Stretch-Based Hyperelastic Material Formulations for Isogeometric

Stretch-Based Hyperelastic Material Formulations for Isogeometric Kirchhoff-Love Shells with Application to Wrinkling Hugo Verhelst, Matthias Möller, Henk den Besten, Angelos Mantzaflaris, Mirek Kaminski

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Hugo Verhelst, Matthias Möller, Henk den Besten, Angelos Mantzaflaris, Mirek Kaminski. Stretch- Based Hyperelastic Material Formulations for Isogeometric Kirchhoff-Love Shells with Application to Wrinkling. 2021. hal-02890963v2

HAL Id: hal-02890963 https://hal.archives-ouvertes.fr/hal-02890963v2 Preprint submitted on 31 May 2021

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H.M. Verhelst1,1,, M. Moller¨ 1, J.H. Den Besten1, A. Mantzaﬂaris1, M.L. Kaminski1

aDelft University of Technology, Department of Maritime and Transport Technology, Mekelweg 2, Delft 2628 CD, The Netherlands bDelft University of Technology, Department of Applied Mathematics, Van Mourik Broekmanweg 6, Delft 2628 XE, The Netherlands cInria Sophia Antipolis - Méditerranée, UniversitéCôted’Azur 2004 route des Lucioles - BP 93, 06902 Sophia Antipolis cedex, France

Abstract Modelling nonlinear phenomena in thin rubber shells calls for stretch-based material models, such as the Ogden model which conve- niently utilizes eigenvalues of the deformation tensor. Derivation and implementation of such models have been already made in Finite Element Methods. This is, however, still lacking in shell formulations based on Isogeometric Analysis, where higher-order continuity of the spline basis is employed for improved accuracy. This paper fills this gap by presenting formulations of stretch-based material models for isogeometric Kirchhoff-Love shells. We derive general formulations based on explicit treatment in terms of derivatives of the strain energy density functions with respect to principal stretches for (in)compressible material models where determination of eigenvalues as well as the spectral basis transformations is required. Using several numerical benchmarks, we verify our formulations on invariant-based Neo-Hookean and Mooney-Rivlin models and with a stretch-based Ogden model. In addition, the model is applied to simulate collapsing behaviour of a truncated cone and it is used to simulate tension wrinkling of a thin sheet. Keywords: isogeometric analysis, Kirchhoff-Love shell, stretch-based strain energy density, arc-length methods, wrinkling

1. Introduction when applied to compressible shells [22, 23]. In the latter case, principal directions and values need to be solved using an eigen- To model phenomena like wrinkling in membranes [1–5] or value problem and a tensor transformation is required. However, the deformation of biological tissues [6–8], thin shell formula- for compressible materials no additional parameters are required. tions with non-linear hyperelastic material models are typically used. These material models are defined using a strain energy With the advent of isogeometric analysis (IGA) [24], new (density) function, which measures the strain energy stored in the spline-based shell formulations have been presented [25–27]. material when deformed [9]. Material models with strain energy The advantage of these formulations is that the geometry is ex- density functions based on the invariants (i.e. invariant-based actly preserved after discretization and that arbitrary continuity models) of the deformation tensor, such as the Neo-Hookean or of the basis functions across element boundaries provides high the Mooney-Rivlin formulations, have been widely used to study convergence rates and allows for achieving necessary continu- wrinkling or deformation of biological tissues. However, for rub- ities in variational formulations, for instance leading to rotation- ber materials or living organs such as the liver, spine, skin, rec- free Kirchhoff-Love shell formulations [25, 28]. These formu- tum, bladder or the aorta, material models defined by the eigen- lations have been used to advance the development of refine- values and eigenvectors of the deformation tensor (i.e. stretch- ment splines [29] and to optimize shell structures [30], amongst based models) such as the Ogden, Sharriff or exponential and other developments. A general hyperelastic isogeometric shell logarithmic models [10–12] provide better accuracy of material formulation has been developed for general compressible and in- characteristics with respect to experimental tests [13–15]. compressible material models [31] and specific formulations for To include hyperelastic material models into shell formula- biological membranes have been obtained [32]. Roohbakashan tions, derivatives of the strain energy density function with re- and Sauer [7] developed formulations to eliminate numerical spect to the components of the deformation tensor are required to through-thickness integration for hyperelastic Kirchhoff-Love define the stress and material tensors. For invariant-based mod- shells. Isogeometric Kirchhoff-Love shell formulations are suc- els, this is generally a straight-forward exercise, since the invari- cessfully used for biomedical applications to model aortic valve ants of the deformation tensor are defined in terms of the com- closure [33] and bioprosthetic heart valve dynamics [34, 35] as ponents of the deformation tensor. However, for stretch-based well as for industrial applications to perform buckling, vibration models, these derivatives result in stress and material tensors de- and nonlinear deformation analyses of composite wind turbine fined in the spectral basis (i.e. in terms of the eigenvectors), blades [36, 37]. However, all advances in [7, 31–34, 36] em- making incorporation of these models non-trivial. The first in- ploy the derivatives of the strain energy density function with corporation of stretch-based material models in the Finite Ele- respect to the components of the deformation tensor, thus appli- ment Method (FEM) was obtained for axisymmetric problems cation of these works is possible for invariant-based models. On [16, 17] and later the extension to generally shaped shells was the other hand, stretch-based models such as the Ogden model re- made [8, 18, 19]. In these works, either closed-form expres- quire specific treatment of the spectral deformation tensor the ex- sions of the tangents of the principal stretches [17, 19] were ob- isting generalized formulations. Contrary to the aforementioned tained, or explicit computation of principal directions and values developments in the FEM context, stretch-based material mod- [16, 20, 21] was performed. In the former case, an unknown els have not yet been considered in isogeometric Kirchhoff-Love stretching parameter is used, which can be eliminated for incom- shell formulations. pressible models [22] and, in fact, imposes numerical difficulties In this paper we present mathematical formulations for the incorporation of stretch-based material models in the isogeometric Kirchhoff-Love shell model for (in)compressible material models. This enables the use of stretch-based material models such as ∗Corresponding Author. Email addresses: [email protected] (H.M. Verhelst), the Ogden model together with the efficient Kirchhoff-Love shell [email protected] (M. Moller),¨ [email protected] (J.H. formulation in isogeometric analysis, for application on wrin- Den Besten), [email protected] (A. Mantzaflaris) kling analysis or biomechanical simulations. The formulations

Preprint submitted to SPM May 25, 2021 hold for material models defined for 3D continua which are in- A¯. The derivative of the inverse and the determinant of a tensor, tegrated over the shell thickness. We employ explicit determi- with respect to one of its components become: nation of the principal directions and values applicable to compressible and incompressible materials. The tensor transforma- ∂ tr A i j ∂ det{A} tion from the spectral to the curvilinear basis - which is needed = a , = det{A}A¯i j and for compatibility with existing codes - implies additional com- ∂Ai j ∂Ai j ¯ (2) putational costs compared to a component-based formulation. ∂A 1n −1 −1 −1 −1o These costs are minimised by using minor and major symme- = − Aik Al j + Ail Ak j . try of the hyperelastic material tensor. Besides comparison with ∂Ai j 2 analytical solutions, the model is applied to simulate structural instabilities: the collapse of a truncated cone [19] and the wrin- 2.2. Shell Coordinate System kling phenomenon in a stretched sheet. These instabilities are The Kirchhoff-Love shell element formulation is based on the captured with (extended) arc-length methods [38, 39], combined Kirchhoff Hypothesis, that is, the cross-section does not shear with IGA [40]. The former simulation reveals the complex col- and orthogonal vectors in the undeformed configuration remain lapse behaviour of the truncated cone when using the arc-length orthogonal after deformation. As a consequence, any point in method; something that was not reported in literature before. For the shell can be represented by a point on the mid-surface and a the latter simulation, this paper reports the first IGA results for contribution in normal direction: this case, compared to results from commercial FEM packages. Following the introduction of notations, preliminary identi- x = r + θ3a , (3) ties and the isogeometric Kirchhoff-Love shell formulation back- 3 grounds (Section2), we derive the stretch-based formulations in- with the shell mid-surface r(θ1, θ2) and the unit normal direction cluding numerical procedures (Section3) and discuss the isogeo- 3 1 2 3 metric Kirchhoff-Love shell implementation aspects (Section4). a3(θ ) for the deformed configuration x(θ , θ , θ ). For the unde- The model is benchmarked with analytical or reference solutions formed configuration x˚, the same relation holds with all quanti- and it is applied to model the collapse behaviour of a truncated ties decorated with a ˚·. The parametrization utilizes surface coor- cone and the wrinkling formation in a stretched thin sheet in Sec- dinates θα and the through-thickness coordinate θ3. Derivatives i tion5. Concluding remarks follow in Section6. with respect to these coordinates are denoted by (·),i = ∂(·)/∂θ . 2. The Kirchhoff-Love Shell Model The covariant basis of the mid-surface is represented by ai Using continuum mechanics and tensor calculus [41–43], the ∂r a × a a = , a = 1 2 , (4) isogeometric Kirchhoff-Love formulations [7, 25, 31, 44] are α α 3 | × | briefly summarized. For more details and elaborate derivations ∂θ a1 a2 reference is made to previous publications. and the first fundamental form is aαβ = aα · aβ. The curvature Firstly, Section 2.1 provides the notations that are used in this α β paper, as well as some preliminary tensor identities. Section 2.2 tensor b = bαβ a ⊗ a is represented by the second fundamental introduces the coordinate system and consequently the curvilin- form of surfaces, which can be obtained using the Hessian of the ear basis that are used for the Kirchhoff-Love shell formulation. surface aα,β or the derivative of the normal vector a3,α In Section 2.3 we provide the formulations of the shell kinematics, where the concepts of deformation and strain are defined. bαβ = a3 · aα,β = −a3,β · aα. (5) Lastly, Section 2.4 provides the variationel formulation of the Kirchhoff-Love shell, without specifying the constitutive rela- The derivative of the normal vector is obtained by Weingarten’s tions, since those are covered in Section3. β β αγ formula a3,α = −bαaβ with bα = a bγβ as the mixed curvature 2.1. Notations and Preliminary Identities tensor [44]. Taking the derivative of Eq. (3), the covariant basis For the ease of reference, the notations and preliminary identi- of the shell coordinate system x can be formulated as follows: ties are based on the ones used in [31]. Lower-case italic quanti- 3 ties (a) represent scalars, lower-case bold quantities (a) denote gα = x,α = aα + θ a3,α, g3 = x,3 = a3. (6) vectors. Upper-case quantities denote two-dimensional quantities; italic and non-bold (A) for matrices, italic and bold for The metric coefficients are constructed by taking the inner- second-order tensors (A). Third-order tensors are not used in the product of these basis vectors, i.e. present work, and fourth-order tensors are represented by calli- graphic capitals (A). The following product operators are de- 3 32 fined: inner product a · b, cross-product a × b and tensor product gαβ = gα · gβ = aαβ − 2θ bαβ + θ a3,α · a3,β, (7) a ⊗ b. Furthermore, we represent covariant basis vectors with j subscripts (ai) and contravariant vectors with superscript (a ). where in the second equality, Eq. (5) is used. Moreover, gα3 = 0 Latin indices take values {1, 2, 3} whereas Greek ones take values and g33 = 1 [25]. Using the definition of the covariant metric gi j, j j j the contravariant metric gi j and basis vectors gi can be found: {1, 2}. By construction, ai · a = δi , with δi the Kronecker delta. i j ⊗ Second- and fourth-order tensors are denoted by A = A ai a j = αβ −1 α αβ i j i jkl i j k l g = [gαβ] , g = g gβ. (8) Ai j a ⊗a and A = A ai ⊗a j ⊗ak ⊗al = Ai jkl a ⊗a ⊗a ⊗a , respectively, where Ai j and Ai jkl denote covariant components and i j i jkl The third contravariant basis vector g3 is again the normal vector A and A denote contravariant components. a since it has unit-length by construction (see Eq. (4)). The Einstein summation convention is adopted to represent 3 tensor operations and when summations are unclear, it is explic- Remark 1. In the isogeometric Kirchhoff-Love shell formula- itly mentioned. In this notation, the trace and determinant of a tions [25, 31], the last term in Eq. (7) is neglected because of the i j 3 2 tensor are defined for tensor A = Ai j a ⊗ a as in [31, 41, 42] thin shell assumption, meaning (θ ) takes small values. How- α ever, the co- and contravariant basis vectors (gα and g , re- i j spectively) are used in the mapping of the stretch-based material tr A = Ai ja and det{A} = Ai j / ai j , (1) matrix onto the contravariant undeformed basis (Section 4.3). To enable an accurate comparison of the invariant-based and i j i j where Ai j denotes the determinant of the matrix A, a = a · a stretch-based formulation, we do not neglect the O((θ3)2) term, −1 and ai j = ai · a j. The inverse of a tensor A is denoted by A or contrary to previous works [7, 31]. 2 2.3. Shell Kinematics its decomposition to membrane and bending contributions (ε and κ, respectively): The deformed and undeformed configurations (x and x˚, respectively) are related to each other by the mid-plane deforma- 1 1 3 tion vector u by r = r˚+u and a3 = a3(r˚+u). However, in both the Eαβ = gαβ − g˚αβ = (aαβ − a˚αβ) − 2θ bαβ − b˚αβ invariant-based and stretch-based forms that are described in this 2 2 (19) paper, the deformations are defined using the undeformed and = εαβ + θ3καβ. deformed geometries. In continuum mechanics, the deformation gradient F and the deformation tensor C are defined as [31, 43]: Remark 2. Following up on Remark1; the contribution of the O((θ3)2) term in Eq. (7) is neglected in the strain tensor and 3 2 dx > its derivatives. The O((θ ) ) term is only included in Eq. (7) to F = = g ⊗ g˚i, C = F F = g · g g˚i ⊗ g˚ j = g g˚i ⊗ g˚ j. (9) dx˚ i i j i j ensure equivalence in comparison of the stretch- and invariant- based formulations. Note that the deformation tensor is defined in the contravariant undeformed basis g˚i ⊗ g˚ j. For Kirchhoff-Love shells, it is 2.4. Variational Formulation known that gα3 = g3α = 0, hence this implies Cα3 = C3α = 0. The shell internal and external equilibrium equations in varia- Since g33 = 1, which implies C33 to be unity and meaning that tional form are derived by the principle of virtual work [25, 31]. the thickness remains constant under deformation. In hyperelas- The variations of internal and external work are defined as: tic Kirchhoff-Love shell formulations, the contribution of C33 is therefore incorporated by static condensation, where the correc- δW(u, δu) = δWint − δWext tion of C33 is performed analytically for incompressible materials Z Z (20) and iteratively for compressible materials. Therefore, we denote = n : δε + m : δκ dΩ − f · δu dΩ , the deformation tensor C and its inverse C¯ as denoted as: Ω Ω α β with δu being the virtual displacement, δε and δκ the virtual C = gαβ g˚ ⊗ g˚ + C33a˚3 ⊗ a˚3, (10) q 1 2 αβ 1 strain components, Ω the mid-surface and dΩ = a˚αβ dθ dθ C¯ = g g˚α ⊗ g˚β + a˚3 ⊗ a˚3. (11) C33 the differential area in the undeformed configuration, mapped to the integration domain Ω∗ = [0, 1]2 using the undeformed mid- From Eqs. (10) and (11), it can be observed that the thickness- plane measure. Furthermore, with slight abuse of notation, the αβ αβ contribution (index 3) is decoupled from the in-plane contribu- tensors n = n g˚α ⊗ g˚β and m = m g˚α ⊗ g˚β denote the shell tions (Greek indices α, β). This is a consequence of the Kirchhoff normal force and bending moment tensors, respectively, with Hypothesis and therefore is only valid for Kirchhoff-Love shells. ˜ α β Z Z Consequently, using the definition C = gαβ g˚ ⊗ g˚ , the trace and αβ αβ 3 αβ 3 αβ 3 determinant of C can be simplified accordingly [41, 42]: n = S dθ , m = θ S dθ . (21) [−t/2,t/2] [−t/2,t/2] ˜ αβ tr C = tr C + C33 = gαβg + C33, (12) Here, S αβ denotes the coefficients of the stress tensor following from the constitutive relations that will be derived in Section3 gαβ det{C} = det{F}2 = J2 = C = J2C = λ2λ2λ2, (13) and t stands for the shell thickness. The total differentials of the 33 0 33 1 2 3 g˚αβ stress resultants are: Z Z αβ αβγδ 3 αβγδ 3 3 where J denotes the Jacobian determinant and J0 is its in-plane dn = C dθ dεγδ + C θ dθ dκγδ , counterpart. Furthermore, the tensor invariants of C simplify to: [−t/2,t/2] [−t/2,t/2] Z Z 2 αβ 2 2 2 dmαβ = αβγδθ3 dθ3 dε + αβγδ θ3 dθ3 dκ . I1 := tr{C} = gαβg˚ + C33 = λ + λ + λ , (14) C γδ C γδ 1 2 3 [−t/2,t/2] [−t/2,t/2] 1 n o (22) I := tr{C}2 − tr C2 = C g g˚αβ + J2 2 2 33 αβ 0 Discretizing the equations using known formulations from 2 2 2 2 2 2 previous publications [25, 31, 44], the solution u is represented = λ1λ2 + λ2λ3 + λ1λ3, (15) by a finite sum of weighted basis functions and the tensors n, m, 2 2 2 ε and κ are linearized around the weights using Gateaux deriva- I3 := det{C} = λ1λ2λ3, (16) tives. The linearized tensors are denoted by (·)0 = ∂(·) in the ∂ur 2 where λi are the principal stretches of the shell and λ are the following, where ur are individual weights of the solution vec- i 0 eigenvalues of the deformation tensor C. The squares of the tor. Note that u denotes the basis functions [31]. Using the eigenvalues are the roots of the characteristic polynomial: discretized system, the residual vector is defined by: Z Z 2 3 − 2 2 2 − int ext ∂ε ∂κ ∂u (λi ) I1(λi ) + I2λi I3 = 0. (17) Rr = Fr −Fr = n : +m : dΩ− f· dΩ , (23) Ω ∂ur ∂ur Ω ∂ur Corresponding eigenvectors are denoted by vi, which are normalized to have unit-length. The eigenvalue decomposition (or and must be equal to the zero vector for the weights u corre- spectral decomposition) of the deformation tensor C can be writ- sponding to the exact solution. To solve the residual equation ten as [41, 42]: R = 0, another linearization is performed, yielding the Jacobian matrix or tangential stiffness matrix K: i j 2 C = Ci jg˚ ⊗ g˚ = λi vi ⊗ vi. (18) int ext Krs = Krs − Krs (24) Z 2 2 Where the Einstein summation convention is used. Since C33 ∂n ∂ε ∂ ε ∂m ∂κ ∂ κ is decoupled by construction, one can immediately see from = : + n : + : + m : dΩ √ ∂us ∂ur ∂ur∂us ∂us ∂ur ∂ur∂us Eqs. (10) and (18) that λ = C and v = a˚ . Ω 3 33 3 3 Z ∂f ∂u For the sake of completeness, we recall the definition of the − · dΩ . α β Green-Lagrange strain tensor E = Eαβ g˚ ⊗ g˚ from [25, 31] and Ω ∂us ∂ur 3 Note that the matrix contains a contribution for the external load Remark 3. From Eq. (18) and Eq. (28), it follows that depending on the solution vector (f(u)). For instance, follower- pressures are defined by f(u) = pn(u), where n is the surface ∂C ∂λ2 v ⊗ v i . normal. In order to solve for nonlinear equation, Newton itera- 2 = i i = (30) tions are performed for solution u and increment ∆u by solving ∂(λi ) ∂C

K∆u = −R. (25) Due to the fact that the eigenvector basis with vi is orthogonal and normalized (i.e. orthonormal), the product the basis vectors 3. Stretch-Based Constitutive Relations vi span the identity tensor: I = vi ⊗ vi. Invariant-based (in)compressible material model formulations Furthermore, it can also be shown that for the material tensor, have been obtained for the strain energy density functions Ψ(C) the following holds [17–19, 21, 41]: in component-form based on [31]. However, when experimental material data fitting is involved a formulation in terms of 1 ∂S ii S j j − S ii stretches (i.e. in terms of the eigenvalues of C, Ψ(λ) with i jkl δ jδl δkδl δlδk − δ j . C = i k + 2 2 ( i j + i j)(1 i ) (31) λ = (λ1, λ2, λ3) might be preferred, meaning that a transforma- λk ∂λk λ j − λi tion to spectral form is required. Therefore, this section provides the main contribution of this paper: the generalized formulations where the indices (i, j, k, l) refer to specific components of the for the implementation of stretch-based material models in the fourth-order material tensor, thus no summation over the indices isogeometric Kirchhoff-Love shell model. Throughout this sec- is applied. The first part of Ci jkl represents the normal compo- tion, reference is made to equations of [31] for comparison pur- nents (diagonal elements) and the second part denotes the shear poses. components (off-diagonal elements). In the formulation of the The section is structured as follows: Section 3.1 provides the component-based counterpart of this equation ([31, eq. 36]) basics for the derivation of the stretch-based constitutive rela- these parts are not explicitly visible, since the spectral form by tions. Thereafter, Section 3.2 and Section 3.3 provide the deriva- definition uses the principal directions of the deformation tensor, tions for incompressible and compressible material models, re- whereas shear and normal contributions are mixed in the curvi- spectively, in the stretch-based formulations. These formulations linear form of the material tensor. Note that for the second part are the novelty of the present paper. of this equation, the case λi = λ j results in an undefined result. 3.1. General Relations Hence, using L’Hopital’s rule, this limit case can be identified:

Assuming Ψ(λ), we derive relations for the stress and material ∂S j j ∂S ii j j ii − j j ii ! tensor in terms of the (normalized) eigenvector bases (Eq. (18)): S − S ∂λ ∂λ 1 ∂S ∂S j j − . lim 2 2 = lim = (32) λ j→λi − λ j→λi 2λ 2λ ∂λ ∂λ 3 3 λ j λi j i j j X i j X i jkl S = S vi ⊗ v j, C = C vi ⊗ v j ⊗ vk ⊗ vl. (26) Since J = λ1λ2λ3, the derivatives of J are: i, j=1 i, j,k,l=1 ∂J J ∂2 J J These equations are valid for 3D continua and hence need to be j = , = (1 − δi ) . (33) modified to incorporate the through-thickness stress components. ∂λi λi ∂λ j∂λ j λiλ j Reading Eq. (10), Cαβ = gαβ but C33 , g33 = 1 to avoid viola- tion of the plane stress condition. To correctly incorporate the 3.2. Incompressible Material Models plane-stress condition (S 33 = 0), the material tensor C is mod- For incompressible materials, the incompressibility condition ified using static condensation, which implies that the material (J = 1) is enforced using a Lagrange multiplier p in the strain tensor Cˆ corrected for plane-stress is defined by [31]: energy density function [31, 41]:

αβ33 33δγ αβγδ αβδγ C C Ψ(λi) = Ψel(λi) − p(J − 1). (34) Cˆ = C − . (27) C3333 where Ψel is the original strain energy density function. Using For incompressible materials, this term is derived analytically us- Eq. (29), the stress tensor becomes: ing the incompressibility condition (J = 1) whereas for com- ! pressible materials, it is corrected iteratively. 1 ∂Ψel ∂p ∂J S ii = − − p . (35) When S and C are known, these tensors are transformed to the λi ∂λi ∂λi ∂λi bases g˚i ⊗ g˚ j and g˚i ⊗ g˚ j ⊗ g˚k ⊗ g˚l, respectively. This will be discussed in Section 4.3. Where again, we do not sum over repeated indices. Compar- The derivative of any scalar function with respect to the defor- ing S ii with the component-based result in [31, eq. 41] shows mation tensor C can be written as a derivative with respect to the that all components can easily be obtained using substitution in stretch by applying the chain rule [41]: Eq. (28). To comply with the plane-stress condition (S 33 = 0), 3 3 3 the equation to be solved for the Langrange multiplier p using ∂(·) X ∂(·) ∂λ2 X ∂(·) X 1 ∂(·) the incompressibility condition (J = 1) denotes: i v ⊗ v v ⊗ v . = 2 = 2 i i = i i (28) ∂C ∂λ ∂C ∂λ 2λi ∂λi ! i=1 i i=1 i i=1 1 ∂Ψ ∂J el − p = 0, (36) From this, it follows that: λ3 ∂λ3 ∂λ3  1 ∂Ψ which implies, using the derivative of J from Eq. (33):  , i = j S i j = λ ∂λ (29)  i i !−1 0, i j ∂J ∂Ψel ∂Ψel , p = = λ3 . (37) ∂λ3 ∂λ3 ∂λ3 which shows that the coefficients of the stress tensor are purely diagonal and we thus refer with S ii, i = 1, ..., 3 to the non-zero It can easily be shown that Eq. (37) is similar to the expression 2 components of S. of p in the component-based form [31, eq. 46] using λ3 = C33. 4 The derivative of the stress tensor with respect to the stretch is Using this result, the in-plane incompressible material tensor can required to find the material tensor, as observed in Eq. (31). From be evaluated as: Eq. (35) it follows that: αα ββ αα αβγδ 1 ∂S β δ S − S γ δ δ γ β = δαδ + (δαδ + δ δ )(1 − δα) ii ! 2 C γ 2 2 β α β ∂S ∂ 1 ∂Ψ 1 ∂ Ψ 1 ∂Ψ λγ ∂λγ λ − λα = = − δ j β i 2  2  ∂λ j ∂λ j λi ∂λi λi ∂λi∂λ j λ ∂λi 1  ∂ Ψel ∂Ψel  β i − λ + 2 δ δδ , (47) 2 2 2  3 2  α γ 1 ∂ Ψ ∂p ∂J ∂p ∂J λ λ ∂λ ∂λ3 = el − − (38) α γ 3 λ ∂λ ∂λ ∂λ ∂λ ∂λ ∂λ i i j i j j i where the second term should be replaced by Eq. (32) if λα = λβ. 2 !! ∂ J j 1 ∂Ψel ∂J 3.3. Compressible Material Models − p − δi − p , ∂λi∂λ j λi ∂λi ∂λi For compressible models, the Jacobian determinant J is not necessarily equal to 1. As a consequence, the deformation gra- where the incompressibility condition (J = 1) is used again and dient F and deformation tensor C are modified such that F and where no summation over repeated indices is applied. Note that C are a multiplicative decomposition of a volume-changing (di- the Kronecker delta δ j covers the case when i = j. The derivative lational) part depending on J and a volume preserving (distor- i tional) part depending on the modified deformation gradient and of p follows from Eq. (37) and reads: deformation tensors, C˙ and F˙ , respectively [45]: − 1 − 2 2 ˙ 3 ˙ 3 ∂p ∂ Ψel 3 ∂Ψel F = J F, C = J C. (48) = λ3 + δi . (39) ∂λi ∂λ3∂λi ∂λ3 The modified deformation gradient and deformation tensor have determinants which are equal to 1 (corresponding to volume Again, this result can be compared to its component-based coun- preservation), meaning: terpart in [31, eq. 47] and using Eq. (28) it can be observed that n o n o ˙ ˙ ˙ ˙ ˙ these equations are similar. Substituting Eqs. (33), (37) and (39) det F = λ1λ2λ3 = 1, det C = 1, (49) and J = 1 into Eqs. (35) and (38) then yields: where the modified principal stretches λ˙ i are defined as: ! − 1 1 ∂Ψel λ3 ∂Ψel ˙ 3 S αα = − , (40) λi = J λi. (50) λα ∂λα λα ∂λ3 Consequently, the invariants of the modified deformation tensor αα " 2 2 ! ∂S 1 ∂ Ψel 1 ∂ Ψel 3 ∂Ψel C˙ become: = − λ3 + δα ∂λ λ ∂λ ∂λ λ ∂λ ∂λ ∂λ −2/3 −4/3 β α α β β 3 α 3 I˙1 = J I1, I˙2 = J I2, I˙3 = 1, (51) 2 ! β 1 ∂ Ψel ∂Ψel ∂Ψel (1 − δ ) 3 α with Ii the invariants of the deformation tensor C. With F˙ , C˙ and − λ3 + δβ − λ3 (41) λα ∂λ3∂λβ ∂λ3 ∂λ3 λαλβ I˙k as defined above, the strain energy density function Ψ(C) for !# 1 ∂Ψ 1 ∂Ψ a compressible material can be described in a decoupled form, − δβ el − λ el . separating the response in a isochoric (or distortional) elastic part α λ ∂λ λ 3 ∂λ α α α 3 Ψiso(λ˙) and an volumetric (or dilational) elastic part Ψvol(J)[41, 42, 45]: Here, we do not apply summation over repeated indices. Com- Ψ(λ) = Ψiso(λ˙) + Ψvol(J). (52) −1 parison with the invariant-based formulation shows that λi in The volumetric elastic part Ψvol is required to be strictly convex front of the second term in Eq. (40) translates to C¯i j in [31, eq. and equal to zero if and only if J = 1 and C˙ = I [41]. 49]. Using these identities, the material tensor can be derived For compressible materials, the plane stress condition is incor- from Eq. (31). For the static condensation term, reference is porated by solving S 33 = 0 for C33 using Newton linearizations made to Eq. (27), hence the components Cαβ33, C33αβ and C3333 [31, 46]: need to be evaluated. From Eq. (31) it follows that: 33 1 3333 S + C ∆C33 = 0, (53)   2 1 ∂S αα 1 ∂2Ψ ∂Ψ αβ33 β  el el  β (n+1) (n) (n) C = δα = − λ3 + 2 δα, (42) where C is incrementally updated by C = C + ∆C with λ ∂λ λ λ2  ∂λ2 ∂λ  33 33 33 33 3 3 3 α 3 3 the increment on iteration n: 3  2  1 ∂S 1  ∂ Ψel ∂Ψel  33 33γδ = δδ = − λ + 2 δδ , (43) S C γ 2  3 2  γ (n) (n) λγ ∂λγ λ λ ∂λ ∂λ C − . 3 γ 3 3 ∆ 33 = 2 3333 (54) 3 2 C(n) 3333 1 ∂S 1 ∂ Ψel ∂Ψel C = = − λ3 + 2 , (44) λ ∂λ λ3 ∂λ2 ∂λ In each iteration, the updated stress tensor S and material tensor 3 3 3 3 3 can be computed and iterations are continued until the plane C stress condition is satisfied within a certain tolerance, i.e. S 33 < such that the static condensation term becomes: tol. When converged, static condensation can be performed for the material tensor using Eq. (27). Rather than using C(0) = 1 2 2 33 1 ∂ Ψel ∂Ψel (0) −2 αβ33 33γδ λ2λ2 λ2 λ3 ∂λ2 + 2 ∂λ [31], C = J is used for incompressible materials, although C C 3 α γ 3 3 β δ 33 0 = − δαδγ (45) 3333 2 the difference for the two approaches is negligible. C 1 ∂ Ψel ∂Ψel 3 λ3 2 + 2 Using Eq. (50), any volumetric strain energy density function λ ∂λ ∂λ3 3 3 for incompressible materials can be transformed to its compress-  2  ible material equivalent by substituting Eq. (50) into Eq. (52) and 1  ∂ Ψel ∂Ψel  β δ = − λ3 + 2 δαδ . (46) by selecting a volumetric component Ψ . Static condensation λ2 λ2  2 ∂λ  γ vol α γ ∂λ3 3 (Eq. (27)) is performed before transforming the material tensor. 5 4. Implementation Aspects undeformed covariant curvilinear basis by: In this section, we recall the assembly of the nonlinear system X3 for isogeometric Kirchhoff-Love shells (Section 4.1) as well as S˜ i j = S pq(v · g˚i)(v · g˚ j), the computation of the eigenvalues and eigenvectors of the de- p q p,q=1 formation tensor C (Section 4.2). Then we provide details about (60) the transformation of the stress and material tensors S and C from X3 ˜ i jkl pqrs i j k l spectral to curvilinear bases (Section 4.3). C = C (vp · g˚ )(vq · g˚ )(vr · g˚ )(vs · g˚ ), 4.1. System Assembly p,q,r,s=1

For the implementation of Kirchhoff-Love shells recall that the where S˜ i j and C˜ i jkl are the coefficients of the stress and material vector of internal forces and the tangential stiffness matrix read tensors in the curvilinear basis. [25, 31]: Obviously, the tensor transformation only needs to be computed for non-zero components of Cpqrs. For incompressible ma- Z ! C int > ∂ε¯ > ∂κ¯ terial models, the plane-stress correction for 33 is applied ana- Fr = n¯ + m¯ dΩ , (55) lytically, which implies that the transformations only need to be Ω ∂ur ∂ur applied for indices ranging from α, β, γ, δ = 1, 2, thus the trans- Z ! 2 4 ¯ 0 ∂ε¯ ¯ 1 ∂κ¯ ∂ε¯ > ∂ ε¯ formation consists of mapping 2 = 16 entries. However, it is Krs = D + D + n¯ (56) known that for hyperelastic materials the contravariant compo- Ω ∂us ∂us ∂ur ∂ur∂us nents of the material tensor, i jkl, posses minor and major sym- ! 2 ! C ∂ε¯ ∂κ¯ ∂ε¯ > ∂ κ¯ metry [41, 42], i.e. + D¯ 1 + D¯ 2 + m¯ dΩ . (57) ∂us ∂us ∂ur ∂ur∂us abcd bacd abdc C = C = C minor symmetry, (61) k th cdab Here, we note that the matrices D¯ , k = 0, 1, 2, are k thickness = C major symmetry, (62) moments of the material tensor represented as a 3 × 3 matrix and n¯ and m¯ are the zero-th and first thickness moments of the so that only six unique components exists for the 2 × 2 × 2 × 2 stress tensor, see [31]. The thickness integrals are, in the present tensor. Furthermore, Eq. (31) implies that the non-zero compo- paper and in [31], computed using numerical through-thickness nents of Ci jkl are of the form Ciiii, Cii j j, Ci ji j and Ci j ji of which integration with four Gaussian points. As discussed in [7], the the last two are equal by virtue of the minor symmetry property. matrices D¯ 1 can differ in the variations of the normal force tensor This implies that the 2 × 2 × 2 × 2 tensor has only four uniquely ¯ ¯ 1111 1122 2222 1212 n and the moment tensor m depending the analytic projected or defined components, namely C , C , C and C . directly decoupled alternatives for thickness integration. For compressible material models, the static condensation 4.2. Eigenvalue Computation term is computed in the spectral basis, i.e. on the tensor C before it is transformed to the covariant undeformed tensor basis. From The eigenvalues of tensor quantity can be computed by solv- Eq. (54) we see that the iterative procedure to find C33 requires ing Eq. (17) or, alternatively, by computing the eigenvalues of the computation of C3333, Cαβ33 and C33αβ, where the last two i ⊗ j are equal by virtue of the major symmetry property. Reusing the the matrix that results from computation√ of C = Ci j g˚ g˚ in- 2 minor and major symmetries, the computation is reduced to four cluding the outer product. Since λ = C33 is decoupled by 3 distinct components, namely 1133, 2233, 1233 and 3333. construction, it suffices to compute λ2 and λ2 by computing the C C C C 1 2 Accordingly, it can be concluded that for incompressible ma- eigenvectors and eigenvalues of the 3 × 3 matrix following from terials four and for compressible materials eight unique compo- computation of C = Cαβg˚α⊗g˚β. This computation results in three nents of the spectral material tensor need to be computed, when eigenpairs (eigenvalues and eigenvector) of which one eigenpair exploiting minor and major symmetry, as well as the nature of contains the zero-vector due to the decoupled construction. The Eq. (31). In summary, the transformations give rise to certain 3 other two eigenpairs (λα ∈ R, vα ∈ R ) are the in-plane principle additional costs, which can be limited, however, by exploiting stretches and their directions. symmetry properties efficiently. 4.3. Tensor Transformation 5. Numerical experiments Since the stretch-based stress and material tensor are derived For benchmarking purposes, the results of four numerical ex- in spectral form (i.e. in the eigenvector space) a transformation periments have been used for verification and validation of the towards the curvilinear basis needs required in order to use these presented formulations for incompressible and compressible ma- entities in further computations. Recall that the spectral forms of terial models. For the uniaxial tension and pressurized balloon S and are: benchmarks (Sections 5.1 and 5.2, respectively), analytical so- C lutions are available, therefore they will serve as verification of 3 3 the stretch-based material model formulations. Combining the X ii X i jkl present method with (extended) arc-length methods, we inves- S = S vi ⊗ vi, C = C vi ⊗ v j ⊗ vk ⊗ vl. (58) tigate the collapsing behaviour of a truncated conical shell [19] i=1 i, j,k,l=1 (Section 5.3) and we simulate wrinkling of a stretched thin sheet (Section 5.4). The invariant-based stress and material tensors are defined in the In order to verify the presented isogemetric Kirchhoff-Love curvilinear basis, as follows: formulation for a stretch-based Ogden material with its FEM couterpart, the conical shell collapse (Section 5.3) is incorporated. Finally, we will apply our approach to model wrinkling X3 X3 S S i j g˚ ⊗ g˚ i jkl g˚ ⊗ g˚ ⊗ g˚ ⊗ g˚ . of a thin sheet subject to tension. Our models have been imple- = i j C = C i j k l (59) mented in the open-source library G+Smo (Geometry + Simula- i, j=1 i, j,k,l=1 tion Modules) [47, 48] and download and installation instructions to reproduce the data presented in the following are provided in Since the strain tensors (c.f. Eq. (19)) are defined in the curvilin- the Supplementary Material. ear basis, it is convenient to define the quantities in the variational In the numerical experiments, compressible and incompress- form (c.f. Eq. (20)) defined in the curvilinear basis. Hence, the ible formulations of the Neo-Hookean (NH), Mooney-Rivlin stretch-based stress and material tensors are transformed to the (MR) and Ogden (OG) material models have been used. The 6 Neo-Hookean models are given by (compressible and incom- L pressible, respectively):

µ − 2 Ψ(C) = J 3 I − 3 + Ψ (J), (63) 1 vol W 2 y µ σt Ψ(C) = (I1 − 3). (64) 2 x The Mooney-Rivlin models are given by [49, 50] (compressible and incompressible, respectively): Figure 1: Geometry for the uniaxial tension case. The filled geometry represents the undeformed configuration and the dashed line indicates the undeformed ge- c1 − 2 c2 − 4 Ψ(C) = J 3 I − 3 + J 3 I − 3 + Ψ (J), (65) ometry. The bottom side of the undeformed sheet is fixed in y-direction and the 2 1 2 2 vol left side of the sheet is fixed in x-direction. The applied load is σt where σ is the c c actual Cauchy stress and t is the thickness of the sheet. Ψ(C) = 1 (I − 3) + 2 (I − 3). (66) 2 1 2 2 Table 1: Residual norms per iteration for the 10th load-step for uniaxial tension For Ogden models, the following formulations are used (com- for all material models in compressible and incompressible forms. For the Neo- Hookean and Mooney-Rivlin models, the iteration residuals are provided for the pressible and incompressible, respectively): stretch-based and invariant-based approaches. For the Ogden model, only the results for the stretch-based formulations are given, since no invariant-based for- N mulation exists. For the Neo-Hookean and Mooney-Rivlin models, results are X µ 1 p − αp αp αp only observed in the last iteration, due to machine precision of the arithmetic. Ψ(λ) = J 3 λ + λ + λ − 3 + Ψ (J), (67) α 1 2 3 vol The Supplementary Material provides instructions to reproduce this table. p=1 p   X3 XN µ  It. Neo-Hookean Mooney-Rivlin Ogden  p αp  Stretch Invariant Stretch Invariant Stretch Ψ(λ) =  (λq − 1). (68)  α  q=1 p=1 p Incompressible 1 2.033 · 10−4 2.033 · 10−4 4.021 · 10−3 3.999 · 10−3 4.442 · 10−2 For all models, the following volumetric part of the strain energy 2 1.129 · 10−6 1.129 · 10−6 2.248 · 10−5 2.253 · 10−5 1.313 · 10−6 density function is adopted: 3 3.575 · 10−11 3.575 · 10−11 7.106 · 10−10 7.229 · 10−10 4.149 · 10−11 4 2.554 · 10−16 6.929 · 10−16 5.088 · 10−16 1.776 · 10−15 1.602 · 10−16 −2 −β Compressible Ψvol = KG(J) = Kβ β log(J) + J − 1 . (69) 1 1.617 · 10−3 1.617 · 10−3 2.100 · 10−3 2.100 · 10−3 5.215 · 10−3 To check consistency of invariant based models (i.e. the NH and 2 2.296 · 10−7 2.296 · 10−7 2.890 · 10−6 2.890 · 10−6 1.759 · 10−7 3 9.443 · 10−13 9.440 · 10−13 1.344 · 10−11 1.344 · 10−11 2.584 · 10−13 MR models), the invariants can be replaced by Eqs. (14) to (16) 4 1.153 · 10−15 1.252 · 10−16 1.115 · 10−15 1.988 · 10−16 1.625 · 10−15 to obtain stretch-based forms, which is thus equivalent to the component-based form from [31]. Unless stated otherwise, for the compressible models β = −2, and for the Mooney-Rivlin model c1/c2 = 7 [50] is used. For the Ogden model the coeffi- iterations (see Table1), showing that the present formulation cients from [51] are re-scaled to the value of µ: provides exactly the same rates of convergence as the invariant- based method. Last but not least, Newton iterations converge 6.300 with optimal speed (second-order convergence rate). µ1 = µ, α1 = 1.3, µ0 5.2. Pressurized Balloon 0.012 µ2 = µ, α2 = 5.0, (70) The response affected pressure of a spherical balloon is used µ0 for benchmarking purposes as well. The analytical pressure for- 0.100 mulation is obtained from [41, eq. 6.132]. The numerical model µ = − µ, α = −2.0, 3 µ 3 results are based on follower pressures, i.e. f = p0a3 where a3 is 0 the unit normal in the current configuration. The balloon is mod- where µ = 4.225. eled as a quarter of a hemi-sphere, of which the bottom point is 0 fixed in all directions, and on the sides a symmetry condition is 5.1. Uniaxial Tension applied by clamping the sides in normal direction and restriction The first benchmark case is uniaxial tension of a material deflections orthogonal to the symmetry boundary (see Fig.3). block. A block with dimensions L × W × t = 1 × 1 × 0.001 [m3] is The geometry is modelled by 2 elements over the height and 2 considered. The shear modulus is µ = E/(2(1 + ν)) where E and elements over the quarter-circumference, both of quadratic order. ν are the Young’s modulus and Poisson ratio, respectively, such For R = 10 [m], t = 0.1 [m] and µ = 4.2255 · 105 [N/m2], a that µ = 1.5·106[N/m2]. The block is modeled by shell elements, perfect agreement is obtained for all presented material models i.e. the L × W plane is considered and all edges are restrained in in comparison to the analytical solutions Fig.4. vertical direction (z = 0). The left edge (x = 0) is restrained in In Table2 we represent the total CPU times related to sys- x direction and on the right edge (x = L) a distributed load σt is tem assembly for different material models for different mesh 4 applied. The bottom edge (y = 0) is restrained in y direction and refinement levels and quadratic order for p0 = 10 . The assem- the top edge (y = B) is free to move (see Fig.1). bly times for both the invariant-based formulations and for the In Fig.2 the results for uniaxial tension are depicted. For both stretch-based formulations are given for the Neo-Hookean and compressible and incompressible materials, the analytical solu- Mooney-Rivlin material models, whereas the stretch-based for- tion for the Cauchy stress are obtained from [41, ex. 1]. The mulation is only available for the Ogden model. The total num- numerical and analytical solutions for incompressible and com- ber of nonlinear iterations is the same in all cases, and so is the pressible materials show a perfect match for all quantities studied number of assembly operations. The table shows that the stretch- (thickness decrease λ3, axial Cauchy stress σ and Jacobian deter- based formulations are slower than the invariant-based formula- minant J). Note that the Jacobian determinant for incompress- tions, which is expected by the requirement for the transforma- ible materials is equal to 1 and hence not shown. The residual tion of the basis of the deformation tensor. It can also be seen norms of the non-linear iteration convergence for the invariant- that the Ogden model requires significantly more CPU time than based and stretch-based Neo-Hookean and Mooney-Rivlin mod- the other models, which is due to the large number of terms in els as well as the stretch-based Ogden model are equal in all the strain energy density function. 7 Compressible Incompressible Table 2: Total CPU assembly times (seconds) for the different material models (invariant-based where applicable) for different mesh sizes (#El.) for the inflated 3

λ 1 1 balloon benchmark. All results are obtained for the incompressible material mod- √ els. 1/ λ 0.8 0.8 0.6 0.6 #El. Neo-Hookean Mooney-Rivlin Ogden Invariant Stretch Invariant Stretch 0.4 0.4 1 0.18 0.13 0.18 0.13 0.41 4 0.42 0.28 0.43 0.29 1.07 0.2 0.2 Thickness decrease 0 5 10 0 5 10 16 1.42 0.93 1.45 0.94 3.95 64 6.19 4.55 6.69 4.35 18.49 256 40.67 26.77 44.10 28.60 119.65 9 ·107 ·10 σ 1 6 0.8 4 0.6 5.3. Conical Shell Collapse 0.4 2 A collapsing conical shell (or frustrum) is presented as a 0.2 benchmark for modelling of strong non-linearities [19]. A con- 0 0 ical shell with height H = 1 [m], top radius r = 1 [m], bottom Axial Cauchy-stress 0 5 10 0 5 10 radius R = 2[m] and thickness t = 0.1[m] as depicted in Fig.5 is Stretch λ considered. Since the reference solution models the frustrum ax-

J isymmetrically, a quarter of the geometry is modelled with 32 quadratic elements over the height and one quadratic element 3 over the quarter-circumference to represent axial symmetry. The CIA corresponding material model is of the Ogden type and has the 2 NH following parameters: MR OG 2 1 µ1 = 6.300 [N/m ], α1 = 1.3,

Jacobian determinant 0 5 10 µ = 0.012 [N/m2], α = 5.0, Stretch λ 2 2 2 µ3 = −0.100 [N/m ], α3 = −2.0, Figure 2: Results for uniaxial tension for compressible (C, left column) and incompressible materials (I, right column); where the first row presents the thick- 2 ness decrease λ3, the second row the axial Cauchy stress or true axial stress σ and implying that µ = 4.225 [N/m ]. Two sets of boundary condi- the last row the Jacobian determinant J for compressible materials; all against tions are considered for this geometry. In both sets the bottom of the stretch λ. The material models that are used are the Neo-Hookean (NH) the Mooney-Rivlin (MR) and the Ogden (OG) material models and comparison is the shell (Γ2) is hinged, hence the displacements are restricted in made to analytical (A) solutions from [41, ex. 1]. The Supplementary Material all directions. The top shell edge (Γ1) is either kept rigid (no x provides instructions to reproduce these figures. and y displacements) or free, referred to as constant or variable radius, respectively [19]. On the top edge, a uniform load p is applied, providing a uniform displacement ∆. Due to symmetry, only one quarter of the geometry is modelled, which means that symmetry boundary conditions are applied on the x = 0 and y = 0 planes (Γ3, Γ4, see Fig.5); restricting in-plane de- Γ1 R formations normal to the boundaries and restricting rotations on the boundary by applying clamped boundary conditions as de- p Γ scribed in [25]. The quarter-conical shell is modelled with 32 2 quartic shell elements over the width. Γ4 z Loads are applied using displacement-control (DC) or arc- y Γ length control. In the former case, displacements are applied on x 3 the top-side of the cone and the deformation of the cone as well as the corresponding load on the top-boundary are computed. In Figure 3: Geometry of the inflated balloon with 4 quadratic elements. Symmetry the latter case, Crisfield’s spherical arc-length procedure [38] is conditions are applied on the boundaries Γ1, Γ2 and Γ4, which means that rotations around these boundaries and displacements in-plane normal to the bound- used with extensions for resolving complex roots [52, 53]. If this aries are fixed. The bottom of the balloon (Γ3) is an edge with a radius of 0.01 method does not converge to an equilibrium point, the step size and is fixed in all directions. is bisected until a converged step is found. After this step, the step size is reset to its original value [40]. Figs.6 and7 present the result of the collapsing conical 6,000

p shell (constant and variable radius, respectively) of the present incomp. ana. study and the reference results from [19]. The results for the 4,000 NH displacement-controlled (DC) solution procedure shows that the MR diﬀerence between the used material models are negligible, since OG the actual strains are relatively small. The results also agree 2,000 with the displacement-controlled reference results of [19], and

Internal pressure minor differences between the results might be a result of FE 0 shear locking as involved for the reference results. Since more steps have been used for the displacement-controlled calcula- 1 2 3 4 5 tions, sharp corners in the curve can be observed for ∆ ∼ 1.9 Stretch λ for constant radius and ∆ ∼ 1.8 for variable radius. An arc-length based calculation was used as well. From the Figure 4: Inflation of a balloon. The vertical axis depicts the internally applied results, one can observe revelation of the collapsing mechanism pressure and the horizontal axis depicts the stretch λ1 = λ2 = λ. The different lines and markers represent different material models, including Neo-Hookean of the conical shell. For both cases (constant and variable radius) (NH), Mooney-Rivlin (MR) and Ogden (OG). The radius of the sphere is R = an almost anti-symmetric pattern in the load-deflection space can 10 [m] and the thickness of the sphere t = 0.1 [m]. The Supplementary Material be observed, which initiates and finishes at the kinks in the curve provides instructions to reproduce this figure. that was found with the displacement-control procedure. For the 8 z

y p −2 x R1 ·10

Γ4 Γ1 H

Γ3 F R2 5

m] B Γ2 / [N

p H Figure 5: Geometry of the collapsing conical shell with 32 quadratic elements over the height. 0 D E 0.4 A I E ALM DC NH OG

Distributed load ALM DC G

m] 0.2 MR OG, Ref. [19] / F NH OG

[N OG − H 5

p MR OG, Ref. [19] B OG C 0 I 0 0.5 1 1.5 2 D Displacement ∆ [m] G −0.2

Distributed load (a) Load-displacement diagram

−0.4 0 0.5 1 1.5 2 Displacement ∆ [m]

−2 A B C D m] ·10 / 4 [N J p 2 0 −2 C A −4 K E F G H 0 . 1 . 2

Distributed load 0 5 1 5 Displacement ∆ [m]

(a) Load-displacement diagram.

I (b) Intermediate states of the frustrum

A B C D Figure 7: Result of the collapsing conical shell with variable radius; (a) load- displacement diagram, (b) undeformed geometries matching with the points in- dicated with capital letters in the diagram. The lines represent solutions obtained using the Arc-Length Method (ALM) and the markers represent solutions obtained by Displacement Control (DC). The material models are Neo-Hookean (NH), Mooney-Rivlin (MR) and Ogden (OG). Since variation between the material models is rather small for the DC solutions, only the results for the OG material model are given. The reference results are obtained from [19]. A movie of the collapse (video 2) and instructions to reproduce the data are given as Sup- E F G H plementary Material.

(1 + ε)L

I J K (b) Intermediate states of the frustrum

Figure 6: Result of the collapsing conical shell with constant radius;(a) load- W P displacement diagram,(b) undeformed geometries matching with the points indi- y cated with capital letters in the diagram. The lines represent solutions obtained using the Arc-Length Method (ALM) and the markers represent solutions ob- x tained by Displacement Control (DC). Note that the solution for the NH and MR models are overlapping on most parts of the path. The material models are Neo- L Hookean (NH), Mooney-Rivlin (MR) and Ogden (OG). Since variation between the material models is rather small for the DC solutions, only the results for the Figure 8: Modeling geometry for the uniaxially loaded restrained sheet. OG material model are given. The reference results are obtained from [19].A movie of the collapse (video 1) and instructions to reproduce the data are given as Supplementary Material. 9 constant-radius shell, Fig. 6a shows two loops of large magni- 2 NH NH, [7] tude. In Figs. 6b and 7b it can be seen that collapsing behaviour 1.5 EA MR MR, [7] of the conical shell consists of states in which multiple waves in / radial direction occur. For both cases, it can be seen that after the P 1 OG loops with the highest force-amplitude, the shell and its collapse- path invert and continue on the path that can be obtained with Force 0.5 displacement-control. 0 To the best of the authors’ knowledge, the collapsing of 0 0.5 1 1.5 2 2.5 3 a conical shell was not investigated before. Complex load- Strain ε displacement paths from Figs. 6a and 7a show that displacement- (a) Benchmarking results comparing to [7]. The reference results are obtained numerically. controlled simulations in this case ignore the collapsing behaviour of the shell with multiple limit points. The authors highly 0.4 NH L/W = 4, [3]

EA MR L/W = 8, [3] encourage further investigations on this benchmark for verifica- / tion and validation. P OG L/W = 2, [3] 0.2 5.4. Wrinkling of a stretched sheet Force As an application of the model, we consider the wrinkling phe- 0 nomenon of a stretched, thin membrane (see Fig. 11). Scaling 0 0.1 0.2 0.3 0.4 0.5 laws based on experiments were first published in [1,2] and an- (b) Benchmarking results for comparison with [3]. Experimental results are depicted by markers for different aspect ratios L/W of the sheet. Numerical results from [3] are not alytical formulations related to this problem were established in included since they are indistinquishable from the present MR results. [54]. Numerical results to this problem have been established for sheets with different aspect ratios β and different dimension- Figure 9: Uniaxial tension of a restrained sheet using incompressible material less thickness α [3–5, 55–59]. In most numerical studies, Neo- models. The dimensionless force is obtained by normalization of the applied Hookean or Mooney-Rivlin models were used to model the wrin- force P by the Young’s modulus E and the cross sectional area A. The Supple- kling phenomenon, since strains usually reach high values (typ- mentary Material provides instructions to reproduce these figures. ically ε ∼ 10 − 50%). In this paper, we model tension wrinkling for the sake of benchmarking using incompressible Neo- 10−1 10−1

Hookean, Mooney-Rivlin and Ogden models and Isogeometric Kirchhoff-Love shells, which is a novelty to the best of the au- 10−3 10−3 thors’ knowledge. In the first part of this section, the model is p = 2 benchmarked on a restrained sheet without wrinkling formation p = 3 and material parameter determination is performed. Thereafter, 10−5 10−5 the results of wrinkling simulations are presented. Relative error p = 4 Relative error − − − − Material test 10 2 10 1 100 10 2 10 1 100 Mesh size h Mesh size h Related to the first benchmark in the work of [7] and on the (a) Neo-Hookean (NH): the orders of conver- (b) Ogden (OG): the orders of conver- experiments of [3], a tensile load is applied on a strip of which gence following from Richardson extrapola- gence following from Richardson extrap- the short edges are fixed and the long edges are free (see Fig.8). tion are 2.11 (p = 2), 2.15 (p = 3) and 2.17 olation are 2.27 (p = 2), 2.40 (p = 3) and Focus is on the non-domensional load versus end-point displace- (p = 4). 2.52 (p = 4). ment in longitudinal (load and displacement) direction. Figure 10: Convergence rate of the restrained sheet under uniaxial tension with Firstly, for the geometric parameters, L = 9[mm], W = 3[mm] values from [7] for different material models (a-b). The error is relative error and t = 0.3 [mm] are used, leading to L/W = 3 and t/W = 0.1. = |εnum − εR|/εR where εnum is the numerical value of the strain and εR is the The material has Poisson’s ratio ν = 0.5 and for the NH material Richardson-extrapolated value of the strain related to the last three meshes, all for a dimensionless force P/EA = 0.5. The orders of convergence following from model a Young’s modulus of E = 30 [kPa] is involved and for the Richardson extrapolation are provided in the captions below the subfigures. the MR material model one of E = 90 [kPa] leading to, µ = The Supplementary Material provides instructions to reproduce the strain data. 10 [kPa] and µ = 30 [kPa], respectively. For the MR model, c1/c2 = 1/2 such that c1 = 1/9 and c2 = 2/9. Scaling according to Eq. (70) is applied for the Ogden material model and 8 × 8 quadratic elements are used. A good match with the results of W = 140 [mm] and t = 0.14 [mm], leading to L/W = 2 and the directly decoupled method of [7] for the incompressible Neo- t/W = 103. The material models are incompressible and for the Hookean and Mooney-Rivlin models can be observed in Fig. 9a. NH material model, a parameter µ = 1.91 · 105 [Pa] is used, Note that the forces in the reference paper are normalized by 5 while for the MR model the parameters c1 = 3.16 · 10 [Pa] E = 3c1 for both the Neo-Hookean and Mooney-Rivlin models, · 5 whereas in the present simulations, the forces are normalized by and c2 = 1.24 10 [Pa] are used. The results are depicted in E = 3µ (since ν = 0.5 in the comparison with [7]). Fig. 9b, from which it can be seen that there is an excellent agreement between the numerical results from [3] (obtained using the In Fig. 10, we provide convergence plots of the present model ABAQUS S4R element) and with the experimental results. In ad- (NH and OG stretch-based models) with respect to the relative dition, the depicted fit for the Ogden material model was found, error in the strains given a nondimensional load of P/EA = 0.5. − The errors are plotted with respect to the Richardson extrapola- using parameters α1 = 1.1 [-], µ1 = 1.0µ0 [Pa], α2 = 7 [-], µ2 = −0.003µ0 [Pa], α3 = −3 [-] and µ3 = −0.4µ0 [Pa] with tion from the three finest meshes, since analytical solutions to the 5 problem are not available. The results obtained for the NH model µ0 = 1.91 · 10 Pa. obtained from the invariant-based form are exactly the same and Wrinkling simulations hence not provided here. The figures show that the convergence of the method is around second-order, independent of the order For the wrinkling simulations, we follow the work of [3] with of the spline basis. Reference papers [7, 31] do not provide esti- the same parameters for the Mooney-Rivlin and Ogden models as mates of the order of convergence for the invariant-based mate- in Fig. 9b. The model setup for the wrinkling simulations is de- rial models or convergence plots for similar simulations. Hence, picted in Fig. 11. The modeling domain is depicted in the shaded further comparison and investigations on the order of conver- area and surrounded by boundaries Γk, k = 1, .., 4. Firstly, the gence for such membrane-dominated responses for shells with boundary Γ1 is free, meaning that no displacement constraints are nonlinear material models are recommended. involved. Furthermore, the boundary at Γ2 is clamped (matching Secondly, we compare our numerical model to the experi- the adjacent control points parallel to the symmetry axes) and mental results from a similar setup as depicted in Fig.8[3]. displacements in y-direction and out-of-plane displacements are The corresponding geometric parameters are L = 280 [mm], restricted. The displacements in x-direction are all equal over Γ2. 10 (1 + ε)L Exp. [3] Γ1 3 FI

Γ4 Γ2 H-L H-L S/R Γ3 t

W P / SHELL181 ) 2 y w MR OG x max( 1 L

Figure 11: Modeling geometry for the wrinkled sheet. 0 0 0.1 0.2 0.3 0.4 0.5 ε Symmetry is imposed over Γ4 by clamping the edges and by restricting deformations orthogonal to the axes (u = 0). Lastly, (a) Strain-amplitude diagram of the tension wrinkling of a thin sheet. The vertical axis rep- x resents the maximum amplitude normalized by the shell thickness t and the horizontal axis anti-symmetry is imposed over Γ3 by restricting displacements represents the strain ε of the sheet. The present model is used to obtain the Mooney-Rivlin in vertical direction and orthogonal to the boundary (u = 0). (MR) and Ogden (OG) results. The fully integrated (FI), Hughes-Liu (H-L) and Hughes-Liu y Selective/Reduced (H-L S/R) results are obtained using LS-DYNA and the SHELL181 re- Similar to [3], we apply a anti-symmetry condition over Γ3 since sults are obtained using ANSYS. the symmetric and anti-symmetric wrinkling patterns can appear at the same critical load [5, 57]. For continuation, Crisfield’s spherical arc-length method [38] is used with an extension for approaching bifurcation points [39], branch switching [60] and complex-root resolving [52, 53], all summarised and applied to IGA in [40]. Furthermore, for comparison, results from LS-DYNA (R11.0) and ANSYS (R19.1) simulations are presented for the same geometry and a Mooney-Rivlin model with the same parameters, however ν = 0.499 in the LS-DYNA simulations since incompressible materials (ν = 0.5) are not implemented. A displacement control approach is employed with an initial perturbation based on the first buckling mode corresponding to a tension load situation, perturbed with a factor of 10−4. In LS-DYNA, the Hughes-Liu, the Hughes-Liu selective/reduced and the fully integrated shell elements are used, all with 4 quadrature points through-thickness and a shear correction factor equal to zero [61]. The results for the ANSYS SHELL181 element [62] are obtained using default options, which includes reduced integration and hour-glassing control. For both the LS-DYNA and ANSYS simulations, mesh refinements were applied until convergence. ε = 0.1 ε = 0.2 ε = 0.1 ε = 0.2 From Fig. 12a large difference between the different solvers (b) Contour plot of out-of-plane displace- (c) Contour plot of out-of-plane displacements w for different strains ε for the MR ments w for different strains ε for the OG and between the material models can be observed. Firstly, it model. model. can be concluded that the MR results from the Isogeometric Kirchhoff-Love shell correspond most with the results obtained Figure 12: Wrinkling formation in a thin sheet subject to tension. The Supple- with LS-DYNA. Additionally, these results show good corre- mentary Material provides instructions to reproduce these figures. spondence with the experimental results both in the low strain regime (until ε ∼ 0.08) as well as towards restabilization of the wrinkles (between ε ∼ 0.2 and ε ∼ 0.3), only the maximum amplitude is slightly underestimated and the restabilization point ingly with limited computational costs due to tensor symmetries. (i.e. the point where the wrinkles vanish again) is predicted too The results from numerical experiments with Neo-Hookean early. Secondly, it can be observed that there is a large difference and Mooney-Rivlin material models, which can be represented between the results from IGA or LS-DYNA and from ANSYS. in terms of invariants as well as in terms of stretches, shows Although different shell options in the FEA libraries have been that identical iteration residuals and correct Newton-convergence varied (e.g. reduced/full integration, shear correction factors), rates have been obtained. This confirms that the stretch-based the origin of these differences is yet unknown to the authors and and invariant-based shell formulations are equivalent. For these requires further investigations. Lastly, significant differences be- models it is also shown that the present formulation leads to tween the Ogden and Mooney-Rivlin results can be observed, al- higher CPU times due to the projection of the stress and mate- though the similarities in the material behaviour in Fig. 9b. From rial tensor; therefore, the advantage of the present formulation this it can be concluded that material fitting possibly needs to be is mainly related to stretch-based material models (e.g. the Og- done using experimental tests of different loading configurations, den model) and not to models that can be expressed explicitly e.g. testing the bending response of the material. in terms of the curvilinear tensor components of the deformation tensor (e.g. the invariant-based Neo-Hookean and Mooney 6. Conclusions and recommendations Rivlin models). The analytical benchmarks have shown very This paper provides mathematical formulations to accurately good agreements confirming that the formulations and imple- and efficiently model thin rubbers and several biological tissues mentation are correct. by combining stretch-based material formulations such as the Employing (extended) arc-length methods in combination Ogden material model and smooth spline formulations of the Iso- with the present model, we investigated the collapsing behaviour geometric Kirchhoff-Love shell. The formulations apply to com- of a truncated conical shell and the wrinkling behaviour of a pressible and incompressible material models and are based on stretched thin sheet. In case of the collapsing truncated coni- an eigenvalue computation to obtain the principal stretches and cal shell, the Ogden model was used in combination with either their direction (i.e. the spectral basis). 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