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1 Ä Preprint Rur University Bocum Lerstul für Tecnisce Mecanik A novel variational algoritmic formulation for wrinkling at finite strains based on energy minimization: application to mes adaption J. Mosler Tis is a preprint of an article accepted by: Computer Metods in Applied Mecanics and Engineering (2007)

2 A novel variational algoritmic formulation for wrinkling at finite strains based on energy minimization: application to mes adaption J. Mosler Lerstul für Tecnisce Mecanik Rur University Bocum Universitätsstr. 150,D Bocum, Germany URL: SUMMARY Tis paper is concerned wit an efficient novel algoritmic formulation for wrinkling at finite strains. In contrast to previously publised numerical implementations, te advocated metod is fully variational. More precisely, te parameters describing wrinkles or slacks, togeter wit te unknown deformation mapping, are computed jointly by minimizing te potential energy of te considered mecanical system. Furtermore, te wrinkling criteria are naturally included witin te presented variational framework. Te proposed metod allows to employ tree-dimensional constitutive models witout any additional modification, i.e., a projection in plane stress space is not required. Analogously to te wrinkling parameters, te non-vanising out-of-plane component of te strain tensor results conveniently from relaxing te respective Helmoltz energy of te membrane. Te proposed framework is very general and does not rely on any assumption regarding te symmetry of te material, i.e., arbitrary anisotropic yperelastic models can be consistently taken into account. Te advantages associated wit suc a variational metod are manifold. For instance, it opens up te possibility of applying standard optimization algoritms to te numerical implementation. Tis is especially important for igly non-linear or singular problems suc as wrinkling. On te oter and, minimization principles provide a suitable basis for a posteriori error estimation and tus, for adaptive finite element formulations. As a prototype, a variational error indicator leading to an efficient -adaption for wrinkling is briefly discussed. Te performance of te wrinkling approac is demonstrated by selected finite element analyses. 1 Introduction Many design concepts in practical engineering are based on membrane structures. Particularly, in applications were weigt is an issue suc as in aerospace industries, ligtweigt membranes are frequently employed. One state-of-te-art example is given by solar sails. Nowadays, muc researc is focused on tose sails wic seem to be a promising way for energy generation in space. Altoug te aforementioned application is relatively new, mecanical models for membranes date back, at least, to te pioneering work by Wagner [1] and Reissner [2]. Since ten, Reissner s so-called tension field teory as been adopted and furter elaborated by several researcers, cf., e.g. [3 7]. For a detailed overview on tis teory, te interested reader is referred to te compreensive work by Steigmann [8] and references cited terein. All te quoted works 1

3 2 J. MOSLER ave in common tat eiter by modifying te considered constitutive law or by enancing te strain tensor, te stresses are altered suc tat compression is avoided (tension field teory). Clearly, te assumption of vanising compressive stresses represents an idealization. Alternatively, it is possible to approximate wrinkles occurring in membranes explicitly. For instance, in [9, 10] wrinkling is modeled by using a sell-type formulation and considering a very small bending stiffness (tin sell). One advantage of suc a metod is tat te assumption known from tension field teory is not required. Furtermore, in some applications, te bending stiffness does play an important role. In suc cases, it can be consistently taken into account by using (bending) sells. For example, in [11, 12] folding in tin-films was analyzed by applying a von Kármán plate teory. Witin te cited works, te unknown deformations and tus, wrinkles or slacks, follow directly from an energy minimization principle. However, if wrinkles are modeled explicitly, a large number of finite elements is required in order to capture te respective deformation adequately. Hence, te numerical effort associated wit tis modeling strategy is relatively ig compared to te approaces described in te previous paragrap. Consequently, tey will not be considered in te remaining part of tis paper. Neglecting bending stiffness, a variational strategy similar to tat in [11, 12] was proposed in a series of papers by Pipkin, cf. [13 15]. In [13], Pipkin analyzed te energy of a membrane for isotropic yperelastic material models. He proved tat te quasi-convexification of te Helmoltz energy wic guarantees, under a few furter conditions, lower semi-continuity of te resulting potential of te considered mecanical system and ence, te existence of solutions (energy minimizers; see [16]), defines a relaxed energy functional wose derivatives yield te membrane stresses. More precisely, te stresses predicted by tis relaxed potential obey te restrictions imposed by tension field teory, i.e., te resulting stress field is associated wit uniaxial tension. Pipkin furter elaborated is ideas in [15]. In tis work, e analyzed wrinkling in arbitrary anisotropic yperelastic solids. Under te assumption tat te two-dimensional Helmoltz energy of te membrane is convex (wit respect to te rigt Caucy-Green tensor), Pipkin generalized is findings for isotropic material models to arbitrary symmetries. Pipkin s metod allows to re-formulate tension field teory witin a variational framework. It sould be noted tat a different approac leading to similar results was proposed by Epstein [17]. Surprisingly, altoug Pipkin s ideas are well-suited for algoritmic formulations, a fully variational numerical framework as not been advocated yet. As a consequence, witin te present paper, te ideas by Pipkin are furter elaborated and implemented in a finite element code resulting in a numerically efficient formulation. In te context of tree-dimensional yperelasto-(visco)plasticity, a similar variational framework was proposed by Ortiz and coworkers [18 20] (see also te more recent works [21, 22]). Rougly speaking, in tose approaces wic are often referred to as variational constitutive updates te state variables follow from relaxing a certain potential. Based on tis relaxed functional, te deformation mapping results from energy minimization as well. In te present paper, a similar numerical strategy for te analysis of wrinkles and slacks in arbitrary yperelastic membranes at finite strains is discussed. In contrast to te works by Pipkin, te non-vanising out-of-plane component of te strain tensor is considered and follows from energy relaxation as well. Consequently, plane stress conditions do not need to be enforced explicitly but are naturally included witin te novel variational framework. As a result, tree-dimensional constitutive models can be directly applied witout any modification required. Furtermore, since witin te proposed variational metod all unknown state variables and te deformation mapping follow from a minimization principle, standard optimization strategies can be adopted for solving te resulting mecanical problem. Tis is especially important for igly non-linear or singular problems suc as wrinkling. An additional appealing property of minimization principles is tat tey provide a natural basis to estimate te quality of te numerical solution and ence, tey represent a natural basis

4 Variational algoritmic formulation for wrinkling at finite strains 3 for mes adaption, cf. [23 25]. As a prototype, a variational -adaption is discussed witin te presented paper. Te paper is organized as follows: In Section 2, te kinematics of wrinkles and slacks are discussed. Based on an intuitive engineering approac it is sown tat most of te different models wic can be found in te literature are essentially equivalent. A novel algoritmic formulation for wrinkling representing one of te key contributions of te present paper is sown in Section 3. Starting wit te fundamentals, criteria signaling te formation of wrinkles and slacks are derived. In case of wrinkling, te respective state variables are computed from a local minimization principle. All linearizations required for applying an optimization strategy suc as Newton s metod are given. Section 4 is concerned wit a novel variational -adaption for membranes. Using te variational structure of te underlying mecanical problem, a pysically and matematically sound error indicator is proposed. Finally, te applicability and versatility of te variational wrinkling algoritm as well as te performance of te novel variational -adaption for membranes are illustrated by means of two numerical examples in Section 5. 2 Kinematics of wrinkling In tis section, te kinematics of wrinkling are discussed. Altoug different models can be found in te literature, it will be sown tat tey are essentially equivalent. More precisely, tey are included witin a general framework proposed by Pipkin [15]. However, since Pipkin s approac is based on a rater matematical argumentation suc as quasi-convexification, an intuitive engineering wrinkling model is presented in Section 2.1 first. Subsequently, it is modified finally resulting in Pipkin s metod. In Section 2.2, te kinematics are sigtly enanced suc tat tey account for a non-vanising out-of-plane strain component. Tis is necessary in order to employ tree-dimensional constitutive models. 2.1 Fundamentals In wat follows, a membrane occupying a region Ω R 3 in its reference undeformed configuration is considered. Since a membrane represents a two-dimensional submanifold in R 3, it can be conveniently caracterized (at least locally) by a cart X : R 2 U M R 3, θ α X implying te introduction of curvilinear coordinates θ 1, θ 2. Te same olds for te deformed configuration, i.e., x : R 2 U K R 3, θ α x. Here, X, x denote position vectors of a point witin te undeformed and te deformed configuration, respectively. Wit tese carts wic are sufficiently smoot diffeomorpisms, i.e, X Diff 1 (U, M) and x Diff 1 (U, K), te deformation mapping connecting X and x is well defined, i.e., ϕ = x X 1. Locally, te deformation is measured by te deformation gradient F := x/ θ α θ α /θ X. Based on F strain measures can be derived. One suc measure is te rigt Caucy-Green tensor C = F T F represented by a symmetric and positive definite second-order tensor. It bears empasis tat witout loss of generality, te deformation gradient F as well as te strain tensor C are defined by 2 2 tensors witin tis subsection, even if a tree-dimensional space is considered. Furter comments are omitted, cf. [14, 26]. In te remaining part of tis paper te notation C > 0 is used to signal tat C is positive definite. Next, focus is on a modified strain tensor reflecting effects due to wrinkling. According to Roddeman et al. [3], wrinkles can be taken into account by modifying te deformation gradient. More precisely, following [3, 4], te standard deformation gradient F is enanced multiplicatively resulting in a relaxed deformation gradient F r, i.e, F r = (1 + ã n n) F, ã 0 (1)

5 4 J. MOSLER Here, n is te wrinkle direction and ã te local cange in lengt of te membrane due to wrinkling. Subsequently, te material response is computed by using F r instead of F. It bears empasis tat Eq. (1) is formally identical to te well-known split in multiplicative plasticity teory (F = F e F p ). Tis can be seen better by applying te Serman-Morrison formula, i.e., F = F w F r, F w = 1 ã 1 + ã n n (2) Altoug bot models look identical, it sould be noted tat tey are significantly different. Wile plasticity is istory-dependent and tus pat-dependent, wrinkling can be computed locally (in space and time). By te principle of material objectivity, te material response depends on F r troug te respective rigt Caucy-Green tensor. From Eq. (1) C r it is obtained as Consequently, by setting Eq. (3) can be re-written as C r := F T r F r = F T F + [ 2ã + ã 2] (n F ) (n F ) (3) a 2 := ( 2ã + ã 2) n F 2 2 0, N = n F n F 2 (4) C r = C + C w, C w = a 2 N N (5) Hence, te additive enancement of te rigt Caucy-Green tensor (5) is equivalent to te multiplicative split F = F r F w. Clearly, Roddeman s model, or equivalently te split (5), are associated wit te kinematics induced by wrinkling. However, in te present paper, wrinkles as well as slacks are to be modeled. Hence, a sligt modification of te kinematics is proposed, i.e, an additive decomposition of te Caucy-Green tensor according to C r = C + C w, C w = a 2 N N + b 2 M M (6) is adopted. Here, N and M span a cartesian coordinate system. Obviously, C w is identical to Eq. (5), if b equals zero. Oterwise, C w corresponds to a slack. Tis will be sown in te next sections. According to te spectral decomposition teorem, any symmetric tensor wic is semipositive definite can be re-written into a format suc as Eq. (6) 2. In te following, te notation C w 0 is used to indicate tat C w is represented by a semi positive definite tensor. For tis reason, Eq. (6) is equivalent to wic as been proposed by Pipkin, cf. [15]. C r = C + C w, C w 0 (7) Remark 1. It bears empasis tat wrinkling is treated as a purely local problem witin tis paper. More precisely, in line wit plasticity teory, te wrinkling parameters (a,α) defining te enanced kinematics according to Eq. (1) or (5) are incompatible, i.e., tey do not derive from a compatible displacement field. Tus, tey are not continuous in general. However, it is obvious tat continuity can easily be enforced.

6 Variational algoritmic formulation for wrinkling at finite strains 5 Remark 2. Similar to Roddeman, Epstein [17] proposed modified kinematics of te type F r = F 1 F, F := Q [ exp( α 2 )n n + m m ], α [0, ) (8) Here, n and m are defined as before, Q SO(2) is an arbitrary proper ortogonal tensor and α is, as will be seen, associated wit te cange in lengt of te membrane due to wrinkling. Witout loss of generality, Q = 1 is considered in wat follows (Q does not affect C). Clearly, by using n n + m m = 1, F can be re-written as F = 1 + [ exp( α 2 ) 1 ] n n (9) }{{} =: α and finally, application of te Serman-Morrison formula yields F 1 = 1 α 1 + α n n (10) Hence, by setting ã := α 1 + α 0 (11) Epstein s assumption is equivalent to Roddeman s model. 2.2 Modification of te kinematics required for applying tree-dimensional constitutive models Clearly, if a fully tree-dimensional material model is to be used, te plane stress condition caracterizing a membrane cannot be guaranteed a priori. As a consequence, te kinematics ave to be sligtly modified. In wat follows, te vector E 3 is defined by E 3 := N M. Wit tis definition, te two-dimensional relaxed strain tensor C r in Eq. (7) is replaced by C r = C + C w + C 33 E 3 E 3, C w 0, C 33 > 0 (12) In contrast to Eq. (7), all tensors in Eq. (12) belong to R 3 3. More precisely, C as well as C w in Eq. (12) are obtained by adding additional zeros to te respective tensors in Eq. (7). Te generally non-vanising component C 33 can be computed from te plane stress condition or as will be sown in tis paper, from relaxing an energy potential. Remark 3. Not every yperelastic tree-dimensional constitutive model can be used. More specifically, it as to fulfill some material symmetry conditions, i.e., te sear components ( ) i3 of te stresses resulting from C r as defined in Eq. (12) ave to vanis. Obviously, isotropic models fulfill tis restriction. 3 A novel wrinkling algoritm based on energy minimization In tis section, a novel wrinkling algoritm is discussed. In contrast to oter implementations wic can be found in te literature, it is based on energy minimization. More precisely, te ideas proposed Pipkin by [15] are furter elaborated and implemented resulting in an efficient finite element code. One of te distinguising features of te resulting metod is tat te principle of energy minimization, togeter wit te kinematics (5) define te formation of wrinkles and slacks completely. Consequently, and in contrast to oter numerical approaces, additional

7 6 J. MOSLER assumptions suc as tose of tension field teory do not ave to be enforced explicitly, but are naturally included. Te fundamentals of te algoritmic formulation are discussed in Subsection 3.1 first. For te sake of clarity, attention is focused on anisotropic yperelastic material models wic fulfill a priori plane stress conditions. Criteria associated wit te formation of slacks and wrinkles are elaborated in Subsection 3.2, wile details about te numerical implementation are sown in Subsection 3.3. Finally, Subsection 3.4 is concerned te modifications of te numerical implementation necessary for applying tree-dimensional constitutive models directly. 3.1 Fundamentals As mentioned before, for te sake of clarity, attention is focused first on anisotropic yperelastic material models wic fulfill a priori plane stress conditions. Suc a model is described in te appendix of tis paper. Hence, in wat follows, C R 2 2. Te respective generalizations will be discussed in Subsection 3.4. Wit te strain energy density Ψ(C) 0, te first Piola-Kircoff stress tensor P and te second Piola-Kircoff stresses S are computed as P = 2 F Ψ C S = 2 Ψ C Furtermore, it is assumed tat te reference configuration is fully unloaded (locally), i.e., (13) Ψ(C = 1) = 0, P(C = 1) = 0 (14) It is well known tat for suc material models te deformation mapping ϕ can be computed by applying te principle of minimum potential energy. Tis can be written as wit I(ϕ) := Ω ϕ = arg inf ϕ Ψ(C) dv Ω I(ϕ), (15) B ϕ dv Ω 2 T ϕ da. (16) Here, B and T represent prescribed body forces and tractions acting at Ω 2. Obviously, ϕ as to comply wit te essential boundary conditions imposing additional restrictions in te resulting minimization problem. Clearly, if principle (15) is directly employed, compressive stresses violating te pysical properties of membranes may occur. Following Section 2, wrinkles and slacks can be taken into account by replacing te standard Caucy-Green tensor by its relaxed counterpart C r := C + a 2 N N + b 2 M M, a 2, b 2 0 (17) Since energy minimization is te overriding principle governing every aspect of te mecanical problem under investigation, it is natural to postulate tat wrinkles and slacks form suc tat tey lead to te energetically most favorable state. Hence, te modified principle of minimum potential energy (ϕ, a 2, b 2, α) = arg inf I(ϕ), (18) wit I(ϕ, a 2, b 2, α) := Ω Ψ(C r ) dv ϕ,a 2,b 2,α Ω B ϕ dv Ω 2 T ϕ da (19)

8 Variational algoritmic formulation for wrinkling at finite strains 7 and te parameterization N = [cosα; sin α], M = [ sin α; cosα] = α N (20) is considered. In te remaining part of tis section it will be sown ow to determine efficiently te wrinkling (slack) parameters a 2, b 2 and α, togeter wit te deformation mapping ϕ based on Eq. (18). Furtermore, te implications resulting from tis variational principle are discussed. Since te wrinkling variables (a 2, b 2 and α) are defined locally, te minimization problem (18) can be decomposed into two classes of optimization problems. First, for a given deformation mapping, te parameters a 2, b 2 and α are computed from te local minimization principle (a 2, b 2, α) := arg inf a 2 >0 b 2 >0 α [0,π] Ψ(C r ), (21) Clearly, tis in turn, defines a relaxed energy functional Ψ r (C) := inf a 2 >0 b 2 >0 α [0,π] Ψ(C r ), (22) depending only on te deformation mapping. Interestingly, Eq. (22) is formally identical to so-called variational constitutive updates suc as proposed by Ortiz et al., cf. [18, 19]. Once te local problem (22) as been solved, te deformation mapping follows from ϕ = arg inf Ψ r (C) dv B ϕ dv T ϕ da (23) ϕ Ω Ω Ω 2 It bears empasis tat bot problems (21) and (23) can be computed by using standard optimization algoritms. Clearly, a spatial discretization of Eq. (23) is required. Te applicability of standard numerical metods is a very appealing feature of te proposed framework, since wrinkling leads usually to igly nonlinear and singular system of equations. Problem (23) can be treated as standard yperelasticity. Consequently, focus is on te solution of Eq. (21) in te remaining part of tis section. Altoug for some classes of yperelastic material models closed form solutions for te relaxed energy can be derived, cf. [13, 27], numerical metods are required in general. Remark 4. It is noteworty tat minimization principle (21) is smoot in a, b and α (if Ψ is smoot in C). As a consequence, te transition between te classical solution (a = b = 0) and wrinkles or slacks is smoot for te variational strategy. Tis improves te numerical robustness of te resulting finite element metod significantly. 3.2 Wrinkling criteria In principle, optimization problem (22) could be implemented directly. However, suc a formulation is not very efficient. For instance, consider a loading state wic does not lead to wrinkling. In suc a case, it is more convenient to derive a criterion signalling in advance tat wrinkles will not occur. Referring to computational plasticity or variational constitutive updates a so-called trial step is usually applied. In te next paragraps, analogous criteria will be derived.

9 8 J. MOSLER Formation of slacks In tis subsection, criteria necessary for te formation of slacks are discussed. Here and encefort, a slack is defined by a relaxed Caucy-Green tensor sowing non-vanising wrinkling parameters in bot directions, i.e., a 2 > 0 and b 2 > 0. Tis definition coincides wit tose wic can be found in te literature. In tis case, te stationarity conditions associated wit Eq.(21) yield Ψ : C r = 0 C r a (24) a S : (N N) = 0 Ψ : C r C r b = 0 b S : (M M) = 0 Ψ : C r C r α = 0 [ a 2 b 2] S : (N M) = 0 Since S belongs to te space of symmetric second order tensors wic can be generated by S = span{n N, M M, (N M) sym }, Eqs. (24)-(26) imply tat slacks are stress free, i.e., S = 0 (te case b = a does not lead to any problems as will be sown in te next paragrap). Tus, if additionally, te pysically sound conditions S(C) = 0 C = 1 and Ψ(C) = 0 C = 1 are assumed, a slack is represented by (25) (26) slack a 2 > 0, b 2 > 0, C r = 1 (27) Te aforementioned assumptions are closely related to te Generalized Coleman-Noll Inequality, cf. [28]. Next, te conditions associated wit te formation of slacks are discussed. For tat purpose, te spectral decomposition of te compatible Caucy-Green tensor is considered, i.e., C = 2 λ 2 i Ñ i Ñ i (28) i=1 Suppose tis C results in a slack. In tis case, te wrinkling-related part of te relaxed Caucy- Green tensor is given by C r = 1 C w = 2 (1 λ 2 i ) Ñ i Ñ i (29) Since C w is a positive definite tensor in case of slacks (a 2 > 0, b 2 > 0), te eigenvalues of C ave to fulfill λ 2 i < 1 (30) Tat is, slacks can only form if te strains are of purely compressive type. Tus, in summary, te following implications old i=1 λ 2 1 < 1, λ 2 2 < 1 slack C r = 1, S = 0, Ψ = 0 (31) Tis condition is equivalent to tat known from te classical strain-based or strain-stress-based wrinkle criteria, cf. [6, 7]. It is noteworty tat witin te proposed variational framework, te criterion associated wit te formations of slacks follows naturally from energy minimization and does not ave to be enforced in an ad-oc manner.

10 Variational algoritmic formulation for wrinkling at finite strains (a) Figure 1: Energy landscape of Ψ(C r (a 2, α)) as a function in a (orizontal direction) and α (vertical direction); a minimum is igligted by wite color. Te respective isotropic stored energy functional is given in Appendix A. Plots for: (a) a compatible Caucy-Green tensor C = diag{2; 0.9}; te minimum is associated wit a 2 = 0 implying tat no wrinkles will occur. (b) C = diag{2; 0.5}; minimum occurs at a 2 > 0 and N points into te direction of te smaller eigenvalue of C, i.e., α = π/2. Hence, a wrinkle will form Formation of wrinkles Analogously to te previous subsection, wrinkling is defined by a relaxed Caucy-Green tensor sowing only one non-vanising wrinkling parameter (a 2 > 0, b 2 = 0). In tis case, te stationarity conditions of Eq.(21) are obtained as (cf. Eqs. (24)-(26)) (b) S : (N N) = 0 S : (N M) = 0 S N = 0 (32) As a consequence, wrinkles are caracterized by a uniaxial stress state. Tis condition known from tension field teory, cf. [8], is usually enforced explicitly. It bears empasis tat witin te variational model it is naturally included. For te derivation of necessary and sufficient conditions associated wit wrinkling, te energy Ψ(C r ) = Ψ(C r (C, a, α)) (33) as a function in a and α is analyzed (for constant C), see Fig. 1. Wrinkles do not occur, if te solution is optimal (stable) from an energetical point of view, i.e., if (0, α) = arg inf a 2 0 α [0,π] Ψ(C r ), C r = C + a 2 N N (34) Oterwise wrinkles form resulting in a reduction of te energy. Obviously, Ψ(C r (a = 0), α) = Ψ(C) (35) and furtermore, according to Eqs. (24) and (26), [ a Ψ(C r ); α Ψ(C r )] a=0 = 0 (36)

11 10 J. MOSLER Hence, te classical solution (a = 0) is associated wit a saddle point (in te a-α diagram). For te derivation of wrinkling conditions, stability of te standard solution (a = 0) is studied. Tis can be done by analyzing te second derivative 2 aψ = S : (N N) + a 2 (N N) : C : (N N), C := 4 Ψ2 C 2 r (37) Hence, for a = 0, 2 a Ψ a=0 = S : (N N) (38) is obtained. Consequently, stability in an energetical sense requires te minimal eigenvalue to be greater tan zero. If one of te eigenvalues is less tan zero wrinkles form (or slacks). Tis is equivalent to te classical (ad-oc) teory. Tus, in summary, te following wrinkling condition olds λ 2 1 > 1, λ 2 2 < λ 2 1, S(C) 0 wrinkling (a 2, α) see Eq. (21)) (39) Remark 5. As mentioned before, witin many membrane models, te wrinkling parameters a and α are computed by applying tension field teory [8]. Consequently, a and α are obtained from te non-linear set of equations R S := S N = 0 (40) Wit a := (a, α), Eq. (40) defines an implicit function of te type (under some matematical restrictions) a = f(c) (41) and tus, C r = g(c) (42) cf. [17]. Wit tis implicitly defined function g, a relaxed stored energy functional can be defined, i.e., Ψ r (C) = Ψ g(c) (43) As a consequence, te kinematical approac C r = C + C w is equivalent to modifying te stored energy functional, cf. [7]. Remark 6. Clearly, Eqs. (32) imply tat te wrinkling tensor C w is coaxial to S. Consequently, if te Helmoltz energy is isotropic in C, C w is coaxial to C as well. Hence, for isotropic material models, te vector N, or equivalently α, is known in advance. As a result, only te wrinkling parameter a as to be computed numerically wic improves te algoritmic efficiency significantly Stable solution witout wrinkles or slacks According to te previous subsection, te classical solution is stable from an energetical point of view, if te stresses are (semi-) positive definite. Hence, te following condition can be derived S 0 no wrinkles or slacks (44)

12 Variational algoritmic formulation for wrinkling at finite strains Numerical implementation of energy-driven wrinkling Details about te numerical implementation are discussed witin tis subsection. In contrast to most works previously publised, te proposed algoritm is directly based on te underlying variational principle governing te formation of wrinkles or slacks. More precisely, te local state variables are computed by minimizing te respective Helmoltz energy. Following Subsections , te local state of a membrane undergoing large deformations can be classified into te following groups: wrinkles, slacks and te classical solution witout any wrinkles or slacks. To identify te type of a given deformation, te largest eigenvalue of te compatible rigt Caucy-Green tensor C is computed first. Subsequently, te criterion signalling te formation of slacks (31) is cecked. In case of slacks, te wrinkling-related part of te Caucy-Green tensor follows directly from a spectral decomposition, cf. Eq. (29). On te oter and, if te largest eigenvalue of C is greater tan one, te possibility of wrinkling as to be analyzed. For tat purpose, te smallest eigenvalue of te stress tensor is calculated by assuming a trial state caracterized by C w = 0. If tis value is greater tan zero, no wrinkles or slacks will form and te trial state represents already te solution to te problem. On te oter and, if inequality (44) does not old, te standard solution (C w = 0) as to be relaxed and te wrinkling strains are computed from te minimization problem (21) wit b 2 = 0. It sould be noted tat for isotropic models, te classification of te type of deformation can be implemented more efficiently. For instance, as stated in [28], if te considered (standard) yperelastic material obeys te strong ellipticity condition, it fulfills te Baker-Ericksen inequalities. Teses inequalities in turn imply tat te greater principal stress occurs always in te direction of te greater principal stretc. Hence, te smaller principal stress can be computed directly witout calculating all components of te stress tensor. Clearly, te classical solution (no wrinkles or slacks) does not require any modification of te standard yperelastic constitutive model. Likewise, in case of slacks, te computation of te wrinkling strains is straigtforward. Hence, te only non-trivial problem is te calculation of te mecanical response, if wrinkles occur. As mentioned before, classical optimization algoritms can be applied to solve te respective minimization problem (21). We ave employed tree different optimization scemes: a conjugate gradient algoritm of Fletcer-Reeves and Polak- Ribiere type [29], a limited memory BFGS metod [30] and a damped Newton s sceme, cf. [31] for furter details. Since for eac of tose metods te derivatives of te function to be minimized are required, tey are summarized witin te next paragraps First derivatives Residuals In case of wrinkling, te non-linear set of equations (21) as to be solved for b = 0. Te respective stationarity condition reads, cf. Subsection 3.2.2, or equivalently, wit te first derivatives given by R S (a) = 0, a := [a 2 ; α] T (45) R P (a) = 0, a := [a 2 ; α] T (46) R S (a) := S(a 2, α) N(α), R P (a) := P(a 2, α) N(α) (47) It sould be noted tat, te deformation gradient F is regular and ence, R S (a) = 0 or R P (a) = 0 is indeed equivalent to vanising true stresses, i.e., σ n = 0. Here, σ denotes te Caucy stresses.

13 12 J. MOSLER Second derivatives If Newton s metod is to be applied, te second derivatives are required as well. Te linearizations of Eq. (45) are given by (for fixed C) R S (a 2 ) = 1 N C : (N N), 2 Ψ2 C := 4 C 2 r (48) R S α = S M a2 N C : α (N N) (49) Consistent linearization - algoritmic tangent moduli Suppose Newton s metod is adopted to solve te resulting global minimization problem (23). In tis case, te linearization of te considered optimization algoritm is required to guarantee an asymptotical quadratic convergent. Assuming te (local) wrinkling problem is converged, te linearizations of Eq. (45) can be calculated (C is not fixed anymore). A straigtforward calculation yields da = 1 2 ( (a 2 )R S α R S ) 1 (N C) : dc, a = [a 2 ; α] (50) For te sake of compactness, te tird-order tensor A is introduced by Wit tis notation, te consistent tangent can be computed by da = A : dc. (51) C T := 2 ds dc = C : [ I sym + (N N) A (1) + 2 a 2 (N M) A (2)] (52) Here, te tensors A (i) are defined according to Clearly, by using te identity [A (i) ] jk = [A] ijk, (53) P = d F Ψ = Cr Ψ : d C C r : F C = F S (54) te linearizations of te first Piola-Kircoff stresses can be computed as well. Tey result in A := dp df = 1 S + [F C T F T ] t, wit [P t ] ijkl = [P] ijlk. (55) Te non-standard dyadic product is defined in Appendix A Modification of te numerical implementation for applying tree-dimensional constitutive models According to Subsection 2.2, if a fully tree-dimensional material model is to be used, te plane stress condition caracterizing a membrane cannot be guaranteed a priori. As a consequence, te kinematics ave to be sligtly modified, i.e., C r = C + C w + C 33 E 3 E 3, C w 0, C 33 > 0 (56)

14 Variational algoritmic formulation for wrinkling at finite strains 13 Hence, instead of Eq. (21) te modified minimization principle reads now (C 33, a 2, b 2, α) := arg inf C 33 >0 a 2 0 b 2 0 α [0,π] Ψ(C r ), (57) Tis can be sown as follows. First, it is noted tat te stationarity conditions wit respect to a, b and α are not affected by te modified kinematics. Hence, tey are identical to tose described in detail in te previous subsections. Furtermore, te respective condition associated wit C 33 reads Ψ C 33 = S : (E 3 E 3 ) = 0 (58) As a consequence, te stresses resulting from energy relaxation fulfill indeed te plane stress condition. It bears empasis tat in case of vanising sear strains ( ) i3, Eq. (58) is equivalent to σ 33 = 0 were σ denotes te Caucy stresses. Remark 7. Te implementation of te tree-dimensional model is almost identical to tat fulfilling a priori plane stress conditions. More precisely, witin te resulting 3D finite element formulation, a minimization wit respect to C 33 is performed first. Subsequently, te wrinkling criteria according to Subsection 3.2 are cecked. In case of slacks, te solution follows directly from Subsection On te oter and, if te formation of wrinkles is signalled, te state variables are computed from Eq. (57). Te respective derivatives necessary for an optimization algoritm are given by Eq. (58) and in Subsection Te second derivatives can be derived similarly. However, for te sake of brevity, tey are omitted ere. 3.5 Modifications necessary for initial stresses Membranes often sow initial stresses, e.g. resulting from prestress. Clearly, tese stresses can be modeled simply by applying te respective loading istory. Alternatively, te Helmoltz energy of te constitutive model can be modified. In te following, S 0 denotes te initial stresses (of second Piola-Kircoff type) and F 0 is a (possibly incompatible) deformation gradient caracterizing te previous loading istory. Hence, wit F = GRADϕ, te effective deformation gradient reads F eff := F F 0 (59) and te resulting free energy is given by Ψ eff (C) := Ψ(C eff ), C eff := F T eff F eff = F T 0 C F 0 (60) Te initial strains F 0 can be computed from te constraint S 0 = 2 F 0 1 Ψ eff C eff Ceff =F T 0 F 0 F T 0 (61) Clearly, in case of wrinkling, te effective rigt Caucy-Green tensor yields now C eff = F T 0 (C + C w) F 0 (62) and te wrinkling parameters can be obtained in te same manner as described before. It bears empasis tat te variational -adaption wic will be presented in te next section can be applied to any minimization problem; in particular to tat discussed ere. As a result, te novel mes adaption allows to take prestress consistently into account.

15 14 J. MOSLER 4 Variational -adaptivity Te proposed numerical framework suitable for te analysis of wrinkles and slacks is directly governed by te underlying variational principle. In case of yperelasticity, tis principle is tat of te minimum potential energy. Te advantages resulting from suc a variational structure are manifold. For instance, it opens up te possibility to apply standard optimization strategies to solve te considered problem as sown in te previous section. Moreover, minimization principles provide a natural basis to estimate te quality of te numerical solution and ence, tey represent a natural basis for mes adaption. Suc variational mes adaptions were advocated in [23] in case of an Arbitrary Lagrangian-Eulerian (ALE) formulations, wile a variational - adaption was presented in [24]. For an overview, te interested reader is referred to [32]. In tis section, te -adaptive finite element metod discussed in [24] is sligtly modified and combined wit te energy driven wrinkling algoritm as proposed in Section 3. It sould be noted tat tis section is not meant to be a compreensive overview on adaptive finite elements metods. For instance, so-called goal-oriented metods are not covered at all. For furter details on mes adaption, te reader is referred to [33 35]. According to [24], te (modified) principle of minimum potential energy (23) provides an unambiguous comparison criterion for test functions: a test function ϕ (1) is better tan anoter ϕ (2) if and only if I(ϕ (1) ) < I(ϕ (2) ). All oter considerations are, in effect, spurious. Tis energy comparison criterion is te basis of variational mes adaption. Based on tis relatively simple observation, an error indicator of te type Ĩ = inf I (ϕ (1) ϕ (1) ) inf ϕ (2) I (ϕ (2) ) (63) can be introduced. Here, ϕ (1) is te deformation mapping spanned by te initial mes and ϕ(2) is associated wit a locally refined triangulation. Accordingly, Ĩ cecks te effect of local mes refinement on te solution. In tis work, attention is confined to simplicial meses and edge bisection (cf [36 38]) as te device for acieving mes refinement, Fig. 2. In wat follows, σ e denotes te transformation tat refines an initial mes T (1) by bisection of edge e resulting in T (2). Obviously, edge bisection generates a nested family of triangulations and ence, te space of admissible deformations is indeed enlarged (ϕ (1) V (1) V (2) ϕ (1) ). As a result, Ĩ(e) 0. Unfortunately, te proposed error indicator is numerically very expensive. More precisely, a global optimization problem as to be solved for every edge. Terefore, te local energy released by mes refinement is estimated by means of a lower bound obtained by relaxing a local patc of elements. Wit ϕ (1) being te solution of te initial mes, i.e., ϕ(1) = arg inf I (ϕ (1) ), te error indicator I (e) as proposed in [24] reads 0 I (e) := I (ϕ (1) ) supp(ϕ (2) inf ϕ (2) V 2, ϕ (2) Ω 1 = ϕ, ϕ(1) )=supp(v (2) (1) /V ) I (ϕ (2) ) Ĩ (64) It is computed for every edge e. Here, ϕ and supp(f) are prescribed deformations acting at 1 Ω and te support of te function f, respectively. According to Eq. (64), te influence of local mes refinement is estimated by relaxing te deformation field only in te refined region supp(v (2) (1) /V ). Clearly tis set is defined by all elements saring te considered edge e. As a consequence, in te numerical implementation, te deformation mapping is fixed and identical to tat of te intial mes outside supp(v (2) /V (1) ), wile in te interior of supp(v (2) /V (1) ) te

16 Variational algoritmic formulation for wrinkling at finite strains σ Figure 2: Mes refinement in two dimensions by applying edge-bisection; σ e denotes te transformation tat refines an initial mes by bisection of edge e nodal displacements are computed by minimizing te respectiev energy. Furter details on te presented variational indicator are omitted. Tey may be found in [24, 32]. Te error measure (64) sows O(n) complexity and terefore, can be computed efficiently. Having calculated I for eac edge witin te triangulation, te edges are sorted accordingly and te limits µ loc (T ) = max I (e), e ρ loc (T ) = min e I (e) (65) can be specified. Finally, te following mes refinement strategy is employed: i) For all edges in T DO: IF I (e) > α ref (µ loc (T ) ρ loc (T )) + ρ loc (T ), mark e for refinement. ii) Apply refinement. Here, α ref (0, 1] is a numerical parameter defining te tresold of te edges to be cut. Remark 8. Witin te numerical examples presented in Section 5 only te energetically most favorable edge is cut at a time. Hence, α ref = 1. Remark 9. Te algoritm presented in tis subsection is not stable. Consequently, degenerated elements can occur (te aspect ratios of te elements created by tis metod can converge to infinity). As a consequence, as sown in [24, 32], if only edges wic ave been marked by te presented error indicator are cut, te resulting meses can be igly anisotropic. However, tis anisotropy is related to te pysics of te considered problem and terefore, te performance of tose meses is superior to tat of teir isotropic counterparts, cf. [24]. It sould be noted tat in some cases, e.g., if iterative solvers are used, te aspect ratio of te elements may need to be maintained. In suc a case, edges can be bisected by means of Rivara s longest-edge propagation pat (LEPP) bisection algoritm [36 39], wic guarantees an upper bound on te element aspect ratio. Te performance of tis metod is sown in Section 5 as well. Remark 10. In te case of linearized elasticity teory, it can be sown tat te proposed error indicator is equivalent to a matematically rigorously derived error estimate based on local Diriclet problems, cf. [40, 41]. Te respective proof can be found in [32]. Hence, in tis case, te advocated error indicator inerits te same properties as te estimate according to [40, 41]. However, it sould be noted tat tis equivalence is not fulfilled in te more general case. For igly non-linear problems, it is usually not even known, if an analytical solution exists, and even if it exists, it is not clear, if it is unique.

17 16 J. MOSLER u 2 u1 100mm Lamé constants λ N/mm 2 µ N/mm 2 200mm Figure 3: Sear test of a membrane: geometry (tickness of te membrane t = 0.2 mm) and material parameters (yperelastic constitutive model according to Appendix A); boundary conditions: First, loading is applied by prescribing a vertical displacements of magnitude u 2 = 1mm and u 1 = 0. Subsequently, u 1 is increased up to u 1 = 10mm by olding fixed u 2 = 1mm. 5 Numerical examples Te applicability and versatility of te variational wrinkling algoritm as well as te performance of te novel variational -adaption are illustrated by means of two numerical examples. Particularly, te convergence rate of te proposed variational mes adaption is igligted. Wile a sear test is analyzed in Subsection 5.1, torsion of a circular membrane is computed in Subsection Sear test Te material parameters, te geometry and te boundary conditions of te first example are sown in Fig. 3. According to Fig. 3, tension is applied during te first loading stage, i.e., u 2 = 1mm and u 1 = 0. Subsequently, u 1 is increased up to a final magnitude of u 1 = 10mm by olding u 2 = 1mm constant. Te mecanical problem is identical in every way to tat treated in [6] wit te sole exception tat a different constitutive model is used. Here, te yperelastic potential described in Appendix A is adopted. However, it sould be noted tat te occurring strains are relatively small and ence, te considered material model predicts almost te same response as te isotropic version of te constitutive law in [6]. Te response of te membrane is baselined by means of a relatively fine discretization ( DoFs) consisting of 6-node quadratic triangle elements, see Fig. 4(a). Five different computations ave been performed. Four of tose are based on te minimization algoritm explained in te previous section. Te resulting local and global optimization problems ave been solved by applying a conjugate gradient approac of Fletcer-Reeves and Polak-Ribiere type [29], a limited memory BFGS metod [30] and a damped Newton s sceme, cf. [31]. Additionally, a semi-analytical metod as been used as well. In tis case, te minimization problem caracterizing te formation of wrinkles and slacks is solved analytically. Te respective equations are summarized in Appendix A. As expected, te predicted mecanical response is independent of te applied solution sceme. Te computed final deformed configuration, togeter wit te distribution of te wrinkling strains, is sown in Fig. 4(a). Here, te larger eigenvalue of te wrinkling strain C w is plotted. According to Fig. 4(a), wrinkles form at te

18 Variational algoritmic formulation for wrinkling at finite strains 17 Reaction force λ max (C w ) 0.55 (a) Displacement u 1 (b) Wriggers et al. uniform, fine 10 Figure 4: Sear test of a membrane: (a) final deformed configuration and distribution of te wrinkling strain λ max (C w ) computed by means of a uniformly fine discretization consisting of 6-node bi-quadratic triangle elements ( DoFs); and (b) resulting load-displacement diagram (u 1 vs. conjugate force) Figure 5: Sear test of a membrane: initial coarse discretization used for te adaptive finite element computations lower left as well as at te upper rigt part of te membrane. Clearly, it is not known, if Fig. 4(a) corresponds to wrinkles or slacks in general. However, for te analyzed structure only a few slacks wic are not sown explicitly develop at te upper left and te lower rigt part of te structure. Consequently, Fig. 4(a) is almost exclusively associated wit wrinkles. It sould be noted tat te distributions of bot te wrinkles as well as te (not sown) slacks agree wit tose reported in [6]. Tis can also be verified by te load-displacement diagram in Fig. 4(b). Te proposed wrinkling algoritm, in conjunction wit a relatively fine discretization, predicts an only marginally softer structural response tan te numerical model presented in [6]. Next, te mecanical problem is re-analyzed by employing te -adaptive sceme as discussed in Section 4. For tat purpose, a computation based on te coarse discretization in Fig. 5 is performed first. Subsequently, te solution is improved by applying two different variational -adaptions: an unconstrained variational -adaption calculation according to Section 4, in wic only te energetically most favorable edge is bisected at eac step; and a constrained variational -adaption calculation, in wic an upper bound on te aspect ratio of te elements is maintained by means of Rivara s longest-edge propagation pat (LEPP) bisection algoritm, cf. Remark 9. Fig. 6 sows te adaptively computed meses, te deformed configurations and te distribution of te wrinkling strain. By comparing te plots of te wrinkling strain in Fig. 6 to tat

19 18 J. MOSLER (a) (b) 0 λ max (C w ) 0.55 Figure 6: Sear test of a membrane: final meses, deformed configurations, togeter wit te distribution of te wrinkling strain λ max (C w ), after: (a) variational -adaption; and (b) variational -adaption combined wit Rivara s longest-edge propagation pat (LEPP) bisection algoritm corresponding to te fine discretization (Fig. 4(a)), te performance of te proposed variational mes adaption becomes evident: Altoug te mes computed by applying te unconstrained (constrained) -adaption consists of only 582 (597) nodes (in contrast to nodes), it predicts an almost identical wrinkling pattern. Te quality of te numerical approximation associated wit te variational mes adaptions can be investigated best by analyzing te convergence in energy. Te respective diagram is sown in Fig. 7. According to Fig. 7, te convergence rate of te adaptive sceme is remarkable. Altoug te final meses resulting from te variational adaption are relatively coarse (582 and 597 nodes), te error in energy (wit reference to te finest uniform discretization) is negligibly small (0.02%). Furtermore, as already noted in [24], constraining te aspect ratio of te elements as a detrimental effect on te rate of convergence. However, for te considered example, te decrease in performance is only marginal compared to te fully unconstrained algoritm. 5.2 Torsion of a circular membrane Te next example is concerned wit a circular membrane subjected to torsion, Fig. 8. A similar problem as been investigated by several researcers, cf. e.g. [5, 7, 26]. Here, te geometry and te material parameters are cosen according to [42] wit te sole exception tat plastic effects are neglected. Te membrane aving a tickness of t = 0.01mm is clamped at te outer as well as at te inner boundary and loading is applied by increasing te angle α at te inner boundary up to α max = 1.8 = rad. Following te previous subsection, tree different computations ave been performed. Start-

20 Variational algoritmic formulation for wrinkling at finite strains uniform -adaption Rivara Energy I # nodes Figure 7: Sear test of a membrane: Convergence in energy resulting from: variational - adaption; and variational -adaption combined wit Rivara s longest-edge propagation pat (LEPP) bisection algoritm α Lamé constants λ N/mm 2 µ N/mm 2 Geometry r a 125 mm r a 45 mm t 0.01 mm r i r a Figure 8: Torsion of a circular membrane: geometry, material parameters (yperelastic constitutive model according to Appendix A) and boundary conditions. Te membrane is clamped at bot radii.

21 20 J. MOSLER ing wit te initially coarse discretization sown in Fig. 9 (top left), te mes is refined subsequently by applying: te unconstrained -adaption discussed in Section 4; and te variational -adaptive sceme combined wit Rivara s metod for maintaining te aspect ration. It sould be noted tat te implemented edge-bisection metod inserts new nodes in suc a way tat te boundary of te initial mes is not modified. Since a bi-quadratic isoparametric formulation is applied ere, te boundaries of te initial mes, and as a result tose of te refined discretizations, are no circles, but quadratic approximations. Consequently, te boundaries Ω are not smoot ( Ω C 1 ) resulting in artificial singularities. However, it bears empasis tat tese effects ave only a weak influence on te numerical solution (difference in energy is less tan 0.5%). Furtermore, te insertion of new nodes at te boundaries can be easily modified suc tat te resulting curves converge to circles. Fig. 9 summarizes te results computed from te different strategies. For te sake of comparison, uniform refinement is considered as well. At te top rigt in Fig. 9 te computed wrinkling distribution predicted by uniformly refining te coarse discretization at te top left in Fig. 9 is sown. Te respective mes wic is not presented consists of nodes. According to tis plot, te wrinkling strain and tus, te strain-energy, displays a rapid variation in te radial direction, wile it varies slowly in te ortogonal direction. In contrast to te uniformly refined triangulation, te discretizations corresponding to te adaptive metods contain only 5001 nodes (eac). Altoug te number of DoFs is relatively small, te wrinkling pattern associated wit te variational -adaption agrees perfectly wit tat of te uniform mes, cf. Fig. 9. Hence, te adaptive metod increases te numerical efficiency significantly. It bears empasis tat te unconstrained -adaption results in igly anisotropic and directional meses tat trace te fine structure of te energy-density field, see Fig. 9. Following te previous subsection, te quality of te numerical scemes is investigated by analyzing te convergence in energy. Te respective diagram is sown in Fig. 10. As evident from Fig. 10, all refinement strategies converge to te same limiting value igligted by te orizontal dased line. Furtermore, te adaptive metods sow a remarkably iger convergence rate. Analogous to te previous subsection, te unconstrained variational -adaption is superior to tat of constrained variational -adaption. In contrast to te sear test analyzed in Subsection 5.1, te wrinkling pattern associated wit te example studied ere is more localized and igly anisotropic, i.e., wrinkles only occur in a relatively small region in te vicinity of te inner boundary and te distribution of te wrinkling strains strongly varies in te radial direction compared to te ortogonal direction. Clearly, tis effect can be better taken into account by te unconstrained -adaption in wic te elements are allowed to sow different lengt scales in ortogonal directions (see Fig. 9). As a consequence, te superiority of unconstrained -adaption is even more pronounced compared to te previous example. 6 Conclusion A fully variational finite element formulation suitable for te analysis of membranes as been presented in tis paper. In contrast to previous numerical models and inspired by te works by Pipkin [13, 15], te state variables, togeter wit te deformation mapping, follow jointly from minimizing an energy functional. Te proposed framework is very general and does not rely on any material symmetry of te yperelastic material model. Te elaborated metod allows to employ arbitrary, fully tree-dimensional yperelastic constitutive models directly. More specifically, plane stress conditions caracterizing a membrane stress state are naturally included witing te variational formulation. Hence, a numerically expensive projection of te material model to plane stress space is not required.

22 Variational algoritmic formulation for wrinkling at finite strains 21 uniform meses left: initial rigt: final unconstrained -adaption -adaption +Rivara 0 λ max (C w ) 0.55 (a) (b) Figure 9: Torsion of a circular membrane: (a) initial mes and discretizations predicted by te (un)constrained variational -adaption; (b) distribution of te wrinkling strain λ max (C w ) resulting from: uniform refinement; and (un)constrained variational -adaption