Lecture 4: Basic Review of Stress and Strain, Mechanics of Beams

Transcription

1 MECH 466 Microelectromechanical Sstems Universit of Victoria Dept. of Mechanical Engineering Lecture 4: Basic Review of Stress and Strain, Mechanics of Beams 1 Overview Compliant Mechanisms Basics of Mechanics of Materials Bending of Beams Stress within Beams Moment of Inertia Appendices: (A) Stress and Strain (B) Poisson s Ratio (C) Stress Tensor (D) Strain Tensor 2

2 Design of Micro-Mechanisms In order to utilize the mechanical aspect of MEMS devices, most MEMS devices must be capable of motion. In other words, most micro-mechanical devices are micromechanisms, and we can appl the concepts of kinematics and dnamics when the are designed. However, there are three fundamental differences between macro-mechanisms and micro-mechanisms. (a) Component Design Limitations (2D shapes onl) -Cannot make: Ball Bearings, Roller bearings, etc... (b) Minimum Feature Size and Tolerance (c) Stiction -High ratio surface adhesion vs. volumetric forces. 3 Compliant Mechanisms In order to overcome most of these problems, man MEMS devices are designed as compliant mechanisms. Compliant mechanisms are a class of mechanisms that do not use an traditional joints (i.e. revolute, slider, prismatic, etc...), but instead use flexible spring like joints to allow their constituent parts to translate and rotate. The simplest example of a compliant mechanism is a common spring: 4

3 Compliant Mechanisms A more complex example of a compliant mechanism is that of a four bar linkage : Consider the benefits of compliant mechanisms in general, and how the appl to micro-devices. 5 Benefits and Limitation of Compliant Mechanisms Benefits: -Single material with no need for joints or lubricants -Built-in spring back -Highl precise, with zero pla/ slop in the mechanism -Lower fabrication cost Limitations -Built-in spring back -Complex to design, often requiring Finite Element Analsis -Must consider applied loads and fatigue life 6

4 Compliant Mechanisms for MEMS Because compliant mechanisms can be made from a single piece of material, the are ideall suited for miniaturization. -Since the have no revolute or sliding joints, there are no internal stiction or friction problems. -Since the hinges are undergoing elastic deflection, the automaticall return to their initial position when applied forces are removed. -The can be scaled down to an scale (even the nano-scale), as long as the material exhibits linear-elastic behavior. Movie of Compliant Active Microgripper 7 Mechanics of Materials, Basic Concepts of Stress and Strain Since compliant mechanisms are used for MEMS devices, there is a significant need to understand the mechanics of materials. The stud of mechanics of materials describes how solid materials will deform (change shape) and how the will fail (break) when subjected to applied forces. Mechanics of materials analsis is based on several basic concepts such as: (a) Newton s Laws of Motion: - (1st Law): Inertia - (2nd Law): F=ma - (3rd Law): Reaction Force (b) Equilibrium Condition (c) Stress and Strain (d) Material Properties 8

5 Definition of Stress and Strain Stress is a measure of: Applied force on a material Area over which that force is applied Normal stress is defined as: Strain is a measure of: Elongation of a material due to an applied force The original length of the material Normal strain is defined as: (*Note: the textbook denotes strain as s ) 9 Relation Between Stress and Strain Hooke s Law defines the relationship between stress and strain, where: The above equation is a simple linear model for the 1-D analsis of materials operating in the elastic region of behavior. If we require a 3D analsis of materials, we must use a more advanced matrix relationship between stress and strain, known as Generalized Hooke s Law. 10

6 Graphical Relation Between Stress and Strain stress brittle material general elastic material Ceramics, Crstal Silicon, Polsilicon Metals, Gold, Aluminum, Certain Plastics Linear Region Linear Region soft rubber strain 11 Movie of Tensile Testing of Steel The relationship of stress and strain for steel can be observed in the following movie: stress ield stress (s) plastic regime fracture point proportional limit ield point strain elastic regime perfect plasticit or ielding strain hardening necking Tensile Testing A36 mild steel, (speed = 4X), [Civil Engineering, Universit of New Mexico] 12

7 Values for E (modulus of elasticit) Some tpical values for E for common MEMS materials are listed below: Material E (GPa) Yield Strength (MPa) Fracture Strength (MPa) Single Crstal Silicon: <100> <110> <111> N/A 600 to 7700 Silicon Nitride (SiN) Polsilicon 120 to 175 N/A 1000 to 3000 <100> <110> N/A 6400 to Silicon Oxide 73 N/A 8400 Silicon Carbide (SiC) 700 N/A Stainless Steel Gold Aluminum Note: Appendix A lists numerous material properties for tpical MEMS materials. 13 Beam Bending For MEMS applications, we analze beams for a number of reasons including: (a) Internal stress at an point (b) Maximum stress and it s location (c) Beam Stiffness (d) Beam Deflection For a majorit of MEMS applications, there are essentiall three general cases for beam bending. Note that for macro-scale beam bending, there ma be dozens of general cases. 14

8 We will consider onl in-plane beam bending (bending about axis that is normal to the page) for simplicit. Case A: Cantilever Beams Beam Bending (i.e. diving board configuration) x Fixed End Free End Case B: Bridge Beam x Fixed End Fixed End 15 Beam Bending Case C: Guided End Beams d x Fixed End Guided End 16

9 (1) Determine all forces and moments using static equilibrium conditions (2) Create diagrams for: -Axial Force -Shear Force -Bending Moment Analsis of Beams: (3) Develop equation for stress at an point in the beam (4) Develop equations for K (stiffness) and d (deflection) for the beam. 17 Example of Beam Axial, Shear and Moment Diagrams: See Class Notes 18

10 Definition of Beam Parameters We will consider simple beams that have: - A straight shape - A vertical axis of smmetr Neutral Axis t h z x l w Examples of vertical smmetr include: Cross-Sections of Various Beams [image from Mechanics of Materials, E. P. Popov] 19 Pure Bending of Beams These are cases where a beam is subjected to a bending moment where we assume: - Deflection as a result of bending is less than 5% of beam length - All plane cross-sections of the beam before bending remain as straight planes after bending x Compression Neutral Axis +M +M Plane Cross-Sections Tension 20

11 Pure Bending of Beams Consider some infinitesimal cubes of material in the beam: t Maximum Compression Zero Stress x define c Point of Interest Maximum Tension 21 Definition of Beam Bending Stress The beam stress formula is given b: - where: Applies to all beams in a state of pure bending. Derivation is available in textbooks on solid mechanics. 22

12 Moment of Inertia The moment of inertia, I, of a beam depends on the geometrical properties of the cross-section area A of a beam. I is defined as: - where: I is relative to the centroid of the cross-section area More generall moment of inertia is defined as Izz: This is known as the parallel axis theorem 23 Examples of Moment of Inertia See Class Notes 24

13 Beam Deflection A unique analtical solution exists for beam deflection, given b: (a) beam geometr (b) loading conditions (c) boundar conditions Generall, beam curvature ρ (rho), can be defined as: Additionall, we can also define ρ as: where v is the beam deflection from the initial position. This approximation is valid when v < 5% of beam length. 25 Beam Deflection Therefore, we can develop the following differential equation, which can be solved for an beam, given the specific beam (a) geometr, (b) loading condition and (c) boundar condition: It is beond the scope of this course to solve these equations. For this course, we can use Appendix B in the textbook, which provides the deflections associated with this equation, for common general cases found with MEMS beams. Example of beam deflection for cantilever beam: d q F 26

14 Beam Stiffness A useful concept in predicting the forces and deflections within MEMS beams is the concept of stiffness. The stiffness model normall associated with springs can be expressed as: Where K is a constant of proportionalit that defines the relation between applied force, F, and the resulting spring deflection, x. 27 Beam Stiffness Given the equation for the tip deflection of a beam, we can define that beam s stiffness as: Example of beam stiffness: Consider the cantilever beam in the previous example: d q F Since: Therefore: 28

15 Calculation of Combined Mechanical Stiffnesses Computation of Stiffness for Springs in Series. Computation of Stiffness for Springs in Parallel See Class Notes 29 For some MEMS applications, the beams that allow the sensor or actuator to move undergo a twisting/torsional action. In these cases, it is useful to review the basic formulas governing the torsion of beams, to determine: (a) Maximum stress and it s location (b) Beam Stiffness (c) Beam Deflection Beam Torsion 30

16 Beam Torsion For Circular Beams The basic assumptions for the torsion of circular beams (a) All sections initiall plane and perpendicular to the lengthwise axis, remain plane after torsion. (b) Following twisting, all cross-sections remain undistorted and have a linear variation of stress from the center of twist (where τx=0) to the outer surface (where τx= τmax). (c) Material is homogeneous and obes Hooke s law. 31 Beam Torsion For Circular Beams The governing equations for circular beam torsion are presented below, without derivation: where: τ - shear stress T - applied torque r - radius from center to point of interest J - polar moment of inertia For circular x-section For deformation, we have: and since where: where: - angle of twist per unit length G - Modulus of shear - angle of twist 32

17 Beam Torsion For Non-Circular Beams The governing equations for non-circular beam torsion depend on the cross-sectional geometr. Derivation of these equations requires advanced knowledge of mechanics, and is beond the scope of this course. Table 6.2 on the left provides equations for the maximum stress, it s location, and the Angle of twist per unit length for various crosssections. 33 Beam Torsion Some FEM (finite element analsis) simulations of the distribution of shear stress due to torsion, for beam cross-sections are shown below: Some FEM simulations of the deformation due to torsion, for beam cross-sections are shown below: 34

18 APPENDICIES: Definitions & Reference Materials: (A) Stress and Strain (B) Poisson s Ratio (C) Stress Tensor (D) Strain Tensor 35 Basic Concepts of Stress and Strain Mechanics of materials describes how solid materials will deform (change shape) and how the will fail (break) when subjected to applied forces. Mechanics of materials analsis is based on several basic concepts such as: (a) Newton s Laws of Motion: - (1st Law): Inertia - (2nd Law): F=ma - (3rd Law): Reaction Force (b) Equilibrium Condition (c) Stress and Strain (d) Material Properties 36

19 Static Equilibrium of Bodies The static equilibrium condition states that all forces and moments applied to a bod are balanced such that there is no net acceleration of the bod. More specificall: -The vector summation of all forces acting on a bod must be equal to zero, and... -The sum of all moments acting must be equal to zero. Therefore, for a 3D bod in space: - these six equations must be satisfied for the bod to be in static equilibrium 37 Definition of Stress and Strain Stress is a measure of: Applied force on a material Area over which that force is applied Normal stress is defined as: Strain is a measure of: Elongation of a material due to an applied force The original length of the material Normal strain is defined as: (*Note: the textbook denotes strain as s ) 38

20 Normal stress: Definition of Stress and Strain Normal strain: normal stress/strain shear stress/strain un-loaded A a rod under no applied force L A loaded F F F dx l L+DL F 39 Definition of Stress and Strain Shear stress: --> Shear strain: normal stress/strain shear stress/strain un-loaded A a rod under no applied force L A loaded F F F dx l L+DL F 40

21 Relation Between Stress and Strain Hooke s Law defines the relationship between stress and strain, where: The above equation is a simple linear model for the 1-D analsis of materials operating in the elastic region of behavior. If we require a 3D analsis of materials, we must use a more advanced matrix relationship between stress and strain, known as Generalized Hooke s Law. 41 Definition of Poisson s Ratio When ou strain a bod along one axis, it will change shape along the other axes. For example, consider the rectangular bod below: If it is in tension, its cross-sectional area will become reduced. We can define Poisson s ratio as: Note for Si: (See Appendix A for other materials) 42

22 Definition of Stress Tensor Consider a solid bod (as shown below) with an arbitrar shape, subjected to a set of arbitrar forces. F2 We wish to analze the state of stress that exists within this solid bod, and ma also want to determine the deformation of the solid bod. F1 F3 43 Definition of Stress Tensor The first step is to define a coordinate sstem that is suitable for analsis. F2 F1 x z F3 44

23 Definition of Stress Tensor If a cross section of this loaded bod is taken, we wish to determine the stress at each and ever point within the interior. F2 Because there are various applied forces with various directions, in general, the stress distribution throughout the solid bod will be non-uniform. F1 For the purposes of analsis we can discretize the solid bod into cubes, and will consider the stress on each cube. x z F3 45 Definition of Stress Tensor Consider a single cube of material from the solid bod. Further, assume that the cube is infinitesimal in size. (1) The external forces F1, F2 and F3 act on the bod, while each infinitesimal cube ma have a set of small local force(s) (i) acting on it. d z dx dz x (2) 46

24 Definition of Stress Tensor We can now define the normal stress on a single face, as shown below. Note the notation used to indicate the stress. It consists of two indices, or subscripts. The first refers to the plane on which the stress acts, and the second refers to the direction of the stress τxx = σxx z τxx = σxx x x (2) z 47 Definition of Stress Tensor We can now define shear stress on the same face. Note that since there are two possible directions we will define two shear stresses. τx τxx = σxx τxz τxx = σxx τx z τxz x z 48

25 Definition of Stress Tensor Similarl, we can define the normal and shear stresses on the faces of the cube that are perpendicular to the -direction: σ τx τz τx τxx = σxx τxz τxx = σxx τx τx τz τxz x z σ 49 Definition of Stress Tensor Lastl, we can define the normal and shear stresses on the faces of the cube that are perpendicular to the z-direction: σ τz τx σzz τx τxx = σxx τxz τz τxx = σxx τx τx τzx τz τxz x σzz z σ 50

26 Definition of Stress Tensor A stress tensor completel defines the state of stress for such a cube, with respect to the chosen x,, and z cartesian coordinates. Note that there are 18 stresses defined on the cube. In order for the cube to be in static equilibrium in translation, we can observe that stresses that are on opposite faces of the cube, and opposite in direction must be equal. Therefore, we onl need to specif 9 stresses to represent the stress tensor, as: 51 Definition of Stress Tensor In addition, for the cube to be in static equilibrium in rotation, we can observe that shear stress acting on the sides of the cube, must all have a net moment of zero acting on the cube: τx τx τx Therefore, we can see that: τx And as a result, we onl need to define six unique stresses to describe the stress tensor. (Note the stress tensor is smmetric) x 52

27 Definition of Strain Tensor In a similar manner, the strain tensor can be derived, and is expressed as: When we appl the rotational equilibrium condition, this will reduce to (which is expressed in common notation): α αx x 53 Tensor Notation as used in Text Since there are onl 6 pieces of unique information contained in the matrices, an alternative method to describe these tensors, is in a single column notation: 54

28 Definition of Stiffness Matrix Using the new tensor notation for stress and strain, we can define the general relationship between stress and strain as: Where C = Stiffness Matrix: 55 Definition of Stiffness Matrix Essentiall, the stiffness matrix C is analogous to the modulus of elasticit, E. However, C encompasses all elasticit information for all normal and shear stresses with respect to all normal and shear strains. 56