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A variational approach to frame-indifferent quasistatic viscoelasticity of rate type. 一种变分法,用于处理与框架无关的速率型准静态粘弹性。
IF 4.3 3区 综合性期刊 Q1 MULTIDISCIPLINARY SCIENCES Pub Date : 2024-08-23 Epub Date: 2024-07-15 DOI: 10.1098/rsta.2023.0307
Yasemin Şengül

The three-dimensional dynamical model for nonlinear viscoelasticity of strain-rate type is investigated in a quasistatic setting under the assumption of higher-order regularity of the deformation, which in the literature is referred to as the case of non-simple materials. The existence of weak solutions is proven using a time-discretization technique while respecting the condition of dynamical frame indifference. Some observations on frame indifference for strain-rate-type stresses are made, and corrections are proposed for some related work in the literature. Finally, a counterexample is given to show that the assumed higher-order regularity is necessary in order to obtain the required compactness.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

在变形的高阶正则性假设下,在准静态环境中研究了应变速率型非线性粘弹性的三维动力学模型,这在文献中被称为非简单材料的情况。利用时间离散化技术证明了弱解的存在,同时尊重了动态框架无偏的条件。对应变速率型应力的框架无关性提出了一些看法,并对文献中的一些相关工作提出了修正。最后,给出了一个反例,以说明假定的高阶正则性对于获得所需的紧凑性是必要的。本文是主题 "非光滑变分问题在力学中的应用 "的一部分。
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引用次数: 0
What is the physical origin of the gradient flow structure of variational fracture models? 变分断裂模型梯度流动结构的物理起源是什么?
IF 4.3 3区 综合性期刊 Q1 MULTIDISCIPLINARY SCIENCES Pub Date : 2024-08-23 Epub Date: 2024-07-15 DOI: 10.1098/rsta.2023.0297
Masato Kimura, Takeshi Takaishi, Yoshimi Tanaka

We investigate a physical characterization of the gradient flow structure of variational fracture models for brittle materials: a Griffith-type fracture model and an irreversible fracture phase field model. We derive the Griffith-type fracture model by assuming that the fracture energy in Griffith's theory is an increasing function of the crack tip velocity. Such a velocity dependence of the fracture energy is typically observed in polymers. We also prove an energy dissipation identity of the Griffith-type fracture model, in other words, its gradient flow structure. On the other hand, the irreversible fracture phase field model is derived as a unidirectional gradient flow of a regularized total energy. We have considered the time relaxation parameter a mathematical approximation parameter, which we should choose as small as possible. In this research, however, we reveal the physical origin of the gradient flow structure of the fracture phase field model (F-PFM) and show that the small time relaxation parameter is characterized as the rate of velocity dependence of the fracture energy. It is verified by comparing the energy dissipation properties of those two models and by analysing a travelling wave solution of the irreversible F-PFM. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

我们研究了脆性材料变分断裂模型梯度流结构的物理特征:格里菲斯型断裂模型和不可逆断裂相场模型。我们假设格里菲斯理论中的断裂能是裂纹尖端速度的递增函数,从而推导出格里菲斯型断裂模型。在聚合物中通常可以观察到断裂能与速度的关系。我们还证明了格里菲斯型断裂模型的能量耗散特性,即其梯度流动结构。另一方面,不可逆断裂相场模型被推导为正则化总能量的单向梯度流。我们认为时间松弛参数是一个数学近似参数,应尽可能选小。但在本研究中,我们揭示了断裂相场模型(F-PFM)梯度流结构的物理起源,并证明了小时间松弛参数的特征是断裂能量的速度依赖性。通过比较这两种模型的能量耗散特性,并分析不可逆 F-PFM 的行波解,验证了这一点。本文是 "非光滑变分问题在力学中的应用 "专题的一部分。
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引用次数: 0
Thin inclusion at the junction of two elastic bodies: non-coercive case. 两个弹性体交界处的薄包裹体:非胁迫情况。
IF 4.3 3区 综合性期刊 Q1 MULTIDISCIPLINARY SCIENCES Pub Date : 2024-08-23 Epub Date: 2024-07-15 DOI: 10.1098/rsta.2023.0296
A M Khludnev

This article addresses an analysis of the non-coercive boundary value problem describing an equilibrium state of two contacting elastic bodies connected by a thin elastic inclusion. Nonlinear conditions of inequality type are imposed at the joint boundary of the bodies providing a mutual non-penetration. As for conditions at the external boundary, they are Neumann type and imply the non-coercivity of the problem. Assuming that external forces satisfy suitable conditions, a solution existence of the problem analysed is proved. Passages to limits are justified as the rigidity parameters of the inclusion and the elastic body tend to infinity.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

本文分析了非强制边界值问题,该问题描述了由薄弹性夹层连接的两个接触弹性体的平衡状态。不等式类型的非线性条件施加在两个体的联合边界上,提供了相互不穿透性。至于外部边界的条件,它们是诺伊曼类型的,意味着问题的非矫顽力性。假设外力满足适当的条件,就能证明所分析问题的解的存在性。当包容体和弹性体的刚度参数趋于无穷大时,进入极限是合理的。
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引用次数: 0
On the numerical corroboration of an obstacle problem for linearly elastic flexural shells. 关于线性弹性挠性壳的障碍物问题的数值确证。
IF 4.3 3区 综合性期刊 Q1 MULTIDISCIPLINARY SCIENCES Pub Date : 2024-08-23 Epub Date: 2024-07-15 DOI: 10.1098/rsta.2023.0306
Xin Peng, Paolo Piersanti, Xiaoqin Shen

In this article, we study the numerical corroboration of a variational model governed by a fourth-order elliptic operator that describes the deformation of a linearly elastic flexural shell subjected not to cross a prescribed flat obstacle. The problem under consideration is modelled by means of a set of variational inequalities posed over a non-empty, closed and convex subset of a suitable Sobolev space and is known to admit a unique solution. Qualitative and quantitative numerical experiments corroborating the validity of the model and its asymptotic similarity with Koiter's model are also presented.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

在这篇文章中,我们研究了一个由四阶椭圆算子支配的变分模型的数值确证,该模型描述了一个线性弹性挠性壳体在不跨越规定的平面障碍物的情况下的变形。所考虑的问题是通过在合适的 Sobolev 空间的非空、闭合和凸子集上提出的一组变分不等式来模拟的,并且已知该问题有一个唯一的解决方案。本文是 "非光滑变分问题在力学中的应用 "专题的一部分。
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引用次数: 0
The homogenized dynamical model of a thermoelastic composite stitched with reinforcing filaments. 用增强丝缝合的热弹性复合材料的均质动力学模型。
IF 4.3 3区 综合性期刊 Q1 MULTIDISCIPLINARY SCIENCES Pub Date : 2024-08-23 Epub Date: 2024-07-15 DOI: 10.1098/rsta.2023.0304
Evgeny M Rudoy, Sergey A Sazhenkov

The dynamical problem of linear thermoelasticity for a body with incorporated thin rectilinear inclusions is studied. It is assumed that the inclusions (i.e. filaments and threads) are parallel to each other and the problem contains a small parameter [Formula: see text], which characterizes the distance between two neighbouring inclusions. Using the two-scale convergence approach, we find the limiting problem as [Formula: see text]. As a result, we get a well-posed homogenized model of an anisotropic inhomogeneous body with effective characteristics inheriting thermomechanical properties of inclusions.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

本课题研究的是含有细直角夹杂物的物体的线性热弹性动力学问题。假设夹杂物(即细丝和细线)相互平行,问题包含一个小参数[公式:见正文],该参数描述了两个相邻夹杂物之间的距离。利用双尺度收敛法,我们发现极限问题为[公式:见正文]。因此,我们得到了一个各向异性非均质体的均质化模型,该模型具有继承夹杂物热力学特性的有效特征。
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引用次数: 0
Exploring the performance of the topological energy method for object and damage detection from noisy and poor databases. 探索拓扑能量法在从嘈杂和贫乏的数据库中检测物体和损伤方面的性能。
IF 4.3 3区 综合性期刊 Q1 MULTIDISCIPLINARY SCIENCES Pub Date : 2024-08-23 Epub Date: 2024-07-15 DOI: 10.1098/rsta.2023.0303
M Pena, S Muñoz, M-L Rapún

In this work, we study the performance of numerical methods based on the computation of topological energies to process data from synthetic and real experiments where only noisy limited-view data corresponding to a few frequencies are available. We show numerical experiments in two problems of practical interest. The first one corresponds to experimental measurements of the electromagnetic scattering produced by different objects extracted from the Fresnel database. The second one is related to synthetic experiments in a simplified model where steel welding joints are acoustically tested to identify possible flaws (air bubbles and inclusions) produced during the welding process. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

在这项工作中,我们研究了基于拓扑能量计算的数值方法在处理合成和实际实验数据时的性能,在这些实验中,只有对应于少数频率的噪声有限视图数据可用。我们展示了两个实际问题的数值实验。第一个问题涉及从菲涅尔数据库中提取的不同物体产生的电磁散射的实验测量结果。第二个问题与简化模型中的合成实验有关,即对钢焊接接头进行声学测试,以识别焊接过程中可能产生的缺陷(气泡和夹杂物)。本文是主题 "非光滑变分问题在力学中的应用 "的一部分。
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引用次数: 0
Long-time behaviour of a porous medium model with degenerate hysteresis. 具有退化滞后的多孔介质模型的长期行为。
IF 4.3 3区 综合性期刊 Q1 MULTIDISCIPLINARY SCIENCES Pub Date : 2024-08-23 Epub Date: 2024-07-15 DOI: 10.1098/rsta.2023.0299
Chiara Gavioli, Pavel Krejčí

Hysteresis in the pressure-saturation relation in unsaturated porous media, owing to surface tension on the liquid-gas interface, exhibits strong degeneracy in the resulting mass balance equation. As an extension of previous existence and uniqueness results, we prove that under physically admissible initial conditions and without mass exchange with the exterior, the unique global solution of the fluid diffusion problem exists and asymptotically converges as time tends to infinity to a possibly non-homogeneous mass distribution and an a priori unknown constant pressure.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

在非饱和多孔介质中,由于液气界面上的表面张力,压力-饱和度关系中的滞后现象在由此产生的质量平衡方程中表现出很强的退化性。作为之前存在性和唯一性结果的扩展,我们证明了在物理上可接受的初始条件下,在不与外部进行质量交换的情况下,流体扩散问题的唯一全局解是存在的,并且随着时间趋于无穷,会渐近收敛到可能非均质的质量分布和先验未知的恒定压力。
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引用次数: 0
Minimization of peak stresses with the shape derivative. 利用形状导数使峰值应力最小化。
IF 4.3 3区 综合性期刊 Q1 MULTIDISCIPLINARY SCIENCES Pub Date : 2024-08-23 Epub Date: 2024-07-15 DOI: 10.1098/rsta.2023.0309
Phillip Baumann, Kevin Sturm

This article is concerned with the minimization of peak stresses occurring in linear elasticity. We propose to minimize the maximal von Mises stress of the elastic body. This leads to a non-smooth shape functional. We derive the shape derivative and associate it with the Clarke sub-differential. Using a steepest descent algorithm, we present numerical simulations. We compare our results to the usual [Formula: see text]-norm regularization and show that our algorithm performs better in the presented tests.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

本文涉及线性弹性中峰值应力的最小化。我们建议尽量减小弹性体的最大 von Mises 应力。这将导致一个非光滑的形状函数。我们推导出形状导数,并将其与克拉克子导数联系起来。我们使用最陡峭下降算法进行了数值模拟。我们将结果与通常的[公式:见正文]正则化进行了比较,结果表明我们的算法在测试中表现更好。
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引用次数: 0
Hard interfaces with microstructure: the cases of strain gradient elasticity and micropolar elasticity. 具有微结构的硬界面:应变梯度弹性和微弹性的情况。
IF 4.3 3区 综合性期刊 Q1 MULTIDISCIPLINARY SCIENCES Pub Date : 2024-08-23 Epub Date: 2024-07-15 DOI: 10.1098/rsta.2023.0308
Michele Serpilli, Raffaella Rizzoni, Frédéric Lebon

As the size of a layered structure scales down, the adhesive layer thickness correspondingly decreases from macro- to micro-scale. The influence of the material microstructure of the adhesive becomes more pronounced, and possible size effect phenomena can appear. This paper describes the mechanical behaviour of composites made of two solids, bonded together by a thin layer, in the framework of strain gradient and micropolar elasticity. The adhesive layer is assumed to have the same stiffness properties as the adherents. By means of the asymptotic methods, the contact laws are derived at order 0 and order 1. These conditions represent a formal generalization of the hard elastic interface conditions. A simple benchmark equilibrium problem (a three-layer composite micro-bar subjected to an axial load) is developed to numerically assess the asymptotic model. Size effects and non-local phenomena, owing to high strain concentrations at the edges, are highlighted. The example proves the efficiency of the proposed approach in designing micro-scale-layered devices.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

随着分层结构尺寸的缩小,粘合剂层厚度也相应地从宏观尺度减小到微观尺度。粘合剂材料微观结构的影响变得更加明显,可能出现尺寸效应现象。本文在应变梯度和微极性弹性的框架下,描述了由两个固体通过薄层粘合而成的复合材料的机械性能。假设粘合层具有与被粘物相同的刚度特性。通过渐近方法,得出了阶 0 和阶 1 的接触定律。这些条件代表了硬弹性界面条件的形式概括。为了对渐近模型进行数值评估,我们开发了一个简单的基准平衡问题(承受轴向载荷的三层复合微棒)。由于边缘的高应变集中,尺寸效应和非局部现象得到了强调。这个例子证明了所提出的方法在设计微尺度层状装置中的效率。本文是 "非光滑变分问题在力学中的应用 "专题的一部分。
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引用次数: 0
Variational inequality for a Timoshenko plate contacting at the boundary with an inclined obstacle. 蒂莫申科板与倾斜障碍物边界接触的变量不等式。
IF 4.3 3区 综合性期刊 Q1 MULTIDISCIPLINARY SCIENCES Pub Date : 2024-08-23 Epub Date: 2024-07-15 DOI: 10.1098/rsta.2023.0298
Victor A Kovtunenko, Nyurgun P Lazarev

A class of variational inequalities describing the equilibrium of elastic Timoshenko plates whose boundary is in contact with the side surface of an inclined obstacle is considered. At the plate boundary, mixed conditions of Dirichlet type and a non-penetration condition of inequality type are imposed on displacements in the mid-plane. The novelty consists of modelling oblique interaction with the inclined obstacle which takes into account shear deformation and rotation of transverse cross-sections in the plate. For proposed problems of equilibrium of the plate contacting the inclined obstacle, the unique solvability of the corresponding variational inequality is proved. Under the assumption that the variational solution is smooth enough, optimality conditions are obtained in the form of equilibrium equations and relations revealing the mechanical properties of integrated stresses, moments and generalized displacements on the contact part of the boundary. Accounting for complementarity type conditions owing to the contact of the plate with the inclined obstacle, a primal-dual variational formulation of the obstacle problem is derived. A semi-smooth Newton method based on a generalized gradient is constructed and performed as a primal-dual active-set algorithm. It is advantageous for efficient numerical solution of the problem, provided by a super-linear estimate for the corresponding iterates in function spaces. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

研究了描述边界与倾斜障碍物侧表面接触的弹性季莫申科板平衡的一类变分不等式。在板的边界上,对中平面的位移施加了迪里夏特类型的混合条件和不等式类型的非穿透条件。新颖之处在于模拟与倾斜障碍物的斜向相互作用,其中考虑到了板中横截面的剪切变形和旋转。对于所提出的板与倾斜障碍物接触的平衡问题,证明了相应变分不等式的唯一可解性。在假设变分求解足够平滑的前提下,以平衡方程和关系的形式获得了最优条件,揭示了边界接触部分的综合应力、力矩和广义位移的力学特性。考虑到由于板与倾斜障碍物接触而产生的互补型条件,得出了障碍物问题的初等二元变分公式。构建了基于广义梯度的半光滑牛顿方法,并将其作为初等-二元主动集算法执行。通过对函数空间中相应迭代的超线性估计,该方法有利于问题的高效数值求解。本文是主题 "非光滑变分问题在力学中的应用 "的一部分。
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引用次数: 0
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