## Abstract

Presented here is a comprehensive model for hook bending behavior under contact loading conditions, motivated by the relevance of this problem to reusable hook attachment systems in nature and engineering. In this work, a large-deflection model that can describe the bending of hooks, taken as precurved cantilevers with uniform initial curvature, was derived and compared with physical testing. Physical testing was performed with stainless-steel and aluminum hooks shaped as semicircular arcs. The force versus displacement behavior exhibited a linear portion for small displacements but at large displacements there was an asymptotic relation where the force approached some limit and remained flat as further displacement occurred. Comparison with testing showed that the model developed in this paper gave good agreement with the physical testing. Surprisingly, in dimensionless form, all parameters to define the hook transform to approximately linear functions of displacement. Using these linear relations, several equations are presented that allow for rapid calculation of the dimensional force versus displacement for a hook.

## 1 Introduction

Hook and loop materials have become a mainstay of everyday life, being used in places where the quick and temporary joining of two surfaces is required. These technologies were based on the observation of structures used by some plants to disperse seeds, evolving long structures with hooked ends that enable temporary attachment to an animal that may carry the seed to some distance away. Many other natural structures rely on hooks for various purposes. The feathers of birds use hooks and barbs to form self-healing surfaces [1,2]. This continues to inspire applications of hook structures for reusable and reversible attachment mechanisms. Interlocking hooks are an active area of research ranging from microfabricated hooks for robots [3] to the development of hooks made of Nitinol for temperature-mediated performance [4] to probabilistic hook arrays for dry adhesive materials [5].

Although hook-based attachment systems have been in use for many decades, the loading behavior of the bending hooks subject to contact constraints is not well described. Despite the pervasiveness of hook structures, previous analytical models rely on the approximation of small displacements, which may introduce inaccuracies when generalizing for all cases of hooks. A model which can accurately accommodate large deformations of compliant curved beams can be of great use in the development of new hook-based attachment systems and other compliant mechanisms. The present work advances this goal by building from previous hook models as well as from models of compliant cantilevers [6–11]. Up to now, large-deflection analysis of compliant beams has focused on beams without initial curvature. The present work applies large-deflection analysis to the problem of cantilevers with nonzero initial curvature and also accommodates sliding contact conditions using frictionless interlocking constraint [11].

This paper builds from past work using Jacobi elliptic integral elastica solutions in order to gain mathematical insight into the behavior of constrained curved cantilevers subject to large deformations. This approach works well with defined end conditions, having been applied to analyze cantilever deflection under defined end loads [8–10], frictionless sliding contact points [11], and large-deformation buckling [12,13]. More broadly, large-deformation analysis of problems in compliant mechanics has recently experienced a small renaissance driven by problems in biology and soft materials [14], mechanical metamaterials and advanced manufacturing (the motivation for this study), computer visualization, and new availability of computational power for numerical investigations. Particularly, models of slender rods and ribbons parametrize curvature along the length of a given slender object [14–16] in order to facilitate computation of large-deformation behavior. Advancements in solution methods and analysis of boundary conditions serve a general discourse in large-deformation mechanics to define tractable problems that provide both mathematical insight and also relevance to science and engineering.

Returning to the examination of deforming hooks, many hooks like those in Velcro and in burdock plants (*Arctium* spp.) are comprised of a straight section and a curved section. In this paper, a model for the large displacement force versus deflection behavior of the curved section of a hook is presented in order to focus on the problem-solving method for nonzero initial curvature. Hooks are commonly used in two ways, one where loading occurs only at the end of the hook and another where the load point slides along the length of the hook as it deforms due to the load. Both scenarios are presented below. The analytical model is verified with physical testing performed with manufactured hooks utilizing a tensile testing machine with custom loading jigs.

## 2 Theory

### 2.1 Beam Geometry and Boundary Conditions.

The hook is modeled as a curved cantilever that extends from the coordinate system origin at point *A* to an endpoint *B* where load ** F** is applied, as in Fig. 1(a). In its initial, unloaded configuration, the cantilever beam is an arc with constant curvature that ends at point

*B*

_{0}. The portion of the beam extending past the loading point is excluded from the bending calculation, as no loading occurs in those regions. A reference coordinate system (

*x*,

*y*) is defined as shown in Fig. 1(a), where point

*A*is located at the origin. It will be assumed that the loading point

*B*will move only in the positive

*x*-direction. The force

**is applied at point**

*F**B*, with components

*Q*and

*P*being its

*x*and

*y*components, respectively. Additional details of the problem geometry are provided in Fig. 1(b), where

*φ*is the angle between the

*x*-axis and the direction of

**.**

*F*The angle *θ*(*s*) is the tangent angle at a distance *s* along the arc length as measured from point *A*. To focus the derivation below on the solution procedure and the solution of greatest interest for modeling most hooks, and also to draw from previous approaches to similar problems [10,11], at point *A*, the cantilever has a clamped boundary condition and *θ*(*s* = 0) = 0. It should be noted that it is possible to solve for an arbitrary angle at point *A*, 0 is chosen here as it simplifies analysis. At the other endpoint, the angle *θ*(*s* = *s*_{B}) = *θ*_{B} is the end tangent angle and ** B** is the tangent vector of the curve at point

*B*.

There have been several recent advancements in modeling the large deflections of cantilevers subjected to end loading [8–11]. Because the hook is taken as a curved cantilever, these large-deflection models can then be adapted to this problem. The main problem is how to account for the sliding contact condition, where the length of the hook increases as it slips past a rigid obstacle in contact at point *B*. Sliding contact conditions were previously introduced by author Brown for large-deflection flat cantilevers in Ref. [11]. Adapting to the curved cantilever analysis and the (*x*, *y*) coordinate system introduced in Fig. 1(a), the contact point is fixed in *y* but undergoes +*x* displacement during a loading experiment.

Because the cantilever is clamped at point *A*, it experiences typical *x* and *y* reaction forces and a reaction bending moment at that point. In the case of the compliant hook in frictionless contact with an applied load, ** F** may only act in a direction normal to the curve at the contact point

*B*, and there is no applied moment at point

*B*. As described below, the bending moment

*M*(

*s*) at a given point along the beam can be found using

**and the displacement vector**

*F***from the given point (**

*R**x*,

*y*) to the endpoint (

*x*

_{B},

*y*

_{B}).

### 2.2 Formulation of Differential Equations for Beam Kinematics.

Solution begins with the Euler–Bernoulli beam theory, where the local curvature *κ*(*s*) = *dθ*/*ds* at a given point *s* along a beam, with initial curvature *κ*_{0}, is proportional to the bending moment *M*(*s*) at that point, as shown in Eq. (1*a*). The position along the beam is bounded as 0 ≤ *s* ≤ *s*_{B} where *s*_{B} is the total arc length of the beam between anchor point *A* and contact point *B*. Here, the beam is taken to be inextensible; that is, the length of the beam does not change or experience deformation. This approach to the calculation of curvature provides the fundamental basis for the analysis of large-deformation models using elliptic integrals. Relating local kinematics to the applied forces via *κ*(*s*) and *M*(*s*) enables eventual determination of overall force versus displacement behavior for a compliant hook.

*M*(

*s*) is determined using Eqs. (1

*b*)–(1

*g*). For in-plane motion,

*M*(

*s*) is defined as the magnitude of the cross product between the vector

**from any point along the beam to the loading point and the applied force**

*R***. Force**

*F***is separated into its**

*F**x*and

*y*components,

*Q*and

*P*

_{,}respectively. With the frictionless contact condition, it is then possible to write

*P*in terms of

*Q*using the direction of the force using Eq. (1), where

*θ*is the deflected cantilever end angle, and

_{B}*φ*is the angle of applied force, which is defined according to Eq. (1

*f*) [11]

*a*)

*b*)

*c*)

*d*)

*e*)

*f*)

*g*)

*b*)–(1

*g*) into Eq. (1

*a*), and expanding, Eq. (2) is obtained as follows. First, combining Eqs. (1

*b*), (1

*c*), and (1

*d*) yields

*M*= −

*Q*(

*m*(

*x*

_{B}−

*x*(

*s*)) + (

*y*

_{B}−

*y*(

*s*))), which is substituted into Eq. (1

*a*), resulting in (

*dθ*/

*ds*) = −(

*Q*/

*EI*)(

*m*(

*x*

_{B}−

*x*) + (

*y*

_{B}−

*y*)) +

*κ*

_{0}. From this, the derivative is taken with respect to

*s*, resulting in (

*d*

^{2}

*θ*/

*ds*

^{2}) = (

*Q*/

*EI*)(

*m*(

*dx*/

*ds*) + (

*dy*/

*ds*)). In this form, the relations (

*dy*/

*ds*) = sin

*θ*and (

*dx*/

*ds*) = cos

*θ*are substituted, resulting in (

*d*

^{2}

*θ*/

*ds*

^{2}) = (2

*Q*/

*EI*) (

*m*cos

*θ*+ sin

*θ*). A factor of 2(

*dθ*/

*ds*) is multiplied on both sides and then integrated, resulting in (

*dθ*/

*ds*)

^{2}= (2

*Q*/

*EI*)(

*m*sin

*θ*− cos

*θ*) +

*C*, where

*C*is an integration constant. After, Eqs. (1

*a*)–(1

*d*) are substituted for

*dθ*/

*ds*and the boundary condition at point

*A*, (

*θ*= 0,

*s*= 0), is applied, the result is $(d\theta /ds)2=(2Q/EI)(msin\theta \u2212cos\theta +(\kappa 02(EI)/2Q))$. Finally, distance

*L*=

*s*

_{B}is introduced and

*L*

^{2}is multiplied on both sides to create a dimensionless equation, and the coefficients are substituted with dimensionless parameters $\lambda =\kappa 02EI/2Q$ and

*β*=

*QL*

^{2}/

*EI*, resulting in Eqs. (2

*a*)–2(

*h*), which relate the coordinates

*x*

_{B},

*y*

_{B}, arc length

*s*

_{B}, and tangent angle

*θ*

_{B}of point

*B*to applied load

*Q*, beam flexural rigidity

*EI*, and initial curvature

*κ*

_{0}. Equations (2

*b*)–(2

*d*) are obtained by rearranging and integrating Eq. (2

*a*) and again applying the relations (

*dy*/

*ds*) = sin

*θ*and (

*dx*/

*ds*) = cos

*θ*.

*a*)

*b*)

*c*)

*d*)

*e*)

*f*)

*g*)

*h*)

For calculation of relationships among (*x*/*L*), (*y*/*L*), (*s*/*L*), and *θ*, substitute *x*, *y*, *s*, and *θ* for *x*_{B}, *y*_{B}, *s*_{B}, and *θ*_{B} in Eqs. (2*b*)–(2*d*), and calculate results for 0 ≤ *θ* ≤ *θ*_{B}. The distance *L* is introduced mainly to retain an analogy to derivations in prior work [10] that used *L* as an invariant distance for nondimensionalization. Here, *y*_{B} is invariant instead, dimensionless equations Eqs. (2*b*)–(2*d*) retain mathematical relationships among dimensionless values. Retaining the distinction between *L* and *s*_{B} enables consideration of the relationships among variables that derive when invariance is applied to one (or more) of them. Equations (2*b*)–(2*d*) are difficult to solve, but have solutions using the Jacobi elliptic integrals and associated Jacobi elliptic functions [8–12,17].

In the present formulation, the term *β* is the nondimensional loading parameter corresponding to applied load *Q*. In order to adapt the convention used by related earlier work [8–11], *β* is chosen to be the *x*-component of the nondimensional applied force, as *α* is the *y*-component of the force. The dimensionless term *λ* appears as a result of the initial curvature. A distinctive feature of the present elliptic integral formulation relative to prior work is that the parameter *λ* is a function of the *β* dimensionless loading parameter. Retaining *β* within the integral renders calculation of end angle *θ*_{B} more difficult, but provides an essential mechanism to accommodate initial beam curvature. In contrast to Zhang and Chen’s general formulation [10], which required a nonzero end moment to account for initial curvature, Eq. (2) operates without this assumption. In the limit when *Q* → 0, *λ* dominates the other terms in the denominator of Eqs. (2*b*)–(2*d*), enabling simplification of these equations by dominating the other terms in the denominator, which then allows moving the denominator outside of the integral. The specified initial curvature conditions are obtained after algebraic simplification.

Under the formulation of Eqs. (2*a*) and (2*b*), a search scheme must be used to obtain the end angle *θ*_{B}. To do this, the right side of Eq. (2*b*) can be integrated for a range of possible *θ*_{B} values, from 0 to initial end angle *θ*_{0} as defined by the initial condition of Fig. 1(a). The correct angle *θ*_{B} value for a given *β* is determined as the angle for which the right side of Eq. (2*b*) equals the left side within a given tolerance. The left side of the equation is the result of the integration of the length of the cantilever from 0 to the total length *s*_{B}. As it is nondimensionalized with the arc length *L* and *L* = *s*_{B}, the value of the left side will always be 1. Once *θ*_{B} is found for a given *β*, it can be used in the computation of Eqs. (2*c*) and (2*d*).

### 2.3 Solution With Jacobi Elliptic Integrals

#### 2.3.1 Solved Elliptic Integrals for Beam Differential Equations.

Using a table of elliptic integrals, Eq. (2) is transformed to Eqs. (3) and (4) [17]. Due to constraints of the Jacobi elliptic functions, the problem is separated into two different cases with different ranges for their validity. Case I is where *λ* > *μ*, and case II is where *λ* < *μ*. This distinction between two separate cases emerges from mathematical manipulation using Jacobi elliptic functions [17]. If the load and end angle are known, then *λ* and *μ* can be calculated to determine the use of either case I or case II; if they are not known, then a numerical scheme must be used as discussed further in this paper. The transition between cases occurs when *λ* = *μ*, or $(\kappa 02(EI)/2)=F$, which can be found from $\lambda =(\kappa 02EI/2Q)=\mu =m2+1=tan2\phi +1=sec\phi =(F/Q)$. Strain energy of a cantilever under pure bending equals $(1/2)\theta 0M=(1/2)(\kappa 0L)(EI\kappa 0)=(1/2)\kappa 02EIL$; therefore, the transition between case I and case II corresponds with the point at which the contact force *F* begins to exceed the strain energy per length that would be required to obtain the initial curvature under pure bending.

*a*)–(3

*c*) and Eqs. (4

*a*)–(4

*c*)

*a*)

*b*)

*c*)

*d*)

*e*)

*f*)

*g*)

*h*)

*i*)

*a*)

*b*)

*c*)

*d*)

*e*)

*f*)

*g*)

*h*)

*i*)

Because Eqs. (3) and (4) are transformations of Eq. (2) and are equivalent to Eq. (2) under case I and case II as described previously, Eqs. (3) and (4) can be applied to the operations described in Sec. 2.2 rather than direct integration of Eq. (2). In Eqs. (3) and (4), integrations are confined to Eqs. (3*d*), (3*e*), (4*d*), and (4*e*) using well-defined elliptic integral formulations. Equations (3*a*) and (4*a*) can be applied to search for endpoint angle *θ*_{B} as described in Sec. 2.2: for any given nondimensional load *β*, the end angle for this condition is not known; therefore, given a value of *β*, first find the value of *θ*_{B} for which (*s*_{B}/*L*) = 1. A search across a range of candidate *θ*_{B} values is required because in Eq. (3*a*), both *f* and *μ* are functions of *θ*, presenting a significant challenge to invert Eq. (3*a*) to find *θ*_{B} as a function of *β*. Similarly, the arc shape can be constructed as described in Sec. 2.2, but using Eqs. (3*b*), (3*c*), (4*b*), and (4*c*) rather than Eqs. (2*c*) and (2*d*).

#### 2.3.2 Application to Hooks With Fixed Contact Position and Constant Arc Length.

While the sliding contact condition is of greater interest for mechanical attachment, analysis of hook deformation based on an invariant arc length *L*, as illustrated in Fig. 2(a), helps to probe the behavior of the dimensionless variables contained within Eqs. (2)–(4), with the result provided in Fig. 2(b), with Δ = (*x*_{B}/*L*) − (*x*_{0}/*L*). This scenario is equivalent to a pinned attachment with free rotation through a hole at a given length of a deforming beam. Here, case I is initially used to solve for the deflection, which eventually transitions to Case II under further loading. Around the region where *λ* ≈ *μ*, the two cases produce close to the same solution. The solution begins by first selecting the initial end angle *θ*_{0} and a range of applied *β* values. These are used to compute the term *λ*, using Eq. (2*g*), corresponding to each value for *β*. For these given values of *λ* and *β*, the end angle *θ*_{B} is found with a search as described earlier, and *θ*_{B} then determines the corresponding *x*_{B}, *y*_{B}, and Δ.

To explore the nondimensionalized force versus displacement mechanical behavior, *β* should be increased incrementally at which point the relation can be constructed as in Fig. 2(b). As *β* is increased due to loading, *λ* and *μ* should be compared with one another to determine which case is valid. Once *λ* is found to be less than *μ*, case II is then used to compute the displacement. For consideration of the sliding contact condition, this same procedure will be used.

As an example, a hook with the initial shape of a quarter circle and corresponding initial end angle $\theta 0=\pi 2$ is used. A diagram showing this can be seen in Fig. 2(a). The *y* and *x* components, *α* and *β* respectively, of the nondimensional load are plotted versus dimensionless displacement Δ in Fig. 2(b). As *β* is known, once θ* _{B}* is found, then

*α*can be found with Eq. (2

*e*). While the case where $\theta 0=\pi 2$ is shown, it is possible to do this with any arbitrary end angle. This in fact is necessary for computation of the sliding contact condition.

#### 2.3.3 Application to Hooks With Sliding Contact Condition.

Solution of the sliding contact conditions uses the same dimensionless calculations, provided previously, as the hook with the end loading contact condition, but in dimensional space, the arc length *L* varies and *y*_{B} is invariant, rather than vice versa for the contact point at a fixed arc length. As the loading point *B* moves, the length of the beam that spans between *A* and *B* also increases as more of the previous beam slides past the contact point. Solution is possible by nondimensionalizing and rescaling the displaced hook, with relevant parameters defined in Eq. (5). It is preferential to rescale using the *y* distance from the loading point *B* to the anchor point *A*, as shown in Fig. 3(a).

*θ*

_{0}used in the calculation of

*λ*(Eq. (2

*g*)) must be larger than

*θ*

_{0}for the unloaded configuration. The parameter

*θ*

_{0}is crucial for solving the hook displacement. It is the nondimensional curvature and is equal to the initial end angle for the hook under no loading. The parameter

*θ*

_{0}is found by dividing the length

*s*

_{B}by the initial hook radius

*r*

_{0}as in Eq. (5

*a*). The invariant

*y*

_{B}of the sliding contact condition implies that

*L*in Eq. (2

*d*) varies, but

*L*is also an input to

*λ*in the right-hand side of Eq. (2

*d*). Because of this, another parameter $\theta *$ is developed to support rescaling in the sliding contact problem. It is defined, Eq. (5

*d*), as the ratio between $\theta 0(1)$ and $\theta 0(2)$, where the superscripts (1) and (2) are added to denote (1) the initial conditions before any loading and (2) the initial end angle of the arc length that has slipped past the contact point after some loading has occurred. The initial curvature of the beam

*κ*

_{0}can be represented with Eq. (5

*a*). To explore the calculation of mechanical behavior under sliding contact, the case of a hook with the unloaded shape of a quarter circle and corresponding initial end angle $\theta 0(1)=\pi 2$ is used as an example. Because of this initial geometry, the invariant loading distance

*y*

_{0}is equal to the initial radius of curvature

*r*

_{0}. Dividing the

*x*-displacement

*δ*by the invariant loading distance

*y*

_{0}results in the nondimensional displacement Δ.

*a*)

*b*)

*c*)

*d*)

A table of correlated values for the sliding contact condition can be established with the following algorithm:

Increment $\theta *$, and by definition therefore increment $(L/r0)=\theta 0(2)$

Increment

*β*and calculate*s*_{B}/*L*for all possible end angles*θ*_{B}after deformation using Eqs. (3*a*) and (4*a*), 0 ≤*θ*_{B}<*π*Apply (

*s*_{B}/*L*) = 1 to determine the corresponding*θ*_{B}for a given input of*β*for a specified value of $\theta *$If interested in beam kinematics,

*x*/*L*and*y*/*L*can be computed additionally here in Step 3

Repeat Steps 2 and 3 to create a set of (

*β*,*θ*_{B}) pairs for given $\theta *$, then apply Eqs. (3*b*), (3*c*), (4*b*), and (4*c*) to generate corresponding values of*x*_{B}/*L*and*y*_{B}/*L*Continue to increment $\theta *$ and perform Steps 1–4 to establish an array of matched $(\theta *,\beta ,\theta B,(xB/L),(yB/L))$ values

Use the matched $(\theta *,\beta ,\theta B,(xB/L),(yB/L))$ values to determine other correlated values of interest such as Δ

The sliding contact constraint

*y*_{B}=*r*_{0}, or equivalently $1=yBLr0L=yBL\theta 0(2)$, can be applied to find the set of $(\theta *,\beta ,$$xBL,yBL)$ values that describe motion under this constraint

Solution of the sliding contact condition requires the computation of many points along the length of the hook. Equations (3*a*) and (4*a*) are only valid until the right side is equal to one, necessitating for the curvature and initial end angle to be varied. As done previously with the purely end-loaded condition, a value for the initial end angle $\theta 0(2)$ is selected, where $\pi 2<\theta 0(2)<\pi $ for the geometry presented. The nondimensional load *β* is then increased from 0 to some finite value when solving Eqs. (3) and (4). Doing this will obtain the parameters $\theta 0(2)$, $yBL$, and $xBL$ for a given load *β*. The dimensionless load *β* should be incremented while comparing $yBL$ with $1/\theta 0(2)$, Eq. (5*a*). This is then repeated for all values of $\theta *$ until the end of the hook section is reached. Once the values for $yBL$ and the radius match, $xBL$ for a given load is obtained, which will then be used for calculation of the displacement Δ. A flowchart illustrating the procedure to solve for Δ can be seen in Fig. 3(b).

For case II, near the end where $\theta *\u22481.925$, the elliptic integrals reach their limit. That is, the elliptic integrals transition from incomplete to complete elliptic integrals, and computing values of $\theta *$ greater than 1.925 lead to an erroneous solution. The apparent linear relation enables extrapolation of the *x _{B}*,

*y*, and

_{B}*κ*parameters to $\theta 0(2)=\pi 2$. This result was then rendered to a dimensional form for a hook with a radius of curvature of 25 mm, thickness of 100

*µ*m (0.004 in), width of 12.7 mm, and Young’s modulus of 200 GPa (Fig. 4(b)). As

*β*and

*δ*are known, they can be converted to dimensional values using Eqs. (2

*f*) and (5

*c*), respectively.

Surprisingly, upon computing Eqs. (3) and (4) for the entire range over which they are valid, a linear relationship between the parameters *β*, $xBL$, $yBL$, $\theta *$ and the dimensionless displacement Δ is obtained. This can be seen in Figs. 4(a) and 4(c).

Due to the computational intensity of calculating Δ as well as the apparent linear relationships depending on this parameter, linear fits were applied among the correlated variables to simplify the future calculation. For all the parameters, results are reported as Eq. (6), with Eq. (6*d*), the result of substituting Eqs. (6*b*) and (6*c*) into Eq. (6*a*). These equations retain the dimensional displacement *δ* and the dimensional force *Q* to enable prediction of the behavior for a real hook.

*a*)

*b*)

*c*)

*d*)

*e*)

The linear fit equations for the parameters shown in Fig. 4 are presented in Eq. (7). The *R*^{2} values for *β*, $xL$, $yL$, and *θ** as functions of Δ are 0.9997, 0.9164, 0.9843, and 0.9977, respectively.

*a*)

*b*)

*c*)

*d*)

*e*)

*a*) and (8

*b*) to facilitate the comparison of sliding contact physical data with theory. Both Eqs. (8

*a*) and (8

*b*) are transformed to remove the explicit dependence on the length of the hook, instead of using an explicit dependence on the initial radius of curvature

*r*

_{0}. In experimentation, it is not convenient to continually measure the length of the beam, replacing arc length with

*r*

_{0}reduces the uncertainty of calculations based on measured data in addition to increasing the ease of use. From the linear fit equations for

*β*and

*θ*

_{B}(Eqs. (7

*a*), (7

*d*), and (8

*a*)) and relating

*r*

_{0}to

*L*in Eq. (8

*d*) (from Eq. (5

*d*) and $\theta 0(1)=\pi 2$), a simple expression defining $\beta *$ is obtained, Eq. (8

*c*), which then provides a quick and simple way to compare nondimensionalized data and theory. These equations are plotted and shown in Fig. 5. In Fig. 5, Eq. (8

*a*) is plotted using the direct outputs from Eqs. (3), (4), and (8

*c*) and uses the linear fit equations in Eq. (7)

*a*)

*b*)

*c*)

*d*)

Importantly, with this result, the force versus displacement of a bending hook can be found using simple functions of readily measurable parameters (*δ*, *r*_{0}, and *Q*), and the time to compute the results is nearly instantaneous relative to the computations inherent to the elliptic model.

Computational resource intensity was briefly examined on a desktop computer using built-in central processing unit (CPU) timer functions in Matlab. Computing Eq. (3) required approximately 6.5 s for a single input of *β* and *θ*_{0}, searching for *θ*_{B} over 1000 increments in the interval of 0 to $\pi 2$. Computation of force versus displacement requires this procedure to be repeated multiple times; therefore, the time to compute a sweep of *β* for a given *θ*_{0} will consume a significant amount of time. If one were to repeat this procedure say 20 times, it would take approximately 2 min for a single *θ*_{0}. Computation of the sliding contact behavior additionally requires sweeping *θ*_{0}, which further significantly contributes to the time. If one were to repeat the single *θ*_{0} procedure for a range of *θ*_{0}, it can be seen that to have good precision the time to compute readily becomes incredibly large. This is in stark contrast to the calculations in Eq. (8) based on polynomial fits. Calculation from an input of 1000 points takes a mere 0.018 s, demonstrating orders of magnitude improvement over the elliptic integral computation.

## 3 Experiments

To verify the analytical methods presented earlier, a test was devised to measure the force versus displacement behavior of curved cantilevers made from metal strips. Experiments were performed on a Shimadzu AGS-X(5kN) mechanical test frame, mounted with custom-fabricated test jigs. Each jig consisted of a clamp to hold a curved strip of metal, fixed to a base that allows movement for accurate positioning. A polished metal bar is suspended from the load cell integrated with the moving crosshead of the tensile tester. The bar is polished to reduce friction. This can be seen in Fig. 6. In Fig. 6(a), the experimental setup with a stainless-steel cantilever can be seen along with the direction of motion of the crosshead. The crosshead of the tensile tester is moved until the hook slips past the bar. Once the hook slips past, the tensile test machine is stopped, and the experimental setup is reset. The hook nearing the slipping point can be seen in Fig. 6(b). Aluminum cantilevers were fabricated to see how cantilevers undergoing plastic deformation would deviate from the purely elastic model. Images of the aluminum cantilevers can be seen in Figs. 6(c) and 6(d) which show one of the cantilever points with an inflection point, and the cantilever once it has snapped past the crosshead, respectively.

The stainless-steel samples were fabricated from 300-series stainless-steel sheets that are 0.004” (0.1 mm) thick. Young’s modulus and yield strength were taken as 212 GPa and 205 Mpa, respectively [18]. The sheets were cut into rectangles that are 1” (25.4 mm) wide, 5.5” (139.7 mm) long and then were formed into curves with a radius of 44 mm. The strips are formed into a curved shape by rolling them around a round mandrel. The excess length past the desired arc length was clamped in a fixed base. The aluminum hooks were made from 3003 aluminum with h14-1/2 hard temper, 0.41 mm in thickness. Young’s modulus and yield strength for the aluminum were 68.9 GPa and 145 Mpa, respectively [19]. Two variations of hooks were manufactured, with six hooks of 19.05 mm wide and 10 hooks of 12.7 mm wide. The hooks had a nominally 70-mm radius of curvature.

The results from the physical testing of the stainless-steel hooks are plotted in Fig. 7(a) and compared to several modeling approaches in Fig. 7(b). Figures 7(c) and 7(d) show the physical testing results from the aluminum hook variations. At small displacements, the force has a nearly linear relation to displacement, and then, the slope of the loading curves gradually decreases. The aluminum samples (but not the steel samples) exhibited permanent deformation after testing. Plastic deformation in the aluminum hooks is clearly visible in the aluminum hooks; in Fig. 6(d), the aluminum hook does not have uniform curvature. Before testing, the aluminum hooks were nearly identical to the stainless-steel hook pictured in Fig. 6(a).

## 4 Discussion and Comparison of Models of Bending

The data from 10 trials of stainless-steel hooks are averaged and plotted along with the elliptic model in Fig. 7(a). Because of manufacturing tolerances and difficulty in measuring the radius accurately, the elliptic model was calculated with the known radius of 43.5 mm ± 1.0 mm (2.3% of the nominal value). This shows that a small change in the initial radius of curvature has a considerable effect on force at higher displacements, but the uncertainty in *r*_{0} effectively bounds the experimental result. It is evident that the elliptic model gives good agreement with the physical data across the entire range.

The data from the aluminum hook trials and corresponding elliptic models are compared in Figs. 7(c) and 7(d). With the plastic bending occurring in the hooks, it can be seen that the elliptic model gives good agreement with data for small displacements. The force given by the aluminum hook decreases and then flattens out without a maximum similar to the case of a fully elastic hook. Plastic deformation appears to initiate at a displacement of 6 mm in the 19.05-mm-wide hooks. This corresponds to a dimensionless displacement of 0.17. Calculating the bending stress at the fixed end of the cantilever using a dimensionless displacement of 0.17 gives a value of 1.30 GPa, which well exceeds the expected yield strength of 150 MPa. With conditions of plastic deformation, it is expected that the effective flexural rigidity of the material reduces in the regions subject to plastic flow, which would imply a reduced force (and bending moment) to achieve a given displacement.

From Fig. 6(c), the inflection point can be seen circled. At the loading point, the hook curves to the right. Toward the base of the hook, this curvature decreases until it is nearly flat when it reaches the inflection point; past this point the hook curves in the opposite direction. As previously mentioned, for full accuracy, the elliptic model should be calculated with accommodation of the inflection point; however, it can be seen that the agreement between the model and the data supports the earlier assumption that the inflection point can be ignored. Surprisingly, the stainless-steel hook does not demonstrate an inflection point. One potential explanation is that this relates to the plastic deformation that occurs in the aluminum hook but not in the stainless-steel hook.

In addition to the elliptic integral method, Eqs. (3*b*)–(3*d*) were solved iteratively using a trapezoidal numerical integration. The values for *β*, *θ*_{B}, *x*_{B}, *y*_{B}, and *δ* are then obtained, which can then be scaled into the hook solution using the approach presented in Sec. 2.3.3. Similar to a solution with the elliptic model, a curvature is selected and then *β* is increased until the appropriate value for *y*_{B} is found. This method took significantly longer to compute than the elliptic approach.

Possible sources of errors in the testing could lie with the design of the experiment, as the bar pulling up on the hook has a non-negligible diameter that slightly changes the position and direction of the applied force during the tests. Small deviations in the placement of the hook relative to the bar as well as friction could have effects that contribute to the discrepancy. Residual stresses may also be introduced into the sample from the bending process. Other improvements to the test could eventually include a method to optically measure curvature, as well as methods to measure and quantify residual stresses in the material after the bending process.

## 5 Conclusion

We applied the Jacobi elliptic functions and Jacobi elliptic integrals to find a solution to the force versus displacement behavior of a curved cantilever under point contact loading. To find a solution to the sliding contact condition, the problem was nondimensionalized, with rescaling based on initial curvature. Surprisingly, in the nondimensional form, all parameters to define the problem of the hook are approximately linear functions of displacement. From this, linear fit equations were found which can be used to calculate the dimensional force versus displacement without requiring lengthy computation with the elliptic approach. Physical testing with hooks made from stainless steel showed the model developed gives good agreement and verified the model was appropriate for the conditions tested. From this work, there is now a simple model to predict the force versus displacement for any curved elastic cantilever under point contact conditions, something not previously available.

Perhaps, the most interesting and surprising result of this work is the apparent linear relation between *β* and Δ, and the result that other parameters necessary to fully define the problem are all linear functions of the dimensionless displacement. It is evident that there is a more profound behavior behind this mechanical system. This suggests that there may be a more fundamental underlying relationship that remains to be uncovered and compliant curved systems present an area of mechanics that should be investigated further. A more general solution could be sought out using techniques such as Cosserat rod theory. Future work should additionally consider the straight section of the hook added to the problem and performance deviations that are to be expected due to manufacturing uncertainties. Inflection points also remain for further consideration. As a result of the present investigation, we have obtained simple equations that can accurately predict the large displacement force versus displacement behavior of hooks, something that has not existed until now. Despite the observed computational intensity, this model, specifically Eqs. (6*e*) and (6*f*), is significantly faster than using finite element models.

## Acknowledgment

The authors thank Dr. Lloyd Hihara, Dr. Tyler Ray, and Dr. Bardia Konh, University of Hawai‘i at Mānoa, for the review of a draft of this paper.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The data sets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper.

## Funding Data

Air Force Office of Scientific Research (Award No. FA9550-20-1-0256; Funder ID: 10.13039/100000181).

Air Force Research Laboratory Summer Faculty Fellowship Program (Grant No. AFRL SFFP; Funder ID: 10.13039/100006602) for summer 2020 and summer 2021.

## Nomenclature

*e*=parameter calculated using incomplete elliptic integrals of the second kind

*f*=parameter calculated using incomplete elliptic integrals of the first kind

*h*=distance from neutral axis to point in cross section of beam

*m*=−cot

*θ*_{B}*r*=radius of curvature or radius of hook

*s*=arc length measured from point

*A**t*=thickness of bending cantilever cross section

*x*=the

*x*-coordinate of a point along the curved cantilever beam*y*=the

*y*-coordinate of a point along the curved cantilever beam*A*=fixed endpoint of hook

*B*=loading point of hook

*C*=integration constant

*D*=parameter used in elliptic integral calculations

*E*=Young's modulus

*I*=moment of inertia of hook cross-section

*L*=total length of arc measured from fixed point to loading point

*M*=magnitude of bending moment at a point along hook

*P*=vertical component of applied force

*Q*=horizontal component of applied force

=*R*vector from any point along the curved beam to the loading point

*B*- $i^$ =
unit vector in the

*x*-direction- $j^$ =
unit vector in the

*y*-direction*R*^{2}=statistical coefficient of determination, based on linear regression fit

*k*_{I}=elliptic modulus for case I

*k*_{II}=elliptic modulus for case II

*r*_{0}=initial radius of curvature

*s*_{B}=arc length measured from

*A*to*B**x*_{B}=the

*x*-coordinate of hook end point*B**x*_{0}=initial

*x*-coordinate of end point*B**y*_{B}=the

*y*-coordinate of hook end point*B**y*_{0}=invariant loading distance, initial

*y*-coordinate of end point*B**B*_{0}=loading point of hook at initial configuration

*α*=dimensionless loading parameter in

*y*-direction*β*=dimensionless loading parameter in

*x*-direction*γ*_{A}=elliptic argument for case II associated with point

*A**γ*_{B}=elliptic Argument for case II associated with point

*B*- Δ =
dimensionless displacement in

*x*direction for hook with end load condition*δ*=dimensional displacement for hook with sliding load condition

- $E$ =
incomplete elliptic integral of the second kind

- $F$ =
incomplete elliptic integral of the first kind

*θ*=tangent angle along the length of the hook

- $\theta *$ =
dimensionless ratio of initial end angle of arc length after some loading divided by the initial end angle at the contact point

*θ*_{B}=angle of the end point

*B**θ*_{0}=initial angle of hook end point

*B**κ*=hook curvature

*κ*_{0}=initial beam curvature

*λ*=dimensionless curvature-to-load parameter

*μ*=$csc\theta B$

*φ*=direction of applied load

*ω*_{A}=elliptic argument for case I associated with point

*A**ω*_{B}=elliptic argument for case I associated with point

*B*

## References

^{®}Nonlinear Mechanics