Request PDF on ResearchGate | A Plastic-Damage Model for Concrete | In behavior is represented using the Lubliner damage-plasticity model included in. behavior of concrete using various proposed models. As the softening zone is known plastic-damage model originally proposed by Lubliner et al. and later on. Lubliner, J., Oliver, J., Oller, S. and Oñate, E. () A Plastic-Damage Model for Concrete. International Journal of Solids and Structures, 25,

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A coupled plastic damage model with two damage scalars is proposed to describe the nonlinear features of concrete.

The constitutive formulations are developed by assuming that damage can be represented effectively in the material compliance tensor. Damage evolution law and plastic damage coupling are described using the framework of irreversible thermodynamics. The plasticity part is developed without using the effective stress concept. A plastic yield function based on the true stress is adopted with two lubkiner functions, one for tensile loading history and the other for compressive loading history.

To couple the damage to the plasticity, the damage parameters are introduced into the plastic yield function by considering a reduction of the plastic hardening rate. The specific reduction factor is then deduced from the compliance tensor of the damaged material. Finally, the proposed model is applied to plain concrete.

Comparison between the experimental data and the numerical simulations shows that the plaatic-damage model is able to describe the main features of the mechanical performances observed in concrete material under uniaxial, biaxial, and cyclic loadings. The mechanical behavior of concrete is unique, due plastc-damage the influence of micromechanisms involved in the nucleation and growth of microcracks and plastic flow.

This behavior is characterized by several features as follows: These features have to be considered in constitutive models of concrete materials.

Ckncrete model these features, several mechanics theories have been used. In general, the comcrete mechanics theories can be used to model the nucleation and growth of microcracks [ plastic-damagee — 3 ], whereas the plasticity theories can be used to model the plastic flow component of the deformation [ 4 ]. Because damage mechanics can be used to describe the progressive weakening of solids due to the development of microcracks, and because plasticity theories have been successfully applied to the modeling of plaxtic-damage processes, several plastic damage models of concrete have been developed through combining damage mechanics and plasticity theories [ 5 — 13 ].

These coupled plastic damage models CPDMs could be formulated in the irreversible thermodynamics framework and can be easily applied to describe the essential nonlinear performances of concrete including the strain softening and the stiffness degradation.

One type of CPDMs is based on the concept of effective stress, which was initially proposed by Kachanov [ 14 ] for metal creep failure. In plastic-damagf models, the plastic yield function is defined in the effective configuration pertaining to the stresses in the undamaged material [ 57 — 9 ]. Many authors used this approach to couple damage to plasticity. Some of these models, developed by the use of two damage scalars and damage energy release rate-based damage criteria, show excellent performance in reproducing the typical nonlinear behavior concrwte concrete materials under different monotonic and plaastic-damage load conditions.

The other concreet of CPDMs is opposite of the above. In these models, the true stress appears in the plastic process, which clearly couples plasticity to damage [ 6101115 — 17 ]. However, considering the influence of damage on plastic flow, it is difficult to develop a plastic yield function that can be used to describe the plastic deformation and damage growth of concrete simultaneously.

Even if several types of expressions for the plastic yield function written in terms of the effective stress have been successfully applied to model some of the typical nonlinearities of concrete such as the volumetric dilation and strength increase under multidimensional compressionthey cannot be directly used in the true stress space. A common approach to solve this problem is to introduce a reduction factor of plastic hardening rate into the plastic yield function.

The plastic yield function is usually expressed by a function of the stress tensor and plastic hardening function, so the damage parameters are included in the plastic yield function with the introduction of the reduction factor in the plastic hardening function.

Several authors applied this approach to develop the CPDM with a true stress space plastic yield function. The influence of damage on the plastic flow is calculated by considering a reduction of the plastic hardening rate. The reduction factor is introduced concrets the plastic yield function for the coupling between the damage evolution and the plastic flow.

Nguyen and Houlsby [ 10 ] proposed a double scalar damage-plasticity model for concrete based on thermodynamic principles. In this model, the plasticity part is based on the true stress using a yield function with two hardening functions, one for the tensile loading history and the other for the compressive loading history. Two reduction factors are introduced into the tensile and the compressive hardening functions to consider the influence of the tensile and compressive damage mechanisms on the plastic flow, respectively.

The model incorporates a plastic yield function written in terms of the true stress. Damage variables are introduced all over the plastic yield function.

For the models mentioned above, the reduction factor of the plastic hardening rate is equal to concrehe damage variable defined in these models, and there is no distinction in the aspect of choosing a reduction factor between uniaxial and biaxial loadings. The plastic hardening rate is generally defined by the equivalent plastic strain, so the sum of the principal damage values in three directions: By introducing such a reduction factor, the widely used plastic yield functions, such as those applied by Lee and Fenves [ 7 ], Wu et al.

In the present work, a coupled plastic damage model is formulated in the framework of thermodynamics. The constitutive formulations are developed by considering an increment in the concrete compliance due to microcrack propagation.

The plasticity part is based on the true stress using a yield function with tensile and compressive hardening functions.

The plasticity yield function widely used in effective stress space is modified plasitc-damage be applied in this study by considering a reduction in the plastic hardening rate. Comparison between the experimental data and the numerical simulations shows that the proposed model is able to represent the main features of mechanical performances observed in concrete material under uniaxial, biaxial, and cyclic loadings. The model presented in this work is thermodynamically consistent and is developed using internal variables to represent the material damage state.

The assumption of small strains will be adopted in this work. In the isothermal conditions, the state variables are composed of the total strain tensorscalar damage variables andplastic strainand internal variables for plastic hardening and. The total strain tensor is decomposed into an elastic part and plastic part: To establish the constitutive law, a thermodynamic potential for the damaged elastoplastic material should be introduced as a function of the internal state variables.

Considering that the damage process is coupled with plastic deformation and plastic luboiner, the thermodynamic potential can be expressed by where and are the elastic and plastic energy for plastic hardening of damaged material, respectively. According to the second principle of thermodynamics, any arbitrary irreversible process moel the Clausius-Duhem inequality as.

Substituting the time derivative of the thermodynamic potential into the inequality yields. Because the inequality must hold for any value of,andthe constitutive equality can be obtained as follows: It is assumed in 6 that the damage concrehe be represented effectively in the material plaatic-damage tensor. This assumption is in line with many published papers. Based on this consideration, several researchers proposed an elastic degradation model in which the material stiffness or compliance was adopted as the damage variable.

The elastic degradation theory was then extended to the multisurface elastic-damage model [ 21 ] and to the combined plastic damage model [ 6151622 ]. Considering that an added compliance tensor is induced by the microcrack propagation, the fourth-order compliance tensor is decomposed as follows [ 23 ]: Microcracks in concrete are induced in two modes: In general, the splitting mode is dominant in tension, whereas the compressive mode is dominant in compression.

To incorporate these modes into the formulation, the stress tensor is decomposed into a positive part and negative part: According to Faria et al. To progress further, evolutionary relations are required for the added compliance tensors and.

Based on the previous work of Yazdani and Karnawat [ 6 ], Ortiz [ 23 ], and Wu and Xu [ 25 ], the added compliance tensors are expressed in terms of response tensors and such that where the response tensors and determine the evolution directions of the added compliance tensors andrespectively. The thermodynamic forces conjugated to the corresponding damage variables are given by where the terms and are the derivatives of the stiffness tensor of the damaged material with respect to the damage mode andrespectively.

Substituting 14 into 13 and calling for 5one obtains. Finally, the rate form of the constitutive equation can be given as. The derivation of the detailed rate equation from 16 requires determining the evolution laws of the damage variables and plastic strains.

Kubliner the basis of the normality structure in the continuum mechanics, the evolution law for the damage variables can be derived by in which and denote the tensile and compressive damage multipliers, respectively. In the special case of elastic damage loading without plastic flowthe damage multipliers and can be determined by calling for the damage consistency condition under static loading: Irreversible plastic deformation will be formed during the deformation process of concrete.

The plastic strain rate can be determined using a plastic yield function with multiple isotropic hardening criteria and a plasticity flow rule. The yield function determines under what conditions the concrete begins to yield and how the yielding of the material evolves as the irreversible deformation accumulates [ 26 ].

### A Coupled Plastic Damage Model for Concrete considering the Effect of Damage on Plastic Flow

According to Shao et al. The plastic hardening functions and can be deduced by the derivative of the thermodynamic potential: The plastic potential is also a function of the stress tensor, the scalar damage variables, and the internal hardening variables.

Note that is the yield function in the associated flow rule. At the given function of the plastic potential, the evolution law of the plastic strain is expressed as follows: In the special case of plastic loading without damage evolution andthe plastic multiplier can be determined from the plastic consistency condition: The numerical algorithm of the proposed constitutive model is implemented in a finite element code. The elastic predictor-damage and plastic corrector integrating algorithm proposed by Crisfield [ 28 ] and then used by Nguyen and Houlsby [ 10 ] is used here for calculating the coupled plastic damage model.

In this algorithm, the damage and plastic corrector is along the normal at plastic-samage elastic trial point, which avoids plastic-dzmage the intersection between the predicting increments of elastic stress and the damage surfaces. Furthermore, this algorithm ensures that the plastic and damage consistent conditions are fulfilled at any stage of the loading process.

For details on the numerical scheme and the associated algorithmic steps, refer to the published lublinfr [ modl28 ].

The following notational convention is used in this section: The elastic trial stress is defined as follows: By solving 1617and 21 in terms of the trial stress, the increments of the equivalent plastic strains andplastic strainand damage variables and can be obtained: The expression for the increments of the plastic strain can be obtained by substituting 27b into Substituting 28 into 27a yields the relationship between the stress and the damage variables, written in incremental form as follows: Substituting 29 into 27c and 27d yields the increments of the damage variables and.

Back substituting the increments of the damage variables into 29 results in the stress increment. Specific functions for concrete are proposed based on the general framework of the coupled plastic damage model given in the previous section. Generally, the parameters and are the functions of the damage scalars, respectively. Their mathematical expressions can be logarithmic, power, or exponential functions.

The specific expressions of and are given as where and are, respectively, the initial damage energy release threshold under tension and compression, and are the parameters controlling the damage evolution rate under tension, and and are the parameters controlling the damage evolution rate under compression. The plastic yield function adopted in this work was first introduced in the Barcelona model by Lubliner et al.

## Mathematical Problems in Engineering

It has proven to be excellent in modeling the biaxial strength of concrete. However, the yield function is usually used in effective stress. Considering that damage induces a reduction in the plastic hardening rate, the damage parameters are then introduced into the yield function by the plastic hardening functions of damaged material [ and ].