## 1 Introduction
With regard to the physical-based models, the seminal works of [PERSON] and [PERSON] [18], [PERSON] [19] and [PERSON] and [PERSON] [20] have stated the relevant role of hydrostatic pressure in the nucleation, growth and coalescence of voids; however, the prediction of shear failure is an issue for this class of models, and modifications have been attempted [21]. In the framework of Continuous Damage Mechanics, [PERSON] [22] proposed a non-linear model for damage accumulation and failure for different materials, and extended the study to SA537 steel under different strain rates [23]. On the other hand, a number of empirical fracture models have also been established in the literature, such as those by [PERSON] and [PERSON] [24], [PERSON] and [PERSON] [8], and [PERSON] and [PERSON] [25], to name a few; more recently, a class of damage models has been derived from the Mohr-Coulomb fracture criterion, based again on the works of [PERSON] [26; 27; 28]. According to this approach, in the context of the several parameters that describe the stress state at a point in a material, it is believed that not only the stress triaxiality, but also the Lode angle, control the evolution of damage up to the final failure of the material. The effectiveness of these models under dynamic conditions has been demonstrated on brittle materials, such as rocks [29], and on ductile metals, such as TRIP steels and AA2024 [11]. A modification of the Bao-Wierzbicki model has been conceived by [PERSON] et al. [30]; specific functions of triaxiality and Lode related parameters were introduced to describe the fracture surface of API X65 steel under static conditions [31]. In this paper, an attempt is made to investigate how two existing damage models, namely, Bai-Wierzbicki [28] and Coppola-Cortese-Folgarati [30], which were not specifically devised for conditions of high strain rates and large temperature variations, can be adapted and calibrated under dynamic loading conditions, or used for materials subjected to different high temperatures when operating. First, proper analytical formulations are selected from the literature that are able to predict the elasto-plastic constitutive behavior ([PERSON] and [PERSON] [32], [PERSON] and [PERSON] [5]), taking into account temperature and strain rate effects. Both plasticity and damage models are intended for use or implementation in Implicit and Explicit Finite Elements Analysis. A calibration strategy based on experimental data and inverse methods is identified and proposed for all models. To facilitate robust tuning, different multiaxial tests are set up and executed, specifically tensile tests on round bars, as well as tests on round notched bars and plane strain tests. For temperature tests, a standard uniaxial machine equipped with a high temperature furnace is used, while for dynamic tests, a Split Hopkinson Bar developed by some of the authors is employed [33; 34]. The specimens' geometries, facilities' setups and test conditions are optimized to characterize the material under investigation at an average strain rate of about 2000 s\({}^{-1}\) and selected temperatures of 600 \({}^{\circ}\)C and 700 \({}^{\circ}\)C. These temperatures fall within the range of so-called warm temperatures for steels (550-850 \({}^{\circ}\)C) [35; 36], which, though well below recrystallization temperatures, are nevertheless high enough to affect the material's mechanical behavior significantly. In this range, it can be assumed that the damage mechanisms do not differ from those that bring a material to failure at room temperature. Due to the limitations imposed by the available facilities, dynamic and temperature effects are studied separately. In addition to the global quantities directly measured during tests, such as load-displacement curves, Finite Element simulations reproducing the experiments are also used to retrieve local non-measurable information at the critical point of each specimen and test.