07 - basics balance equations - closed and open systems
07 - balance equations 1 where are we??? 2
final projects - me337 2013 continuum mechanics of growth
kinematic equations for finite growth
• cerebral aneurysm growth: sheila • muscle growth: katrina, jaqi balance equations for open systems • brain tumor growth: cesare, zhuozhi • left vs right arms of athletes: ronaldo, maria, lyndia • airway wall remodeling: mona constitutive equations for living tissues • corneal growth / keratoconus: yanli • bone adaptation in birds: jan, david • neuronal growth in fish: andrew fe analyses for biological structures
continuum- & computational biomechanics
homework 01 - due thu in class 3 where are we??? 4 potato - kinematics kinematics of finite growth
• nonlinear deformation map [1] consider an elastic body at time ,unloaded &stressfree with • spatial derivative of - deformation gradient with
kinematic equations 5 kinematics of growth 6
kinematics of finite growth kinematics of finite growth
[1] consider an elastic body at time ,unloaded &stressfree [1] consider an elastic body at time ,unloaded &stressfree [2] imagine the body is cut into infinitesimal elements each of [2] imagine the body is cut into infinitesimal elements each of which is allowed to undergo volumetric growth which is allowed to undergo volumetric growth [3] after growing the elements, may be incompatible
kinematics of growth 7 kinematics of growth 8 kinematics of finite growth potato - kinematics of finite growth
[1] consider an elastic body at time ,unloaded &stressfree [2] imagine the body is cut into infinitesimal elements each of which is allowed to undergo volumetric growth • incompatible growth configuration & growth tensor [3] after growing the elements, may be incompatible
[4] loading generates compatible current configuration rodriguez, hoger & mc culloch [1994]
kinematics of growth 9 kinematics of growth 10
balance equations balance equations balance equations of balance equations of mass, mass, linear momentum, angular momentum momentum, angular momentum and energy, sup- and energy apply to all material bodies. plemented with an entropy inequality each one gives rise to a field equation, constitute the set of conservation laws. holing on the configurations of a body in a the law of conservation of mass/matter sufficiently smooth motion and a jump states that the mass of a closed system of condition on surfaces of discontinuity. substances will remain constant, regardless like position, time and body, the concepts of the processes acting inside the system. of mass, force, heating and internal energy the principle of conservation of momentum which enter into the formulation of the states that the total momentum of a balance equations are regarded as having closed system of objects is constant. primitive status in continuum mechanics. chadwick continuum mechanics [1976]
balance equations 11 balance equations 12 potato - balance equations potato - balance equations
isolation isolation
[1] isolation of subset from [1] isolation of subset from [2] characterization of influence of remaining body through phenomenological quantities - contact fluxes , &
balance equations 13 balance equations 14
potato - balance equations potato - balance equations
isolation isolation
[1] isolation of subset from [1] isolation of subset from [2] characterization of influence of remaining body through [2] characterization of influence of remaining body through phenomenological quantities - contact fluxes , & phenomenological quantities - contact fluxes , & [3] definition of basic physical quantities - mass, linear and [3] definition of basic physical quantities - mass, linear and angular momentum, energy angular momentum, energy [4] postulation of balance of these quantities
balance equations 15 balance equations 16 generic balance equation balance of mass
general format balance of mass … balance quantity … density … flux … no mass flux … source … no mass source … production … no mass production
continuity equation balance equations - closed systems 17 balance equations - closed systems 18
balance of (linear) momentum compare
balance of momentum … linear momentum density … momentum flux - stress … momentum source - force … no momentum production
equilibrium equation mass point balance equations - closed systems 19 balance equations - closed systems 20 balance of (internal) energy balance of entropy
balance of internal energy balance of entropy … internal energy density … entropy density … heat flux … entropy flux … heat source … entropy source … no heat production … entropy production energy equation entropy `inequality internal mechanical power thermal external power
balance equations - closed systems 21 balance equations - closed systems 22
dissipation inequality thermodynamic systems • dissipaltion inequality isolated system thermo- dynamical system which is not allowed to • identification have any interaction with its environment. • with legendre-fenchel transform enclosed by a rigid, adiabatic, imperme- able membrane. • free energy • definition of stress and entropy (e.g., neo hooke’s law)
• thermodynamic restriction (e.g., fourier’s law)
balance equations - closed systems 23 balance equations 24 thermodynamic systems thermodynamic systems adiabatic closed system closed system thermodynamic thermodynamic system which is allowed to system which is allowed to exchange exchange exclusively mechanical work, mechanical work and heat, typically typically , with its environ- and , with its ment. enclosed by a deformable, adiabatic, environment. enclosed by a deformable, impermeable membrane. characterized through diathermal, impermeable membrane. charac- its state of deformation . terized through its state of deformation and temperature .
balance equations 25 balance equations 26
thermodynamic systems open system thermodynamics open system thermodynamic system which is allowed to exchange mechanical work, heat and mass, typically , and with its environment. enclosed by a deformable, diathermal, permeable membrane. characterized through its state of deformation , temperature and density .
”…thermodynamics recognizes no special role of the biological…” bridgman,’the nature of thermodynamics’, [1941]
balance equations 27 balance equations - open systems 28 why do we need open systems if we have porous media? balance of mass
theory of open systems theory of porous media mass flux • cell movement (migration) • constituents spatially separated • local superposition of constituents mass source • overall behavior priliminary deter- • consideration of mixture of mul- • cell growth (proliferation) mined by one single constituent tiple constituents • cell division (hyperplasia) • exchange of mass, momentum, • exchange of mass, momentum, • cell enlargement (hypertrophy) energy and entropy with environ- energy and entropy amongst biological equilbrium ment constituents cowin & hegedus [1976], beaupré, orr & carter [1990], harrigan & hamilton [1992], jacobs, levenston, beaupré, simo & carter [1995], huiskes [2000], carter & beaupré [2001]
balance equations - open systems 29 balance equations - open systems 30
balance of mass balance of mass
the model does not take explicitly into account that the body is growing due to the absorption of some other materials. in absence of some of the vital constituents , no growth is possible. conversely, when part of the material dies, some of the bricks contained in the cellular membrane can be re-used by other cells. in this respect, an approach using mixture theory might by useful. simulation of cell growth - cahn-hilliard equation kuhl & schmid [2006], wells, kuhl & garikipati [2006] ambrosi & mollica [2002]
balance equations - open systems 31 balance equations - open systems 32 generic balance equation balance of (linear) momentum
• volume specific version
general format • subract weighted balance of mass … balance quantity … flux … source • mass specific version … production
mechanical equilbrium
balance equations - open systems 33 balance equations - open systems 34
example of open systems - rocket propulsion example of open systems - rocket propulsion
balance of mass a saturn v rocket like the one that took men to the moon has a with ejection mass of 2.500.000 kg at liftoff. it goes straight up vertically and burns fuel at a uniform rate of 16.000 kg/s for a balance of momentum - volume specific duration of 2 minutes. the exhaust speed of gas from the saturn v is 3.0 km/s!
balance of momentum - mass specific what is the speed of the rocket immediately after the with combustion ceases? you should include the effect of gravity near the surface of the earth, but you can neglect air balance of momentum - rocket head-ejection resistance.!
plot the burnout velocity as a function of time over the propulsive force range of 0 to 120 seconds to see the increase in speed of the velocity of ejection rocket with time. !
example - rocket propulsion 35 example - rocket propulsion 36 example of open systems - rocket propulsion example of open systems - rocket propulsion
with gravity
integration
velocity
example - rocket propulsion 37 example - rocket propulsion 38
example of open systems - rocket propulsion balance of (internal) energy
velocity • volume specific version
• subract weighted balance of mass
• mass specific version
energy equilbrium velocity
example - rocket propulsion 39 balance equations - open systems 40 balance of entropy dissipation inequality • dissipaltion inequality • volume specific version • identification • subract weighted balance of mass • free energy • definition of stress and entropy
• mass specific version • thermodynamic restrictions
”… a living organism can only keep alive by continuously drawing from its environment negative entropy ‘inequality ’ entropy. it feeds upon negative entropy to compensate the entropy increase it produces by living. ” schrödinger ”what is life?” [1944]
balance equations - open systems 41 balance equations - open systems 42
open systems - dissipation inequality example of open systems - the galileo giant
with definition of arguments of free energy
evaluation of dissipation inequality
„...dal che e manifesto, che chi volesse mantener in un vastissimo gigante le proporzioni, provides guidelines for the appropriate choice of the che hanno le membra in un huomo ordinario, bisognerebbe o trouar materia molto piu constitutive equations dura, e resistente per formarne l'ossa o vero ammettere, che la robustezza sua fusse a proporzione assai piu fiacca, che negli huomini de statura mediocre; altrimente thermodynamically conjugate pairs crescendogli a smisurata altezza si vedrebbono dal proprio peso opprimere, e cadere…” galileo,”discorsi e dimostrazioni matematiche”, [1638]
balance equations - open systems 43 example - the galileo giant 44 example of open systems - the galileo giant example of open systems - the galileo giant
the tallest man in medical history is consider and compare the two cases: robert pershing wadlow. he was born at alton, illinois, on february 22, 1918. calculale the vertical displacement and he was 2.72m / 8ft 11.1”, tall. " the energy for a giant with a constant " bone mineral density along the his weight was 222.71kg. his shoe height!
size was 47cm / 18.5”, and his hands calculate the vertical displacement and measured 32.4cm / 12.75”. his arm the bone mineral density for a giant span was 2.88m / 9 ft 5.75”, and his with a constant energy along the peak daily food consumption was height! 8000 calories. guiness world records [2010] guiness world records [2010]
example - the galileo giant 45 example - the galileo giant 46
example of closed systems - the galileo giant example of open systems - the galileo giant density displacement energy • balance equation closed systems
• neo hookean free energy
• stress from dissipation inequality
• quasi-static case
• constant gravity load
• from balance eqn linear stress
• closed system, constant density
example - the galileo giant 47 example - the galileo giant 48 example of open systems - the galileo giant example of open systems - the galileo giant
• balance equations open systems density displacement energy
• free energy
• stress from dissipation inequality
• quasi-static case
• constant gravity load
• from balance eqn linear stress
• open system, varying density
example - the galileo giant 49 example - the galileo giant 50