Learning Objectives By the end of this section, the student should be able to:

Transcription

1 Mechanics of Materials stress and strain flexural l beam bending analysis under simple loading conditions EECE Learning Objectives By the end of this section, the student should be able to: describe the stress strain relationship for elastic materials calculate beam deflection for simple loading conditions describe comb drive actuators and how they are used in siliconbased microsystems urther Reading Chang Liu, oundations of MEMS, Chapter 3: Review of Essential Electrical and Mechanical Concepts EECE

2 Mechanical design issues for MEMS devices Example: MEMS accelerometer uses a loaded cantilever beam (or other design) to sense inertial motion (acceleration) electrical circuitry to transduce the proof mass vibration into an electrical signal for amplification A MEMS accelerometer is a key component of many automobile airbag sensors. The device consists of a center region, the proof mass, which is connected by springs to the surrounding structure. The interdigitated fingers are capacitors used by on chip electronics (not shown) to sense acceleration. EECE Movable electrodes in the form of interdigitated fingers are attached to the proof mass. The fixed and moving electrodes form a bank of parallel connected l tdcapacitors. If an acceleration is applied to the chip, the proof mass will move under an inertial force against the chip frame. This changes the finger distances and therefore the total capacitance, which is read using on chip signal EECE 300 processing 011 electronics. 4

3 Mechanical design issues for MEMS devices Example: nanoscale cantilever mass sensor small amount of mass binds to cantilever (single bacterium, single virus particle) change in mass changes the resonance frequency μm A small gold dot rests on a silicon cantilever. The dot is a test mass for studying how the cantilever can be used to measure the masses of tiny particles, including viruses, with attogram precision. ILIC, et al. (004) Attogram detection using nanoelectromechanical oscillators. Journal of Applied Physics, EECE 95, Stress and strain EECE

4 Normal stress and strain Mechanical stresses are either normal stress or shear stress. If the stress acts in a direction perpendicular to the cross section, it is called a normal stress. The normal stress σ is defined as the force applied to a given area A. The SI unit of stress is N/m, or Pa. A normal stress can be tensile or compressive. The unit elongation is the strain. or a rod with uniform cross sectional area subjected to axial loading, the normal strain is given by L + ΔL EECE Poisson s ratio In reality, the applied longitudinal stress along the x axis not only produces a longitudinal elongation in the direction of the stress, but a reduction of the cross sectional area as well. The material must try to maintain constant atomic spacing and bulk volume. Poisson s ratio is defined das the ratio between transverse and longitudinal lelongations: ν ε y = = ε x ε z ε x lateral strain Δd ν = = axial strain d 0 L0 ΔL cross section L L + ΔL EECE

5 Elastic behavior Hooke s Law and springs When a material experiences a stress, the usual result is a deformation resulting in a strain on the material. A common example is a spring which is either stretched or compressed. This is an example of Hooke s Law, a simple elastic response to an external force. When = 0, the Δx goes back to zero as long as the force remained in the elastic limit. There is no permanent deformation in the elastic limit. Atoms are held together with atomic forces. If we imagine interatomic force acting as springs to provide restoring force when atoms are pulled apart or pushed together, the modulus of elasticity is the measure of the stiffness of the interatomic spring near the equilibrium point. slope = k EECE Δx 9 Young s modulus Under small deformation, the stress and the strain terms are proportional to each other according to Hooke s law: The proportion constant E is called the modulus of elasticity or Young s modulus. It is an intrinsic property of a material. It is a constant for a given material, irrespective of the shape and dimensions of the mechanical element. The Young s modulus is a measure of the stiffness of a material. It is measured in stress units (force/area). E [=] Pa i Clicker: a. k ~ L b. k ~ 1/L EECE

6 Longitudinal stress and strain A cylindrical silicon rod is pulled on both ends with a force of 10 mn. The rod is 1mm long and 100 μm in diameter. ind the stress and strain in the longitudinal direction of the rod, given that E = 130 x 10 9 N/m. 3 a. ε = i Clicker: 6 b. ε = c. ε = d. ε = EECE Elastic behavior of materials elastic regime stress σ (/A) silicon 160 GPa gold 78 GPa Δσ bones 18 GPa slope = E = Δ ε silicone rubber ε silicone rubber ~ 500 kpa brain tissue ~ 3 kpa strain ε (δl/l) /) EECE

7 Elastic limit and plastic behavior of materials At some point when applying a stress, the material will reach a maximum strength at which point a deformation is produced. This is the yield strength. This is the elastic limit of the material. Beyond this point, the material is plastically deformed. If the load is removed, the material retains this deformation at zero load. EECE Shear stress and strain The magnitude of the shear stress is defined as τ = A The unit of τ is N/m. Shear stress has no tendency to elongate or shorten the element in the x, y, and z directions. Instead, the shear stresses produce a change in the shape of the element. Shear strain, defined as the extent of rotational displacement, is γ = ΔX X L tanθ θ for small θ The shear strain is unitless. Itis the angular displacement expressed in radians. θ EECE

8 Shear modulus of elasticity The shear stress and strain are related to each other by a proportional constant, called the shear modulus of elasticity G. G= τ γ The unit of G is N/m. The value of G depends on the material, not the shape and dimensions of an object. or a given material, E, G, and Poisson s ratio are related by: G= E (1 +ν ) EECE lexural beam bending analysis lexural beams are commonly encountered in MEMS as spring support elements. It is important to calculate the bending of a beam under simple loading conditions, analyze induced internal stress, and determine the resonant frequency associated with the l element. EECE

9 Possible boundary conditions A flexural beam can be classified according to the combination of the two mechanical boundary conditions associated with it. EECE Beam segment in pure bending When a beam is loaded by force, stresses and strains are created throughout the interior of the beam. The loads acting on a beam cause the beam to bend (or flex), deforming its axis into a curve. Consider a portion of a beam (A B) in pure bending. The cross section of the beam is symmetric about the y axis. It is assumed that the cross sections of the beam, such as sections mn and pq, remain plane and normal to the longitudinal axis. The lower part of the beam is in tension and the upper part is in compression. EECE

10 Beam segment in pure bending When a beam is loaded by force, stresses and strains are created throughout the interior of the beam. Somewhere between the top and bottom of the beam is a surface in which longitudinal lines do not change in length. This surface, indicated by st, is called the neutral surface of the beam. The intersection i between the neutral surface with any cross sectional plane such as line tu is the neutral axis of the cross section. EECE Deflection of beam under pure bending The general method for calculating the curvature of the beam under small displacement is to solve a second order differential equation of a beam: y EI = M ( x) x where M(x) represents the bending moment at the cross section at location x. y represents the displacement at location x. The x axis runs along the longitudinal direction of the cantilever. EECE

11 Cantilever beam under small deflections Beam segment under pure bending: y top is in tension neutral axis ε = 0 bottom is in compression ε x ε max t/ ε max t/ y EECE Bending To calculate the magnitude of stresses at any location in the beam: At any section, the distributed stress contributes to distributed force, which subsequently gives rise to a reaction moment with respect to the neutral axis. basic concept of moment: force distance t A ( ) t w h = σ ( ) ( ) M = d h h = h da h Assuming that the magnitude of stress is linearly related to h and is the highest at the surface (denoted by σ max ), thisequation can be rewritten as: h t t max max M = t σ max da h = t h da = I w h= t/ t/ w h= t / σ σ The term I is called the moment of inertia associated with a particular cross section. EECE

12 Bending strain and beam curvature Radius of curvature geometric connec on to strain dθ R ε max ( ) also, for small deflections = R+ t/ dθ Rdθ t / 1 d y Rdθ = = R R dx EECE Curvature and strain Combining the curvature and moment results: t / ε max = σ = Eε R h t t max max M = t max t w σ da h= h da I h= t/ t/ = w h= t/ σ σ 1 d y R = dx EI y = M ( x) x where M(x) represents the bending moment at the cross section at location x. y represents the displacement at location x. The x axis runs along the longitudinal direction of the cantilever. EECE

13 Cantilever beam Goal: find relation between tip deflection y(x = L) and applied load. Clamped end: y = 0, dy/dx = 0 at x = 0 x x = L Assume tip deflection is small compared to length of beam. Shear stresses negligible. M( x) = EI y x ( ) M ( x) = L x EECE Cantilever beam bending i Clicker: deflection at beam tip: a. y( L) = L EI b. y ( L ) = L 3EI 3 c. y( L) = L 3EI i Clicker: slope at beam tip: a. y ( L) = L EI b. y( L) = L 3EI 3 c. y( L) = L 3EI EECE

14 Review: fin nding mom ent of inert tia taken from E. Popov, Engineering g Mechanics of Solids EECE inding the spring constant Beams are the most frequently encountered spring element in MEMS. These microbeams serve as mechanical springs for sensing and actuation. The stiffness of these beams is a frequently encountered design concern. The stiffness is characterized dby the spring constant (or force constant). The mechanical spring constant is the ratio of the applied force and the resultant displacement: k = x or a cantilever with a point loading on the free end, the maximum deflection occurs at the free end. or a fixed fixed bridge with a loading force in the center of the span, the center has the largest deflection. EECE

15 Spring constant for cantilever A fixed free beam with a rectangular cross section is one of the most common scenarios encountered in MEMS. The free end of the beam will reach a certain bent angle, The resultant vertical displacement equals The spring constant of the cantilever is: EECE Liu, oundations of MEMS 9 Spring constant for cantilever The spring constant of the cantilever is: 3EI Ewt k = = = 3 3 x l 4l 3 The spring constant: decreases with increasing length proportional to the width strongly influenced by change in thickness due to the term t 3. The stiffness of the cantilever depends on the direction of the bending. If the force is applied longitudinally, the constant would be very different. The beam provides compliance in one direction and resistance to movement in another. EECE

16 Springs i Clicker: a. b. 3 Ea b k a = 3 4 L 3 Eab k = a 3 4L b i Clicker: a. b. 3 Ea b k b = 3 4L 3 Eab kb = 3 4L b L a a EECE Spring systems parallel connected springs In many applications, two or more springs may be connected to form a spring system. In the parallel case: Same displacement load is shared and the spring constant is the sum of the individual spring constants. L c a / b / i Clicker: a. k = k k 1 b. k = a k a EECE

17 Spring systems serially connected springs In many applications, two or more springs may be connected to form a spring system. In the serial case: Same load deflec ons add L total a L c b Lc L c i Clicker: a. k = k k 1 b. k = a k a EECE Moments of inertia of two beams example Consider two cantilever beams of the same length and material: one has a cross section of 100 μm by 5 μm, and a second one has a cross section of 50 μm by 8 μm. Which one is more resistant to flexural bending (i.e., stiffer)? 1 L 1 w 1 t 1 i Clicker: a. beam 1 is stiffer b. beam is stiffer w t L EECE

18 End constraints and loading conditions EECE Liu, 300 oundations 011 of MEMS 35 End constraints and loading conditions EECE Liu, 300 oundations 011 of MEMS 36

19 Vertical translational plates ixed guided springs are often used to support rigid plates and facilitate their translation. Often, a plate is supported by two or more such beams. In these cases, one end of the beam is fixed, with all degrees of freedom limited. Another end of the spring can move in the vertical ldirection, i but no angular displacement is allowed because it is connected to the stiff translational plate, which remains parallel to the substrate under allowable plate movement. EECE Liu, oundations of MEMS 37 Vertical translational plate example ind the expression of the force constant associated with the plate. Start with the basic formula for the spring constant of a single fixed guided beam under a transverse loading force. The maximum displacement x occurs at the guided end of the beam: EECE

20 Vertical translational plate example The expression for the spring constant (force constant) of each single fixed guided beam is: If a force is applied to a plate supported by n cantilevers with equal dimensions and force constants, each spring shares 1/n th of the total force load. The total force constant experienced by the spring is nk. The force constant associated with each fixed guided beam is: or the plate supported by two fixedguided beams, the equivalent force constant is or the plate supported by four fixedguided beams, the equivalent force constant is EECE Capacitance Sensor Response A parallel capacitor with an area (A) of μm is supported by four cantilever beams. The plate is made of polycrystalline silicon that is t = μm thick. The distance between the bottom of the plate and the substrate is d = 1 μm. Each cantilever beam is l = 400 μm, w = 0 μm, t = μm. ind the relative change of capacitance under an acceleration of 1 g. EECE

21 Capacitance Sensor Response A parallel capacitor with an area (A) of μm is supported by four cantilever beams. The plate is made of polycrystalline silicon that is t = μm thick. The distance between the bottom of the plate and the substrate is d = 1 μm. Each cantilever beam is l = 400 μm, w = 0 μm, t = μm. ind the relative change of capacitance under an acceleration of 1 m/s. EECE Capacitance Sensor Response EECE

22 Microneedles for painless drug delivery N.-T. Nguyen and S. T. Wereley, undamentals And Applications of Microfluidics, Second ed: EECE Artech 300 House, Molded polysilicon microneedles D. V. McAllister, M. G. Allen, and M. R. Prausnitz, "Microfabricated Microneedles for Gene and Drug EECE Delivery," Annual Review of Biomedical Engineering, vol., pp ,

23 Solid square Solid rectangle 1 4 I = W 1 3 I = WH 1 1 Hollow square I 1 1 W w H h Hollow rectangle 4 4 = ( ) I = ( WH wh ) 1 Solid circle I π R 4 Hollow circle 4 = I = ( R 4 r 4 ) π 4 EECE Thin annulus I 3 = π R t 45 Buckling of a microneedle If the needle s length is relatively long compared to its width, the first failure mode is buckling. Thecritical buckling force: b π EI = 4L b EECE

24 Parylene microneedle buckling A microneedle made of parylene C has channel dimensions 00 μm 00 μm mm. The parylene layer is deposited over 4 hours at a rate of 5 μm/hr (over a sacrificial material). The Young s modulus is 3GPa 3. GPa. Calculate the critical buckling force for this needle. EECE Bending of a microneedle As a rough estimate, the bending stiffness of a needle can be modeled by the spring constant k: or a hollow square needle: EECE

25 Parylene microneedle bending A microneedle made of parylene C has channel (inner) dimensions 00 μm 00 μm mm. The parylene layer is deposited over 4 hours at a rate of 5 μm/hr (over a sacrificial material). The Young s modulus is 3. GPa. Calculate the tip deflection under a force of 15 mn at the tip. EECE Polysilicon microneedle If the previous needle were made of polysilicon, what are the tip deflection and critical buckling force? Assume all other parameters are the same, and Young s modulus of polysilicon is 150 GPa. i Clicker: a. the tip deflection is directly proportional to Young s modulus. b. the tip deflection is inversely proportional to Young s modulus. i Clicker: a. the critical buckling force is directly proportional to Young s modulus. b. the critical buckling force is inversely proportional to Young s modulus. EECE

26 Microneedle mechanical design Insertion force The area A of contact surface at the needle tip is considered the determining factor for the insertion force i. The puncture toughness G p which is the work per area needed to initiate a crack: G p W = A Insertion force (static case, wherethe insertion speed is slow andthe kinetic energy of the needle is negligible) 1 i,max = 0 + Gp A χ 0 and χ determined experimentally. 0 initial force χ characteristic insertion length EECE Insertion force of a microneedle The tip radius of a microneedle is 50 μm. Assume a puncture toughness of the skin of 30 kj/m, a characteristic insertion length of 150 μm, and an initial force 0 = 0.1 N. Determine the force required for puncturing the skin. EECE

27 Differential thermal expansion If two materials are in close contact during temperature changes (heating or cooling) and their thermal expansion coefficients are not equal, then differential expansion (during heating) or differential contraction (during cooling) can occur. This can happen during thin film deposition. i The film is deposited dstress free at T dep, but then at room temperature the film is under a thermal mismatch strain. ( ) T ε = γ γ Δ 1 T EECE Beam bending intrinsic stress EECE

28 Thermal coefficient of expansion At the temperature increases, most solids expand in volume. The lateral dimensions of the object increases in all directions. The volumetric thermal expansion coefficient α is the ratio between the relative change of volume to the degree of temperature variation, The linear expansion coefficient i γ is the change of only one dimension of an object due to temperature variation, The percentage change in length in an unconstrained object (free to expand at both ends) due to the increase in temperature is given by If the object is fixed at both ends, then the thermal expansion produces a stress given by EECE Thermal Expansion EECE

29 Thermal actuator Assume that you have a silicon beam that is 100 μm long, and 1μm square. You heat it by 100K. How much force do you get if you constrain it? How much elongation if you allow it to expand? Linear thermal coefficient of expansion for silicon is /K. E Si = 160 GPa. Area= ε = γ ΔT = σ = Ε ε = = A σ = δl= ε L= EECE Thermal Actuators Use thermal expansion for actuation Very effective and high force output per unit area Cold arm Actuator translates in this direction Current output pad Hot arm Cascaded thermal actuators for high force Current input pad EECE J. Judy, 300 UCLA

30 Bimetallic beam actuator radius of curvature Bimetallic actuation ti uses the difference in thermal coefficient i of expansion of two bonded d solids. This principle is often called thermal bimorph actuation. Bimetallic actuators offer an almost linear deflection dependence on heating power. Their disadvantages are high power consumption and slow response. The resulting force is proportional to the difference between the thermal expansion coefficients of the two materials and the temperature difference. EECE Bimetallic beam actuator The displacement of the beam tip is calculated as: ( ) y L L R The radius of curvature is: ρ = ( bet ) ( ) ( bet + betbet t1 + 3tt 1 + t) 6 ( γ γ ) Δ Tb E t b E t ( t + t ) where b 1 and b are the widths of the two material layers. The equivalent tforce on the tip of the beam 3 ( EI ) y ( L ) beam (cantilever beam with free end) is: The flexural rigidity of the composite beam is: ( EI ) beam = = ( bet ) ( ) ( bet + betbet t1 + 3tt 1 + t) 1 ( bet + bet ) L 3 EECE

31 Bimetallic beam actuator Assuming that the two layers have the same width b, the radius of curvature and the equivalent tip force can be simplified to: t 1 t1e 1 t 1 te t t te t t1e1 1+ t ρ = 6 1 ( 1) T 1 t γ γ Δ + t 3b 4L t + t te te ( γ γ ) 1 = 1 Δ 1 1 T EECE Design of a thermomechanical valve with bimetallic actuator A thermomechanical microvalve has a rigid square seat of μm. The valve seat is suspended on four flexures. Each flexure is 500 μm long and 00 μm wide. The flexure is made of 10 μm silicon and μm aluminum. The silicon heater is integrated in the flexure. The aluminum layer is evaporated at 400 C. If a normally closed microvalve is to be designed at 5 C, what is the maximum gap between valve opening and the surface of the valve seat wafer? Si: t = = μm, γ K, E 1 = 170 GPa Al: t = μm, γ = K 1, E = 70 GPa EECE

32 Thermomechanical valve EECE Thermomechanical valve EECE

33 Thermomechanical valve: minimum opening/closing pressure If the gap between valve opening and surface of the valve seat is 5 μm, how large should the inlet pressure be to open the valve? What is the temperature difference for opening the valve at zero inlet pressure? The valve opening is μm. EECE