The present paper describes a simplified model for isotropic damage developed on continuum damage theory. The model characterizes the distinct behaviour of concrete in tension and compression using a unified equivalent strain, which governs the growth of damage. In reality, concrete exhibits distributed micro-cracking pattern under multi-directional loads. Therefore, the model is extended to incorporate damage induced anisotropy inherently for a reliable representation of damage, and captures the direction of possible damage evolution. Moreover, the model takes the inelastic behaviour of material into account based on the extended version of Lubliner failure criterion. The model is validated with experimental data.
Constitutive modellingThe growth and coalescence of microcracks in concrete generally lead to the formation and propagation of macro-cracks and ultimately to fracture. Thus, the deformation behaviour of concrete is influenced. Nowadays, continuum damage models are serving as efficient tools for engineers to understand the deformation behaviour of brittle materials such as concrete, rock, etc. This paper discusses isotropic and anisotropic damage models within the context of continuum damage mechanics.
Isotropic damageAccording to continuum damage mechanics, the stress-strain relation for the elasticity behaviour of concrete based on the principle of energy equivalence and adopted damage evolution law [1] are written aswhere, σ and ε are the second-order tensors of Cauchy stress and the strain; C refers the fourth-order elasticity tensor. The constitutive behaviour of the material is influenced by isotropic damage, denoted by a scalar variable D, which usually satisfies the condition (0 ≤ D ≤ 1). The damage evolution law (1b) exhibits exponential softening. The model parameters β 1 and β 2 control the initial rate of damage growth and decay in stress ensuring residual stresses respectively. The growth of damage D is governed by the history deformation parameter κ. There are several alternatives for history parameters which are discussed [1]. But in order to account for both tensile and compressive behaviour of concrete, the damage evolution is considered to be driven by a local measure of equivalent strain eq using a damage loading surface f d due to Lubliner failure criterion [2].Here, I 1 , J 2 , σ max , and H(σ max ) are as defined in [5]. Moreover, D and κ are monotonically increasing parameters. If f d < 0, the material behaves elastically; and when f d = 0, the damage initiates. The damage grows ifḟ d = 0. Thus, the model satisfies Kuhn-Tucker loading/unloading conditions. The initial elastic domain is set by the initial threshold value κ 0 = κ 0t H + κ 0c (1 − H) in tension and compression respectively.
Anisotropic damageAlthough the isotropic damage model satisfactorily describes the damage behaviour of concrete as a first approximation, the experimental investigations prove that the internal degradation of the material depending on the direction of loading applied is inherently anisotropic. T...