# A Thermomechanical Coupling Constitutive Model of Concrete Including Elastoplastic Damage
## 1 Introduction
The compressive behaviors of concrete are heavily conditioned by passive confinement offered by transverse reinforcements, consisting of internal hoops and additional external strengthening, such as fiber reinforced polymer (FRP) wraps and steel jackets [21; 22]. When the mechanical properties of concrete under high temperature conditions are studied by theorical analysis, a thermomechanical coupling constitutive model should be used. The constitutive models developed based on the classical plasticity theory can usually satisfy the stability postulate (e.g., [PERSON]'s stability postulate and [PERSON]'s postulate). [PERSON] et al. [23] developed a multiaxial constitutive model based on the combination of elastoplasticity and damage theories. The isotropic damage was assumed. This model was applied to the numerical simulation of concrete structural members of a fire situation [24]. [PERSON] et al. [25] proposed a concrete constitutive model considering coupled effects of high temperature and high strain rate by modifying the Drucker-Prager constitutive model. [PERSON] et al. [26] presented a non-linear thermo-viscoelastic rheological model based on fractional derivatives for high temperature creep in concrete. [PERSON] et al. [27] established a dynamic damage constitutive model combining statistical damage and a viscoelastic model for the description of the dynamic mechanical properties of basalt-fiber-reinforced concrete exposed to different temperatures. The thermomechanical coupling constitutive models based on the classical plasticity theory may not strictly satisfy the laws of thermodynamics. The development of plastic or irrecoverable strains of concrete exhibits features that are foreign to the classical theory of thermodynamics. A thermomechanical coupling constitutive model should be developed based on thermodynamics and consider the strictness of the theory foundation. [PERSON] et al. [28] proposed a constitutive model concerning concrete cracks based on the theory of thermodynamics. [PERSON] and [PERSON] [29] presented energy-based coupled elastoplastic damage theories. The proposed formulation employs irreversible thermodynamics and internal state variable theory for ductile and brittle materials. [PERSON] et al. [30] modeled a computational analysis of hygro-thermal and mechanical behavior of concrete structures at high temperature including the heat and mass transfer processes. [PERSON] et al. [31] built a computational model for the thermomechanical analysis of concrete structures at high temperature using the finite element method. [PERSON] and [PERSON] [32] defined the total Helmholtz free energy and the elastoplastic damage energy release rate based on the damage mechanism of concrete and developed a series of elastoplastic damage constitutive models that satisfy the principles of thermodynamics. [PERSON] et al. [33] proposed a multiphase thermo-hydro-mechanical model for concrete at high temperature. [PERSON] and [PERSON] [34] proposed a thermodynamically based constitutive model for isotropic homogeneous thermoplastic polymers under arbitrary multiaxial and non-monotonic loadings. [PERSON] et al. [35] developed a confinement-dependent load-induced thermal strain constitutive model for concrete subjected to temperatures up to 500 \({}^{\circ}\)C. [PERSON] et al. [36] proposed a stress-strain model to describe the residual tensile stress-strain relationship of concrete considering the effect of cyclic damage and high temperature. [PERSON] et al. [37] presented a two-phase computational model for determining the response of prestressed hollow-core concrete slab exposed to natural fire. [PERSON] et al. [38] developed an elastoplastic damage constitutive model of concrete considering the effects of dehydration and pore pressure at high temperatures. [PERSON] et al. [39] presented a moisture-dependent thermomechanical constitutive model for concrete able to capture the effect of the moisture content of the material on its mechanical behavior under compressive loads and high transient temperatures.