Voorblad bij tentamen

Transcriptie

1 Voorblad bij tentamen Vaknaam: Mechanica BWT Vakcode: 8MB00 Datum: Begintijd: 09:00 Eindtijd: 12:00 Aantal pagina s: 11 (inclusief voorblad en formuleblad) Aantal vragen: 10 Aantal te behalen punten/normering per vraag: aangegeven per vraag Wijze van vaststellen eindcijfer: totaal aantal punten gedeeld door 10 Wijze van beantwoording vragen: open vragen Inzage: Overige opmerkingen: Formuleblad zit achter de vraagstukken Instructies voor studenten en surveillanten Toegestane hulpmiddelen (mee te nemen door student): Notebook X Rekenmachine Grafische rekenmachine X Dictaat/boek 1 A4-tje met aantekeningen X Woordenboek(en). Zo ja, welke: Engels - NL Let op: toiletbezoek is alleen onder begeleiding toegestaan binnen 15 minuten na aanvang en 15 minuten voor het einde mag de tentamenruimte niet worden verlaten, tenzij anders aangegeven er dient altijd tentamenwerk (volledig ingevuld tentamenpapier: naam, studentnummer e.d.) te worden ingeleverd tijdens het tentamen dienen de huisregels in acht te worden genomen aanwijzingen van examinatoren en surveillanten dienen opgevolgd te worden etui ligt niet op tafel onderling worden geen hulpmiddelen geleend/uitgewisseld Tijdens het maken van schriftelijke tentamens wordt onder (poging tot) fraude in ieder geval verstaan: gebruik van andermans ID-bewijs/campuskaart mobiele telefoon of enige andere media dragende devices liggen op tafel of zijn opgeborgen in de kleding (poging tot) gebruik van ongeoorloofde bronnen en hulpmiddelen, zoals internet, mobiele telefoon e.d. het gebruik van een clicker die niet je eigen clicker is ander papier voor handen hebben dan door de TU/e is verstrekt, tenzij anders aangegeven toiletbezoek (of naar buiten lopen) zonder toestemming of begeleiding Behorende bij Regeling centrale tentamenafname TU/e

2 2

3 Examination: Mechanics for BWT: 8MB00 January 26, 2017, 09:00-12:00 Question 1 (10 points) Relative to a Cartesian basis { e x, e, e y z } three vectors are defined: a ex 3e b ey ez c e 4e x y z Calculate the result of the following tensor-vector products: c ab c and ba Question 2 (10 points) In an experimental setup a metal frame A-C is placed on top of a femur bone in such a way that it can freely rotate at the femoral head at point B and is attached to the trochanter T using a cable C-T. When a vertical force F is applied at point A the frame is in horizontal position (aligned with the x-axis) and the cable is positioned at an angle =30 degrees relative to the frame. A free-body diagram of this loaded position is drawn below. Calculate the reaction forces H H, V H, and the force in the cable F T in terms of F, a and b 3

4 Question 3 (10 points) A strip of material with length l is clamped at point A and loaded by a constant distributed load q and by a compressive force F at its other end B. Calculate the reaction force at point A in terms of q, F and l. Question 4 (10 points) A rectangular membrane ABCD of 40 [mm] by 30 [mm] and 1 [mm] thickness is homogeneously loaded by a force of F x=6 [N] in the x-direction and a force of F y=20 [N] in the y-direction. Specify the full three-dimensional stress tensor Calculate the normal stress vector working on face AC of the triangular part ADC 4

5 Question 5 (10 points) For a loaded tissue the stresses in a specific point are determined to be as indicated in the figure below (units MPa). Make a sketch of Mohr's circles for the stress state in this point and indicate the stress values at the axes. calculate the maximum shear stress magnitude that can occur in the material for this stress state Question 6 (10 points) In a rotating-disc oxygenator of a heart-lung machine the velocity of a blood cell at position x xe x ye y ze z as defined relative to a Cartesian basis { e,e,e } can be described as: v e x e z z x y z with the angular velocity and a constant. The oxygen concentration within the oxygenator can be described as: Cxyzt (,,, ) Cbz ct 0 Calculate an expression of the material time derivative of the oxygen concentration: Cxyzt (,,, ) for the cell when moving through the oxygenator. 5

6 Question 7 (10 points) In an experiment a piece of tissue with volume V 0 is placed in a cylindrical chamber with a diameter of 20 [mm] that is filled with water. The water in the chamber is pressurized by applying a force F =500 [N] to a piston on top and as a result of this pressure the volume of the tissue changes to V. It is known that the tissue has a Young's modulus of 10 [MPa] and a Poisson's ratio of 2/5. The deformations can be considered small. Calculate the change in volume (V- V 0)/V 0 of the tissue when the load is applied. Question 8 (10 points) A piece of material of 3 [mm] by 4 [mm] and 1 [mm] thickness in size is subjected to a homogenous state of deformation described by the deformation tensor: F ee 2ee ee ee x x x y y y z z In the undeformed state, a fiber of the material located in the x-y plane is oriented along the diagonal of the material as indicated in the figure below. Use the Cauchy-Green deformation tensor to calculate the length of the fiber in the deformed state. 6

7 Question 9 (10 points) A membrane with dimensions l x0, l y0 and l z0 in the x-, y- and z-direction respectively, is compressed in the x-direction while it can expand freely in the y-direction. Its dimension in the z-direction is kept constant. The mechanical behavior of the membrane is described as incompressible using the following material model: σ pi GBI with T B F F the left Cauchy-Green tensor, p the pressure and G the shear modulus. Calculate the stress tensor in terms of G and x ( lx / lx0 ) for this deformation. Question 10 (10 points) In a micro-fluidics device a Newtonian fluid flows between 2 plates that are placed at a distance w as indicated in the figure below. The relationship between the stress tensor and the deformation rate tensor D for such fluids can be described as: σ pi 2 D, with the viscosity of the fluid and p the pressure. In the experiment, the velocity v is well described as: v y w v0ex with v 0 a constant. The fluid flow profile is constant in the z-direction and the pressure p is measured in the experiment, such that a two-dimensional description will be sufficient. Calculate the two-dimensional deformation rate tensor D Calculate the two-dimensional stress tensor in terms of p, w and and the coordinate y. 7

8 Answers: 1) ab c 4ex 12e ba c e e y z y a a a 2) HH 3 F; VH 1 F; FT 2 F b b b 3) N ql F 4) 1 1 σ ee ee 5 2 x x y y 49 s 3 e 4 e 625 n x y 5) 6) C bc 7) 1 xx yy zz 10 V 3 xx yy zz ~9.5% V 10 8) l 41 ~ 6.4 9) 1 1 σ G e e G1 e e 2 x 2 x x 2 z z x x y w 10) σ pi v0 e e e e 2 2 x y y x 8