^{*}

Edited by: Nicola Maria Pugno, University of Trento, Italy

Reviewed by: Lorenzo Bardella, Universitá degli Studi di Brescia, Italy; Stefano Vidoli, Sapienza University of Rome, Italy

This article was submitted to Mechanics of Materials, a section of the journal Frontiers in Materials

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Folding a strip of paper generates extremely localized plastic strains. The relaxation of the residual stresses results in a ridge that joins two flat faces at an angle known as the

Folds in thin strips are extremely localized curvatures that occur over a small length-scale. In the limit of such length to zero, the shape of the strip transforms into one with a sharp corner (Lechenault et al.,

Examples of patterns generated in confined conditions (Wang and Zhao,

Several researchers explored the mechanical snap-back and snap-through instabilities of thin structures constrained by controlled boundary conditions, both on displacements and rotations.

In Beharic et al. (

Cazzolli and Dal Corso (

From the literature survey, there appears to be a considerable amount of research in confined thin sheets, especially in determining their stability. Very little yet exists on confined folded sheet. By fold, in this paper, we mean a sharp corner in the deformation (Lobkovsky,

Despite the abundance of solutions of the Euler's

A sharp corner translates into a discontinuity of the rotation of the strip. Using a numerical meshfree method initially developed for cracks, our recently published work (Barbieri et al.,

In this paper, we are particularly interested in how constraining a thin strip with a fold with finite rotational stiffness leads to non-zero curvatures. Also, we study how such curvatures can be tuned by modifying the applied rotations at the boundaries (

Curvature induced by hyperstatic confinement, and change in curvature (tuning) through applied rotations. _{0} and θ_{1}.

To examine which boundary conditions create curvatures in folded strips, we carried out semi-analytical simulations. Assuming invariance in the width, we modeled the strip as a plate under cylindrical bending. Therefore, instead of the Föppl-von Kármán equations, we modeled the strip as a planar, linearly elastic, unshearable, and inextensible rod, according to the Euler's theory of the

The equilibrium equations for the

with boundary conditions

where (·)′ = d/d^{2}) the bending stiffness, _{0} the applied horizontal force at _{0} the applied vertical force at _{0} the bending moment at _{0} the rotation at _{0}, _{0} at

The _{f}. _{0}, rest opening angle _{0} and discontinuity [[

We pass to the dimensionless form using the following normalizations:

resulting in

and the following system of non-linear ordinary differential equations [the

with boundary conditions

where

with θ_{F0} such that

and

where am is the

with

with

The Cartesian coordinates of the deformation are

with

where dn is the

where

where _{max} = _{max}) and

and

The strain energy of the rod is

Let us now assume that a fold exists at 0 < _{f} < 1. The fold is a discontinuity 〚·〛 on the rotation, with the following constitutive model (Lechenault et al.,

with _{f} and _{f}) is the bending moment, assumed continuous in the absence of concentrated moments applied at _{f}. The angle _{0} is the _{0} = 0, there is no fold; when _{0} < 0 the fold is called a _{0} > 0 the fold is called a _{0}. The moment at the fold is calculated as the product of the force measured by the dynamometer and the maximum height of the profile _{f}).

Experimental setup used in Lechenault et al. (

The geometrical meaning of _{0} is shown in _{0} by the simple relation

The rest opening angle is a mechanical property related to the yield stress of the material of the strip (Lechenault et al.,

where ϵ_{p} is the plastic strain, σ_{Y} the yield stress and _{0} appears to be independent from the thickness of the strip (Lechenault et al.,

To tune the curvatures, we will further assume that at least one of the following quantities is assigned:

which renders one or all of _{0}, _{0}, _{0} unknowns to be determined. The procedure to compute such unknowns is a _{f}: let us call this solution μ^{−}, θ^{−}, ^{−}, and ^{−}. Then, the values

the procedure is iterated with a non-linear solver until

In this section we report the results of the _{0} = _{0} = 0.

Let us examine the case where only

Therefore

_{f} and the rest opening angle _{0}.

Deformations for _{f} = 1/5 and κ = 2, and for _{f} = 1/2 and κ = ∞, both with rest opening angle _{1} = 0 (isostatic confinement). Blue continuous line:

In addition to condition (29), we consider cases where the boundary conditions are provided by a couple of rotations applied respectively at the start and the end of the strip:

We also impose that the end lies in the same plane as the start of the strip, therefore _{1} = 0. Under boundary conditions (31),

Firstly, we consider the case of a fold with infinite rotational stiffness. By changing the applied rotations, we investigate the changes in sign of the curvatures at both sides of the fold (

Thirdly, we isolate the effect of the rotational stiffness by varying κ for different strips under the strips under the same couple of applied rotations and the same position of the fold. We consider anti-symmetrical applied rotations.

Non-uniqueness of the deformation for a sheet with an infinitely stiff fold positioned at _{f} = 0.5, with rest angle

There exist multiple solutions, depending on the initial moment _{0} and shear force _{0}. This non-uniqueness is a consequence of the non-linearity of the _{0}, _{0} and _{0} (Equation 13).

In the proceeding of the paper, we will always refer to the solution with the lowest strain energy. In fact, Equation (23) states that for the same θ_{0} and _{1}, the strain energy grows with _{0} and _{0}

Having resolved this disambiguation, we now examine the changes in sign of the curvatures (_{0} and _{1} ranging from −|_{0}| to |_{0}|, with an increment of 1°. The fold has position _{0} = 1/2 and infinite rotational stiffness. With respect to the signs of the curvatures, there exist three regions: one where both curvatures are positive, one where both are negative and one where the curvatures have different signs.

Signs of the curvatures at the left _{0} = μ(0) and at the right μ(1) of the fold. The rest angle is _{f} = 1/2.

_{1} = −_{0}.

Sign of the shear force _{0} for a fold with rest angle _{f} = 1/2.

_{f} = 1/2, rest angle _{1} = 0 and symmetrical applied rotations _{0} increases as κ decreases. When κ reaches a critical value κ_{c}, the discontinuity on the rotation disappears; below κ_{c}, the fold transitions from a mountain into a valley. In the limit of zero rotational stiffness, the fold becomes a perfect hinge: the deformation is equivalent to one of a flat rod, under isostatic boundary conditions, containing a fold with a rest opening angle equal to twice the applied rotation at the end.

Deformations for variable rotational stiffness for symmetrical applied rotations

Instead, when the applied rotations are equal (anti-symmetrical rotations), the limit configuration for zero rotational stiffness is the solution of the

Deformations for variable rotational stiffness for anti-symmetrical applied rotations

In this section, we derive a simplified analytical model to explain the numerical results obtained in section 3.

In this case, the solution of the system (5) is

with δ being the

is the

The angle θ_{0} is given by

meaning that the deformation depends only upon the position of the fold _{f} and the rest angle _{0}, as anticipated in

For simplicity, we will also assume _{1} unassigned, making _{0} = 0. In addition, we will assume, in first instance, that the fold is infinitely stiff, meaning κ → ∞. In this case, the

and the solutions for the beam are

Assigning the following boundary conditions

the two unknowns _{0} and _{0} are given by

For _{f} = 1/2, the curvature will be uniform (_{0} = 0) if

regardless of the rest angle of the fold. Such straight line in Equation (41) is observable in both

Therefore, the curvature will be identically zero if both _{0} = _{0} = 0, which happens for

The curvature in Equation (38a) is a linear function in

The values of μ(0) and μ(1) are

Curvature signs due to hyperstatic confinement through applied rotations. The fold has infinite rotational stiffness. Comparison between the

Interestingly, the values in Equation (42) hold also for folds with finite rotational stiffness. In fact, when κ is finite, the solutions are

where

with

In addition, for _{f} = 1/2, θ_{0} = −_{1}, also

The

Curvature signs due to hyperstatic confinement through applied rotations. The fold has finite low rotational stiffness κ = 1/2. Comparison between the

The denominator in Equation (46) is always positive

preventing the appearance of any singularities.

The discontinuity in rotation at the fold is given by

and is showed in

Discontinuity on the rotation at the fold for variable rotational stiffness. Comparison between values from

The discontinuity becomes zero for

For _{f} = 1/2:

therefore, when the applied rotations are equal, then the critical rotational stiffness is zero, as in

When θ_{0} is of the same sign of _{0}, then _{c} > 0 as in _{0} = −_{1} and _{0} of the opposite sign, then there is no critical value, and the solution is an undeformed roof-like deformation, as determined by Equation (42).

Discontinuity on the rotation at the fold for variable rotational stiffness. Comparison between values from

It must be remarked, however, that even though a critical rotational stiffness for fold disappearance exists, the linear relationship in (24) might not hold for small rotational stiffness.

A thin sheet containing a fold assumes a roof-like undeformed state when confined isostatically. Instead, if confined hyperstatically, it will possess one or more curvatures. Such curvatures change according to the rotations applied at the ends of the sheet. Using the Euler's

We considered a fold as a discontinuity in the rotations, equal to the supplementary angle to its dihedral angle. The mechanical behavior of the fold depends upon two material properties: the rest opening angle and the rotational stiffness. The rest opening angle is a material property closely related to the yield stress; the rotational stiffness contributes to increasing the dihedral angle of the fold when subjected to a bending moment.

For infinite rotational stiffness, the dihedral angle depends only on the rest opening angle. For the signs of the curvatures before and after the fold, we can identify three regions: one where both curvatures are positive, one where they are both negative and one when they are opposite in sign. We presented a map of such occurrences according to the applied rotations; for small rotations and displacements, the numerical results agree with a simplified model based on the

For folds with finite rotational stiffness, the two regions with same-sign curvatures shrink. Interestingly, there exists a critical value of the rotational stiffness when the fold disappears, meaning that the discontinuity on the rotation becomes zero. In particular, when the applied rotations are opposite in signs and the fold is a mountain, the critical value of the rotational stiffness is different from zero, and represents a transition from mountain to valley. In the limit of zero rotational stiffness, the sheet transforms into an undeformed valley with rest angle equal to twice the applied rotation at the end.

All datasets generated for this study are included in the manuscript and/or the supplementary files.

EB designed the research, conducted the analysis, and wrote the manuscript.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.