By Jacek Tejchman, Jerzy Bobiński
The booklet analyzes a quasi-static fracture technique in concrete and bolstered concrete through constitutive versions formulated inside of continuum mechanics. a continuing and discontinuous modelling procedure was once used. utilizing a continual process, numerical analyses have been played utilizing a finite point technique and 4 assorted better continuum types: isotropic elasto-plastic, isotropic harm and anisotropic smeared crack one. The types have been outfitted with a attribute size of micro-structure via a non-local and a second-gradient concept. so that they may perhaps safely describe the formation of localized zones with a definite thickness and spacing and a comparable deterministic dimension impression. utilizing a discontinuous FE strategy, numerical result of cracks utilizing a cohesive crack version and XFEM have been offered that have been additionally correctly regularized. Finite aspect analyses have been played with concrete parts less than monotonic uniaxial compression, uniaxial rigidity, bending and shear-extension. Concrete beams less than cyclic loading have been additionally simulated utilizing a coupled elasto-plastic-damage technique. Numerical simulations have been played at macro- and meso-level of concrete. A stochastic and deterministic dimension impression was once rigorously investigated. in relation to bolstered concrete specimens, FE calculations have been performed with bars, slim and brief beams, columns, corbels and tanks. Tensile and shear failure mechanisms have been studied. Numerical effects have been in comparison with effects from corresponding personal and identified within the clinical literature laboratory and full-scale tests.
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Additional info for Continuous and Discontinuous Modelling of Fracture in Concrete Using FEM
2002), Huerta et al. (2003) and Jirásek (2004a). e. material brittleness. The material softening starts when the when the equivalent strain measure reaches the initial threshold κ0 (material hardening is neglected). The parameter β determines the rate of the damage growth (larger value of β causes a faster damage growth). In one dimensional problems, for ε → ∞ (uniaxial tension), the stress approaches the value of (1 − α )Eκ 0 (Fig. 4b). 1 Local Models for Monotonic Loading 57 The constitutive isotropic damage model for concrete requires the following 5 material constants: E, υ, κ0, α and β (Eq.
30) where ψ is the dilatancy angle (ψ≠φ). The increments of plastic strains dε ij were p calculated as d ε ijp = d κ1 ∂g1 d κ1 ∂q ∂p = + tanψ ( ). 31) b) Fig. 1 Drucker-Prager criterion in the space of principal stresses (a) and on the plane q-p (b) In turn, in the tensile regime, a Rankine criterion was used with the yield function f2 using isotropic hardening and softening defined as (Bobiński 2006, Marzec et al. 2007, Majewski et al. 2008) (Fig. 32) where σi – principal stresses, σt – tensile yield stress and κ2 – softening parameter (equal to the maximum principal plastic strain ε1p ).
The total strains ε ij were decomposed into the elastic ε ije and inelastic crack strains ε ijcr ε ij = ε ije + ε ijcr . 53) Between the concrete stresses and cracked strains, the following relationship was valid (in a local coordinate system) cr cr σ ij = Cijkl ε kl cr with the secant cracked stiffness matrix Cijkl (defined only for open cracks). Fig. 1 Local Models for Monotonic Loading 61 Fig. 6 Relationships between normal crack stress versus normal crack strain in softening range during loading, unloading and reloading (de Borst 1986) cr The matrix Cijkl was assumed to be diagonal.