3 Balance equations ME338A CONTINUUM MECHANICS

Transcription

1 ME338A CONTINUUM MECHANICS lecture otes 1 thursy, may 1, 28 Basic ideas util ow: kiematics, i.e., geeral statemets that characterize deformatio of a material body B without studyig its physical cause ow: balace equatios, i.e., geeral statemets that characterize the cause of motio of material body B Basic strategy / recipe [1] isolatio of arbitrary subset P S of the body S [2] characterizatio of the ifluece of the remaiig body B\P S o P S through pheomeological quatities, i.e., the cotact stress t ad the cotact heat flu q [3] defiitio of basic physical quatities, i.e., mass M, liear mometum I, momet of mometum D, ad eergy E of subset P S [4] postulate of balace of these quatities reders global balace equatios for subset P S [5] localizatio of global balace equatios reders local balace equatios at poit P S 75

2 3.1 Cocept of stress Cosider a part P B Bcut off from the referece body B ad its spatial couterpart P S Sclosed by the respective bouries P B ad P S. I the deformed cofiguratio, we itroduce the tractio vector t T S Δ f t = lim Δa Δa = d f (3.1.1) that acts o the surface elemet of P S ad represets the mechaical actio d f of the rest of the body at the viciity P S \So P S. B P B T PB Cauchy s postulate N F F t t P S P S The tractio vector t at poit ad time t ca be epressed eclusively i terms of the poit, the time t, ad the ormal T S to the taget plae to a imagiary surface P S passig through this poit. t = t (, t; ) (3.1.2) Cauchy s lemma The tractio vectors actig o opposite sides of a surface are 76 S equal i magitude ad opposite i sig. t(, ) = t(, ) (3.1.3) Cauchy s theorem The spatial tractio vector t depeds liearly o the spatial ormal to the surface P S through the Cauchy or true stress tesor σ, t(, t; ) := σ(, t) (3.1.4) The Cauchy stress tesor σ ca thus be uderstood as a mappig trasformig ormals T S oto taget vectors t T S. T S T S, σ := (3.1.5) t = σ Cauchy tetrahedro The above represeatio ca be illustrated i terms of the Cauchy tetrahedro. The equilibrium of all tractio vectors actig o the tetrahedro yields the followig epressio. t() = t( i ) i = t(e i ) i = t i i (3.1.6) With the help of the surface theorem, the area fractios ca be related as follows. i = i i = e i i = e i = cos (e i, ) (3.1.7) The tractio vectors t the take the atural iterpretatio as a liear map of the correspodig surface ormals. t() =t i i = t i cos (e i, ) =t i [e i ] =[t i e i ] (3.1.8) 77

3 t 1 2 = e 2 t 2 e 3 e 1 e 2 1 = e 1 t e 3 σ 33 σ 13 σ 23 σ 32 σ 22 σ 31 e 1 e 2 σ12 σ 11 σ 21 3 = e 3 A compariso with t = σ yields the iterpretatio of secod order stress tesor as σ = t i e i. For Barry, we ca epress the Cauchy stress as σ = t i e j = σ ij e i e j ad rewrite the Cauchy theorem (3.1.4) i ide represetatio. t = σ ij e i e j k e k = σ ij k δ jk e i = σ ij j e i = t i e i. (3.1.9) The matri represetatio of tesor coordiates of σ ij ca the be epressed as σ 11 σ 12 σ 13 t t 1 [σ ij ]= σ 21 σ 22 σ 23 = t t 2 (3.1.1) σ 31 σ 32 σ 33 givig rise to the followig geometric iterpretatio i terms of the tractio vectors o the tetrahedral surfaces. t t 3 t 1 = [ σ 11 σ 12 σ 13 ] t t 2 = [ σ 21 σ 22 σ 23 ] t (3.1.11) t 3 = [ σ 31 σ 32 σ 33 ] t 78 t 3 I our otatio, the first ide of the stress tesor represets the directio of the correspodig tractio vector ad the secod ide idicates the surface ormal. The diagoal etries of the matri of the stress compoets represet the ormal stresses, the off-diagoal terms are related to the shear stress. The pressure p correspods to the average of all ormal stresses. p = 1 3 tr(σ) =1 3 [ σ 11 + σ 22 + σ 33 ] (3.1.12) Kirchhoff stress Aother spatial stress measure, the Kirchhoff stress tesor, also kow as the weighted Cauchy stress tesor, is defied as τ := J σ (3.1.13) ad widely used i the spatial descriptio of stress power terms i the referece volume. 79

4 First Piola Kirchhoff stress Now let us cosider aother spatial tractio vector T T S defied through the force equality T := t by scalig the spatial force term (t ) through the referece area elemet. Based o this defiitio, we itroduce the Piola stress, sometimes also referred to as the first Piola-Kirchhoff stress, by the referece Cauchy theorem T := P N (3.1.14) leadig to P N = σ. Usigtheareamap,weobtai the relatio P = τ F t = J σ F t (3.1.15) betwee the Piola stress ad the spatial Cauchy stress ad Kirchhoff stress. Notice that similar to the deformatio gradiet F, P is a two poit tesor. It possesses the followig geometrical mappig properties. T P := B T S, (3.1.16) N T = P N. Piola trasform / Piola idetity The trasformatio ( ) B = J( ) S F t (3.1.17) is called the Piola trasform. It ca be used to trasform the spatial Cauchy stress tesor ito the two poit Piola stress. It is widely employed i trasformig the objects actig o a spatial surface oto their material couterparts. The immediate outcome of the Piola trasformatio is the Piola idetity that also implies the equality Div (JF t )=. J div( ) S = Div ( ) B = Div (J( ) S F t ) (3.1.18) 8 Secod Piola Kirchhoff stress The Lagragea stress vector T T B may be defied through the pull-back of the spatial stress vector T T S. T = ϕ (t) =F 1 T T B, T A =(F 1 ) A at a The last fumetal stress measure we will itroduce for ow is the secod Piola-Kirchhoff stress tesor S, whichis defied by T := S N (3.1.19) yieldig T S := B T B, (3.1.2) N T = S N. The secod Piola-Kirchhoff stress tesor S does ot possess a real physical iterpretatio, yet it is ofte used i computatioal applicatios sice ulike P, it has a symmetric structure ad is thus easy to store computatioally. Pull back ad push forward We ca epress the secod Piola-Kirchhoff stress tesor i terms of the other stress tesors S := ϕ (P) = F 1 P, S AB = (F 1 ) A a PaB, S := ϕ (τ) = F 1 τ F t, S AB = (F 1 ) A a τab (F 1 ) B b as the pull-back of the two-poit ad spatial objects. Apparetly the coverse push-forward relatios do also hold for the spatial stress tesors as show i the diagram below. τ = J σ = ϕ (P) =P F t ad τ = ϕ (S) =F S F t, 81

5 T T B T B F S P τ = Jσ N F t Physical iterpretatio of stress measures B t T S T S Let d f be a force elemet i the spatial cofiguratio S d f = t = σ (3.1.21) d f = σ. (3.1.22) The Cauchy stress σ relates a force elemet d f of the spatial cofiguratio to a surface elemet of the spatial cofiguratio, it is thus a purely spatial quatity. Let d f be a force elemet i the spatial cofiguratio S d f = t = σ = σ = J σ F t = P (3.1.23) d f = P. (3.1.24) 82 e 1 e 3 df d f t e 2 The first Piola-Kirchhoff stress P relates a force elemet d f of the spatial cofiguratio to a surface elemet of the material cofiguratio, it is thus a two-poit quatity. Let df be a force elemet i the material cofiguratio B df = F 1 dp = F 1 P = J F 1 σ F t = S (3.1.25) df = S. (3.1.26) The secod Piola-Kirchhoff stress S relates a force elemet df of the material cofiguratio to a surface elemet of the material cofiguratio, it is thus a purely material quatity. The Cauchy stress is ofte called the true stress, because this is the stress that you ca actually measure i a eperimet. The first Piola-Kirchhoff stress relates to measurig the specime geometry before the test whereas the secod Piola-Kirchhoff stress ca ot be measured at all. 83 B S

6 Eample Holzapfel, p. 113, eample 3.1 A deformatio of a body is described by 1 = = = (3.1.27) The Cauchy stress tesor for a certai poit of the body is give by its matri represetatio as σ = 5 kn/cm2 (3.1.28) Determie the spatial / Cauchy tractio vector t ad the material / Piola tractio vector T actig o the plae characterized by the outward ormal = e 2 i the curret cofiguratio. Solutio From the give deformatio, we fid the compoets of the deformatio gradiet ad its iverse. 6 2 F = 1 2 F 1 = 1 6 (3.1.29) For this deformatio J = det(f) =1, i.e., the deformatio is isochoric / volume preservig. The compoets of the Piola tesor read. P = J σ F t = 1 kn/cm2 (3.1.3) 84 I order to fid the outward ormal N i the referece cofiguratio, we recall Naso s formula N = 1/J Ft. Hece, with the traspose of F ad J = 1 ad kowig that = e 2, we fid that N = 1 2 e 1 (3.1.31) thus, N = e 1. Fially, with Cauchy s theorem, t = σ = 5 kn/cm2 T = P N = 1 kn/cm2 (3.1.32) i.e., t = 5 e 2 ad T = 1 e 2, respectively. As ca be see, t ad T have the same directio. The magitude of T is twice that of t, because the deformed area is half the udeformed area. 85

7 3.2 Cocept of heat flu Similar to the cocept of stress, we cosider a part P B B cut off from the referece body B ad its spatial couterpart P S Sclosed by the respective bouries P B ad P S. I the deformed cofiguratio, we itroduce the heat flu q that acts o the surface elemet of P S ad represets the thermal actio of the rest of the body at the viciity P S \S o P S. B P B Q PB N F F t q P S P S S Cauchy type theorem The heat flu vector q depeds liearly o the spatial ormal to the surface P S through the Cauchy or true heat flu vector q. q(, t; ) := q(, t) (3.2.1) We ca the use the Piola trasform ( ) B = J( ) S F t to obtai the material heat flu vector Q Q = J q F t (3.2.2) q := Q N. (3.2.3) 86