Plastic model with non-local damage applied to concrete

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82P. GRASSL AND M. JIRÁSEKwhere(x s ðR s Þ¼ 1 þ A sR 2 s if R s 51pffiffiffiffiffið20Þ1 3A s þ 4A s R s if R s 51A s is a model parameter determined from the softening response in uniaxial compression. Thedimensionless variable R s ¼ ’e pV =’e pV is defined as the ratio between the ‘negative’ volumetricplastic strain rate’e pV ¼ X3I¼1h ’e pI i ð21Þand the volumetric plastic strain rate ’e pV ¼ d : _e p : Since this ratio depends only on the flowdirection @g p =@%r; R s can be considered as a function of the effective stress. In (21), ’e pI are theprincipal components of the rate of plastic strains and hi denotes the McAuley brackets(positive-part operator). For uniaxial tension, for instance, all three principal plastic strain ratesare non-negative, and so ’e pV ¼ 0; R s ¼ 0 and x s ¼ 1: This means that under uniaxial tensileloading we have k d ¼ k p k 0 ; same as for the simple one-dimensional model from Section 4. Onthe other hand, under compressive stress states the negative principal plastic strain rates lead toa ductility measure x s larger than one and the evolution of damage is slowed down. It should beemphasized that the flow rule for this specific model is constructed such that the volumetric partof plastic strain rate at the ultimate yield surface cannot be negative.The damage-plastic model contains 15 parameters, but only 6 of them are actually calibratedfor different concrete types, namely Young’s modulus E; Poisson’s ratio n; tensile strength f t ;compressive strength f c ; parameter e f of the damage law (14), and parameter A s in the ductilitymeasure (20) of the damage model. The remaining parameters are set to their default valuesspecified in Reference [24].The response of the local model has been compared to several material tests in Reference [24].It has been shown that the model can describe the failure of concrete under tension and uniaxial,biaxial and triaxial compression. Here, only two selected examples are reproduced: uniaxialtension with unloading tested by Gopalaratnam and Shah [37] (Figure 10) and uniaxialcompression with unloading tested by Karsan and Jirsa [38] (Figure 11).The non-local extension of the model is straightforward. The local variable k d is in thedamage law (14) replaced by the ‘over-non-local’ variable #k d defined in (10). Note that,for m > 1; #k d can be negative, and so it is important to include in (14) the definition of g d fornegative arguments.6.1. Three-point bending testThe first structural example with the non-local damage-plastic model for concrete is a threepointbending test of a single-edge notched beam reported by Kormeling and Reinhardt [39]; seeFigure 12.The experiment is modelled by triangular plane-strain finite elements. For this type ofanalysis, local stress–strain relations with strain softening are known to result in meshdependentload–displacement curves. The capability of the non-local damage-plastic model toovercome this pathological dependence is assessed by three analyses on three meshes withelement sizes of 7.5, 5 and 2:5 mm; see Figure 12 for the coarse mesh.Copyright # 2005 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 30:71–90
PLASTIC MODEL WITH NON-LOCAL DAMAGE 834constitutive responseexperimentsstress [MPa]32100 0.2 0.4strain [mm/m]Figure 10. Model response under uniaxial tension with unloading compared to experimental resultsreported by Gopalaratnam and Shah [37].0-10stress [MPa]-20-30constitutive responseexperiments-4 -3 -2 -1 0strain [mm/m]Figure 11. Model response under uniaxial compression with unloading compared to experimental resultsreported by Karsan and Jirsa [38].The material parameters are E ¼ 20 GPa; n ¼ 0:2; f t ¼ 2:4 MPa; f c ¼ 24 MPa; e f ¼ 0:001025and A s ¼ 2: The other parameters of the local damage-plastic model are set to their defaultvalues [24]. The non-local parameters are taken as R ¼ 25 mm and m ¼ 1:The load–displacement curve obtained on the fine mesh is in a reasonable agreement with theexperiment, cf. Figure 13, and the results do not suffer by pathological sensitivity to the meshsize, cf. Figure 14. The contour plots of the damage variable and the increment of the volumetricplastic strain at three stages of the analysis (marked by hollow circles in Figure 14) on the finemesh are presented in Figure 15. The plastic strain tends to fully localize and the damage isdistributed over distance R around the plastic zone.Copyright # 2005 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 30:71–90
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