## 1 Introduction  
During geothermal energy extraction and oil and gas production, the cement stone is widely used to fill the annulus void between the formation and the casing [1, 2]. The structural integrity of cement stones directly affects the safety and stability of underground engineering [3, 4, 5]. One of the main problems affecting the durability of cementitious composites is acid corrosion [6, 7, 8]. In wells with high sulfur gas fields or where deep sour gas condensate is present as a production companion, sulfuric acid is easily generated in the acidic environment formed by the dissolution of acid gases such as H\({}_{2}\)S. The acidic solution will react chemically with cement, changing the mechanical properties of cementie and seriously affect the operating life of the well. In addition, due to the length and complexity of in situ testing, sulfuric acid can be used to simulate the acidic environment by continuously soaking the specimens [9, 10, 11, 12]. Besides, the cement stone is under a three-dimensional (3D) stress state (\(\sigma_{1}>\sigma_{2}>\sigma_{3}\)) due to the coupling effect of geostress and engineering disturbance [13, 14, 15]. Since the cement stone is classified as an alkaline cementing material, the presence of acidic fluid in the formation would affect the structure and mechanical behavior of cement stone [16, 17]. Thus, it is imperative to study the mechanical properties of cement stone subjected to acid corrosion under three-dimensional stress states to ensure the stability of the wellbore. Various triaxial compression tests were conducted to examine the mechanical properties of cement stone. The brittleness and energy dissipation laws of cement stone were explored by carrying out triaxial compression and triaxial cyclic loading tests. As the confining pressure and temperature increased, the mechanical properties of cement stone changed from strain softening to finally ideal plasticity [1]. The plastic deformation performance of cement stone under various confining pressures was analyzed [2]. The basic mechanical parameter of cement stone was discussed, and a negative correlation was found between cohesion, the modulus of elasticity, and temperature [3]. The effect of cyclic loading conditions on the mechanical properties of cement stone was investigated, and it was observed that the cement sheath gradually enters the plastic yield state with an increase in the fracturing stage [4]. In addition, the enhancement of the mechanical properties of the cement stone by fibers was also studied [5]. Expansion cracking formed by calcium carbonate was reduced by the addition of active fillers [18]. The above-mentioned studies are limited to the two-dimensional (2D) mechanical characteristics of cement stones. Therefore, various true triaxial tests were conducted to study the 3D mechanical properties of alkaline cementing material, mainly including concrete [19; 20; 21; 22] and cement stone [14; 23]. The 3D deformation properties and failure criteria of aggregate concrete are analyzed [19; 24]. The impact of a 3D stress path on the damage performance of concrete was carried out, and it was found that different combinations of hydrostatic and bias stresses lead to different damage scenarios for cement [21]. The failure mode of concrete subjected to true triaxial compression was analyzed [22]. The 3D strength characteristics and energy evolution of cement stone was discussed [14]. Additionally, the 3D constitutive models for cement stone and concrete were also proposed [20; 23]. Apart from the failure mode, energy, strength, and deformation properties discussed above, the crack stress thresholds of geomaterial are critical to understanding the failure process of geomaterial [25], which mainly includes crack initiation stress (the initial point of the stable crack growth stage), crack damage stress (the initial point of the unstable crack growth stage), and peak strength [25].