Phase-Field Drucker-Prager Model for Concrete

Abstract

An efficient, accurate and minimal-parameter constitutive model for a simulation of concrete structures behavior can be developed based on Drucker-Prager elastic-plastic yield criterion: √ϕ = αI_1 + J_2D − k, where k = σ_0 + h ε̄p, σ0 is the initial yield stress, h is the hardening parameter, α is the material parameter, and ε̄_p is the equivalent plastic strain [1]. Total free energy consists of the elastic-plastic and the fracture contribution as [2]: ψ = ψ_ep + ψ_f , where ψ_ep = g(1/2σ : ε_e + σ_0 ε̄_p + 1/2 h ε̄^2 p ), ψ_f = G_v (d + l_c^2 |∇d|^2 ), σ is the stress tensor and ε_e is the elastic strain tensor. The phase-field damage evolution law is defined as: G_v [d − l_c^2 ∇^2 d] + g′ H_max = 0, where the degradation function is g = (1 − d)^2 , l_c is the characteristic length, d is the damage variable, and H_max = ψ_ep − ψ_cr is the maximal total strain energy. The fracture energy G_v is calculated in a relation to the tension and compression strength and the threshold value of critical total strain energy ψ_cr = G_v /2. The results of uniaxial tension and compression tests are presented in Fig.1.