The elasto-plastic exquisite corpse: A Suquet legacy

Pierre Marie Suquet's pioneering work on plasticity paved the way for the mathematical theory of plasticity as we know it today. In this contribution we propose to review the most recent advances on that front and to illustrate how those impact classical problems of quasi-static elasto-plastic evolutions. Most notably, we exhibit new flow rules and derive conditions that prohibit the onset of plastic slips. From these, we obtain new uniqueness results for such evolutions.     

Anisotropic responses, constitutive modeling and the effects of strain-rate and temperature on the formability of an aluminum alloy

Finite deformation anisotropic responses of AA5182-O, over a wide range of strain-rates (10⁻⁴ to 10⁰ s⁻¹) and temperatures (293-473 K) are presented. The plastic anisotropy parameters were experimentally determined from tensile experiments using specimens from sheet material. Using the experimental results under plane stress conditions, the anisotropy coefficients for Barlat’s yield function (YLD96) were calculated at different strain-rates and temperatures. The correlations obtained from YLD96 are in good agreement with the observed experimental results. The strain-rate sensitivity of AA5182-O alloy changed from negative at 293 K to positive at 473 K. Khan-Huang-Liang (KHL) constitutive model is shown to correlate the observed strain-rate and temperature dependent responses reasonably well. The material parameters were obtained from the experimental responses along the rolling direction (RD) of the sheet. Marciniak and Kuckzinsky (M-K) theory was used to obtain the theoretical strain and stress-based forming limit curves (FLCs) at different strain-rates and temperatures. The experimental result from the published literature is compared with the FLCs from the current study.     

Inversion and quasi-static problems in linear viscoelasticity

Within the theory of linear viscoelasticity, we seek solutions to the inversion problem of the constitutive equation respectively inL ² and in the spaceS of the tempered distributions. Successively we study the quasi-static problem inS. Both problems admit one and only one solution if the relaxation function satisfies Graffi's inequality. Finally we show that the inversion problem and the quasi-static one are deeply connected and that every counterexample about the existence or uniqueness of the solutions for the first problem also provides a counterexample for the latter.

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Sommario Nell'ambito della viscoelasticità lineare, si studia il problema dell'invertibilità della equazione costitutiva rispettivamente inL ² e nello spazioS delle distribuzioni temperate. Successivamente si studia il problema quasi statico inS. Entrambi i problemi ammettono una ed una sola soluzione se la funzione di rilassamento soddisfa la disuguaglianza di Graffi. Infine si mostra che i due problemi sono strettamente correlati in quanto ogni problema quasi statico è riconducibile ad un problema di invertibilità.

     

On the time decay of solutions in porous-elasticity with quasi-static microvoids

In this paper we investigate the temporal asymptotic behavior of the solutions of the one-dimensional porous-elasticity problem with porous dissipation when the motion of microvoids is assumed to be quasi-static. This question has been recently studied in the general dynamical case. Thus, the natural question is to know if the assumption of quasi-static motion for the microvoids implies significant differences in the behavior of the solutions from the results obtained in the general dynamical case. It is worth noting that this assumption involves a qualitative change in the system of equations to be analyzed because it arises from the combination of a parabolic equation with an hyperbolic one, rather different from the well-known system of the thermo-elastic problem. First, we study the coupling of elasticity with porosity and we show that if only porous dissipation is present, the decay of solutions is slow, but if viscoelasticity is added, then the solutions decay exponentially. After that, we introduce thermal effects in the system and we show that while temperature brings exponential stability to the solutions, microtemperature does not.     

Formal solution of quasi-static problems

When the quasi-static problem is defined by a set of differential equations complemented by initial and boundary conditions, the resulting quasi-static solutions may exhibit a limited reach over the time domain. On the other hand, the infinity of equilibrium paths that can be obtained in a general non-linear problem also indicates that a proper definition of the quasi-static solution must be provided. In inelasticity problems, this infinite number of equilibrium paths occur even when no dissipative bifurcations are present. In the present paper, a general solution for quasi-static problems in Solid Mechanics is defined and explored. Special attention is addressed to material non-linearities though geometric non-linearities are also covered by the definition. Earlier concepts of path and state stability are recovered in order to reduce the number of solutions to those that are physically acceptable. The important link with the original dynamic problem is accounted for by enforcing a preferential load direction. The resulting definition relies on a time-objective criterion with straightforward applicability to the most common numerical models. In the final part of the paper, simple 1D problems are used to illustrate some of the concepts introduced in the present developments.     

Recovering convexity in non-associated plasticity

We quickly review two main non-associated plasticity models, the Armstrong–Frederick model of nonlinear kinematic hardening and the Drucker–Prager cap model. Non-associativity is commonly thought to preclude any kind of variational formulation, be it in a Hencky-type (static) setting, or when considering a quasi-static evolution because non-associativity destroys convexity. We demonstrate that such an opinion is misguided: associativity (and convexity) can be restored at the expense of the introduction of state variable-dependent dissipation potentials.     

扫描近场光学显微镜的光耦合偶极子模型

为了研究扫描近场光学显微镜中探针和粗糙样品表面的耦合相互作用,提出了一种光耦合偶极子模型.在该模型中,探针和样品突起都由光极化偶极子表示,在准静态电磁场近似的情况下样品表面的诱导极化效应由影像偶极子表示,应用偶极子辐射理论可以得到系统的自洽场方程.此模型提供了一种直观分析扫描近场光学显微镜中探针和样品相互作用机理的方法.在此基础上,进一步讨论了金属样品的近场成像特点和其特有的局域光学共振现象.数值结果表明：不同于一般的介质样品,金属样品的近场图像与入射光频率直接相关,改变入射光的频率,获得的样品近场图像的形状和对比度都会发生变化.特别是当入射光频率处于样品极化共振范围内时,金属纳米粒子的极化率会出现光极化共振,这样就可以获得样品粒子的最大有效尺寸,为提高系统的分辨率提供了一条重要途径.