Universität Karlsruhe (TH) - am Institut für Baustatik
Universität Karlsruhe (TH) - am Institut für Baustatik Universität Karlsruhe (TH) - am Institut für Baustatik
and thus from (50) ¯B δv R = N ε F T −1 ∫ N T σ B δv R dA = 0 . (56) Ω e Thus we conclude for a unique solution that the material tangent matrix must be at least positive semidefinite δv T k m δv ≥ 0 ∀ δv , (57) with k m according to (51). The equal sign holds for the admissible rigid body motions. For the present quadrilateral element with n u = 20 degrees of freedom and n r =6rigidbody modes eq. (57) imposes the following restrictions on N σ and N ɛ : (i) The columns of N σ and N ε are linear independent, or equivalent F is positive definite. (ii) The number of stress and strain parameters n ε must fulfill n ε ≥ n u − n r = 14. (iii) In view of (55)and (56) the constraint ker[ ¯B] =ker[B] yields the requirement that the rank of matrix G must be equal to n ε . The conditions are fulfilled for the present element. It possesses with 6 zero eigenvalues which correspond to the rigid motions the correct rank. C Numerical integration of the stress resultants The relation between the strains at a layer point with coordinate ξ 3 and the shell strains (4) is rewritten in matrix notation ⎡ ⎤ ε 11 ⎡ ⎤ ⎡ E 11 1 0 0 ξ 3 ⎤ ε 0 0 0 0 22 E 22 0 1 0 0 ξ 3 0 0 0 2ε 12 2E 12 = 0 0 1 0 0 ξ 3 0 0 κ 11 ⎢ ⎥ ⎢ ⎥ ⎣ 2E 13 ⎦ ⎣ 0 0 0 0 0 0 1 0 ⎦ κ 22 (58) 2E 23 0 0 0 0 0 0 0 1 2κ 12 ⎢ ⎥ ⎣ γ 1 ⎦ γ 2 E m = A ε The kinematic shell model assumes inextensibility in thickness direction. Thus the plane stress condition has to be enforced which then yields the thickness strains, see e.g. [41]. Therefore the reduced vector S m =[S 11 ,S 22 ,S 12 ,S 13 ,S 23 ] T with components of the Second Piola–Kirchhoff stresses is introduced. Inserting (58) in the internal virtual work expression of the body δW i = ∫ ∫ Ω (ξ 3 ) δET mS m ¯μdξ 3 d Ω yields the vector of the stress resultants ∂εW = (h/2) ∫ (−h/2) A T S m ¯μ dξ 3 (59) where ¯μ denotes the determinant of the so–called shifter tensor. 26
The plane stress condition S 33 (E 33 ) = 0 is iteratively enforced. For this purpose the increment of the Second Piola–Kirchhoff stress tensor is written as follows [ ] [ dS m C dS 33 = mm C m3 ][ ] dEm C 3m C 33 dE 33 (60) dS = ¯C dE where ¯C denotes the linearized Second Piola–Kirchhoff stress tensor which is determined within a three–dimensional stress analysis at the considered layer point. The Taylor series of the plane stress condition is aborted after the linear term and set to zero S 33 (E (i) 33 +ΔE (i) 33 )=S 33(i) + ∂S33(i) ΔE (i) ∂E (i) 33 =0 with 33 and the solution yields the update formula ∂S 33(i) ∂E (i) 33 = C 33(i) (61) E (i+1) 33 = E (i) 33 − S33(i) C 33(i) (62) where i denotes the iteration number. Thus the nonlinear scalar equation S 33 (E 33 )=0is iteratively solved for the unknown thickness strains using Newton´s scheme. One obtains the stress vector S m and the tangent matrix ¯C with submatrices according to (60). For constitutive equations with a linear relation between S and E only one iteration step is necessary. This is e.g. the case for isotropy when using the so–called St.Venant Kirchhoff law. Finally the linearization of the stress resultants (59) considering the kinematic equation (58) yields C = ∂2 W ∂ε 2 = (h/2) ∫ (−h/2) A T ∂Sm ∂E m ∂E m ∂ε ¯μ dξ3 = (h/2) ∫ (−h/2) A T ˜Cmm A ¯μ dξ 3 (63) where ˜C mm = C mm − C m3 (C 33 ) −1 C 3m is obtained by static condensation of dE 33 in (60). Summarizing the algorithm provides an interface to arbitrary nonlinear three–dimensional material laws. It requires the computation of S and ¯C which is a standard output of any nonlinear three–dimensional stress analysis. It is important to note that in the Hu–Washizu functional (19) the independent strains ε enter into the constitutive model, and thus in (58). The thickness integration in (59) and(63) is performed numerically by summation over layers and with two Gauss integration points for each layer. The numerical computations show that four layers are sufficient when considering inelastic material behaviour. Since we use an orthogonal basis system at the element center ¯μ = 1 holds only at the center. The numerical tests however show that ¯μ = 1 can be set for the whole element and convergence against the correct solution is given. In case of a constant matrix ˜C mm , or a constant matrix for each layer in laminated shells, an analytical thickness integration is possible. 27
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and thus from (50)<br />
¯B δv R = N ε F T −1 ∫<br />
N T σ B δv R dA = 0 . (56)<br />
Ω e<br />
Thus we conclude for a unique solution that the material tangent matrix must be at least<br />
positive semidefinite<br />
δv T k m δv ≥ 0 ∀ δv , (57)<br />
with k m according to (51). The equal sign holds for the admissible rigid body motions. For<br />
the present quadrilateral element with n u = 20 degrees of freedom and n r =6rigidbody<br />
modes eq. (57) imposes the following restrictions on N σ and N ɛ :<br />
(i) The columns of N σ and N ε are linear independent, or equivalent F is positive definite.<br />
(ii) The number of stress and strain par<strong>am</strong>eters n ε must fulfill n ε ≥ n u − n r = 14.<br />
(iii) In view of (55)and (56) the constraint ker[ ¯B] =ker[B] yields the requirement that the<br />
rank of matrix G must be equal to n ε .<br />
The conditions are fulfilled for the present element. It possesses with 6 zero eigenvalues which<br />
correspond to the rigid motions the correct rank.<br />
C<br />
Numerical integration of the stress resultants<br />
The relation between the strains at a layer point with coordinate ξ 3 and the shell strains (4)<br />
is rewritten in matrix notation<br />
⎡ ⎤<br />
ε 11<br />
⎡ ⎤ ⎡<br />
E 11 1 0 0 ξ 3 ⎤<br />
ε<br />
0 0 0 0<br />
22<br />
E 22<br />
0 1 0 0 ξ 3 0 0 0<br />
2ε 12<br />
2E 12<br />
=<br />
0 0 1 0 0 ξ 3 0 0<br />
κ 11<br />
⎢ ⎥ ⎢<br />
⎥<br />
⎣ 2E 13 ⎦ ⎣ 0 0 0 0 0 0 1 0 ⎦<br />
κ 22<br />
(58)<br />
2E 23 0 0 0 0 0 0 0 1<br />
2κ 12<br />
⎢ ⎥<br />
⎣ γ 1 ⎦<br />
γ 2<br />
E m = A ε<br />
The kinematic shell model assumes inextensibility in thickness direction. Thus the plane<br />
stress condition has to be enforced which then yields the thickness strains, see e.g. [41].<br />
Therefore the reduced vector S m =[S 11 ,S 22 ,S 12 ,S 13 ,S 23 ] T with components of the Second<br />
Piola–Kirchhoff stresses is introduced.<br />
Inserting (58) in the internal virtual work expression of the body δW i = ∫ ∫<br />
Ω (ξ 3 ) δET mS m ¯μdξ 3 d Ω<br />
yields the vector of the stress resultants<br />
∂εW =<br />
(h/2) ∫<br />
(−h/2)<br />
A T S m ¯μ dξ 3 (59)<br />
where ¯μ denotes the determinant of the so–called shifter tensor.<br />
26
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