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 hardening functions, one for tensile loading history and the other for compressive loading history.
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The lirst view is often held with royard to ductile crystalline solids, cspcciillly metals. The broitdcst arelt of success of pklsticity theory with conorctc is the treatment of reinforced concrete see Chcn, for 3 survey of the results und other situations in which the material acts primarily in compression.
Rushid Suidan and Schnobrich In spite of the success of this :tpproitch in solving numcrotjs problems, it presents some inconvenient fcitturcs that limit its usefulness. Some of these limitations could bo avoided if a single constitutivc model could be used that governs the non-linear behnvior of concrete.
The variable K can be non-dimensionalized so that its maximum value is unity. As was also said. Numerous improvements to these yield criteria have been proposed in recent years : Chen and Chen Ottosen Chen 19SZ.
Podgbrski Fnrdisand Chen A first attempt of the authors to obtain suitable yield criteria for concrete in the form of eqn 2 was to modify the Mohr- Coulomb criterion to tit enpcrimsntal data. Numerical results obtained in the finite-clement analysis of various concrctc structures using tbc framework of standard plasticity theory wcrc cncnuraging and motivated the present rcscarch.
A new yield criterion of the form 2. Howcvcr, unlike the usual plasticity models with isotropic hardcniny, c is not nocsssarily taken simply as :I function of K.
Unianial curves a-f : a tension: b compresson 2. DeJhirion of K Uniasial stress states. Let us further assume that the areas under these curves arc finite and equal to. For the tension test. The arguments have been advanced both on physical grounds and on the basis of the mesh-sensitivity of numerical solutions obtained by means of the finite-element method. In problems involving tensile cracking. G, may be identified with the specific fracture energy Cc, defined as the energy required to form a unit area of crack.
For the characteristic length I, various approaches have been proposed : hLilnt and Oh , Crisfeld Ccrvcra el ul. Not so much attention has been paid to the corresponding compressive problem. Compressive failure may occur through several mechanisms-crushing, shearing. Now, gco is mesh-independent and is therefore a material property. For gv, we postulate. G,, is an assumed material property chosen in such a way that the numericai analysis of a standard compression test gives results that coincide with experimental data, Muffiu.
In attempting to extend the preceding definitions to multiaxial stress states. More specificsliy. In order that the value of K correspondins to the peak stress may be the same in the biaxial as in the uniaxial compression case. Such an cquntion mny bc JccIuccd from the more general equation. The special cuscs of pure tension and compression follow obviously from cqn In particular. A possible form is [ 44 k t7.
With the hofp of eqns 12 and I 3. In biaxial tests on concrete and gromaterials it is usually found Kupfer et II. The same result is nob found in triaxiai compression tests, at least at sufficiently large hydrostatic pressures; under these conditions it is found that the hardening goes on indefinitely.
In order words, while the yield surface however defined is closed. The so-celled crrp model DiMaggio and Sandlcr, has been used to describe this discontinuity. Equation 2. It is an essential feature of this model that the same function F a.
As was discussed in Section 2. As a rule. With these forms. This form uill be adopted in the present work for the yield surface. IO :trirl I. IO, yicltling z bctwccn 0. Once r is known. The paramrtcr 7 appears only in triaxial compression. Let TM and CM dcsignatc. Let us define the present model then yields that is. The meridians thus described are therefore straight.
Typical values range from about 0. From eqn f6 we obtain. The form taken by tho proposed yitld surface on ditrcrcnt planes of the stress space is shown in Fig. The singular points of the yield surfaccarc the following : a those where the maximum normal stress changes direction. It is not unreasonable to extend this rule to ali cases corresponding to compression meridians-that is.
The other set of singular points is of great importance because it includes biaxial and, as a special case. In this case there is no symmetry argument to dictate a choice of 8.
Andanaes et al. Jf it is the former. This indicates that. This resuft will he xlopt4 as 3 working rule for the dctinition of f at the singuhtr points bcfonging to category h.
Uniaxial compression is rcprcscntcd by :I point that is the intcrscction of the just- discusscd singularity locus with a cornprcssion mcriclian. A-; L? I , respectively. The plastic degradation variables 4, are those associated with plastic deformation. It will conscqucntly bc assumed. Now, from the chain rule, and cqns 19 anti 7-l , We define the operator C, by 27, note that C, is ;I nonlinerrr operator in that it depends on the direction of d.
Note that we are omitting any dependence of the yield criterion on the degradation variables. As we said above. The tangent stiffness, as a piecewise linear operator, is symmetric if C, is symmetric crrrtlif C,g is proportional to Cff.
Kupfer et al. Equation 28 leads to the operator C, given by Thus Ck is symmetric if and only if k is proportional or equal. However, the majority of investigators of the multiaxial behavior of concrete see Cedolin tr rrl.
Instead, they have found that the bulk modulus depends primarily on the volume strain, and the shear modulus on the octahedral shear strain, n the elastic range. Let us consider. It can be shown that these functions can be obtained, in monotonic radial loading, from the following forms of 6, and 42 : 3. Such conseyucnces of the associated flow rufc as uniqueness of solutions to boun- dary-v:llue problems and the theorems of limit analysis ilow largely from the MPD axiom, which hits also been helpful in clarifying the form taken by the associated flow rule in large- dcformntion plasticity Lubliner, f, In its most general form, the MPD axiom may bc expressed as follows.
Note that fftc 6, arc incfudcd among the z,. Direction of cracking. Cracking is the most important external mrmifestation of damage in a concrete struc- ture. In order to obtain a graphic representation of this kind of damage.
Other criteri:t for defining onset and dirccticms ofcracking as the hwiizttiotr c.
A plastic damage model for concrete structure cracks with two damage variables
We'd like to understand how you use our websites in order to improve them. Register your interest. A new plastic—damage constitutive model based on the combination of damage mechanics and classical plastic theory was developed to simulate the failure of concrete. In order to explain different material behaviors of concrete under tensile and compressive loadings, the plastic yield criterion, the different kinematic hardening rule for tension and compressive and the isotropic flow rule were established in the effective stress space. Meanwhile, two different empirical damage evolution equations were adopted: one for compression and the other for tension.
A Coupled Plastic Damage Model for Concrete considering the Effect of Damage on Plastic Flow
We'd like to understand how you use our websites in order to improve them. Register your interest. Based on the concepts of continuum damage theory, a new plastic damage model for concrete crack failure is developed through studying the basic damage mechanics. Two damage variables, tensile damage variable for tensile damage and shear damage variable for compressive damage, are adopted to represent the influence of microscopic damage on material macro-mechanics properties under tensile and compressive loadings.
A plastic-damage model for concrete
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