Multi-scale simulation of plant tissue deformation using a model for individual cell mechanics

We present a micro-macro method for the simulation of large elastic deformations of plant tissue. At the microscopic level, we use a mass-spring model to describe the geometrical structure and basic properties of individual plant cells. The macroscopic domain is discretized using standard finite elements, in which the macroscopic material properties (the stress-strain relation) are not given in analytical form, but are computed using the microscopic model in small subdomains, called representative volume elements (RVEs), centered around the macroscopic quadrature points. The boundary conditions for these RVEs are derived from the macroscopic deformation gradient. The computation of the macroscopic stress tensor is based on the definition of virial stress, as defined in molecular dynamics. The anisotropic Eulerian elasticity tensor is estimated using a forward finite difference approximation for the Truesdell rate of the Cauchy stress tensor. We investigate the influence of the size of the RVE and the boundary conditions. This multi-scale method converges to the solution of the full microscopic simulation, for both globally and adaptively refined finite element meshes, and achieves a significant speedup compared to the full microscopic simulation.

[1]  J. Kirkwood,et al.  The Statistical Mechanical Theory of Transport Processes. IV. The Equations of Hydrodynamics , 1950 .

[2]  J. D. Eshelby.  The determination of the elastic field of an ellipsoidal inclusion, and related problems , 1957, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[3]  C. H. Hertz,et al.  On the Relation between Turgor Pressure and Tissue Rigidity. II. Theoretical Calculations on Model Systems , 1958 .

[4]  G. Taylor,et al.  The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid , 1958, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[5]  S. Shtrikman,et al.  A variational approach to the theory of the elastic behaviour of polycrystals , 1962 .

[6]  K. Tanaka,et al.  Average stress in matrix and average elastic energy of materials with misfitting inclusions , 1973 .

[7]  T. Hughes.  Equivalence of Finite Elements for Nearly Incompressible Elasticity , 1977 .

[8]  A. Bensoussan,et al.  Asymptotic analysis for periodic structures , 1979 .

[9]  Medhat A. Haroun,et al.  Reduced and selective integration techniques in the finite element analysis of plates , 1978 .

[10]  P. Cundall,et al.  A discrete numerical model for granular assemblies , 1979 .

[11]  J. Woolley.  Maintenance of air in intercellular spaces of plants. , 1983, Plant physiology.

[12]  Nuri N. Mohsenin,et al.  RHEOLOGICAL CHARACTERIZATION OF APPLE CORTEX , 1987 .

[13]  Yousef Saad,et al.  Hybrid Krylov Methods for Nonlinear Systems of Equations , 1990, SIAM J. Sci. Comput..

[14]  J. C. Simo,et al.  A CLASS OF MIXED ASSUMED STRAIN METHODS AND THE METHOD OF INCOMPATIBLE MODES , 1990 .

[15]  Subra Suresh,et al.  Deformation of metal-matrix composites with continuous fibers: geometrical effects of fiber distribution and shape , 1991 .

[16]  V. M. Puri,et al.  Characterization of nonlinear creep behavior of two food products , 1991 .

[17]  C. Miehe,et al.  Numerical computation of algorithmic (consistent) tangent moduli in large-strain computational inelasticity , 1996 .

[18]  W. Brekelmans,et al.  Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling , 1998 .

[19]  M. C. Jarvis,et al.  Intercellular separation forces generated by intracellular pressure , 1998 .

[20]  Homer F. Walker,et al.  NITSOL: A Newton Iterative Solver for Nonlinear Systems , 1998, SIAM J. Sci. Comput..

[21]  K. Spring.  Routes and mechanism of fluid transport by epithelia. , 1998, Annual review of physiology.

[22]  Heh Han Meijer,et al.  Effective properties of a viscoplastic constitutive model obtained by homogenisation , 1999 .

[23]  J. Schröder,et al.  Computational homogenization analysis in finite plasticity Simulation of texture development in polycrystalline materials , 1999 .

[24]  J. Chaboche,et al.  FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials , 2000 .

[25]  N. Kikuchi,et al.  Simulation of the multi-scale convergence in computational homogenization approaches , 2000 .

[26]  D. Hepworth,et al.  A method of calculating the mechanical properties of nanoscopic plant cell wall components from tissue properties , 2000 .

[27]  Fpt Frank Baaijens,et al.  An approach to micro-macro modeling of heterogeneous materials , 2001 .

[28]  P. Lintilhac,et al.  An insight into cell elasticity and load-bearing ability. Measurement and theory. , 2001, Plant physiology.

[29]  T. J. Delph,et al.  Stress calculation in atomistic simulations of perfect and imperfect solids , 2001 .

[30]  V. Kouznetsova,et al.  Multi‐scale constitutive modelling of heterogeneous materials with a gradient‐enhanced computational homogenization scheme , 2002 .

[31]  Hanns-Christof Spatz,et al.  Micromechanics of plant tissues beyond the linear-elastic range , 2002, Planta.

[32]  Min Zhou,et al.  A new look at the atomic level virial stress: on continuum-molecular system equivalence , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[33]  Jozef Keckes,et al.  Cell-wall recovery after irreversible deformation of wood , 2003, Nature materials.

[34]  D. M. Bruce,et al.  Mathematical modelling of the cellular mechanics of plants. , 2003, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[35]  V. Kouznetsova,et al.  Multi-scale second-order computational homogenization of multi-phase materials : a nested finite element solution strategy , 2004 .

[36]  Jenny Blewett,et al.  A micromanipulation method to measure the mechanical properties of single tomato suspension cells , 2004, Biotechnology Letters.

[37]  D. Roose,et al.  Computation of Permeability of Textile Reinforcements , 2005 .

[38]  Bart Nicolai,et al.  Micromechanical behaviour of onion epidermal tissue , 2005 .

[39]  O. C. Zienkiewicz,et al.  The Finite Element Method: Its Basis and Fundamentals , 2005 .

[40]  M. Ostoja-Starzewski.  Material spatial randomness: From statistical to representative volume element☆ , 2006 .

[41]  Antoine Gloria,et al.  An Analytical Framework for the Numerical Homogenization of Monotone Elliptic Operators and Quasiconvex Energies , 2006, Multiscale Model. Simul..

[42]  Jacob Fish,et al.  A mathematical homogenization perspective of virial stress , 2006 .

[43]  Bart Nicolai,et al.  Micromechanics : Simulating the elastic behavior of onion epidermis tissue , 2006 .

[44]  W. Bangerth,et al.  deal.II—A general-purpose object-oriented finite element library , 2007, TOMS.

[45]  M. Geers,et al.  An enhanced multi‐scale approach for masonry wall computations with localization of damage , 2007 .

[46]  Weiqing Ren,et al.  Seamless multiscale modeling of complex fluids using fiber bundle dynamics , 2007 .

[47]  T. Newman,et al.  Modeling cell rheology with the Subcellular Element Model , 2008, Physical biology.

[48]  O. C. Zienkiewicz,et al.  The Finite Element Method for Solid and Structural Mechanics , 2013 .