A finite-strain corotational triangular shell element

Transcription

1 A finite-strain corotational triangular shell element Federica Caselli, Paolo Bisegna Department of Civil Engineering and Computer Science, University of Rome Tor Vergata, Italy {caselli, Keywords: Shell finite elements, corotational formulation, finite strain, hyperelastic materials. SUMMARY. An original three-node facet-shell element suited for the analysis of thin hyperelastic shells undergoing large displacements, large rotations, fleure and large membrane strains is presented. 1 INTRODUCTION Shell structures are widely used in engineering practice, e.g., in structural, aerospace and biomedical applications, and the development of efficient computational models for their analysis well into the nonlinear regime has been receiving continuous attention in the last decades. The finite element method has promoted the most significant advances in computational shell mechanics. Different types of finite elements have been developed for the analysis of shell structures, including solid 3D elements, continuum-based (or degenerated) shell elements, 2D elements based on a shell theory, flat elements. The latter are especially attractive due to their simplicity. In particular, flat triangular elements combined with a corotational formulation have gained increasing attention in recent years [1]. The corotational formulation (see e.g. [2]) is based on the idea of separating rigid body motions from strain producing ones and has had an ecellent track record for problems with arbitrarily large displacements and small strain response. In fact, in those cases, eisting high-performance linear elements can be reused as core elements in the geometrically nonlinear contet, after large rigid body motions have been filtered out by the corotational machinery. The restriction to small strain has represented a long standing limitation for the corotational approach. In order to allow the etension to large strain problems, the correct identification of the element rigid motion plays a crucial role. In particular, as first pointed out by Crisfield [3], a polar decomposition based corotational approach is required. Recently, Rankin [4] eploited elementindependent corotation based on polar decomposition at the element centroid for the analysis of structures undergoing large strain response with standard off-the-shelf elements originally designed for small strain. The pivotal issue in [4] is the link between corotation and the Biot strain measure: in case of 2D or 3D continua, the strain computed in the framework of a corotational formulation from a small-strain approimation approaches the Biot strain. However, this link is eact only in the limit of mesh refinement. In this work, a new strategy for the etension to large strain response is proposed, resulting into a simple and effective three-node triangular shell element able to handle geometric nonlinearities (large displacements, large rotations and large membrane strains), as well as material nonlinearities. Its main ingredients are: i) a polar-decomposition based corotational formulation eploiting original closed-form formulas for its efficient implementation and rigorously accounting for distributed loads [5, 6]; ii) a kinematic description of the core-element displacement field inspired by the theory of small deformation superimposed on large [7]. The efficacy of the resulting shell element is demonstrated by means of numerical eamples of hyperelastic response with huge strains. The proposed element represents an effective tool for the analysis of problems involving rubber-like or biological materials and comple features such as contact and fluid-structure interaction. 1

2 2 FINITE-STRAIN COROTATIONAL SHELL ELEMENT 2.1 Polar decomposition based corotational framework The corotational approach is here briefly reviewed. Three-node triangular shell elements with 6 degrees of freedom per node are considered. Let u i and ϑ i respectively denote the displacement and rotation vector of the typical node V i and let R i denote the rotation tensor related to ϑ i. The element nodal parameters u i, ϑ i are collected into the 18 1 vector a = {u 1 ; ϑ 1 ; u 2 ; ϑ 2 ; u 3 ; ϑ 3 }, (1) where the semicolon symbol denotes column stacking. Moreover, let p and u(p) respectively denote the typical point in the reference configuration and the element displacement field. In the derivation of corotational finite elements, the deformation f(p) = p + u(p) is multiplicatively decomposed as follows f = r f, (2) where r is a rigid transformation, characterised by a reference point G, a translation vector t, and a rotation tensor R, that is, r(p) = G + t + R[p G]. (3) Hence, the transformation f, implying the same deformational motion as f, is obtained from the latter after filtering out the rigid motion r. Denoting by u(p) = f(p) p (4) the filtered displacement field, equations (2) and (3) yield [ ] u = G + t + R p + u G p. (5) Then, a suitable interpolation is chosen for u, characterising the core-element formulation where a is the filtered counterpart of the element nodal parameters a u = u(a, p), (6) a = {u 1 ; ϑ 1 ; u 2 ; ϑ 2 ; u 3 ; ϑ 3 }, (7) to be used for the computation of core-element nodal residual vector and consistent material tangent stiffness tensor. The filtered nodal displacements u i are obtained by imposing the interpolation conditions u(a, V i ) = u i and u(a, V i ) = u i in (5). The following epression is derived: [ ] u i = R T V i + u i (G + t) (V i G). (8) The filtered nodal rotation vectors ϑ i are given by ϑ i = a log R i, (9) where R i = R T R i are the filtered nodal rotation tensors, a( ) denotes the ais of a skewsymmetric tensor, and log( ) denotes the tensor logarithm. Of course, what makes the corotational approach work is a correct identification of the rigid body motion [4], and most of the literature on corotation concerns this choice either directly or 2

3 (a) V3 (b) V1 V2 f V3 f V 3 V3 V2 V1 y V2 V 1 V1 V 2 y Figure 1: Polar decomposition based corotational filter. (a) Overall transformation f and (b) deformational transformation f. Reference and transformed triangles are shown in cyan and red, respectively. It is noted that all points Vi and V i in (b) are coplanar. reference V, X fo intermediate V o, o deformed V, f Figure 2: Representation of the deformational transformation f : it is decomposed into a transformation f o, characterised by arbitrarily large membrane strains and taking into account thickness etensibility, and a superimposed transformation f, keeping the nodes of the element fied and characterised by small bending and drilling rotations, i.e., f = f f o. indirectly. Whereas the rotation tensor of the polar decomposition of the element motion varies from point to point, any form of corotation attempts to provide a rotation tensor R that is a close analogue to the local rotation tensor in an average sense over a given element. Clearly, in the limit of a fine grid, if R is a good approimation for some point in the element, then the effects of rigid-body motion can be ignored for the whole element. Several proposals have been put forth in the literature for identification of the rigid body motion, such as, for eample, the side alignment, according to which R is such that RT maps the triangle joining the deformed nodes onto the plane of the reference triangle with alignment of one side. As the strain becomes large, however, standard methods fail to pass the finite-stretch patch test (i.e., if a group of elements is subjected to a uniform stretch, no spurious deformational rotations should appear). This is demonstrated by the presence of a finite rotation R = I even for a displacement field representing a pure deformation. For three-node triangular shell elements the rigid body motion can be identified by means of the polar decomposition of the gradient of the homogeneous transformation mapping the element nodes Vi onto their current positions Vi. This approach is adopted here. A detailed description of the method can be found in [5], which also provides a MATLAB implementation of the corotational procedure as a supplementary material. Figure 1 shows an eample of the filtering operation. 3

4 2.2 Finite-strain core element formulation: a small-on-large approach The core-element formulation (6) relevant to the deformational transformation f (Figure 1(b)) is here specified. The basic idea is to eploit the corotational framework and the plate-like geometry of the element to decompose the transformation f into a transformation f o characterised by arbitrarily large membrane strains, also accounting for thickness variation, and a superimposed transformation f keeping the nodes of the element fied and characterised by small drilling and bending rotations (Figure 2), that is, f = f f o. (10) In some sense, this approach etends the concept of corotation to deformations: the large average membrane deformation is etracted from the total deformation and the remaining small deformation is handled with high-performance small-strain elements. Accordingly, from equations (2) and (10), the overall decomposition f = r f f o (11) is adopted. The transformation f o maps the reference configuration V into an intermediate configuration V o, and the transformation f maps the intermediate configuration V o into the deformed configuration V. Denoting by u o and u the displacement fields of f o and f, respectively, and denoting by X, o, and the position vectors of the typical point p in the reference, intermediate and deformed configurations, respectively, from (10) and (4) it turns out that and with o = f o (X) = X + u o (X), = f ( o ) = o + u ( o ), (12) = f(x) = f (f o (X)) = X + u(x), (13) u(x) = u o (X) + u ( o (X)). (14) From (10), and (12), the core-element deformation gradient F = X f turns out to be with and Introducing the relevant Green-Lagrange strain tensors F = F F o, (15) F o = I + H o, F = I + H, (16) H o = X u o, H = ou. (17) E o = 1 2 (F ot F o I), E = 1 2 (F T F I), (18) the core-element Green-Lagrange strain tensor E specifies as E = 1 2 (F T F I) = 1 2 [F ot (I + 2E )F o I] = E o + F ot E F o, (19) where the last term represents the pull back of E to the reference configuration. 4

5 For convenience, the orthonormal frame {e, h, n} defined by e = V 2 V 1 V 2 V 1, h = n e, n = e (V 3 V 1 ) e (V 3 V 1 ), (20) with origin O at the centroid of the reference triangle V 1 V 2 V 3, is introduced, and the coordinates of the position vectors X, o, and are denoted by X = (X, Y, Z), o = ( o, y o, o ), = (, y, ). (21) Material symmetry with respect to the shell midsurface is assumed. The displacement field u is described as the composition of: a membrane displacement field u o with thickness etensibility, where u o (X) e = u o (X, Y ), (22) u o (X) h = v o (X, Y ), (23) u o (X) n = o Z, (24) o = λ Z Z (25) and the transversal stretch λ Z is obtained as a nonlinear function of X u o (X, Y ) and X v o (X, Y ) by imposing the plane-stress condition S(E o Z=0 )n n = 0 on the midsurface (here and in the following S denotes the second Piola-Kirchhoff stress tensor); a superimposed plate displacement field u, u ( o ) e = u ( o, y o ) + ϑ y( o, y o ) o, (26) u ( o ) h = v ( o, y o ) ϑ ( o, y o ) o, (27) u ( o ) n = w ( o, y o ), (28) where the Kirchhoff hypothesis is assumed, so that ϑ and ϑ y depend on w. In the present corotational framework [5], the deformational transformation f maps the reference triangle nodes, that is the points V i, into the points V i = V i + u i. (29) Since the latter belong to the plane containing the reference triangle V 1 V 2 V 3 (Figure 1(b)), the following conditions can be imposed u o (X) X=Vi O = V i V i, (30) u ( o ) o =V i O = 0, (31) that is, u o takes care of mapping the points V i into their deformed counterparts V i. In turn, the latter are kept fied by u. It is noted that H is not in general symmetric because u implies drilling-type rotations, related to u and v, and bending-induced rotations, related to ow. The smaller the element sie, the smaller these rotations are. Indeed, in the limit of mesh refinement, drilling-type 5

6 rotations reduces as the element-wise constant rotation field provided by the corotational machinery approaches the local rotation field, and bending-induced rotations reduces as the intermediate configuration approaches the deformed configuration. No approimations are made on the Green-Lagrange strain tensor E o corresponding to u o, so that arbitrarily large membrane strains can be handled. Conversely, the Green-Lagrange strain tensor E corresponding to u is approimated according to the von Kármán theory (see e.g. [8]) E = 1 2 (H + H T + H T H ) ε m + ε b + ε vk, (32) where ε m and ε b are the linear membrane and bending strains and ε vk is the von Kármán-type nonlinear contribution. Thanks to the corotational approach, the rotations implied by the displacement gradient H can be made sufficiently small with a sufficiently fine mesh, hence the simplification of the nonlinear strain-displacement relation enforced by approimation (32) is justified, provided that ε b is small compared to the unity. The following interpolations are eploited. The components u o (X, Y ) and v o (X, Y ) of the displacement field u o (X) are approimated with the constant strain triangle (CST) interpolation of the nodal in-plane displacements. The membrane strain ε m is obtained by means of the Assumed Natural DEviatoric Strain (ANDES) template [9] interpolation of the nodal drilling rotations, assuming null nodal translations as a consequence of (31). The bending and von Kármán strains, ε b and ε vk, are obtained by the Discrete Kirchhoff Trianlge (DKT) [10] interpolation of the nodal bending rotations. It is worth noticing that an hidden dependance from the nodal in-plane displacements is inherited by the superimposed strains ε m, ε b and ε vk : in fact, they are defined on the intermediate configuration V o which, in turn, depends on the underlying membrane field u o. is 2.3 Equilibrium weak formulation The weak formulation of the equilibrium equations at the element level in a material description q δa = q i δa + q l δa = S : δe dv l δu da, (33) V where q is the nodal residual vector and q i and q l are the nodal internal-force and load vectors, respectively. Moreover, E = (F T F I)/2 is the Green-Lagrange strain tensor relevant to the overall deformation gradient F = X f; l denotes the applied forces per unit reference area, assumed to act on the midsurface A. From equations (2) and (3) it turns out that F = RF. Hence, the core-element Green-Lagrange strain tensor E coincides with E. As a consequence, the internal-force contribution is recast as q i δa = S(E) : δe dv = S(E) : δe dv = q i δa, (34) V where q i is the nodal internal-force vector work-conjugated to the filtered nodal parameters. It is pointed out that the last equality is the core-element equilibrium weak formulation, entailing the interpolation (6) for the relevant finite-element implementation. V A 6

7 From (19), writing the weak form in terms of the intermediate configuration V o, it results that q i δa = where V and the relationship S : δe dv = ( S : sym[δf ot (I + 2E )F o ] + F ot δe F o) 1 V J o dv o = o σ o : sym[(i + 2E ) o δu o ] dv o + σ o : δe dv o, (35) V o V o σ o = 1 J o F o SF ot, J o = det F o, (36) o δu o = (δf o )F o 1 (37) has been employed. From the first of (36), the tensor σ o represents a push forward of the second Piola-Kirchhoff stress tensor S to the intermediate configuration, scaled by the reciprocal of the Jacobi determinant J o of the deformation gradient F o. The first term on the right hand side of (35) accounts for the virtual work epended by σ o as a consequence of variations of u o keeping u fied, whereas the second term represents the virtual work epended by σ o as a consequence of variations of u. The element computations provide the nodal internal-force vector at the core-element level, that is, q i. In order to lift the nodal internal-force vector to the overall level, the crucial ingredient is the so-called projector operator Π, able to etract the deformational part from incremental displacements: δa = Π δa. (38) Indeed, from the virtual work identity (34), it turns out that q i = Π T q i. (39) Analogously, the variation δπ of the projector operator is required to lift the core-element tangent stiffness tensor to the overall level in a consistent way. In this work, a highly efficient implementation is obtained by eploiting simple closed-form formulas for the computation of the projector operator and its variation, providing a rigorous, eplicit algorithm for the computation of the nodal residual vector and of the consistent tangent stiffness tensor. A thorough description can be found in [5] where the treatment of distributed loads is also developed. The linearisation of (35) and the relevant finite-element implementation is detailed in [7]. 3 NUMERICAL RESULTS In order to validate the present formulation an etensive numerical campaign has been conducted. Some eamples are reported in the following, other benchmark tests can be found in [7]. The numerical eperiments have been performed by means of an in-house MATLAB code. The full Newton-Raphson method was used in the nonlinear solution procedure, the simultaneous 0.5% force tolerance and 1% displacement tolerance was employed as convergence criterion, and the automatic load incrementation scheme described in [11] was followed.. 7

8 (a) (b) 8 y C Load /10 (psi) present 2 Rankin 2006 ABAQUS 1 Huges Carnoy 1983 Oden Vertical displacement at disk centre /10 (in) Figure 3: Pressuried rubber disk. (a) Reference configuration and (b) load deflection curves. (a) (b) (c) Figure 4: Pressuried rubber disk. Deformed configuration at (a) 17, (b) 50, and (c) 70 psi. Transversal stretch ( ) present ABAQUS Vertical displacement at disk centre /R ( ) Figure 5: Pressuried rubber disk. Transversal stretch at the centre of the disk. 8

9 A y Figure 6: Snap-through of a fibre-reinforced cylindrical arc. Reference configuration (in white) and some deformed configurations. 3.1 Pressuried rubber disk A rubber disk of radius R = 7.5 in and thickness h = 0.5 in, hinged around its outside edge, is subjected to a uniform pressure p so that it bulges into a spherical shape (Figures 3(a) and 4). This eample was considered, for eample, in [12, 13, 4], and constitutes a benchmark problem of the commercial code ABAQUS [14] for hyperelastic response under strains of the order of 400% or more. The problem is aisymmetric and one-eight of the disk was modelled in the analysis, with the mesh depicted in Figure 3(a). The incompressible Mooney-Rivlin strain energy function was adopted: W = c 10 (I 1 3) + c 01 (I 2 3) + p( I 3 1), (40) where I i are the invariants of the right Cauchy-Green tensor, the scalar p is a Lagrange multiplier enforcing the incompressibility condition, and the following values were considered for the material constants: c 10 = 80 psi and c 01 = 20 psi [12, 13]. The Riks method was used in the solution procedure. The number of load increments and the total number of consumed iterations were NINC=7 and NITER=25, respectively. The -displacement at the centre of the disk C is plotted in the load deflection curve of Figure 3(b) and the results in [12, 13, 4, 14] are reported for comparison. An ecellent agreement between the curve relevant to the present solution and the one provided in [4] is found. Only the latter curves continue at very high displacements. Three deformed configurations, corresponding to a load of 17, 50 or 70 psi, are depicted in Figure 4, showing how etreme this test is. Figure 5 shows the transversal stretch arising from the displacement field u o, that is, λ Z in (25), at the centre of the disk against the -displacement. The transversal stretch reported in [14] is also plotted for comparison. A remarkable thinning is revealed. This problem clearly demonstrates the capability of the method proposed here in computing very large strain response. 3.2 Snap-through of a fibre-reinforced cylindrical arc A cylindrical arc made of a fibre-reinforced transversely isotropic material is subjected to a compressive pressure load p (Figure 6). Two fibre orientations are considered: fibres aligned along the aial direction, that is, along the y-ais, and fibres aligned along the circumferential direction. The dimensions of the arc are typical of a piece of heart valve leaflet tissue: height 0.5 cm, length 2 cm, width 0.1 cm, and thickness 0.1 cm. The material was modelled with the transversely isotropic 9

10 (a) 2000 circumferential aial (b) circumferential aial Load [kpa] 1000 Load [kpa] Vertical displacement at point A [cm] Vertical displacement at point A [cm] Figure 7: Snap-through of a fibre-reinforced cylindrical arc. (a) Load deflection curves and (b) their oom. incompressible strain energy function proposed in [15]: W = c 0 [ep c 1(I 1 3) 2 +c 2 (I 4 1) 2 1] + p(j 1), (41) with material parameters c 0 = 188 kpa, c 1 = 5.07 and c 2 = 2.33 reproducing the behaviour of the aortic valve leaflets documented in the eperimental data of [16]. In (41), I 4 represents the square of the stretch in fiber direction and the relevant term is active only if the fibres are under tension, that is, I 4 > 1, to mimic the behaviour of collagen fibres. Owing to symmetry one-half of the structure was considered in the analysis, performed with the Riks method. The -displacement and the -displacement were restrained at the bottom of the arc, while the -rotation and the -rotation were restrained at all nodes. The number of load increments and the total number of consumed iterations were NINC=41 and NITER=189, respectively, for the case of aially aligned fibres, and NINC=32 and NITER=148, respectively, for the case of circumferential fibres. Figure 6 portraits the reference configuration and some deformed configurations, for the case that the fibres are aligned in the circumferential direction. The downward displacements at point A is plotted in the load deflection curves of Figure 7(a), for both fibre orientations. A oomed plot is shown in Figure 7(b). With both fibre arrangements, an initial bending-dominated snap-through phase is followed by a membrane-dominated phase, when the arc has reversed its curvature. In this final phase, a much stiffer response is noticed when the fibres are aligned in the circumferential direction. A similar multi-modal behaviour characterises the gross biomechanics of the aortic valve leaflets. 4 CONCLUSIONS An original triangular shell finite element based on a polar decomposition based corotational formulation and on a small-on-large description of the core-element displacement field has been presented. The element turned out to be effective for thin hyperelastic shells undergoing large displacements, large rotations, fleure and large membrane strains. Its simplicity in terms of geometry and node and degrees of freedom configuration makes it attractive for applications involving comple features such as contact and fluid-structure interaction. References [1] Gal, E. and Levy, R., Geometrically nonlinear analysis of shell structures using a flat triangular shell finite element, Arch. Comput. Method Eng., 13(3), (2006). 10

11 [2] NourOmid, B. and Rankin, C. C., Finite rotation analysis and consistent linearisation using projectors, Comput. Meth. Appl. Mech. Eng., 93(3), (1991). [3] Moita, G. F. and Crisfield, M. A., A finite element formulation for 3-D continua using the co-rotational technique, Int. J. Numer. Meth. Eng., 39(22), (1996). [4] Rankin, C. C., Application of linear finite elements to finite strain using corotation, in Proc. 47th AIAA/ASME/ASCE/ASC Structures, Structural Dynamics, and Materials Conference (AIAA paper No AIAA ), Newport, Rhode Island (2006). [5] Caselli, F. and Bisegna, P., Polar decomposition based corotational framework for triangular shell elements with distributed loads, Int. J. Numer. Meth. Eng., 95(6), (2013). [6] Caselli, F. and Bisegna, P., A corotational shell finite element for aortic valve modeling, in Proc. 6th European Congress on Computational Methods in Applied Sciences and Engineering, Wien, Austria (2012). [7] Caselli, F., Polar decomposition based corotational framework for nonlinear analysis of hyperelastic shell structures with application to biomechanics, PhD Thesis, Rome (2013). [8] Fung, Y. C., Foundations of Solid Mechanics, Prentice-Hall, Englewood Cliffs, NJ (1965). [9] Felippa, C. A., A study of optimal membrane triangles with drilling freedoms, Comput. Meth. Appl. Mech. Eng., 192(16-18), (2003). [10] Bato, J. L., Bathe, K. J. and Ho, L. W., A study of three-node triangular plate bending elements, Int. J. Numer. Meth. Eng., 15(12), (1980). [11] Se, K. Y., Liu, X. H. and Lo, S. H., Popular benchmark problems for geometric nonlinear analysis of shells, Finite Elem. Anal. Des., 40(11), (2004). [12] Oden, J. T., Finite Elements of Nonlinear Continua, McGraw-Hill, New York (1972). [13] Hughes, T. J. R. and Carnoy, E., Nonlinear finite element shell formulation accounting for large membrane strains, Comput. Meth. Appl. Mech. Eng., 39(1), (1983). [14] ABAQUS Benchmarks Manual. Version 6.5. [15] Holapfel, G. A., Sommer, G., Gasser, C. T. and Regitnig, P., Determination of the layer specific mechanical properties of human coronary arteries with non-atherosclerotic intimal thickening, and related constitutive modelling, Amer. J. Physiol. Heart Circ. Physiol., 289(5), H2048 H2058 (2005). [16] Stradins, P., Lacis, I., Oolanta R., Burina, B., Ose, V., Feldmane, L. and Kasyanov, V., Comparison of biomechanical and structural properties between human aortic and pulmonary valve, Eur. J. Cardiothorac. Surg., 26(3), (2004). 11