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Generalised tangent stabilised nonlinear elasticity: An automated framework for controlling material and geometric instabilities

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2025年 436卷

作者： Poya, Roman Ortigosa, Rogelio Gil, Antonio J. Kim, Theodore Bonet, Javier Meshing Framework Simulat &Test Solut Siemens Digital Indus Software Cambridge England Multiphys Simulat & Optimisat Lab Tech Universityof Cartagena Cartagena Murcia Spain Swansea Univ Fac Sci & Engn Zienkiewicz Ctr Modelling Data & AI Bay Campus Swansea Wales Yale Univ Yale Sch Engn & Appl Sci Comp Graph Grp New Haven CT USA Ctr Int Metodes Numer Engn CIMNE Barcelona Spain

Tangent stabilised large strain isotropic elasticity was recently proposed by Poya et al. (2023) wherein by working directly with principal stretches the entire eigenstructure of constitutive and geometric/initial stiffness terms were found in closed-form, giving fresh insights into exact convexity conditions of highly non-convex functions in discrete settings. Consequently, owing to these newly found tangent eigenvalues an analytic tangent stabilisation was proposed (for common non-convex strain energies that exhibit material and/or geometric instabilities) bypassing incumbent numerical approaches routinely used in nonlinear finite element analysis. This formulation appears to be extremely robust for quasi-static simulation of complex deformations even with no load increments and time stepping while still capturing instabilities (similar to dynamic analysis) automatically in ways that are infeasible for path-following techniques in practice. In this work, we generalise the notion of tangent stabilised elasticity to virtually all known invariant formulations of nonlinear elasticity. We show that, closed- form eigen-decomposition of tangents is easily available irrespective of invariant formulation or integrity basis. In particular, we work out closed-form tangent eigensystems for isotropic Total Lagrangian deformation gradient (F)-based and right Cauchy-Green (C)-based as well as Updated Lagrangian left Cauchy-Green (b)-based formulations and present their exact convexity conditions postulated in terms of their corresponding tangent and geometric stiffness eigenvalues. In addition, we introduce the notion of geometrically stabilised polyconvex large strain elasticity for models that are materially stable but exhibit geometric instabilities for whom we construct their geometric stiffness in a spectrally-decomposed form analytically. We further extend this framework to the case of transverse isotropy where once again, closed- form tangent eigensystems are found for co

关键词： nonlinear elasticity Large strain Material instability Geometric instability Tangent stabilisation Finite elements

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Mechanical characterization of nonlinear elasticity of growing intestinal organoids with a microinjection method

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ACTA BIOMATERIALIA 2025年 196卷 271-280页

作者： Xiu, Jidong Xue, Rui Duan, Xiaocen Yao, Fangyun Liu, Xiaozhi Meng, Fanlu Xiong, Chunyang Huang, Jianyong Peking Univ Coll Engn Dept Mech & Engn Sci Beijing 100871 Peoples R China Fifth Cent Hosp Tianjin Tianjin Key Lab Epigenet Organ Dev Premature Infan Tianjin 300450 Peoples R China Univ Chinese Acad Sci Wenzhou Inst Wenzhou 325000 Zhejiang Peoples R China

Mechanical properties of intestinal organoids are crucial for intestinal development, homeostatic renewal, and pathogenesis. However, characterizing these properties remains challenging. Here, we developed a microinjection-based method to quantify the growth time-dependent nonlinear elasticity of intestinal organoids. With aid of the neo-Hookean hyperelastic constitutive model, we discovered that the global elastic modulus of intestinal organoids increased linearly during the early stages of culture, followed by a sharp rise, indicating a time-dependent nonlinear hardening behaviour during growth. The global modulus of intestinal organoids was found to correlate with the cell phenotype ratio, revealing a significant relationship between mechanical properties and biological phenotypes. Furthermore, we developed a biomechanical model on the basis of the unsteady Bernoulli equation to quantitatively explore the global mechanical responses of intestinal organoids, which showed good agreement with the experimental data. The work not only elucidated the mechanical response and modulus characteristics of small intestinal organoids from a biomechanical perspective, but also presented a new microinjection-based methodology for quantifying the mechanical properties of organoids, offering significant potential for various organoid-related applications. Statement of significance: Mechanical properties of intestinal organoids are essential for intestinal development, homeostatic renewal, and pathogenesis. However, how to quantitatively characterize their global mechanical properties remains challenging. Here, we developed a new microinjection-based experimental platform to quantify spatiotemporal dynamics of mechanical responses and global elasticity of intestinal organoids. Unlike traditional nanoindentation methods, the proposed characterization technique can quantitatively measure the global mechanical properties of organoids, which is crucial for detecting the inherent relat

关键词： Intestinal organoid Microinjection Mechanical characterization nonlinear elasticity

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nonlinear elasticity modeling of biogels

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JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS 2010年第3期58卷 300-310页

作者： Wu, M. S. Kirchner, H. O. K. Nanyang Technol Univ Sch Mech & Aerosp Engn Singapore 639798 Singapore INM Leibniz Inst New Mat D-66123 Saarbrucken Germany

A second-order nonlinear elastic model is developed for biogels, whereby the elastic constants are estimated from the free energy densities for polymer networks and polymer-water mixtures. For typical values of the constants, important physical insights are revealed: a biogel stiffens under tension, contracts longitudinally under torsion (inverse Poynting effect), and experiences a negative normal stress under simple shear. This approach of extracting nonlinear elastic constants works for any constitutive law, and has the advantage of exploring physical phenomena without recourse to computationally intensive simulations. (C) 2009 Elsevier Ltd All rights reserved.

关键词： Biogel nonlinear elasticity Inverse Poynting effect Negative normal stress

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nonlinear elasticity of monolayer hexagonal crystals: Theory and application to circular bulge test

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EUROPEAN JOURNAL OF MECHANICS A-SOLIDS 2018年 68卷 117-132页

作者： Delfani, M. R. KN Toosi Univ Technol Fac Civil Engn POB 15875-4416 Tehran Iran

Graphene and graphene-like materials possessing monolayer hexagonal crystals have become one of the focal points of scientific researches. In this paper, an elasticity peculiar to these materials is formulated that is capable of taking both geometrical and material nonlinearities into consideration. A thin plate is proposed as a continuum. model for such materials but, after postulating certain assumptions consistent with their physical nature, it is revealed that the thin plate behaves analogous to a membrane with nonzero bending rigidity. By making use of concepts from differential geometry of surfaces, interesting properties related to deformation field of the thin plate are proved. As an application to the proposed, theory, the elastic behavior of a circular specimen of a monolayer hexagonal crystal subjected to bulge test is examined and, through a perturbation analysis, a series solution is obtained for its deformation field.

关键词： Monolayer hexagonal crystal nonlinear elasticity Differential geometry of surfaces Circular bulge test

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nonlinear elasticity in a Deforming Ambient Space

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JOURNAL OF nonlinear SCIENCE 2016年第6期26卷 1651-1692页

作者： Yavari, Arash Ozakin, Arkadas Sadik, Souhayl Georgia Inst Technol Sch Civil & Environm Engn Atlanta GA 30332 USA Georgia Inst Technol George W Woodruff Sch Mech Engn Atlanta GA 30332 USA Skytree Inc San Jose CA 95110 USA

In this paper, we formulate a nonlinear elasticity theory in which the ambient space is evolving. For a continuum moving in an evolving ambient space, we model time dependency of the metric by a time-dependent embedding of the ambient space in a larger manifold with a fixed background metric. We derive both the tangential and the normal governing equations. We then reduce the standard energy balance written in the larger ambient space to that in the evolving ambient space. We consider quasi-static deformations of the ambient space and show that a quasi-static deformation of the ambient space results in stresses, in general. We linearize the nonlinear theory about a reference motion and show that variation of the spatial metric corresponds to an effective field of body forces.

关键词： Geometric mechanics nonlinear elasticity Deforming ambient space

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nonlinear elasticity in a meta-film comprising nano-helices

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THIN SOLID FILMS 2020年第0期695卷 137749-000页

作者： Kaneko, Ryota Yukishita, Tetsuya Sumigawa, Takashi Kitamura, Takayuki Kyoto Univ Dept Mech Engn & Sci Grad Sch Engn Nishikyo Ku Kyoto 6158540 Japan

The aim of this study is to fabricate a meta-film exhibiting nonlinear elastic behavior on the basis of the significant deformability of spring-shaped nano-helices using glancing angle deposition. The unique mechanical properties of this film were examined by extracting a single nano-helix and examining this element using in situ electron microscopy while applying a tensile load. The helix showed nonlinear reversible behavior and the relationship between load, F [nN], and displacement, delta [nm], was determined to be F = 4.1 delta + 0.0041 delta(2). A nonlinear finite element analysis confirmed that this nonlinearity originated from the large deformation permitted by the spring shape. Both the stiffness and degree of nonlinearity of the meta-film could potentially be adjusted over a wide range by modifying the geometry and the density of the nano-helices.

关键词： Nano-helix nonlinear elasticity Meta-film Glancing angle deposition Tensile test Amorphous silicon

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nonlinear elasticity and friction of liquid-crystalline polymer monolayers

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JOURNAL OF CHEMICAL PHYSICS 1998年第2期109卷 827-833页

作者： Subbotin, A ten Brinke, G Kulichikhin, VG Hadziioannou, G Russian Acad Sci Inst Petrochem Synth Moscow 117912 Russia Univ Groningen Polymer Chem Lab NL-9747 AG Groningen Netherlands Univ Groningen Ctr Mat Sci NL-9747 AG Groningen Netherlands

In the present paper we consider nonlinear elasticity and friction of grafted persistent chains, which are highly stretched in the normal to the surface direction due to orientational interactions. We examine the normal and the lateral forces both in equilibrium and under shear sliding when the monolayer is confined by a bare surface. We show that in the confined monolayer in equilibrium the tilted orientation of the director becomes stable. In the sliding regime the friction force passes through a maximum value. The additional normal force in the sliding regime, when the distance between the surfaces is fixed, is also considered. We show that this force is attractive for small velocities and becomes repulsive for high velocities after the friction force passes through the maximum value. (C) 1998 American Institute of Physics.

关键词： .lateral orientational Monolayers stretched sliding Persistent tilted grafted liquid-crystalline normal force nonlinear elasticity friction force Laboratory of Polymer Chemistry and Materials Science Centre bare surface

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nonlinear elasticity and buckling in the simplest soft-strut tensegrity paradigm

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INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS 2018年第Nov.期106卷 80-88页

作者： Palumbo, Stefania Carotenuto, Angelo Rosario Cutolo, Arsenio Deseri, Luca Fraldi, Massimiliano Univ Trento Dept Civil Environm & Mech Engn Trento Italy Univ Napoli Federico II Dept Struct Engn & Architecture Via Claudio 21 I-80125 Naples Italy Univ Pittsburgh Dept Mech Engn & Mat Sci Pittsburgh PA 15260 USA Univ Napoli Federico II Interdisciplinary Res Ctr Biomat Naples Italy

Tensegrity systems occur when self-equilibrated states are achieved through the interplay of pretensed (cables) and precompressed (struts) elements. The principles that govern these types of structures have been widely observed in many living systems across the scales and recently recognized, with soft or buckling bars, in the cytoskeleton as well as within single protein architectures as associated with key cellular and subcellular processes. To properly model these mechanical phenomena, some limitations dictated by the mostly linear approaches -used in literature when dealing with tensegrity structures - need to be overcome. To this aim, the present work provides a novel 2-element soft-tensegrity paradigm that includes, for the first time, (neo-Hookean) finite hyperelasticity for cable and strut, the latter potentially undergoing both contraction and buckling at each prestressed equilibrium stage. It is finally shown that constitutive properties, instability and bar deformability cooperate to determine unusual form-finding results, providing peculiar overall mechanical responses as external forces are applied.

关键词： Soft-bar tensegrity nonlinear elasticity Buckling instability

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nonlinear elasticity with limiting small strain for cracks subject to non-penetration

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MATHEMATICS AND MECHANICS OF SOLIDS 2017年第6期22卷 1334-1346页

作者： Itou, Hiromichi Kovtunenko, Victor A. Rajagopal, Kumbakonam R. Tokyo Univ Sci Dept Math Tokyo Japan Graz Univ Inst Math & Sci Comp Graz Austria Russian Acad Sci Lavrentyev Inst Hydrodynam Siberian Div Novosibirsk Russia Texas A&M Univ Dept Mech Engn College Stn TX USA

A major drawback of the study of cracks within the context of the linearized theory of elasticity is the inconsistency that one obtains with regard to the strain at a crack tip, namely it becoming infinite. In this paper we consider the problem within the context of an elastic body that exhibits limiting small strain wherein we are not faced with such an inconsistency. We introduce the concept of a non-smooth viscosity solution which is described by generalized variational inequalities and coincides with the weak solution in the smooth case. The well-posedness is proved by the construction of an approximation problem using elliptic regularization and penalization techniques.

关键词： nonlinear elasticity limiting small strain nonlinear crack with non-penetration variational inequality generalized solution regularization penalization

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nonlinear elasticity of Blood Vessels and Vascular Grafts

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ACS BIOMATERIALS SCIENCE & ENGINEERING 2024年第6期10卷 3631-3654页

作者： Wang, Xiaofeng Li, Kecheng Yuan, Yuan Zhang, Ning Zou, Zifan Wang, Yun Yan, Shujie Li, Xiaomeng Zhao, Peng Li, Qian Zhengzhou Univ Natl Ctr Int Res Micronano Molding Technol Sch Mech & Safety Engn Zhengzhou 450001 Peoples R China Zhejiang Univ State Key Lab Fluid Power & Mechatron Syst Hangzhou 310027 Peoples R China

The transplantation of vascular grafts has emerged as a prevailing approach to address vascular disorders. However, the development of small-diameter vascular grafts is still in progress, as they serve in a more complicated mechanical environment than their counterparts with larger diameters. The biocompatibility and functional characteristics of small-diameter vascular grafts have been well developed;however, mismatch in mechanical properties between the vascular grafts and native arteries has not been accomplished, which might facilitate the long-term patency of small-diameter vascular grafts. From a point of view in mechanics, mimicking the nonlinear elastic mechanical behavior exhibited by natural blood vessels might be the state-of-the-art in designing vascular grafts. This review centers on elucidating the nonlinear elastic behavior of natural blood vessels and vascular grafts. The biological functionality and limitations associated with as-reported vascular grafts are meticulously reviewed and the future trajectory for fabricating biomimetic small-diameter grafts is discussed. This review might provide a different insight from the traditional design and fabrication of artificial vascular grafts.

关键词： Blood vessels nonlinear elasticity vasculargrafts mechanical properties

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