Short noteElastic Hugoniot curve of one-dimensional Wilkins model with general Grüneisen-type equation of state
Introduction
It's well known that the Hugoniot curve, which describes shock relation between states ahead and behind a shock, plays very important role in the study of shock transition theory in gas dynamics and elastic-plastic solid. In gas dynamics, the Hugoniot curve is defined in the specific volume and pressure plane . Its characters, including monotonicity, curvature, relation with adiabatic curve etc., have been strictly proved [13], [6], [14], [2] for general equation of state (EOS). The basic properties of shock transition, such as the increase of entropy, compressibility of shock and uniqueness behind a shock, can be obtained from the analysis of Hugoniot curve.
In elastic-plastic solid, the Rankine-Hugoniot condition is based on τ and the Cauchy stress and the theory of shock transition becomes very complicated. For example, for a given elastic state ahead of shock, the shock structure might include single elastic wave, elastic-plastic two-wave, or a single merge wave [12], [9]. In which some theoretical problems, such as the existence of two-wave structure, have not been solved completely, see appendix. This makes it more imperative to research properties of Hugoniot curve. There are already some works on the subject [5], [12], [3], [10], but most of them focus on some special cases, such as special EOS, modified model or special initial state. For general EOS and arbitrary initial state before a shock, the properties of the Hugoniot curve have not been given yet.
In this paper, we investigate the properties of the Hugoniot curve for Wilkins model [17] of elastic-plastic solid with general Grüneisen-type EOS. This work is of great significance for studying the stress wave structures, which can provide better help for the design of numerical algorithms. This model is a hypo-elastic one and has some well known flaws [5], [11]. One is that it is not in conservative form, which brings difficulty for the mathematical analysis of shocks. Another drawback is that its constitutive law can not naturally satisfy the principle of material objectivity, and thus Jaumann type derivatives have to be introduced to restore the principle of frame-indifference. The third is its thermodynamical inconsistency, which results in the loss of entropy criterion to discontinuity. Despite these deficiencies, this model is largely used in engineering science due to its accuracy in the domain of solid dynamics for many materials. The motivation of the work is to provide a theoretical basis for constructing exact or approximate Riemann solvers in a Godunov method, thus we have to handle arbitrary Riemann initial values created in the calculation procedure, which might cause the variation of wave structures, refer to the details in the appendix. In this short note, we only discuss the case of a single elastic shock, which is a typical representative of other wave types. Different from gas dynamics with positive pressure, the signs of the Cauchy stress and hydrostatic pressure p in solid are uncertain. Such uncertainty results in difficulties to prove the properties of the Hugoniot curve. The general EOS and thermodynamically inconsistency of Wilkins model [5], [11] intensify difficulty of solving the problem. Our aim is to overcome the obstacles from complex EOS and obtain the property of the Hugoniot curve. The key work is to find two inequality relations of some physical quantities, and simplify the proof process, which might offer help to future researchers who have the capability to finally solve this kind of problem.
Section snippets
Governing equations
The governing equations of one-dimensional elastic-perfectly plastic solid are as follows: where and are the state vector and flux vector respectively. ρ and u are the density and velocity. is the specific total energy and e is the specific internal energy. We remark here that the elastic energy is not included in the total energy in this paper. It is because its contribution can be neglected for most materials based on the analysis
Properties of the Hugoniot curves of elastic and plastic shocks
The relations between the fluxes and variables of states on both sides of the strong discontinuity are the Rankine-Hugoniot conditions. That is to say, for an elastic (plastic) shock with velocity S, the fluxes and states before and behind the shock front satisfy . Here, the variables with subscript ‘A’ are the initial states before the shock front. Through a simple derivation from these relations, we can obtain two famous curves that characterize the state behind the shock in the
Conclusion
Based on some natural assumptions, the characters of the Hugoniot curves in Wilkins model with general Grüneisen-type EOS are obtained for arbitrary initial value ahead of a shock front. We demonstrate that both the elastic and plastic Hugoniot curves are monotonic decreasing and convex downward. After the shock, the Cauchy stress and specific volume thus always decrease. Different from gas dynamics, the specific internal energy and specific entropy of solid material do not necessarily increase
CRediT authorship contribution statement
Xiao Li: Conceptualization, Formal analysis, Investigation, Methodology, Writing – original draft, Writing – review & editing. Jiayin Zhai: Writing – review & editing. Zhijun Shen: Funding acquisition, Resources, Supervision, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This project was supported by the National Natural Science Foundation of China (11971071, U1630249) and the Science Challenge Project (No. JCKY2016212A502).
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