Finite Elements in Elasticit Section : Review of Elasticit Stress & Strain 2 Constitutive Theor 3 Energ Methods
Section : Stress and Strain Stress at a point Q : F = lim A 0 A F = lim A 0 A F = lim A 0 A Stress matri [ ( Q) ] = Stress vector ( ( Q) ) = : Stress and Strain (cont) Stresses must satisf equilibrium equations in pointwise manner: Strong Form 2
: Stress and Strain (cont) Stresses act on inclined surfaces as follows: S nˆ ( Q) = S = n n nˆ = [ ( Q) ]( nˆ ) ( Q) nˆ n inˆ ( Q) 2 2 = S : Stress and Strain (cont) Strain at a pt Q related to displacements : ( ) ( ) Q :,, Q :,, Displacement functions (,, ), (,, ), (,, ) u v w defined b: ( ) ( ) ( ) = + u,, ; = + v,, ; = + w,, 3
: Stress and Strain (cont) Normal strain relates to changes in sie : ε Q D QD Q D d = ; QD d D Q ( ) ( ) ( ) ( ) ( +, ) (, ) ( ) Q D = = + d + u + d, + u, = d + u + d, u, ε = d ε u d u u Q v = ( Q ) w ε = ( Q ) : Stress and Strain (cont) Shearing strain relates to changes in angle : (, ) (, ) v + d u + d v u γ = α + β = + = + d d γ γ w u = + w v = + ( Q) ( Q) ( Q) ( Q) ( Q) ( Q) 4
: Stress and Strain (cont) Sometimes FEA programs use elasticit shearing strains : ε = γ ε = γ ε = γ 2 2 2 Strains must satisf 6 compatibilit equations: 2 2 2 γ ε ε = + 2 2 Eg: (usuall automatic for most formulations) Section 2 : Constitutive Theor For linear elastic materials, stresses and strains are related b the Generalied Hooke s Law : { } ( ) ( ) [ ] ( ) ( ) = C ε ε + o o ε c c c c c c ε c c c c c c ε c c c c c c = = = γ c c c c c c γ c c c c c c γ c c c c c c ( ) ;( ε) ; [ C] 2 3 4 5 6 2 22 23 24 25 26 3 23 33 34 35 36 4 24 34 44 45 46 5 25 35 45 55 56 6 26 36 46 56 66 Elasticit matri; ( ) ( ε) o o = residual stresses = residual strains 5
2 : Constitutive Theor (cont) For isotropic linear elastic materials, elasticit matri takes special form: [ C] ν ν ν 0 0 0 ν ν ν 0 0 0 E ν ν ν 0 0 0 = 0 0 0 2 0 0 ( 2ν )( ν ) ( ν ) + 2 0 0 0 0 2 ( 2ν ) 0 0 0 0 0 0 2 ( 2ν ) E = Young's modulus, ν = Poisson's ratio 2 : Constitutive Theor (cont) Special cases of GHL: Plane Stress : all out-of-plane stresses assumed ero ε ν 0 E = ν 0 2 γ ν 0 0 2 ( ν ) ν Note: ε = ( ε + ε ) required ν ( ) = ;( ε) = ε ;[ C] Plane Strain : all out-of-plane strains assumed ero ε ( ) = ;( ε) = ε ;[ C] γ ( ) Note: = ν + required ν 0 ν 2 ν ν = 0 E ν 2 0 0 ν 6
2 : Constitutive Theor (cont) Other constitutive relations: Orthotropic : material has less smmetr than isotropic case FRP, wood, reinforced concrete, Viscoelastic : stresses in material depend on both strain and strain rate Asphalt, soils, concrete (creep), Nonlinear : stresses not proportional to strains Elastomers,, ductile ielding, cracking, 2 : Constitutive Theor (cont) Strain Energ Energ stored in an elastic material during deformation; can be recovered completel Work done during : ( )( ) dw = F + df dl FdL F = A ; dl = dε L o o ( ε )( ) dw = d A L o o ( ) ε final W = A L dε o o εo If all eternal work is stored, ε final ( ) U = W = V dε o εo 7
2 : Constitutive Theor (cont) Strain Energ Densit : strain energ per unit volume In general, final U = U V = dε U = o ε Volume ε o UdV ε final ε final ε final γ final γ final γ final U= dε + dε + dε + dγ + dγ + dγ εo εo εo γ o γ o γ o Section 3 : Energ Methods Energ methods are techniques for satisfing equilibrium or compatibilit on a global level rather than pointwise Two general tpes can be identified: Methods that assume equilibrium and enforce displacement compatibilit (Virtual force principle, complementar strain energ theorem, ) Methods that assume displacement compatibilit and enforce equilibrium (Virtual displacement principle, Castigliano s st theorem, ) Most important for FEA! 8
3 : Energ Methods (cont) Principle of Virtual Displacements (Elastic case): (aka Principle of Virtual Work, Principle of Minimum Potential Energ) Elastic bod under the action of bod force b and surface stresses T Appl an admissible virtual displacement δu Infinitesimal in sie and speed Consistent with constraints Has appropriate continuit Otherwise arbitrar PVD states that δwe δwi for an admissible is equivalent to static equilibrium = δu 3 : Energ Methods (cont) Eternal and Internal Work: ( δu) So, PVD for an elastic bod takes the form [ ]( ) δw = b δudv + T δuda = b δudv + nˆ δuda e i volume surface volume surface δw = δu = U dv volume volume { δε δε δε δγ δγ δγ } = + + + + + [ ]( ) ( ) ( ) b δudv + nˆ δuda = δε dv volume surface volume dv 9
3 : Energ Methods (cont) Recall: Integration b Parts b a b ( ) ( ) = ( ) ( ) ( ) ( ) f g d f g g f d In 3D, the corresponding rule is: g f f,,,, dv f,, g,, n da g,,,, dv ( ) ( ) = ( ) ( ) ( ) ( ) volume surface volume a b a 3 : Energ Methods (cont) Take a closer look at internal work: ( δ u) δε = δε = δ δ { } dv ( u) n da u dv volume surface volume ( δ v) δε = { δε } dv = ( δ v) nda δ v dv volume surface volume ( δ w) δε = { δε } dv = ( δ w) nda δ w dv volume surface volume ( δ v) ( δu) δγ = + δγ dv = δv n da δ v dv + δu n da δ u dv { } ( ) ( ) volume surface volume surface volume δγ dv = δ w n da δ w dv δ v n da δ v dv { } ( ) + ( ) volume surface volume surface volume δγ dv = δ w n da δ w dv δu n da δ u dv { } ( ) + ( ) volume surface volume surface volume 0
3 : Energ Methods (cont) + + n δ u δ u δwi = n d + + i δ v dv δ w i δ v A surface n δ w volume [ ]( nˆ ) δuda + + e surface [ ]( ˆ ) [ ]( ˆ ) δw = δw n δuda A δudv = b δudv + n δuda surface volume volume surface { } A + b δudv = 0 for an arbitrar δu volume A + b = 0 A B reversing the steps, can show that the equilibrium equations impl δw= δw δwi= δwe is called the weak form of static equilibrium i e 3 : Energ Methods (cont) n i= Raleigh-Rit Rit Method : a specific wa of implementing the Principle of Virtual Displacements Define total potential energ Π = Wi We ; PVD is then stated as δπ = δwi δwe = 0 Assume ou can approimate the displacement functions as a sum of known functions with unknown coefficients Write everthing in PVD in terms of virtual displacements and real displacements (Note: stresses are real, not virtual!) Using algebra, rewrite PVD in the form ( ) ( ) unknown virtual coefficient * equation involving real coefficients = 0 i Each unknown virtual coefficient generates one equation to solve for unknown real coefficients i
3 : Energ Methods (cont) Raleigh-Rit Rit Method: Eample Given: An aial bar has a length L,, constant modulus of elasticit E,, and a variable cross-sectional sectional area given b the π function A( ) = A ( +β sin o ( L )), where β is a known parameter Aial forces F and F 2 act at = 0 and = L, respectivel, and the corresponding displacements are u and u 2 Required: Using the Raleigh-Rit Rit method and the assumed displacement function u( ) = u ( L) + u2 ( L), determine the equation that relates the aial forces to the aial displacements for this element 3 : Energ Methods (cont) Solution : ) Treat u and u 2 as unknown parameters Thus, the virtual displacement is given b ( ) ( ) δ u( ) = δu L + δu2 L 2) Calculate internal and eternal work: δw = Fδu + F δ u e 2 2 (no bod force terms) δwi = δε dv = δε A( ) d ε bar bar u = = u ( ) + u ( + ) = u2 u * L 2 * L L ( L ) δε = and = E * δ u2 δ u u2 u L 2
3 : Energ Methods (cont) (Cont) : = L 2) u2 u δ u2 δ u π δwi = E *( L )* L * Ao { + β sin ( L )} d = 0 { } u2 u δ u2 δ u 2βL ( L ) L o { π } β 2 ( L )( π ) = E * * * A * L + u2 u 2 u2 u 2β { } { ( L )( π )} δw = δu * EA + δu * EA + i o o 3) Equate internal and eternal work: u2 u 2β u2 u 2β { o ( L )( π )} δ o ( L )( π ) u u2 2β ( )( ) L π EAo 2β ( + ) u2 u 2β π ( )( ) L = u L π 2 2 2 { } Fδu + Fδu = δu * EA + u * EA + For δu : F = EAo + u F For δu2 : F2 = EAo + 2 F2 3