Investigation of mechanical responses to the tactile perception of surfaces with different textures using the finite element method

Abstract

Tactile perception is essential for humans to perceive the world, and it usually results in mechanical responses from the finger. In this study, a nonlinear, viscoelastic, and multilayered finite element model of the finger was developed. The relationship between the mechanical responses within the finger and tactile perception while the finger scanned different surface textures was studied. The results showed that the sensitivity of tactile perception is affected by the peak value of von Mises stress, which is itself determined by the shape and density of a given texture. The von Mises stress varies periodically with time, and this variation depends on the periodicity of the texture. Displacement signals around Pacinian corpuscles have periodic variation. The period of displacement decreases as the density of the texture increases. The spectral centroid increases as the spacing of the texture decreases. The related mechanisms are discussed in this article.

Introduction

In daily life, people perceive and communicate with the world in many ways, such as through vision, hearing, taste, touch, and smell. Early in the 1990s, Srinivasan et al., ¹ a pre-eminent leader in the field of tactile perception, proposed that tactile perception results from the combination of a stimulus pattern applied to the skin and the mechanical responses of receptors to the stimulus. As shown in Figure 1, mechanical receptors and nerve endings are distributed throughout the layers of the skin, which are critical for human tactile perception. The friction between the skin and a contact surface exerts a mechanical stimulus on the skin (e.g. vibration, compression, and/or tension) that mechanical receptors within the skin layers convert into an afferent impulse within their corresponding nerve endings. Then, through a route comprising the nerve endings, spine, brain stem, and cerebral cortex, this tactile information is processed by the central nervous system, resulting in tactile perception. In this way, humans can recognize the geometry, hardness, humidity, and texture of objects. ²

Figure 1.

Schematics illustrating the process of human tactile perception when the finger is moving on a surface.

The finger is an important tool for tactile perception for humans, and it is one of the most sensitive parts of the human body. Because there is currently no direct method of measuring the stress or strain within the finger, researchers are developing reliable models to study the tactile responses of the finger to different stimuli.

Studies in the field of tactile biomechanics focus mainly on the relationship between tactile perception and mechanical responses within the skin when the skin is exposed to different external stimuli. Developing finger models is the most common strategy to research this relationship. In general, finger models can be classified into two categories: ³ physical models and structure models. Physical models are built based on quasi-viscoelastic constitutive equations, whereas structure models are mainly built using finite element methods.

Phillips and Johnson ⁴ simplified the fingertip to a homogeneous and isotropic elastic medium that is infinite in extension. The model is a smooth and flat surface, based on continuum mechanics theory and Hooke’s law. The deflection reported by this model closely approximates empirical observations in the cases of small deformations. However, it is clear that the skin is neither homogeneous nor isotonic, and according to Hooke’s law, the deflection reported by the model will be very large when the force becomes large enough. Srinivasan ² improved Phillips’ model and represented the fingertip as a flexible membrane with a negligible thickness enclosing an incompressible fluid (the “waterbed” model). This model has been applied to predict the deflection of the fingertip surface in response to static and linear loads. However, it cannot accurately predict a deflection profile 3 mm away from the load. Hu ⁵ and Maeno et al.^6,7 built a multilayer finite element model of the fingertip based on the material properties and geometrical parameters of the finger. The cross section of the fingertip in this model is a multilayered spheroid that includes the epidermis, dermis, subcutaneous tissue, and bone. Responses of the mechanical receptors in different skin layers to external stimuli were analyzed. Shao et al. ⁸ also built a two-dimensional finite element model of the finger. The variance of the stress within the epidermis and dermis with time and position was analyzed when the finger was scanned over a textured surface.

Katz ⁹ observed that coarse textures could be distinguished based on their static contours by simply pressing down on an object, whereas fine textures required a sliding motion to generate vibrations. Hollins et al. ¹⁰ and Akay ¹¹ supported the idea that lateral movements and vibrations play a critical role in the perception of fine textures. Zhang et al. ¹² conducted an experiment on the combination of friction signals and subjective touch sensation. A UMT-II multi-specimen tribometer was used to load different normal force values on the index finger to simulate touch force. When sample texture spacing was increased, volunteers made more accurate judgments of the texture spacing in different samples.

Previous models were rarely built based on both viscoelastic materials and dynamic perception of motion and were thus insufficient to fully characterize the mechanisms behind tactile perception. In this article, a nonlinear, viscoelastic, multilayered finite element model of the finger was developed. The influence of texture shape on the tactile perception of the finger was studied. Based on the responses of mechanical receptors to external stimulation, the mechanism of tactile perception was analyzed. This study advances our understanding of the perception mechanisms of skin in terms of different surface textures.

Methods

Finite element model of the finger

ABAQUS/CAE v6.12 was used to construct the nonlinear, viscoelastic, multilayered finite element model of the finger with fingerprints, as shown in Figure 2. The finger was set to have a width of 16 mm and a height of 12 mm, corresponding to the typical adult index finger. ¹³ The finger model was assumed to be composed of skin layers (corresponding to the epidermis and dermis of human skin), subcutaneous tissue, and bone. ² The thicknesses of the epidermis and dermis layers were assumed to be 0.7 ¹⁴ and 0.8 mm, ¹⁵ respectively. The stress relaxation function is defined using the Prony series in ABAQUS ¹⁶

$$g\left(t\right)=\frac{G\left(t\right)}{G_0}=1-\sum _{i=1}^{N_G}{g}_i(1-e^{-t/{\tau }_i})$$

(1)

where G₀ and G(t) are the instantaneous and time-dependent moduli, respectively; g_i and τ_i (i = 1, 2, 3, …, N_G) are the stress relaxation parameters, respectively; and N_G is the number of terms used in the stress relaxation function. Prony series parameters (g_i, τ_i) were obtained by nonlinear fitting of the g(t) curve. The viscoelastic properties of the finger were set according to these parameters.

Figure 2.

(a) Finite element model of the finger and (b) meshing and dimension of the finite element model.

Volume density function of strain energy Q can be used to characterize the hyperelastic deformation behavior of subcutaneous tissue, and the instantaneous elastic stress response can be obtained by the derivative of Q

$${\sigma }_{0,\mathrm{ij}}=\frac{2}{J}{Y}_{\mathrm{ir}}\frac{\partial Q}{\partial C_{\mathrm{rs}}}{Y}_{\mathrm{sj}}^T,i,j,r,s=1,2,3$$

(2)

where Y and C_rs are the deformation gradient tensor and the Cauchy–Green deformation tensor, respectively. Because the subcutaneous tissue of the finger can be observed as an almost incompressible elastomer, Q can be expressed as the nonlinear elastic deformation energy density of Ogden

$$Q=\sum _{i=1}^{N_n}\frac{2{\tau }_i}{{\alpha }_i^2}[{\epsilon }_1^{{\alpha }_i}+{\epsilon }_2^{{\alpha }_i}+{\epsilon }_3^{{\alpha }_i}-3+\frac{1}{\beta }(J^{-{\alpha }_i\beta }-1)]$$

(3)

where J = ε₁ε₂ε₃ is the change rate of volume, β = u/(l − 2u), u is Poisson’s ratio, ε_i (i = 1, 2, 3) is the major stretch rate, τ is the shear modulus, and α is the property constant of the material. In ABAQUS, hyperelasticity can be expressed by the Ogden model that requires the shear modulus τ and the property constant of material α to be inputted into ABAQUS.

Table 1 shows the material parameters of various skin layers and the perceived samples used in the simulation.

Table 1.

Material parameters of skin layers and perceived samples used in the simulation.^3,5,6

	Linear elastic		Hyperelastic		Viscoelastic
	Young’s modulus (MPa)	Poisson’s ratio	τ	α	gi		τi
					1	2	1	2
Epidermis	0.08	0.48	–	–	–	–	–	–
Dermis	0.05	0.48	–	–	0.148	0.252	2.123	9.371
Subcutaneous tissue	–	–	−4.895e−2	5.511	0.148	0.252	2.123	9.371
Bone	17000	0.3	–	–	–	–	–	–
Perceived sample	206	0.3	–	–	–	–	–	–

The plane strain element was used to mesh the model because deformation was negligible when the finger slid across the x-direction out of the modeled plane. The finger model had 2394 nodes and 1926 elements. The elements included 1880 linear quadrilateral elements (4 nodes quad linear reduced-integration element, CPS4R) and 46 linear triangular elements (3 nodes triangle linear element, CPS3). The algorithm of the medial axis was chosen during the mesh generation. Several smaller mesh sizes were tried, and each produced nearly the same results as the chosen mesh size, indicating the results were converging. Because the various skin layers were in contact with each other, binding constraints were applied between the skin layers. Coupled constraints of movement between the finger and the reference point were constructed by establishing a reference point RP-1 along the bone edge, as shown in Figure 2. Then, the load and boundary conditions were applied in terms of the reference point RP-1.

The simulation was performed in two steps: first, a static loading process (including two analysis steps) and then a dynamic process (including one analysis step). In the first analysis step, the perceived sample was fixed by constructing constraints for all degrees of freedom of its base line. The constraints of rotation and x-direction displacement of the finger were constructed through the reference point RP-1, and a load of 0.01 N was applied linearly through the reference point RP-1 to establish a stable contact between the finger and the perceived sample. In the second analysis, the preset load was continually applied through the reference point RP-1. In the third analysis, the constraint of x-direction displacement of finger was released, and the boundary condition of a certain velocity was applied at the same time. Then, the finger scanned over the surface of the perceived sample at a certain speed for a preset normal load. The formula of finite sliding was defined between the finger and the perceived sample. The Coulomb friction contact model was used to determine the dynamic coefficient of friction between the finger and the perceived sample. For the Coulomb friction model, it is assumed that the tangential force is proportional to the normal force at each contact node. The equation is as follows

$${\tau }_{\mathrm{crit}}=\mu \times p$$

(4)

where ${\tau }_{\mathrm{crit}}$ is the critical shear stress, $\mu$ is the coefficient of friction, and p is the normal contact pressure. The dynamic friction coefficient was set at 0.5 in this model.

Merkel disks and Meissner’s corpuscles are located at the interface between the epidermis and dermis and are marked as “▪” in Figure 2. Pacinian corpuscles and Ruffini endings are located at the interface between the dermis and soft tissue and are marked as “•” in Figure 2. In this model, friction and vibration induced by the finger scanning over the perceived surface would stimulate these mechanical receptors, and in a human body, the perceived information would be converted to the corresponding action potentials, which would then be transmitted to the cerebral cortex. These are the components of tactile perception.

Simulation of tactile perception of different surface textures

Different surface textures induce different tactile sensations. In this study, the simulation of tactile perception when the finger scanned over a perceived sample surface with different texture shapes was examined. Three texture shapes (rectangle, triangle, and circle) with the same period were chosen and are shown in Figure 3. The simulation of tactile perception when the finger scanned over the sample surface with same texture shape but different texture spacing was also studied. The samples with different spacing of the rectangle texture are shown in Figure 4.

Figure 3.

Rectangular, triangular, and circular textures with the same period.

Figure 4.

Rectangular textures with different spacing.

Results and discussion

Simulation of tactile perception for different types of textures

Time and frequency domains of von Mises stress around Meissner’s corpuscles and Merkel disks in response to the finger scanning different types of textures (rectangular, triangular, and circular) are shown in Figure 5.

Figure 5.

Contact area, time domain, and frequency domain of the von Mises stress for (a) rectangular texture, (b) circular texture, and (c) triangular texture.

As Figure 5 shows, the numbers of von Mises stress peaks were the same when the finger scanned three types of textures because their periods were consistent and corresponded to a peak of 5 Hz in the frequency domain. However, the peak values of von Mises stress in descending order were 7733 (triangular texture), 3263 (circular texture), and 2352 Pa (rectangular texture). Therefore, the values of the von Mises stress peaks were influenced by the types of textures that were scanned, which affected the sensitivity of tactile perception.

Because the contact between the finger and the triangle texture was very small, some stress concentration points were formed. High von Mises stress peaks appeared, and Meissner’s corpuscles and Merkel disks were stimulated strongly. The contact between the finger and the circular and rectangular textures was a surface contact. However, the contact area between the finger and circular texture was smaller than that between the finger and the rectangular texture, and therefore, the von Mises stress peak was larger for the circular texture than for the rectangular texture. The difference in contact area resulted in different contact pressures and generated variations in the deformation and von Mises stress peaks. These differences enable differentiation of tactile perception of various textures.

Simulation of tactile perception for different spacing of textures

The time domain and frequency domain of the von Mises stress around Meissner’s corpuscles and Merkel disks when the finger scanned rectangular textures with different spacing are shown in Figure 6. The data show that the peak values of von Mises stress for 1-, 0.8-, and 0.5-mm texture spacing were 2352, 2248, and 1812 Pa, respectively, suggesting that the peak values of von Mises stress decrease with decreasing texture spacing. This correlation occurs because the contact area between the finger and the perceived sample will increase and the deformation will decrease as texture spacing decreases. In addition, the von Mises stress varied periodically with time, which is controlled by the periodicity of the textures. When the finger contacted two adjacent bulges of the textures, as shown in Figure 7(a), the contact center lay in the hollow. In this case, the finger was squeezed by the textures, and a larger deformation was generated. The von Mises stress became concentrated around Meissner’s corpuscles and Merkel disks, and stress peaks appeared due to the deformation. When the finger moved to the next condition, as shown in Figure7(b), the contact center moved to the projection of the textures. In this case, the contact area was reduced, the deformation of the finger recovered, and the von Mises stress decreased. The von Mises stress was lowest when the contact center and the center of projection coincided.

Figure 6.

Time domain and frequency domain of the von Mises stress for different spacing of the rectangular textures: (a) spacing of 1 mm, (b) spacing of 0.8 mm, and (c) spacing of 0.5 mm.

Figure 7.

Contour of the Von Mises stress at conditions (a) A and (b) B.

Figure 8 shows the variation of the contact area during this process. To study the effect of texture periodicity on tactile perception, the numbers of texture periods and von Mises stress peaks when the finger scans textures with different spacing are shown in Table 2. The finger scan time and speed were set as 1 s and 10 mm/s, respectively, so the sliding distance was 10 mm. For example, in Table 2, 1-mm texture spacing corresponded to a 2-mm texture period, as shown in Figure 4. There were five texture periods within the sliding distance of 10 mm. The number of von Mises stress peaks was 4.5, as shown in Figure 6. Table 2 shows that the number of von Mises stress peaks within the finger depended on the period and consistency of the texture. The stress peaks corresponded to condition A, and the stress valleys corresponded to condition B in Figures 7 and 8. Based on the frequency domain of the von Mises stress in Figure 8, the frequency peaks of the von Mises stress fell at approximately 5, 6.5, and 10 Hz, and the respective periods were 0.2, 0.15, and 0.1 s. These results corresponded to the texture periods. Periodic deformation and recovery aroused von Mises stress with different frequencies around the mechanical receptors. In a human body, the action potentials generated by this mechanism would be transmitted to the brain, resulting in tactile perception of the texture.

Figure 8.

Variation in the contact area during the sliding process.

Table 2.

Numbers of texture period and von Mises stress peaks.

Texture spacing (mm)	Numbers of texture period	Numbers of von Mises stress peaks
1.0	5.0	4.5
0.8	6.25	6.0
0.5	10.0	10.0

Figure 9 shows the displacement signals and vibration signals around Pacinian corpuscles when the finger scanned rectangular textures with different spacing. The displacement signals around Pacinian corpuscles exhibited periodic variation, with the period of displacement decreasing as the density of the texture increased. Displacement signals were transformed into vibration signals using differential operations, as shown in Figure 9.

Figure 9.

Displacement and vibration signals around Pacinian corpuscles when the finger scans rectangular textures with different spacing: (a) spacing of 1 mm, (b) spacing of 0.8 mm, and (c) spacing of 0.5 mm.

Characteristic features of the spectral centroid (SC) extracted from the tactile vibration signals were used to characterize the fineness.^17–19 The SC was calculated from the vibration signals according to the following equation

$$\mathrm{SC}=\frac{\sum _i(\mathrm{fft}{(x_i)}^2·f_i)}{\sum _i\mathrm{fft}{\left(x_i\right)}^2}$$

(5)

where x_i is the amplitude of the vibration signals, f_i is the frequency, and i is the index of the sample number in the time domain. fft(x_i) indicates that the signal is analyzed based on the Fourier transform, and i is the index of the point of the fft. According to this equation, a larger value of SC indicates a fine texture, and a smaller value indicates a relatively coarse texture.

To verify the effectiveness of this characteristic feature of the SC, the values of SC were calculated when the finger scanned textures with different spacing, as shown in Table 3. The values of SC increased as the spacing of the texture decreased, verifying that the fine texture had a larger SC. This result agrees with the established characteristics of the SC as well as our previous experimental studies.^18–20

Table 3.

Values of the SC when the finger scanned textures with different spacing.

Texture spacing (mm)	1.0	0.8	0.5
SC (Hz)	111.39	113.02	116.92

SC: spectral centroid.

Conclusion

In this study, the influence of surface texture on the mechanical responses of tactile perception was studied using the finite element method. The conclusions from this study are detailed below.

The values of von Mises stress peaks were influenced by the shape and density of the textures. The peak values of von Mises stress in descending order were 7733 (triangular texture), 3263 (circular texture), and 2352 Pa (rectangular texture). The peak values of the von Mises stress decreased as the spacing of the texture decreased. The peak values of von Mises stress in descending order were 2352 (1-mm spacing), 2248 (0.8-mm spacing), and 1812 Pa (0.5-mm spacing).

The von Mises stress and displacement signals around Pacinian corpuscles vary periodically with time. The periodic deformation and recovery induced by different frequency and peak values of von Mises stress around mechanical receptors enable humans to differentiate between surfaces with different textures. The values of SC increased as the spacing of the texture decreased, verifying that the fine texture had a larger SC.