# Non-Linear Creep-Relaxation Constitutive Damage Model for Aging Concrete
## 1 Introduction  
Concrete is one of the most important construction materials in use all over the world. It is an attractive material due to its versatility, easy fabrication, and low cost. Accordingly, concrete has been used in the construction of different types of structures, such as bridges, buildings, and dams, among others. However, due to its composite nature and quasi-brittle behavior, its mechanical behavior is difficult to predict under different loading conditions [1]. Particularly, in the case of bridges, which are subjected to a combined cyclic and sustained loading, recent studies [2, 3] have shown that the current state of the art on models that account for purely sustained loading (creep) underestimate the long-term deflection, especially in long-span bridges. This underestimation has been attributed to the combined effect of sustained cyclic loading, known as cyclic creep. To account for such a phenomenon, a theory of cyclic creep based on linear elastic fracture mechanics was developed in [4]. It was found that cyclic creep has a negative impact on the long-term deflection of medium-span bridges (bridges with spans between 40 and 80 m). However, the theory has not been developed under the assumptions of the thermodynamic framework; moreover, concrete is assumed as a linear elastic material, which is an inaccurate representation of material behavior. As an alternative to linear elastic fracture mechanics, Continuum Damage Mechanics (CDM) theory has been shown to have potential for the characterization of concrete mechanical behavior. Numerous studies in CDM theory for concrete under diverse conditions have been developed. For instance, damage models for brittle materials under monotonic, quasi-static, cyclic, and dynamic loading behavior were developed in [5; 6; 7]. In [8], a model based on mixture theory (separating the mortar and aggregate) and CDM was developed; the unilateral effect of cracks is primarily considered in this model. Some experiments and models related to the unilateral effect of cracks in concrete were conducted in [9; 10]. CDM-bounding surface models for concrete under cyclic and monotonic loading were developed in [11; 12]. Damage models coupled with plasticity were proposed in [13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23]; some of the features of these models were slip-type strain, multiaxial cyclic and monotonic loading behavior, isotropic and anisotropic damage, and the inclusion of damage tensor variables to account for compressive, tensile, and shear damage. [PERSON] et al. [24] reformulated the theory of CDM for brittle materials by treating them as elastic damage materials. Following this line of thought, a model for an elastic-brittle material with anisotropic damage was developed [25]; in this model, a representation theorem was first introduced to define the thermodynamic potential. The relationship between fracture and damage was studied in [26; 27]. Damage models for concrete under hysteretic and fatigue behavior were developed in [28; 29; 30; 31]. A damage model based on CDM was presented in [32]; in this model, the effects of initial damage are treated. Subsequently, a method to correctly determine the damage functions in quasi-brittle materials was obtained in [33]. The separation of hydrostatic and deviatoric components of damage were treated and discussed in [34]. In [35], a model for cyclic loading based on an isotropic damage variable was developed; stiffness recovery, inelastic strains, and frictional sliding were some of its features. Reviews of CDM and plasticity applied to concrete can be found in [36; 37; 38], and more recently in [39], where multiscale modeling, phase-field modeling, rate-dependent models, and fatigue were discussed. One of the main challenges with the creep characterization of concrete is the change in the mechanical viscoelastic properties as concrete ages--a concept known as aging. [PERSON] was one of the first to recognize the aging of concrete and developed some early models [40; 41].