**1. Introduction**

With the application of high-strength concrete in civil engineering, transportation, water conservancy, municipal engineering, and other engineering fields, it is of great significance to study the deformation law and failure characteristics of high-strength concrete subject to complex stress states aiming to improved scientific designs of concrete buildings (structures) and to the guarantee of their safety [1,2]. High-strength concrete is a type of composite material composed of sand, stone, cementing material, and water, mixed based on specific analogies. Owing to the incompleteness of vibration, incomplete hydration reaction, and temperature effects, there are a large number of discontinuity and irregular shape cracks and joints in the interior of high-strength concrete. The deformation and mechanical characteristics of high-strength concrete are obviously nonlinear and discontinuous. This leads to the complexity of micro-structure and macro-strength evolution of high-strength concrete at different ages and complex stress conditions. It is difficult to effectively judge

**Citation:** Zhang, L.; Cheng, H.; Wang, X.; Liu, J.; Guo, L. Statistical Damage Constitutive Model for High-Strength Concrete Based on Dissipation Energy Density. *Crystals* **2021**, *11*, 800. https://doi.org/ 10.3390/cryst11070800

Academic Editors: Cesare Signorini, Antonella Sola, Sumit Chakraborty, Valentina Volpini and Ing. José L. García

Received: 24 May 2021 Accepted: 28 June 2021 Published: 8 July 2021

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the strength change and failure behavior of concrete materials by using classical elastic– plastic theory. Numerous research studies have shown that the failure of materials is a state instability phenomenon driven by energy, and the transmission and transformation of energy are the fundamental reasons for the deformation and failure of materials [3,4]. Therefore, the energy method based on thermodynamic theory is an effective way to study the constitutive relationships and failure behaviors of concrete materials.

The constitutive relationship constitutes the basis for the study of the relationship between material deformation and load. Ever since the proposition of the parallel bar system (PBS) model for the concrete uniaxial tensile process in 1982 by Krajcinovic [5], statistical damage theory has gradually become a new research hotspot of concrete damage mechanics. Yang et al. [6] proposed a statistical damage constitutive model of multi-size polypropylene fiber concrete under impact load and obtained the statistical parameters based on the particle swarm optimization algorithm. Bai et al. [7] assumed that there are two damage mechanisms of fracture and yield in the meso-structures of quasi-brittle materials, and established a triaxial orthotropic statistical damage model for concrete that can predict its constitutive behavior at complex loading environments. Based on Weibull and lognormal statistical distribution theory and Lemaitre's strain equivalent principle, Liang Hui et al. [8] established a sectional uniaxial compression damage constitutive model of concrete materials by introducing the strain-rate factor. Cervera [9] established a rate-independent isotropic damage constitutive model of concrete, and used the model to conduct seismic analysis of concrete dams. Zhou [10] investigated the compression behavior of coral aggregate concrete (CAC) at uniaxial and triaxial loading, and proposed a constitutive model for coral aggregate concrete subjected to uniaxial and triaxial compression, wherein the suggested models correlated well with the test results. Wu [11] established the plastic damage constitutive relation with the internal variables based on the continuum damage mechanics, and proposed an energy release rate-based plastic-damage model for concrete. The aforementioned studies showed that the constitutive behavior of concrete materials can be studied from the perspective of statistics, but at present, most studies use the mechanical or deformation parameters of concrete materials as the basis for establishing statistical damage constitutive equations, and few studies have introduced the energy dissipation density parameter into the constitutive relationship.

To study the energy evolution law and damage constitutive behavior of high-strength concrete subjected to complex stress states, conventional triaxial compression tests at different confining pressures were conducted with ZTCR-2000 rock triaxial testing system. The evolution law of input energy, elastic strain energy and dissipation energy with axial strain and confining pressure were analyzed. Based on the continuum damage theory and non-equilibrium statistical method, a statistical damage constitutive model was established for high-strength concrete based on the use of the ratio of dissipation energy density of concrete to the dissipation energy density that corresponded to peak stress.
