# Lab.09.pptx - 10/19/11 9. EIGENVECTORS, EIGENVALUES,...

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Unformatted text preview: 10/19/11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN I Main Topics A Mo?va?on B Inverse [A] ­1 of a real matrix A C Determinant |A| of a real matrix A D Eigenvectors and eigenvalues (Principal direc?ons and principal values) E Diagonaliza?on of a matrix F Quadra?c forms and the principal axes theorem (maxima/minima in 2D) G Strain ellipses, strain ellipsoids, and principal strains Appendix 1 10/19/11 GG303 1 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN A forward deforma?on deforms a unit circle to a strain ellipse Objec?ve: To describe size and shape of strain ellipse 10/19/11 GG303 2 1 10/19/11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN •  For irrota?onal deforma?on, the axes which transform to become the principal strain axes (dashed) are not rotated in either the forward deforma?on or in the inverse deforma?on 10/19/11 GG303 3 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN •  For rota?onal deforma?on, the inverse deforma?on to retro ­deform the strain ellipse back to a unit circle causes lines along the principal strain axes in the strain ellipse to be rotated. •  The angular diﬀerence between the orienta?on of the principal strain axes (solid) and the retro ­ deformed axes (dashed) in the ini?al state is the rota?on. 10/19/11 GG303 4 2 10/19/11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN II Mo?va?on A Eigenvectors and eigenvalues provide simple, elegant, and clear ways to solve many scien?ﬁc problems [e.g., geometry, strain, stress, curvature (shapes of surfaces)] B To ﬁnd the magnitudes and direc?ons of the principal strains using linear algebra rather than graphical means or calculus (e.g., Ramsay and Huber, 1984) 10/19/11 GG303 5 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN ⎡ ⎤ III Inverse [A] ­1 of a real matrix A [ A ] = ⎢ a b ⎥ cd ⎣ ⎦ A [A][A] ­1 = [A] ­1[A] = I, where I = iden?ty matrix B [A] and [A] ­1 must be nxn C Inverse [A] ­1 of a 2x2 matrix [ A ]−1 = 1 ⎡ d −b ⎤ 1 ⎡ d −b ⎤ ⎢ ⎥= ⎢ ⎥ ad − bc ⎣ − c a ⎦ A ⎣ − c a ⎦ D Inverse [A] ­1 of a 3x3 matrix also requires determinant to be non ­zero 10/19/11 GG303 6 3 10/19/11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN III Determinant |A| of a real matrix |A| A A number that provides scaling informa?on on a square matrix B Determinant of a 2x2 matrix ⎡a b⎤ A=⎢ ⎥ , A = ad − bc ⎣c d⎦ Akin to: Cross product (an area) Scalar triple product (a volume) C Determinant of a 3x3 matrix: ⎡a b ⎢ A=⎢ d e ⎢g h ⎣ 10/19/11 c⎤ e ⎥ f ⎥ , A = a h i⎥ ⎦ f i −b d f g i de gh +c GG303 7 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN III Determinant (cont.) D Geometric meanings of the real matrix equa?on AX = B = 0 1 |A| ≠ 0 ; a [A] ­1 exists b Describes two lines (or 3 planes) that intersect at the origin c X has a unique solu?on 2  |A| = 0 ; a [A] ­1 does not exist b Describes two co ­linear lines that that pass through the origin (or three planes that intersect a line or plane through the origin) c X has no unique solu?on 10/19/11 GG303 Intersecting lines have non-parallel normals AX = B = 0 nx(1) nx(2) ny(1) ny(2) x y = d1=0 d2=0 |A| = nx(1) * ny(2) - ny(1) * nx(2) ≠ 0 n1 x n2 ≠ 0 Parallel lines have parallel normals AX = B = 0 nx(1) nx(2) ny(1) ny(2) x y = d1=0 d2=0 |A| = nx(1) * ny(2) - ny(1) * nx(2) = 0 n1 x n2 = 0 8 4 10/19/11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN III Eigenvectors and eigenvalues A Eigenvalue problems involve the matrix equa?on AX = λX 1  A is a (known) square matrix (nxn) 2  X is a non ­zero direc?onal eigenvector (nx1) 3  λ is a number, an eigenvalue 4  λX is a vector (nx1) 5  AX is a vector (nx1) 6  Eigenvectors (X) maintain their orienta(on when mul?plied by matrix A 7  If X is a unit vector, λ is the length of AX 10/19/11 GG303 9 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN III Eigenvalue problems, eigenvectors and eigenvalues (cont.) B Eigenvectors have corresponding eigenvalues, and vice ­versa C In Matlab, [v,d] = eig(A), ﬁnds eigenvectors (v) and eigenvalues (d) 10/19/11 GG303 10 5 10/19/11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN III Eigenvalue problems, eigenvectors and eigenvalues (cont.) D Examples ⎡ 1 0 ⎤⎡ x ⎤ ⎡ x 1  Iden?ty matrix (I) ⎢ 0 1 ⎥ ⎢ y ⎥ = ⎢ y ⎣ ⎢ ⎦⎣ ⎥ ⎦ ⎢ ⎣ ⎤ ⎡x⎤ ⎥ = 1⎢ ⎥ ⎥ ⎢y⎦ ⎥ ⎦ ⎣ All vectors in the xy ­plane maintain their orienta?on when operated on by the iden?ty matrix, so all vectors are eigenvectors, and all vectors maintain their length, so all eigenvalues for I equal 1 10/19/11 GG303 11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN III Eigenvalue problems, eigenvectors and eigenvalues (cont.) D Examples 2  A matrix for rota?ons in the xy plane ⎡ cos ω ⎢ ⎣ − sin ω ⎡x⎤ cos ω ⎤ ⎡ x ⎤ ⎥⎢ y ⎥ = λ ⎢ y ⎥ cos ω ⎦ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ All non ­zero real vectors rotate; a 2D rota?on matrix has no real eigenvectors and hence no real eigenvalues 10/19/11 GG303 12 6 10/19/11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN III Eigenvalue problems, eigenvectors and eigenvalues (cont.) D Examples 3  A 3D rota?on matrix a The only vectors that are not rotated are along the axis of rota?on b The real eigenvectors of a 3D rota?on matrix give the orienta?on of the axis of rota?on c The rota?on does not change the length of vectors, so the real eigenvalues equal 1 10/19/11 GG303 13 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN III Eigenvalue problems, eigenvectors and eigenvalues (cont.) D Examples Eigenvalue ⎡ ⎤ A 4 = ⎢ 0 2 ⎥ 20 ⎣ Eigenvector ⎦ ⎡ 1 ⎤ ⎡ 0 2 ⎤⎡ 1 ⎤ ⎡ 2 ⎤ ⎡1⎤ A⎢ ⎥=⎢ ⎥⎢ ⎥=⎢ ⎥ = 2⎢ ⎥ ⎣ 1 ⎦ ⎣ 2 0 ⎦⎣ 1 ⎦ ⎣ 2 ⎦ ⎣1⎦ ⎡ 1 ⎤ ⎡ 0 2 ⎤ ⎡ 1 ⎤ ⎡ −2 ⎤ ⎡1⎤ A⎢ ⎥=⎢ ⎥⎢ ⎥=⎢ ⎥ = −2 ⎢ ⎥ ⎣ −1 ⎦ ⎣ 2 0 ⎦ ⎣ −1 ⎦ ⎣ 2 ⎦ ⎣ −1 ⎦ 10/19/11 GG303 Eigenvalue Eigenvector 14 7 10/19/11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN III Eigenvalue problems, eigenvectors and eigenvalues (cont.) Eigenvalues D Examples ⎡ Eigenvectors 3⎤ A 5 = ⎢ 9 1 ⎥ 3 ⎣ ⎦ ⎡ −3 0.1 ⎤ ⎡ 9 3 ⎤ ⎡ −3 0.1 ⎤ ⎡ −30 0.1 ⎤ ⎡ −3 0.1 ⎤ ⎥=⎢ ⎥=⎢ ⎥ = 10 ⎢ ⎥ A⎢ ⎥⎢ ⎢ − 0.1 ⎥ ⎣ 3 1 ⎦ ⎢ − 0.1 ⎥ ⎢ −10 0.1 ⎥ ⎢ − 0.1 ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎡ 0.1 A⎢ ⎢ −3 0.1 ⎣ 10/19/11 ⎤ ⎡ 9 3 ⎤⎡ 0.1 ⎥=⎢ ⎥⎢ ⎥ ⎣ 3 1 ⎦ ⎢ −3 0.1 ⎦ ⎣ ⎤ ⎡0⎤ ⎡ 0.1 ⎥=⎢ ⎥ = 0⎢ ⎥ ⎣0⎦ ⎢ −3 0.1 ⎦ ⎣ GG303 ⎤ ⎥ ⎥ ⎦ 15 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN III Eigenvalue problems, eigenvectors and eigenvalues (cont.) E Alterna?ve form of an eigenvalue equa?on 1 [A][X]=λ[X] Subtrac?ng λ[IX] = λ[X] from both sides yields: 2 [A ­Iλ][X]=0 (same form as [A][X]=0) F Solu?on condi?ons and connec?ons with determinants 1 Unique trivial solu?on of [X] = 0 if and only if |A ­Iλ|≠0 2 Eigenvector solu?ons ([X] ≠ 0) if and only if |A ­Iλ|=0 10/19/11 GG303 16 8 10/19/11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN III Eigenvalue problems, eigenvectors and eigenvalues (cont.) J Characteris?c equa?on: |A ­Iλ|=0 1 The roots of the characteris?c equa?on are the eigenvalues 10/19/11 GG303 17 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN III Eigenvalue problems, eigenvectors and eigenvalues (cont.) J Characteris?c equa?on: |A ­Iλ|=0 ⎡ ⎤ 2 Eigenvalues of a general 2x2 matrix A = ⎢ a b ⎥ ⎣c d⎦ a A − I λ = a − I c b b =0 d−λ ( a − λ ) ( d − λ ) − bc = 0 c λ 2 − ( a + d ) λ + ( ad − bc ) = 0 d λ1 , λ2 = 10/19/11 (a+d) = tr(A) (ad ­bc) = |A| ( a + d ) ± ( a + d )2 − 4 ( ad − bc ) 2 GG303 18 9 10/19/11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN III Eigenvalue problems, eigenvectors and eigenvalues (cont.) J Characteris?c equa?on: |A ­Iλ|=0 3 Eigenvalues of a symmetric 2x2 matrix a λ1 , λ2 = b λ1 , λ2 = c λ1 , λ2 = d 10/19/11 ( a + d ) ± ( a + d )2 − 4 ( ad − b 2 ) ⎡a b⎤ A=⎢ ⎥ ⎣b d⎦ 2 ( a + d ) ± ( a + 2ad + d )2 − 4 ad + 4b 2 2 ( a + d ) ± ( a − 2ad + d )2 + 4b 2 λ1 , λ2 = 2 ( a + d ) ± ( a − d )2 + 4 b 2 2 GG303 Radical term cannot be nega?ve. Eigenvalues are real. 19 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN III Eigenvalue problems, eigenvectors and eigenvalues (cont.) L Dis?nct eigenvectors (X1, X2) of a symmetric 2x2 matrix are perpendicular (X1 • X2 = 0) 1a AX1 =λ1X1 1b AX2 =λ2X2 AX1 parallels X1, AX2 parallels X2 (property of eigenvectors) Donng AX1 by X2 and AX2 by X1 can test whether X1 and X2 are orthogonal. 2a X2•AX1 = X2•λ1X1 = λ1 (X2•X1) 2b X1•AX2 = X1•λ2X2 = λ2 (X1•X2) 10/19/11 GG303 20 10 10/19/11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN L Dis?nct eigenvectors (X1, X2) of a symmetric 2x2 matrix are perpendicular (X1 • X2 = 0) (cont.) The lep sides of 2a and 2b are equal 3a 3b 10/19/11 ⎡ x1 ⎤ ⎡ a b ⎤ ⎡ x2 ⎤ ⎡ x1 ⎤ ⎡ ax2 + by2 ⎢ ⎥• ⎢ ⎥=⎢ ⎥• ⎢ ⎥⎢ ⎢ y1 ⎥ ⎣ b d ⎦ ⎢ y2 ⎥ ⎢ y1 ⎥ ⎢ bx2 + dy2 ⎣ ⎦ ⎣ ⎦⎣ ⎦⎣ = ax1 x2 + bx1 y2 + by1 x2 + dy1 y2 ⎤ ⎥ ⎥ ⎦ ⎡ x2 ⎤ ⎡ a b ⎤ ⎡ x1 ⎤ ⎡ x2 ⎤ ⎡ ax1 + by1 ⎢ ⎥• ⎢ ⎥=⎢ ⎥• ⎢ ⎥⎢ ⎢ y2 ⎥ ⎣ b d ⎦ ⎢ y1 ⎥ ⎢ y2 ⎥ ⎢ bx1 + dy1 ⎣ ⎦ ⎣ ⎦⎣ ⎦⎣ = ax1 x2 + by1 x2 + bx1 y2 + dy1 y2 ⎤ ⎥ ⎥ ⎦ GG303 21 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN L Dis?nct eigenvectors (X1, X2) of a symmetric 2x2 matrix are perpendicular Since the lep sides of (2a) and (2b) are equal, the right sides must be equal too. Hence, 4  λ1 (X2•X1) =λ2 (X1•X2) Now subtract the right side of (4) from the lep 5  (λ1 – λ2)(X2•X1) =0 • The eigenvalues generally are diﬀerent, so λ1 – λ2 ≠ 0. • This means for (5) to hold that X2•X1 =0. • Therefore, the eigenvectors (X1, X2) of a symmetric 2x2 matrix are perpendicular 10/19/11 GG303 22 11 10/19/11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN IV Diagonaliza?on of real symmetric nxn matrices A A real symmetric matrix [A] can be diagonalized (converted to a matrix with zeros for all elements oﬀ the main diagonal) by pre ­ mul?plying by the inverse of the matrix of its eigenvectors and post ­ mul?plying by the matrix of its eigenvectors. The resul?ng diagonal matrix [Λ] contains eigenvalues along the main diagonal. [X 1 : X2 ] −1 ⎡ λ1 [ A ][ X1 : X2 ] = ⎢ 10/19/11 ⎢0 ⎣ 0⎤ ⎥ = [ Λ] λ2 ⎥ ⎦ GG303 23 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN IV Diagonaliza?on (cont.) B Proof “Bookshelf” stack the eigenvectors to form a square matrix S 1 [ S ] = [ X1 : X2 ] Post ­mul?ply S by A, recalling that AX = λX ⎡ λ1 0 ⎤ ⎥ = [ S ][ Λ ] 2 [ A ][ S ] = [ A ][ X1 : X2 ] = [ λ1 X1 : λ2 X2 ] = [ X1 : X2 ] ⎢ ⎢ 0 λ2 ⎥ ⎣ ⎦ Eigenvector matrix 10/19/11 GG303 Eigenvalue matrix 24 12 10/19/11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN IV Diagonaliza?on (cont.) B Proof “Bookshelf” stack the eigenvectors to form a square matrix S [ 1 [ S ] = X1 : X 2 Post ­mul?ply S by A, recalling that AX = λX 2 [ A ][ S ] = [ S ][ Λ ] Eigenvalue matrix Eigenvector matrix 10/19/11 Pre ­mul?ply both sides of (2) by S ­1 to get Λ 3 ⎡ S −1 ⎤ [ A ][ S ] = ⎡ S −1 ⎤ [ S ][ Λ ] = [ Λ ] ⎣⎦ ⎣⎦ Post ­mul?ply both sides of (2) by S ­1 to get A −1 −1 4 [ A ][ S ] ⎡ S ⎤ = [ S ][ Λ ] ⎡ S ⎤ = [ A ] ⎣⎦ ⎣⎦ So if we can ﬁnd the eigenvectors (the principal direc?ons) we can ﬁnd the principal values of A GG303 25 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN Λ = ⎡ 2 0 ⎤ ⎢ ⎥ IV Diagonaliza?on (cont.) C Example (see III.D) ⎡0 2⎤ ⎥ ⎣2 0⎦ ⎣ 0 −2 ⎦ [ A] = ⎢ ⎡ ⎤ [ S ] = [ X1 : X2 ] = ⎢ 1 −1 ⎥ ⎣1 1 ⎦ [X 1 ⎡ 1 −1 ⎤ : X2 ] = ⎢ ⎥ ⎣1 1 ⎦ ⎡ 0 2 ⎤ ⎡ 1 −1 ⎤ ⎡ 2 2 ⎤ ⎡ 1 −1 ⎤ ⎡ 2 0 ⎤ ⎥⎢ ⎥=⎢ ⎥ = [ SΛ ] = ⎢ ⎥⎢ ⎥ ⎣ 2 0 ⎦ ⎣ 1 1 ⎦ ⎣ 2 −2 ⎦ ⎣ 1 1 ⎦ ⎣ 0 −2 ⎦ [ A ][ S ] = ⎢ 1 ⎡ 1 1 ⎤⎡ 2 2 ⎤ ⎡ 2 0 ⎤ ⎡ S −1 ⎤ [ A ][ S ] = ⎢ ⎥⎢ ⎥=⎢ ⎥ = [ Λ] ⎣⎦ 2 ⎣ −1 1 ⎦ ⎣ 2 −2 ⎦ ⎣ 0 −2 ⎦ ✔ ⎡ 2 2 ⎤1⎡ 1 1 ⎤ ⎡ 0 2 ⎤ ⎥⎢ ⎥=⎢ ⎥ = [ A ] ✔ ⎣ 2 −2 ⎦ 2 ⎣ −1 1 ⎦ ⎣ 2 0 ⎦ [ A ] = [ S Λ ] ⎡ S −1 ⎤ = ⎢ ⎣⎦ S and S ­1 are analogous (or iden?cal) to rota?on matrices, so diagonaliza?on apparently can occur if the reference frame is rotated 10/19/11 GG303 26 13 10/19/11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Principal Axes Theorem (Spectral Theorem) in Two Dimensions A A “quadra?c (second ­order) form” with no linear terms (e.g., ax2 2bxy + dy2) in one reference frame (e.g., the xy frame) can be wri:en in a simpler form (e.g., λ1x’2 + λ2y’2 ) in a diﬀerent reference frame such that it has a clear geometric meaning and takes advantage of symmetry. The x’ and y ’ axes are the principal axes (eigenvectors), and λ1 and λ2 are the principal values (or eigenvalues). 10/19/11 y x’ λ1 y’ λ2 x GG303 27 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Principal Axes Theorem (Spectral Theorem) in Two Dimensions B If λ1 >0 and λ2 >0, then the new quadra?c form is that of an ellipse. The semi ­axes of an ellipse are the maximum and minimum lengths of line segments connec?ng the center of an ellipse to its perimeter, providing a way to ﬁnd maxima and minima (and their direc?ons) without resor?ng to calculus. 10/19/11 GG303 y x’ λ1 y’ λ2 x 28 14 10/19/11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Principal Axes Theorem (Spectral Theorem) in Two Dimensions C General Example: Equa?on of an ellipse centered at the origin 1 ax 2 + 2bxy + dy 2 = 1 ⎡ a b ⎤⎡ x ⎤ T ⎥ ⎢ y ⎥ = [ X ] [ A ][ X ] = 1 ⎥ ⎣ b d ⎦⎢ ⎣ ⎦ T T T 3 [ X ] [ A ][ X ] = [ X ] ⎡ S ΛS −1 ⎤ [ X ] = [ X ] ⎡ S ΛS T ⎤ [ X ] = 1 ⎣ ⎦ ⎣ ⎦ 2 [ x y ] ⎢ 4 [ X ] [ A ][ X ] = ⎡ X T S ⎤ [ Λ ] ⎡ S T X ⎤ = ⎡ S T X ⎤ [ Λ ] ⎡ S T X ⎤ = 1 ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ T T 10/19/11 GG303 29 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Principal Axes Theorem (Spectral Theorem) in Two Dimensions C General Example: Equa?on of an ellipse centered at the origin (cont.) T T 5 [ X ] [ A ][ X ] = ⎡ S X ⎤ [ Λ ] ⎡ S X ⎤ = [ X ′ ] [ Λ ][ X ′ ] = 1 ⎣ ⎦ ⎣ ⎦ T T T So the equa?on of an ellipse can be wriwen as 2 2 ⎛ ⎞⎛ ⎞ ⎡ λ1 0 ⎤ ⎡ x ⎤ ⎜ x ′ ⎟ + ⎜ y′ ⎟ = 1 2 2 ⎢ ⎥⎢ 6 [ x′ y′ ] 0 λ ⎥ = λ1 x ′ + λ2 y′ = ⎜ 1 ⎟ ⎜1 ⎟ y⎥ ⎢ 2 ⎥⎣ ⎦ ⎜ ⎣ ⎦⎢ λ1 ⎟ ⎜ λ2 ⎟ ⎝ ⎠⎝ ⎠ 10/19/11 GG303 30 15 10/19/11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Principal Axes Theorem (Spectral Theorem) in Two Dimensions D Meaning of solu?on of [F][dX] = [λ][dX] if F is a symmetric 2x2 matrix 1  Solu?ons for λ give the maximum and minimum distance from the origin 2 The eigenvector solu?ons for dX give the direc?ons of the principal axes of the ellipse and the direc?ons of lines that maintain their orienta?on as a unit circle deforms to an ellipse 10/19/11 GG303 31 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN VI Strain ellipses, strain ellipsoids, and principal strains (homogeneous strain) A The strain ellipse and strain ellipsoid 1 A unit circle/sphere in the undeformed state deforms to an ellipse/ellipsoid called the strain ellipse/ellipsoid 2 The strain ellipse characterizes 2 ­D strain at a posi?on in space and a point it ?me; it can vary with x,y,z, and t (?me). 3 A shape change due to non ­ deforma?onal processes (e.g., chemical precipita?on) is not a strain. 10/19/11 GG303 32 16 10/19/11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN VI Strain ellipses, strain ellipsoids, and principal strains A The strain ellipse and strain ellipsoid 4 Characteriza?on of the strain ellipse a An ellipse has a major semi ­ axis (a) and a minor semi ­axis (b). An ellipse can be characterized by the (rela?ve) length, orienta?on, and rota?on of these axes 10/19/11 GG303 33 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN VI Strain ellipses, strain ellipsoids, and principal strains A The strain ellipse and strain ellipsoid 4 Characteriza?on of the strain ellipse b The eigenvectors of a symmetric F ­matrix give the principal axes of the strain ellipse, but the eigenvectors of a nonsymmetric F ­ matrix, represen?ng irrota?onal strain, do not (see diagram on next page). To ﬁnd the principal strains from a nonsymmetric F ­ matrix, the rota?on of eigenvectors needs to be accounted for; one seeks the eigenvalues and eigenvectors for a symmetric matrix that 10/19/11 GG303 corresponds to F. 34 17 10/19/11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN B The reciprocal strain ellipse 1 For homogeneous strain, a unit circle in the undeformed state“retro ­ deforms” to an ellipse (the reciprocal strain ellipse). 2 For homogeneous strain, a unit sphere in the undeformed state“retro ­ deforms” to an ellipsoid (the reciprocal strain ellipsoid). 10/19/11 GG303 35 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN B The reciprocal strain ellipse (cont.) 3  The eigenvectors for a deforma?on with a symmetric F ­matrix will be in the orienta?on of the axes of the strain ellipse, but this will not be the case for deforma?on described by a non ­symmetric F ­matrix . 4  In general the axes of a unit circle that transform to the axes of the strain ellipse will be stretched and rotated. 5  Another way to say this is that the axes of reciprocal strain generally will have an orienta?on diﬀerent from the axes of the strain ellipse. The rota?on of the axes of the strain ellipse refers to the rota?on needed to bring the axes of the reciprocal strain ellipse into coincidence with the axes of the strain ellipse. 10/19/11 GG303 36 18 10/19/11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN C Strain ellipse for general homogeneous rota?onal strain • Decomposi?on of F = VR* by method of Ramsay and Huber (for 2D). Consider the eﬀect of an irrota?onal strain that follows a pure rigid rota?on of the object (not a rigid rota?on of the reference frame) ⎡ a b ⎤ ⎡ A B ⎤ ⎡ cos ω F=⎢ ⎥=⎢ ⎥⎢ ⎣ c d ⎦ ⎣ B D ⎦ ⎣ sin ω 10/19/11 − sin ω ⎤ * ⎥ = VR cos ω ⎦ GG303 37 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN C Strain ellipse for general homo. rota?onal strain ⎡ a b ⎤ ⎡ A B ⎤ ⎡ cos ω ⎥=⎢ ⎥⎢ ⎣ c d ⎦ ⎣ B D ⎦ ⎣ sin ω 1 F = ⎢ a b ⎤ ⎡ A cos ω + B sin ω ⎥=⎢ ⎣ c d ⎦ ⎣ B cos ω + D sin ω 2 ⎡ ⎢ − sin ω ⎤ * ⎥ = VR cos ω ⎦ − A sin ω + B cos ω ⎤ ⎥ − B sin ω + D cos ω ⎦ c ­b = (A+D)sinω, and a+d = (A+D)cosω c−b 3 a + d = a n ω If c=b (F is symmetric), then ω=0 t From 3 one can obtain R*. 10/19/11 GG303 38 19 10/19/11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN C Strain ellipse for general homo. rota?onal strain Post ­mul?plying both sides of (1) by [R*] ­1 = R*T yields V F = VR* F[R*] ­1 = VR*[R*] ­1 = VR*[R*]T = V ⎡ a b ⎤ ⎡ cos ω ⎢ ⎥⎢ ⎣ c d ⎦ ⎣ sin ω 10/19/11 −1 ⎡ a b ⎤ ⎡ cos ω − sin ω ⎤ ⎥ =⎢ ⎥⎢ cos ω ⎦ ⎣ c d ⎦ ⎣ − sin ω GG303 sin ω ⎤ ⎡ A B ⎤ ⎥=⎢ ⎥=V cos ω ⎦ ⎣ B D ⎦ 39 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN G Closing comments 1 Our solu?ons so far depend on knowing the displacement ﬁeld. 2 With satellite imaging we can get an approximate value for the displacement ﬁeld at the surface of the Earth for current deforma?ons 3 Evalua?ng strains for past deforma?ons require certain assump?ons about ini?al sizes and shapes of bodies or the displacement ﬁeld. 4 Alterna?ve approach: formula?on and solu?on of boundary value problems to solve for the displacement and strain ﬁelds. 5 References a Ramsay, J.G., and Huber, M.I., 1983, The techniques of modern structural geology, volume 1: strain analysis: Academic Press, London, 307 p. (See equa?ons of sec?on 5, p. 291). b Ramsay, J.G., and Lisle, M.I., 1983, The techniques of modern structural geology, volume 3: applica?ons of con?nuum mechanics in structural geology: Academic Press, London, 307 p. (See especially sessions 33 and 36). c Malvern, L.E., 1969, Introduc?on to the mechanics of a con?nuous medium: Pren?ce ­Hall, Englewood Cliﬀs, New Jersey, 713 p. (See equa?ons 4.6.1, 4.6.3 a, 4.6.3b on p. 172 ­174).) 10/19/11 GG303 40 20 10/19/11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN Appendix 2 Geometric meaning of a real symmetric 2 x 2 F matrix −1 1a [ X ′ ] = [ A ][ X ] 1b [ A ] [ X ′ ] = [ X ] If the points (x,y) lie along a unit circle 2a ⎡x⎤ T 2b [ x y ] ⎢ y ⎥ = 1 2c [ X ] [ X ] = 1 x 2 + y 2 = 1 ⎢ ⎣ ⎥ ⎦ Subs?tu?ng (1b) into (2c) yields ⎡ 3a ⎡[ A ] [ X ′ ] ⎤ ⎡ [ A ] [ X ′ ]⎤ = 1 3b [ X ′ ] ⎦ ⎣ ⎦ ⎣ ⎣ −1 T −1 10/19/11 T T −1 ⎡ A −1 ⎤ ⎤ ⎡[ A ] [ X ′ ]⎤ = 1 ⎣ ⎦ ⎦⎣ ⎦ GG303 41 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN Appendix 2 Geometric meaning of a real symmetric 2 x 2 F matrix For a symmetric matrix ⎡a b⎤ ⎥ ⎣b d⎦ [ A] = ⎢ [ A ] = [ A ]T [ A ]−1 = 1 ⎡ d −b ⎤ ⎥ 2⎢ A ⎣ −b a ⎦ By inspec?on [A ­1] = [A ­1]T, so T −1 −1⎦ ⎣ T ⎣ −1 ⎦ ⎦ ⎣ −1 ⎦ 4 ⎡[ X ′ ] ⎡ A ⎤ ⎤ ⎡[ A ] [ X ′ ⎤ = ⎡[ X ′ ] ⎡ A ⎤ ⎤ ⎡[ A ] [ X ′ ]⎤ = 1 ⎣ ⎦ ⎦ ⎣ ⎣ T 10/19/11 GG303 42 21 10/19/11 9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN Appendix 2 Geometric meaning of a real symmetric 2 x 2 matrix 2 2 ⎡ ⎡ − db − ba ⎤ ⎡ x ′ ⎤ ⎤ 5 1 ⎢[ x ′ y′ ]T d + b ⎥⎢ ⎥⎥ = 1 2 2 2 A⎢ ⎥⎦ ⎢ ⎥ ⎢ y′ ⎦ ⎥ ⎣ − db − ba a + b ⎦ ⎣ ⎣ ( ) 6 1 2 2 2 2 2 2 2 x ′ ( d + b ) − 2 x ′y′ ( db + ba ) + y′ ( a + b ) = 1 A Since the terms mul?plying just x’ and just y’ are both posi?ve, this equa?on describes an ellipse. For a real, symmetric matrix, the eigenvectors of the matrix coincide with the major and minor axes of the ellipse. 10/19/11 GG303 43 22 ...
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