By Jacek Tejchman, Jerzy Bobiński
The booklet analyzes a quasi-static fracture strategy in concrete and strengthened concrete via constitutive versions formulated inside continuum mechanics. a continual and discontinuous modelling technique used to be used. utilizing a continuing technique, numerical analyses have been played utilizing a finite point technique and 4 varied more advantageous continuum versions: isotropic elasto-plastic, isotropic harm and anisotropic smeared crack one. The versions have been outfitted with a attribute size of micro-structure through a non-local and a second-gradient conception. so that they may possibly competently describe the formation of localized zones with a undeniable thickness and spacing and a comparable deterministic dimension impression. utilizing a discontinuous FE technique, numerical result of cracks utilizing a cohesive crack version and XFEM have been awarded which have been additionally adequately regularized. Finite aspect analyses have been played with concrete parts less than monotonic uniaxial compression, uniaxial stress, bending and shear-extension. Concrete beams lower than cyclic loading have been additionally simulated utilizing a coupled elasto-plastic-damage process. Numerical simulations have been played at macro- and meso-level of concrete. A stochastic and deterministic dimension impact used to be rigorously investigated. in terms of strengthened concrete specimens, FE calculations have been performed with bars, narrow 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 medical literature laboratory and full-scale tests.
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Extra resources for Continuous and Discontinuous Modelling of Fracture in Concrete Using FEM
A crack is initiated if the major principal stress exceeds the tensile strength. The crack direction is usually assumed to be orthogonal to the principal tensile major stress. Between stresses and strains in the crack plane z’-y’ (Fig. 51) where the tangent modulus C represents the relation between the normal crack strain increment and normal stress increment during loading, S is the secant modulus of the unloading branch (Fig. 6), G is the elastic shear modulus and β is the shear stiffness reduction factor (the term βG account for effects like aggregate interlock).
34) where κ 0 denotes the initial value of κ when damage begins. If the loading function f is negative, damage does not develop. During monotonic loading, the parameter κ grows (it coincides with ε ) and during unloading and reloading it remains constant. To define the equivalent strain measure ε , different criteria can be used. In the book, we applied 4 different equivalent strain measures ε . 35) are the principal values of the effective stress e σ ieff = σ ijkl ε kl . 37) 56 3 Continuous Approach to Concrete with σ 1eff > σ 2eff and a non-negative coefficient c.
5) where Ceijkl is the elastic stiffness tensor e Cijkl = λδ ij δ kl + μ (δ ik δ jl + δ il δ jk ) . 7) and δij is a Kronecker delta. 8) ∂σ ij with g as the potential function and dλ as the positive factor of proportionality. If f=g, the flow rule is associated. 9) During plastic deformation, a stress state remains on the boundary of the elastic/plastic region f (σ ij + dσ ij , κ + d κ ) = 0 . 10 may be rewritten in a rate form as (similarly to Eq. 2) df df d σ ij + dκ = 0 . 11 are known as consistency conditions and allow to determine the magnitude of the plastic strain increment.