Physics of Elastic Continua - Stress, Strain, and Waves
In this lecture, we delve into the physics of elastic continua, focusing on stress, strain, and waves in elastic media. Topics include deformation components, rotation of material, effects of strain on vectors, and more. The discussion continues with a detailed look at deformation and its components, emphasizing the relationship between various parameters. Join us as we explore the fascinating world of elastic continua and their underlying principles.
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Presentation Transcript
PHY 711 Classical Mechanics and Mathematical Methods 11-11:50 AM MWF Olin 107 Plan for Lecture 34 Physics of elastic continua Chap. 13 in F & W 1. Stress and strain 2. Waves in elastic media 11/21/2016 PHY 711 Fall 2016 -- Lecture 34 1
11/21/2016 PHY 711 Fall 2016 -- Lecture 34 PHY 711 Fall 2013 -- Lecture 34 2 2
Brief introduction to elastic continua r1 r1 deformation reference = + = = + r r ( ) u r r r ( ) u r ' ' 1 1 1 2 2 2 r ( ) ( ) + r r r r r r u ' ' ( ) + 2 1 2 1 2 1 1 11/21/2016 PHY 711 Fall 2016 -- Lecture 34 3
Brief introduction to elastic continua -- continued Deformation components: = + u x u x 1 2 1 2 u x u x u x j j + + i i i j j i j i O ij ij rotation of material elastic strain tensor 11/21/2016 PHY 711 Fall 2016 -- Lecture 34 4
Brief introduction to elastic continua -- continued r1 r1 reference = r deformation = r + + r r ( ) u r r r u r ' ' ( ) 1 1 1 2 2 2 ( ) ( ) = + + r r r r ( ) u r ' ' 2 1 2 1 2 1 1 3 ( ) + ' ' x x x x x x 2 1 2 1 2 1 i i i i ij j j = 1 j Effects of strain on a vector: a = + = a' a' ( ) + + a' a x y z 11 21 + 31 ( ) ' 1 a a 11 11/21/2016 PHY 711 Fall 2016 -- Lecture 34 5
Deformation b b a a ( ( ) = = + + + + + + a' a x y z a 11 21 31 ) b' b x y z b 12 22 32 = = = a b for 0 co s ab 2 ab ( ) + = = a b ' ' 2 cos ' ab ab 21 12 12 cos cos ( ) ( ) ( ) ) ( ) = + = sin sin cos ' cos ( ) ( in sin s = 2 2 12 12 2 11/21/2016 PHY 711 Fall 2016 -- Lecture 34 6
Brief introduction to elastic continua -- continued Deformation components: = + u x u x 1 2 1 2 u x u x u x j j + + i i i j j i j i X O ij ij ( ) ( ) ( ) ( ) = + = 1 Tr + u ' (1 ) V V V = V = a b c a b c ' ' ' ' V dV V d ( ) = + + = == = u Tr 11 22 33 11/21/2016 PHY 711 Fall 2016 -- Lecture 34 7
Elastic stress tensor 3 th n A component of force acting on surface T d A i dA d ij j = 1 j = Generalization of Ho oke's law, F : kx x 2 u x u x j = u + La me' coefficients : i T ij ij j i ( ) = or : Tr T ij ij ij 11/21/2016 PHY 711 Fall 2016 -- Lecture 34 8
Elastic stress tensor -- continued ( ) = Tr 2 T ij ij ij 2 3 ( ) T ( ) = 3 + Note that: Tr Tr p V bulk modulus= K V 1 ( ) = Inverse Hooke's law: T Tr T ij ij ij 2 3 2 + 3 11/21/2016 PHY 711 Fall 2016 -- Lecture 34 9
Stress tensor -- continued 2 3 + In terms of bulk modulus: = K 1 ( ) = T Tr T ij ij ij 2 3 2 + 3 1 K 1 1 3 ( ) ( ) = Tr T T Tr T ij ij ij ij 9 2 11/21/2016 PHY 711 Fall 2016 -- Lecture 34 10
1 1 K 1 1 3 ( ) ( ) ( ) = = T Tr T Tr T T Tr T ij ij ij ij ij ij 2 3 2 9 2 + 3 = Example -- hydrostatic pressure: T dp ij ij ( ) T = Tr 3 dp dp dp K = ij ij ij 2 3 3 + 3 dV V dp K ( ) = = Note that: Tr p V = K V 11/21/2016 PHY 711 Fall 2016 -- Lecture 34 11
1 1 K 1 1 3 ( ) ( ) ( ) = = T Tr T Tr T T Tr T ij ij ij ij ij ij 2 3 2 9 2 + 3 = dp ij zz = Example -- uniaxial pressure: T ij 0 otherwise 1 E = in terms of Young's modulus T zz zz + 9 K K = E 3 1 K 1 = = d p xx yy 9 6 1 3 2 + K K yy = = = Poisson ratio: xx 2 3 zz zz 11/21/2016 PHY 711 Fall 2016 -- Lecture 34 12
= dp ij z z = Example -- uniaxial pressure: T ij 0 otherwise transverse contributions: 1 K 1 = = + T xx yy zz 9 6 Poisson's ratio: 1 3 2 3 2 + K = = xx K zz Relationships between elastic constants: 1 31 2 1 21 + E = K E = 11/21/2016 PHY 711 Fall 2016 -- Lecture 34 13
1 K 1 1 ( ) ( ) = Tr T T Tr T ij ij ij ij 9 2 3 Shear modulus for otherwise f or f T T = xy yx T ij 0 = = xy yx 11/21/2016 PHY 711 Fall 2016 -- Lecture 34 14
Values of bulk modulus K for elemental materials -- 11/21/2016 PHY 711 Fall 2016 -- Lecture 34 15
Values of Youngs modulus E for elemental materials -- 11/21/2016 PHY 711 Fall 2016 -- Lecture 34 16
Values of Poisson ratio for elemental materials -- 11/21/2016 PHY 711 Fall 2016 -- Lecture 34 17
Values of shear modulus for elemental materials -- 11/21/2016 PHY 711 Fall 2016 -- Lecture 34 18
Dynamical equations of elastic continuum 2 u 1 3 ( ) = + + + 2 u u f K 2 t In the absence of external forces, this reduces to two decoupled wave equations representing longitudinal and transverse modes: = where 0 and l = + = + u u u l t = u u 0 t 1/2 4 3 K 1/2 = and c c l t 11/21/2016 PHY 711 Fall 2016 -- Lecture 34 19
Typical velocities of longitudinal sound waves Material cl (m/s) http://www.engineeringtoolbox.com/sound-speed-solids-d_713.html air: cl=343 m/s water: cl=1433 m/s 11/21/2016 PHY 711 Fall 2016 -- Lecture 34 20
from: https://pangea.stanford.edu/courses/gp262/Notes/5.Elasticity.pdf 11/21/2016 PHY 711 Fall 2016 -- Lecture 34 21
