The true stress versus true plastic strain diagrams, also known as material flow curves, characterize the plastic behaviors of materials. Among the available methods for measuring flow curves, the tensile test is the most commonly used due to its benefits of no friction and high precision. Although there are various approaches to obtaining flow curves from tensile tests, the classical approach for extracting true stresses and true plastic strains by measuring the nominal stress-strain curves is the most widely used in practice.
From a glance, it seems to be an easy task to extract flow curves from nominal stresses and strains. Interestingly, there are several unresolved points. Firstly, the flow curve measurement in the homogenous plastic deformation region of materials with no yield point often calculates the true stress by assuming a constant volume during elastic-plastic straining while neglecting the elastic volume changes. This approach is reasonable for materials with relatively low strength and high bulk modulus. For high-strength and low bulk modulus materials, it leads to errors because the higher the strength, and the lower the bulk modulus, the greater the elastic volume change. Secondly, for materials with pronounced yield point phenomenon and Lüders bands, the emergence of a complicated 3D stress state at the edge of the Lüders bands makes it very difficult to determine the real inherent material behavior. Therefore, developing effective strategies for solving these two points is highly desirable.
On this account, Rainer Schwab who is Professor Emeritus of Materials Science and Engineering and Mr. Anton Harter at Karlsruhe University of Applied Sciences proposed new approaches for extracting true stresses and strains from nominal stresses and strains during tensile testing. Their main objective was to solve the aforementioned points. To achieve this, the authors focused on (i) finding an exact solution for extracting true stresses and true plastic strains from the nominal stresses and strains in the uniform plastic deformation region and (ii) finding the real inherent flow curve and real upper yield strength in the Lüders region of materials with pronounced yield point phenomenon. For each of the two points, analytical solutions were derived. Their research work is currently published in the journal Strain.
Concerning point (i), the authors derived a complete set of exact analytical solutions for true stresses and strains with remarkable simplicity and beauty which completely accounts for the elastic volume changes. This set of exact solutions was cross-checked using finite element simulations and first- and zeroth-order approximations; perfect agreement was obtained. Concerning point (ii), a new macroscopic analytical method was used to determine the real material’s behavior in the Lüders region. This method is characterized by a high true upper yield point, typical strain hardening, and triaxiality of the stress state at the edges of the Lüders bands. This method was validated experimentally and through simulations and good agreement was obtained.
In summary, the study by Professor Rainer Schwab and Mr. Anton Harter addressed two fundamentals points that limit the accurate determination of flow curves in standard tensile testing. The exact solutions for extracting true stresses and true plastic strains in homogenous plastic deformation regions as well as inherent flow curves in the Lüders region, were determined. Both points were successfully validated, and the derived analytical solutions are in good agreement with the simulation and experimental results. In a statement to Advances in Engineering, Professor Rainer Schwab, the corresponding author, explained that although nature may behave in a complex manner, the underlying basic principles are often very simple. For instance, in the case of tensile tests, exact solutions may be derived with simple (mathematically beautiful) governing equations, and complicated plastic deformation behaviors can be explained with a few basic ideas.
Schwab, R. and Harter, A., 2021. Extracting true stresses and strains from nominal stresses and strains in tensile testing. Strain, 57(6), p. e12396.