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Tailorable elasticity of cantilever using spatio-angular functionally graded biomimetic scales

Abstract

Cantilevered beams are of immense importance as structural and sensorial members for a number of applications. Endowing tailorable elasticity can have wide ranging engineering ramification. Such tailorability could be possible using some type of spatial gradation in the beam’s material or cross section. However, these often require extensive additive and subtractive material processing or specialized casts. Herein, we demonstrate an alternative bio-inspired mechanical pathway, which is based on exploiting the nonlinearity that would arise from a functionally graded (FG) distribution of biomimetic scales on the surface using an analytical approach. This functional gradation is geometrically sourced and could arise from either spatial or angular gradation of scales. We analyze such FG cantilever beams under different loading conditions including point loading at the free end, uniform traction, linearly distributed traction, and concentrated moment loading at the free end. In comparison with uniformly distributed scales for all cases of the loading addressed, we find significant differences in bending stiffness for both spatial and angular gradations. Spatial and angular functional gradations share some universality but also sharp contrasts in their effect on the underlying beam. We also quantify the landscape of spatio-angular tailorability on stiffness gains. We compare our models with select experiments for validation. This highlights that a combination of both types of gradation in the structure can be used to alter stiffness and therefore offer a pathway to tailor the elasticity of a cantilever beam relatively easily. These results demonstrate an architected framework for designing and optimizing scale-covered FG beams.

Introduction

Cantilevered beams arise in a number of diverse engineering applications spanning an enormous variety of length scales [1,2,3,4,5,6,7,8,9,10]. In several of these applications, tailoring elasticity can be of tremendous significance since it can be used to design the response according to stimulus or guard against unwanted instabilities. To this end, functional gradation (FG) is a useful strategy. FG materials are high-performance composite materials consisting of two or more constituent phases with variation in composition. This gradation can lead to a desired enhancement in the thermal/mechanical properties, compared with their conventional counterparts. This makes them ideal for various engineering applications including biomedical [11, 12], cellular structures [13,14,15,16,17,18], soft robotics [19, 20], and several others [21, 22].

However, typical FG materials could be difficult to fabricate requiring extensive materials processing such as directional solidification [23], specialized machining paths, or even additive manufacturing [23, 24]. An alternative exists in pursuing surface-based strategy such as biomimetic scales. Such dermal scales are a pervasive feature within Kingdom Animalia. Their advantages extend well into a variety of important functions, which enhance survivability, such as protection, camouflaging, and locomotion [25,26,27,28,29]. In nature, certain fishes possess remarkably periodic scale distribution, for instance, Elasmobranchs [30, 31] and Teleosts [32,33,34,35]. However, more often, organisms display a large variation in scale distribution within their own bodies [36,37,38,39]. This is primarily due to both physiological factors [40,41,42] and in response to the functional requirements [43,44,45] resulting in varying density of scales and even their arrangements. Members of the order Squamata and Crocodilia display axial functional scale gradation, to improve locomotion [43]. Their scale density increases near the head and tail in order to enable a better spatial mobility [43]. This gradation also varies among species, for instance snakes, boas, pythons, and many vipers have small irregularly-arranged head scales in contrast to the large systematic head scales of most advanced snakes [43]. Interestingly, scaly features with functional gradation are also common in hair and furs of mammals [46].

At the laboratory scale, biomimetic scale–covered structures can be fabricated by embedding plate-like structures into a soft substrate [47,48,49]. Extensive work has been conducted on the kinematics and mechanics of biomimetic scale–covered structures from the point of view of localized loading such as armor applications [50] as well as fostering nonlinearities in global deformations in bending and twisting stemming from scales engagement [48, 51,52,53,54,55,56,57]. In these studies, the uniformly distributed scales are considered. This considerably simplifies the design and analysis but unfortunately also precludes the many advantages that come from functional gradation.

In this paper, we study the effect of spatio-angular FG of scales in altering the behavior of biomimetic scale–covered cantilevered beam under different loading conditions including point loading at the free end, uniform traction, linearly distributed traction, and concentrated moment loading at the free end. Motivated by purely qualitative experiments, we investigate the behavior in detail using an analytical model aided by computational investigations. Our results disclose significant differences in bending stiffness for both spatial and angular gradations for all cases of the loading addressed. We found that there are some striking similarities among traction or point loading conditions. Interestingly, for concentrated moment loads, we find the behavior to be qualitatively different. This is in contrast to traditional FG or composite materials. We also quantify and compare the combined spatio-angular tailorability for all four load cases to elucidate the full landscape of variability.

Material and methods

Kinematics of cantilever FG biomimetic scale-covered beams

Scales on the surface can demonstrate functional gradation in two distinct geometrical variables—spatial (scale distribution) and angular (scale inclination) (Fig. 1a). We demonstrate the effect of these gains using qualitative experiments of beam deflection (Fig. 1 c and d). These figures illustrate a noticeable difference in deflection between two similar FG biomimetic scale–covered samples, one with uniform distribution and the other linear FG, under their own self weight. These samples were fabricated through adhering 3D printed scales (polylactic acid (PLA)) with an elastomer known as vinylpolysiloxane (VPS). Both samples were clamped at one side. The deflection was measured due to the weight of the fabricated samples. The FG scale sample demonstrates a decrease of 30% in deflection, compared with uniform scaly sample (a direct observation from the scaled mat).

Fig. 1
figure1

a Three biomimetic scale covered beams (from top to bottom) in which scales are uniformly distributed (top), linearly placed along the beam (middle), and uniformly distributed with linear variation in angle along the beam (bottom). b A 3D SolidWorks model of a substrate with embedded scales arranged linearly along the length of the beam and a schematic diagram of two adjacent scales. c The deflection of a cantilever biomimetic scale covered beam (with scales equally spaced) due to beams self-weight. d A self-weight deflection of a cantilever biomimetic scale covered beam in which the scales are linearly distributed with the same initial inclination angle 𝜃0 = 5. The dimensions of the beam and scales are 220 mm (length) × 25 mm (width) × 10 mm (height) and 35 mm × 25 mm × 1 mm, respectively

For our analytical model, the FG biomimetic scale–covered beam is composed of an underlying substrate and partially embedded scales on its surface (Fig. 1b). The substrate is of length LB and height h, while each individual scale of thickness D is consisted of an exposed part of length l and embedded portion L, which makes the total length of scale be ls = l + L. We refer to the overlapping ratio of scales as η = l/d, where d is the spacing between two adjacent scales [48]. Note that as we introduce spatial gradation, η will vary with position. In such FG beams, all scales would start from an initial inclination angle 𝜃0 measured with respect to the beam centerline and increase after engagement in a highly nonlinear fashion [48, 51, 57]. We measured the Young’s modulus of the of VPS and PLA via tensile test using an MTS Insight®; and found them to be 1.5 MPa and 2.86 GPa, respectively.

This high contrast in the modulus along with the assumption Dls and Lh (Shallow embedding) allows us to model the scales as rigid plates embedded in a semi-infinite elastic media. Accordingly, the scales rotation is modeled as a linear torsional spring [48, 51]. We further impose small strains in the beam, and hence Euler-Bernoulli assumptions remain applicable.

We start our model by first acknowledging the lack of global periodicity in FG scaled systems due to the nonperiodic engagement of scales and nonuniformity of bending caused by the boundary and loading conditions [57]. This prevents us from using the same form of kinematic relationship developed earlier [48, 51]. Also, addressing the kinematics of the FG system is considerably more challenging than uniformly distributed scale case considered earlier in literature which relied on periodic engagement [48, 51, 55]. We therefore use an alternative approach developed more recently [57] which can better capture the nonuniformity that arises in FG system. We impose a deformation on the underlying substrate in the form y(x) = γf(x), where f(x) is a shape function and γ is a dimensionless constant determined by the beam material, geometry, and load. For a beam without scales, for a given loading condition, the shape of the deflected beam would be a function y(x). For instance, for a cantilever beam of bending rigidity EI , length LB, and subjected to a point load of p0 at its free end, the deflection is given by y(x) = p0h/6EI(3LBx2x3). As the load is increased, the deflection also increases but the overall shape, which is captured by the function f(x) = 3LBx2x3, remains the same. Therefore, we can isolate the loading effect on deflection using a form y(x) = γf(x) where γ = p0h/6EI is the normalized load factor. Clearly, for different boundary condition or loads, we will have different γ and f(x). This approach is similar to the one commonly used in literature [58, 59]. Next , we assume that in spite of adding the scales, the corresponding shape function would not change for the given loading but instead would only vary in magnitude. This would result in our scale-covered FG beam deflection to be y(x) = γf(x) with a different γ. The engagement of scales can then be tracked by means of the distance parameter Δi. The distance parameter is the perpendicular distance between the right extremity of one of the scales to the adjacent scale (see Fig. 2a). This distance parameter can be written as [57]:

$$ {\varDelta}_{i}= {1 \over l}((y_{i + 1}^{L} - y_{i + 1}^{R})(x_{i + 1}^{L} - {x_{i}^{R}}) - (x_{i + 1}^{L} - x_{i + 1}^{R})(y_{i + 1}^{L} - {y_{i}^{R}})),i = 1,..,{N_{s}} - 1 $$
(1)

where Ns is the total number of scales, \({x_{i}^{L}}\) and \({y_{i}^{L}}\) refer to the position of the left extremity of the i th scale, measured with respect to the global coordinates (Fig. 2a), while \({x_{i}^{R}}\) and \({y_{i}^{R}}\) denote the position of right extremity of the i th scale, also measured with respect to the global coordinates (Fig. 2a). The engagement occurs when Δi ≤ 0 where i = 1,..., Ne, and Ne would be the number of scales in contact. This results in a set of 3Ne − 2 nonlinear equations after the condition of scales interaction is met. These equations consist of constraints on the fixed length of scales, the geometry of engagement, and balance of moments about the base of all engaged scales (Fig. 2b). In general, the unknowns after scales engagement will be 𝜃i, i = 1 : Ne and \({x_{i}^{R}},{y_{i}^{R}}, i=1:2N_{e}-2\) to obtain the equilibrium configuration of the scale-covered structure. Consequently, the derived equations consist of Ne − 1 equations based on the fixed length of rigid scale (similar to Eq.2) , 2Ne − 2 equations derived based on the slope of each scale (similar to Eqs. 3 and 4), and one equation based on the moment balance. The last equation is derived from the moment balance of the two engaged scales located at the far right. Thus, balancing the moment about points B and C(Fig.2b) yields Eq.5 that can be utilized for finding the normal force between any two consecutive scales, except the case when Ne = 2 or i = 1, for which \({N_{i}} = {{{K_{B}}\left ({{\theta _{i}} - {\theta _{0}}} \right )} \over {l\cos \limits \left ({{\alpha _{i}}} \right )}} = {{{K_{B}}\left ({{\theta _{i + 1}} - {\theta _{0}}} \right )} \over {{r_{i}}}}\), where KB is the spring constant of the linear torsional spring (rigid scales rotation) which models the resistance of the substrate to rotation of the embedded scale. The stiffness KB is analytically approximated as KB = CBED2(L/D)n where E is the modulus of elasticity of the substrate and CB, n are constants with values 0.86, 1.75, respectively [48, 57]. Note that αi = 𝜃i+ 1 + ψi+ 1𝜃iψi and \({r_{i}} = \sqrt {{{\left ({{x_{i}^{R}} - {x_{i + 1}^{L}}} \right )}^{2}} + {{\left ({{y_{i}^{R}} - \gamma f({x_{i + 1}^{L}})} \right )}^{2}}} \) with i = 1 : Ns − 1. For more details about the derivation of these equations, the reader is referred to [57].

$$ {\left( {{x_{i}^{R}} - {{x_{i}^{L}}}} \right)^{2}} + {\left( {{y_{i}^{R}} - \gamma f({{x_{i}^{L}}})} \right)^{2}} = {l^{2}}. $$
(2)
$$ \tan \left( {{\theta_{i}} + {\psi_{i}}} \right) = {{{y_{i}^{R}} - \gamma f\left( {{{x_{i}^{L}}}} \right)} \over {{x_{i}^{R}} - {{x_{i}^{L}}}}}, $$
(3)
$$ \tan \left( {{\theta_{i + 1}} + {\psi_{i + 1}}} \right) = {{{y_{i}^{R}} - \gamma f\left( {{x_{i + 1}^{L}}} \right)} \over {{x_{i}^{R}} - {x_{i + 1}^{L}}}}, \text{and} $$
(4)
$$ {N_{i + 1}} = {{{K_{B}}\left( {{\theta_{i + 1}} - {\theta_{0}}} \right) + {N_{i}}{r_{i}}} \over {l\cos\left( {{\alpha_{i + 1}}} \right)}} = {{{K_{B}}\left( {{\theta_{i + 2}} - {\theta_{0}}} \right)} \over {{r_{i + 1}}}}. $$
(5)
Fig. 2
figure2

a The geometry of scales before engagement at a configuration of a deformed underlying substrate. b The FBD of a scale when it is in contact with two neighboring scales shown as the dotted lines for clarity (adapted from [57]). Here, subscript index refers to scale number, superscript R and L are right and left side of a scale, respectively, N is the normal reaction between neighboring scales, Δi is the perpendicular distance between adjacent scales, ψ is the angular rotation of the substrate at the base location of scales, and finally 𝜃 denotes the inclination angle of scales

The above nonlinear equations are then solved numerically to ensure equilibrium at each step of deformation. The outcome of solving these equations is the orientation of each scale as the underlying substrate progressively deforms into an arc.

Mechanics of cantilever FG biomimetic scale-covered beams

The resultant bending mode of FG system can be envisioned as a combination of substrate deformation and scales rotation. The strain energy due to the deformation of the underlying substrate can be written as \({\varOmega }_{B}= {\int \limits }_{0}^{L_{B}} {1 \over 2} EI\kappa ^{2} dx\), where EI is the bending rigidity of the beam and κ is the instantaneous curvature. The scales energy is modeled as \({\varOmega }_{scales}={\sum }_{i=1}^{N_{e}}{1\over 2} K_{B} (\theta _{i}-\theta _{0} )^{2}\). Here, 𝜃i is the scales rotational displacement evaluated using the kinematic approach illustrated earlier, Ne is the number of scales in contact, and KB is again the spring constant of the linear torsional spring (rigid scales rotation). The total potential energy is π = Ωbeam + ΩscalesH(Δi) − W where W is the external work done by the applied load and H(Δi) is the Heaviside step function. The final deflection shape is obtained using energy minimization principle based on γ [59]. In the differential form, the variation in energy can now be written as \({d{\varOmega }_{beam}\over d\gamma }+{d{\varOmega }_{scales}\over d\gamma } H({\varDelta }_{i} )={dW \over d\gamma }\), which allows us to compute the beam deflection under given load. Here, we study the effect of beam deflection under four cases of loading conditions. We start with a point load at the free end of the beam, p0. For this case as mentioned earlier, the deformation of the plain beam is \(y(x)={{p_{0} h^{2}}\over {6EI}} (3L_{B}x^{2}-x^{3})\) [58], while the work done is W = p0y(LB). Here \(\gamma = {p_{0} h^{2}\over 6EI}\) and the tip deflection is \(y_{tip} = {{p_{0}{L_{B}^{3}}}\over {3EI}}\). Once biomimetic scales start interacting during deformation, they add stiffness to the structure, which would require an additional amount of load to obtain an equivalent deflection similar to a plain beam. This equivalent load is derived utilizing the variational-energetic equation described earlier and can be written as:

$$ p = {p_{0}} + {h^{2} \over {{2{L_{B}^{3}}}}}\sum\limits_{i = 1}^{{N_{e}}} {k_{B}}\left( {\theta_{i} - {\theta_{0}}} \right){{d\theta } \over {d\gamma }}. $$
(6)

The second loading case is a uniformly distributed traction, w0 across the span of the beam. For this case, the deformation of the plain beam is \(y(x)={{w_{0} h^{3}}\over {24EI}} (4L_{B}x^{3}-x^{4}-6{L_{B}^{2}} x^{2} )\) [58], while the work done is \(W = {\int \limits }_{0}^{L_{B}} {w_{o}}y(x)dx\). This indicates that \(\gamma = {{w_{0} h^{3}}\over {24EI}}\) and the tip deflection is \(y_{tip} = {{w_{0}{L_{B}^{4}}}\over {8EI}}\). We now use the variational-energetic equation to obtain an equivalent load (only applicable after scales engagement), which results in a deflection equivalent to unscaled beam under the same type of loading. The equivalent load is

$$ w = {w_{0}} + {5h^{3} \over {{6{L_{B}^{5}}}}}\sum\limits_{i = 1}^{{N_{e}}} {k_{B}}\left( {\theta_{i} - {\theta_{0}}} \right){{d\theta } \over {d\gamma }}. $$
(7)

Another loading case investigated in this work is a linearly distributed traction, v0, across the span of the beam. This loading causes a plain beam to deform according to the formula \(y(x)={{v_{0} h^{3}}\over {120EI}} (10{L_{B}^{2}}x^{2}-10L_{B}x^{3}+5x^{4}-{x^{5} \over L_{B}})\) [58], and the work done is \(W = {\int \limits }_{0}^{L_{B}} v_{o}(1-{x \over L_{B}})y(x)dx\). Here, \(\gamma ={{v_{0} h^{3}}\over {120EI}}\) and the tip deflection is \(y_{tip} = {{v_{0}{L_{B}^{4}}}\over {30EI}}\). The equivalent load after scales interaction for such loading type can be expressed as

$$ v = {v_{0}} + {2.1h^{3} \over {{{L_{B}^{5}}}}}\sum\limits_{i = 1}^{{N_{e}}} {k_{B}}\left( {\theta_{i} - {\theta_{0}}} \right){{d\theta } \over {d\gamma }}. $$
(8)

The last loading case addressed is a concentrated moment loading at the free end of the scaled beam, M0. The deformation of a plain beam under such type of loading is \(y(x)={{M_{0} h}\over {2EI}} (x^{2} )\) [58], and the work done is \(W = {\int \limits }_{0}^{{\kappa }^{\prime }} {M_{o}}d\kappa \) where \(\kappa = {{d^{2}y(x)} \over {dx^{2}}}\). These formulas imply that \(\gamma ={{M_{0} h}\over {2EI}}\) and the tip deflection is \(y_{tip} = {{M_{0}{L_{B}^{2}}}\over {2EI}}\). Scales interaction causes an increase in the stiffness of the underlying substrate as compared with a plain beam. Thus, the equivalent moment required to match the deflection of a plain beam after scales engagement is derived as:

$$ M = {M_{0}} + {h \over {{L_{B}}}}\sum\limits_{i = 1}^{{N_{e}}} {k_{B}}\left( {\theta_{i} - {\theta_{0}}} \right){{d\theta } \over {d\gamma }}. $$
(9)

Note that for all four loading cases investigated, d𝜃/dγ is numerically evaluated using central finite difference technique. Additionally, the tip deflection of the beam will deviate from linearity due to the highly nonlinear regime brought about scales sliding [48]. Equations 678, and 9 now serve as a proxy to track the tip deflection of the cantilever FG biomimetic scale–covered beam when scales sliding commences. For this paper, we fix the beam geometry parameters as LB = 1000 mm and h = 50 mm, while the scales parameters as L = 7 mm, ls = 250, 𝜃0 = 3, and D = 0.1 mm. The modulus of elasticity of the beam is taken to be E = 1.5 MPa.

Verification

The results of this work are numerically verified through carrying out finite element (FE) simulations utilizing ABAQUS (Dassault systems). In our model, we made an assembly of two parts, substrate and scales, both of 2D deformable shell type. Thereafter, we impose rigidity on the scales obviating any need for material property on scales. Thus, our model consists of a linear elastic substrate (with isotropic material properties E = 1.5 MPa and Poisson ratio ν = 0.42) and embedded rigid scales. We used a static linear step to match the corresponding kinematics of Euler-Bernoulli beam. The contact was employed via self-contact option through the entire geometry and then imposing frictionless sliding between every two neighboring scales along with node to surface discretization method to prevent any penetration between any contacted scales [60]. The load was assigned to the unscaled side of the beam based on the loading condition. For instance, the case of point loading was employed through applying a force in Y-direction to the right edge, uniform and linearly traction cases were performed as force per unit length to the bottom edge, and the case of concentrated moment was conducted via applying a moment to the mid-point of the right edge. The boundary conditions were the same for all four cases of loading and applied by constraining the displacement of the built-in edge of the beam, Ux, y, z = 0. Mesh convergence was obtained via varying the global size parameter [60] of our beam starting from 5 with progressive reduction to 1 for sufficient mesh convergence. We further used two types of elements, in the family of plane stress with quadratic geometric order, due to the complexity of top surface of our scaled beam. Biquadratic element CPS8 [60] was used for the area that is far from the embedded scales part and a triangular quadratic element CPS6M [60] for the other areas. The total number of elements in the entire model were 59312 for CPS8 and 10346 for CPS6M.

Results and discussion

To investigate the interplay of FG type with loading, we calculate the tip deflection with four different types of loading conditions—point load, uniform traction, linearly decreasing traction, and concentrated tip moment. These loads are moment-equivalent normalized. In other words, the total applied moment at the built-in end of the cantilevers is considered to be same for all loading cases. This leads to, for the case of point load \(\bar {P} = 2p_{0} L_{B} h/3EI\) where p0 is force magnitude, for uniform traction \( \bar {q} = w_{0} {L_{B}^{2}} h/4EI\) where w0 is applied traction, for linearly decreasing traction \(\bar {q}_{0}=v_{0} {L_{B}^{2}} h/15EI\) where v0 is the maximum traction, and for moment \(\bar {M} = M_{0} h/EI\) where M0 is the magnitude of applied moment.

Interplay of loading and spatial scale distribution

In this section, we investigate the interplay between four distinct loads on the cantilever with the form of functional gradation. For comparison, we use four different classes of distribution functions, exponential increasing (from the cantilever base), logarithmic decreasing, linear increasing, and a nonmonotonic sinusoidal. The spatial position of the first scale (x0) and the last scale (xf) in the reference configurations are fixed for all functions at \({x_{0} \over L_{B}} =0.01\) and \({x_{f} \over L_{B}} =0.98\), respectively. The total number of scales has been fixed to Ns = 20 scales and recall that LB = 1000 mm. Using these information, the position of the i th scale for the exponentially increasing function would be xi = aebi where a = 7.85 mm and b = 0.24. Similarly for the logarithmic decreasing the position function would be \(x_{i}=x_{0}+a \log i\) with a = 323.8 mm. For the linear increasing, the position function would be xi = x0 + d0 + α(i − 2) where d0 = 4mm and α = 5.11 mm and i > 1. Finally, for the sinusoidal function, \(x_{i}=a \sin \limits {({{\pi i}\over {N_{s}}})}+b\) where a = 569 mm and b = − 79 mm. We compare these functional gradation functions with a uniform beam with same number of scales. We use these information to plot the normalized tip deflection versus the normalized applied load for the functions and loading cases, as shown in Fig. 3. The dark green dots in the figure illustrate the results from FE simulations for select cases, which are an excellent match with our analytical results. Clearly, having more scales near the built-in end (exponential and linear) significantly increases the stiffness when compared with a uniform scale distribution, whereas the stiffness is lesser than the uniform case for logarithmically position function, which has more scales near the free end. For the nonmonotonic sinusoidal distribution, we find the stiffness to be lesser than the uniform scale distribution case but more than monotonic decreasing logarithmic function. All of these cases show more stiffness when compared with unscaled beam as expected. This is true for all loads.

Fig. 3
figure3

The tip deflection (horizontal axis) of a FG cantilever biomimetic scale–covered beam (with 20 scales) normalized by the thickness of the beam under different loading conditions. The function utilized to distribute scales along the beam starting from the fixed edge (see inset) is described textually for each curve. The nondimensionalized loading cases are a point load at the free end of the beam, b uniformly distributed load, c linearly distributed load, and d concentrated moment at the free end. In all four cases, the green dots represent the FE simulations. Also, the dimensions of the beam and scales are 1000 mm (length) × 50 mm (width) × 1 mm (height) and 250 mm × 1 mm × 0.1 mm, respectively while the inclination angle is 𝜃0 = 3

However, there is a perceptible difference between the point load/traction (Fig. 3a–c) and moment case (Fig 3d) in terms of relative difference in stiffness. For the moment case, the divergence between the functions which have more scales at the built-in end (exponential and linear) is much lesser. Similarly, the divergence in stiffness between logarithmic, which has more scales at the free end and plain beam (no scales), is also much higher for the moment case when compared with the traction. Also, note that the stiffness gain for nonmonotonic (sinusoid) is very close to the uniform distribution scale beam. This stark difference in behavior is because of the differences in the kinematics of scale engagement due to the fundamentally different loading types. The concentrated moment produces a different type of deformation (more uniform curvature) than the point loads (Fig 3a) and traction (Fig. 3b, c) resulting in different post-engagement kinematics. These similarities and divergences in behavior draw an important distinction from traditional material. Here, the role of load and the distribution interplay critically in these materials and can be used for various degrees of tailorability.

Effect of spatial gradient on beam deflections

In this section, we focus on the role of spatial gradient on the deformation behavior of the cantilever beam. For brevity, we chose linear FG which yields a single scalar gradient parameter. This allows us to investigate the two diverging cases—mononotic increasing scale spacing (positive gradient) and monotonic decreasing scale spacing (negative gradient) using a single scalar index, that of the gradient. A linear-spatial gradation of scales is imposed in the form of di = d0 + δd(i − 2), i > 1, where d0 is the spacing between the first (left side of the beam) and the second scale, index i denotes the scale number, and δd is the gradient and could be positive or negative. Note that δd = 0 corresponds to uniform distribution of scales. We first investigate the load-displacement relationship (Fig. 4), for various load conditions. We find that positive gradient leads to greater stiffness gains when compared with either negative gradient or uniform scale beams for all load cases. Similarly, negative gradients lead to loss of stiffness for all load cases when compared with the uniform scale distribution. For the moment loading case (Fig. 4d), we again find significant more gains in stiffness when compared with plain beams for all types of scale distribution. This is primarily due to the difference in the deformation behavior in the moment loading case. In this case, the relative differences in gradations are muted (see Fig. 4d), when compared with the point load (Fig. 4a), and/or traction (Fig. 4b, c) cases. The relative nonuniformity in loading did not much affect the load-displacement behavior as seen in the figures.

Fig. 4
figure4

Nondimensional tip deflection (horizontal axis) of a FG cantilever biomimetic scale–covered beam (with 20 scales) under different loading conditions. The scales are distributed linearly with different linear gradation. The nondimensionalized loading cases are a point load at the free end of the beam, b uniformly distributed load, c linearly distributed load, and d concentrated moment at the free end. In all four cases, the dimensions of the beam and scales are 1000 mm (length) × 50 mm (width) × 1 mm (height) and 250 mm × 1 mm × 0.1 mm, respectively while the inclination angle is 𝜃0 = 3

These differences can be further probed for different loads to investigate the role of load in mediating the deformation response of FG beam. This is investigated in Fig. 5. Here, we plot the normalized tip deflection with spatial gradient of scales for increasing load intensity for all four loading cases. These plots confirm that higher positive gradients lead to greater stiffness (lower tip deflection) when compared with negative gradients. This is true for all load cases shown in Fig. 5. We find that the tailorability (positive/negative gradient contrast) through FG is relatively low for lower loads and increases sharply as loads are increased. In this, there is also significant difference between point loads, traction, and moment loading. For moment loading (Fig. 5d), the effect of FG on enhancing stiffness yields diminishing returns for increasing positive gradients. This indicates that the kinematics clearly dictate FG behavior, even for these relatively small deflections. At the same time, the relative similarity of deflections can be seen in Fig. 5 a, b, and c.

Fig. 5
figure5

Nondimensional tip deflection of a FG cantilever biomimetic scale–covered beam (with 20 scales) for all possible distribution of linear gradation of scales under different loading cases. The nondimensionalized loading cases are a point load at the free end of the beam, b uniformly distributed load, c linearly distributed load, and d concentrated moment at the free end. In all four cases, the dimensions of the beam and scales are 1000 mm (length) × 50 mm (width) × 1 mm (height) and 250 mm × 1 mm × 0.1 mm, respectively while the inclination angle is here fixed 𝜃0 = 3

Effect of angular gradient on beam deflections

Angular gradation effect on cantilever beams is considered via imposing a linear variation of the initial inclination angle of scales. This is expressed through 𝜃0i = 𝜃01 + δ𝜃(i − 1) where 𝜃01 is the initial inclination angle of the first scale (again left side of the beam), δ𝜃 is the gradient and could be positive or negative, and i is the scale number. Here, we address all possible choices of varying scales orientation in the range of 3𝜃0 ≤ 25. Note that we investigate only a small range of inclination angles to assure interaction of scales. We plot the tip deflection of the FG cantilever beams for different angular gradations (Fig. 6). We find that for all types of loading, there is a broad similarity in behavior for negative gradation. The tip deflections do not change as long as the gradation is negative for all load magnitudes (Fig. 6). There is a sharp change in tip deflection as the gradient becomes positive. This is the case the scales are closer to the surface than the end. This sharp divergence in behavior, universal across loading conditions, is a result of initial successful engagement of scales near the built-in end. Sparse scales simply do not engage enough at the higher end of curvature (built-in end) to affect the tip deflection. As the scales are made to be closer to the surface of the substrate towards the built-in end, they engage much earlier as deflection proceeds and increases stiffness of the beam. However, again in spite of the similarity, there is also noted divergence between point load/traction (Fig. 6a–c) and moment loading (Fig. 6d). The stiffness gains from positive FG is much more pronounced for the moment case due to uniform curvature throughout the span of the beam.

Fig. 6
figure6

Nondimensionalized tip deflection of a FG cantilever biomimetic scale–covered beam (with 20 scales equally spaced) for possible angular gradation of scales in the range 𝜃0 ≥ 3 under different loading conditions. The nondimensionalized loading cases are a point load at the free end of the beam, b uniformly distributed load, c linearly distributed load, and d concentrated moment at the free end. In all four cases, the dimensions of the beam and scales are 1000 mm (length) × 50 mm (width) × 1 mm (height) and 250 mm × 1 mm × 0.1 mm, respectively while the spacing between scales is d = 51 mm

Therefore, angular functional gradation offers interesting study in contrast. On one hand, angular gradation is much easier from a fabrication point of view and can result in dramatic contrasts in mechanical response in FG beams. However, for higher scale angles, the engagement occurs at a much later stage and geometric and material nonlinearities might be overpowering. This is also one of the key limitation of angular gradation. The envelope of tailorability is in principle high with angular gradation but at the same time, the overall bracket of angular variation is limited since very high angles are not that useful due to late engagement. In addition, angles also determine the thresholds of engagement. Therefore, if scales are at higher angles where the curvatures are high, their gradient effects are minimal. This can be clearly seen in Fig. 6. This makes the mechanics of this system rich in geometrically dictated nonlinearities.

Spatio-angular gradation of scales

The effect of combining both spatial and angular gradations on the bending stiffness of our cantilever FG biomimetic scale–covered beam is presented here. We map the landscape of stiffness over both spatial and angular orientations using a phase diagram for each loading case (Fig. 7). Here, we track the nondimensional tip deflection of our FG cantilever scaled beam for positive gradients. Note that negative gradients δd < 0 is not considered here because they do not contribute to decreasing deflection with regard to uniformly scaled beam. The phase diagrams clearly highlight the entire landscape of stiffness gains from both spatial and angular gradation of scales. They are continuous with increase in angular gradations for point load/traction cases (Fig. 7a–c). For the moment loading case (Fig. 7d), there is an initial transition to high stiffness for angular FG and then no further gains take place. For all load cases, higher spatial gradation which means more scales at the built-in end leads to greater gains in stiffness. However, these gains are most pronounced for point load/traction conditions. This shows a very interesting pattern, that is moment loading is sensitive to angular gradations whereas point load/traction respond to both spatial and angular FG. By controlling the spatio-angular gradients, a rich landscape of tailorable behavior can be possible.

Fig. 7
figure7

Contour plot of the nondimensionalized tip deflection of a FG cantilever biomimetic scale–covered beam (with 20 scales ) spanned by positive linear spacing and linear angular gradation of scales in the range 𝜃0 ≥ 3 under different loading conditions. The nondimensionalized loading cases are a point load at the free end of the beam, \(\overline P =2 \times 10^{-2} \), b uniformly distributed load, \(\overline q =2 \times 10^{-2} \), c linearly decreasing load, \(\overline q_{0}=2 \times 10^{-2} \), and d concentrated moment at the free end, \(\overline M =2 \times 10^{-2} \). In all four cases, the dimensions of the beam and scales are 1000 mm (length) × 50 mm (width) × 1 mm (height) and 250 mm × 1 mm × 0.1 mm, respectively

Experemental validation

To validate both theoretical and numerical simulation results, we have performed point load experimental tests using an MTS Insight®;. We connected a 220 N high-performance braided fishing line to a 100 N load cell (Fig. 8a) to simulate point load. The MTS was used to obtain load-displacement characteristic for two types of scales covered beams—uniform distribution and Linear FG. The substrates were 3D printed using an Ultimaker S5 with a flexible filament known as TPU 95A (thermoplastic polyurethane) while the scales were thin sheets of galvanized steel plates (with Young’s modulus of 200 GPa) to ensure rigidity in scales as compared with the substrate. The sample geometries were rectangular with dimensions 200 mm (length) × 25 mm (width) × 10 mm (height) for the substrate and 350 mm × 25 mm × 1 mm for the scales. The Young’s modulus of TPU 95A was measured via carrying out cantilever bending tests in small deformation using MTS Insight®;, and found to be 30 MPa using the strength of material \(k = {{3EI}\over {{L_{B}^{3}}}}\). The scales and substrate were firmly adhered using super glue cyanoacrylate. Furthermore, the exposed parts of the scales, which would slide over each other, were covered using a Scotch tape and wiped with oil to minimize frictional effects during sliding between scales. The results with experimental error bars as uncertainties from different tests are illustrated in Fig. 8b, which plots normalized load \(\bar {P} = {{2p_{0} L_{B} h}\over {3EI}}\) vs tip deflection. The figure shows a slight difference in the initial slope between the plain beam and the two cases of scales covered beams even before scales engagement, indicating a difference in effective stiffness of the sample. This deviation in stiffness is mainly due to the embedded part of scales, which add stiffness to the substrate before scales interaction and has been discussed earlier in literature [54, 56]. We see a broadly similar trends in the response between our theory and the experiment. To resolve the issue of the initial stiffness due to inclusion effect of the embedded parts, we use an effective modulus for the structure based on the scales volume fraction ϕ, previously presented in literature [54]. For a small volume fraction ϕ ≤ 0.25, the effective modulus is approximately Eeff = E(4ϕ + 1). In our case, ϕ = 0.0780 which leads to Eeff ≈ 1.3E. Note that the increase in stiffness was not included in our FE simulations because the scale volume fraction was (theoretical limit of very small volume fraction), ϕ = 2.8 × 10− 4 to highlight stiffness gains solely from geometrical effects. The results when these inclusion effects are considered are presented in Fig. 8c. We observe excellent agreement between our theory and the experimental results in the presentation of the load-displacement characteristics. Thus, our analytical model could be used for carrying out design as well as understanding the mechanism for synthesizing substrates with tailorable elasticity.

Fig. 8
figure8

a An experimental setup for a point load deflection of scales covered FG cantilever beam using an MTS Insight®;. (Top) perspective view and (bottom) side view of the same experiment. b The tip deflection (horizontal axis) of a plain cantilever beam and FG cantilever biomimetic scale covered beam (with 13 scales) normalized by the thickness of the beam under a point loading applied at the free end of the beam. The results are obtained for the case E = 30 MPa. c A similar plot to Fig. 8b but here Eeff = 1.3E. In all figures, the spacing between scales for the uniform distribution is d = 13 mm while for FG is d0 = 8 mm and δd = − 1.36. Additionally, the inclination angle of scales is 𝜃0 = 10

Conclusions

In this work, we explore geometrical tailorability of elasticity brought about by controlling the distribution and orientation of scales on a slender substrate subject to different loading conditions. These include point load at the free end, uniform traction, linearly decreasing traction, and concentrated moment at the free end. We utilized a model, FE computations, and experiments in this work. We found that there were several similarities in mechanical responses between these loading. Subtle but important distinctions existed between concentrated moment and the other load cases indicating the critical interplay of deformation with scale kinematics. Although scales sliding would inevitably lead to substrate stiffening, the degree of stiffening is strongly dependent upon the sliding kinematics and the underlying substrate deformation, which is sensitive to specific loading types. Thus, if the scales are arranged in such a way as to alter the sliding kinematics, it changes the individual scales rotation with deformation. This change in scale rotation response leads to a change in the stored elastic energy from the scale-substrate elastic interaction. It is this phenomenon which gives tailorability to the system. Finally, we also quantified the entire landscape of stiffness tailorability dependent on both spatial and angular FG. These two FG strategies yielded different types of tailorabilities indicating the many possibilities for structural design.

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Correspondence to Ranajay Ghosh.

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Ali, H., Ebrahimi, H. & Ghosh, R. Tailorable elasticity of cantilever using spatio-angular functionally graded biomimetic scales. Mech Soft Mater 1, 10 (2019). https://doi.org/10.1007/s42558-019-0012-2

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Keywords

  • Biomimetic scales
  • Functionally graded material
  • Nonlinear elasticity
  • Stiffness tailorability