Hierarchic parametrization (5p-hier.)

The approach of adding a difference vector onto the director of the undeformed con-figuration is most commonly used in FEA due to reduced continuity requirements on the applied function spaces. In this section, alternatively, a hierarchic Reissner-Mindlin shell model is derived, which imposes the transverse shear on the rotated director of the 3-parameter Kirchhoff-Love model of Section 4.3. Mechanically, both continuous models (5p-stand. and 5p-hier.) yield the same result. With regard to finite element discretiza-tion, however, the hierarchic parametrization of the current director will remove certain locking effects to be specified in more detail in Chapter 5.

The continuity requirements on the displacement functions for the proposed hierarchic shell model (5p-hier.) are identical to those of the 3-parameter formulation of the pre-vious section, i.e. C¹, which can, as mentioned before, be naturally satisfied with the higher-continuity NURBS discretizations of Chapter 3.

The idea of splitting the total deformation of the director of shear deformable struc-tures into a component related to bending and an additional independent component due to shear is quite natural and was frequently used in classical theories on beams,

plates and shells for more than fifty years. The textbookBaşar and Krätzig(1985), although not the first scientific work on this topic, describes in an illustrative manner the split of the entire rotation of the shell director in a contribution with respect to the deformed shell normal (Kirchhoff-Love) and an additional rotation related to shear.

The original motivation and teaching purpose inBaşar and Krätzig(1985) was to derive Kirchhoff-Love theories from shear deformation theories by simply removing the transverse shear contribution. For FEA, this approach, moreover, offers the possibil-ity of an independent parametrization of the shear deformations. This consequently avoids incompatibilities of the discrete function spaces in the kinematic equations for the transverse shear and thus transverse shear locking.

Kinematics

The imposition of a difference vectorw^5p-hier., which exclusively accounts for shear de-formations, on the rotated directora^3p3 of the Kirchhoff-Love formulation is displayed schematically in Figure 4.3 and defined in Equation (4.21).

e1

e₂ e₃

A₁

A2

A3

a₁

a2

a^5p-hier.₃ Φ×A3

a^3p₃ w^5p-hier.

w^5p-stand.

θ¹

θ² θ³= 0

r x^5p

Figure 4.3:Deformed configuration – 5-parameter models.

a^5p-hier.3 = a^3p3

*+,-membrane+

bending

+ w^5p-hier.

* +, -transverse shear

=a^5p-stand.3

(4.21)

The hierarchic difference vectorw^5p-hier.is a function of the in-plane convective coor-dinatesθ^αonly, as described for the standard Reissner-Mindlin model. Inextensibility of the director can therefore be again defined by expressing the components ofw^5p-hier.

with respect to the covariant base vectorsAα

w^5p-hier.= ˜w¹A1+ ˜w²A2. (4.22) If the vectorsa3^3p and a^5p-hier.3 are normalized, their differencew^5p-hier. exactly corre-sponds to the shear angleγ. For a shell thickness oft, the hierarchic difference vector is related to the shear angle and half of the shell thickness. For thin shells witht→0, the solution of the shear-deformable Reissner-Mindlin shell asymptotically converges to-wards the Kirchhoff-Love solution, whereas the removal ofw^5p-hier.from Equation (4.21) directly leads to the3-parameter Kirchhoff-Love model. This concept has been applied in a similar way inLong et al.(2012) in the context of smooth subdivision surfaces for accurately modeling both thin and thick shells.

The entire displacement field of the hierarchic Reissner-Mindlin shell consists of contri-butions of the Kirchhoff-Love model and the hierarchic difference vectorw^5p-hier.. Partial derivation ofu^5p-hier., required for computing the Green-Lagrange strain tensor coeffi-cients, yields

u^5p-hier._,α =u^3p_,α+θ³w^5p-hier._,α ,

u^5p-hier.,3 =u^3p,3+w^5p-hier.. (4.23)

With the covariant base vectors of Equation (4.2), the linearized Green-Lagrange strain tensorε^5p-hier.is defined

ε^5p-hier. =ε^5p-hier._ij Gⁱ⊗G^j, with

ε^5p-hier._ij =1 2

&

u^5p-hier.,i ·Gj+u^5p-hier.,j ·Gi '.

(4.24)

The computation of the strain tensor coefficients follows directly from Equations (4.2), (4.12), (4.22), (4.23) and (4.24), again neglecting quadratic contributions inθ³

ε^5p-hier.11 =ε^3p11 +θ^3&w^5p-hier.,1 ·A1 ',

2ε^5p-hier.₁₂ = 2ε^3p12+θ^3&w^5p-hier.,1 ·A2+w^5p-hier.,2 ·A1 ',

ε^5p-hier.22 =ε^3p22 +θ^3&w^5p-hier.,2 ·A2 ',

2ε^5p-hier.13 =w^5p-hier.·A1, 2ε^5p-hier.₂₃ =w^5p-hier.·A2, ε^5p-hier.₃₃ = 0.

(4.25)

Important to notice in Equation (4.25) are the transverse shear strain coefficientsε^5p-hier._α3 , which only consist of the constant componentsw^5p-hier.·Aαthrough the thickness of the shell. Linear terms do not show up due to the inextensibility constraint.

The discrete hierarchic shell element (5p-hier.) with pure displacement formulation to be derived in Chapter 5 with equal order interpolation of both the mid-surface dis-placements and the difference vectors will be free from transverse shear locking. The condition of zero transverse shear strains for the case of pure bending can be easily established by setting the hierarchic difference vector to zero.

The standard model of Section 4.4.1, on the other hand, has additional contributions with first derivatives of the mid-surface displacement fieldvin the strain tensor coeffi-cientsε^5p-stand.α3 – see Equation (4.19). This may lead to transverse shear locking in the discrete finite element model with equal order interpolation of both the mid-surface and difference vector displacements. Further details with regard to the discrete shell models and the problem of locking are provided in Chapter 5.

Constitutive law

The constitutive law of the hierarchic 5-parameter shell model is identical to the standard formulation of Section 4.4.1 and will therefore not be repeated here.

Internal virtual work

The internal virtual work of the hierarchic 5-parameter Reissner-Mindlin shell can now be derived with the strain tensor of Equation (4.24) and the modified material tensorC^5p

δΠ5p-hier., int

Pvw =

.

Ω

δ^&ε^5p-hier.'T:C^5p:ε^5p-hier.dΩ. (4.26)

4.5 3D shell model (7p)

For the previously presented thin to moderately thick shell models with three or five parameters, respectively, several simplifications and assumptions on the kinematics in thickness direction of the shell body have been defined, which also require a modification of the material law to ensure an asymptotically correct formulation. For thick shells, however, effects in thickness direction become more and more pronounced and have to be accounted for.

In this section, two 7-parameter shell formulations are derived, which represent an ex-tension of the 5-parameter Reissner-Mindlin models of Section 4.4. The 3D shells incor-porate extensibility of the director in thickness direction and enable the application of three-dimensional constitutive laws without the need of modifications.