GG612 Lecture 3. Outline

Transcription

1 GG612 Lecture 3 Strain 11/3/15 GG611 1 Outline Mathema8cal Opera8ons Strain General concepts Homogeneous strain E (strain matri) ε (infinitesimal strain) Principal values and principal direc8ons 11/3/15 GG

2 Main Theme Representa8on of strain at a point in a clear concise manner 11/3/15 GG611 3 Vector Conven8ons X = ini8al posi8on X = final posi8on U = displacement X U X 11/3/15 GG

3 Matri Inverses [AA] - 1 = [A] - 1[ A] = [I] [AB] - 1 = [B - 1 ][A - 1 ] Proof [AB]B - 1 A - 1 =A[I]A - 1 =AA - 1 =[I] [AB][AB] - 1 = [I] The two lex sides of the equa8ons above are equal [AB]B - 1 A - 1 =[AB][AB] - 1 Dropping the [AB] terms on both sides iel [B - 1 A - 1 ]= [AB] /3/15 GG611 5 a b = [a T ][b] a 1 a 2 a n [AB] T = [B T ][A T ] b 1 b 2 " b n Matri Transposes = a 1 b 1 + a 2 b a n b n A matri [A] is smmetric if [A] T = [A] Smmetric Not smmetric Proof A nq = AB = a 1 a 2 " ; B = qm a n a 1 b 1 a1 b 2 # a 1 b m a 2 b 1 a2 b 2 # a 2 b m " " " " a n b 1 an b 1 # a n b m [ AB] T = b 1 B T A T b = 2 " b m [ AB] T = B T A T b 1 b2 # b m a 1 b 1 a2 b 1 # a n b 1 a 1 b 2 a2 b 2 # a n b 1 " " " " a 1 b m a2 b m # a n b m a 1 a2 # a n = Each of the n rows of [A] is row vector with q components. Each of the m columns of [B] is a column vector with q components These match b 1 a 1 b1 a 2 # b 1 a n b 2 a 1 b2 a 2 # b 2 a n " " " " b m a 1 bm a 1 # b m a 1 11/3/15 GG

4 Rota8on Matri [R] Rota8ons change the orienta8ons of vectors but not their lengths (or the square of the lengths) X X = X X X X = X X X = RX X X = [RX] [RX] X X = [RX] T [RX] X X = [X T R T ] [RX] [X T ] [X]= [X T R T ] [RX] [X T ][I][X]= [X T ][R T ] [R][X] [I] = [R T ] [R] But [I] = [R - 1 ] [R], so [R T ] = [R - 1 ] X X 11/3/15 GG611 7 R = cosθ sinθ Rota8on Matri [R] 2D Eample sinθ ; X cosθ [ ] = [ R] [ X] X θ z X = cosθ sinθ sinθ cosθ = cosθ + sinθ = sinθ + cosθ = R T = cosθ sinθ sinθ cosθ RR T = cosθ sinθ sinθ cosθ cosθ sinθ sinθ = cosθ 1 1 R T = R 1 11/3/15 GG

5 General Concepts Deforma8on = Rigid bod mo8on + Strain Rigid bod mo8on Rigid bod transla8on Treated b matri addi8on [X ] = [X] + [U] Rigid bod rota8on Changes orienta8on of lines, but not their length Ais of rota8on does not rotate Treated b matri mul8plica8on [X ] = [R] [X] Transla8on Transla8on + Rota8on Transla8on + Rota8on + Strain 11/3/15 GG611 9 Normal strains General Concepts Change in line length Etension (elonga8on) = Δs/s Stretch = S = s /s Quadra8c elonga8on = Q = (s /s ) 2 Shear strains Change in right angles Dimensions: Dimensionless 11/3/15 GG

6 Homogeneous Strain Parallel lines to parallel lines (2D and 3D) Circle to ellipse (2D) Sphere to ellipsoid (3D) [ X ] = [ F] [ X] Parametric eqn. of an ellipse = a b c d Matri of constants = cosθ = sinθ X X 11/3/15 GG Homogeneous strain Matri Representa8on (2D) [ X ] = [ F] [ X] = a b c d [ X] = [ F] 1 [ X ] = a b c d 1 11/3/15 GG

7 d = d + d d = d + d d d = Matri Representa8ons: Posi8ons (2D) [ d X ] = [ F] [ dx] d d 11/3/15 GG d d = Matri Representa8ons: Posi8ons (2D) = a b c d [ X ] = [ F] [ X] d d If deriva8ves are constant (e.g., at a point), then the equa8ons are linear in and, and d, d, d, and d can be replaced b,,, and, respec8vel. 11/3/15 GG

8 du = u u d + d dv = d + d du dv = Matri Representa8ons Displacements (2D) u u [ du ] = [ J u ][ dx] d d 11/3/15 GG u v Matri Representa8ons Displacements (2D) If deriva8ves are constant, then du, dv, d, and d can be replaced b u, v,, and, respec8vel. u = u + u v = + = U u u If deriva8ves are constant (e.g., at a point), [ ] = [ J u ][ X] then the equa8ons are linear in and 11/3/15 GG

9 Matri Representa8ons Posi8ons and Displacements (2D) U = X X U = FX X = FX IX U = [ F I ] X [ F I ] J u [ F] = [ J u ] = a b c d a 1 b c d 1 11/3/15 GG ε (Infinitesimal Strain Matri, 2D) J u = u u ε = 1 2 J + J T u u ω = 1 2 J J u u T u u u = 1 + u u + u 2 + u u u 2 u J u = ε + ω ε is smmetric ω is an8- smmetric Linear superposi8on 11/3/15 GG

10 ε = ε (Infinitesimal Strain Matri, 2D) Meaning of components u Pure strain without rota8on = u du dv = ε ε ε ε 1 = ε ε First column in ε: rela8ve displacement vector for unit element in - direc8on ε is displacement in the - direc8on of right end of unit element in - direc8on 11/3/15 GG ε = ε (Infinitesimal Strain Matri, 2D) Meaning of components u Pure strain without rota8on = u du dv = ε ε ε ε 1 = ε ε Second column in ε: rela8ve displacement vector for unit element in - direc8on ε is displacement in the - direc8on of upper end of unit element in - direc8on 11/3/15 GG

11 ε = ε (Infinitesimal Strain Matri, 2D) Meaning of components u Pure strain without rota8on = u ε 11 = ε = elonga8on of line parallel to - ais ε 12 = ε (Δθ)/2 ε 21 = ε (Δθ)/2 ε 22 = ε = elonga8on of line parallel to - ais Δθ 2 = (ψ 2 ψ 1 ) 2 = u Posi8ve angles are counter- clockwise Shear strain > if angle between + and + aes decreases 11/3/15 GG ω = ω (Infinitesimal Strain Matri, 2D) Meaning of components u Pure rota8on without strain = u du dv ω z ω z 1 = ω z ω z ω z << 1 radian First column in ω: rela8ve displacement vector for unit element in - direc8on ω 12 is displacement in the - direc8on of right end of unit element in - direc8on 11/3/15 GG

12 ω = ω (Infinitesimal Strain Matri, 2D) Meaning of components u Pure rota8on without strain = u du dv ω z ω z 1 = ω z ω z ω z << 1 radian Second column in ω: rela8ve displacement vector for unit element in - direc8on ω 21 is displacement in the nega%ve - direc8on of upper end of unit element in - direc8on 11/3/15 GG ω = ω 11 = ω 12 = - ω z ω 21 = ω z ω 22 = ω (Infinitesimal Strain Matri, 2D) Meaning of components u Pure rota8on without strain = u ω z = (ψ 2 + ψ 1 ) 2 = 1 2 u 11/3/15 GG

13 Principal Values (eigenvalues) and Principal Direc8ons (eigenvectors) Mathema(cs 1 [X] = [F][X], where [X] is an posi8on vector and [F] is a matri of constants Meaning / / [F] converts We seek the lengths of the semi- aes of the ellipse and their direc8ons The lengths of the semi- aes of the ellipse give the principal stretches if a unit circle is transformed to the ellipse 11/3/15 GG Principal Values (eigenvalues) and Principal Direc8ons (eigenvectors) Mathema(cs 2 d(x X) = X dx = Meaning Ma/min lengths of X are where X and its tangent(s) dx are perpendicular 11/3/15 GG

14 Principal Values (eigenvalues) and Principal Direc8ons (eigenvectors) Mathema(cs 2 d(x X) = X dx = 3 [F][X] = λ[x] (Eigenvector equa8on) 4 If [F] = [F] T, a) X i X j =, etc. b) X 1 X ma, X n X min Meaning Ma/min lengths of X are where X and its tangent(s) dx are perpendicular For solu8ons of (3), [X] (eigenvector) is not rotated b [F] but can be stretched, where λ is the stretch (eigenvalue) If [F] is smmetric, then its eigenvectors are perpendicular its eigenvectors are in direc8ons of maimum/minimum X (from 2 and 4) 11/3/15 GG Principal Values (eigenvalues) and Principal Direc8ons (eigenvectors) Mathema(cs 5 If [F] [F] T, let [B] = [F] T [F] a) [B] = [[F] T [F]] = [[F] T [F]] T b) [B][X] = λ 2 [X] = Q [X] (Eigenvector equa8on) c) [F] = [R][U], where [U] = [C] ½, [C] = [F] T [F], and [R] = [F][U] - 1 d) [F] = [V][R], where [V] = [B] ½, [B] = [F] [F T ], and [R] = [V] - 1 [F] Meaning Even if [F] is not smmetric, then [B] = [F] T [F] and [C] = [F] [F] T are smmetric The principal values of [F] T [F] include the maimum and minimum quadra8c elonga8ons The principal direc8ons are the eigenvectors of [C] F can be decomposed into the product of a smmetric stretch matri ([U] or [V]), and a rota8on matri [R] 11/3/15 GG

15 Eample 1: [F] is not smmetric >> F = [1 2; 1] F = >> [fvec,fval] = eig(f) fvec = fval = 1 1 >> C=F'*F C = >> [uvec,uval] = eig(u) uvec = uval = >> [cvec,cval] = eig(c) cvec = cval = >> principal_direc8ons = R*cvec >> U=C^(1/2) U = >> R = F*inv(U) R = principal_direc8ons = The two principal stretches are given b the eigenvalues of [C]. 2 The eigenvectors of [F] are the vectors that [F] does not rotate. 3 The eigenvectors of [C] must be rotated to give the principal direc8ons of the ellipse. 11/3/15 GG Eample 2: [F] is not smmetric >> F = [1 2; 1] F = >> [fvec,fval] = eig(f) fvec = fval = 1 1 >> B=F*F B= >> V=B^(1/2) U = >> R = inv(v)*f R = >> [vvec,vval] = eig(v) vvec = vval = >> [bvec,bval] = eig(b) bvec = bval = >> principal_direc8ons = bvec principal_direc8ons = The two principal stretches are given b the eigenvalues of [B]. 2 The eigenvectors of [F] are the vectors that [F] does not rotate. 3 The eigenvectors of [B] give the principal direc8ons of the ellipse directl. 11/3/15 GG

16 Summar of Strain Quan88es for describing strains are dimensionless Strain describes changes in distance between points and changes in right angles Strain at a point can be represented b the orienta8on and magnitude of the principal stretches Smmetric strains and stress are smmetric: eigenvalues are orthogonal and do not rotate Asmmetric strain matrices involve rota8on Infinitesimal strains can be superposed linearl Finite strains involve matri mul8plica8on The same final deforma8on can be achieved b different paths 11/3/15 GG