10/15/12 1 9. Eigenvectors, Eigenvalues, and Finite

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN I Main Topics A Movaon B Inverse [A]-1 of a real matrix A C Determinant |A| of a real matrix A D Eigenvectors and eigenvalues (Principal direcons and principal values) E Diagonalizaon of a matrix F Principal axes theorem (maxima/minima) in 2D G Strain ellipses, strain ellipsoids, and principal strains

H Principal values and principal direcons for symmetric and non-symmetric matrices * Appendices 1 and 2

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN

A forward deformaon deforms a unit circle to a strain ellipse

Objecve: To describe size and shape of strain ellipse

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• For irrotaonal deformaon, the axes which transform to become the principal strain axes (dashed) are not rotated in either the forward deformaon or in the inverse deformaon

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• For rotaonal deformaon, the inverse deformaon 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 orientaon of the principal strain axes (solid) and the retro- deformed axes (dashed) in the inial state is the rotaon.

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II Movaon A Eigenvectors and eigenvalues provide simple, elegant, and clear ways to solve many scienﬁc problems [e.g., geometry, strain, stress, curvature (shapes of surfaces)] B To ﬁnd the magnitudes and direcons of the principal strains using linear algebra rather than graphical means or calculus (e.g., Ramsay and Huber, 1984)

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN

-1 ⎡ a b ⎤ III Inverse [A] of a real matrix A [A] = ⎢ ⎥ ⎣ c d ⎦

A [A][A]-1 = [A]-1[A] = I, where I = identy matrix B [A] and [A]-1 must be nxn C Inverse [A]-1 of a 2x2 matrix

−1 1 ⎡ d −b ⎤ 1 ⎡ d −b ⎤ [A] = ⎢ ⎥ = ⎢ ⎥ ad − bc ⎣ −c a ⎦ A ⎣ −c a ⎦ D Inverse [A]-1 of a 3x3 matrix also requires determinant to be non-zero

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN IV Determinant |A| of a real matrix |A| A A number that provides scaling informaon on a square matrix B Determinant of a 2x2 matrix ⎡ a b ⎤ Akin to: A = ⎢ ⎥, A = ad − bc Cross product (an area) ⎣ c d ⎦ Scalar triple product (a volume) C Determinant of a 3x3 matrix: ⎡ a b c ⎤ ⎢ ⎥ e f d f d e A = d e f , A = a − b + c ⎢ ⎥ h i g i g h ⎢ g h i ⎥ ⎣ ⎦ 10/15/12 GG303 7

9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN

IV Determinant (cont.) Intersecting lines have non-parallel normals D Geometric meanings of the real AX = B = 0 matrix equaon AX = B = 0 (1) (1) nx ny x d1=0 1 |A| ≠ 0 ; (2) (2) = nx ny y d2=0 a [A]-1 exists

b Describes two lines (or 3 (1) (2) (1) (2) |A| = nx * ny - ny * nx ≠ 0 planes) that intersect at n x n ≠ 0 the origin 1 2 c X has a unique soluon 2 |A| = 0 ; Parallel lines have parallel normals a [A]-1 does not exist AX = B = 0 (1) (1) b Describes two co-linear nx ny x d1=0 (2) (2) = lines that that pass nx ny y d2=0 through the origin (or three planes that intersect (1) (2) (1) (2) a line or plane through |A| = nx * ny - ny * nx = 0 the origin) n1 x n2 = 0 c X has no unique soluon

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Eigenvectors and eigenvalues A Eigenvalue problems involve the matrix equaon AX = λX 1 A is a (known) square matrix (nxn) 2 X is a non-zero direconal 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 orientaon when mulplied by matrix A 7 If X is a unit vector, λ is the length of AX

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V 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)

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Eigenvalue problems, eigenvectors and eigenvalues (cont.) D Examples ⎡ 1 0 ⎤ ⎡ x ⎤ ⎡ x ⎤ ⎡ x ⎤ = = 1 1 Identy matrix (I) ⎢ ⎥ ⎢ y ⎥ ⎢ y ⎥ ⎢ y ⎥ ⎣ 0 1 ⎦ ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ All vectors in the xy-plane maintain their orientaon when operated on by the identy matrix, so all vectors are eigenvectors, and all vectors maintain their length, so all eigenvalues for I equal 1

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Eigenvalue problems, eigenvectors and eigenvalues (cont.) D Examples 2 A matrix for rotaons in the xy plane ⎡ cosω cosω ⎤ ⎡ x ⎤ ⎡ x ⎤ = λ ⎢ ⎥ ⎢ y ⎥ ⎢ y ⎥ ⎣ −sinω cosω ⎦ ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ All non-zero real vectors rotate; a 2D rotaon matrix has no real eigenvectors and hence no real eigenvalues

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Eigenvalue problems, eigenvectors and eigenvalues (cont.) D Examples 3 A 3D rotaon matrix a The only vectors that are not rotated are along the axis of rotaon b The real eigenvectors of a 3D rotaon matrix give the orientaon of the axis of rotaon c The rotaon does not change the length of vectors, so the real eigenvalues equal 1

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Eigenvalue problems, eigenvectors and eigenvalues (cont.) D Examples ⎡ 0 2 ⎤ Eigenvalue 4 A = ⎢ ⎥ ⎣ 2 0 ⎦ Eigenvector

⎡ 1 ⎤ ⎡ 0 2 ⎤ ⎡ 1 ⎤ ⎡ 2 ⎤ ⎡ 1 ⎤ A ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ = 2 ⎢ ⎥ Eigenvalue ⎣ 1 ⎦ ⎣ 2 0 ⎦ ⎣ 1 ⎦ ⎣ 2 ⎦ ⎣ 1 ⎦ Eigenvector ⎡ 1 ⎤ ⎡ 0 2 ⎤ ⎡ 1 ⎤ ⎡ −2 ⎤ ⎡ 1 ⎤ A ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ = −2 ⎢ ⎥ ⎣ −1 ⎦ ⎣ 2 0 ⎦ ⎣ −1 ⎦ ⎣ 2 ⎦ ⎣ −1 ⎦

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Eigenvalue problems, eigenvectors and eigenvalues (cont.) Eigenvalues D Examples ⎡ 9 3 ⎤ Eigenvectors 5 A = ⎢ ⎥ ⎣ 3 1 ⎦

⎡ −3 0.1 ⎤ ⎡ 9 3 ⎤ ⎡ −3 0.1 ⎤ ⎡ −30 0.1 ⎤ ⎡ −3 0.1 ⎤ A ⎢ ⎥ = ⎢ ⎥ = ⎢ ⎥ = 10 ⎢ ⎥ ⎢ 3 1 ⎥ ⎣⎢ − 0.1 ⎦⎥ ⎣ ⎦ ⎣⎢ − 0.1 ⎦⎥ ⎣⎢ −10 0.1 ⎦⎥ ⎣⎢ − 0.1 ⎦⎥

⎡ 0.1 ⎤ ⎡ 9 3 ⎤ ⎡ 0.1 ⎤ ⎡ 0 ⎤ ⎡ 0.1 ⎤ A ⎢ ⎥ = ⎢ ⎥ = = 0 ⎢ ⎥ ⎢ 3 1 ⎥ ⎢ 0 ⎥ ⎣⎢ −3 0.1 ⎦⎥ ⎣ ⎦ ⎣⎢ −3 0.1 ⎦⎥ ⎣ ⎦ ⎣⎢ −3 0.1 ⎦⎥

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Eigenvalue problems, eigenvectors and eigenvalues (cont.) E Alternave form of an eigenvalue equaon 1 [A][X]=λ[X] Subtracng λ[IX] = λ[X] from both sides yields: 2 [A-Iλ][X]=0 (same form as [A][X]=0) F Soluon condions and connecons with determinants 1 Unique trivial soluon of [X] = 0 if and only if |A-Iλ|≠0 2 Eigenvector soluons ([X] ≠ 0) if and only if |A-Iλ|=0

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Eigenvalue problems, eigenvectors and eigenvalues (cont.) J Characterisc equaon: |A-Iλ|=0 1 The roots of the characterisc equaon are the eigenvalues

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Eigenvalue problems, eigenvectors and eigenvalues (cont.) J Characterisc equaon: |A-Iλ|=0 ⎡ a b ⎤ 2 Eigenvalues of a general 2x2 matrix A = ⎢ ⎥ ⎣ c d ⎦ a − I b a A − Iλ = = 0 c d − λ b (a − λ)(d − λ) − bc = 0 (a+d) = tr(A) (ad-bc) = |A| c λ 2 − (a + d)λ + (ad − bc) = 0

2 (a + d) ± (a + d) − 4(ad − bc) d λ ,λ = 1 2 2

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN V Eigenvalue problems, eigenvectors and eigenvalues (cont.) J Characterisc equaon: |A-Iλ|=0 ⎡ a b ⎤ 3 Eigenvalues of a symmetric 2x2 matrix A = ⎢ ⎥ ⎣ b d ⎦ 2 (a + d) ± (a + d) − 4(ad − b2 ) a λ ,λ = 1 2 2 2 (a + d) ± (a + 2ad + d) − 4ad + 4b2 b λ ,λ = 1 2 2 2 a + d ± a − 2ad + d + 4b2 Radical term ( ) ( ) cannot be c λ1,λ2 = 2 negave. 2 (a + d) ± (a − d) + 4b2 Eigenvalues are d λ ,λ = real. 1 2 2 10/15/12 GG303 19

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V Eigenvalue problems, eigenvectors and eigenvalues (cont.)

K Disnct 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)

Dong 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)

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K Disnct eigenvectors (X1, X2) of a symmetric 2x2 matrix are perpendicular (X1 • X2 = 0) (cont.) The le sides of 2a and 2b are equal

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ x1 ⎡ a b ⎤ x2 x1 ax2 + by2 ⎢ ⎥ • ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ • ⎢ ⎥ y b d y y bx + dy 3a ⎣⎢ 1 ⎦⎥ ⎣ ⎦ ⎣⎢ 2 ⎦⎥ ⎣⎢ 1 ⎦⎥ ⎣⎢ 2 2 ⎦⎥

= ax1x2 + bx1y2 + by1x2 + dy1y2 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ x2 ⎡ a b ⎤ x1 x2 ax1 + by1 3b ⎢ ⎥ • ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ • ⎢ ⎥ y b d y y bx + dy ⎣⎢ 2 ⎦⎥ ⎣ ⎦ ⎣⎢ 1 ⎦⎥ ⎣⎢ 2 ⎦⎥ ⎣⎢ 1 1 ⎦⎥

= ax1x2 + by1x2 + bx1y2 + dy1y2

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K Disnct eigenvectors (X1, X2) of a symmetric 2x2 matrix are perpendicular (cont.) Since the le 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 le

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

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN VI Diagonalizaon 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- mulplying by the inverse of the matrix of its eigenvectors and post- mulplying by the matrix of its eigenvectors. The resulng diagonal matrix [Λ] contains eigenvalues along the main diagonal. ⎡ ⎤ −1 λ1 0 [X1 : X2 ] [A][X1 : X2 ] = ⎢ ⎥ = [Λ] 0 λ ⎣⎢ 2 ⎦⎥

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN VI Diagonalizaon (cont.) B Proof “Bookshelf” stack the eigenvectors to form a square matrix S

1 [S] = [X1 : X2 ] Eigenvector matrix Post-mulply S by A, recalling that AX = λX ⎡ ⎤ λ1 0 2 [A][S] = [A][X1 : X2 ] = [λ1X1 : λ2 X2 ] = [X1 : X2 ]⎢ ⎥ = [S][Λ] 0 λ ⎣⎢ 2 ⎦⎥ Eigenvector Eigenvalue matrix matrix 10/15/12 GG303 24

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN VI Diagonalizaon (cont.) Pre-mulply both sides of (2) by B Proof S-1 and then use (2) to get Λ “Bookshelf” stack the −1 −1 eigenvectors to form a 3 ⎣⎡S ⎦⎤[A][S] = ⎣⎡S ⎦⎤[S][Λ] = [Λ] square matrix S

1 [S] = [X1 : X2 ] Post-mulply both sides of (2) by S-1 and again then use (2) to get A Post-mulply S by A, −1 −1 recalling that AX = λX 4 [A][S]⎣⎡S ⎦⎤ = [S][Λ]⎣⎡S ⎦⎤ = [A] 2 [A][S] = [S][Λ] So if we can ﬁnd the eigenvectors (the principal direcons) we can Eigenvector Eigenvalue ﬁnd the principal values of A matrix matrix 10/15/12 GG303 25

9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN ⎡ 2 0 ⎤ Λ = ⎢ ⎥ 0 −2 VI Diagonalizaon (cont.) ⎡ 0 2 ⎤ ⎣ ⎦ [A] = ⎢ ⎥ C Example (see III.D) ⎣ 2 0 ⎦ ⎡ 1 −1 ⎤ ⎡ ⎤ X1 : X2 = 1 −1 [ ] ⎢ 1 1 ⎥ [S] = [X1 : X2 ] = ⎢ ⎥ ⎣ ⎦ ⎣ 1 1 ⎦ ⎡ 0 2 ⎤ ⎡ 1 −1 ⎤ ⎡ 2 2 ⎤ ⎡ 1 −1 ⎤ ⎡ 2 0 ⎤ [A][S] = ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ = [SΛ] = ⎢ ⎥ ⎢ ⎥ ⎣ 2 0 ⎦ ⎣ 1 1 ⎦ ⎣ 2 −2 ⎦ ⎣ 1 1 ⎦ ⎣ 0 −2 ⎦

−1 1 ⎡ 1 1 ⎤ ⎡ 2 2 ⎤ ⎡ 2 0 ⎤ ⎣⎡S ⎦⎤[A][S] = ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ = [Λ] ✔ 2 ⎣ −1 1 ⎦ ⎣ 2 −2 ⎦ ⎣ 0 −2 ⎦

−1 ⎡ 2 2 ⎤ 1 ⎡ 1 1 ⎤ ⎡ 0 2 ⎤ [A] = [SΛ]⎣⎡S ⎦⎤ = ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ = [A] ✔ ⎣ 2 −2 ⎦ 2 ⎣ −1 1 ⎦ ⎣ 2 0 ⎦ S and S-1 are analogous (or idencal) to rotaon matrices, so diagonalizaon apparently can occur if the reference frame is rotated 10/15/12 GG303 26

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN VII Principal Axes Theorem (Spectral Theorem) in Two Dimensions A A “quadrac (second-order) y form” with no linear terms x’ 2 2 (e.g., ax + 2bxy + dy ) in one λ reference frame (e.g., the xy y’ 1 frame) can be wrien in a λ 2 2 simpler form (e.g., λ1x’ + 2 λ2y’ ) in a diﬀerent reference frame such that it has a clear x 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).

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN VII Principal Axes Theorem in Two Dimensions

B If λ1 >0 and λ2 >0, then the y new quadrac form is that x’ of an ellipse. The semi-axes y’ λ1 of an ellipse are the maximum and minimum λ2 lengths of line segments connecng the center of an x ellipse to its perimeter, providing a way to ﬁnd maxima and minima (and their direcons) without resorng to calculus.

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN VII Principal Axes Theorem in Two Dimensions C General Example: Equaon of an ellipse centered at the origin 1 ax2 + 2bxy + dy2 = 1

⎡ a b ⎤ ⎡ x ⎤ T 2 x y = X A X = 1 [ ]⎢ ⎥ ⎢ y ⎥ [ ] [ ][ ] ⎣ b d ⎦ ⎣⎢ ⎦⎥ T T −1 T T 3 [X] [A][X] = [X] ⎣⎡SΛS ⎦⎤[X] = [X] ⎣⎡SΛS ⎦⎤[X] = 1

T T T T T T 4 [ X] [A][X] = ⎣⎡X S⎦⎤[Λ]⎣⎡S X ⎦⎤ = ⎣⎡S X ⎦⎤ [Λ]⎣⎡S X ⎦⎤ = 1

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN VII Principal Axes Theorem in Two Dimensions C General Example: Equaon of an ellipse centered at the origin (cont.)

T T T T T 5 [X] [A][X] = ⎣⎡S X ⎦⎤ [Λ]⎣⎡S X ⎦⎤ = [X′] [Λ][X′] = 1

So the equaon of an ellipse can be wrien as 2 2 ⎛ ⎞ ⎛ ⎞ ⎡ 0 ⎤ ⎡ ⎤ λ1 x 2 2 ⎜ x′ ⎟ ⎜ y′ ⎟ 6 [x′ y′]⎢ ⎥ ⎢ ⎥ = λ1x′ + λ2 y′ = + = 1 0 λ y ⎜ 1 ⎟ ⎜ 1 ⎟ ⎣⎢ 2 ⎦⎥ ⎣⎢ ⎦⎥ ⎜ ⎟ ⎜ ⎟ ⎝ λ1 ⎠ ⎝ λ2 ⎠

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN VII Principal Axes Theorem in Two Dimensions D Meaning of soluon of [F][dX] = [λ][dX] if F is a symmetric 2x2 matrix 1 Soluons for λ give the maximum and minimum distance from the origin 2 The eigenvector soluons for dX give the direcons of the principal axes of the ellipse and the direcons of lines that maintain their orientaon as a unit circle deforms to an ellipse 10/15/12 GG303 31

9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN VIII 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 posion in space and a point it me; it can vary with x,y,z, and t (me). 3 A shape change due to non- deformaonal processes (e.g., chemical precipitaon) is not a strain.

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN VIII Strain ellipses, strain ellipsoids, and principal strains (cont.) A The strain ellipse and strain ellipsoid 4 Characterizaon of the strain ellipse a An ellipse has a major semi-axis (a) and a minor semi-axis (b). It can be characterized by the (relave) length, orientaon, and rotaon of these axes

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VIII Strain ellipses, strain ellipsoids, and principal strains (cont.) A The strain ellipse and strain ellipsoid 4 Characterizaon 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, represenng irrotaonal strain, do not. c To ﬁnd the principal strain axes from a nonsymmetric F- matrix, the rotaon of the axes needs to be accounted for. d We seek eigenvalues and eigenvectors for a symmetric matrix related to F.

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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; not shown).

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN B The reciprocal strain ellipse (cont.) 3 The eigenvectors for a deformaon with a symmetric F-matrix will be in the orientaon of the axes of the strain ellipse, but this will not be the case for deformaon 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.

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN B The reciprocal strain ellipse (cont.) 5 Another way to say this is that the axes of reciprocal strain generally will have an orientaon diﬀerent from the axes of the strain ellipse. The rotaon of the axes of the strain ellipse refers to the rotaon needed to bring the axes of the reciprocal strain ellipse into coincidence with the axes of the strain ellipse.

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN C Strain ellipse for general homogeneous rotaonal strain • Decomposion of F = VR* by method of Ramsay and Huber (for 2D). Consider the eﬀect of an irrotaonal strain that follows a pure rigid rotaon of the object (not a rigid rotaon of the reference frame)

⎡ a b ⎤ ⎡ A B ⎤ ⎡ cosω −sinω ⎤ * F = ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ = VR ⎣ c d ⎦ ⎣ B D ⎦ ⎣ sinω cosω ⎦

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN C Strain ellipse for general homo. rotaonal strain

⎡ a b ⎤ ⎡ A B ⎤ ⎡ cosω −sinω ⎤ * 1 F = ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ = VR ⎣ c d ⎦ ⎣ B D ⎦ ⎣ sinω cosω ⎦

⎡ a b ⎤ ⎡ Acosω + Bsinω −Asinω + Bcosω ⎤ 2 ⎢ ⎥ = ⎢ ⎥ ⎣ c d ⎦ ⎣ Bcosω + Dsinω −Bsinω + D cosω ⎦ c-b = (A+D)sinω, and a+d = (A+D)cosω

c − b = tanω 3 If a + d c=b (F is symmetric), then ω=0

From 3 one can obtain R*.

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN C Strain ellipse for general homo. rotaonal strain Post-mulplying both sides of (1) by [R*]-1 = R*T yields V, the symmetric “part” of F.

F = VR*  F[R*]-1 = VR*[R*]-1 = VR*[R*]T = V

−1 ⎡ a b ⎤ ⎡ cosω −sinω ⎤ ⎡ a b ⎤ ⎡ cosω sinω ⎤ ⎡ A B ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ = V ⎣ c d ⎦ ⎣ sinω cosω ⎦ ⎣ c d ⎦ ⎣ −sinω cosω ⎦ ⎣ B D ⎦

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IX Symmetric and non-symmetric matrices A Review: Deformaon gradient ⎡ ∂x′ ∂x′ ⎤ tensor (F) ⎢ ⎥ ⎡ F F ⎤ ∂x ∂y 1 Consists of posion F = ⎢ 11 12 ⎥ = ⎢ ⎥ [ ] ⎢ ⎥ derivaves (dimensionless ⎢ F21 F22 ⎥ ∂y′ ∂y′ ⎣ ⎦ ⎢ ⎥ constants) ∂x ∂y ⎣⎢ ⎦⎥ 2 Is not necessarily T ⎡ ⎤ symmetric: [F]≠[F] ⎡ dx′ ⎤ F11 F12 ⎡ dx ⎤ 3 Transforms an inial ⎢ ⎥ = ⎢ ⎥⎢ ⎥ dy′ ⎢ F21 F22 ⎥ dy change-in-posion vector ⎣⎢ ⎦⎥ ⎣ ⎦⎣⎢ ⎦⎥ to a ﬁnal change-in- posion vector 4 Describes state of deformaon at a point

9. Strain and Eigenvectors

IX Symmetric and non- symmetric matrices B If [F] is not symmetric, then vectors that F operates on that don’t rotate (e.g., a horizontal vector at right) are not necessarily the longest and shortest

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9. Strain and Eigenvectors

IX Symmetric and non- symmetric matrices C Treatment for non- symmetric F 1 Start with the 2 L f deﬁnion of 2 = Q quadrac L0 elongaon Q   2 Express using dot X′ • X′ products   = Q X • X 3 Clear the     denominator X′ • X′ = X • X Q ( )

9. Strain and Eigenvectors     IX Symmetric and non-symmetric X′ • X′ = X • X Q matrices ( ) T T C Treatment for non-symmetric F [FX] [FX] = [X] [X]Q 4 Replace X’ with [FX] T T T 5 Re-arrange both sides [X] [F] [FX] = [X] Q[X] 6 Both sides of this equaon T lead oﬀ with [X] , which T cannot be a zero vector, so it ⎣⎡F F⎦⎤[X] = Q[X] can be dropped from both sides to yield an eigenvector " A X = λ X " equaon [ ][ ] [ ] 7 The eigenvalues of [FTF] are the principal quadrac elongaons

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9. Strain and Eigenvectors

IX Symmetric and non-symmetric matrices D General Case 1 Since [FTF] is symmetric, the principal quadrac elongaons are T perpendicular ⎣⎡F F⎦⎤[X] = Q[X] 2 The principal stretches are the square

roots of the principal quadrac 2 elongaons (Q); they are the square L f L f T roots of the eigenvalues of [F F] Q = 2 ; S = ⇒ Q = S L L 3 The direcons of the principal 0 0 quadrac elongaons (Q) and principal stretches (S) are the same; they are the square roots of the eigenvalues of [FTF] 4 The axes of principal strain rotate

9. Strain and Eigenvectors

IX Symmetric and non-symmetric matrices FT F X Q X E Special Case: [F] is symmetric ⎣⎡ ⎦⎤[ ] = [ ] 1 [FTF] = [F 2 ] because F = FT 2 The principal stretches (S) again are ⎡F 2 ⎤[X] = Q[X] the square roots of the principal ⎣ ⎦ quadrac elongaons (Q) (i.e., the L2 L square roots of the eigenvalues of Q = f ; S = f ⇒ Q = S L2 L [F 2]) 0 0 3 The principal stretches (S) also are F X = S X the eigenvalues of [F ], directly [ ][ ] [ ] 4 The direcons of the principal stretches (S) are the eigenvectors of [F ], and of [FTF] = [F 2 ]! 5 The principal axes do not rotate

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN X Closing comments 1 Our soluons 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 deformaons 3 Evaluang strains for past deformaons require certain assumpons about inial sizes and shapes of bodies or the displacement field. 4 Alternave approach: formulaon and soluon of boundary value problems to solve for the displacement and strain fields. 5 The deformaon gradient matrix F has strain and rotaon intertwined; the two can be separated using matrix mulplicaon. In the symmetric infinitesimal matrix [ε], the rotaon is already separated. 6 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 equaons of secon 5, p. 291). b Ramsay, J.G., and Lisle, M.I., 1983, The techniques of modern structural geology, volume 3: applicaons of connuum mechanics in structural geology: Academic Press, London, 307 p. (See especially sessions 33 and 36). c Malvern, L.E., 1969, Introducon to the mechanics of a connuous medium: Prence-Hall, Englewood Cliffs, New Jersey, 713 p. (See equaons 4.6.1, 4.6.3 a, 4.6.3b on p. 172-174).)

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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

⎡ x ⎤ T 2 2 x y = 1 X X = 1 2a x + y = 1 2b [ ]⎢ y ⎥ 2c [ ] [ ] ⎣⎢ ⎦⎥ Substung (1b) into (2c) yields

1 T 1 T T 1 ⎡ A − X′ ⎤ ⎡ A − X′ ⎤ = 1 ⎡ X′ ⎡A−1 ⎤ ⎤ ⎡ A − X′ ⎤ = 1 3a ⎣[ ] [ ]⎦ ⎣[ ] [ ]⎦ 3b ⎣[ ] ⎣ ⎦ ⎦ ⎣[ ] [ ]⎦

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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 ⎤ T −1 1 ⎡ d −b ⎤ [A] = ⎢ ⎥ [A] = [A] [A] = 2 ⎢ ⎥ ⎣ b d ⎦ A ⎣ −b a ⎦ By inspecon [A-1] = [A-1]T, so

T T 1 T 1 ⎡ X′ ⎡A−1 ⎤ ⎤ ⎡ A − X′ ⎤ = ⎡ X′ ⎡A−1 ⎤⎤ ⎡ A − X′ ⎤ = 1 4 ⎣[ ] ⎣ ⎦ ⎦ ⎣[ ] [ ]⎦ ⎣[ ] ⎣ ⎦⎦ ⎣[ ] [ ]⎦

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9. EIGENVECTORS, EIGENVALUES, AND FINITE STRAIN Appendix 2 Geometric meaning of a real symmetric 2 x 2 matrix

⎡ ⎡ 2 2 ⎤ ⎡ ⎤⎤ 5 1 T d + b −db − ba x′ 2 ⎢[x′ y′] ⎢ ⎥ ⎢ ⎥⎥ = 1 2 2 y′ A ⎣⎢ ⎣⎢ −db − ba a + b ⎦⎥ ⎣⎢ ⎦⎥⎦⎥

6 1 2 2 2 2 2 2 2 x′ (d + b ) − 2x′y′(db + ba) + y′ (a + b ) = 1 A ( )

Since the terms mulplying just x’2 and just y’ 2 are both posive, this equaon describes an ellipse. For a real, symmetric matrix, the eigenvectors of the matrix coincide with the major and minor axes of the ellipse.

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