# A meshless method for the nonlinear von Kármán plate with multiple folds of complex shape

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## Abstract

We present a meshless discretisation method for the solution of the non-linear equations of the von Kármán plate containing folds. The plate has Mindlin–Reissner kinematics where the rotations are independent of the derivatives of the normal deflection, hence discretised with different shape functions. While in cracks displacements are discontinuous, in folds, rotations are discontinuous. To introduce a discontinuity in the rotations, we use an enriched weight function previously derived by the authors for cracks (Barbieri et al. in Int J Numer Methods Eng 90(2):177–195, 2012). With this approach, there is no need to introduce additional degrees of freedom for the folds, nor the mesh needs to follow the folding lines. Instead, the folds can be arbitrarily oriented and have endpoints either on the boundary or internal to the plate. Also, the geometry of the folds can be straight or have kinks. The results show that the method can reproduce the sharp edges of the folding lines, for various folding configurations and compare satisfactorily with analytical formulas for buckling or load-displacement curves from reference solutions.

## Keywords

Meshless Meshfree von Kármán plate Folds## 1 Introduction

Folded structures are omnipresent in various fields of science and engineering. For example, they manifest as chevron and detachment folds in geology [1, 2, 3], *origami* mechanisms in mechanical engineering [4], origami mechanical metamaterials [5], roofs in civil engineering [6], and even in the arts [7] and architecture [8, 9].

A vast literature exists on *origami* and folded structures, and a thorough review would be outside the bounds of the present work, nor is in the scopes of this article. The interested reader can refer to [10, 11].

However, it is worth noticing that most of the simulations of origami or folded structures consider plates as infinitely stiff (or *mechanisms*), and the deformations are essentially rigid motions [12, 13] or *bar-hinge* models [14].

Instead, for deformable plates, one has to solve the governing equations for plates, either for small or large normal deflections *w*. To introduce the folding lines, one needs to enforce a discontinuity on the rotations of the plate. For this purpose, various researchers introduced *internal hinge lines*. Numerous works attempted analytical or semi-analytical solutions for the buckling and vibrations of linear elastic plates with internal hinges. In [15], Xiang and co-workers used the Levy’s method (separation of variables) with an assumed trigonometric solution and a *domain decomposition* method to study the buckling of rectangular plates with through-width internal hinges. The idea is to divide the domain into two parts, provide different solutions for each sub-domain, and then *patch* the solutions together by imposing the continuity of the displacements, moments and shear forces, but not for the rotations. The subdivision is possible because, in these examples, the folding lines are straight and intersect the boundary of the plate.

This method was successfully used for linear elastic plates. Examples include buckling loads [16], natural vibrations [17] for a first-order shear deformation plate theory. In [18], Grossi used a combination of the Ritz method with Lagrangian multipliers for anisotropic plates and also with curved internal hinge lines [19]. Nonetheless, these methods become less general and cumbersome for von Kármán plates with multiple folding lines that do not allow a domain decomposition. In these cases, numerical methods are easier to use and more versatile.

Several examples of studies of folded plates can be found in the literature, especially in Civil Engineering. It suffices to think that a folded plate is a common architectural building block, for example used for roofs or windows [20]. However, in these cases, the fold is often created as a stress-free ridge in the initial mesh. We will not consider these cases in this paper, as we focus on stress-free flat plates that develop ridge-like deformation *a posteriori* as the result of applied loads.

Notable examples in the meshfree literature include [21, 22], where the fold geometry is introduced *a priori*, by joining different plates through boundary conditions. A meshfree method discretises each plate, and the folded plate is an assembly of individual plates.

In this work, instead, we present a method that can handle folding lines that are internal to the plate and does not need to consider a folded plate as an assembly of several plates. In fact, in some cases, it is not possible to separate the domain, for example for complicated folding lines, or folds having a *finite* length.

The sketch in Fig. 1 presents the idea of this paper: a fold is a line of discontinuity (indicated with symbol \(\llbracket \cdot \rrbracket \)) for the rotation fields, like cracks are discontinuities for the displacement fields. We hereby define as *infinite* folds those lines extending fully across the domain and *finite* as those folds with extremes within the interior of the domain. In the Mindlin–Reissner kinematics, rotations are discretised with independent shape functions from the displacements. Therefore, introducing a discontinuity in the rotations will generate a folding line. For this reason, the same machinery developed for cracks applies to folding. In this paper, we use a method for cracks in 2D for meshfree methods developed by one of the authors [23].

## 2 The von Kármán plate

*E*being the Young’s modulus and \(\nu \) is the Poisson ratio. The Second Piola-Kirchhoff stress is given by

## 3 Weak form of the equations

## 4 The discretization of the weak form

We will use a meshfree setting, namely the Reproducing Kernel Particle Method (RKPM) [25], combined with the intrinsic enrichment presented in [23, 26, 27] to introduce discontinuities in the rotations.

The mid-plane \(\mathcal {S}_0\) is discretized with a cover of *N* overlapping spheres \( \mathcal {Q}_I \subset \mathcal {S}_0\) of variable radii \(r_I, \ I=1,\ldots ,N\), such that \(\mathcal {S}_0\subset \bigcup _{I=1}^{N} \mathcal {Q}_I \). We call nodes the centres of these spheres \(\mathbf X _I\), and we consider *H* as an average measure of the distance between two neighbouring nodes. Because of the overlap, \(H<r_I, \ I=1,\ldots ,N\).

Following a Bubnov-Galerkin method, we approximate test and trial functions as a linear combination of compact support RKPM shape functions.

*k*,

*r*is the average of the radii \(r_I\). In the following, we use the empirical rule for the support radii \(r = k + 1\) to avoid singularity of the moment matrix.

### 4.1 Weight function enrichment

*weight function enrichment*, where

*S*being the axis perpendicular to segment \(\mathfrak {s}\) (Fig. 3). Applying function

*e*in Eq. (44) to the nodes \(\mathbf X _I \in \mathcal {S}_0^{-}\) would simply zero their weight functions. To avoid such occurrence, the weight functions are modified with the complement to 1 of

*e*

*d*is defined in Eq. (43).

*n*folding lines, denoted by \(\mathfrak {s}_1, \, \mathfrak {s}_2, \, \ldots \mathfrak {s}_n\). Let \(\bar{e}_1, \, \bar{e}_2, \, \ldots \bar{e}_n\) be the respective enrichments. Then, these enrichments apply iteratively to the \(\omega \), as shown in Fig. 5:

*H*, to avoid singularity of the moment matrix (Eq. (24)).

### 4.2 Discretised equations of motion

*X*and

*Y*axis in the reference configuration and \(n_X,\, n_Y\) are the components of the normal unit vector of curve \(\mathcal {L}^0_t\). Finally,

where *l* is the arc-length parameter, and \(\varDelta \mathbf d = \mathbf d -\mathbf d _0\), where \(\mathbf d _0\) is a previously converged state. We solve the nonlinear equations with a quasi-Newton *trust-region* algorithm for sparse problems [29], with a provided sparsity pattern for the Jacobian to speed up the iterations. Both Eqs. (30) and (37) are normalized, so that the error in each iteration is expressed as a percentage. The convergence is usually achieved within machine precision in 4 or 5 iterations.

## 5 Numerical examples

### 5.1 Simply supported plate without and with folds

We first solved a classic benchmark example (Fig. 6) for the von Kármán plate, with the aim of identifying the optimal number of nodes and the order of the basis in Eq. (23). All the quantities are appropriately normalised to obtain a dimensionless problem: \(L = 1\, \hbox {m},\, E=1\, \hbox {Pa}, \, \nu =0.316, \, h/L=0.1\). All four edges are simply supported, \(u_0=v_0=w_0 = 0\). A normal uniform pressure *p* is applied on the mid-plane. The direction of the load is fixed, i.e. it does not follow the deformed normal direction of the plate.

*p*, the vertical deflection is smaller than the one for the plate with no folds. This stiffening agrees with the intuition: a flat thin sheet of paper will bend under its own weight. But, if the sheet of paper contains folds, it will bend less (or not bend at all) (Fig. 11).

### 5.2 Verification of the method

*roof-like*deformation (Eq. (50) and Fig. 12a) with no curvature; the second solution (Eq. (53) and Fig. 12b) has a uniform non-zero curvature with a cusp at the fold line. For both cases, we computed the \(\mathcal {L}_2\) norm of the error:

*w*being the exact solution given in “Appendix C”. We also evaluated the error for the rotations

The order of reproducibility of the shape functions is 1 and the order of quadrature is 1. The number of quadrature cells per shorter side of the plate is always 4 times the number of nodes (per shorter side of the plate). We kept the same *H* in both axes: for example, if *n* is the number of nodes on the shorter side (in this case *b*, *y*-axis), the number of nodes in the *x*-axis is \(L/b\,n\).

All the quantities are appropriately normalised to obtain a dimensionless problem: \(L = 1\, \hbox {m},\, b=h=0.1\, \hbox {m}, \ E=1\, \hbox {Pa}, \, \nu =0.3, \, h/L=0.1\). The dihedral angle of the fold (the amplitude of the discontinuity in the rotations) was set at \(\varphi _0=\pi /10\).

Figure 13 shows the percentage error for different nodal spacing *H*: even with a small number of nodes (3) along the *y*-axis \((H=b/2)\) the percentage error is \(0.1965\%\) for the displacements and \(0.1699\%\) for the rotations.

Figure 14 shows the error in the displacements and rotations for the second case (Eq. (53)) considered for the verification. Also in this case, the errors are very small: for example, with \((H=b/2)\) the percentage error is \(0.5785\%\) for the displacements and \(0.5176\%\) for the rotations.

### 5.3 Rectangular plate with folds in the middle

Next, we show more explicitly the effect of the enrichment in a plate by inserting one or more folds in the middle. We construct an ad-hoc example as in Fig. 15. Firstly, we consider a flat rectangular plate, with a fold straight line in the middle. On this line, all mid-plane displacements are fixed Also, at the left (\(X=0\)) and right edge (\(X=L\)) symmetric rotations \(\bar{\varphi _{1}}\) are applied. For this example, \(b/L = 0.4\) and \( h/L = 0.1\). The material properties are the same as in Sect. 5.1. Figure 16a shows the deformation of the plate without a fold: the rotation field \(\varphi _{1}(X,Y)\) is smooth and continuous, and the deformation shows no kinks. Instead, with a fold in the middle, the rotation field is discontinuous (Fig. 16b) and the deformation shows a ridge.

The enrichment allows the introduction of multiple and kinked folds, and Fig. 17 show the same example with two inclined parallel infinite folds separated by *L* / 8 and a central fold composed by two lines with a kink in the middle.

### 5.4 Buckling of plates with folds

*L*/ 4. On these last threefolds, the transverse deflection is fixed. The plate contains an initial imperfection on the right and left edges, as an initial applied rotation \(\alpha =5^{\circ }\). A compressive load \(\lambda \) is applied on the right edge. The presence of the folds increases the critical Euler’s load, since the effective length is the distance between the vertical fold lines

*L*/ 4

with *I* being the minimum area moment of inertia of the cross section. Figure 19 shows the load-displacement curve, with the load approaching asymptotically the critical load. The deformation stages are displayed in Fig. 20, where the fold lines create a shiny reflection in the deformed mid-plane surface.

## 6 Conclusions, limitations and future directions

This paper demonstrated a meshless approach to the simulations of folds in nonlinear von Kármán plates. The folding lines in this paper can be infinite (extending throughout the mid-plane) or finite (with end-points internal to the mid-plane) folds. We showed examples of plates containing multiple folds of straight or kinked shape.

The main conclusion of the article is that with the same methods developed for cracks, it is possible to simulate folding of plates. In fact, folds are discontinuities in the rotations, just like cracks are discontinuities in the displacements. Therefore, if one has a method for reproducing discontinuous shape functions, it is possible to use such method to discretise the rotation fields.

For this reason, we believe that this paper can bridge the fields of computational fracture mechanics and computational structural instabilities. Imagine all the vast literature of numerical methods for fracture mechanics that can now be used to simulate folding of structures, which includes the realm of *origami* mechanics.

Possible future directions include the application of methods such eXtended Finite Element (XFEM) [31], cracking particles [32], phase fields [33, 34] and isogeometric analysis [35, 36] to folding.

This approach, of course, is possible if the shape functions approximating the rotations are different from the ones used for displacements. This is the case of thick plates, which possess Mindlin–Reissner kinematics. For thin plates, therefore, a possible future direction could be the introduction of discontinuities directly into the derivatives of the shape functions. Examples include intrinsic [37] or extrinsic enrichments [38, 39].

Using Mindlin–Reissner kinematics for thin plates might generate shear locking issues. However, owing to the merits of the RKPM approximation, the use of a higher-order reproducibility allowed good accuracy even to thin plates, with aspect ratio \(h/L=0.01\). Of course, we do not claim that this is a solution to the shear locking issue: a more thorough analysis of this problem is outside the scopes of the paper. We refer to the works of [40, 41, 42] for a detailed analysis of the shear locking issue in meshfree methods.

## Notes

### Acknowledgements

Ettore Barbieri’s work was supported by JSPS KAKENHI Grant No. JP18K18065.

## Supplementary material

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