13. BASIC CONCEPTS OF KINEMATICS AND DEFORMATION

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1 AND DEFORMATION I Main Topics (see chapters 14 and 18 of Means, 1976) A Fundamental principles of cononuum mechanics B PosiOon vectors and coordinate transformaoon equaoons C Displacement vectors and displacement equaoons D DeformaOon E Homogeneous and inhomogeneous strain 10/3/12 GG303 1 AND DEFORMATION TransiOon From ParOcle Mechanics to ConOnuum Mechanics Newton s Pendulum hup:// r.jpg Sheep Mountain AnOcline, Wyoming hup:// 10/3/12 GG

2 II Fundamental principles of cononuum mechanics A Number of parocles is sufficiently large that the concept of bulk material behavior is meaningful hup:// diamond.jpg hup:// gemstone.jpg 10/3/12 GG303 3 II Fundamental principles of cononuum mechanics B Relates natural world to the realm of mathemaocs C DensiOes of mass, momentum, and energy exist (no holes ) 10/3/12 GG

3 II Fundamental principles of cononuum mechanics D Examples of cononuous properoes 1 Density ρ = lim ΔV 0 Δm ΔV So certain deriva-ves have to exist 10/3/12 GG303 5 II Fundamental principles of cononuum mechanics D Examples of cononuous properoes (cont.) 2 Hydraulic conducovity ("permeability") E Scale mauers Note that the concept of derivaoves becomes difficult at certain scales 10/3/12 GG

4 10/3/ BASIC CONCEPTS OF KINEMATICS AND DEFORMATION II Fundamental principles of cononuum mechanics F Variability 1 Heterogeneity: material property depends on posioon 10/3/12 beg.utexas.edu GG303 Cathedral Peak, CA BASIC CONCEPTS OF KINEMATICS AND DEFORMATION II Fundamental principles of cononuum mechanics F Variability 2 Anisotropy: material property depends on orientaoon Hand sample of gneiss hup://en.wikipedia.org/wiki/file:gneiss.jpg 10/3/12 GG

5 III PosiOon vectors and coordinate transformaoon equaoons A X = inioal (undeformed) posioon vector B X = final (current, or deformed) posioon vector (at Ome Δt) C Coordinate transformaoon equaoons 1 X = f(x) Lagrangian: final posioon a funcoon of inioal posioon 2 X = g(x ) Eulerian: inioal posioon a funcoon of final posioon 10/3/12 GG303 9 IV Displacement vector (U) A U = X - X 1 x- component: u x or u 2 y- component: u y or v 3 z- component: u z or w B Lagrangian U(X): displacement in terms of inioal posioon C Eulerian U(X ): displacement in terms of final posioon 10/3/12 GG

6 V DeformaOon: rigid body mooon + change in size and/or shape A Rigid body translaoon 1 No change in the length of line connecong any points 2 All points displaced by an equal vector (equal amount and direcoon); no displacement of points relaove to one another 3 [X'] = [U] +[X] matrix addioon (U is a constant) 10/3/12 GG V DeformaOon: rigid body mooon + change in size and/or shape B Rigid body rotaoon 1 No change in the length of line connecong any points 2 All points rotated by an equal amount about a common axis; no angular displacement of points relaove to one another 3 [X'] = [a][x] matrix muloplicaoon; rows in [a] are dir. cosines! 10/3/12 GG

7 V DeformaOon: rigid body mooon + change in size and/or shape C Change in size and /or shape (distoroonal strain) 1 At least some line segments connecong points in a body change lengths (i.e., the relaove posioons of points changes) 2 u is not a constant throughout the body (i.e., u varies) 10/3/12 GG C Change in size and /or shape (distoroonal strain) cont. 3 Change in linear dimension A Extension (or elongaoon): ε ε = ΔL = L L 1 0 B Stretch: S S = L 1 = + L 1 = 1+ ε C QuadraOc elongaoon: λ λ = L 1 = S 2 2 D All are dimensionless 10/3/12 GG

8 C Change in size and /or shape (distoroonal strain) cont. 3 Shear strain: γ a Describes change in right in angle between originally perpendicular lines b γ = tanψ For small Ψ, tanψ Ψ c Dimensionless 10/3/12 GG V DeformaOon: rigid body mooon + change in size and/or shape (cont.) D Change in volume (dilaoonal strain) 1 DilaOon (Δ) Δ = ΔV = V 1 V 0 V 0 V 0 2 Dimensionless 10/3/12 GG

9 D Change in volume (dilaoonal strain) cont. 3 Example Δ = ΔV = V V 1 o V o V o V 0 = a o b o 0 = a b c o o o 0 0 c o V 1 = a b 1 0 = 0 0 c 1 a 0 (1+ ε 1 ) b 0 (1+ ε 2 ) c 0 (1+ ε 3 ) = a 1 b 1 c 1 V 1 = a 0 b 0 c 0 0 (1+ ε 2 ) 0 (1+ ε 1 ) (1+ ε 3 ) = a 0 b 0 c 0 S 1 S 2 S 3 = a 0 b 0 c 0 0 S 2 0 S S 3 Δ = V 1 V 0 V 0 = a 0 b 0 c 0 S 1 S 2 S 3 a 0 b 0 c 0 a 0 b 0 c 0 = S 1 S 2 S 3 1 ε 1 + ε 2 + ε 3 For small strains (ε<<1) 10/3/12 GG VI Homogeneous and inhomogenous strain 10/3/12 GG

10 VI Homogeneous and inhomogenous strain 10/3/12 GG