07 - basics balance equations - closed and open systems 07 - balance equations 1 where are we??? 2 final projects continuum mechanics of growth kinematic equations for finite growth • tendon growth: harrison, brandon, mohammed, matthew • tendon growth and remodeling: peter balance equations for open systems • muscle growth: alex! • skin growth and healing: beth, ann, armen • benign vocal fold nodule and polyp growth: corey constitutive equations for living tissues • cerebral aneurysm growth: jina • growth of swelling gels: hardik, xi, ill • bone growth in martial arts: kevin, alison, safwan, kamil fe analyses for biological structures continuum- & computational biomechanics where are we??? 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 [3] after growing the elements, may be incompatible • incompatible growth configuration & growth tensor [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 duration balance of momentum - volume specific 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 range of propulsive force 0 to 120 seconds to see the increase in speed of the rocket velocity of ejection 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