Instability-driven shape forming of fiber reinforced polymer frames

Thin ﬁ ber reinforced polymer (FRP) composites are widely implemented in adaptive and morphing structures. However, realization of the necessary complex 3 ‐ dimensional FRP structures requires the use of expensive molds thereby limiting the design space and ﬂ exibility. Using the elastic strain energy of pre ‐ stretched membranes holds potential for addressing this challenge. In this work, a novel manufacturing technique for fabricating 3 ‐ dimensional FRP structures moldlessly is presented where pre ‐ stretched membranes are used to drive out ‐ of ‐ plane buckling instabilities of FRP composite shells. To explore the potential of this approach, a simple square frame design is investigated. An analytical model based on high deformation beam buckling theory is developed for understanding the parameters driving the out ‐ of ‐ plane behavior of these structures. Experimental and ﬁ nite element results are used for model validation and reveal excellent agreement, with errors less than 10% over a large portion of the design space. Analytical and ﬁ nite element models demonstrate that the out ‐ of ‐ plane deformation can be tailored by varying the structure ’ s geometric and material parameters. A new design space for FRP composite laminates is characterized, enabling highly ﬂ exible design. The manufacturing and modeling techniques can be extended to other geometries for the realization and analysis of arbitrarily complex surfaces.


Complex shape-forming of FRP structures
Complex 3D topologies, such as those proposed for mechanical metamaterial concepts, have sparked the advancement of a broad range of engineering applications in the field of adaptive, morphing, and deployable structures.The implementation of thin fiber reinforced polymer (FRP) composite laminates, already widely used in deployble and adaptive structures due to their excellent specific stiffness, strength, and extremely low achievable curvatures [1][2][3][4][5], into complex topologies is limited by manufacturing complexity.The realization of 3-dimensional structures with FRP laminates requires molds, thereby increasing cost and reducing design flexibility.As an alternative, research has focused on manufacturing routes based on self-shaping mechanisms for the realization of curved structures from flat FRP laminates.
A well-studied self-shaping concept relies on the differential thermal expansion and anisotropy of the FRP constituents to create thermal residual stresses.This approach is utilized, for example, for the fabrication of anti-symmetric thin FRP laminates, where internal bending-moment reactions are generated by tailoring fiber orientation, layup stacking sequence, and material properties to generate 3D shells [6,7].The dependency of the curved shape on geometrical and material parameters has been well analyzed and modeled in literature [8][9][10][11].This method has been utilized for the realization of complex shapes such as corrugated thin-walled laminates [12] and applied to adaptive structures [13,14].However, singly or doubly curved shapes obtained by the thermal effect suffer from low curvaturesdrastically restricting the range of application of the thin FRP laminates generated by this process.
Another class of self-shaping concepts, of particular interest for the scope of this work, relies on elastic instabilities [15].The key element of such concepts is the combination of anisotropic material distribution and actuation mechanisms.For example, reversible wrinkle patterns have been created on silk/polydimethylsiloxane bilayer structures [16] and controlled buckling patterns have been investigated by combining thin metal films with elastomeric polymers [17].In this case, surface instability is induced by thermal contraction of one of the constituents.Other studies realized self-folding patterns from an initially planar configuration by using hydrophilic materials [18] or a light-activated polystyrene sheet [19].
The need for an external stimulus for shape-forming can be eliminated by using the elastic strain energy of pre-stretched membranes.Researchers have shown how to take advantage of surface instabilities in order to generate wrinkles [20][21][22] and fabricate structured polymer surfaces with functional micro-or nano-patterns [23].For example, Al-Rashed et al. [24] presented different buckling patterns created on a bi-axially pre-stretched bilayer thin shell.Similarly, the concept has been adopted in order to augment the functionality of electronic devices; highly stretchable sensors and foldable silicon integrated circuits have been manufactured by using a thin pre-strained elastomeric substrate bonded with thin silicon ribbon elements [25][26][27].Ruslan et al. [28] utilized pre-stretched elastic sheets combined with an anisotropic distribution of disconnected rigid tiles and presented an approximate model that allows the realization of targeted doubly curved surfaces fabricated as a flat piece.Furthermore, architects combined 3D printed polylactide (PLA) with a pre-stretched textile for the realization of self-shaping shoes [29] or building covers [30].
It has been shown that utilizing elastic strain energy enables the fabrication of complex 3D topologies.Therefore, applying this approach to shaping of FRP laminates holds great potential.Combining thin FRP shells with pre-stretched membranes can expand the design space of composite laminates, enabling the fabrication of highly curved and complex structures without the need for a mold.
In this paper, an investigation on instability-driven shape forming of FRP frames is presented for the first time, with focus on a structure consisting of a frame-like thin composite shell bonded to a bi-axially pre-stretched membrane resulting in out-of-plane deformation upon the release of the pre-strain.Previous work modeled the deformation of buckled films on compliant substrates [20,24]; however, no research has been conducted for predicting the deflection of slender and rigid elements on pre-stretched membranes.This work characterizes the attainable out-of-plane displacement and predicts the buckling response of the composite frame.Moreover, the limits of this geometrical design space are explored through a combined analytic, finite element, and experimental approach that describes the shape-forming phenomenon.This study develops a framework for the analysis of arbitrary combinations of slender composite elements shaped by a prestretched membrane.In fact, the presented fabrication and modeling technique can be extended to other geometries, enabling the manufacturing and characterization of arbitrary 3D surfaces.

Description of the structure
The potential of pre-stretched membranes to shape FRP laminates is investigated through a simple geometryan FRP frame structure with a high degree of symmetry.The FRP frame structure under study consists of four strips made from a symmetric layup of thin carbon fibre prepreg bonded to a bi-axially pre-stretched thermoplastic polyurethane (TPU) foil.All strips have a rectangular shape with length, L, width, w, and a common layup.They are connected with each other in a frame-like structure with overlapping resembling a plain-weave pattern, selected for its rotational symmetry, as shown in Fig. 1a.The pre-stress in the membrane is created by applying the corresponding mechanical strains prior to bonding.The bonding is performed in the flat configuration (x-y plane) and the shape-forming mechanism arises when the pre-stress is released.Fig. 1b is a schematic representation of the deformed state of the FRP frame.The out-of-plane buckling of the four laminates leads to the final equilibrium shape of the frame.Each laminate shows the same deformation, and the structure is anti-symmetric along the two lines connecting the mid-points of opposite laminates (red lines in Fig. 1b).Moreover, we observe that along these lines the out-of-plane deformation (in the z-direction) is constant.

Materials
For the purposes of this investigation, ultra-thin prepreg with an areal weight of 40 g=m 2 composed of the ThinPreg 513 epoxy resin and M40J carbon fibre from NTPT has been selected to manufacture the strips.The proposed fabrication technique can be applied to arbitrary layups.The choice of the materials and layup of the laminates is instead designed to fulfill three requirements.First, the bending stiffness of the laminate must be much higher than that of the membrane, so as to guarantee that the deformation along the lateral direction of each strip is negligible compared to that along the out-of-plane direction.Second, and equally important, the layup needs to be selected to guarantee an initially flat configuration and avoid undesired bendtwist coupling in the final state.Third, the material needs to be chosen such that plastic deformation is avoided considering the large deformation that the strips need to withstand.
A symmetric ½0=90=0 layup is used to avoid bend-twist coupling and reduce the likelihood of matrix failure due to transverse loading.Table 1 provides the material data.The membrane consists of Convestro's Platilon U2102A Highly Elastic Polyurethane film with a thickness of t m ¼ 25 μm.The two components are bonded together with 130 μm thick 3M High-Strength Acrylic Adhesive 300LSE double sided adhesive tape.
The elastic parameters of the TPU foil have been determined using uni-axial tensile testing.The strain range of interest for the purpose of this work was up to around 10% as prototypes with higher pre-strains showed membrane wrinkling in their buckled state.Details about the wrinkling phenomenon will be given in Section 4.4.Measurements were performed in two material directions: along the machine direction, X, and transverse to it, Y.The foil is manufactured by pulling the material in one direction, therefore, testing in two orthogonal directions was required to characterize any anisotropy resulting from the process.Following the uni-axial test setup utilized in [33], the specimens have a total width of 10 mm and length of 40 mm.Tests were performed at ambient conditions and at a strain rate of 3 mm/min utilizing the in-house testing machine from the Institute of Mechanical Systems at ETH Zuerich.The machine has a load cell with a 100 N capacity and custom-made clamps with a surface area of 10 × 10 mm 2 , equipped with sandpaper to prevent the slipping of the test specimens [34].The specimens were ink patterned and Digital Image Correlation was utilized for in-plane strain analysis.The post processing analysis was performed by utilizing the code developed in [33].
Six samples were tested in the X direction and six in the Y direction.The stress-strain relationship in the two directions resulted in a 5% anisotropy and was therefore deemed insignificant.The results presented here refer to the X direction.
Fig. 2a represents the evolution of strains, ɛ y , perpendicular to the test direction as a function of the applied strain, ɛ x .The experimental value of the Poisson ratio was determined to be ν ¼ 0:48, obtained through linear fitting of the test data for all six samples.As this value is sufficiently close to 0.5, the material has been considered to be incompressible.
Fig. 2b shows the corresponding Cauchy stress, σ x , versus the applied strain, ɛ x , for the same specimens.A Neo-Hookean material law for incompressible hyperelastic material behavior has been selected.The strain-energy function for this material model is given by, where μ is the shear modulus and I 1 the first invariant of the deformation tensor.The fitting curve derived from Eq. 1, taken as the average over all six samples, is shown in Fig. 2b and the derived material constants are summarized in Table 2.
An incompressible Neo-Hookean material law is typical for modeling such elastic foils.In fact, this model has been recommended for a strain range below 15% for polyurethane elastomers in [35].Furthermore, Gent suggests in [36] that this model is typically accurate for strains below 20%.The almost linear fitting curve can be interpreted as the mean curve between the tangent at zero strain and the secant between the two extremities of the tested strain range.This model gives a sufficiently good approximation of the material behavior for the interest of this work as will be proved with the experimental validation in Section 4.1.However, it is not recommended to utilize the here provided data for a strain range higher than the one tested.

Fabrication technique
The FRP frames were manufactured utilizing the materials presented in the previous section.The FRP strips were cured on a flat plate following the overlapping scheme illustrated in Fig. 1a, with the cure cycle recommended by the manufacturer (2 h at 120 C,

Modeling of shape forming instabilities
In this section, two different modeling approaches for characterizing the equilibrium deformation of FRP frames are presented.The lam-Table 1 Material properties for NTPT Thinpreg 513/M40J carbon fiber reinforced prepreg used to manufacture the experimental specimens [31,32].inates undergo a deflection similar to that of buckled beams under compression.Therefore, we developed an analytical model that uses the high deformation elastica theory from Euler [37] for understanding the mechanism driving the shape-forming.This model predicts the deformation of the frame by minimizing the total energy of the structure.After introducing the analytical model in Section 3.1, a Finite Element (FE) model is presented in Section 3.2 for more accurate predictions.An analytical model is desirable because it can reduce the computational complexity especially if used to model structures with larger assemblies of beams.

Analytical model
The total energy of the system can be expressed as the sum of the energies of the two constituents: the frame and the membrane.The structure possesses anti-symmetry along the two lines connecting mid-points of opposite laminates (Fig. 4), and consequently only one quarter of the structure is modeled.We consider as a reference an L corner, as depicted in Fig. 5, consisting of the halves of two FRP strips and a quarter of the membrane.The deformations of the complete structure are computed through the use of symmetry.Noting that the dominant deformation mode of the laminate is by bending about its axis of lower second-moment of inertia, we model each half of the laminated strip as a beam with equivalent bending stiffness EI ¼ w d11 , where E is the elastic modulus, I is the second moment of area of the beam's cross-section and, d 11 is a component of the compliance matrix for the FRP strips expressed as [38]: where ɛ 0 and κ are the mid-plane strains and curvatures and N and M are the force and moment resultants per unit composite width, respectively.The beams are clamped at their intersection and the curvature is assumed to be zero at the other 'free' end (i.e. at the middle of the FRP strips).Two identical compressive forces F are applied at the end of both beams to model the forces applied by the pre-stretched membrane (Fig. 5a).Both beams undergo the same deformation along their length.Therefore, only the deflection of one beam is computed.The beams' mid-planes are assumed to be inextensible and pure bending deformation is considered.Any transverse deformation of the beams, for example twisting, is neglected.Given a compressive force, F, higher than the critical buckling load, F cr , the deflection can be the computed using the elastica theory from Euler [37].For this buckling case [37]: A summary of high deflection buckling theory from Timoshenko is given in Eqs. ( 3)-( 9) [37].The variable conventions used are illustrated in Fig. 5a.The variable s is defined from the free end of the beam along the length of the beam itself and consequently s ∈ ½0; L=2.The angle of the beam at a certain value of s is θðsÞ.The curvature of the beam can be expressed as dθ ds , therefore, the exact differential equation for the deflection can be written as: At the free end of the beam θ ¼ α and it can be observed that the bending moment is zero, therefore, dθ ds ¼ 0. Introducing the constant k 2 ¼ F EI , integrating Eq. 3 and imposing the boundary conditions at the upper end of the beam, the following equation is obtained: To simplify the obtained expressions, the notation p ¼ sin α 2 À Á is used, and ϕ is introduced such that, with ϕ varying from 0 to π=2 as θ varies from 0 to α.Finally, recalling that the change in length of the beam is neglected and that at the lower end of the beam θ ¼ 0, the shape of the elastic curve can be derived.A given point along the beam with ϕ ¼ ϕ has coordinates ðxð ϕÞ; zð ϕÞÞ: and The total deflection, z a , of the free end of the beam is given by substituting ϕ ¼ π=2 in Eq. 7. It is then possible to express z a as the following: Similarly, the in-plane projection of the beam length, x a , can be written as: where EðpÞ indicates the complete elliptic integral of second kind.The projection ðx a ; z a Þ is illustrated in Fig. 5a.Due to symmetry, the deformation of the beam along the y-axis is equivalent and given by ðy a ¼ x a ; z a Þ.
In order to relate the compressive force, F, with the force exerted by the membrane on the FRP strips, the buckling solution given by the elastica theory is used as an input variable for computing the membrane energy.In particular, for a given F, the membrane is modeled as a 3D surface, f ðF; x; yÞ, that matches the shape of the deformed beams in the L corner.In Fig. 5b, we show an example of the fitting function.This function can be expressed as the following: and is defined on the interval ðx; yÞ ∈ ð½x 0 ; x a ; ½y 0 ; y a Þ.The amplitude, a 0 , is calculated so that the deflection of the membrane at the Lcorner coincides with the deflection of the beams at s ¼ L=2 À w: The coordinates ðx 0 ; z 0 Þ are derived from Eqs. 6 and 7 with θ 0 ¼ θðs ¼ L=2 À wÞ.The deformation of the beams is identical, therefore it holds that x 0 ¼ y 0 .
This 3D surface has been designed so that the two following properties are satisfied.First, the surface approximates the deformation of the beams along two sides.Second, the surface has constant height, z a , along the other two sides.The membrane in the manufactured prototypes indeed satisfies both of these properties (blue lines in Fig. 4).For a force, F, lower than F cr , no deflection occurs in the beams (x a ¼ L=2; z 0 ¼ z a ¼ 0) and, consequently, the membrane has constant height equal to zero.
Having approximated the deformed membrane, we neglect shear strains in the membrane and assume that the in-plane strains, ɛ x and ɛ y , are uniform along x and y, respectively, and can be expressed in the form: Here, for a given F, L x ðF; y À Þ is the arc length of the membrane for a fixed y ¼ y and L y ðF; xÞ is the arc length for a fixed x ¼ x.The side length of the membrane prior to stretching is denoted L 0 .Therefore, it holds that: where ɛ is the bi-axial prestrain applied to the membrane.
The presented model for the frame and membrane allows to characterize the deformation of both constituents of the structure.Therefore, the strain energy of the whole model can be computed as the sum of the energy of the sub-components.Both the energy of the laminates and the membrane can be written as a function of F. In particular, we can write the bending strain energy of the entire frame as [37]: where the coefficient 8 is a direct consequence of the previously mentioned assumption of symmetry and dθ ds is given by Eq. 4. Recalling Eq. 1, the strain energy for a Neo-Hookean material for the entire membrane is [39]: where I 1 is the first invariant of the deformation tensor and it is equal to: and λ i are the principal stretches of the membrane and can be expressed as: From the incompressibility assumption, λ 3 is derived by solving: The total energy of the frame, E t , is given by the sum of E b and E m : The solution is then computed with the software Mathematica 12.1 [40] by numerically minimizing the E t as a function of F. With an iterative procedure, the total energy is calculated starting from F ¼ F cr and increasing the value of F until the minimum of E t is reached.

FE model
Although the shape-forming phenomena can be physically understood with the help of an analytical model, the nature of the deformation and the interaction of the laminate strips in the corners as well as their twisting is not accounted for.Thus, a finite element analysis (FEA) is performed in order to fully characterize the deformation.FEA is implemented in the commercial software ABAQUS© 6.14-1 standard [41].The structure is modeled by employing reduced integration, S4R shell elements for the FRP laminates and M3D4R membrane elements for the TPU foil.After performing a mesh dependency study, both constituents are modeled with 1 mm elements.The FRP strips' material behavior is computed using the built-in composite layup tool in ABAQUS and each ply is modeled with lamina properties taken from the manufacturer [31,32] and listed in Table 1.The membrane is modeled using a hyperelastic Neo-Hookean material model (Table 2).The adhesive layer is included in the layup of the laminates with constant thickness 130 μ m and assumed Young's modulus of 1 MPa.Provided the modulus of the laminate is much higher than the one of the adhesive, the influence of the adhesive parameters on the solution has been observed to be minimal.
All the components are connected along the common boundaries using TIE constraints.In particular, the FRP strips are bonded to one another in the overlapping corners' surfaces (scheme in Fig. 1a).Each FRP strip is tied to the membrane along the surface corresponding to where the physical bonding occurs.
The model begins in an initially flat configuration.The prestreching of the membrane is defined as a predefined stress field in the initial state of the simulation.For an incompressible Neo-Hookean material under biaxial extension, it holds that the in-plane Cauchystresses σ x ; σ y are: where λ ¼ 1 þ ɛ.A geometrically nonlinear static analysis is adopted to obtain the buckled shape.Fig. 6 shows the FRP frame with the imposed boundary conditions (BCs).Both BCs are imposed to a set of 4 nodes of the elements at the corner of the frame.Fig. 6 is an example of the simulated equilibrium deformed state of an FRP frame with L ¼ 120 mm; w ¼ 20 mm and ɛ ¼ 0:05.In order to perform a comparison between the different models, for the post processing of the FE results we define z max as the maximum out-of-plane displacement along the midline of a strip (dashed line in Fig. 6).For the analytical model, z max ¼ 2z a due to symmetry.Observing that the strips are slightly twisted in the FE results and that the analytical model assumes pure bending deformation, the definition of z max along the midline ensures a better comparison between the two models.

Design space of FRP frames
The potential of the shape-forming phenomenon of FPR frames is now analyzed.First, we perform an investigation on the total energy of the structure to compare the analytical model to FEA (Section 4.1).Second, we compare the two models to experimental results (Section 4.2).Third, the design space of achievable deformations is inves-tigated (Section 4.3) and, to conclude, the range of validity of the analytical model is derived (Section 4.4).

Energy verification
Before presenting the effects of the model parameters on the shapeforming of FRP structures, a verification of the computed strain energy is conducted.Fig. 7 illustrates the dependency of the strain energy of the membrane, E m , the bending strain energy of the laminates, E b , and the total energy of the structure, E t , on the out-of-plane deformation of the strips, z max , as the two models converge to a minimum energy configuration for a particular geometry with L ¼ 120 mm; w ¼ 20 mm, and ɛ ¼ 0:05.The energy profiles are shown for both the analytical and the FE model for a structure with same geometry, material, and prestretching of the one illustrated in Fig. 6.For z max ¼ 0:0 mm (initially flat configuration), the total energy is equal to the energy stored in the pre-stretched membrane (ɛ x ¼ ɛ y ¼ ɛ) and the FRP strips are strain-free.Once the pre-tensioning is released, buckling occurs and consequently E b increases.The increase of the deflection of the frame corresponds to a decrease of the strains of the membrane.The solution for both models is found at the minimum of the total energy that is highlighted with a red dot in the figure.Fig. 7 also illustrates the non-linear relationship between E b and z max as expected from the Elastica theory.
It can be observed that the discrepancy between the energies of the two models is increasing with larger z max .The difference between the strain energies of the FRP strips, E b , is due to twisting of the strips that is not considered in the analytical model.The error for the beams' energy is, however, much smaller than for the membrane.The discrepancy between E m predicted analytically and via FE is a consequence of the assumption of uniform membrane strains in the analytical model.In fact, from the FE results it has been observed that non-uniform strains of the membrane are present for high out-of-plane displacements.Fig. 8a and b show the elastic strains (ɛ x and ɛ y ) of the membrane in the two main material directions for the same structural parameters as in Fig. 6.As expected, the strains are equal to the prestretching ɛ in the proximity of the frame and decrease towards the center of the structure.The in-plane shear strains, ɛ xy , are plotted in Fig. 8c.The shear strains in the membrane are non-zero in the FE simulations but neglected in the beam-based model, since ɛ xy are one or two orders of magnitude lower than ɛ x and ɛ y in a large portion of the membrane.Even though it is clear that these assumptions of the analytical model have an influence on the energy curves, the results are overall in good agreement.We conclude that the beam-based  model is capable of capturing the mechanics behind the shape-forming phenomenon.In fact, for the illustrated case, the error between the two predicted z max at equilibrium is less then 3%.

Comparison with physical prototypes
Frame prototypes were manufactured with the materials and methods introduced in Section 2. It was observed that the manufactured prototypes are symmetrically bi-stable.Specifically, it is possible to snap upwards and downwards opposite corners of the frame to switch between the two configurations.The two configurations have identical buckled shape and, consequently, identical z max .In the FE simulation is possible to predict the other stable state by applying the boundary conditions to the other two opposing corners of the FRP frame (Fig. 6).The total energies of both equilibrium configurations are identical.This phenomenon is not investigated further in this work.
In Fig. 9, the curves that represent the dependency of the out-ofplane displacement, z max , on membrane pre-strain, ɛ, are shown.The figure confirms a good agreement for both analytical (an) and finite elements (FE) models.The curves are plotted for two different values of w.Frames with wider FRP strips show lower out-of-plane displacement for the same pre-strain ɛ.Increasing the width of the strips by keeping constant the total length L, implies that the overall free area of the membrane diminishes, resulting in a lower global out-of-plane deformation.Both models predict that at very low pre-strains the frame does not buckle and that, as expected, higher pre-stretching leads to higher out-of-plane displacement of the frame.At low prestrains the behavior of the curves is highly non-linear, but this nonlinearity becomes less pronounced for higher ɛ.Also noteworthy is the fact that the out-of-plane deformation is more sensitive to the variable ɛ for low strains, which holds important implications for the required fabrication accuracy to achieve a desired out-of-plane displacement.
Experimental data points (exp) are also shown in Fig. 9.A total of 10 prototypes have been fabricated.As for the FE results, the z max of the experiments is taken as the maximum out-of-plane displacement along the strip midline (dashed line in Fig. 6).The uncertainty of the experimental data stems from the imprecision of the prestretching of the membrane, which was performed manually.The analytical and FE predictions generally fall within the range of error of the experimental data.Therefore, the agreement between the experimental data and the values predicted by both models is good.

Parametric study
The presented models can be used to explore the limits of the shape-forming mechanism to understand the maximum achievable out-of-plane deformation and guide the design of such structures.The parameters influencing this mechanism are the bending stiffness, width, and length of the laminates as well as the thickness and elastic modulus of the membrane.Therefore, we define the relevant nondimensional design parameters as follows: 2 L 2 t m ρ and β are, respectively, the aspect ratio and the stiffness ratio.Assuming a given pre-strain, ɛ, in the membrane, we can study the out-of-plane displacement of the structure due to the variation of ρ

9.
Plot of z max as a function of ɛ for both analytical (an) and FE models (FEA).Experimental data points are given by the manufactured prototypes with constant L ¼ 120 mm.Lines and experimental data are given for two different FRP strip widths, w. and β.The effect of the design parameters on the out-of-plane displacement of the laminates, z max , is presented for a constant pre-strain of ɛ ¼ 0:05 in Fig. 10.This plot has been derived utilizing the FE model presented in Section 3.2.The dashed line and the shaded region define when the pre-strain is not sufficient to buckle the FRP strips.The definition of this limit case corresponds to when there is no change of the sign of the curvature along the longitudinal direction of each strip.This is equivalent to no inflection points.Consequently, we observe that all buckled frames (z max > 0) possess one inflection point in the middle of each strip.
Fig. 10 shows a non-linear dependence of the out-of-plane displacement on ρ and β.The isolines follow a parabolic shape with a negative gradient.For high ρ and β no deformation occurs whereas by decreasing either of these two parameters, z max is rapidly increasing.This can be understood intuitively as more slender and softer frames yield higher out-of-plane displacements.As noted previously, we again see that the sensitivity of the z max to ρ and β is higher for low out-of plane displacements.We can also observe that the higher is the stiffness ratio, the lower is the maximum z max achievable by the structure by varying the ρ.However, increasing the ɛ would allow higher out-ofplane displacements for those cases too.
Through this parametric study, a design methodology for FPR frames is presented.Given, for example, a target deformation, the proposed manufacturing technique allows to select the other design parameters in order to fulfill either geometrical (ρ) or stiffness (β) requirements.Considering that the motivation for the proposed instability-driven shape-forming method is to induce high out-ofplane deformations, we demonstrated that in this out-of-plane displacement regime the design shows low sensitivity to ɛ; ρ, and β.This is favorable for the fabrication process as lower precision is required to achieve a desired displacement.
As a last observation, it holds that to the out-of-plane buckling of the structure corresponds an in-plane shrinking.Considering the projection of the laminates on the plane determined by the structure prior to buckling, we define as L p the projected length of one side.For the analytical model it holds that L p ¼ 2x a .We define as the global inplane strain ɛ in as: This variable is equal in the x-and y-directions of the structure because of symmetry.The dependency of ɛ in on ρ and β for a prestretching of ɛ ¼ 0:05 is plotted in Fig. 11.Interestingly, the total inplane strain of an FRP shell for low values of ρ and βcan be higher than the actual pre-stretching applied at the membrane, although in these cases the membrane tends to wrinkle.In-plane shrinking and out-ofplane deformation are proportional as the lines of Fig. 10 and Fig. 11 follow similar trends.

Range of validity of the analytical model
The comparison between the analytical and FE model results presented in Section 4.2 was performed only for a specific combination of β and ρ.For completeness, the comparison is extended throughout the entire design space.We show in Fig. 12 the percent error between the z max predicted by the two models.In a large area, the results show less than 10% error, indicating that the analytical model can successfully capture the mechanics of the structure.However, we observe that for low displacements, z max , the solutions are diverging significantly.Looking at the results of the FE model, we note that the FRP strips are exhibiting significant twisting for these cases.Comparing the strains ɛ x in the membrane (Fig. 12, panel a) with those in a nontwisted structure (Fig. 8a), we observe that they are not symmetric.Therefore, the error between the model stems from the assumption of pure bending in the beam-based model.Moreover, we observe an increase of the error for very low values of ρ and β where the out-ofplane deformations are highest.In those cases, we observe wrinkling in the membrane in the area close to the middle of the FRP strips (Fig. 12, panel b).Analyzing the wrinkling phenomenon of the membrane in this limit case is out of the scope of this work as wrinkling is not desirable.This phenomenon is not taken into account by the analytical model and not fully characterized by the FE model.Other modeling techniques have been suggested for characterizing the wrinkling phenomenon and can be potentially implemented in the model described in Section 3.2 [42].
To conclude, the beam-based analytic model, despite its simplifying assumptions, is excellent for predicting the out-of-plane displacement of the FRP frames over a large area of the design space and can significantly reduce the computational complexity and the time required for predictions.This enables the analytic model to be used in future studies for the prediction of arbitrary 3D surfaces shaped by the novel technique.

Conclusion and outlook
In this paper, a novel class of self-shaping composite structures has been introduced.The structures are manufactured by combining thin FRP strips with pre-stretched membranes.The results show that strain  This work characterizes the properties that can be exploited for shape-forming and their dependency on the structure's geometric and material parameters.The interaction between the two constituents has been modeled through both an analytical model based on the elastica solution of Euler [37] and finite element analysis.The results have been validated with experiments and the novel mold-less manufacturing technique has been illustrated.In particular, a parametrization of the frame has been proposed which allows to describe the shapeforming phenomenon as a function of only three variables: namely the aspect ratio, the stiffness ratio, and the pre-strain.It has been demonstrated that both the out-of-plane deformation and the in-plane shrinking of the structure can be tailored by combining those three parameters.
The proposed beam-based analytical model successfully predicts the deformed shape and models of the shape-forming mechanism.It has been demonstrated that the model predicts out-of-plane displacements within 10% of FEA in a large portion of the design space, with larger errors only when the model assumptions are violated, in particular when the FRP strips twist or the membrane wrinkles.This holds great potential because the model can be expanded, for example to periodic geometries or even non-periodic arrangements of buckled beams, being particularly favorable because of higher computational efficiency compared to FEA.Therefore, the developed modeling technique is a useful tool for predicting the out-of-plane deformation of arbitrary structures.
The results show that instability-driven shape forming of FRP frames enables the realization of 3-dimensional structures fabricated moldlessly and that the technique allows for a highly flexible design.In fact, the design parameters can be selected in order to fulfill stiffness and/or geometrical requirements.In this work the investigation has been carried out on a specific geometrical configuration, however, the developed manufacturing route can be expanded to other geometries or to periodic structures.The concept can be further implemented for manufacturing arbitrary 3-dimensional surfaces made feasible by modern CNC machines that can cut the FRP to the required shape.This feasibility study successfully opens a new design space and, therefore, novel application possibilities such as adaptive, morphing, and deployable structures for thin FRP composite laminates.

Fig. 2 .
Fig.2.Uni-axial test results for six samples of the TPU foil and fitting functions.

Fig. 3 .
Fig. 3. Pre-stretching rig used for bi-axially pre-stretching the TPU foil.The FRP frame is placed in the center of the membrane area.

Fig. 5 .
Fig. 5. Schematic representation of the beam model of a quarter of an FRP frame.The red dashed lines represent the symmetry lines of the structure.(a) Details about the notation used; (b) example of membrane modeled with the function f ðF; x; yÞ.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 6 .
Fig. 6.FE result of an FRP frame in its deformed state with L ¼ 120 mm; w ¼ 20 mm; ɛ ¼ 0:05.U z indicates the out-of-plane displacement.The dashed line represents the strip midline.A schematic representation of the boundary conditions is included.

Fig. 10 .
Fig. 10.Contour plot showing z max lines at the variation of ρ and β for ɛ ¼ 0:05.Results computed via the FE model.

Fig. 11 .
Fig. 11.Contour plot showing ɛ in lines at the variation of ρ and β for ɛ ¼ 0:05.Results computed via the FE model.

Fig. 12 .
Fig. 12. Percent error between the z max predicted by the FE and the analytical model at the variation of ρ and β for ɛ ¼ 0:05.(a) Elastic strain ɛ x calculated by the FE model for a FRP structure with ρ ¼ 0:233 and β ¼ 0:024.(b) Out-ofplane displacements U z calculated by the FE model for a FRP structure with ρ ¼ 0:133 and β ¼ 0:011.

Table 2
Neo-Hookean incompressible material constants of the TPU foil.R 2 is the coefficient of determination.