Strain-Displacement Relation for Plates

1 Strain-Displacement Relation for Plates 1.1 1.1.1 1-D Strain Measure Engineering Strain Engineering strain ε is defined as the relative displacement: ε= ds − ds0 ds0 (1) where ds0 is the increment of initial lenght and ds is the increment of current length. 1.1.2 Green-Lagrangian Strain Instead of comparing the length, one can compare the square of lengths: ds2 − ds20 2ds20 ds − ds0 ds + ds0 = ds0 2ds0 E= (2) Where ds → ds0 , the second term is Eq. (2) tends to unity, and the Green strain measure and the engineering strain become identical. Equation (2) can be put into an equivalnet form: (3) ds2 − ds20 = 2Eds20 which will now be generalized to the 3-D case. 1.2 1.2.1 3-D Strain Measure Derivation of Green-Lagrangian Strain Tensor for Plates Let define the following quanties: • a = [ai ]: vector of the initial (material) coordinate system • x = [xi ]: vector of the current (spatial) coordinate system • u = [ui ]: displacement vector where the index i = 1, 2, 3. The relation between those quantities is: xi = ai + ui dxi = dai + dui (4) du da dx u x a O Now, the squares of the initial and the current length increment can be written in terms of ai and ui : ds20 = dai daj δ ij (5) ds2 = dxi dxj δ ij (6) = (dai + dui ) (daj + duj ) δ ij where the Kronecker tensor δ ij reads: ¯ ¯ ¯1 0 0¯ ¯ ¯ δ ij = ¯¯ 0 1 0 ¯¯ ¯0 0 1¯ (7) The vector u can be considered as a function of: • the initial (material) coordinate system, u (a), which leads to Lagrangian descripion, or • the current (spatial) coordinates, u (x), which leads to the Eulerian descrip tion In structural mechanics, the Lagrangian description is preferable: ui = ui (ai ) ∂ui dui = dak = ui ,k dak ∂ak ∂uj duj = dal = uj ,l dal ∂al (8) Let us calculated the difference in the length square: ds2 − ds20 = (dai + dui ) (daj + duj ) δ ij − dai daj δ ij (9) Using Eq. (8) and the definition of δ ij , the difference in the length square can be transformed into: ds2 − ds20 = (duj dai + dui daj + dui duj ) δ ij (10) = (uj ,l dal dai + ui ,k dak daj + ui ,k dak uj ,l dal ) δ ij = [uj ,l (daj δ jl ) dai + ui ,k (dai δ ik ) daj + ui ,k uj ,l (dai δ ik ) (daj δ jl )] δ ij = (uj ,i +ui ,j +ui ,k uj ,k ) dai daj = 2Eij dai daj where, by analogy with the 1-D case, the Lagrangian or Green strain tensor Eij is defined: 1 (11) Eij = (ui ,j +uj ,i +uk ,i uk ,j ) 2 In the case of small displacement gradient (uk ,i ¿ 1), the second nonlinear term can be neglected leading to the defintion of the infinitesimal strain tensor: εij = 1 (ui ,j +uj ,i ) 2 (12) From the defintion, the strain tensor is symmetric εij = εji , which can be seen by intechanign the indices i for j and j for i. In the moderately large deflection theory of structures, the nonlinear terms are important. Therefore, Eq. (11) will be used as a starting point in the development of the general theory of plates. Components of Green-Lagrangian Strain Tensor Let define the following range convention for indices: • Greek letters: α, β, ... = 1, 2 • Roman letters: i, j, ... = 1, 2, 3 With this range convention, the Roman letters are also written as: i = α, 3 (13) j = β, 3 The Lagrangian or Green strain tensor can be expressed: E11 E12 # E13 # E22 # E23 = Eij = E21 " " " # " " E31 E32 # E33 Eαβ " Eα 3 # # E3β # " # " # E33 where Eαβ is the in-plane component of strain tensor, Eα3 and E3β are out-of-plane shear components of strain tensor, and E33 is the through-thickness component of strain tensor. Similarily, displacement vector can be divided into two components: u1 u uα ui = u2 = v = " " " u3 w w where uα is the in-plane components of the displacement vector, and u3 = w is the out-of-plane components of the displacement vector and also called as the trans verse displacement. Initial Undeformed Configuration a z u x x uα u3 middle surface η ξ Deformed Configuration Assumptions of the von Karman Theory The von Karman thoery of mod erately large deflection of plates assumes: 1. The plate is thin. The thickness h is much smaller than the typical plate dimension, h ¿ L. 2. The magnitude of the transverse deflection is of the same order as the thickness of plate, |w| = O (h). In practice, the present theory is still a good engineering approximation for deflections up to ten plate thickness. 3. Gradients of in-plane displacements uα ,β are small so that their product or square can be neglected. 4. Love-Kirchhoff hypothesis is satisfied. In-plane displacements are a linear function of the z−coordiate (3-coordinate). uα = u◦α − z u3 ,α (14) where u◦α is the displacement of the middle surface, which is independent of z−coordinate, i.e. u◦α ,3 = 0; and u3 ,α is the slope which is negative for the "smiling" beam. z 5. The out-of-plane displacement is independent of the z−coordiante, i.e. u3 ,3 = 0. 1.2.2 Specification of Strain-Displacement Relation for Plates In the theory of moderately large deflections, the strain-displacement relation can be specified for plates. In-Plane Terms of the Strain Tensors From the general expression, Eq. (11), the 2-D in-plane componets of the strain tensor are: Eαβ = 1 (uα ,β +uβ ,α +uk ,α uk ,β ) 2 (15) Here, consider the last, nonlinear term: uk ,α uk ,β = u1 ,α u1 ,β +u2 ,α u2 ,β +u3 ,α u3 ,β (16) = uγ ,α uγ ,β +u3 ,α u3 ,β In the view of the Assumption 3, the first term in the above equation is zero, uγ ,α uγ ,β ' 0. Therefore, the 2-D in-plane components of strain tensor reads: Eαβ = 1 (uα ,β +uβ ,α +w,α w,β ) 2 (17) where w = u3 . Introducing Eq. (14) into Eq. (17), i.e. applying Love-Kirchhoff hypothesis, one gets: ¡ ¢ ¤ 1£ (18) Eαβ = (u◦α − z w,α ) ,β + u◦β − z w,β ,α +w,α w,β 2 ¢ 1¡ ◦ = uα ,β +u◦β ,α −2 z w,αβ +w,α w,β 2 ¢ 1¡ ◦ 1 = uα ,β +u◦β ,α − z w,αβ + w,α w,β 2 2 From the definiton of the curvature, one gets: καβ = −w,αβ (19) Now, Eq. (18) can be re-casted in the form: ◦ Eαβ = Eαβ + z καβ (20) ◦ is composed of a linear and a where the strain tensor of the middle surface Eαβ nonlinear term: ¢ 1 1¡ ◦ ◦ Eαβ = (21) uα ,β +u◦β ,α + w,α w,β 2 2 In the limiting case of small displacements, the second term can be neglected as compared to the first term. In the classical bending theory of plate, the in-plane displacements are assumed to be zero uα = 0 so that strains are only due to the curvatue: (22) Eαβ = z καβ where καβ ¯ ¯ ¯ ¯ ¯ ¯ κ11 κ12 ¯ ¯ ¯ ¯ ¯ = − =¯ ¯ κ21 κ22 ¯ ¯ ¯ ∂2w ∂x2 ∂2w ∂x∂y ∂2w ∂x∂y ∂2w ∂y2 ¯ ¯ ¯ ¯ ¯ = −w,αβ ¯ ¯ ¯ (23) In the above equation, κ11 and κ22 are curvatures of the cylindrical bending, and κ12 is the twist which tells how the slope in the x−direction changes with the y−direction: µ ¶ ∂ ∂w κ12 = ∂y ∂x for a cylinder κ12 = 0 Interpretation of the linear terms: be expressed in the followings: 1 2 ³ ´ u◦α ,β +u◦β ,α Each component can 1 (u1 ,1 +u1 ,1 ) = u1 ,1 = 2 1 = (u2 ,2 +u2 ,2 ) = u2 ,2 = 2 ε11 = ε22 ε12 |if 1 1 ε12 = (u1 ,2 +u2 ,1 ) = 2 2 1 du1 u2 =0 = 2 dy µ du1 dx du2 dy du1 du2 + dy dx (24) (25) ¶ (26) u2 ≡ 0 ε11 ε12 x x x y y ε22 y u1 Therefore, ε11 and ε22 are the tensile strain in the two directions, and ε12 is the change of angles, i.e. shear strain. Interpretation of the nonlinear term: 12 w,α w,β Let α = 1 and β = 1. Then, the nonlienar term reads: ¯ µ ¶ ¯ 1 1 dw dw 1 dw 2 ¯ w,α w,β ¯ = = (27) 2 dx dx 2 2 dx α=1,β=1 One can also obtain the same quantity by the defintion of 1-D Green-Lagrangian strain: ¡ 2 ¢ µ ¶ µ ¶ ds0 + dw2 − ds20 1 dw 2 1 dw 2 ds2 − ds20 E= ' = = (28) 2 ds0 2 dx 2ds20 2ds20 dx x ds0 = dx ds0 z, w ds ds 2 = ds02 + dw2 dw Thus, the conclusion is that the nonlinear term 12 w,α w,β represents the change of length of the plate element due to finite rotations. Out-Of-Plane Terms of the Strain Tensors Refering to the definition intro duced in Section 1.2.1, there are three other componets of the strain tensor: E3β , Eα3 and E33 . Using the general expression for the components of the strain tensor, Eq. (11), it can be shown that the application of Assumption 4 and 5 lead to the following expressions: 1 (u3 ,β +uβ ,3 +uk ,3 uk ,β ) 2 1 = [u3 ,β +uβ ,3 + (u1 ,3 u1 ,β +u2 ,3 u2 ,β +u3 ,3 u3 ,β )] 2 1 = [u3 ,β −u3 ,β + (−u3 ,1 u1 ,β −u3 ,2 u2 ,β )] 2 1 = (−u3 ,1 u1 ,β −u3 ,2 u2 ,β ) 2 1 = − w,γ uγ ,β 2 E3β = 1 (uα ,3 +u3 ,α +uk ,α uk ,3 ) 2 1 = [uα ,3 +u3 ,α + (u1 ,α u1 ,3 +u2 ,α u2 ,3 +u3 ,α u3 ,3 )] 2 1 = [−u3 ,α +u3 ,α + (−u1 ,α u3 ,1 −u2 ,α u3 ,2 )] 2 1 = (−u1 ,α u3 ,1 −u2 ,α u3 ,2 ) 2 1 = − w,γ uγ ,α 2 Eα3 = (29) (30) 1 (u3 ,3 +u3 ,3 +uk ,3 uk ,3 ) 2 i 1h = u3 ,3 + (u1 ,3 )2 + (u2 ,3 )2 + (u3 ,3 )2 2 i 1h = (u1 ,3 )2 + (u2 ,3 )2 2 i 1h = (−u3 ,1 )2 + (−u3 ,2 )2 2 1 = w,γ w,γ 2 E33 = (31) The above are all second order terms which vanish for small deflection theory of plates. In the theory of moderately larege deflection of plates, the out-of-plate shear strains as well as the through-thickness strain is not zero. Therefore, an assumption "plane remains plane," expressed by Eq. (14), does not mean that "normal remains normal." The existance of the out-of-plane shear strain means that lines originally normal to the middle surface do not remain normal to the deformed plate. However, the incremental work of these strains with the corresponding stresses is negligible: E3β σ 3β , Eα3 σ α3 and E33 σ 33 , are small (32) because the corresponding stress σ 3β , σ α3 and σ 33 are small as compared to the in-plane stress σ αβ . One can conclude that the elastic strain energy (and even plastic dissipation) is well approximated using the plane strain assumption: Z Z 1 1 σ ij εij dz ' σ αβ εαβ dz (33) 2 2 h h