Analytical Mechanics: Elastic Deformation

Size: px
Start display at page:

Download "Analytical Mechanics: Elastic Deformation"

Transcription

1 Analytical Mechanics: Elastic Deformation Shinichi Hirai Dept. Robotics, Ritsumeikan Univ. Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 1 / 60

2 Agenda Agenda 1 Stress-strain Relationship Piecewise Linear Approximation Dynamic Deformation 2 Three-dimensional Elastic Deformation Stress-strain Relationship Piecewise Linear Approximation Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 2 / 60

3 Stress-strain Relationship Displacement vector P(x, y) u(x, y) v(x, y) arbitrary point inside object displacement of point P(x, y) along x-axis displacement of point P(x, y) along y-axis displacement vector u = [ u v Note: vector u depends on x and y. ] Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 3 / 60

4 Stress-strain Relationship Cauchy strain B 1 1 P(x,y) A 1+ v y P u y 1+ B u x A v x Deformation of small square region u v = extension along x-axis, = extension along y-axis x y u v = shear rotation, = shear + rotation y x Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 4 / 60

5 Stress-strain Relationship Cauchy strain normal strain component along x-axis at point P: ε xx = u x normal strain component along y-axis at point P: shear strain at point P: ε yy = v y 2ε xy = u y + v x Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 5 / 60

6 Stress-strain Relationship Stress-strain relationship pseudo strain vector (strain vector): ε = ε xx ε yy 2ε xy σ xx σ yy σ xy normal stress component along x-axis at point P normal stress component along y-axis at point P a shear stress component at point P σ = σ xx σ yy σ xy Note: σ T ε represents energy density (energy per unit volume) Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 6 / 60

7 Stress-strain Relationship Stress-strain relationship in linear elastic material Stress-strain relationship: σ = σ(ε) Stress-strain relationship in linear elastic material: σ = Dε where 3 3 matrix D is referred to as elasticity matrix Elasticity matrix of an linear isotropic material + 2µ 0 D = + 2µ µ where and µ denote Lamé s constants Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 7 / 60

8 Stress-strain Relationship Stress-strain relationship in linear elastic material Elasticity matrix of an linear isotropic material: D = I + µi µ where I = , I µ = Lamé s constants characterize linear isotropic elasticity: = νe (1 + ν)(1 2ν), µ = E 2(1 + ν) where Young s modulus E and Poisson s ratio ν Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 8 / 60

9 Stress-strain Relationship Potential energy potential energy density of linear elatic material: S 1 2 σt ε = 1 2 εt Dε potential energy stored in an elastic object: 1 1 U = 2 σt ε h ds = 2 εt Dε h ds where h denotes the constant thickness of the object Note: h ds denotes volume of small region S Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 9 / 60

10 Piecewise Linear Approximation Cover of region by triangles region S cover by triangles P i nodal point of a triangle [ x i, y i ] T coordinates of point P i P i P j P k a triangle consisting of nodal points P i, P j, P k Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 10 / 60

11 Piecewise Linear Approximation Potential energy U i,j,k : Potential energy stored in P i P j P k U = all triangles U i,j,k How to compute U i,j,k? Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 11 / 60

12 Piecewise Linear Approximation Signed area Signed area positive if the triangular loop is counter clockwise negative if the loop is clockwise Signed area of triangle OP i P j : OP i P j = 1 2 x i y i x j y j = 1 2 (x iy j y i x j ) Signed area of triangle P i P j P k : P i P j P k = 1 2 x j x i x k x i y j y i y k y i = 1 2 {(x iy j x j y i ) + (x j y k x k y j ) + (x k y i x i y k )} Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 12 / 60

13 Piecewise Linear Approximation Shape functions two-dimensional shape function N i,j,k (x, y) on triangle P i P j P k : N i,j,k (x, y) = (y j y k )x (x j x k )y + (x j y k x k y j ) 2 P i P j P k Note that { 1 at point Pi N i,j,k (x, y) = 0 at point P j and P k Note: P(x, y) be an arbitrary point within the triangle: N i,j,k (x, y) = PP jp k P i P j P k assume that function N i,j,k (x, y) vanishes outside P i P j P k Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 13 / 60

14 Piecewise Linear Approximation Shape functions Any function u(x, y) can be linearly approximated inside P i P j P k : u(x, y) = u i N i,j,k (x, y) + u j N j,k,i (x, y) + u k N k,i,j (x, y) where u i = u(x i, y i ), u j = u(x j, y j ), u k = u(x k, y k ) Partial derivatives of N i,j,k (x, y): N i,j,k x = y j y k 2 P i P j P k, N i,j,k y = (x j x k ) 2 P i P j P k. Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 14 / 60

15 Piecewise Linear Approximation Approximating displacement vector displacement vector u in region P i P j P k : u = N i,j,k u i + N j,k,i u j + N k,i,j u k collective vector: u i,j,k = u i u j u k collective vectors: γ u = u i u j u k, γ v = v i v j v k, γ = [ γu γ v ] Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 15 / 60

16 Piecewise Linear Approximation Strain components partial derivatives inside = P i P j P k : u ξ = a T γ u, u η = b T γ u, v ξ = a T γ v, v η = b T γ v, where a = 1 2 y j y k y k y i y i y j, b = 1 2 normal and shear strain components inside : x j x k x k x i x i x j ε xx = a T γ u, ε yy = b T γ v 2ε xy = b T γ u + a T γ v Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 16 / 60

17 Potential energy density of potential energy of an isotropic linear elastic material: 1 2 εt Dε = 1 2 εt (I + µi µ )ε = 1 2 (ε xx + ε yy ) µ { 2ε 2 xx + 2ε 2 yy + (2ε xy ) 2} potential energy stored in element P i P j P k : where G i,j,k G i,j,k µ = 1 2 = 1 2 U i,j,k = G i,j,k P i P j P k (ε xx + ε yy ) 2 h ds P i P j P k 2 ( ε 2 xx + ε 2 yy + µg i,j,k µ ) h ds P i P j P k (2ε xy ) 2 h ds Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 17 / 60

18 Potential energy computing G i,j,k : where H aa = H ab = G i,j,k = 1 2 are constant matrices. [ γ T u γ T v = 1 2 γt H i,j,k γ aa T h ds = aa T h ab T h ds = ab T h, ] [ H aa H ba H ab H bb H bb = H ba = ] [ γu γ v ] bb T h ds = bb T h ba T h ds = ba T h Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 18 / 60

19 Potential energy computing G i,j,k µ : where G i,j,k µ = 1 2 [ γ T u γ T v = 1 2 γt H i,j,k µ γ Hµ aa = 2H aa + H bb, H ab µ = H ba, are constant matrices. Hµ ba = H ab ] [ Hµ aa Hµ ab Hµ ba Hµ bb ] [ γu γ v Hµ bb = 2H bb + H aa ] Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 19 / 60

20 Potential energy where with U i,j,k = G i,j,k G i,j,k + µg i,j,k µ = 1 2 γt H i,j,k γ G i,j,k µ = 1 2 γt H i,j,k µ γ γ = u i u j u k v i v j v k Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 20 / 60

21 Permutation u i u i v i u i,j,k = u j = u j u k v j u k v k Converting a quadratic form with respect to γ into another with respect to u i,j,k : γ T H i,j,k γ = ui,j,kj T i,j,k u i,j,k Permutating rows and columns of H i,j,k J i,j,k yields J i,j,k : = H i,j,k ([1, 4, 2, 5, 3, 6], [1, 4, 2, 5, 3, 6]) Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 21 / 60

22 Potential energy where with J i,j,k U i,j,k = G i,j,k G i,j,k + µg i,j,k µ = 1 2 ut i,j,kj i,j,k u i,j,k G i,j,k µ = 1 2 ut i,j,kj i,j,k µ u i,j,k = H i,j,k ([1, 4, 2, 5, 3, 6], [1, 4, 2, 5, 3, 6]) J i,j,k µ = H i,j,k µ ([1, 4, 2, 5, 3, 6], [1, 4, 2, 5, 3, 6]) partial connection matrices Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 22 / 60

23 Nodal elastic forces a set of nodal forces applied to P i, P j, P k : where f i,j,k = K i,j,k u i,j,k K i,j,k = J i,j,k + µjµ i,j,k stiffness matrix and µ are physical parameters. and Jµ i,j,k are geometric; they include no physical parameters. J i,j,k Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 23 / 60

24 Example y P4 P5 P6 1 T2 T4 T1 T3 P1 1 P2 1 P3 x square divided into 4 triangles: T 1 = P 1 P 2 P 4, T 2 = P 2 P 5 P 4, T 3 = P 2 P 3 P 5, T 4 = P 3 P 6 P 5 thickness h is constantly equal to 2 Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 24 / 60

25 Example T 1 = P 1 P 2 P 4 : a = [ 1, 1, 0 ] T, b = [ 1, 0, 1 ] T H 1,2,4 = J 1,2,4 = Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 25 / 60

26 Example T 1 = P 1 P 2 P 4 : a = [ 1, 1, 0 ] T, b = [ 1, 0, 1 ] T Hµ 1,2,4 = J 1,2,4 µ = Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 26 / 60

27 Example T 2 = P 2 P 5 P 4 : a = [ 0, 1, 1 ] T, b = [ 1, 1, 0 ] T J 2,5,4 = J 2,5,4 µ = Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 27 / 60

28 Example T 3 = P 2 P 3 P 5 : a = [ 1, 1, 0 ] T, b = [ 1, 0, 1 ] T J 2,3,5 = J 1,2,4, Jµ 2,3,5 = Jµ 1,2,4 T 4 = P 3 P 6 P 5 : a = [ 0, 1, 1 ] T, b = [ 1, 1, 0 ] T J 3,6,5 = J 2,5,4, Jµ 3,6,5 = Jµ 2,5,4 Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 28 / 60

29 Example adding the contribution of matrix J 1,2,4 to connection matrix J : J = Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 29 / 60

30 Example adding the contribution of matrix J 2,5,4 to connection matrix J : J = Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 30 / 60

31 Example adding the contribution of matrix J 2,3,5 to connection matrix J : J = Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 31 / 60

32 Example adding the contribution of matrix J 3,6,5 to connection matrix J : J = Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 32 / 60

33 Example computing J µ yields: J µ = Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 33 / 60

34 Example potential energy: u N = u 1 u 2 u 3 u 4 u 5 u 6 U = 1 2 ut NKu N where K = J + µj µ Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 34 / 60

35 Example P4 P1 T1 T2 P5 P2 T3 P6 T4 p P3 edge P 1 P 4 is fixed on a rigid wall uniform pressure p = [ p x, p y ] T is exerted over edge P 3 P 6 Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 35 / 60

36 Example work done by the pressure W = [ (P3 P 6 h/2) p (P 3 P 6 h/2) p ] T [ u3 u 6 ] = f T extu N where f ext = 0 0 (P 2 P 5 h/2) p 0 0 (P 2 P 5 h/2) p Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 36 / 60

37 Example u 1 = 0 and u 4 = 0 are integrated into A T u N = 0, where I 2 2 O O O A = O O O I 2 2 = O O O O Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 37 / 60

38 Example Variational principle in statics under geometric constraints minimize J(u N, A ) = 1 2 u N T Ku N f T extu N T A A T u N where A is a collective vector consisting of four Lagrange multipliers [ J u N = Ku N f ext A A = 0, J A = A T u N = 0 ] [ ] K A un = A T A [ fext 0 ] Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 38 / 60

39 Dynamic Deformation Kinetic energy ρ: density of an object at point P(x, y) kinetic energy inside = P i P j P k : 1 T i,j,k = P i P j P k 2 ρ ut u h ds total kinetic energy of the object: T = P i P j P k T i,j,k velocity of any point within the triangle: u = u i N i,j,k + u j N j,k,i + u k N k,i,j. Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 39 / 60

40 Dynamic Deformation Kinetic energy assume that density ρ is constant: T i,j,k = 1 [ ] u T 2 i u j T u k T Mi,j,k u i u j u k where total kinetic energy: M i,j,k = ρh P ip j P k 12 2I 2 2 I 2 2 I 2 2 I 2 2 2I 2 2 I 2 2 I 2 2 I 2 2 2I 2 2 T = 1 2 u N T M u N matrix M is referred to as an inertia matrix Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 40 / 60

41 Dynamic Deformation Dynamic equation Lagrangian under geometric constraints L = T U + W + T A A T u N = 1 2 u N T M u N 1 2 u N T Ku N + f T extu N + T A A T u N, where A is a set of Lagrange multipliers a set of Lagrange equations of motion: L d L = 0 u N dt u N a set of motion equations of nodal points: Ku N + f ext + A A Mü N = 0 Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 41 / 60

42 Dynamic Deformation Dynamic equation Equation for stabilizing constraints: (A T ü N ) + 2α(A T u N ) + α 2 (A T u N ) = 0 introducing velocity vector v N = u N : u N = v N, M v N A A = Ku N + f ext, A T v N = A T (2αv N + α 2 u N ) u N = v N, [ ] [ ] [ ] M A vn Ku A T = N + f ext A T (2αv N + α 2 u N ) A Note: the second linear equation is solvable, implying that we can compute v N. Thus, we can sketch u N and v N using any ODE solver. Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 42 / 60

43 Dynamic Deformation Example (E = 30, ν = 0.35 and c = 20, ν vis = 0.35) s 10 s s 30 s Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 43 / 60

44 Three-dimensional Elastic Deformation Stress-strain Relationship 3D Cauchy strain C C B 1 1 B A P 1 A P Deformation of small cubic region Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 44 / 60

45 Three-dimensional Elastic Deformation Stress-strain Relationship 3D Cauchy strain u v w = ext. along x, = ext. along y, = ext. along z, x y z u v = shear in xy rot ard z, = shear in xy + rot ard z, y x v w = shear in yz rot ard x, = shear in yz + rot ard x, z y w u = shear in zx rot ard y, = shear in zx + rot ard y x z Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 45 / 60

46 Three-dimensional Elastic Deformation Stress-strain Relationship 3D Cauchy strain normal strain components: ε xx = u x, shear strain components: 2ε yz = v z + w y, strain vector: ε yy = v y, 2ε zx = w x + u z, ε = ε xx ε yy ε zz 2ε yz 2ε zx 2ε xy ε zz = w z 2ε xy = u y + v x Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 46 / 60

47 Three-dimensional Elastic Deformation Stress-strain Relationship Stress-strain relationship stress vector: σ = σ xx σ yy σ zz σ yz σ zx σ xy stress-strain relationship in linear elastic material: σ = Dε Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 47 / 60

48 Three-dimensional Elastic Deformation Stress-strain Relationship Stress-strain relationship elasticity matrix of an isotropic material: where and 1 = [ 1 O I = O O, I = D = I + µi µ ] [ 2I O, I µ = O I ], O = Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 48 / 60

49 Three-dimensional Elastic Deformation Piecewise Linear Approximation Signed volume signed volume of tetrahedron OP i P j P k : OP i P j P k = 1 x i x j x k 6 y i y j y k z i z j z k signed volume of tetrahedron P i P j P k P l : P i P j P k P l = 1 x j x i x k x i x l x i 6 y j y i y k y i y l y i z j z i z k z i z l z i Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 49 / 60

50 Three-dimensional Elastic Deformation Piecewise Linear Approximation Shape functions 3D shape function N i,j,k,l (x, y, z) on tetrahedron P i P j P k P l : N i,j,k,l (x, y, z) = PP { jp k P l 1 at point Pi = P i P j P k P l 0 within P j P k P l linear approximation of function u(x, y, z) inside P i P j P k P l : u(x, y, z) = u i N i,j,k,l (x, y, z) + u j N j,k,l,i (x, y, z) + u k N k,l,i,j (x, y, z) + u l N l,i,j,k (x, y, z) Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 50 / 60

51 Three-dimensional Elastic Deformation Piecewise Linear Approximation Shape functions N i,j,k,l x N i,j,k,l y where 1 PP j P k P l = = 1 1 a j,k,l, P i P j P k P l x 6 P i P j P k P l = 1 1 N i,j,k,l b j,k,l, = 1 1 c j,k,l 6 P i P j P k P l z 6 P i P j P k P l a j,k,l = y j z j b j,k,l = z j x j c j,k,l = x j y j y k z k z k x k x k y k + y k z k + z k x k + x k y k y l z l z l x l x l y l + y l z l + z l x l + x l y l y j z j z j x j x j y j Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 51 / 60

52 Three-dimensional Elastic Deformation Potential energy potential energy in = P i P j P k P l : 1 U i,j,k,l = P i P j P k P l 2 εt Dε dv collective vectors: and γ u = u i u j u k u l, γ v = u i,j,k,l = v i v j v k v l, γ w u i u j u k u l = w i w j w k w l, γ = γ u γ v γ w Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 52 / 60

53 Three-dimensional Elastic Deformation Potential energy partial derivatives: where a = 1 6 u ξ = a T γ u, u η = b T γ u, u ζ = c T γ u v ξ = a T γ v, v η = b T γ v, v ζ = c T γ v w ξ = a T γ w, w η = b T γ w, w ζ = c T γ w a j,k,l a k,l,i a l,i,j a i,j,k, b = 1 6 b j,k,l b k,l,i b l,i,j b i,j,k, c = 1 6 c j,k,l c k,l,i c l,i,j c i,j,k Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 53 / 60

54 Three-dimensional Elastic Deformation Potential energy normal strain components: shear strain components: ε xx = a T γ u, ε yy = b T γ v, ε zz = c T γ w 2ε yz = c T γ v + b T γ w, 2ε zx = a T γ w + c T γ u, 2ε xy = b T γ u + a T γ v density of potential energy of an isotropic linear elastic material: 1 2 εt Dε = 1 2 (ε xx + ε yy + ε zz ) µ { 2ε 2 xx + 2ε 2 yy + 2ε 2 zz + (2ε yz ) 2 + (2ε zx ) 2 + (2ε xy ) 2} Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 54 / 60

55 Three-dimensional Elastic Deformation Potential energy potential energy in = P i P j P k P l : where G i,j,k,l G i,j,k,l µ = 1 2 = 1 2 U i,j,k,l = G i,j,k,l {ε xx + ε yy + ε zz } 2 dv + µg i,j,k,l µ { 2ε 2 xx + 2ε 2 yy + 2ε 2 zz + (2ε yz ) 2 + (2ε zx ) 2 + (2ε xy ) 2} dv Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 55 / 60

56 Three-dimensional Elastic Deformation Potential energy G = 1 2 [ γ T u γ T v γ T w ] H aa H ba H ca H ab H bb H cb H ac H bc H cc γ u γ v γ w = 1 2 γt H i,j,k,l γ G µ = 1 2 H aa = aa T, H bb = bb T, H cc = cc T H ab = (H ba ) T = ab T, H bc = (H cb ) T = bc T, [ γ T u γ T v γ T w ] Hµ aa = 2H aa + H bb + H cc, H ab µ = H ba, H ba µ = H ab, H aa µ H ab µ H ac µ Hµ ba Hµ bb Hµ bc Hµ ca Hµ cb Hµ cc γ u γ v γ w = 1 2 γt H i,j,k,l µ γ Hµ bb = 2H bb + H cc + H aa, H bc µ = H cb, H cb µ = H bc, Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 56 / 60

57 Three-dimensional Elastic Deformation Permutation γ = u i u j u k u l v i v j v k v l w i w j w k w l, u i,j,k,l = u i v i w i u j v j w j u k v k w k u l v l w l Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 57 / 60

58 Three-dimensional Elastic Deformation Potential energy where U i,j,k,l = G i,j,k,l G i,j,k,l = 1 2 ut i,j,k,l J i,j,k,l + µg i,j,k,l µ u i,j,k,l with J i,j,k,l G i,j,k,l µ = 1 2 ut i,j,k,l J i,j,k,l µ u i,j,k,l = H i,j,k,l ([1, 5, 9, 2, 6, 10, 3, 7, 11, 4, 8, 12], [1, 5, 9, 2, 6, 10, 3, 7, 11, 4 J i,j,k,l µ = H i,j,k,l µ ([1, 5, 9, 2, 6, 10, 3, 7, 11, 4, 8, 12], [1, 5, 9, 2, 6, 10, 3, 7, 11, 4 partial connection matrices Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 58 / 60

59 Three-dimensional Elastic Deformation Simulating Viscoelastic Deformation Report #6 due date : Jan. 31 (Thurs.) Simulate the deformation of a rectangular viscoelastic object shown in the figure. The bottom surface is fixed to the ground. Uniform pressure is applied to the middle of the top surface downward for a while, then the pressure is released. Use appropriate values of geometrical and physical parameters of the object. P13 P14 P5 P6 p P15 P9 P10 P11 P7 P16 P12 P8 P1 P2 P3 P4 Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 59 / 60

60 Summary Summary 2D elastic deformation stress-strain relationship and potential energy piecewise linear approximation of potential energy formulating static deformation based on variational principle formulating dynamic deformation based on variational principle 3D elastic deformation piecewise linear approximation in 3D space Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 60 / 60

Analytical Mechanics: Elastic Deformation

Analytical Mechanics: Elastic Deformation Analytical Mechanics: Elastic Deformation Shinichi Hirai Dept. Robotics, Ritsumeikan Univ. Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation / 59 Agenda Agenda

More information

Finite Element Method in Geotechnical Engineering

Finite Element Method in Geotechnical Engineering Finite Element Method in Geotechnical Engineering Short Course on + Dynamics Boulder, Colorado January 5-8, 2004 Stein Sture Professor of Civil Engineering University of Colorado at Boulder Contents Steps

More information

UNIVERSITY OF SASKATCHEWAN ME MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich

UNIVERSITY OF SASKATCHEWAN ME MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich UNIVERSITY OF SASKATCHEWAN ME 313.3 MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich A CLOSED BOOK EXAMINATION TIME: 3 HOURS For Marker s Use Only LAST NAME (printed): FIRST

More information

Basic Equations of Elasticity

Basic Equations of Elasticity A Basic Equations of Elasticity A.1 STRESS The state of stress at any point in a loaded bo is defined completely in terms of the nine components of stress: σ xx,σ yy,σ zz,σ xy,σ yx,σ yz,σ zy,σ zx,andσ

More information

Mechanics PhD Preliminary Spring 2017

Mechanics PhD Preliminary Spring 2017 Mechanics PhD Preliminary Spring 2017 1. (10 points) Consider a body Ω that is assembled by gluing together two separate bodies along a flat interface. The normal vector to the interface is given by n

More information

SEMM Mechanics PhD Preliminary Exam Spring Consider a two-dimensional rigid motion, whose displacement field is given by

SEMM Mechanics PhD Preliminary Exam Spring Consider a two-dimensional rigid motion, whose displacement field is given by SEMM Mechanics PhD Preliminary Exam Spring 2014 1. Consider a two-dimensional rigid motion, whose displacement field is given by u(x) = [cos(β)x 1 + sin(β)x 2 X 1 ]e 1 + [ sin(β)x 1 + cos(β)x 2 X 2 ]e

More information

PEAT SEISMOLOGY Lecture 2: Continuum mechanics

PEAT SEISMOLOGY Lecture 2: Continuum mechanics PEAT8002 - SEISMOLOGY Lecture 2: Continuum mechanics Nick Rawlinson Research School of Earth Sciences Australian National University Strain Strain is the formal description of the change in shape of a

More information

The Finite Element Method

The Finite Element Method The Finite Element Method 3D Problems Heat Transfer and Elasticity Read: Chapter 14 CONTENTS Finite element models of 3-D Heat Transfer Finite element model of 3-D Elasticity Typical 3-D Finite Elements

More information

Measurement of deformation. Measurement of elastic force. Constitutive law. Finite element method

Measurement of deformation. Measurement of elastic force. Constitutive law. Finite element method Deformable Bodies Deformation x p(x) Given a rest shape x and its deformed configuration p(x), how large is the internal restoring force f(p)? To answer this question, we need a way to measure deformation

More information

3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1

3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is

More information

Variational principles in mechanics

Variational principles in mechanics CHAPTER Variational principles in mechanics.1 Linear Elasticity n D Figure.1: A domain and its boundary = D [. Consider a domain Ω R 3 with its boundary = D [ of normal n (see Figure.1). The problem of

More information

Problem " Â F y = 0. ) R A + 2R B + R C = 200 kn ) 2R A + 2R B = 200 kn [using symmetry R A = R C ] ) R A + R B = 100 kn

Problem  Â F y = 0. ) R A + 2R B + R C = 200 kn ) 2R A + 2R B = 200 kn [using symmetry R A = R C ] ) R A + R B = 100 kn Problem 0. Three cables are attached as shown. Determine the reactions in the supports. Assume R B as redundant. Also, L AD L CD cos 60 m m. uation of uilibrium: + " Â F y 0 ) R A cos 60 + R B + R C cos

More information

HIGHER-ORDER THEORIES

HIGHER-ORDER THEORIES HIGHER-ORDER THEORIES THIRD-ORDER SHEAR DEFORMATION PLATE THEORY LAYERWISE LAMINATE THEORY J.N. Reddy 1 Third-Order Shear Deformation Plate Theory Assumed Displacement Field µ u(x y z t) u 0 (x y t) +

More information

NDT&E Methods: UT. VJ Technologies CAVITY INSPECTION. Nondestructive Testing & Evaluation TPU Lecture Course 2015/16.

NDT&E Methods: UT. VJ Technologies CAVITY INSPECTION. Nondestructive Testing & Evaluation TPU Lecture Course 2015/16. CAVITY INSPECTION NDT&E Methods: UT VJ Technologies NDT&E Methods: UT 6. NDT&E: Introduction to Methods 6.1. Ultrasonic Testing: Basics of Elasto-Dynamics 6.2. Principles of Measurement 6.3. The Pulse-Echo

More information

MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso

MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso Table of Contents Thermodynamics Derivation Hooke s Law: Anisotropic Elasticity

More information

A short review of continuum mechanics

A short review of continuum mechanics A short review of continuum mechanics Professor Anette M. Karlsson, Department of Mechanical ngineering, UD September, 006 This is a short and arbitrary review of continuum mechanics. Most of this material

More information

Introduction to Seismology Spring 2008

Introduction to Seismology Spring 2008 MIT OpenCourseWare http://ocw.mit.edu 12.510 Introduction to Seismology Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Stress and Strain

More information

Content. Department of Mathematics University of Oslo

Content. Department of Mathematics University of Oslo Chapter: 1 MEK4560 The Finite Element Method in Solid Mechanics II (January 25, 2008) (E-post:torgeiru@math.uio.no) Page 1 of 14 Content 1 Introduction to MEK4560 3 1.1 Minimum Potential energy..............................

More information

Lecture 8. Stress Strain in Multi-dimension

Lecture 8. Stress Strain in Multi-dimension Lecture 8. Stress Strain in Multi-dimension Module. General Field Equations General Field Equations [] Equilibrium Equations in Elastic bodies xx x y z yx zx f x 0, etc [2] Kinematics xx u x x,etc. [3]

More information

MEC-E8001 FINITE ELEMENT ANALYSIS

MEC-E8001 FINITE ELEMENT ANALYSIS MEC-E800 FINIE EEMEN ANAYSIS 07 - WHY FINIE EEMENS AND IS HEORY? Design of machines and structures: Solution to stress or displacement by analytical method is often impossible due to complex geometry,

More information

NONLINEAR CONTINUUM FORMULATIONS CONTENTS

NONLINEAR CONTINUUM FORMULATIONS CONTENTS NONLINEAR CONTINUUM FORMULATIONS CONTENTS Introduction to nonlinear continuum mechanics Descriptions of motion Measures of stresses and strains Updated and Total Lagrangian formulations Continuum shell

More information

Fundamentals of Linear Elasticity

Fundamentals of Linear Elasticity Fundamentals of Linear Elasticity Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research of the Polish Academy

More information

CIV-E1060 Engineering Computation and Simulation Examination, December 12, 2017 / Niiranen

CIV-E1060 Engineering Computation and Simulation Examination, December 12, 2017 / Niiranen CIV-E16 Engineering Computation and Simulation Examination, December 12, 217 / Niiranen This examination consists of 3 problems rated by the standard scale 1...6. Problem 1 Let us consider a long and tall

More information

3.2 Hooke s law anisotropic elasticity Robert Hooke ( ) Most general relationship

3.2 Hooke s law anisotropic elasticity Robert Hooke ( ) Most general relationship 3.2 Hooke s law anisotropic elasticity Robert Hooke (1635-1703) Most general relationship σ = C ε + C ε + C ε + C γ + C γ + C γ 11 12 yy 13 zz 14 xy 15 xz 16 yz σ = C ε + C ε + C ε + C γ + C γ + C γ yy

More information

Mechanical Properties of Materials

Mechanical Properties of Materials Mechanical Properties of Materials Strains Material Model Stresses Learning objectives Understand the qualitative and quantitative description of mechanical properties of materials. Learn the logic of

More information

16.20 HANDOUT #2 Fall, 2002 Review of General Elasticity

16.20 HANDOUT #2 Fall, 2002 Review of General Elasticity 6.20 HANDOUT #2 Fall, 2002 Review of General Elasticity NOTATION REVIEW (e.g., for strain) Engineering Contracted Engineering Tensor Tensor ε x = ε = ε xx = ε ε y = ε 2 = ε yy = ε 22 ε z = ε 3 = ε zz =

More information

Constitutive Relations

Constitutive Relations Constitutive Relations Dr. Andri Andriyana Centre de Mise en Forme des Matériaux, CEMEF UMR CNRS 7635 École des Mines de Paris, 06904 Sophia Antipolis, France Spring, 2008 Outline Outline 1 Review of field

More information

NONLINEAR WAVE EQUATIONS ARISING IN MODELING OF SOME STRAIN-HARDENING STRUCTURES

NONLINEAR WAVE EQUATIONS ARISING IN MODELING OF SOME STRAIN-HARDENING STRUCTURES NONLINEAR WAE EQUATIONS ARISING IN MODELING OF SOME STRAIN-HARDENING STRUCTURES DONGMING WEI Department of Mathematics, University of New Orleans, 2 Lakeshore Dr., New Orleans, LA 7148,USA E-mail: dwei@uno.edu

More information

Consider an elastic spring as shown in the Fig.2.4. When the spring is slowly

Consider an elastic spring as shown in the Fig.2.4. When the spring is slowly .3 Strain Energy Consider an elastic spring as shown in the Fig..4. When the spring is slowly pulled, it deflects by a small amount u 1. When the load is removed from the spring, it goes back to the original

More information

. D CR Nomenclature D 1

. D CR Nomenclature D 1 . D CR Nomenclature D 1 Appendix D: CR NOMENCLATURE D 2 The notation used by different investigators working in CR formulations has not coalesced, since the topic is in flux. This Appendix identifies the

More information

HIGHER-ORDER THEORIES

HIGHER-ORDER THEORIES HIGHER-ORDER THEORIES Third-order Shear Deformation Plate Theory Displacement and strain fields Equations of motion Navier s solution for bending Layerwise Laminate Theory Interlaminar stress and strain

More information

Inverse Design (and a lightweight introduction to the Finite Element Method) Stelian Coros

Inverse Design (and a lightweight introduction to the Finite Element Method) Stelian Coros Inverse Design (and a lightweight introduction to the Finite Element Method) Stelian Coros Computational Design Forward design: direct manipulation of design parameters Level of abstraction Exploration

More information

Macroscopic theory Rock as 'elastic continuum'

Macroscopic theory Rock as 'elastic continuum' Elasticity and Seismic Waves Macroscopic theory Rock as 'elastic continuum' Elastic body is deformed in response to stress Two types of deformation: Change in volume and shape Equations of motion Wave

More information

4 NON-LINEAR ANALYSIS

4 NON-LINEAR ANALYSIS 4 NON-INEAR ANAYSIS arge displacement elasticity theory, principle of virtual work arge displacement FEA with solid, thin slab, and bar models Virtual work density of internal forces revisited 4-1 SOURCES

More information

M5 Simple Beam Theory (continued)

M5 Simple Beam Theory (continued) M5 Simple Beam Theory (continued) Reading: Crandall, Dahl and Lardner 7.-7.6 In the previous lecture we had reached the point of obtaining 5 equations, 5 unknowns by application of equations of elasticity

More information

3D Elasticity Theory

3D Elasticity Theory 3D lasticity Theory Many structural analysis problems are analysed using the theory of elasticity in which Hooke s law is used to enforce proportionality between stress and strain at any deformation level.

More information

Mechanics of Earthquakes and Faulting

Mechanics of Earthquakes and Faulting Mechanics of Earthquakes and Faulting www.geosc.psu.edu/courses/geosc508 Overview Milestones in continuum mechanics Concepts of modulus and stiffness. Stress-strain relations Elasticity Surface and body

More information

CIVL4332 L1 Introduction to Finite Element Method

CIVL4332 L1 Introduction to Finite Element Method CIVL L Introduction to Finite Element Method CIVL L Introduction to Finite Element Method by Joe Gattas, Faris Albermani Introduction The FEM is a numerical technique for solving physical problems such

More information

Continuum Mechanics and the Finite Element Method

Continuum Mechanics and the Finite Element Method Continuum Mechanics and the Finite Element Method 1 Assignment 2 Due on March 2 nd @ midnight 2 Suppose you want to simulate this The familiar mass-spring system l 0 l y i X y i x Spring length before/after

More information

Artificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik. Robot Dynamics. Dr.-Ing. John Nassour J.

Artificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik. Robot Dynamics. Dr.-Ing. John Nassour J. Artificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik Robot Dynamics Dr.-Ing. John Nassour 25.1.218 J.Nassour 1 Introduction Dynamics concerns the motion of bodies Includes Kinematics

More information

ME FINITE ELEMENT ANALYSIS FORMULAS

ME FINITE ELEMENT ANALYSIS FORMULAS ME 2353 - FINITE ELEMENT ANALYSIS FORMULAS UNIT I FINITE ELEMENT FORMULATION OF BOUNDARY VALUE PROBLEMS 01. Global Equation for Force Vector, {F} = [K] {u} {F} = Global Force Vector [K] = Global Stiffness

More information

Continuum Mechanics. Continuum Mechanics and Constitutive Equations

Continuum Mechanics. Continuum Mechanics and Constitutive Equations Continuum Mechanics Continuum Mechanics and Constitutive Equations Continuum mechanics pertains to the description of mechanical behavior of materials under the assumption that the material is a uniform

More information

Lecture notes Models of Mechanics

Lecture notes Models of Mechanics Lecture notes Models of Mechanics Anders Klarbring Division of Mechanics, Linköping University, Sweden Lecture 7: Small deformation theories Klarbring (Mechanics, LiU) Lecture notes Linköping 2012 1 /

More information

20. Rheology & Linear Elasticity

20. Rheology & Linear Elasticity I Main Topics A Rheology: Macroscopic deformation behavior B Linear elasticity for homogeneous isotropic materials 10/29/18 GG303 1 Viscous (fluid) Behavior http://manoa.hawaii.edu/graduate/content/slide-lava

More information

Example 3.7 Consider the undeformed configuration of a solid as shown in Figure 3.60.

Example 3.7 Consider the undeformed configuration of a solid as shown in Figure 3.60. 162 3. The linear 3-D elasticity mathematical model The 3-D elasticity model is of great importance, since it is our highest order hierarchical model assuming linear elastic behavior. Therefore, it provides

More information

General elastic beam with an elastic foundation

General elastic beam with an elastic foundation General elastic beam with an elastic foundation Figure 1 shows a beam-column on an elastic foundation. The beam is connected to a continuous series of foundation springs. The other end of the foundation

More information

Nonconservative Loading: Overview

Nonconservative Loading: Overview 35 Nonconservative Loading: Overview 35 Chapter 35: NONCONSERVATIVE LOADING: OVERVIEW TABLE OF CONTENTS Page 35. Introduction..................... 35 3 35.2 Sources...................... 35 3 35.3 Three

More information

EFFECTS OF THERMAL STRESSES AND BOUNDARY CONDITIONS ON THE RESPONSE OF A RECTANGULAR ELASTIC BODY MADE OF FGM

EFFECTS OF THERMAL STRESSES AND BOUNDARY CONDITIONS ON THE RESPONSE OF A RECTANGULAR ELASTIC BODY MADE OF FGM Proceedings of the International Conference on Mechanical Engineering 2007 (ICME2007) 29-31 December 2007, Dhaka, Bangladesh ICME2007-AM-76 EFFECTS OF THERMAL STRESSES AND BOUNDARY CONDITIONS ON THE RESPONSE

More information

A Study on Numerical Solution to the Incompressible Navier-Stokes Equation

A Study on Numerical Solution to the Incompressible Navier-Stokes Equation A Study on Numerical Solution to the Incompressible Navier-Stokes Equation Zipeng Zhao May 2014 1 Introduction 1.1 Motivation One of the most important applications of finite differences lies in the field

More information

Symmetric Bending of Beams

Symmetric Bending of Beams Symmetric Bending of Beams beam is any long structural member on which loads act perpendicular to the longitudinal axis. Learning objectives Understand the theory, its limitations and its applications

More information

MEC-E8001 Finite Element Analysis, Exam (example) 2018

MEC-E8001 Finite Element Analysis, Exam (example) 2018 MEC-E8 inite Element Analysis Exam (example) 8. ind the transverse displacement wx ( ) of the structure consisting of one beam element and point forces and. he rotations of the endpoints are assumed to

More information

COMPRESSION AND BENDING STIFFNESS OF FIBER-REINFORCED ELASTOMERIC BEARINGS. Abstract. Introduction

COMPRESSION AND BENDING STIFFNESS OF FIBER-REINFORCED ELASTOMERIC BEARINGS. Abstract. Introduction COMPRESSION AND BENDING STIFFNESS OF FIBER-REINFORCED ELASTOMERIC BEARINGS Hsiang-Chuan Tsai, National Taiwan University of Science and Technology, Taipei, Taiwan James M. Kelly, University of California,

More information

EE C247B ME C218 Introduction to MEMS Design Spring 2017

EE C247B ME C218 Introduction to MEMS Design Spring 2017 247B/M 28: Introduction to MMS Design Lecture 0m2: Mechanics of Materials CTN 2/6/7 Outline C247B M C28 Introduction to MMS Design Spring 207 Prof. Clark T.- Reading: Senturia, Chpt. 8 Lecture Topics:

More information

Continuum mechanics V. Constitutive equations. 1. Constitutive equation: definition and basic axioms

Continuum mechanics V. Constitutive equations. 1. Constitutive equation: definition and basic axioms Continuum mechanics office Math 0.107 ales.janka@unifr.ch http://perso.unifr.ch/ales.janka/mechanics Mars 16, 2011, Université de Fribourg 1. Constitutive equation: definition and basic axioms Constitutive

More information

Mathematical Background

Mathematical Background CHAPTER ONE Mathematical Background This book assumes a background in the fundamentals of solid mechanics and the mechanical behavior of materials, including elasticity, plasticity, and friction. A previous

More information

Unit 13 Review of Simple Beam Theory

Unit 13 Review of Simple Beam Theory MIT - 16.0 Fall, 00 Unit 13 Review of Simple Beam Theory Readings: Review Unified Engineering notes on Beam Theory BMP 3.8, 3.9, 3.10 T & G 10-15 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics

More information

CH.3. COMPATIBILITY EQUATIONS. Multimedia Course on Continuum Mechanics

CH.3. COMPATIBILITY EQUATIONS. Multimedia Course on Continuum Mechanics CH.3. COMPATIBILITY EQUATIONS Multimedia Course on Continuum Mechanics Overview Introduction Lecture 1 Compatibility Conditions Lecture Compatibility Equations of a Potential Vector Field Lecture 3 Compatibility

More information

BACKGROUNDS. Two Models of Deformable Body. Distinct Element Method (DEM)

BACKGROUNDS. Two Models of Deformable Body. Distinct Element Method (DEM) BACKGROUNDS Two Models of Deformable Body continuum rigid-body spring deformation expressed in terms of field variables assembly of rigid-bodies connected by spring Distinct Element Method (DEM) simple

More information

1 Hooke s law, stiffness, and compliance

1 Hooke s law, stiffness, and compliance Non-quilibrium Continuum Physics TA session #5 TA: Yohai Bar Sinai 3.04.206 Linear elasticity I This TA session is the first of three at least, maybe more) in which we ll dive deep deep into linear elasticity

More information

Stress analysis of a stepped bar

Stress analysis of a stepped bar Stress analysis of a stepped bar Problem Find the stresses induced in the axially loaded stepped bar shown in Figure. The bar has cross-sectional areas of A ) and A ) over the lengths l ) and l ), respectively.

More information

Optimal thickness of a cylindrical shell under dynamical loading

Optimal thickness of a cylindrical shell under dynamical loading Optimal thickness of a cylindrical shell under dynamical loading Paul Ziemann Institute of Mathematics and Computer Science, E.-M.-A. University Greifswald, Germany e-mail paul.ziemann@uni-greifswald.de

More information

Game Physics. Game and Media Technology Master Program - Utrecht University. Dr. Nicolas Pronost

Game Physics. Game and Media Technology Master Program - Utrecht University. Dr. Nicolas Pronost Game and Media Technology Master Program - Utrecht University Dr. Nicolas Pronost Soft body physics Soft bodies In reality, objects are not purely rigid for some it is a good approximation but if you hit

More information

2 Introduction to mechanics

2 Introduction to mechanics 21 Motivation Thermodynamic bodies are being characterized by two competing opposite phenomena, energy and entropy which some researchers in thermodynamics would classify as cause and chance or determinancy

More information

Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis

Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis uke University epartment of Civil and Environmental Engineering CEE 42L. Matrix Structural Analysis Henri P. Gavin Fall, 22 Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods

More information

Elements of Rock Mechanics

Elements of Rock Mechanics Elements of Rock Mechanics Stress and strain Creep Constitutive equation Hooke's law Empirical relations Effects of porosity and fluids Anelasticity and viscoelasticity Reading: Shearer, 3 Stress Consider

More information

Topic 5: Finite Element Method

Topic 5: Finite Element Method Topic 5: Finite Element Method 1 Finite Element Method (1) Main problem of classical variational methods (Ritz method etc.) difficult (op impossible) definition of approximation function ϕ for non-trivial

More information

ABHELSINKI UNIVERSITY OF TECHNOLOGY

ABHELSINKI UNIVERSITY OF TECHNOLOGY ABHELSINKI UNIVERSITY OF TECHNOLOGY TECHNISCHE UNIVERSITÄT HELSINKI UNIVERSITE DE TECHNOLOGIE D HELSINKI A posteriori error analysis for the Morley plate element Jarkko Niiranen Department of Structural

More information

Constitutive Relations

Constitutive Relations Constitutive Relations Andri Andriyana, Ph.D. Centre de Mise en Forme des Matériaux, CEMEF UMR CNRS 7635 École des Mines de Paris, 06904 Sophia Antipolis, France Spring, 2008 Outline Outline 1 Review of

More information

Rock Rheology GEOL 5700 Physics and Chemistry of the Solid Earth

Rock Rheology GEOL 5700 Physics and Chemistry of the Solid Earth Rock Rheology GEOL 5700 Physics and Chemistry of the Solid Earth References: Turcotte and Schubert, Geodynamics, Sections 2.1,-2.4, 2.7, 3.1-3.8, 6.1, 6.2, 6.8, 7.1-7.4. Jaeger and Cook, Fundamentals of

More information

BHAR AT HID AS AN ENGIN E ERI N G C O L L E G E NATTR A MPA LL I

BHAR AT HID AS AN ENGIN E ERI N G C O L L E G E NATTR A MPA LL I BHAR AT HID AS AN ENGIN E ERI N G C O L L E G E NATTR A MPA LL I 635 8 54. Third Year M E C H A NICAL VI S E M ES TER QUE S T I ON B ANK Subject: ME 6 603 FIN I T E E LE ME N T A N A L YSIS UNI T - I INTRODUCTION

More information

Math 4263 Homework Set 1

Math 4263 Homework Set 1 Homework Set 1 1. Solve the following PDE/BVP 2. Solve the following PDE/BVP 2u t + 3u x = 0 u (x, 0) = sin (x) u x + e x u y = 0 u (0, y) = y 2 3. (a) Find the curves γ : t (x (t), y (t)) such that that

More information

CHAPTER 5. Beam Theory

CHAPTER 5. Beam Theory CHPTER 5. Beam Theory SangJoon Shin School of Mechanical and erospace Engineering Seoul National University ctive eroelasticity and Rotorcraft Lab. 5. The Euler-Bernoulli assumptions One of its dimensions

More information

AA242B: MECHANICAL VIBRATIONS

AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS 1 / 57 AA242B: MECHANICAL VIBRATIONS Dynamics of Continuous Systems These slides are based on the recommended textbook: M. Géradin and D. Rixen, Mechanical Vibrations: Theory

More information

16.21 Techniques of Structural Analysis and Design Spring 2003 Unit #5 - Constitutive Equations

16.21 Techniques of Structural Analysis and Design Spring 2003 Unit #5 - Constitutive Equations 6.2 Techniques of Structural Analysis and Design Spring 2003 Unit #5 - Constitutive quations Constitutive quations For elastic materials: If the relation is linear: Û σ ij = σ ij (ɛ) = ρ () ɛ ij σ ij =

More information

Name (Print) ME Mechanics of Materials Exam # 1 Date: October 5, 2016 Time: 8:00 10:00 PM

Name (Print) ME Mechanics of Materials Exam # 1 Date: October 5, 2016 Time: 8:00 10:00 PM Name (Print) (Last) (First) Instructions: ME 323 - Mechanics of Materials Exam # 1 Date: October 5, 2016 Time: 8:00 10:00 PM Circle your lecturer s name and your class meeting time. Gonzalez Krousgrill

More information

Chapter 3 Stress, Strain, Virtual Power and Conservation Principles

Chapter 3 Stress, Strain, Virtual Power and Conservation Principles Chapter 3 Stress, Strain, irtual Power and Conservation Principles 1 Introduction Stress and strain are key concepts in the analytical characterization of the mechanical state of a solid body. While stress

More information

By drawing Mohr s circle, the stress transformation in 2-D can be done graphically. + σ x σ y. cos 2θ + τ xy sin 2θ, (1) sin 2θ + τ xy cos 2θ.

By drawing Mohr s circle, the stress transformation in 2-D can be done graphically. + σ x σ y. cos 2θ + τ xy sin 2θ, (1) sin 2θ + τ xy cos 2θ. Mohr s Circle By drawing Mohr s circle, the stress transformation in -D can be done graphically. σ = σ x + σ y τ = σ x σ y + σ x σ y cos θ + τ xy sin θ, 1 sin θ + τ xy cos θ. Note that the angle of rotation,

More information

Midterm Examination. Please initial the statement below to show that you have read it

Midterm Examination. Please initial the statement below to show that you have read it EN75: Advanced Mechanics of Solids Midterm Examination School of Engineering Brown University NAME: General Instructions No collaboration of any kind is permitted on this examination. You may use two pages

More information

MECHANICS OF MATERIALS

MECHANICS OF MATERIALS CHATR Stress MCHANICS OF MATRIALS and Strain Axial Loading Stress & Strain: Axial Loading Suitability of a structure or machine may depend on the deformations in the structure as well as the stresses induced

More information

Professor Terje Haukaas University of British Columbia, Vancouver The M4 Element. Figure 1: Bilinear Mindlin element.

Professor Terje Haukaas University of British Columbia, Vancouver   The M4 Element. Figure 1: Bilinear Mindlin element. Professor Terje Hakaas University of British Colmbia, ancover www.inrisk.bc.ca The M Element variety of plate elements exist, some being characterized as Kirchhoff elements, i.e., for thin plates, and

More information

CONSTITUTIVE RELATIONS FOR LINEAR ELASTIC SOLIDS

CONSTITUTIVE RELATIONS FOR LINEAR ELASTIC SOLIDS Chapter 9 CONSTITUTIV RLATIONS FOR LINAR LASTIC SOLIDS Figure 9.1: Hooke memorial window, St. Helen s, Bishopsgate, City of London 211 212 CHAPTR 9. CONSTITUTIV RLATIONS FOR LINAR LASTIC SOLIDS 9.1 Mechanical

More information

NONLINEAR STRUCTURAL DYNAMICS USING FE METHODS

NONLINEAR STRUCTURAL DYNAMICS USING FE METHODS NONLINEAR STRUCTURAL DYNAMICS USING FE METHODS Nonlinear Structural Dynamics Using FE Methods emphasizes fundamental mechanics principles and outlines a modern approach to understanding structural dynamics.

More information

Analytical Mechanics: Variational Principles

Analytical Mechanics: Variational Principles Analytical Mechanics: Variational Principles Shinichi Hirai Dept. Robotics, Ritsumeikan Univ. Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Variational Principles 1 / 71 Agenda

More information

Lecture 7: The Beam Element Equations.

Lecture 7: The Beam Element Equations. 4.1 Beam Stiffness. A Beam: A long slender structural component generally subjected to transverse loading that produces significant bending effects as opposed to twisting or axial effects. MECH 40: Finite

More information

Soft Bodies. Good approximation for hard ones. approximation breaks when objects break, or deform. Generalization: soft (deformable) bodies

Soft Bodies. Good approximation for hard ones. approximation breaks when objects break, or deform. Generalization: soft (deformable) bodies Soft-Body Physics Soft Bodies Realistic objects are not purely rigid. Good approximation for hard ones. approximation breaks when objects break, or deform. Generalization: soft (deformable) bodies Deformed

More information

FINAL EXAMINATION. (CE130-2 Mechanics of Materials)

FINAL EXAMINATION. (CE130-2 Mechanics of Materials) UNIVERSITY OF CLIFORNI, ERKELEY FLL SEMESTER 001 FINL EXMINTION (CE130- Mechanics of Materials) Problem 1: (15 points) pinned -bar structure is shown in Figure 1. There is an external force, W = 5000N,

More information

CH.11. VARIATIONAL PRINCIPLES. Continuum Mechanics Course (MMC)

CH.11. VARIATIONAL PRINCIPLES. Continuum Mechanics Course (MMC) CH.11. ARIATIONAL PRINCIPLES Continuum Mechanics Course (MMC) Overview Introduction Functionals Gâteaux Derivative Extreme of a Functional ariational Principle ariational Form of a Continuum Mechanics

More information

Energy Considerations

Energy Considerations Physics 42200 Waves & Oscillations Lecture 4 French, Chapter 3 Spring 2016 Semester Matthew Jones Energy Considerations The force in Hooke s law is = Potential energy can be used to describe conservative

More information

3D and Planar Constitutive Relations

3D and Planar Constitutive Relations 3D and Planar Constitutive Relations A School on Mechanics of Fibre Reinforced Polymer Composites Knowledge Incubation for TEQIP Indian Institute of Technology Kanpur PM Mohite Department of Aerospace

More information

3. Numerical integration

3. Numerical integration 3. Numerical integration... 3. One-dimensional quadratures... 3. Two- and three-dimensional quadratures... 3.3 Exact Integrals for Straight Sided Triangles... 5 3.4 Reduced and Selected Integration...

More information

MHA042 - Material mechanics: Duggafrågor

MHA042 - Material mechanics: Duggafrågor MHA042 - Material mechanics: Duggafrågor 1) For a static uniaxial bar problem at isothermal (Θ const.) conditions, state principle of energy conservation (first law of thermodynamics). On the basis of

More information

Linearized theory of elasticity

Linearized theory of elasticity Linearized theory of elasticity Arie Verhoeven averhoev@win.tue.nl CASA Seminar, May 24, 2006 Seminar: Continuum mechanics 1 Stress and stress principles Bart Nowak March 8 2 Strain and deformation Mark

More information

THREE-DIMENSIONAL SIMULATION OF THERMAL OXIDATION AND THE INFLUENCE OF STRESS

THREE-DIMENSIONAL SIMULATION OF THERMAL OXIDATION AND THE INFLUENCE OF STRESS THREE-DIMENSIONAL SIMULATION OF THERMAL OXIDATION AND THE INFLUENCE OF STRESS Christian Hollauer, Hajdin Ceric, and Siegfried Selberherr Institute for Microelectronics, Technical University Vienna Gußhausstraße

More information

Using MATLAB and. Abaqus. Finite Element Analysis. Introduction to. Amar Khennane. Taylor & Francis Croup. Taylor & Francis Croup,

Using MATLAB and. Abaqus. Finite Element Analysis. Introduction to. Amar Khennane. Taylor & Francis Croup. Taylor & Francis Croup, Introduction to Finite Element Analysis Using MATLAB and Abaqus Amar Khennane Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup, an informa business

More information

Other state variables include the temperature, θ, and the entropy, S, which are defined below.

Other state variables include the temperature, θ, and the entropy, S, which are defined below. Chapter 3 Thermodynamics In order to complete the formulation we need to express the stress tensor T and the heat-flux vector q in terms of other variables. These expressions are known as constitutive

More information

The Generalized Interpolation Material Point Method

The Generalized Interpolation Material Point Method Compaction of a foam microstructure The Generalized Interpolation Material Point Method Tungsten Particle Impacting sandstone The Material Point Method (MPM) 1. Lagrangian material points carry all state

More information

Chapter 3 Variational Formulation & the Galerkin Method

Chapter 3 Variational Formulation & the Galerkin Method Institute of Structural Engineering Page 1 Chapter 3 Variational Formulation & the Galerkin Method Institute of Structural Engineering Page 2 Today s Lecture Contents: Introduction Differential formulation

More information

Getting started: CFD notation

Getting started: CFD notation PDE of p-th order Getting started: CFD notation f ( u,x, t, u x 1,..., u x n, u, 2 u x 1 x 2,..., p u p ) = 0 scalar unknowns u = u(x, t), x R n, t R, n = 1,2,3 vector unknowns v = v(x, t), v R m, m =

More information

ENGN 2340 Final Project Report. Optimization of Mechanical Isotropy of Soft Network Material

ENGN 2340 Final Project Report. Optimization of Mechanical Isotropy of Soft Network Material ENGN 2340 Final Project Report Optimization of Mechanical Isotropy of Soft Network Material Enrui Zhang 12/15/2017 1. Introduction of the Problem This project deals with the stress-strain response of a

More information

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 13

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 13 Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras (Refer Slide Time: 00:25) Module - 01 Lecture - 13 In the last class, we have seen how

More information