Modelling and Analysis

The field of structural mechanics and dynamics of aerospace systems normally covers hundreds of theoretical studies. These studies cover linear and nonlinear models and various shapes and boundary configurations of the analyzed concepts, and also numerous analysis methods including closed form, asymptotic expansions, and numerical methods (finite element method, boundary element method, finite difference method, etc). However, given the difficulty of modelling and analysing complex gossamer systems such as solar sails, the current general trend is to use computer- generated numerical models to predict the behaviour of these ultralight concepts. Classical closed form solutions are then normally left as a rapid tool to give an order of magnitude assessment of the validity of the results for simplified versions of the problem. Of all numerical computational methods normally volume-discretisation methods are used, which subdivide the whole domain of

the boundary value problem into simpler parts. Of these methods, the finite element method (FEM) is the most widely accepted technique given the powerful commercial structural analysis software available that utilises it. Finite element analysis (FEA), as it is normally known, can accurately represent complex geometry, include dissimilar material properties, easily represent the total solution, and capture local effects. All these are inherent problems that need to be solved when analysing gossamer structures. Three commercial FEA software dominate the field of lightweight space structures modelling. These are: ABAQUS, which is mainly utilised in research centres and universities; MSC, NX or NEi NASTRAN, mainly used in industry; and ANSYS, used both in industry, and research centres/academia.

In this review we will differentiate between the modelling and analysis of solar sail components, mainly thin-film membrane structures and deployable booms, and that of the whole gossamer sail structure. Normally the latter follows the former, once the modelling approach of the simpler problem has been validated.

Thin-Film Membranes

According to [Jenkins, 2001], to the applied mechanician, membranes may mean a surface (thin film) with zero bending rigidity, resulting in nonexistent compressive solutions. The lack of bend- ing rigidity of membranes, caused by extreme thinness and /or low elastic modulus, leads to an essentially under-constrained structure that has equilibrium configurations only for certain loading fields. Under other loading conditions large rigid-body deformations can take place. In addition, these same characteristics lead to an inability to sustain compressive stresses. Time-dependent and nonlinear behaviour are also common features of typical membrane materials.

As stated in [Jenkins et al., 2004], for the present case, the experimental measurements must be considered in the context of high-fidelity computational models of the sail structure. The sails themselves are membrane structures, and as such are delicately constrained, and cannot withstand compressive or transverse loads without some initial prestress. Under sufficient compressive loads to overcome any initial tensile prestress, the membrane wrinkles out of plane. Wrinkling may have thrust degradation effects on solar sail [Murphey et al., 2002], may cause hot spots where membrane overheating can occur, and will likely cause significant departure from unwrinkled dynamic response behavior. Furthermore, slack directions and areas in the sail represent load-carrying indeterminacy that needs to be minimized to increase confidence in structural integrity under mission conditions. Because membrane structures change shape in partial response to applied loads, the total response of the sail is complicated and not intuitive.

Since the beginning of the 21st century there has been extensive research to effectively model using FE codes thin-film membranes and their behaviour for different geometries, boundary con- ditions, and load cases. For the case of solar sail membranes, normally research efforts have been routed towards solving three fundamental uncertainties: the wrinkled state of the membrane and its effects on the propulsive force; the sail shape under gravity conditions, as this can be considered as a several orders of magnitude scaled-up deformation when compared to that of sail in orbit under uniform solar radiation pressure; and the deployed vibration characteristics of the sail.

• Wrinkling

As stated in [Tessler et al., 2005] structural wrinkles are local post-buckling patterns that are manifested by geometrically large transverse deformations whose magnitudes are much larger than the membrane thickness. Membrane wrinkling prediction started in the 1920s and 1930s in [Wagner, 1929, Reissner, 1938]. They established a tension field theory, which as further developed in [Stein and Hedgepeth, 1961] in the 1960s to handle more general problems. The first finite element solution was the Iterative Membrane Properties (IMP) method developed in [Miller and Hedgepeth, 1982, Miller et al., 1985]. This method uses tension field theory with membrane elements of negligible compressive strength and bending stiffness, that incorporate wrinkling through a recursive stiffness- modification procedure that consists of changing the stiffness matrix of the element that is deemed to be wrinkled using a variable Poisson’s ratio in the element formulation. In [Adler et al., 2000] this method was implemented as a user-defined material (UMAT) sub-routine in ABAQUS using the so-called combined wrinkling criterion. This is a combined stress/strain condition that has to be satisfied for a wrinkle to exist. In [Johnston, 2002] the latter method was used to study the static and structural dynamics behaviour of NGST space telescope (now JWST) sunshield, that consists of several parallel membranes layers. Also, in [Blandino et al., 2002] the IMP method implemented with ABAQUS’ M3D4 AND M3D3 membrane elements with a UMAT material formulation was successfully used to model wrinkling of a square membrane for symmetric mechanical loads applied at the corners. These methods are valid for accurately predicting the stress distribution of the membrane, including wrinkled regions, and also the extent of these regions. However, they cannot give accurate details of the wrinkling patterns, i.e amplitude and wavelength, with out-of-plane deformation of membrane structures.

With this in mind, researchers studied membranes as very thin shell structures that now have non-negligible bending stiffness (out-of-plane) as well as membrane stiffness (in-plane). In order to overcome convergence problems in the analysis using geometrically nonlinear shell models, given the large elastic deformations and rigid-body motions encountered with only small amounts of strain energy involved during the onset of buckling (wrinkle formation), several approaches have been proposed. In [Lee and Lee, 2002] a modified quadratic shell element was created that fictitiously modified the Shear and Young’s Moduli to enable locking-free shell analysis of very thin shells. Also, an artificial damping term was introduced to circumvent numerical ill-conditioning due to stability issues in the nonlinear equilibrium equations formulated.

However, commercial FE codes, mainly ABAQUS, have been used in shell-based analysis to simulate the onset of wrinkling in tensioned membranes and the growth and characteristics of wrinkles. S4R5 four-node Mindlin-type quadrilateral shell elements are normally the primary choice found in literature given their small strain, large displacement, reduced integration of the transverse shear energy, and updated Lagrangian frame of reference features [ABAQUS, 2013]. Three main approaches have been proposed that differ in the way the initiation of wrinkling by out-of-plane deformation is triggered during the geometrically nonlinear analysis. In [Wong and Pellegrino, 2006a, Wong and Pellegrino, 2006c] out-of-plane geometric imperfections imposed at the nodes are introduced as a superposition of the first several eigenvectors of the tangent stiffness matrix of the membrane, scaled by a small percentage of the thickness. In [Leifer and Belvin, 2003]

small magnitude forces of opposing directions and zero resultant were utilised. In [Tessler et al., 2005, Papa and S., 2005, Tessler and Sleight, 2007] randomly distributed out-of-plane deflections of similar magnitudes of those imposed in [Yong and Pellegrino, 2002] were used. Wrinkling of orthotropic viscoelastic membranes has also been studied for applications where creep compliance and relaxation modulus are important parameters [Deng and Pellegrino, 2012]. In general, finite element analysis using thin shell elements has been shown to be able to replicate real physics experimentation with an accuracy better than 25%. Nevertheless, analytical models such as the differential equation proposed in [Epstein, 2003] for the number and amplitude of wrinkles, and in [Wong and Pellegrino, 2006b] for the location and pattern of wrinkles with preliminary estimates of their wavelength and amplitude, have also been proposed for simple geometries and loading conditions.

The corner areas of a solar sail membrane are regions of severe stress concentrations, which are difficult and computationally expensive to analyse with volume-discretization methods such as FEA. In [Tessler et al., 2005] is was successfully proposed that the corner regions should be trun- cated and the corner loads replaced by statically equivalent distributed loads along the truncated lines. These removes the stress concentrations, improving the corner mesh quality, and, hence, element performance. In [Tessler and Sleight, 2007] numerical studies demonstrated that excessive mesh refinement in regions of stress concentration may actually be disadvantageous to achieving wrinkled equilibrium states, causing the nonlinear solution to be biased towards the membrane response and totally discarding the very low-energy bending response that is necessary to cause wrinkling-deformation patterns. Also, it was demonstrated that relatively small changes in the size of the truncated corner region produced distinctly different wrinkling deformations, including their patterns, wavelength, and depth. This aspects brings the importance of precise modeling of such regions into focus.

A practical method to minimize wrinkling on solar sails has been to produce a shear-compliant border along the sail edges that allows significant tension and shear to be introduced in the film without producing wrinkling in the areas outside of this compliant boarder area [Talley et al., 2002]. This method was successfully introduced in the ATK-ABLE ground demonstrator design. How- ever, the modelling of the thermoformed 3D-shaped strips of the shear-compliant regions can be challenging. In [Leifer, 2007], rather than modelling the full geometric details, two simplified and efficient approaches were developed. The model that uses an orthotropic material gave qualitatively good results but is highly sensitive to the ratio of the Young’s moduli calculated. The James Webb Space Telescope (JWST) sunshield will be one of the largest gossamer sail structures ever construc- ted. Therefore, thermally lap-welded seams are used to form the large area from independent rolls of reflective polymer films. In [Fellini and Kropp, 2008] both the thermoformed shear-compliant boarders and the seams are modelled in ABAQUS. Micro-wrinkling patterns near the seam regions are effectively captured in the analysis, but the quantitative FEA results for the wavelength and maximum amplitude of the puckers involve errors of up to 40% of the experimental measured values for highly loaded test cases. Therefore, future work on modelling the thermoformed seam regions of large membranes is needed, especially if the parallel seams are closely spaced.

• Static gravity sag

The problem of determining the gravity-induced deformation on the lightweight sail membrane (gravity sag) using finite element codes was produced in [Taleghani et al., 2005a, Sleight et al., 2006, Johnston et al., 2006, Black et al., 2007, Sakamoto et al., 2007]. Except for the latter publication the rest utilised commercial FE software to carry out the nonlinear analysis of the membrane structure under different preloads, boundary conditions and orientations with respect to the gravity field. The horizontal orientation, in which the gravity vector is directed approximately normal to the sail surface, has been normally chosen to reflect the loading geometry of actual solar sails in orbit. Although the solar radiation force is six orders of magnitude smaller than the force of gravity, the induced deformations can be readily scaled down for the case of lightly preloaded orbit structures once the modelling approach has been validated. It is worth mentioning that in [Black et al., 2007] it is stated that largest source of discrepancy between predicted and measured data is membrane slack found during experimentation, and thus, more attention should be paid in incorporating this geometric slack in the “simple” models, rather than in developing very complex numerical models of the membrane. On these studies it was found that FEA qualitatively predicts large membranes gravity sag. Quantitavely, predictions for peak gravity-induced deformations were generally within 10% of the measured values.

• Vibration behaviour

The study of vibrating membranes goes back at least three centuries [Rayleigh, 1864]. Many theoretical studies of membrane vibrations exist in the literature. With the advent of computers, numerical models have also been extensively utilised to study the dynamic characteristics of simple and complex membrane structures. However, many less experimental studies can be found in the open literature [Jenkins et al., 2004], and even the simplest classical cases have not been thoroughly investigated. The extreme flexibility and lightness of membranes and their structural compliance drive the requisite for accurate non-contacting measurement methods, which has restricted the spread of experimental work. In addition, according to [Jenkins and Korde, 2012] the spread of information spatially across the membrane depends on the membrane tension and local curvature, the frequency content of the disturbance, and other factors such as damping, which makes the problem even more challenging. Mode localisation is also common in membranes, in [Jenkins and Korde, 2012] it was shown that this tendency appears to be due to their inherent stiffness-inertia related dynamics. For the present case, given that solar sail membranes normally have very low frequencies (<1 Hz) of the first few fundamental modes, the vibration measurements are even more difficult. Also, particularly challenging is their extremely large sizes, stiffness variations between the booms and sails, as well as the coupling between the sail and the boom dynamics. Model/test correlation efforts for solar sails ground demonstrators will be presented later in the section. The effects of prestress [Kukathasan and S., 2002], wrinkling [Kukathasan and S., 2003], seams [Jenkins and Kondareddy, 1999], manufacturing variability, air loading in non-vacuum condition tests, or thermal loading on the vibration response of sail membranes are still opened questions in the gossamer structures field.

Booms

As previously reviewed there are several types of booms that have been proposed for gossamer sails. These include: flexible shell structures, coilable booms, and inflatables. The modelling of these structures for solar sailing applications have been solely produced with commercial FE software given the complexity of geometries involved and the analysis carried out.

In FE codes truss or rod elements are long slender structural members that can only transmit axial force. They are usually employed to model line-like structures that support loading only along the axis or the centerline of the element. No moments or forces perpendicular to the centerline are supported. The only parameter required to specify the element is the cross-section’s area. Beam elements are used to model structures in which one dimension (the length) is significantly greater than the other two, and in which the longitudinal stress is most important. The advantage of using beam elements comes from the simplification that is achieved by assuming that the member’s deformation can be estimated entirely from variables that are functions of position along the beam axis only. For beam theory to produce acceptable results, the cross-section dimensions should be less than 1/10 of the structure’s typical axial dimension (distance between supports or gross change in cross-section, wavelength of the highest vibration mode of interest, etc) [ABAQUS, 2013]. Beam elements have deformations that include axial stretch, curvature change (bending), and torsion; adding flexibility associated with transverse shear deformation, and, in some cases, even warping (non-uniform out-of-plane deformation of the cross-section). Beam elements are specified by providing either the shape and dimensions of the cross-section or its area and moments of inertia. On the contrary, shell elements are used to model structures in which one dimensions (the thickness) is significantly smaller than the other dimensions, and the stress in the thickness direction are negligible. Shell elements are specified by providing the element’s section properties that define the thickness and material properties associated (laminate lay-up in case of a composite structure).

To evaluate the static and dynamic response of the booms in a deployed state normally: beam elements have been utilised to model inflatable booms [Sleight et al., 2005, Sleight et al., 2006]; shell elements have been used for modelling flexible thin-shell booms [Sickinger et al., 2006, Roybal et al., 2007]; and coilable masts have been generally represented as a 3D truss structure, where longerons have been modelled using beam elements, and diagonals and battens have been modelled using truss or rod elements that support only axial loading [Taleghani et al., 2005a]. However, to model the triangular truss structure attached to one side of the inflatable boom in the L’Garde solar sail GSD, beam elements were utilised to model the rigid V-shaped spreader bars (battens), and cable or truss elements without compression behaviour to model the flexible spreader system lines (third batten, longerons and diagonals) [Sleight et al., 2005].

In [Stanciulescu et al., 2007] static and dynamic analysis of a truss-like isogrid boom for solar sails were performed using FE codes. It was shown that for slender structures, using an analysis with a simplified equivalent beam of constant cross-section is very accurate and much more efficient, as long as the cross-sectional properties of the beam is evaluated to be averages of the properties of the isogrid system. Several researchers have modelled the booms with simplified equivalent beam formulations for computational efficiency, when performing full system solar sail structural

analysis [Taleghani et al., 2003, Sleight, 2004, Banik et al., 2008].

There is a particular type of flexible thin-shell structure that has attracted a great amount of modelling research given the widespread of applications it provides: tape-springs. Over the last decade there has been renewed interested in studying the folding behaviour of tape-springs, which is analysed analytically, numerically, and experimentally in [Seffen and Pellegrino, 1999, Seffen, 2000, Yee et al., 2004, Yee and Pellegrino, 2005, Silver et al., 2005, Walker and Aglietti, 2006, Walker and Aglietti, 2007, Soykasap, 2007, Hoffait et al., 2010, Mallikarachchi and S., 2011, Guinot et al., 2012, Bourgeois et al., 2012] for hinge applications that involve folding/deployment of i.e. rolling hinges [Watt and Pellegrino, 2002], hexapods [Aridon et al., 2008], tubular booms [Mobrem and Adams, 2006, Mallikarachchi, 2011], trusses [Pollard and Murphey, 2006], antennas [Soykasap et al., 2008] or reflectors [Soykasap et al., 2012, Zajac et al., 2013]. This research has created a vast amount of knowledge on the behaviour of tape-springs during the folding, unfolding and locking phenomenae.

According to [Guinot et al., 2012], the modelling of tape springs can be classified into two main approaches: nonlinear shells (2D); and discreted articulated bars (1D). However, this classification can be also extended to include the modelling of other flexible shell structures designed for space applications. The former approach consists in a full computation of the shell model in the framework of large displacements and rotations. This was first tackled in the 1960s and 1970s by Chu [Chu and Krishnamoorthy, 1967], who led the work of studying the buckling and postbuckling behavior of open cylindrical shell structures [Chu and Turula, 1970, Turula and Chu, 1970]. They used Donnell’s theory [Donnell, 1934] applied to open section cylinders. Based on these differential equations, a set of nonlinear finite difference equations were obtained and solved numerically by the Newton-Raphson method [Scarborough, 1966]. In [Yang and Guralnick, 1976] experimental validation of this modelling approach was shown. Recently [Silver et al., 2005] used the same model but implemented in Matlab with a much finer mesh and using Riks’ arc-length method [Riks, 1972] to address the snap-through behavior of tape-springs. In order to generalise the investigation, many of the geometric parameters were reduced to non-dimensional parameters, so that future parametric studies could be performed. Nevertheless, the normal trend has been to use commercial FE software to accurately solve the static and dynamic problems for any geometry, boundary condition or loading configuration proposed [Sickinger et al., 2004, Roybal et al., 2007, Banik