Modification of Concrete Damaged Plasticity model. Part II: Formulation and numerical tests

A refined model for elastoplastic damaged material is formulated based on the plastic potential introduced in Part I [1]. Considered model is an extension of Concrete Damaged Plasticity material implemented in Abaqus [2]. In the paper the stiffness tensor for elastoplastic damaged behaviour is derived. In order to validate the model, computations for the uniaxial tests are performed. Response of the model for various cases of parameter’s choice is shown and compared to the response of the CDP model.


Introduction
An alteration to the Abaqus' Concrete Damaged Plasticity model is made in attempt to improve its flexibility. The new potential shown in Part I [1] is introduced to the equations of elasticity and plasticity theory in order to derive the stiffness tensor for elastoplastic damaged behavior of material. As mentioned in Part I, the extended model requires input of twelve parameters and four material functions.
To confirm validity of this material model, its consistency with basic experiments has to be proven. The objective of the second part of the present paper is to show and analyze numerical results of the uniaxial loading test for the proposed model concerning improved potential. In order to achieve this goal, a simple forward-Euler algorithm was created using Mathematica [3]. The influence of free parameters 1 m , 2 m , 3 m and 4 m on the uniaxial test results is considered. The first two parameters indicate the meridians of Lubliner's condition used as asymptotes of the potential, the third one -location of the potential's apex and the last one -the curvature in the vertex region and the zone of transition between potential's asymptotes (near 1 0 V ). The results of basic tests are compared for various sets of parameters' values.
As it has been noted in Part I, the difference between the new and original Abaqus potential [2] allows more flexibility when it comes to reflecting material properties. Examples of numerical results are shown in comparison to those acquired using the CDP model.

Formulation of the model
Considered elastoplastic damage model for concrete is based on the Lubliner's two-surface failure criterion with isotropic damage described by one scalar damage variable. Plasticity and damage surfaces are the same. The non-associative flow rule is proposed, which is based on the plastic potential described in Part I [1]. Constitutive equations are formulated in the effective stress space [4]. The basic equations and assumptions of the model are as follows.
x The relation between nominal stress tensor and effective stress tensor is: where d is a damage parameter.
x Additive decomposition of the tensor of strain rate into elastic ( E ε E ε ) and plastic ( P ε P ε ) parts and the Hooke's constitutive law for elastic behavior hold: where 1 3 2 K G C I I 1 I I is the stiffness tensor for isotropic elastic undamaged material.
x The dissipation during plastic flow is positive: (3) x The yield criterion is Lubliner's condition [5,6]: where: x The plastic potential ) is described in Part I. The non-associative flow rule and the consistency condition state: x Plastic hardening is prescribed by the following formulae: where P E is a hardening modulus and , , , 0 compressive damage parameters and 0 E is an initial (undamaged) Young modulus. x The formulas for damage parameters are: where T H and C H are material parameters, and T s and C s reflect both material properties and the current effective stress state. See reference [2] for more details.
The Lubliner's yield criterion (4) contains four material constants: values of the compressive yield limit, the compressive biaxial yield limit, the compressive triaxial yield limit and the tensile yield limit of material. To define the plastic potential one needs additionally to determine constants 1 m , 2 m , 3 m and 4 m . The other equations contain four more material constants and four material functions concerning hardening and evolution of damage parameters. Full description of the proposed model of elastoplastic material with isotropic damage requires determining twelve parameters (four of which are chosen arbitrarily) and four material functions, compare Tab. 1 and [7,8,9,10].
uniaxial tension and compression, biaxial compression, triaxial compression

Stiffness tensor
The stiffness tensor for the introduced material model ( EP C ) defines the relation between effective stress rate tensor and strain rate tensor: To derive a formula for the stiffness tensor one has to investigate several cases. It is essential to consider separately two Lubliner's surfaces (for compression 1 0 V d , and ), as well as special cases for the tensile and compression meridians only.
Taking into consideration Hooke's law with the additive split of strain rate tensor (2), the flow rule and consistency condition (6) for 0 /3 T S , one arrives at: where: f and s f are functions of r [1] and min p is the minimum eigenvalue of gradient p . Plastic multiplier is given by the equation: The stiffness tensor (11) and the plastic multiplier (12) for the tensile meridian can be determined by finding limits of expressions (11) and (12)

Uniaxial test
Uniaxial test is a basic experiment determining properties of concrete. It is frequently used to adjust values of the material models' parameters. During the primary stage of loading material is elastic then plastic deformation emerges. Along with plasticity, damage occurs which is observed as the degradation of initial elastic moduli. As the loading reaches a certain point, decrease of volume appears.
In the computations a cyclic change of normal component of stress tensor XX V is assumed. Applied load program is shown on Fig. 1(a) (Fig. 1(b),(c)) which is the outcome of the assumed constitutive law. The development of irreversible deformation begins in the first cycle of loading. Plastic components of the strain tensor become to dominate over elastic parts as the number of cycles grows, see Fig. 1(b). The relative volume (volumetric strain) grows for the initial stages of analysis but then it decreases ( Fig. 1(d)). The damage parameter d grows during the plastic flow, see Fig. 1(e). Its development is responsible for the change of initial Young modulus, which is depicted on Fig. 1(c). d reaches the value of 0.25, which is usually near the point of complete destruction of material.
Plastic dissipation during the process is controlled to ensure thermodynamic consistency of the model. In the presented analysis dissipation is positive and increases as the yielding advances, see Fig. 1(f).
The difference between the original and the new potential and influence of parameters For both tests, regardless of i m selection, the XX H component of strain tensor remains unchanged. It also matches the result for the CDP model. Furthermore, the modification of potential does not seem to influence damage parameter, which is only slightly different between models and does not vary for distinct values of i m .  For uniaxial compression test, the difference between original potential and new potential for variuos values of 1 m and 2 m is depicted on Fig. 3. An essential dissimilarity is observed in values of the lateral strain YY ZZ H H , which implicates the difference of relative volume ( tr V H ε ). As it is shown on Fig. 2(a), incerease of 1 m implicates growth of V H , while increase of 2 m causes V H to decrease (Fig. 2(b)). Parameters 1 m and 2 m govern the slope of the plastic potential's asymptotes, so their great impact on computation results is apparent. However, 2 m affects plastic strain level stronger than 1 m , because it controlls the compressive surface, where the considered test is located. Plastic lateral strain (or V H ) for the CDP model is appreciably smaller than for the modified model. For the uniaxial compression test parameters 3 m and 4 m do not have any noticeable impact on the obtained results. The lack of influence is easily explicable, for the parameters control mainly the region near the potential's apex, whereas the test is located on compression cone. The results of uniaxial tension test are depicted on Fig. 4. It is shown that they depend on 1 m , 2 m and 4 m , while the influence of 3 m is negligible, compare Fig. 4(c). For lower values of 1 m (Fig. 4(a)) and low values of 4 m (Fig. 4(d)) negative lateral component of plastic stress is obtained which implicates that one should choose the tensile meridian to Assumed values of the material parameters are chosen arbitrarily to depict typical properties of concrete. To reflect properties of particular type of concrete and prove quantitative compatibility of the model it is necessary to propose a proper calibration procedure.
Strain-stress curves for concrete subjected to uniaxial load include decreasing branch not shown in the simulation. It is possible to model material behavior in this area assuming an appropriate law of hardening (softening) instead of (7).

Conclusions
The results of a simple simulation of the uniaxial cyclic test prove that the proposed model properly reflects basic features of material behavior, although it is necessary to perform more complex numerical computations to ensure its qualitative and quantitative compatibility with the observed results of experiments.
Another important issue to be addressed is the calibration of the free parameters included in the definition of potential. A preliminary analysis shows that the uniaxial tests results are affected mainly by 1 m , 2 m and 4 m . As for 3 m , another test should be performed to determine its advisable value, preferebly located in the triaxial tension zone.
Extended flexibility of the presented potential allows for more accurate calibration of the material response. This feature is desirable when considering structures where material failure can occur both by tension and compression in some regions. Original Abaqus CDP model cannot cover such cases correctly.