## 1 Introduction  
Over the years, the requirements, in terms of mechanical performance, of engineering materials have become increasingly more stringent in order to meet the high safety and quality standards expected of modern products and processes. In this scenario, in addition to the identification of the elastic limits related to a safe design under normal operation conditions, it is mandatory to investigate the material's elasto-plastic behavior, ductility, and ultimate strength under all possible operating conditions. This is crucial to assessing the structural integrity of components and mechanical systems under prospective accidental overloads, which might take the material beyond its elastic limits and eventually up to failure. This design necessity has led to the development of many theoretical and numerical models [1] that try to explain and predict material behavior in terms of both constitutive laws, i.e., stress-strain relationships, and damage evolution, i.e., the combination of conditions determining progressive material degradation up to failure. In this context, the effects of temperature and strain rate can be considered as well, within the framework of visco-plasticity for constitutive relations, and by validating or adapting the existing formulations with regard to ductile damage. Contemporary visco-plastic models are more reliable and consolidated. They can be divided into phenomenological and physical models. The models belonging to the latter category are usually developed by starting from the thermodynamics and kinetics of a dislocation slip [2]. Single- and poly-crystal plasticity approaches have been used [3]. [PERSON] et al. [4] developed a flow stress model for Body-Centered Cubic (BCC) materials whose yield stress is decomposed into athermal, thermally activated and viscous drag components. The thermal activation of dislocation motion is the basis of the Zerilli-Armstrong model, which accounts for a material's microstructure by describing two different flow-stress laws for BCC and Face-Centered Cubic (FCC) materials [5]; this model is one of the most commonly adopted formulations in dynamic applications, probably because it is relatively simple and capable of accounting for strain rate and temperature effects in a coupled manner. Many empirical models have been conceived as well, such as those by [PERSON] [6] and [PERSON] [7]; one that is widely used is the Johnson-Cook model [8], which addresses the hardening effect of strain rate and the softening effect of temperature with uncoupled multiplicative terms. Ductile damage has been extensively investigated as well, but mainly under quasi-static conditions [9], while fewer studies have been conducted so far that assess the prediction accuracy of damage models when dynamic conditions or temperature variations are involved. The influence of strain rate on the damage of Advanced High-Strength Steels (AHSS) has been investigated by [PERSON] [10] and [PERSON] [11]. [PERSON] [12] and [PERSON] [13] studied Dual-Phase (DP) and Transformation-Induced Plasticity (TRIP) steels, observing a reduction in ductility for DP steels at high strain rates. On the contrary, the ductility of TRIP steel increases with strain rate. DP and TRIP steels were also studied by [PERSON] et al. [14], who noticed a decrease in anisotropy at high strain rates. [PERSON] et al. [15] addressed the issue of calibrating a ductile damage model for two API pipeline steels under hydrogen-assisted degradation. [PERSON] et al. analyzed the damage evolution pf X52 pipeline steel in the presence of defects [16]. [PERSON] et al. explored how the TRIP effect, and thus tensile strength, are affected by different combinations of temperature and strain rate [17]. As regards damage modeling performed under static loading conditions at room temperature, plenty of theories have appeared in the literature, which are divided into phenomenological and physical theories.