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On the well posedness of static boundary value problem within the linear dilatational strain gradient elasticity
Zeitschrift für angewandte Mathematik und Physik ( IF 2 ) Pub Date : 2020-10-10 , DOI: 10.1007/s00033-020-01395-5
Victor A. Eremeyev , Sergey A. Lurie , Yury O. Solyaev , Francesco dell’Isola

In this paper, it is proven an existence and uniqueness theorem for weak solutions of the equilibrium problem for linear isotropic dilatational strain gradient elasticity. Considered elastic bodies have as deformation energy the classical one due to Lamé but augmented with an additive term that depends on the norm of the gradient of dilatation: only one extra second gradient elastic coefficient is introduced. The studied class of solids is therefore related to Korteweg or Cahn–Hilliard fluids. The postulated energy naturally induces the space in which the aforementioned well-posedness result can be formulated. In this energy space, the introduced norm does involve the linear combination of some specific higher-order derivatives only: it is, in fact, a particular example of anisotropic Sobolev space. It is also proven that aforementioned weak solutions belongs to the space \(H^1(div,V)\), i.e. the space of \(H^1\) functions whose divergence belongs to \(H^1\). The proposed mathematical frame is essential to conceptually base, on solid grounds, the numerical integration schemes required to investigate the properties of dilatational strain gradient elastic bodies. Their energy, as studied in the present paper, has manifold interests. Mathematically speaking, its singularity causes interesting mathematical difficulties whose overcoming leads to an increased understanding of the theory of second gradient continua. On the other hand, from the mechanical point of view, it gives an example of energy for a second gradient continuum which can sustain externally applied surface forces and double forces but cannot sustain externally applied surface couples. In this way, it is proven that couple stress continua, introduced by Toupin, represent only a particular case of the more general class of second gradient continua. Moreover, it is easily checked that for dilatational strain gradient continua, balance of force and balance of torques (or couples) are not enough to characterise equilibrium: to this aim, externally applied surface double forces must also be specified. As a consequence, the postulation scheme based on variational principles seems more suitable to study second gradient continua. It has to be remarked finally that dilatational strain gradient seems suitable to model the experimentally observed behaviour of some material used in 3D printing process.



中文翻译:

线性膨胀应变梯度弹性内静态边值问题的适定性

本文证明了线性各向同性膨胀应变梯度弹性平衡问题的弱解的存在唯一性定理。认为弹性体具有经典的变形能量,这是由于Lamé导致的,但又增加了一个附加项,该附加项取决于膨胀梯度的范数:仅引入一个额外的第二个梯度弹性系数。因此,所研究的固体类别与Korteweg或Cahn-Hilliard流体有关。假定的能量自然地诱发了一个空间,可以在其中形成上述良好的姿势结果。在这个能量空间中,引入的范数确实仅涉及某些特定的高阶导数的线性组合:实际上,它是各向异性Sobolev空间的一个特定示例。\(H ^ 1(div,V)\),即\(H ^ 1 \)函数的散度属于\(H ^ 1 \)的空间。所提出的数学框架对于在概念上以坚实的基础为基础,需要使用数值积分方案来研究膨胀应变梯度弹性体的特性。如本文所研究的,它们的能量具有多种利益。从数学上讲,其奇异性引起有趣的数学困难,其克服导致对第二梯度连续性理论的更多理解。另一方面,从机械角度来看,它给出了第二个梯度连续体的能量示例,该第二梯度连续体可以承受外部施加的表面力和双重力,但不能承受外部施加的表面偶。通过这种方式,证明了Toupin引入的耦合应力连续性,仅代表第二梯度连续性的更一般类别的特定情况。此外,很容易检查出,对于连续应变梯度的扩张,力的平衡和扭矩(或偶数)的平衡不足以表征平衡:为此,还必须指定外部施加的表面双力。结果,基于变分原理的假设方案似乎更适合于研究第二个梯度连续体。最后必须指出,膨胀应变梯度似乎适合模拟在3D打印过程中使用的某些材料的实验观察到的行为。还必须指定外部施加的表面双力。结果,基于变分原理的假设方案似乎更适合于研究第二个梯度连续体。最后必须指出,膨胀应变梯度似乎适合模拟在3D打印过程中使用的某些材料的实验观察到的行为。还必须指定外部施加的表面双力。结果,基于变分原理的假设方案似乎更适合于研究第二个梯度连续体。最后必须指出,膨胀应变梯度似乎适合模拟在3D打印过程中使用的某些材料的实验观察到的行为。

更新日期:2020-10-11
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