A framework of elastic–plastic damaging model for concrete under multiaxial stress states

TL;DR: In this article, a constitutive model for concrete characterized by a combined plastic-hardening-damage-fracture dissipative criterion developed within the framework of the simple material model is presented.

About: This article is published in International Journal of Plasticity.The article was published on 2006-12-01 and is currently open access. It has received 106 citations till now. The article focuses on the topics: Strain hardening exponent & Damage mechanics.

Summary

Introduction

• Submitted on 30 Oct 2013 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not.
• Scalar models with two damage variables have also been proposed, in an attempt to distinguish between tension and compression damage mechanisms (Mazars, 1986; Faria et al., 1998; Comi and Perego, 2001; Marfia et al., 2004).
• The two energy functionals ruling the model depend on few material parameters, to which can be given mechanical meanings.
• This does not mean that they are applicable in all cases, since, given the diversity of phenomena that must be modelled, and the ever increasing findings of research, better and more realistic forms can surely be suggested, possibly problem-oriented.

2. Experimental behavior of concrete under multiaxial stress states

• The main experimental observation that the model attempts to reproduce is the lack of a clearly defined elastic response except for very small deformations; actually, damage and cracks form and develop from very low values of the load, well before the peak stress is attained.
• One major problem in constructing a full triaxial model for concrete is that many experimental observations apply to uniaxial behavior, and indeed a variety of models, empirically founded, exists for the uniaxial response, that are of great utility in the practical analysis of frame members.
• After the classical experiments of Kupfer and Gerstle (1973), extensive but, unfortunately, not exhaustive data have recently been furnished for instance by Mier (1984) and Sfer et al. (2002).
• In triaxial compression two main collapse mechanisms have been observed.
• The other, characterized by the formation of many small cracks, is typical of stress states obtained superimposing an uniaxial compression on a hydrostatic compression state (stress states lying close to the compression meridian).

3. A constitutive framework for plain concrete

• This allows a much more robust numerical implementation of the model, that is one of the main concerns for practical applications.
• The model, in this way, does not make a strong distinction between plasticity and damage, that are thoroughly coupled, in principle, in any yield function used.
• Damage variables are responsible for phenomenologically reproducing the strength and the stiffness reduction: two different damage mechanisms associated with tensile and compressive strain processes are enclosed, related to two different scalar damage variables.
• In Section 3.2, the internal energy potential is defined and specifically in Sections 3.2.1 and 3.2.2, the elastic and hardening potentials are, respectively introduced.

3.1. Field variables and generalized elastic constitutive equations

• It is important to underline that the deformations are regular functions throughout the whole structural domain.
• Note also that the internal variables are all scalar functions.
• Indeed kinematic hardening is not considered important for concrete.
• Only the latter property is included in the analysis, adopting two different scalar damage variables associated with tensile and compressive strain processes.
• By standard thermodynamic arguments the driving forces s, dual to the kinematic variables ge, are obtained differentiating the free energy s 2 oge eðgeÞ where the symbol o denotes sub-differentiation, in order to account for the common case of non-smooth energy functionals.

3.2.1. Elastic energy potential

• A general methodology for bimodular elastic materials was introduced by Curnier et al. (1993).
• As an alternative choice, the separating interface can be defined by the eigenvalues of the elastic strain tensor, often splitting it into its tensile and compressive components by means of a polar decomposition (Lubarda et al., 1994; Papa and Taliercio, 1996; Carlson and Hoger, 1986).
• The last choice seems more appropriate for a non-isotropic damage model.
• In the following the interface tr ee 0 is adopted, that is sufficient for illustrating the potentiality of the constitutive framework proposed.
• G þðx1e ;x2eÞ ¼ G ðx1e ;x2eÞ, so that only three parameters can be assigned independently.

3.2.2. Hardening potentials

• The hardening potential in (3) is given by the sum of two contributions.
• The underlying assumption for the hardening potential (6) is that, after a rather low level of stress, irreversible phenomena such as microcracks occur, causing irreversible strains as well as a decay of the tangent stiffness.
• Thanks to the structure of formula (6) it is possible to model the non linear ascending branch of the stress strain behavior of concrete, without introducing ad hoc rules.
• Moreover, the onset of damage, observed well before the peak stress is reached, is correctly reproduced.
• According to experimental observations (see Section 4) it has been defined as (Resende, 1987) W2ðaveÞ ¼.

3.2.3. Generalized elastic relations

• Therefore, a different development for the tensile damage driving force f1 is obtained according to whether a strain process with positive or negative trace is considered (and in any case with trace different from zero).
• The compressive damage force f2 develops, on the contrary, in the same way in both subdomains.

3.3.1. The dissipation functional

• As stated after Eq. (2), constitutive equations for the inelastic behavior are specified assigning the conjugate dissipation potential.
• The dual potential proposed is the indicator function of a suitable generalized elastic domain dcðsÞ ¼ indKðsÞ KðsÞ ¼ s : gðsÞ 6 0f g ð9Þ g(s) being the yield function.
• Therefore the dissipation is uniquely defined once a form for the yield function is chosen.
• A strain rate dependent behavior is also considered for processes whose time scale is comparable with a characteristic relaxation time of the material.
• The model so obtained is a generalized viscoplastic one, including a Perzyna- type formulation for the strain rate _gp ¼ l gðsÞ s0 þ ogðsÞ ð11Þ the bracket being the overstress function.

3.3.2. Generalized elastic domain

• The generalized elastic domain K is defined in the extended space of stresses, thermodynamic forces and conjugated damage variables.
• The hardening and damage variables evolve according to the elastic constitutive equations.
• When a proportional loading is applied, the static variables (r, v, f2) follow a complex path in the generalized stress space; the Ottosen limit state is obtained when this path intersects the limit surface g1.
• This aspect will be discussed in Section 4.
• The expression of this yield function is quite primitive since experimental information is insufficient.

4. Predictions of the model for simple loading histories and parameters estimate

• Initial elastic moduli G0 and k0 (or initial Young modulus E0 and Poisson’s ratio m0); initial hardening modulus H0; damage rate exponents n1 and n2, also known as Namely, Internal energy.
• B1, b2, k1, k2 coefficients; uniaxial elastic limit stress in compression fc; damage rate kf, also known as Ottosen-like criterion.
• Maximum degree of compaction a; compaction rate D; limit elastic hydrostatic stress rcm, also known as Hydrostatic cap.
• Not all of them can be easily correlated to a single material property.
• The model has been calibrated and compared to classical experimental data available in the literature.

4.1. Tensile processes

• Note that both the damage static variables f1 and f2 are different from zero even in the elastic range when x1e ¼ x2e ¼.
• It is possible to demonstrate the identity: Ef ¼ f0 ð14Þ Fig. 2(a) presents the plot of the two terms of the fracture energy as a function of the damage exponent n1.
• Therefore, the parameters a and b affect the shape of the limit envelope as well as the compression limit value of the strength predicted by the fracture criterion g2.
• Contour plots of the ratio between the compressive and the tensile limit stresses are reported in Fig. 2(b) as a function of the parameters a and b.
• The regions highlighted in the plot refer to ratios equal to 10, 20, 30, respectively.

4.2. Compressive processes

• As will be shown in the next section, initially only the plastic criterion g1 is activated and the variables x2 and a develop.
• A numerical uniaxial compression test is presented in Fig. 4(a), where both the axial and the lateral strains are reported vs. the axial stress.
• The transition from the linear to the non-linear behavior is ruled by the initial hardening modulus H0.
• Therefore a value near to 30 50E0 has been used in the numerical simulations.
• The values of the damage parameters n2 and kf essentially affect the peak stress, the post-peak region, the amplitude of the area below the r e curve and the unloading modulus as well.

4.3. Hydrostatic compression

• Da ln a ka ( In this kind of loading condition the kinematic damage variables are constantly null, be- cause only the cap g3 is activated, that is no damage develops in hydrostatic compression.
• Furthermore, the damage forces f1 and f2 do not evolve during the entire process.
• A with slope equal to the elastic unloading: orm oev k!a ¼ K0.
• Then the maximum void compaction a can be estimated reading on the rm ev graph of Fig. 10 the value of the residual volumetric strain corresponding to the point of the plastic branch with a slope near K0.
• From (17) the estimation of the material parameter D is obtained: D ¼ K0Kp0 K0 Kp0 a2 Fig. 11(a) presents a comparison of the model prediction with the experimental hydrostatic compression test by Green and Swanson (1973).

5. Multiaxial stress states and failure envelope

• A numerical procedure has been used for calculating the evolution of the stresses up to the peak for several multiaxial load paths.
• Fig. 12 compares the elastic and the failure (peak) domain for biaxial stress states.
• In Fig. 15(a) compression meridians (# 60 ) for elastic and limit envelopes are compared with the experimental results of triaxial tests by Mier (1984).
• (b) Deviatoric sections of the failure envelope.
• This is in qualitative accordance with experimental findings and to an extent justifies the choice of an associative model.

6. Conclusions

• The applications reported in Sections 4 and 5 suggest that following the methodology described in Section 3 some significant aspects of concrete mechanical behavior can be reproduced by a single elastic plastic-damage model.
• In the model only isotropic damage has been considered for the sake of simplicity; in this way damage is spread over a finite zone and its directionality is accounted for by the evolution of the stress state.
• Other aspects of the model need better specification, like the form of the limit function for hydrostatic compression.
• Finally, a systematic optimization procedure should be developed for the parameter identification.
• Numerical issues, such as strain localization and computational procedures will be reported in a forthcoming paper that deals with applications to plain and reinforced concrete structure as well.

Figures

Fig. 13. Yield criteria on the meridian plane.

Fig. 14. Ottosen criterion: deviatoric sections for present model and isotropic model for different values of the hydrostatic stress.

Fig. 16. Ottosen criterion. (a) Biaxial compression envelopes for fixed values of the damage variable x2e . (b) Iso damage compressive meridian sections.

Fig. 1. Uniaxial cyclic tensile process. (a) Axial strain vs. axial stress. (b) Lateral strain vs. axial stress.

Fig. 10. Typical hydrostatic compression curve.

Fig. 17. Reinforced concrete wall. (a) Mesh, material data and loading condition. (b) Tensile damage distribution after the peak load.

Fig. 12. Biaxial elastic and limit(peak) domain. Experimental data by Mier (1984).

Frequently asked questions

Q1. What have the authors contributed in "A framework of elastic-plastic damaging model for concrete under multiaxial stress states" ?

Contrafatto et al. this paper presented a framework of elastic-plastic damaging model for concrete under multiaxial stress states. 

Q2. What is the underlying assumption for the hardening potential?

The underlying assumption for the hardening potential (6) is that, after a rather low level of stress, irreversible phenomena such as microcracks occur, causing irreversible strains as well as a decay of the tangent stiffness. 

Q3. What is the generalized elastic domain K?

The generalized elastic domain K is defined in the extended space of stresses, thermodynamic forces and conjugated damage variables. 

Q4. What is the main observation that the model attempts to reproduce?

The main experimental observation that the model attempts to reproduce is the lack of a clearly defined elastic response except for very small deformations; actually, damage and cracks form and develop from very low values of the load, well before the peak stress is attained. 

Q5. What is the dimension of the reference volume to which the model applies?

That is the dimension of the reference volume to which the model applies is large enough so that the energy flux through its boundaries is due only to the mechanical work of the stresses. 

Q6. What is the main problem in constructing a full triaxial model for concrete?

One major problem in constructing a full triaxial model for concrete is that many experimental observations apply to uniaxial behavior, and indeed a variety of models, empirically founded, exists for the uniaxial response, that are of great utility in the practical analysis of frame members. 

Q7. What is the effect of the damage on the elastic moduli?

damage affects the elastic moduli for tensile or compressive stress states differently, since a discontinuity is observable in loading unloading stress paths. 

Q8. What is the simplest definition of the dual potential?

The dual (complementary) potential proposed is the indicator function of a suitable generalized elastic domaindcðsÞ ¼ indKðsÞ KðsÞ ¼ s : gðsÞ 6 0f g ð9Þg(s) being the yield function. 

Q9. What is the reason for the deviation in the numerical simulation of lateral strain from experimental data?

The necessity of some correction on the experimental data is one reason for the deviation in the numerical simulation of lateral strain from experimental data for high values of the axial strain. 

Q10. What are the parameters that affect the peak stress, the post-peak region, the ampli?

The values of the damage parameters n2 and kf essentially affect the peak stress, the post-peak region, the amplitude of the area below the r e curve and the unloading modulus as well. 

Q11. What is the kinematic damage of the cap g3?

Da ln a ka(In this kind of loading condition the kinematic damage variables are constantly null, be-cause only the cap g3 is activated, that is no damage develops in hydrostatic compression. 

Q12. What is the common choice when a bimodular material model is adopted?

The choice commonly assumed when a bimodular material model is adopted (Comi and Perego, 2001; Faria et al., 1998) is coupled with an isotropic damage model. 

Q13. What is the simplest choice for separating the elastic energy?

The simplest choice directly derived by the original Curnier proposal, is to take, as separating interface, the deviatoric plane tr ee 0 in the space of the elastic strain.