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1 Eamination: Mechanics for MWT: 8TB00 April 8, 014, 9:00-1:00 Code 8TB00, MWT bachelor Biomedical Engineering Eindhoven University of Technology This is a closed book eam. The use of a laptop and/or electronic calculator is not allowed. This eam consist of 10 questions. The total score that can be obtained is 100 points, and the number of points per question is indicated. Question 1 (10 points): A person is holding a weight of 100 [N] in his hand. A simplified representation of the situation is simplified as drawn in the figure below, S represents the shoulder, E the elbow and H the hand. The attachment points of the muscle M are measured relative to the elbow as indicatedd (units [dm]). The weight of the arm itself can be neglected. Make a free body diagram of the lower arm (E-H) and calculate the force in the muscle when holding the weight Question (10 points): A ligament in the foot attaches at point A to one bone at point B to another bone. The coordinates of these points are measured relative to a cartesian coordinate system as: A B 1e ey 4 e 6e y 1

2 In the initial situation the ligament is just stretched while the force is zero. During a movement point A moves to a new position A* and point B moves to a new position B*. The new positions are indicatedd relative to the same coordinate system by position vectors: A* B* e 7 e 3e 8e y y The elastic constant of the ligament is c=100 [N]. Calculate the force in the ligament in the new position. Question 3 (10 points): The mechanical behavior of a tissue is described by a Kelvin-Voigt model. In an eperiment, a sample of this material is loaded at constant strain rate until a strain of 0 is reached after seconds. Following, the strain is kept constant for another seconds. Finally, the strain is gradually reduced to zero again during seconds. At the start of the eperiment the sample was unloaded. Make a sketch of the force in the spring during this eperiment and indicate the etreme values of the force. Also make a sketch of the force in the damper during this eperiment and indicate the etreme values of the force Question 4 (10 points):

3 A bar with variable cross-section is loaded by an aial force F as indicated in the figure. The d du displacement field can be found by solving the equation: EA 0, but solving this equation d d requires two boundary conditions. Specify the two boundary conditions for the case drawn below Question 5 (10 points): A thin membrane is subject to a state of stress. At some point in the membrane the state of stress can be represented by a combination of normal and shear stresses in the - and y-direction as indicated in the figure below for a very smalll rectangular volume located at that point (all values in units of [N/mm ]). Calculate the full stress tensor at his point and the Von Mises stress 3

4 Question 6 (10 points): The state of stress at a specific point of a material is given by the stress tensor (in units of [N/mm ]): σ 4ee e It is observed that for this state of stress failure occurs along a plane with unit normal vector: 1 n e e y Calculate the magnitude of the shear stress at this plane of failure. Question 7 (10 points): A sample of a material with fibers is cut in such a way that the fibers are oriented at 60 degrees with respect to the sample -ais (see figure below). The sample is subjected to a homogeneous state of strain described by the strain tensor: 3 ε (eey eye ) 1000 Calculate the strain in the fibers due to this state of strain 4

5 Question 8 (10 points): In an eperimental setup for tissue engineering constructss with cardyomyocyte cells are attached to the top of micro-posts (see figure). These posts can be considered elastic beams with a spacing and a length of 1 [mm], a diameter of 0.1 [mm] and a second moment of area I= 5E-6 [Nmm 4 ]. During culturing the cardyomyocyte cells contract several times per second. In the non-contracted state, it can be assumed that the cells do not eert any force to the posts. The length of the construct then is 1 [mm]. In the contracted state, the length of the construct is reduced to 0.8 [mm] and consequently the posts are loaded in bending. Assume that the maimum contracted length of the construct is l c = 0. 4 [mm] and that the elastic constantt of the muscle construct c= 0 E-6[N]. a) Calculate the force eerted by the muscle construct during contraction b) Calculate the maimum stresss in the posts during contraction 5

6 Question 9 (10 points): In a well-known animal eperiment to study bone remodeling, the central part of the ulna of a turkey is cutted on two sides (figure a). Following, pins are placed at both sides of the cutted part such that forces can be applied to the isolated bone in-vivo. In one of these eperiments, pure torsion is applied to the bone. The situation can be represented as in figure b below, where the bone is considered a hollow cylindrical beam with outer radius R o and inner radius R i =R o / and a polar moment of inertia 4 4 I Ro Ri. The central ais of the bone is aligned with the e ais while the pins are aligned with the e z ais. The forces are aligned with the e y ais and act at a distance e from the central ais. Calculate the maimum stress in the bone as a function of e, R o and F. Question 10 (10 points): A circular membranee is used to culture cells. To subject these cells to a mechanical stimulus, the membrane is stretched. During stretching the radius of the membranee is increased by 10%. The membrane has a Young s modulus E=8 [MPa] and a Poisson s ration of v=1/4. Calculate the change in thicknesss of the membrane during stretching. 6

7 Formuleblad behorende bij het vak mechanica voor MWT 8TB00 Vector rekenen Inproduct (inner product): ab. a b cos Uitproduct (vector product): c ab ; c a b sin Tripelproduct (triple product): a b c a b c Dyadisch product (dyadic product): ab p a b p Bewerkingen waarbij een vectorbasis wordt gebruikt: vector notatie Inproduct (inner product): ab a b a b a b y y z z Uitproduct (vector product) ab ab ab e ab ab e ab ab e y z z y z z y y y z Dyadisch product (dyadic product): ab abee +abyeey abzeez abee y y abee y y y y+abee y z y z abee abee abee z z z y z y z z z z Bewerkingen waarbij een vectorbasis wordt gebruikt: kolom notatie Inproduct (inner product): T abab ab y y ab z z Uitproduct (vector product) ab y z ab z y ab azb abz ab y ab y Dyadisch product (dyadic product): ab ab y ab z T ab ab y ab y y ab y z ab z ab z y ab z z

8 Matri operaties Ab c AB C Ia a en: IA AI A 1 A A I Krachten en momenten De component F t van een kracht F in de richting van een eenheidsvector e wordt gegeven door: F Fe e t Moment bepalen t.o.v. punt P, terwijl kracht F aangrijpt in Q: M F Q P Evenwicht (equilibrium) F 0 M 0 Beweging en massatraagheid (movement and inertia) Kracht voor het versnellen van een massa: F ma Moment voor een hoekversnelling (angular acceleration) van een puntmassa op afstand R: M mr Moment voor een hoekversnelling (angular acceleration) van een object rond zijn as: M I Vezels (fibers) en spieren (muscles) Relatie tussen kracht en verlenging voor een vezel l F c 1c 1c l0 Relatie tussen kracht en verlenging voor een geactiveerde spier l F c 1c 1 lc c Visco-elastisch materiaalgedrag (visco-elastic behavior) Elastisch gedrag: 1 F c c F F Visceus gedrag: F c F F

9 Heavyside function: 0alst0 Ht () 1alst0 Kruip functie (creep function) J(t): Als Ft () Ht () is voorgeschreven, dan volgt de rek de kruipfunctie: () t J() t Relaatie functie (relaation function) G(t): Als () t H() t is voorgeschreven, dan volgt de kracht de relaatiefunctie: Ft () Gt () Proportionaliteit (proportionality): Als Ft () FHt 0 () is voorgeschreven, dan volgt de rek: () t F0 J() t Als () t 0H() t is voorgeschreven, dan volgt de kracht: Ft () 0Gt () Superpositie (superposition): Bij een stap in de belasting op t=0 en een tweede stap op t=t 1 volgt voor t > t 1 : () t F J() t FJ( t t ) Ft () Gt () Gt ( t) Spanning (stress) en rek (strain) in een 1-D continuum Spanning: N A Rek: du d Materiaalgedrag: E Evenwichtsvergelijking: d du EA q 0 d d Spanning (stress) en rek (strain) in een 3-D continuum Spanningstensor (stress tensor): Als dyadische producten: σ ee ee ee ee ee ee ee ee ee yy y y zz z z y z z y z z z y y y In matrivorm: y z y yy z zz Rektensor (strain tensor): Als dyadische producten: ε ee ee ee ee ee ee ee ee ee yy y y zz z z y z z y z z z y y y In matrivorm: y z y yy z zz

10 Met normaalrekken (normal strains) en schuifrekken (shear strains): u u y uz Normaalrekken: ; yy ; zz y z 1du du y 1du duz 1duy duz Schuifrekken: y ; z ; dy d dz d dz dy a : ε a a Grootte van de rek in richting van eenheidsvector Materiaalgedrag: wet van Hook (Hook s law): 1 1 v v v E E 1 1 v v v E E 1 1 v v v E E yy zz yy yy zz z z zz yy zz y y E 1 E v vyy v zz 1 v 1 v 1 v E E yy v 1vyy vzz z z 1v1v 1v E E 1 y y zz v vyy v zz 1v1v 1 v Theorema van Cauchy: geeft de spanningvector s op een vlak met normaal n Als dyadisch product: s σ n In matrivorm: s y z n s n y y yy y s z z zz n z Von Mises equivalente spanning: 1 3 M yy yy zz zz y z V Volumerek: tr() ε yy zz Buiging en torsie Pure buiging: Torsie: Normaalpanning: M t I Schuifspanning: Hoekverdraaing: T r J T met GJ E G (1 v)

11 1) F M =50 [N] ) F ) 4) ul () 0; du EA F d 5) σ 5ee 5(e e e e ) ; 10 [N/mm ] y y 0 y y M 6) ss [N/mm ] 3 3 7) 00 8) Mamimum stress in the micro-posts is = =0. [N/mm ] 64Fe 9) 3 15 R 10) zz