by CDPM [60], CDPM1 [29], CDPM2 [61], and ECDPM are shown in Fig. 12. The material and model parameters were set as E = 15 GPa, n = 0.20, f cu = 10 MPa, f tu = 1 MPa, G FI = 0.02 N/mm, A s = 15, and l char = 100 mm. The predicted concrete responses in the tensile regime by all the models are almost identical. In all the models, the concrete reaches its designated ultimate tensile strength, and the capacity continues to decrease in the post-peak region due to accumulated tensile damage. Nevertheless, the models predict different compressive responses. To wit, for CDPM1 (denoted by the magenta-colored dashed line) [29], the initial compressive stiffness of the concrete is equal to the reduced tensile stiffness when the loading branch changes and the stress enters into the compressive regime because of the employment of only a single damage variable. This results in a significant loss of compressive strength (σ C ≈ -2 MPa). Because only a single damage variable is employed in CDPM1, it is not possible to control both the deterioration in stiffness and that in compressive strength independently with the model. In contrast, ECDPM allows such control (denoted by the red dashed line) and the reduction of the ultimate compressive strength (here to σ C ≈ -7 MPa) due to tensile damage, for which the reduction is not purely dictated by the loss of stiffness, unlike in CDPM1. This feature of ECDPM enables the implicit consideration of the closure of tension cracks and the subsequent recovery of compressive stiffness (see the inflection point in the ECDPM curve at σ ≈ -1 MPa in Fig. 12) in a phenomenological manner as opposed to, for example, a micro-mechanics manner wherein cracks are individually modeled and tracked.
As for CDPM (denoted by the blue dashed line) [60] and CDPM2 (denoted by the black dashed line) [61], the concrete reaches its monotonic ultimate compressive strength regardless of the tensile damage it has already suffered in the predictions by these models because the models employ uncoupled damage variables for tension and for compression, and the initial compressive stiffnesses are not affected by the tensile damage and remain intact.
Test data on plain concrete wherein the sample is cycled through tensile and compressive strains such as those applied in the virtual test shown in Fig. 12 do not currently exist in the open literature. As such, in order to fully examine and validate ECDPM and to compare it with CDPM1, one has to either carry out such cyclic tests on plain concrete specimens, or rely on structural componentlevel tests (e.g., on reinforced concrete shear walls, beams, and beam-column joints) for which data are available in extensive databases [76,77].

Summary and conclusions
An ECDPM was proposed for simulating the cyclic behavior of plain concrete under multiaxial loading conditions. This model combines the theories of plasticity and continuum damage mechanics like many of its predecessors; however, it crucially features two damage parameters and a strain-based split weight factor to capture the influence of tensile damage on the compressive response and vice-versa. The performance of ECDPM was evaluated using data sets from a number of experiments including a uniaxial tension test and uniaxial, biaxial, and triaxial compression tests under monotonic and cyclic loading conditions.
The results indicate that ECDPM uniformly exhibits high accuracy and fidelity against the test data with a single exception for lateral strains under low confinement pressures (which is estimated here as not being problematic for common earthquake engineering applications), and is generally superior to the prior CDPM [60], CDPM1 [29], and CDPM2 [61] continuum damage plasticity models. Specifically, ECDPM appears to accurately predict the cyclic behavior of plain concrete, the impacts of lateral confinement on strength and ductility, as well as the tensile and compressive softening responses in the post-peak regions. Future studies are needed to further examine the relative accuracies of these models, especially in predicting system/component-level behavior, for which ECDPM appears to hold better promise. The current lack of experimental data from plain concrete specimens under cyclic tension-compression strain cycles presents a lack of opportunities for model validation and is a major challenge for the development of more accurate and sophisticated continuum-based constitutive models than ECDPM. Despite the high accuracy of ECDPM at the material level, its performance can be further examined at the structural level by simulating the cyclic behavior of RC structures subjected to a combination of axial loads and reversed cyclic loading. This can be considered as a future work which can provide detailed information on the cyclic response of RC structures, such as the damage evolution in concrete and stress distribution in the reinforcement, during earthquakes and provide better perspectives on the seismic design of RC structures.