The UMATHT subroutine in ABAQUS lets users define their own constitutive model for a heat transfer analysis. It's used to define temperature-dependent thermal material behaviour. However, the code you posted appears to be for plastic deformation, not heat transfer, so it looks like it might be for a UMAT subroutine, not UMATHT.

The hardening curve equation that you provided is a common one:

Sf=Sy*(1.d0+E*eqplas/Sy)**xn

It appears to be using an isotropic hardening rule (specifically, a power-law hardening model) where 'Sy' is the initial yield stress, 'E' is the Young's modulus, 'eqplas' is the equivalent plastic strain, and 'xn' is the hardening exponent. 'Sf' is the flow stress. This formulation calculates the yield stress as a function of the equivalent plastic strain.

In the loop following this, Newton's method is used to solve for 'Smises' and 'deqpl' until a certain tolerance is met or the maximum number of iterations ('newton') is reached. 'Smises' is the von Mises stress, and 'deqpl' is the increment in equivalent plastic strain.

Please note that while this power-law hardening model is quite common, it may not be applicable or accurate for all materials. It's critical to use experimental data or data from the literature to calibrate the model parameters ('Sy', 'E', and 'xn') for the specific material you are studying.

For a better understanding of the UMAT or UMATHT subroutines and their intricacies, it would be useful to refer to the ABAQUS documentation and user manual or to reach out to Dassault Systèmes' support. Additionally, consulting with an expert in the field or someone with deep experience in using ABAQUS and user subroutines can be very beneficial.

In the code snippet you shared, the hardening curve is based on the Ramberg-Osgood model, which is commonly used to represent the nonlinear stress-strain behavior of materials. Here is a breakdown of the formulation:

This code snippet suggests that the UMATHT subroutine implements the Ramberg-Osgood hardening model for the material behavior. It calculates the yield stress based on the specified hardening curve equation and uses Newton's method to update the plastic strain and stress until convergence is achieved.

To gain a deeper understanding of the complete UMATHT subroutine and its implementation, it would be helpful to review the entire subroutine and examine how these calculations are integrated into the overall material behavior model.