247B/M 28: Introduction to MMS Design Lecture 0m2: Mechanics of Materials CTN 2/6/7 Outline C247B M C28 Introduction to MMS Design Spring 207 Prof. Clark T.- Reading: Senturia, Chpt. 8 Lecture Topics: Stress, strain, etc., for isotropic materials Thin films: thermal stress, residual stress, and stress gradients Internal dissipation MMS material properties and performance metrics Dept. of lectrical ngineering & Computer Sciences University of California at Berkeley Berkeley, CA 94720 Lecture Module 7: Mechanics of Materials 2 Vertical Stress radients Variation of residual stress in the direction of film growth Can warp released structures in z-direction lasticity 3 4 Copyright 207 Regents of the University of California
247B/M 28: Introduction to MMS Design Lecture 0m2: Mechanics of Materials CTN 2/6/7 Normal Stress (D) Strain (D) If the force acts normal to a surface, then the stress is called a normal stress z z x z y x y x y Differential volume element 5 6 The Poisson Ratio Shear Stress & Strain (D) 7 8 Copyright 207 Regents of the University of California 2
247B/M 28: Introduction to MMS Design Lecture 0m2: Mechanics of Materials CTN 2/6/7 2D and 3D Considerations 2D Strain In general, motion consists of Important assumption: the differential volume element is in static equilibrium no net forces or torques (i.e., rotational movements) very must have an equal in the opposite direction on the other side of the element For no net torque, the shear forces on different faces must also be matched as follows: xy = yx rigid-body displacement (motion of the center of mass) rigid-body rotation (rotation about the center of mass) Deformation relative to displacement and rotation Area element experiences both displacement and deformation Stresses acting on a differential volume element xz = zx Must work with displacement vectors Differential definition u x x u x x u x of axial strain: x x yz = zy x 9 2D Shear Strain x 0 Volume Change for a Uniaxial Stress Stresses acting on a differential volume element Copyright 207 Regents of the University of California 2 3
247B/M 28: Introduction to MMS Design Lecture 0m2: Mechanics of Materials CTN 2/6/7 Isotropic lasticity in 3D Important Case: Plane Stress Isotropic = same in all directions The complete stress-strain relations for an isotropic elastic Common case: very thin film coating a thin, relatively rigid substrate (e.g., a silicon wafer) solid in 3D: (i.e., a generalized Hooke s Law) x y z y y z x z z x y x xy yz zx xy yz zx At regions more than 3 thicknesses from edges, the top surface is stress-free z = 0 et two components of in-plane stress: x ( )[ x ( y 0)] y ( )[ y ( x 0)] Basically, add in off-axis strains from normal stresses in other directions 3 Important Case: Plane Stress (cont.) Net non-zero inplane force (that we just analyzed) where [ ( )] At free edge, in-plane force must be zero 4 Film must be bent back, here Discontinuity of stress at the attached corner stress concentration There s no Poisson contraction, so the film is slightly thicker, here and where Biaxial Modulus = dge Region of a Tensile ( >0) Film Symmetry in the xy-plane x = y = Thus, the in-plane strain components are: x = y = x ( )[ ] 5 Copyright 207 Regents of the University of California Peel forces that can peel the film off the surface 6 4
247B/M 28: Introduction to MMS Design Lecture 0m2: Mechanics of Materials Linear Thermal xpansion T As a Function of Temperature CTN 2/6/7 As temperature increases, most solids expand in volume Definition: linear thermal expansion coefficient Linear thermal expansion coefficient Remarks: T values tend to be in the 0-6 to 0-7 range Can capture the 0-6 by using dimensions of strain/k, where 0-6 K - = strain/k In 3D, get volume thermal expansion coefficient = d x T [Kelvin - ] dt V 3 V T For moderate temperature excursions, T can be treated as a constant of the material, but in actuality, it is a function of temperature 7 T [Madou, Fundamentals of Microfabrication, CRC Press, 998] Cubic symmetry implies that is independent of direction 8 Copyright 207 Regents of the University of California 5