Derivation of Constitutive Equations for Plates

2 Derivation of Constitutive Equations for Plates 2.1 Definitions of Bending Moment and Axial Force Hook’s law in plane stress reads: E [(1 − ν) εαβ + ν εγγ δ αβ ] 1 − ν2 σ αβ = (34) In terms of components: E (εxx + ν εyy ) 1 − ν2 E σ yy = (εyy + ν εxx ) 1 − ν2 E σ xy = εxy 1+ν σ xx = (35) Here, strain tensor can be obtained from the strain-displacement relations: εαβ = ε◦αβ + z καβ (36) Now, define the tensor of bending moment: Mαβ ≡ Z h 2 − h2 σ αβ z dz (37) and the tensor of axial force (membrane force): Nαβ ≡ 2.2 2.2.1 Z h 2 − h2 σ αβ dz (38) Bending Energy Bending Moment Let us assume that ε◦αβ = 0. The bending moment Mαβ can be calculated: E Mαβ = 1 − ν2 Z h 2 − h2 [(1 − ν ) εαβ + ν εγγ δ αβ ] z dz £ ¤ E = (1 − ν) ε◦αβ + ν ε◦γγ δ αβ 2 1−ν + = Z E [(1 − ν) καβ + ν κγγ δ αβ ] 1 − ν2 h 2 −h 2 Z z dz h 2 −h 2 Eh3 [(1 − ν) καβ + ν κγγ δ αβ ] 12 (1 − ν 2 ) z 2 dz (39) Here, we define the bending rigidity of a plate D as follows: D= Eh3 12 (1 − ν 2 ) (40) Now, one gets the moment-curvature relations: Mαβ = D [(1 − ν) καβ + ν κγγ δ αβ ] Mαβ where M12 = M21 due to symmetry. ¯ ¯ ¯ M11 M12 ¯ ¯ ¯ =¯ M21 M22 ¯ (41) (42) M11 = D (κ11 + ν κ22 ) (43) M22 = D (κ22 + ν κ11 ) M12 = D (1 − ν) κ12 2.2.2 Bending Energy Density One -Dimensional Case Here, we use the hat notation for a function of certain argument such as: M11 = M̂11 (κ11 ) (44) = D κ11 Then, the bending energy density Ūb reads : Z κ̄11 Ūb = M̂11 (κ11 ) dκ11 0 Z κ̄11 =D κ11 dκ11 (45) 0 1 = D (κ̄11 )2 2 1 Ūb = M11 κ̄11 2 (46) M11 D κ11 dκ11 General Case General definition of the bending energy density reads: I Ūb = Mαβ dκαβ (47) κ κ11 κ22 0 Calculate the energy density stored when the curvature reaches a given value κ̄αβ . Consider a straight loading path: καβ = η κ̄αβ (48) dκαβ = κ̄αβ dη Mαβ η=1 Mαβ η η=0 καβ Mαβ = M̂αβ (καβ ) = M̂αβ (η κ̄αβ ) = η M̂αβ (κ̄αβ ) καβ (49) where M̂αβ (καβ ) is a homogeneous function of degree one. I Ūb = M̂αβ (καβ ) dκαβ Z 1 = η M̂αβ (κ̄αβ ) κ̄αβ dη 0 Z 1 = M̂αβ (κ̄αβ ) κ̄αβ η dη (50) 0 1 = M̂αβ (κ̄αβ ) κ̄αβ 2 1 = Mαβ κ̄αβ 2 Now, the bending energy density reads: D [(1 − ν) κ̄αβ + ν κ̄γγ δ αβ ] κ̄αβ 2 D = [(1 − ν) κ̄αβ κ̄αβ + ν κ̄γγ κ̄αβ δ αβ ] 2 i D h = (1 − ν) κ̄αβ κ̄αβ + ν (κ̄γγ )2 2 Ūb = (51) The bending energy density expressed in terms of components: D 2 D = 2 D = 2 D = 2 D = 2 Ūb = n h i o (52) (1 − ν) (κ̄11 )2 + 2 (κ̄12 )2 + (κ̄22 )2 + ν (κ̄11 + κ̄22 )2 n h i o (1 − ν) (κ̄11 + κ̄22 )2 − 2 κ̄11 κ̄22 + 2 (κ̄12 )2 + ν (κ̄11 + κ̄22 )2 nh i h io (κ̄11 + κ̄22 )2 − 2 κ̄11 κ̄22 + 2 (κ̄12 )2 − ν −2 κ̄11 κ̄22 + 2 (κ̄12 )2 n h io (κ̄11 + κ̄22 )2 − 2 κ̄11 κ̄22 + 2 (κ̄12 )2 − ν −2 κ̄11 κ̄22 + 2 (κ̄12 )2 n h io (κ̄11 + κ̄22 )2 + 2 (1 − ν) −κ̄11 κ̄22 + (κ̄12 )2 n h io ¯b = D (κ̄11 + κ̄22 )2 − 2 (1 − ν) κ̄11 κ̄22 − (κ̄12 )2 U 2 2.2.3 (53) Total Bending Energy The total bending energy is the integral of the bending energy density over the area of plate: Z Ub = Ūb dA S (54) 2.3 2.3.1 Membrane Energy Axial Force Assume that καβ = 0. The axial force can be calculated: E Nαβ = 1 − ν2 E = 1 − ν2 Z h 2 − h2 Z h 2 − h2 E + 1 − ν2 Z [(1 − ν) εαβ + ν εγγ δ αβ ] dz (55) £ ¤ (1 − ν) ε◦αβ + ν ε◦γγ δ αβ dz h 2 − h2 [(1 − ν ) καβ + ν κγγ δ αβ ] z dz ¤ E £ = (1 − ν) ε◦αβ + ν ε◦γγ δ αβ 2 1−ν Z E + [(1 − ν ) καβ + ν κγγ δ αβ ] 1 − ν2 ¤ Eh £ = (1 − ν ) ε◦αβ + ν ε◦γγ δ αβ 2 1−ν h 2 − h2 Z dz h 2 z dz − h2 Here, we define the axial rigidity of a plate C as follows: C= Eh 1 − ν2 (56) Now, one gets the membrane force-extension relation: Nαβ = C h i (1 − ν) ε◦αβ + ν ε◦γγ δ αβ Nαβ where N12 = N21 due to symmetry. ¯ ¯ ¯ N11 N12 ¯ ¯ ¯ =¯ N21 N22 ¯ N11 = C (ε◦11 + ν ε◦22 ) N22 = C (58) (59) (ε◦22 N12 = C (1 2.3.2 (57) + ν ε◦11 ) − ν) ε◦11 Membrane Energy Density Using the similar definition used in the calculation of the bending energy density, the extension energy (membrane energy) reads: I (60) Ūm = Nαβ dε◦αβ Calculate the energy stored when the extension reaches a given value ε̄◦αβ . Consider a straight loading path: ε◦αβ = η ε̄◦αβ dε◦αβ = ¯ε◦αβ (61) dη ¡ ¢ Nαβ = N̂αβ ε◦αβ ¡ ¢ = N̂αβ η ¯ε◦αβ ¡ ¢ = η N̂αβ ε̄◦αβ (62) ³ ´ where N̂αβ ε◦αβ is a homogeneous function of degree one. Ūm = Z ε̄◦αβ 0 = Z 0 1 ¡ ¢ N̂αβ ε◦αβ dε◦αβ (63) ¡ ¢ η N̂αβ ε̄◦αβ ε̄◦αβ dη 1 ˆ ¡ ◦ ¢ ◦ = N αβ ε̄αβ ε̄αβ 2 1 = Nαβ ε̄◦αβ 2 Now, the extension energy reads: £ ¤ ¯m = C (1 − ν) ε̄◦ + ν ε̄◦γγ δ αβ ε̄◦ U αβ αβ 2 h i ¡ ¢ C 2 = (1 − ν) ε̄◦αβ ε̄◦αβ + ν ε̄◦γγ 2 (64) The extension energy expressed in terms of components: C 2 C = 2 C = 2 C = 2 Ūm = n h i o (1 − ν) (¯ε◦11 )2 + 2 (ε̄◦12 )2 + (ε̄◦22 )2 + ν (ε̄◦11 + ¯ε◦22 )2 (65) n h i o (1 − ν) (¯ε◦11 + ¯ε◦22 )2 − 2 ¯ε◦11¯ε◦22 + 2 (ε¯◦12 )2 + ν (¯ε◦11 + ¯ε◦22 )2 n h io (¯ε◦11 + ¯ε◦22 )2 − 2 ¯ε◦11¯ε◦22 + 2 (ε¯◦12 )2 − ν −2 ¯ε◦11¯ε◦22 + 2 (ε¯◦12 )2 n h io (¯ε◦11 + ¯ε◦22 )2 + 2 (1 − ν) −¯ε◦11¯ε◦22 + (ε¯◦12 )2 n h io ¯m = C (¯ε◦ + ¯ε◦ )2 − 2 (1 − ν) ¯ε◦ ¯ε◦ − (¯ε◦ )2 U 11 22 11 22 12 2 (66) 2.3.3 Total Membrane Energy The total membrane is the integral of the membrane energy density over the area of plate:: Z Um = Ūm dS (67) S