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 Two-dimensional Elastic Deformation Stress-strain Relationship Piecewise Linear Approximation Approximating Potential Energy Dynamic Deformation 2 Three-dimensional Elastic Deformation Stress-strain Relationship Piecewise Linear Approximation Approximating potential energy 3 Summary Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 2 / 59

Stress-strain Relationship Displacement vector P(ξ, η) u(ξ, η) v(ξ, η) arbitrary point inside object displacement of point P(ξ, η) along ξ-axis displacement of point P(ξ, η) along η-axis displacement vector u = [ u v Note: vector u depends on ξ and η. ] Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 3 / 59

Stress-strain Relationship Cauchy strain B P(ξ,η) A + v η P u η + B u ξ A v ξ Deformation of small square region u v = extension along ξ-axis, = extension along η-axis ξ η u v = shear rotation, = shear + rotation η ξ Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 4 / 59

Stress-strain Relationship Cauchy strain normal strain component along ξ-axis at point P: ε ξξ = u ξ normal strain component along η-axis at point P: shear strain at point P: ε ηη = v η 2ε ξη = u η + v ξ Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 5 / 59

Stress-strain Relationship Stress-strain relationship pseudo strain vector (strain vector): ε = ε ξξ ε ηη 2ε ξη σ ξξ σ ηη σ ξη normal stress component along ξ-axis at point P normal stress component along η-axis at point P a shear stress component at point P σ = σ ξξ σ ηη σ ξη Note: σ T ε represents energy density (energy per unit volume) Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 6 / 59

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 λ + 2µ λ 0 D = λ λ + 2µ 0 0 0 µ where λ and µ denote Lamé s constants Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 7 / 59

Stress-strain Relationship Stress-strain relationship in linear elastic material Elasticity matrix of an linear isotropic material: D = λi λ + µi µ where I λ = 0 0 0 0 0, I µ = 2 0 0 0 2 0 0 0 Lamé s constants characterize linear isotropic elasticity: λ = νe ( + ν)( 2ν), µ = E 2( + ν) where Young s modulus E and Poisson s ratio ν Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 8 / 59

Stress-strain Relationship Potential energy potential energy density of linear elatic material: S 2 σt ε = 2 εt Dε potential energy stored in an elastic object: 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 / 59

Piecewise Linear Approximation Cover of region by triangles region S cover by triangles P i nodal point of a triangle [ ξ i, η 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 0 / 59

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 / 59

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 = 2 ξ i η i ξ j η j = 2 (ξ iη j η i ξ j ) Signed area of triangle P i P j P k : P i P j P k = 2 ξ j ξ i ξ k ξ i η j η i η k η i = 2 {(ξ iη j ξ j η i ) + (ξ j η k ξ k η j ) + (ξ k η i ξ i η k )} Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 2 / 59

Piecewise Linear Approximation Shape functions two-dimensional shape function N i,j,k (ξ, η) on triangle P i P j P k : N i,j,k (ξ, η) = (η j η k )ξ (ξ j ξ k )η + (ξ j η k ξ k η j ) 2 P i P j P k Note that { at point Pi N i,j,k (ξ, η) = 0 at point P j and P k Note: P(ξ, η) be an arbitrary point within the triangle: N i,j,k (ξ, η) = PP jp k P i P j P k assume that function N i,j,k (ξ, η) vanishes outside P i P j P k Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 3 / 59

Piecewise Linear Approximation Shape functions Any function u(ξ, η) can be linearly approximated inside P i P j P k : u(ξ, η) = u i N i,j,k (ξ, η) + u j N j,k,i (ξ, η) + u k N k,i,j (ξ, η) where u i = u(ξ i, η i ), u j = u(ξ j, η j ), u k = u(ξ k, η k ) Partial derivatives of N i,j,k (ξ, η): N i,j,k ξ = η j η k 2 P i P j P k, N i,j,k η = (ξ j ξ k ) 2 P i P j P k. Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 4 / 59

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 5 / 59

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 = 2 η j η k η k η i η i η j, b = 2 normal and shear strain components inside : ξ j ξ k ξ k ξ i ξ i ξ j ε ξξ = a T γ u, ε ηη = b T γ v 2ε ξη = b T γ u + a T γ v Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 6 / 59

Approximating Potential Energy Potential energy density of potential energy of an isotropic linear elastic material: 2 εt Dε = 2 εt (λi λ + µi µ )ε = 2 λ(ε ξξ + ε ηη ) 2 + 2 µ { 2ε 2 ξξ + 2ε 2 ηη + (2ε ξη ) 2} potential energy stored in element P i P j P k : where G λ = 2 G µ = 2 U i,j,k = λg λ + µg µ (ε ξξ + ε ηη ) 2 h ds P i P j P k P i P j P k 2 ( ε 2 ξξ + ε 2 ηη ) h ds + 2 P i P j P k (2ε ξη ) 2 h ds Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 7 / 59

Approximating Potential Energy Potential energy computing G λ : where L aa = L ab = G λ = 2 γt Lγ = 2 are constant matrices. [ γ T u γ T v ] [ L aa L ab L ba L bb ] [ γu aa T h ds = aa T h L bb = bb T h ds = bb T h ab T h ds = ab T h, L ba = ba T h ds = ba T h γ v ] Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 8 / 59

Approximating Potential Energy Potential energy computing G µ : where G µ = 2 γt Mγ = 2 [ γ T u γ T v ] [ M aa M ab M ba M bb ] [ γu γ v ] M aa = 2L aa + L bb, M bb = 2L bb + L aa M ab = L ba, M ba = L ab are constant matrices. Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 9 / 59

Approximating Potential Energy Permutation matrix permutation converting u i,j,k into γ: u i u j u k v i v j v k = γ = Pu i,j,k u i v i u j v j u k v k P : permutation matrix Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 20 / 59

Approximating Potential Energy Potential energy G λ = 2 γt Lγ = 2 (Pu i,j,k) T L(Pu i,j,k ) = 2 ut i,j,kj λ i,j,ku i,j,k G µ = 2 γt Mγ = 2 (Pu i,j,k) T M(Pu i,j,k ) = 2 ut i,j,kj µ i,j,k u i,j,k where Ji,j,k λ = P T LP, J µ i,j,k = P T MP partial connection matrices Potential energy U i,j,k = λg λ + µg µ = 2 ut i,j,kk i,j,k u i,j,k where K i,j,k = λj λ i,j,k + µj µ i,j,k stiffness matrix Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 2 / 59

Approximating Potential Energy 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 = λj λ i,j,k + µj µ i,j,k λ and µ are physical parameters. Ji,j,k λ and J µ i,j,k are geometric; they include no physical parameters. Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 22 / 59

Approximating Potential Energy Example η P3 P4 P5 T T3 T0 T2 P0 P P2 ξ square divided into 4 triangles: T 0 = P 0 P P 3, T = P P 4 P 3, T 2 = P P 2 P 4, T 3 = P 2 P 5 P 4 thickness h is constantly equal to 2 Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 23 / 59

Approximating Potential Energy Example T 0 = P 0 P P 3 : a = [,, 0 ] T, b = [, 0, ] T 0 0 0 0 L = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 J 0,,3 λ = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 24 / 59

Approximating Potential Energy Example T 0 = P 0 P P 3 : a = [,, 0 ] T, b = [, 0, ] T 3 2 0 2 2 0 0 0 0 M = 0 0 0 0 3 2 0 0 0 0 0 2 0 2 3 2 0 3 0 2 Jµ 0,,3 = 2 0 2 0 0 0 0 0 0 0 0 2 0 0 0 2 Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 25 / 59

Approximating Potential Energy Example T = P P 4 P 3 : a = [ 0,, ] T, b = [,, 0 ] T 0 0 0 0 0 0 0 0 J,4,3 λ = 0 0 0 0 0 0 0 0 0 0 0 0 J,4,3 µ = 0 0 0 2 0 2 0 0 0 3 2 2 3 0 0 0 2 0 2 0 0 0 Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 26 / 59

Approximating Potential Energy Example T 2 = P P 2 P 4 : a = [,, 0 ] T, b = [, 0, ] T J,2,4 λ = J 0,,3 λ, Jµ,2,4 = Jµ 0,,3 T 3 = P 2 P 5 P 4 : a = [ 0,, ] T, b = [,, 0 ] T J 2,5,4 λ = J,4,3 λ, Jµ 2,5,4 = Jµ,4,3 Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 27 / 59

Approximating Potential Energy Example adding the contribution of matrix J 0,,3 λ to connection matrix J λ : 0 0 0 0 0 0 0 0 0 0 0 0 J λ = 0 0 0 0 0 0 0 0 Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 28 / 59

Approximating Potential Energy Example adding the contribution of matrix J,4,3 λ to connection matrix J λ : 0 0 0 0 0 0 0 0 0 0 0 0 J λ = 0 0 0 0 0 0 0 0 0 0 0 0 Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 29 / 59

Approximating Potential Energy Example adding the contribution of matrix J,2,4 λ to connection matrix J λ : 0 0 0 0 2 0 0 0 0 0 2 0 0 2 0 0 J λ = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 2 Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 30 / 59

Approximating Potential Energy Example adding the contribution of matrix J 2,5,4 λ to connection matrix J λ : 0 0 0 0 2 0 0 0 0 0 2 0 0 2 0 0 0 0 J λ = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 0 2 0 0 0 0 0 0 Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 3 / 59

Approximating Potential Energy Example computing J µ yields: 3 2 0 3 0 2 2 0 6 2 0 2 6 0 0 4 2 0 3 0 0 J µ = 0 3 0 0 2 0 3 0 2 0 0 2 0 0 3 2 0 2 6 2 0 4 0 0 6 0 2 3 2 0 3 Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 32 / 59

Approximating Potential Energy Example potential energy: u N = u 0 u u 2 u 3 u 4 u 5 U = 2 ut NKu N where K = λj λ + µj µ Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 33 / 59

Approximating Potential Energy Example P3 P0 T0 T P4 P T2 P5 T3 p P2 edge P 0 P 3 is fixed on a rigid wall uniform pressure p = [ p ξ, p η ] T is exerted over edge P 2 P 5 Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 34 / 59

Approximating Potential Energy Example work done by the pressure W = [ (P2 P 5 h/2) p (P 2 P 5 h/2) p ] T [ u2 u 5 ] = 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 35 / 59

Approximating Potential Energy Example u 0 = 0 and u 3 = 0 are integrated into A T u N = 0, where 0 0 0 0 0 0 0 0 0 0 I 2 2 O 0 0 0 0 O O 0 0 0 0 A = O O O I 2 2 = 0 0 0 0 0 0 0 O O 0 0 0 O O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 36 / 59

Approximating Potential Energy Example Variational principle in statics under geometric constraints minimize J(u N, λ A ) = 2 u N T Ku N f T extu N λ T A A T u N where λ A is a collective vector consisting of four Lagrange multipliers [ K J u N = Ku N f ext Aλ A = 0, J λ A = A T u N = 0 A A T 0 4 4 ] [ un λ A ] = [ fext 0 ] Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 37 / 59

Dynamic Deformation Kinetic energy ρ: density of an object at point P(ξ, η) kinetic energy inside = P i P j P k : 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 38 / 59

Dynamic Deformation Kinetic energy assume that density ρ is constant: T i,j,k = [ ] 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 2 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 = 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 39 / 59

Dynamic Deformation Dynamic equation Lagrangian under geometric constraints L = T U + W + λ T A A T u N = 2 u N T M u N 2 u N T Ku N + f T extu N + λ T A A T u N, where λ 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 + Aλ A Mü N = 0 Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 40 / 59

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 Aλ 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 ) λ 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 4 / 59

Dynamic Deformation Example (E = 30, ν = 0.35 and c = 20, ν vis = 0.35) 80 80 60 60 40 40 20 20 0 0-20 0 20 40 60-20 0 20 40 60 0 s 0 s 80 80 60 60 40 40 20 20 0 0-20 0 20 40 60-20 0 20 40 60 20 s 30 s Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 42 / 59

Dynamic Deformation Question Report #5 due date : Jan. (Wed.) 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. p P2 P3 P4 P8 P9 P0 P4 P5 P6 P5 P P7 P0 P P2 P3 Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 43 / 59

Three-dimensional Elastic Deformation Stress-strain Relationship 3D Cauchy strain C C B B A P A P Deformation of small cubic region Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 44 / 59

Three-dimensional Elastic Deformation Stress-strain Relationship 3D Cauchy strain u v w = ext. along ξ, = ext. along η, = ext. along ζ, ξ η ζ u v = shear in ξη rot ard ζ, = shear in ξη + rot ard ζ, η ξ v w = shear in ηζ rot ard ξ, = shear in ηζ + rot ard ξ, ζ η w u = shear in ζξ rot ard η, = shear in ζξ + rot ard η ξ ζ Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 45 / 59

Three-dimensional Elastic Deformation Stress-strain Relationship 3D Cauchy strain normal strain components: ε ξξ = u ξ, shear strain components: 2ε ηζ = v ζ + w η, strain vector: ε ηη = v η, 2ε ζξ = w ξ + u ζ, ε = ε ξξ ε ηη ε ζζ 2ε ηζ 2ε ζξ 2ε ξη ε ζζ = w ζ 2ε ξη = u η + v ξ Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 46 / 59

Three-dimensional Elastic Deformation Stress-strain Relationship Stress-strain relationship stress vector: σ = σ ξξ σ ηη σ ζζ σ ηζ σ ζξ σ ξη stress-strain relationship in linear elastic material: σ = Dε Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 47 / 59

Three-dimensional Elastic Deformation Stress-strain Relationship Stress-strain relationship elasticity matrix of an isotropic material: D = λi λ + µi µ where [ O I λ = O O ] [ 2I O, I µ = O I ] and =, I = 0 0 0 0 0 0, O = 0 0 0 0 0 0 0 0 0 Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 48 / 59

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 = ξ i ξ j ξ k 6 η i η j η k ζ i ζ j ζ k signed volume of tetrahedron P i P j P k P l : P i P j P k P l = ξ j ξ i ξ k ξ i ξ l ξ i 6 η j η i η k η i η l η i ζ j ζ i ζ k ζ i ζ l ζ i Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 49 / 59

Three-dimensional Elastic Deformation Piecewise Linear Approximation Shape functions 3D shape function N i,j,k,l (ξ, η, ζ) on tetrahedron P i P j P k P l : N i,j,k,l (ξ, η, ζ) = PP { jp k P l at point Pi = P i P j P k P l 0 within P j P k P l linear approximation of function u(ξ, η, ζ) inside P i P j P k P l : u(ξ, η, ζ) = u i N i,j,k,l (ξ, η, ζ) + u j N j,k,l,i (ξ, η, ζ) + u k N k,l,i,j (ξ, η, ζ) + u l N l,i,j,k (ξ, η, ζ) Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 50 / 59

Three-dimensional Elastic Deformation Piecewise Linear Approximation Shape functions N i,j,k,l ξ N i,j,k,l η where PP j P k P l = = a j,k,l, P i P j P k P l ξ 6 P i P j P k P l = N i,j,k,l b j,k,l, = c j,k,l 6 P i P j P k P l ζ 6 P i P j P k P l a j,k,l = η j ζ j b j,k,l = ζ j ξ j c j,k,l = ξ j η j η k ζ k ζ k ξ k ξ k η k + η k ζ k + ζ k ξ k + ξ k η k η l ζ l ζ l ξ l ξ l η l + η l ζ l + ζ l ξ l + ξ l η l η j ζ j ζ j ξ j ξ j η j Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 5 / 59

Three-dimensional Elastic Deformation Approximating potential energy Potential energy potential energy in = P i P j P k P l : 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 / 59

Three-dimensional Elastic Deformation Approximating potential energy Potential energy partial derivatives: where a = 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 = 6 b j,k,l b k,l,i b l,i,j b i,j,k, c = 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 / 59

Three-dimensional Elastic Deformation Approximating potential energy Potential energy normal strain components: shear strain components: ε ξξ = a T γ u, ε ηη = b T γ v, ε ζζ = c T γ w 2ε ηζ = c T γ v + b T γ w, 2ε ζξ = a T γ w + c T γ u, 2ε ξη = b T γ u + a T γ v density of potential energy of an isotropic linear elastic material: 2 εt Dε = 2 λ(ε ξξ + ε ηη + ε ζζ ) 2 + 2 µ { 2ε 2 ξξ + 2ε 2 ηη + 2ε 2 ζζ + (2ε ηζ ) 2 + (2ε ζξ ) 2 + (2ε ξη ) 2} Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 54 / 59

Three-dimensional Elastic Deformation Approximating potential energy Potential energy potential energy in = P i P j P k P l : where G λ = 2 G µ = 2 U i,j,k,l = λg λ + µg µ. {ε ξξ + ε ηη + ε ζζ } 2 dv P i P j P k P l { 2ε 2 ξξ + 2ε 2 ηη + 2ε 2 ζζ + (2ε ηζ ) 2 + (2ε ζξ ) 2 + (2ε ξη ) 2} dv Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 55 / 59

Three-dimensional Elastic Deformation Approximating potential energy Potential energy G λ = 2 γt Lγ = 2 [ γ T u γ T v γ T w ] L aa L ab L ac L ba L bb L bc L ca L cb L cc γ u γ v γ w L aa = aa T, L bb = bb T, L cc = cc T L ab = L T ba = ab T, L bc = L T cb = bc T, L ca = L T ac = ca T G µ = 2 γt Mγ = [ ] M aa M ab M ac γ u γ T 2 u γv T γw T M ba M bb M bc γ v M ca M cb M cc γ w M aa = 2L aa + L bb + L cc, M bb = 2L bb + L cc + L aa, M cc = 2L cc + L aa + L bb, M ab = L ba, M ba = L ab, M bc = L cb, M cb = L bc, M ca = L ac, M ac = L ca Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 56 / 59

Three-dimensional Elastic Deformation Approximating potential energy Permutation matrix permutation converting u i,j,k,l into γ: γ = Pu i,j,k,l 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 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 / 59

Three-dimensional Elastic Deformation Approximating potential energy Potential energy potential energy G λ = 2 ut i,j,kj λ i,j,k u i,j,k, J λ i,j,k = P T LP G µ = 2 ut i,j,kj µ i,j,k u i,j,k, J µ i,j,k = P T MP where U i,j,k,l = 2 ut i,j,k,l K i,j,k,l u i,j,k,l K i,j,k,l = λj λ i,j,k,l + µj µ i,j,k,l Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 58 / 59

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 59 / 59