This paper reviews the flow behavior and mathematical modeling of various metals and alloys at a wide range of temperatures and strain rates. Furthermore, it discusses the effects of strain rate and temperature on flow behavior. Johnson–Cook is a strong phenomenological model that has been used extensively for predictions of the flow behaviors of metals and alloys. It has been implemented in finite element software packages to optimize strain, strain rate, and temperature as well as to simulate real behaviors in severe conditions. Thus, this work will discuss and critically review the well-proven Johnson–Cook and modified Johnson–Cook-based models. The latest model modifications, along with their strengths and limitations, are introduced and compared. The coupling effect between flow parameters is also presented and discussed. The various methods and techniques used for the determination of model constants are highlighted and discussed. Finally, future research directions for the mathematical modeling of flow behavior are provided.

The aim of this paper is to establish a reliable model that provides the best fit to the specific behavior of the flow stresses of the 10%Cr steel alloy at the time of hot deformation. Modified Johnson–Cook and strain-compensated Arrhenius-type (phenomenological models), in addition to two Artificial Neural Network (ANN) models were established with the view toward investigating their stress prediction performances. The ANN models were trained using Scaled Conjugate Gradient (SCG) and Levenberg–Marquardt (LM) algorithms. The prediction accuracy of the established models was evaluated using the following well-known statistical parameters: (a) correlation coefficient (R), (b) Average Absolute Relative Error (AARE), (c) Root Mean Squared Error (RMSE), and Relative Error (RE). The results showed that both of the modified Johnson–Cook and strain-compensated Arrhenius models could not competently predict the flow behavior. On the contrary, the results indicated that the two proposed ANN models precisely predicted the flow stress values and that the LM-trained ANN provided a superior performance over the SCG-trained model, as it yielded an RMSE of as low as 0.441 MPa.

This work presents modifications for two constitutive models for the prediction of the flow behavior of titanium-based alloys during hot deformation. The modified models are the phenomenological-based Fields-Backofen and the physical-based Zerilli-Armstrong. The modifications are derived and suggested by studying the hot deformation of titanium-based alloy Ti55531. The predictability of the modified models along with the original Fields-Backofen and another modified Zerilli-Armstong models is assessed and evaluated using the well-known statistical parameters correlation coefficient (R), Average Absolute Relative Error (AARE), and Root Mean Square Error (RMSE), for the Ti55531 alloy, and validated with other two different titanium-based alloys SP700 and TC4. The results show that the modified Fields-Backofen gives the best performance with R value of 0.996, AARE value of 3.34%, and RMSE value of 5.64 MPa, and the improved version of the modified Zerilli-Armstrong model comes in the second-best place with R value of 0.992, AARE value of 3.52%, and RMSE value of 9.15 MPa for the Ti55531 alloy.