03 - kinematic equations - large deformations and growth 03 - kinematic equations where are we??? 1 2 final projects - me337 2010 • mechanically driven growth of skin: chris, adrian, xuefeng • muscle growth: brandon, robyn, esteban, ivan, jenny • cardiac growth review: manuel • cardiac growth in response to training: holly, tyler • cardiac growth in response to heart attack:amit • cardiac or arterial growth: andrew • cardiac growth in response to medical devices: kyla, andrew • bone growth in response to medical devices: chinedu • impact of obesity on osteoarthritis: abhishek, chris • tumor growth: apoorva • facial volume aging: jonathan • idiopathic scoliosis or rhubarb growth: anusuya • driving forces for different types of growth: james homework 01 - due thu in class homework 01 - due thu in class 3 4
growth, remodeling and morphogenesis growth, remodeling and morphogenesis growth which is defined as added mass, can occur through cell division remodeling involves changes in (hyperplasia), cell enlargement material properties. These changes, which (hypertrophy), secretion of extracellular often are adaptive, may be brought about by matrix, or accretion @ external or internal alterations in modulus, internal structure, surfaces. negative growth (atrophy) can strength, or density. for example, bones, occur through cell death, cell shrinkage, and heart muscle may change their internal or resorption. in most cases, hyperplasia structures through reorientation of and hypertrophy are mutually exclusive trabeculae and muscle fibers, respectively. processes. depending on the age of the organism and the type of tissue, one of these two growth processes dominates. taber „biomechanics of growth, remodeling and morphogenesis“ [1995] taber „biomechanics of growth, remodeling and morphogenesis“ [1995] introduction introduction 5 6 growth, remodeling and morphogenesis growth, remodeling and morphogenesis morphogenesis is the generation of animal form. usually, the term refers to embryonic development, but wound healing mathematical descriptions of growth and organ regeneration are also in plants and animals have been published morphogenetic events. morphogenesis since the 1940s. most of these analyses are contains a complex series of stages, each purely kinematic and many borrow from the of which depends on the previous stage. methods of continuum mechanics to describe during these stages, genetric and growth rates and velocity fields. during environmental factors guide the spatial- the last quarter century, mechanical temporal motions and differentiation theories of growth have been formulated. (specification) of cells. a flaw in any one stage may lead to structural defects. taber „biomechanics of growth, remodeling and morphogenesis“ [1995] taber „biomechanics of growth, remodeling and morphogenesis“ [1995] introduction introduction 7 8
scaling growth tip growth time lapse sequence of a growing lily pollen tube. note that the morphology of the tube is drawn by the expanding tip and does not change behind it. tip growth is a common mode of cell morphogenesis observed in root hairs, fungal hyphae, pollen tubes, and many unicellular algae. these organisms have cell walls with distinct polymer compositions and structures. sir d‘arcy thompson “on growth and form“ [1917] dumais, long, shaw (2004) kinematics of growth kinematics of growth 9 10 tip growth tip growth unlike diffusely growing cells that expand over their entire surface or large portions of it, cell wall scanning electron microscope of growing lily pollen grains germinated in vitro. the spherical objects expansion in pollen tubes is confined to the apex of the cell. this highly polarized mechanism is are the pollen grains, the cylindrical objects are the pollen tubes, or cellular protuberances growing called tip growth. pollen tubes have the function to rapidly grow and deliver the sperm cells from from the grains (left). brightfield microscopy of the apical region of a lily pollen tube. the outermost kroeger, geitmann, grant [2008] the pollen grain to the ovule. end of the tube is filled mainly with delivery vesicles. kroeger & geitmann [2012] kinematics of growth kinematics of growth 11 12
surface growth surface growth skalak, farrow, hoger [1997] prusinkiewicz & de reuille “constraints of space in plant development“ [2010] kinematics of growth kinematics of growth 13 14 surface growth surface growth prusinkiewicz & de reuille “constraints of space in plant development“ [2010] prusinkiewicz & de reuille “constraints of space in plant development“ [2010] kinematics of growth kinematics of growth 15 16
suggested reading continuum mechanics continuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter. the fact that matter is made of atoms and that it commonly has some sort of heterogeneous microstructure is ignored in the simplify- ing approximation that physical quantities, such as energy and momentum, can be handled in the infinitesimal limit. differential equations can thus be employed in solving malvern le: introduction to the mechanics of a continuous medium, prentice hall, 1969 problems in continuum mechanics. chadwick p: continuum mechanics - concise theory and problems, dover reprint, 1976 bonet j, wood rd: nonlinear continuum mechanics for fe analysis, cambridge university press, 1997 holzapfel ga: nonlinear solid mechanics, a continuum approach for engineering, john wiley & sons, 2000 introduction to continuum mechanics introduction to continuum mechanics 17 18 continuum mechanics continuum mechanics continuum mechanics is continuum hypothesis the branch of mechanics concerned with the we assume that the characteristic length stress in solids, liquids and gases and the scale of the microstructure is much smaller deformation or flow of these materials. the than the characteristic length scale of the adjective continuous refers to the simpli- overall problem, such that the properties fying concept underlying the analysis: we at each point can be understood as averages disregard the molecular structure of matter over a characteristic length scale and picture it as being without gaps or empty spaces. we suppose that all the example: biomechanics mathematical functions entering the theory are continuous functions. this hypothetical continuous material we call a continuum. the continuum hypothesis can be applied when analyzing tissues malvern „introduction to the mechanics of a continuous medium“ [1969] introduction to continuum mechanics introduction to continuum mechanics 19 20
the potato equations the potato equations • kinematic equations - what‘s strain? • kinematic equations - why not ? general equations that characterize the deformation inhomogeneous deformation » non-constant of a physical body without studying its physical cause finite deformation » non-linear inelastic deformation » growth tensor • balance equations - what‘s stress? • balance equations - why not ? general equations that characterize the cause of equilibrium in deformed configuration » multiple stress measures motion of any body • constitutive equations - how are they related? • constitutive equations - why not ? material specific equations that complement the set finite deformation » non-linear of governing equations inelastic deformation » internal variables introduction to continuum mechanics introduction to continuum mechanics 21 22 kinematic equations kinematic equations kinematic equations de- scribe the motion of objects without the kinematics is the study of motion consideration of the masses or forces that per se, regardless of the forces causing bring about the motion. the basis of kine- it. the primitive concepts concerned are matics is the choice of coordinates. the position, time and body, the latter 1st and 2nd time derivatives of the posi- abstracting into mathematical terms tion coordinates give the velocities and intuitive ideas about aggregations of accelerations. the difference in placement matter capable of motion and deformation. between the beginning and the final state of two points in a body expresses the nu- merical value of strain. strain expresses itself as a change in size and/or shape. Chadwick „Continuum mechanics“ [1976] kinematic equations kinematic equations 23 24
potato - kinematics potato - kinematics • transformation of line elements - deformation gradient • nonlinear deformation map with with • uniaxial tension (incompressible), simple shear, rotation • spatial derivative of - deformation gradient with kinematic equations kinematic equations 25 26 potato - kinematics potato - kinematics • transformation of volume elements - determinant of • temporal derivative of - velocity (material time derivative) with • temporal derivative of - acceleration • changes in volume - determinant of deformation tensor with kinematic equations kinematic equations 27 28
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