# Non-Linear Creep-Relaxation Constitutive Damage Model for Aging Concrete
## 1 Introduction
Afterward, concrete aging was assumed to be related to the process of solidification of hydrated matter (cement gel) [42], and through this assumption, a theory for aging creep on concrete was formulated, i.e., the theory of solidification [42; 43; 44; 45]. Unfortunately, the long-term estimations, obtained through [42], showed divergence in their graphical representation. This divergence was attributed to the utilization of the mean distance between capillary pores used in the creep function. Hence, the use of a different rheological model that considers a viscous flow strain component in the total strain was proposed in [43]. This viscous flow component in the strain is due to the migration of calcium silicate hydrate (CSH) particles inside the matrix [46]. This migration is the product of the sustained loading or drying of concrete. The solidification theory in [43; 44] provides the theoretical framework for the derivation of the ACI B3 model [47]. The theory of solidification proposed in [43; 44] accounted for the non-linear regime, i.e., for stresses larger than half the strength of concrete, where irreversible viscoplastic strain prevails. However, the non-linear theory was based only on curve fitting instead of a solid theoretical framework. The thermoviscoelasticity of the solidification theory was presented in [45]. A nonlinear model based on [PERSON] and [PERSON]'s theory of solidification was developed in [48]; in the model, an isotropic damage variable and a rate-type constitutive equation were used. One of the shortcomings was the function utilized for accounting damage in the viscoelastic part of the strain at high levels of stress based on empirical observations. Subsequently, a constitutive relation for brittle materials (concrete and masonry) based on rheological models and CDM was developed in [49]; the aging of concrete was absent in this constitutive relation. More recently, a new type of creep damage model has been developed in [50] to establish the relationship between creep parameters and time under different stresses. In a recent study [51], viscoelastic materials were mathematically modeled considering molecular chain-network configurations and the temperature effects on the mechanical properties were characterized based on the equivalent relationship between temperature and frequency. Moreover, in [52], the mechanical behavior of a viscoelastic damper under monotonic and cyclic loading, focusing on the cyclic stability of mechanical properties, was explored. In related research, and in the lead author's doctoral dissertation [53], a non-linear constitutive model for plain concrete under creep and relaxation was developed, and is presented in this paper. The model is based on thermodynamic constitutive theory, and it can account for creep under high levels of stress. In addition, the model has the potential to be utilized to account for cyclic creep and tertiary creep, since the damage is driven by a loading-unloading condition. The model is based on the Bazant solidification theory [43; 45] and the CDM theory. The model is developed from the strain-based tensor formulation, meaning that the material functions that characterize the model are relaxation functions. Anisotropic damage is considered by using a second-order damage tensor variable, whose principal directions coincide with those of the strain tensor. The Helmholtz free energy is utilized as a measure of energy storage and thermodynamic potential. The Helmholtz free energy is defined based on the analogy of the representation theorem used by [PERSON] and [PERSON] [25]. Finally, it has been assumed that the formulations used for brittle-elastic damageable materials [24] can also be applied to viscoelastic materials. Hence, the material is treated as viscoelastic damageable material.