Nonlinear mechanics of surface growth for cylindrical and spherical elastic bodies

Abstract In this paper we formulate the initial-boundary value problems of accreting cylindrical and spherical nonlinear elastic solids in a geometric framework. It is assumed that the body grows as a result of addition of new (stress-free or pre-stressed) material on part of its boundary. We construct Riemannian material manifolds for a growing body with metrics explicitly depending on the history of applied external loads and deformation during accretion and the growth velocity. We numerically solve the governing equilibrium equations in the case of neo-Hookean solids and compare the accretion and residual stresses with those calculated using the linear mechanics of surface growth.

[1]  Oliver M. O’Reilly,et al.  On the equations of motion for rigid bodies with surface growth , 2004 .

[2]  Alain Goriely,et al.  Riemann–Cartan Geometry of Nonlinear Dislocation Mechanics , 2012 .

[3]  A. Drozdov.  Continuous accretion of a composite cylinder , 1998 .

[4]  P. D. Washabaugh,et al.  Stresses in rotating spheres grown by accretion , 2005 .

[5]  A. McCulloch,et al.  Stress-dependent finite growth in soft elastic tissues. , 1994, Journal of biomechanics.

[6]  Mitsuru Sato,et al.  Stress formation in solidifying bodies. Solidification in a round continuous casting mold , 1998 .

[7]  学術文献普及会.  Memoirs of the unifying study of the basic problems in engineering sciences by means of Geometry , 1955 .

[8]  R. V. b. Southwell,et al.  An introduction to the theory of elasticity for engineers and physicists , 1936 .

[9]  Keiichi Takamizawa,et al.  Kinematics for bodies undergoing residual stress and its applications to the left ventricle , 1990 .

[10]  Arash Yavari,et al.  Geometric nonlinear thermoelasticity and the time evolution of thermal stresses * , 2017 .

[11]  Arash Yavari,et al.  A geometric theory of thermal stresses , 2009, 0912.1298.

[12]  H. P. Yagoda.  Resolution of a Core Problem in Wound Rolls , 1980 .

[13]  V. V. Metlov.  On the accretion of inhomogeneous viscoelastic bodies under finite deformations , 1985 .

[14]  Alain Goriely,et al.  Riemann–Cartan Geometry of Nonlinear Dislocation Mechanics , 2012, Archive for Rational Mechanics and Analysis.

[15]  Colin B. Brown,et al.  Slab avalanching and the state of stress in fallen snow , 1972 .

[16]  B. Bilby,et al.  Continuous distributions of dislocations. Ill , 1956, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[17]  A. V. Manzhirov,et al.  Reference configurations of growing bodies , 2013 .

[18]  Alain Goriely,et al.  Growth and instability in elastic tissues , 2005 .

[19]  R. Ogden.  Non-Linear Elastic Deformations , 1984 .

[20]  Alain Goriely,et al.  A Geometric Theory of Nonlinear Morphoelastic Shells , 2016, J. Nonlinear Sci..

[21]  J. C. Simo,et al.  Stress tensors, Riemannian metrics and the alternative descriptions in elasticity , 1984 .

[22]  Alain Goriely,et al.  The geometry of discombinations and its applications to semi-inverse problems in anelasticity , 2014, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[23]  A. V. Manzhirov,et al.  The mathematical theory of growing solids: Finite deformations , 2012, Doklady Physics.

[24]  Carl Eckart,et al.  The Thermodynamics of Irreversible Processes. IV. The Theory of Elasticity and Anelasticity , 1948 .

[25]  Panayiotis Papadopoulos,et al.  A continuum theory of surface growth , 2010, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[26]  Marcelo Epstein,et al.  Thermomechanics of volumetric growth in uniform bodies , 2000 .

[27]  Arash Yavari,et al.  On the origins of the idea of the multiplicative decomposition of the deformation gradient , 2017 .

[28]  Arkadas Ozakin,et al.  Covariance in linearized elasticity , 2008 .

[29]  Aleksey D. Drozdov,et al.  Viscoelastic structures : mechanics of growth and aging , 1998 .

[30]  Vyacheslav E. Naumov,et al.  Mechanics of Growing Deformable Solids: A Review , 1994 .

[31]  Colin B. Brown,et al.  Gravitational stresses in accreted bodies , 1963, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[32]  Alain Goriely,et al.  Nonlinear Elastic Inclusions in Anisotropic Solids , 2013, Journal of Elasticity.

[33]  Alain Goriely,et al.  The twist-fit problem: finite torsional and shear eigenstrains in nonlinear elastic solids , 2015, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[34]  A. V. Manzhirov.  The general non-inertial initial-boundaryvalue problem for a viscoelastic ageing solid with piecewise-continuous accretion☆ , 1995 .

[35]  Albert Edward Green,et al.  General theory of small elastic deformations superposed on finite elastic deformations , 1952, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[36]  R Skalak,et al.  Kinematics of surface growth , 1997, Journal of mathematical biology.

[37]  K. Grosh,et al.  A continuum treatment of growth in biological tissue: the coupling of mass transport and mechanics , 2003, q-bio/0312001.

[38]  R. Bullough,et al.  Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry , 1955, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[39]  Keiichi Takamizawa.  Stress-Free Configuration of a Thick-Walled Cylindrical Model of the Artery: An Application of Riemann Geometry to the Biomechanics of Soft Tissues , 1991 .

[40]  Arash Yavari,et al.  Circumferentially-symmetric finite eigenstrains in incompressible isotropic nonlinear elastic wedges , 2016 .

[41]  Nicole Propst,et al.  Mathematical Foundations Of Elasticity , 2016 .

[42]  Arash Yavari,et al.  A Geometric Theory of Growth Mechanics , 2009, J. Nonlinear Sci..

[43]  S. Lychev.  Universal deformations of growing solids , 2011 .

[44]  Alain Goriely,et al.  Weyl geometry and the nonlinear mechanics of distributed point defects , 2012, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[45]  Jonas Stålhand,et al.  Theory of residual stresses with application to an arterial geometry , 2007 .