{"title":"Complementary Energy Theorem for Thin Composite Plates in Postbuckling","authors":"S. V. Selyugin","doi":"10.1134/S0025654424602957","DOIUrl":null,"url":null,"abstract":"<p>The thin composite von Kármán plates in postbuckling are considered. Using the first Piola stress tensor and the displacement gradient tensor, the complementary energy variational theorem is proven. The Kirchhoff assumptions are adopted. The plate lay-up is symmetric and pointwise. According to the theorem, at the actual stress state of the plate the complementary energy (as a functional of the internal forces and of the moments) reaches its stationary value. The stationary feature of the actual state is valid as compared to other feasible states satisfying the static equilibrium and the static boundary conditions. The theorem is a consent of the static variational principle. The principle leads to the linear relations between forces/moments, created by the corresponding first Piola stress tensor components, and the 2D-strains/curvatures. An illustrative plate example is given.</p>","PeriodicalId":697,"journal":{"name":"Mechanics of Solids","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mechanics of Solids","FirstCategoryId":"5","ListUrlMain":"https://link.springer.com/article/10.1134/S0025654424602957","RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MECHANICS","Score":null,"Total":0}
引用次数: 0
Abstract
The thin composite von Kármán plates in postbuckling are considered. Using the first Piola stress tensor and the displacement gradient tensor, the complementary energy variational theorem is proven. The Kirchhoff assumptions are adopted. The plate lay-up is symmetric and pointwise. According to the theorem, at the actual stress state of the plate the complementary energy (as a functional of the internal forces and of the moments) reaches its stationary value. The stationary feature of the actual state is valid as compared to other feasible states satisfying the static equilibrium and the static boundary conditions. The theorem is a consent of the static variational principle. The principle leads to the linear relations between forces/moments, created by the corresponding first Piola stress tensor components, and the 2D-strains/curvatures. An illustrative plate example is given.
考虑了后屈曲中的薄复合 von Kármán 板。利用第一皮奥拉应力张量和位移梯度张量,证明了补能变分定理。采用基尔霍夫假设。板的铺设是对称和点对点的。根据该定理,在板的实际应力状态下,互补能(作为内力和力矩的函数)达到其静止值。与满足静态平衡和静态边界条件的其他可行状态相比,实际状态的静止特征是有效的。该定理是对静态变分原理的认可。该原理导致了由相应的第一皮奥拉应力张量分量产生的力/力矩与二维应变/曲率之间的线性关系。给出了一个示例板。
期刊介绍:
Mechanics of Solids publishes articles in the general areas of dynamics of particles and rigid bodies and the mechanics of deformable solids. The journal has a goal of being a comprehensive record of up-to-the-minute research results. The journal coverage is vibration of discrete and continuous systems; stability and optimization of mechanical systems; automatic control theory; dynamics of multiple body systems; elasticity, viscoelasticity and plasticity; mechanics of composite materials; theory of structures and structural stability; wave propagation and impact of solids; fracture mechanics; micromechanics of solids; mechanics of granular and geological materials; structure-fluid interaction; mechanical behavior of materials; gyroscopes and navigation systems; and nanomechanics. Most of the articles in the journal are theoretical and analytical. They present a blend of basic mechanics theory with analysis of contemporary technological problems.