Mathematical models for the plastic region in the tensile test

Question

The tensile test of a material consists of subjecting a standardized specimen to an increasing axial tensile stress until it breaks. During its performance in the laboratory, we can plot a stress-strain curve, which usually has the following form:

For the elastic zone, a linear relationship is found between the applied stress and the deformation undergone by the specimen. This relationship is known as Hooke's Law. On the other hand, for the plastic regime, I cannot find any mathematical model to explain it. In some books, I have found the Ramberg-Osgood equation, but they indicate that it applies mainly in the elastic limit environment.

Do you know of any model for the plastic deformation region (strain hardening, necking...)?

Answer 1

Plasticity is still an actively researched area. The Ludwik-Hollomon equation is one model that is used for the strain-hardening region since it captures the convex shape of the curve using a power law: $$\sigma = K \epsilon^{n} \tag{1}$$ Here, $n$ is known as the strain hardening coefficient or strain hardening exponent. This equation only captures the relationship between the stress and strain. To model the effect of strain rate ($\dot{\epsilon}$), another factor of $\dot{\epsilon}^m$ can be added to give: $$\sigma = K \epsilon^{n}\dot{\epsilon}^{m}\tag{2}$$ Here, $m$ is known as the strain rate sensitivity exponent. This is only a simple empirical model, and there are more (empirical or semi-empirical) models that have been proposed to capture the physics in the problem better. These models are referred to as hardening laws, and one example is Voce hardening.

In the region beyond the Ultimate Tensile Stress (UTS), necking occurs in the material, and the behaviour of the stress-strain curve depends on the heterogeneity in the sample. This phenomenon makes the UTS a point of geometric instability. Therefore, after the onset of necking, the engineering stress-strain curve doesn't possess a lot of fundamental value.

Also, note that the stress-strain curve in the question is highly exaggerated and idealised. In most engineering materials, there is no distinguishable "perfectly plastic" region. In fact, because of strain hardening, it is also not possible to "define" a yield stress purely based on how the curve changes from a linear region to a non-linear region. Rather, yield stress is defined as the stress at which one observes 2% permanent strain.

Answer 2

The other answer is great and comes from a viewpoint of macroscopic plasticity. I'd just like to note that another perspective on plasticity exists, a multiscale view based on following atomistic mechanisms up through the length scales to aim for an understanding of plasticity that is increasingly based on physical mechanisms. The enormous range of length scales makes this a daunting task, and it's therefore still an area of active research.

Much of the research focuses on the evolution of dislocation populations, as the motion of these defects controls the plasticity of metals in many cases. At the very smallest scales, molecular dynamics simulations have been used to study the motion of individual dislocation mechanisms and even some dislocation populations [1]. These are limited to a very high strain rate due to the small time steps required during the numerical integration of Newton's laws to describe the motion of atoms.

At a higher length scale, dislocations can be represented as discrete objects [2] or as part of a continuum [3]. The behaviors of dislocation populations determined through these methods can give an idea of the stress-strain response for a single crystal. Often these rules of strength evolution at the slip system scale are then incorporated into the crystal plasticity finite element method, in which multiple grains of difference orientations and morphologies can be considered in a representative volume element to give a final, overall stress-strain curve [4]. (More commonly, crystal plasticity models are fit to an existing stress-strain curve, to be used only on the same material system.)

Unfortunately, a multiscale approach does not give you a unified mathematical model to describe plasticity. There are important physics to consider at multiple length scales, and you'd have to incorporate an even wider variety of mechanisms if you want to predict failure. Most researchers recognize that all computational methods are compromises and choose the best tool to investigate their specific question. Cited are some recent papers on each topic, not necessarily the best review articles to start out with.

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[1] Zepeda-Ruiz, L. A.; Stukowski, A.; Oppelstrup, T.; Bertin, N.; Barton, N. R.; Freitas, R.; Bulatov, V. V. Atomistic Insights into Metal Hardening. Nat. Mater. 2020. https://doi.org/10.1038/s41563-020-00815-1.

[2] Akhondzadeh, Sh.; Sills, R. B.; Bertin, N.; Cai, W. Dislocation Density-Based Plasticity Model from Massive Discrete Dislocation Dynamics Database. Journal of the Mechanics and Physics of Solids 2020, 145, 104152. https://doi.org/10.1016/j.jmps.2020.104152.

[3] Xu, S.; Smith, L.; Mianroodi, J. R.; Hunter, A.; Svendsen, B.; Beyerlein, I. J. A Comparison of Different Continuum Approaches in Modeling Mixed-Type Dislocations in Al. Modelling Simul. Mater. Sci. Eng. 2019, 27 (7), 074004. https://doi.org/10.1088/1361-651X/ab2d16.

[4] Nguyen, K.; Zhang, M.; Amores, V. J.; Sanz, M. A.; Montáns, F. J. Computational Modeling of Dislocation Slip Mechanisms in Crystal Plasticity: A Short Review. Crystals 2021, 11 (1), 42. https://doi.org/10.3390/cryst11010042.