Abstract
In the non-linear finite element analysis (NFEA) of concrete materials, continuum damage mechanics (CDM) provides a powerful framework for the derivation of constitutive models capable of describing the mechanical behavior of such materials. The internal state variables of CDM can be introduced to the elastic analysis of concrete to form elastic-damage models (no inelastic strains), or to the elastic-plastic analysis in order to form coupled/uncoupled elastic-plastic-damage models. Experimental evidence that is well documented in literature shows that the susceptibility of concrete to damage and failure is distinguished under deviatoric loading from that corresponding to hydrostatic loading. A reduction factor is usually introduced into a CDM model to reduce the susceptibility of concrete to hydrostatic stresses/strains. In this work, the effect of a hydrostatic stress/strain reduction factor on the performances of two NFEA concrete models will be studied. These two (independently published) models did not provide any results showing such effect. One of these two models is an elastic-damage model, whereas the other is an uncoupled elastic-plastic-damage model. Simulations and comparisons are carried out between the performances of the two models under uniaxial tensile and compressive loading conditions. Simulations are also provided for the uncoupled elastic-plastic-damage model under the following additional loading conditions: biaxial tension and biaxial compression, uniaxial cyclic loading, and varying ratios of triaxial compressive loadings. These simulations clearly show the effect of the reduction factor on the numerically depicted behaviors of concrete materials. To have rational comparisons, the hydrostatic stress reduction factor applied to each model is chosen to be a function of the internal state variables common to both models. Therefore, once the two models are calibrated to simulate the experimental behaviors, their corresponding reduction factors are readily available at every increment of the iterative NFEA procedures.
1 Introduction
Concrete is a complex highly non-linear composite material with different mechanical behaviors under different patterns of loading. Furthermore, the material properties of concrete are averaged and homogenized rather than accurately determined. These factors, among many others, render the mechanical analysis of such a quasi-brittle material an everlasting challenge.
Several approaches have been applied in the field of numerical modeling of concrete failure, resulting in different categories of constitutive models, such as continuum damage mechanics (CDM) models (isotropic and anisotropic), fracture energy models, smeared crack models, and others. Within the framework of CDM, there are a number of ways to incorporate the damage-related thermodynamic state variables into the non-linear finite element analysis (NFEA). Restricting this discussion to isotropic damage, some models coupled damage to the elastic analysis of concrete materials, with no consideration of inelastic strains, to form elastic-damage (ED) models ([1–6]; and others). ED models were termed unable to reproduce the unloading slopes during cyclic simulations, unable to capture irreversible strains during plastic flow, and unable to provide an appropriate dilatancy control under multiaxial states of loading. Nevertheless, ED models are quite noticeable in the literature of engineering mechanics.
Other models introduced damage mechanics variables into the elastic-plastic NFEA of concrete materials, to form either coupled or uncoupled elastic-plastic-damage (EPD) models. In the uncoupled models, damage is associated with the elastic analysis, whereas the plastic constitutive equations, although present, remained in the effective (undamaged) configuration. Examples of such models are those of Yazdani and Schreyer [7], Lee and Fenves [8], Shen et al. [9], Contrafatto and Cuomo [10], Voyiadjis and Taqieddin [11], Taqieddin and Voyiadjis [12], just to mention a few. These models are superior as they exclude the shortcomings of the ED models mentioned earlier.
In the coupled EPD models, the damage variables appear in the elastic as well as the plastic constitutive equations and evolution laws ([13–18]; among others). These models exhibit complex yield criteria, evolution laws, implementation procedures, and algorithms; however, they are capable of simulating specific material behaviors that are usually ignored in simpler models.
In case of an interest in anisotropic damage mechanics models, and for thoroughness, the reader is referred to the following contributions: Ju [7], Yazdani and Schreyer [19], Meschke and Lackner [20], Carol et al. [21], Voyiadjis et al. [22], Voyiadjis et al. [23], and references therein.
For such a quasi-brittle material as concrete, experimental evidence [24] showed that hydrostatic pressure affects the material yield and failure strength. Some researchers proposed different damage rules to characterize damage in the deviatoric and volumetric modes of the response ([25, 26]; just to mention a few), whereas others applied a hydrostatic stress/strain reduction factor to their constitutive models ([4, 11]; and others).
The main purpose of this work is to compare the effect of a hydrostatic stress/strain reduction factor on the performances of two isotropic damage models. One of these models is the ED model proposed by Tao and Phillips [4], whereas the other is the uncoupled EPD model proposed by Voyiadjis and Taqieddin [11]. These two models were selected for comparison because both of them defined the hydrostatic stress/strain reduction factor in terms of the same thermodynamic internal state variables. This should not be confused with the fact that the internal state variables themselves are functions of different material properties associated with each model.
The subsequent sections of this paper will introduce, in a brief manner, each of these two models as well as their constitutive equations and material properties relevant to this work. For the full derivations of these models, their numerical integration techniques, and their applications and outcomes, the reader is referred to the already published works of Tao and Phillips [4], Voyiadjis and Taqieddin [11], and Taqieddin and Voyiadjis [12].
2 ED model
Within the ED model of Tao and Phillips [4], the Helmholtz free energy (HFE) function is defined in the damaged configuration of the material in terms of the elastic strain equivalence hypothesis ([27–29]; and references therein) and presented as follows:
where ψe is the total free energy density function (per unit volume), entities with an overbar
where
Under purely isothermal conditions, the second law of thermodynamics states that any irreversible process within a material behavior should satisfy the Clausius-Duheim inequality. Applying standard thermodynamic arguments [30], the following statements are derivable:
where
To distinguish between the different contributions of hydrostatic and deviatoric stress/strain components to damage, and in other words, to reduce the susceptibility of the hydrostatic part to damage, Tao and Phillips [4] defined a hydrostatic stress/strain reduction factor χ. They wrote the HFE function, Eq. (1), in terms of the total strain tensor and the hydrostatic mean strain as follows:
The hydrostatic stress/strain reduction factor χ can take different mathematical forms. Depending on the state of loading, this factor can range from a linear function to a highly non-linear exponential or power function. Tao and Phillips [4] presented the following definition of χ:
where c and d are two material constants that are calibrated with experimental results, and with units that render the factor χ dimensionless.
The thermodynamic conjugate forces in tension Y+ and compression Y- are now readily available using Eqs. (4) and (5) as follows:
By paying attention to Eq. (6), it becomes obvious that Eqs. (7) and (8) are non-linear equations that require local iterations during the numerical integration scheme.
Damage initiation under tension or compression is triggered when the thermodynamic conjugate force in tension or compression, respectively, becomes greater than a specified threshold. This is translated into the following two damage criteria:
where
in which a± and b± are four material constants (in tension or compression) to be calibrated by means of uniaxial tensile and compressive experiments of concrete.
This concludes a short introduction to the ED model. The integration procedure of the constitutive equations is thoroughly explained in the work of Tao and Phillips [4], and will not be discussed here for brevity. In this work, the ED model is implemented (by the authors) into a FORTRAN user-defined material subroutine (UMAT) and linked to the NFEA software ABAQUS in order to study the effect of the stress/strain reduction factor χ on the numerical results. The same concrete material properties used by Tao and Phillips [4] are adopted here to simulate the required effect. Uniaxial tension and uniaxial compression numerical results are simulated over a single two-dimensional (2D) plane-strain verification finite element and presented in Figures 1 and 2.
Uniaxial tension verification results (ED model).
(A) Tensile σ11-ε11 curve. (B) Relation between Y+ and φ+. (C) Relation between ε11 and χ+. (D) Relation between φ+ and χ+.
Uniaxial compression verification results (ED model).
(A) Compressive σ11-ε11 curve. (B) Relation between Y- and φ-. (C) Relation between ε11 and χ-. (D) Relation between φ- and χ-.
The experimental results shown in Figure 1A are the tensile test outcomes of Gopalaratnam and Shah [32], whereas the experimental compressive test results shown in Figure 2A are those reported by Karsan and Jirsa [33].
3 Uncoupled EPD model
In such a constitutive model, damage mechanics formulations appear in the elastic domain of the material, whereas plasticity remains in the effective (undamaged) stress space [19]. Therefore, the strain tensor is additively decomposed into elastic and plastic strain tensors:
and the Cauchy stress tensor is written using the effective stress concept and the equivalent strain hypothesis as follows:
This form of the stress tensor leads to the following incremental constitutive equation
which requires a special technique known as the elastic-predictor plastic and damage correctors to perform the numerical integration procedure.
In the EPD model of [11] a multi-hardening pressure-sensitive effective-stress-space plasticity yield criterion is introduced in addition to the ED formulations previously discussed in the ED model (with some adjustments). This plasticity yield criterion is a modification to a criterion first introduced by Lubliner et al. [34] and later adopted by Lee and Fenves [8], Wu et al. [35], and others. It is expressed in terms of the invariants of the effective stress tensor, material hardening functions, and material constants as follows:
where
where
where
The equivalent plastic strains in tension and compression (κ+, κ-) are defined by the following two expressions:
and
where
where
The EPD model of Taqieddin [11] is consistently derived within the framework of irreversible thermodynamics. The HFE function is defined in terms of a suitable set of elastic and plastic internal state variables and presented as follows:
Applying the internal state variable procedure of Coleman and Gurtin [30], followed by the Lagrange minimization procedure (calculus of functions of several variables), the following thermodynamic laws relating the internal state variables to their corresponding conjugate forces are derivable. These laws are lumped here in a single equation for shortness:
where
which facilitates the following derivative:
where αp is a parameter chosen to provide proper dilatancy [8, 35].
The EPD model also incorporates fracture energy-based coefficients [8, 35, 36, 37] to achieve a reasonable degree of discretization insensitivity in numerical calculations. Although these fracture energy-based coefficients greatly influence the NFEA of real-life simulations, they have no effect on verification tests where the dimensions of the verification element are normalized and used in all simulations.
This concludes a very brief introduction to the EPD model of Voyiadjis and Taqieddin [11]. Many details were overlooked for conciseness, especially those related to the numerical integration procedure. The reader is referred to the work of Voyiadjis and Taqieddin (2009) [11] and Taqieddin and Voyiadjis [12], for a comprehensive coverage of all numerical integration aspects.
In this work, the EPD model is also implemented into a UMAT subroutine and linked to ABAQUS to study the effect of the stress/strain reduction factor χ on the numerical results. The same concrete material properties used by Taqieddin [11] are adopted here to simulate the required effect. Uniaxial tension and uniaxial compression numerical results are simulated over the same verification finite element described in the ED model. These results are presented in Figures 3 and 4. The experimental results in these figures are the same as those demonstrated in the ED model.
Uniaxial tension verification results (EPD model).
(A) Tensile σ11-ε11 curve. (B) Relation between Y+ and φ+. (C) Relation between ε11 and χ+. (D) Relation between φ+ and χ+.
Uniaxial compression verification results (EPD model).
(A) Compressive σ11-ε11 curve. (B) Relation between Y- and φ-. (C) Relation between ε11 and χ-. (D) Relation between φ- and χ-.
It should be mentioned at this point that comparing the effect of the hydrostatic stress/strain reduction factors of both models under multiaxial cases of loading is not valid owing to the inability of the ED model to reproduce such results without altering the material properties and model parameters; therefore, the comparison between the two models discussed in this work was deliberately confined within the framework of uniaxial tests. However, the EPD model is capable of producing results of more elaborate loading scenarios. Results of the biaxial tension and biaxial compression verification tests of the EPD model are shown in Figures 5 and 6, respectively. These biaxial results are compared with their uniaxial counterparts produced by the same EPD model. The material and model parameters used in these biaxial tests are the same as those used by Voyiadjis and Taqieddin [11]. The biaxial experimental results shown in Figures 5A and 6A are those of Kupfer et al. [38].
Uniaxial and equibiaxial tension verification results (EPD model).
(A) Tensile σ11-ε11 curves. (B) Relations between Y+ and φ+. (C) Relations between ε11 and χ+. (D) Relations between φ+ and χ+.
Uniaxial and equibiaxial compression verification results (EPD model).
(A) Compressive σ11-ε11 curves. (B) Relations between Y- and φ-. (C) Relations between ε11 and χ-. (D) Relations between φ- and χ-.
The degradation of the elastic stiffness and its recovery under cyclic loading have not been presented previously by the authors of the EPD model under consideration, and will be discussed here. It is well known that concrete exhibits non-linearity after the onset of microcracking, and different softening behaviors are distinguished under tension and compression (see Figures 3A and 4A). Under cyclic loading conditions, however, the degradation mechanisms involve the opening and closing of previously formed microcracks. Some recovery of the elastic stiffness as the load changes sign during a uniaxial cyclic test is verified experimentally, and is more evident as the load changes from tension to compression, causing opened/tensile cracks to close, resulting in compressive stiffness recovery. However, the cracks are not healed [39], i.e., damage variables are considered as non-decreasing material quantities and are stored at every increment of the analysis. To condense such important behavioral information into a numerical expression, Eq. (2) of the ED model is modified and presented here for the EPD model as follows:
where
Cyclic (tension-compression) verification results (EPD model).
(A) Uniaxial σ11-ε11 curve. (B) Relations between ε11 and χ±.
*Amplitude, name=cyc, value=relative, definition=tabular
0., 0., 1., 1., 2., -8., 3., -1.
where “relative” refers to the 0.0006 tensile strain.
Figure 7B shows the stress/strain reduction factors χ±, plotted against the cyclic uniaxial strains in tension and compression.
The effect of the hydrostatic stress/strain reduction factor is also studied under triaxial compressive loading conditions. It is observed during this study that the EPD model, in its current format, cannot depict the large variation in stress values under different levels of confinements. A drastic change has to be made to the definition of the compressive thermodynamic conjugate force in order to suppress damage action and boost plastic hardening during increasing levels of confinement. This is accomplished by adding a maximum compressive principal stress-related term to the definition of Y-, similar to that introduced by Lubliner et al. [34], to result in the following expression:
where γ is a triaxial stress state coefficient that has to be calibrated with experimental data. It should be noted here that Eq. (28) is non-linear, and therefore, any change in the expression of Y- will result in a change in the magnitude of χ-. For illustration, Figure 8 shows the effect of different values of γ on the compressive stress/strain curve of Figure 4A.
Stress/strain curves at different values of γ.
Compression under confinement is tested using the EPD model, with Eq. (28), and compared with the experimental outcomes of Green and Swanson [40], in Figure 9A. The material parameters used in the simulations are
3D compression verification results (EPD model).
(A) Stress/strain under different confinement levels. (B) Reduction factor at different confinement levels.
Figure 9B shows the variation in the hydrostatic stress/strain reduction factor χ- as the confinement level is increased, plotted against the compressive damage variable.
4 Comparisons and conclusions
Many conclusions can be drawn on the basis of the comparison carried out between the two models; nevertheless, the objective of this work will be the main focus of the comparisons. Figures 10A and 11A show the simulated uniaxial stress/strain results of the ED and EPD models under tension and compression, respectively, plotted against their experimental counterparts. Figures 10B and 11B show the strains plotted against the reduction factors in the tension and the compression verification tests, respectively.
Comparison of the tensile test results.
(A) Tensile σ11-ε11 curve. (B) Relation between ε11 and χ+.
Comparison of the compressive test results.
(A) Compressive σ11-ε11 curve. (B) Relation between ε11 and χ-.
In the case of uniaxial tension verification test, and although the two models are similarly capable of reproducing the uniaxial experimental results in Figure 10A, the magnitudes of the reduction factors are quite different for the two models. It is obvious from Figure 10B that less reduction is needed in the case of the EPD model when compared with the ED model. However, when comparing the magnitudes on the abscissa of Figure 10B, it can be seen that the values of the reduction factors are very small, which is justified by the fact that all the energy is consumed in fracturing the material under uniaxial tension and thus no noticeable reduction is required.
When considering the compression verification test (Figure 11B), it can be seen that the magnitudes of the reduction factors in compression are less divergent from each other and are many folds higher than those of the tensile test. The fact that the EPD model required less reduction than the ED model is observed again under uniaxial compression.
These results are justified by looking at Eq. (5), where the HFE function is written in terms of the combined damage variable Φ, the effective fourth-rank elasticity tensor
Another related outcome of the comparisons is the difference in effect of the hydrostatic stress on the tensile and compressive simulations in general. Although many mathematical forms of the reduction factor are possible, the form of χ chosen here to be identical for tension and compression, Eq. (6), clearly shows that the hydrostatic pressure effect is more dominant under compression than under tension – a result consistent with the literature of engineering mechanics.
As for the biaxial verification tests (Figures 5 and 6), and although there is a reduction in strength in the case of biaxial tension (accompanied by damage increase) and an increase in strength in the case of biaxial compression (accompanied by a damage decrease), the effect of the hydrostatic stress/strain reduction factor on the behavior of the material is strongly observed, and its diversion from the uniaxial effect is clearly noticed, especially as the damage spreads in the concrete material.
With respect to the cyclic verification test, the results clearly exhibit the ability of the EPD model to depict stiffness degradation under cyclic tension and compression. The results also demonstrate that the model is capable of simulating stiffness recovery when the material is not fully damaged. The effect of the reduction factor on the simulation is shown in Figure 7B. Even when the tension side of this figure is magnified too many folds, the size and magnitude of the tension loops are very small compared with the compressive ones, clearly reflecting the importance of the reduction factor under compressive loading. The variation in size of the compression loops, however, reflects the demand of higher reduction factors as damage propagates in the material.
Under a 3D compression test, and although the additional term in Eq. (28) greatly enhances the compressive behavior under confinement, the effect of the hydrostatic stress/strain reduction factor is still observed in Figure 9B. As the level of confinement increases, the amount of damage associated with the same reduction factor is decreased, indicating additional resistance of the concrete material when confinement is incrementally provided.
In conclusion, the effect of the hydrostatic stress/strain reduction factor under tensile loadings is observed to be negligible, where almost the entire amount of energy is spent in fracturing the material. Under compressive loading conditions, however, the reduction factor takes an important role in controlling the amount of damage affecting the concrete material.
References
[1] Mazars J. Thèse de Doctorat d’ Etat, L.M.T., Université Paris, France, 1984.Search in Google Scholar
[2] Mazars J, Pijaudier-Cabot G. J. Eng. Mech. ASCE 1989, 115, 345–365.10.1061/(ASCE)0733-9399(1989)115:2(345)Search in Google Scholar
[3] Willam K, Rhee I, Beylkin G. Meccanica 2001, 36, 131–150.10.1023/A:1011905201001Search in Google Scholar
[4] Tao X, Phillips DV. Cement Concrete Compos. 2005, 27, 716–726.Search in Google Scholar
[5] Júnior FS, Venturini WS. Adv. Eng. Softw. 2007, 38, 538–546.Search in Google Scholar
[6] Khan AR, Al-Gadhib AH, Baluch MH. Int. J. Damage Mech. 2007, 16, 361–398.Search in Google Scholar
[7] Yazdani S, Schreyer HL. J. Eng. Mech. ASCE 1990, 116, 1435–1450.10.1061/(ASCE)0733-9399(1990)116:7(1435)Search in Google Scholar
[8] Lee J, Fenves GL. J. Eng. Mech. ASCE 1998, 124, 892–900.10.1061/(ASCE)0733-9399(1998)124:8(892)Search in Google Scholar
[9] Shen X, Yang L, Zhu F. Adv. Struct. Eng. 2004, 7, 461–467.Search in Google Scholar
[10] Contrafatto L, Cuomo M. Int. J. Plasticity 2006, 22, 2272–2300.10.1016/j.ijplas.2006.03.011Search in Google Scholar
[11] Voyiadjis GZ, Taqieddin ZN. Int. J. Struct Changes Solids – Mech. and App. 2009, 1, 31–59.Search in Google Scholar
[12] Taqieddin ZN, Voyiadjis GZ. Int. J. Struct Changes Solids – Mech. App. 2009, 1, 187–209.Search in Google Scholar
[13] Abu-Lebdeh TM, Voyiadjis GZ. J. Eng. Mech. 1993, 119, 1465–1484.Search in Google Scholar
[14] Luccioni B, Oller S, Danesi R. Comput. Methods Appl. Mech. Eng. 1996, 129, 81–89.Search in Google Scholar
[15] Gatuingt F, Pijaudier-Cabot, G. 15th ASCE Engineering Mechanics Conference, June 2–5, Columbia University: New York, 2002.Search in Google Scholar
[16] Salari MR, Saeb S, Willam KJ, Patchet SJ, Carrasco RC. Comput. Methods Appl. Mech. Eng. 2004, 193, 2625–2643.Search in Google Scholar
[17] Shao JF, Jia Y, Kondo D, Chiarelli AS. Mech. Mater. 2006, 38, 218–232.Search in Google Scholar
[18] Taqieddin ZN, Voyiadjis GZ, Almasri AH. J. Eng. Mech. ASCE 2012, 138, 530–541.10.1061/(ASCE)EM.1943-7889.0000344Search in Google Scholar
[19] Ju JW. Int. J. Solids Struct. 1989, 25, 803–833.Search in Google Scholar
[20] Meschke G, Lackner R. In: Computational Plasticity. Owen, DRJ, et al., Eds. CIMNE: Barcelona, 1997, pp. 1543–1550.Search in Google Scholar
[21] Carol E, Rizzi I, Willam K. Int. J. Solids Struct. 2001, 38, 491–518.Search in Google Scholar
[22] Voyiadjis GZ, Taqieddin ZN, Kattan PI. Int. J. Plasticity 2008, 24, 1946–1965.10.1016/j.ijplas.2008.04.002Search in Google Scholar
[23] Voyiadjis GZ, Taqieddin ZN, Kattan PI. Int. J. Damage Mech. 2009, 18, 603–638.Search in Google Scholar
[24] Resende L. Comput. Methods Appl. Mech. Eng. 1987, 60, 57–93.Search in Google Scholar
[25] Resende L, Martin JB. Comput. Methods Appl. Mech. Eng. 1984, 42, 1–18.Search in Google Scholar
[26] Papa E, Talierco A. Eng. Fract. Mech. 1996, 55, 163–179.Search in Google Scholar
[27] Steinmann P, Miehe C, Stein E. Comput. Mech. 1994, 13, 458–474.Search in Google Scholar
[28] Lemaitre J, Chaboche JL. Mechanics of Solid Materials, 2nd ed. Cambridge University Press: Cambridge, 1998.Search in Google Scholar
[29] Voyiadjis GZ, Kattan PI. Advances in Damage Mechanics: Metals and Metal Matrix Composites, with an Introduction to Fabric Tensors, 2nd ed. Elsevier: Oxford, 2006.Search in Google Scholar
[30] Coleman BD, Gurtin ME. J. Chem. Phys. 1967, 47, 597–613.Search in Google Scholar
[31] Kachanov LM. Izv. Akad. Nauk USSR Otd. Tech. 1958, 8, 26–31 (in Russian).Search in Google Scholar
[32] Gopalaratnam VS, Shah SP. ACI J. 1985, 82, 311–323.Search in Google Scholar
[33] Karsan ID, Jirsa JO. J. Struct. Div. ASCE 1969, 95, 2535–2563.10.1061/JSDEAG.0002418Search in Google Scholar
[34] Lubliner J, Oliver J, Oller S, Onate E. Int. J. Solids Struct. 1989, 25, 299–326.Search in Google Scholar
[35] Wu JY, Li J, Faria R. Int. J. Solids Struct. 2006, 43, 583–612.Search in Google Scholar
[36] Feenstra PH, Borst RD. Int. J. Solids Struct. 1996, 33, 707–730.Search in Google Scholar
[37] Nguyen GD, Houlsby GT. Int. J. Numer. Anal. Methods Geomech. 2008, 32, 353–389.Search in Google Scholar
[38] Kupfer HB, Hildorf HK, Rusch H. ACI J. 1969, 66, 656–666.Search in Google Scholar
[39] Voyiadjis GZ, Shojaei A, Li G, Kattan P. Int. J. Damage Mech. 2012, 21, 391–414.Search in Google Scholar
[40] Green SJ, Swanson SR. AFWL-TR-72-244. US Air Force Weapons Laboratory: Kirtland Air Force Base, NM, 1973.Search in Google Scholar
©2013 by Walter de Gruyter Berlin Boston
This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
