IEEE TRANSACTIONS ON HAPTICS, VOL. ***, NO. ***, *** 2022

1

Surfaces with finger-sized concave feel softer
Koki Inoue∗ , Shogo Okamoto∗† , Yasuhiro Akiyama∗ and Yoji Yamada∗

Abstract—The judgment of elastic softness is determined not only bymechanical parameters related to hardness, such as the elastic modulus
and stiffness, but also by macroscopic surface features. This study
experimentally demonstrates that objects with a finger-sized concavewith a depth of 1—3 mm feel softer than flat surfaces made of the samematerials when they are pushed by a finger. In Experiment 1, participants
judged the surfaces of a rigid material with thumb-sized concaves to besofter than the flat and convex surfaces. Experiment 2 used rubbers ofvarious elastic moduli, and the softness of a concave object with a Young’s
modulus of 0.55 MPa was subjectively equal to that of a flat object withan average Young’s modulus of 0.23 MPa. Furthermore, the softness ofa convex object was subjectively equal to that of a 1.68 MPa flat object.
The contact phenomena between a finger pad and concave or convex
objects are different from those between a finger pad and flat objects,
and they influence the softness judgment. Such phenomena include therelationship between the pressing force and contact area. These results
provide insights into surface design and improve comprehension of theperceptual principles of softness.

I. I NTRODUCTION
When humans touch an object, their fingertips and the object surface deform; based on this, the object’s softness is judged [1], [2]. The
various definitions of softness include elastic or deformable softness[3], and the perception of elastic softness is formed by multiple
principles. One principle is based on the relationship between theobject’s surface displacement and the pressing force [4], [5], [6].
The surface displacement is primarily judged by proprioceptive cues;
however, tactile cues may also mediate it [7]. Another hypotheticalprinciple is based on the coupling of the finger pressing force and thecontact area [8]. The curves of the pressing force and contact area areconsidered to determine the perceived softness of objects [8], [9]. The
pressure distribution in the contact area may also affect the softness
judgement. When the object is soft and includes a large deformationand contact area, the pressure is widely distributed, and its maximum
value is relatively small. This principle, based on the force and contact
area, has been leveraged to develop tactile softness displays [10],[11]. These various principles function simultaneously [2], [12], [13].
Along with these studies, in recent years, softness perception has been
intensively studied, including exploratory strategies [14], [15], pleasantness [16], softness of complex objects [5], and individuality [17],[18].
The interaction between the pressing force and surface deformation, including the contact area, underlies softness perception;
therefore, macro- and mesoscopic surface topological features that
influence finger deformation may change the perceived softness whentouching surfaces of hard materials. For example, in [19], [20], carinterior surface grains were designed to induce soft impressions. Theirstudy claimed that to enhance these effects, the preferred depth andradius of curvature of microscopic and mesoscopic grains were in
the ranges of 60–350 µm and 0.5–2.0 mm, respectively; however,they did not exhibit experimental data on how softness perception
was affected by these grains. Furthermore, earlier studies mostly∗ Department of Mechanical Systems Engineering, Nagoya University,
Furo-cho, Chikusaku, Nagoya 464–8603, Japan† Department of Computer Sciences, Tokyo Metropolitan University, Hino,
Tokyo 191-0065, Japan
This study was in part supported by MEXT Kakenhi (20H04263,
21H05819).

Fig. 1: Experimental scenario; pushing flat and hard plates with a concave or convex at the center.

employed flat surfaces in studying the perception of elastic softness
whereas non-flat objects such as fruits [5], spheres [21] and tumors(e.g. [22]) were used in other studies. The present study investigates
the perceived effects of concaves as large as finger pads, which areapproximately 10–100 times larger than those investigated in [19],[20]. The contact between a finger pad and such a finger-sized
concave leads to a contact area greater than that made by a flat
surface, which may affect the perceived softness.
This study includes two experiments. In Experiment 1, we prepared
hard plastic plates with a finger-sized concave or convex. The sizesof the concaves were designed to resemble the average human first,
third, and fifth finger pads. Participants pushed the center of theseconcaves and convexes to rank them on the basis of their perceived
softness. We expected that the surface with concaves, into which thefinger pads fit, would feel softer than a flat surface. In contrast, aconvex surface would feel harder than a flat surface, considering ahypothetical principle based on the relationship between the pressing
force and contact area [8]. In Experiment 2, the softness perceived
by touching a concave or convex elastic surface was matched withthe softness of flat surfaces made of materials with different elastic
moduli. The results of this study will help with surface design andprovide insights into the comprehension of softness perception.
II. M ETHODSThis study was approved by the Institutional Review Board of the
School of Engineering, Nagoya University (#20-17).
A. Experiment 1: Ranking task of concave and convex hard surfaces
1) Stimuli: We used 16 types of specimens made of hard plastic.
They were made by a 3D printer (Form3, Formlabs Inc., MA, USA)using a hard plastic resin (nominal Young’s modulus: 1.6 GPa).
The specimens were cylindrical, as shown in Figs. 1 and 2. Nine
specimens had an ellipsoidal concave in the center (Samples 1–9).
The concaves on Samples 1–3, 4–6, and 7–9 resembled the sizes ofthe fifth, third, and first fingers, respectively, with respect to hand dataof Japanese people [23]. Three specimens had a spherical concave inthe center (Samples 10–12), and three specimens had an ellipsoidal
convex in the center (Samples 13–15). Furthermore, one specimen
was flat with no concave or convex (Sample 16). The designed values

IEEE TRANSACTIONS ON HAPTICS, VOL. ***, NO. ***, *** 2022

Top view

2

z


y

40
x



Front view




y

ℎ

10

z

x

Fig. 2: Specimen parameters manipulated in the experiment. The upper
figure shows the top view, and the lower figure, the front view.
The unit of length is in millimeters.
TABLE I: Parameters of surface shapes used in the experiment. The
unit of measurement is in millimeters. Samples 1–12 wereconcave surfaces, Samples 13–15 were convex surfaces, andSample 16 was flat.
Sample no.
abhSample no.
abhSample no.
a
b
h

1
11.9
5.6
1
7
15.2
8.3
1
13
11.9
5.6

2
11.9
5.6
2
8
15.2
8.3
2
14
14.5
6.8

3
11.9
5.6
3
9
15.2
8.3
3
15
15.2
8.3

−2

−2

−2

4
14.5
6.8
1
10
7.5
7.5
1
16
0
0
0

5
14.5
6.8
2
11
7.5
7.5
2

6
14.5
6.8
3
12
7.5
7.5
3

of the variable parameters shown in Fig. 2 are listed in Table I. The
height and axes of the 3D-printed objects used in the experimentwere close to these values with an average relative error of 0.4%.
2) Tasks: The participants pushed all 16 specimens, which wererandomly arranged on the desk by the experiment facilitator, andranked them based on their perceived softness, as shown in Fig. 1.
They could freely compare the specimens before completing the task.
No time limit was set, and the specimens could be touched manytimes before completing the ranking task. In a single trial, only
one of the first, third, and fifth fingers were used. The participants
touched the flat surface and the centers of the concave and convex
with their finger pads such that their fingernails did not touch thespecimens. Furthermore, they pushed the surface without sliding
motions. Softness typically describes a deformable characteristic ofmaterials or objects; however, we used solid plastic that could not be
deformed by finger forces. Nonetheless, the participants conducted
the tasks without any questions about the softness. No instructions
were given about the force used for pressing. The participants wereallowed to assign tied ranks to more than one specimen when theirsoftness was perceived as equal.
The experiment facilitator provided instructions on which finger
would be used in the task in a randomized order. Only one task
was conducted using two of the three fingers; however, two tasks
were performed with the remaining finger, which was also randomly
selected. These two tasks tested using the same finger were used

for participant screening purposes, as mentioned in Section II-A4.
The screening task, which used the finger used in the first task, wasconducted after the other three normal tasks. There was a break of afew minutes after each task. During the experiment, the participants
wore smoke sunglasses; however, they were not fully blinded andcould locate the specimens.
3) Participants: Twenty university students in their 20s (mean andstandard deviation: 23.2 ± 1.2 years old, 4 females) participated in
the study after providing written informed consent. They were notaware of the objectives of the study.
4) Data Analysis: To screen the participants, Spearman’s rankcorrelation coefficient was calculated between the results of two
ranking tasks conducted by an individual using the same finger. The
correlation coefficients were significant at a level of 0.05 for allparticipants, and the values ranged between 0.54–0.98 with a medianof 0.88. Hence, we did not screen out any participants and used theresults of the first trial of the two tested using the same finger forthe latter analysis.
We tested the differences in the ranks assigned to the 16 types of
specimens using the Friedman test. If the differences had a significance level of 0.05, as post-hoc tests, we then applied a pairwise
Wilcoxon signed-rank test for the pairs between the specimen thatwas ranked as softest and the other 15 types of specimens. For these
post-hoc tests, the significance level was adjusted using Bonferroni’s
method with a factor of 15.
In contrast to our expectations, effects of the concave depth hwere not clearly observed. As a post-hoc test, we grouped the same
type of concaves, irrespective of the h values, and compared them.
The specimens resembling the fifth finger, i.e., Samples 1–3, those
resembling the third finger, i.e., Samples 4–6, and those resembling
the first finger, i.e., Samples 7–9, were grouped together. Furthermore,
spherical concaves, i.e., Samples 10–12, and convexes, i.e., Samples
13–15, were also combined into two groups. The flat specimen(Sample 16) was not merged with any other groups. The ranks ofthese six groups were compared using the Friedman test and post-hoc
Wilcoxon signed-rank tests, similar to the process described above.
For the post-hoc comparison, the significance level was adjusted bya factor of 5.

B. Experiment 2: Softness matching task using rubber specimens
Participants compared a rubber object that had a concave or convex
surface with flat objects made of rubber with different hardness values. They then selected a flat object whose softness was subjectively
equal to that of the concave or convex object.
1) Rubber specimen stimuli: Fourteen flat rubber cylinders withdifferent hardness values were used as comparison stimuli. The
cylinders were 10 cm in diameter and 5 cm in height. These cylinders
were made by blending two of the five types of commercial silicone
rubber (KE-1415, KE-1417, KE-26, Shin-Etsu Chemical Co., Ltd.,
Japan; ELASTOSIL M8520, ELASTOSIL M4470, Wacker Asahikasei Silicone Co., Ltd., Germany). The hardness of the cylinders wastested using a Shore A durometer (GS-719N Teclock Co., Ltd., Japan)
and the results were transformed to Young’s moduli following [24].
The results ranged between 0.20–3.00 MPa, as listed in Table II.
We used a cylinder with a concave as a concave reference stimulus;
the size of the concave was the same as that of Sample 8 in Table I.
A cylinder with a convex was used as a convex reference stimulus;
the convex size was identical to that of Sample 15. These reference
stimuli were made of rubber G in Table II, with a Young’s modulus
of 0.55 MPa. The molded rubber concave and convex were slightly
smaller than their original designs by 4.0% on average.

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3

TABLE II: Elastic moduli of all samples in Experiment 2.
Sample nameElastic modulus (MPa)Sample nameElastic modulus (MPa)Sample nameElastic modulus (MPa)

A
0.20
F
0.47
K
1.23

B
0.23
G
0.55
L
1.50

C
0.27
H
0.61
M
1.95

D
0.33
I
0.73
N
3.00

E
0.39
J
0.94

2) Tasks: Participants compared one of the two reference stimuli,i.e., concave or convex objects, and 14 comparative stimuli, i.e., flat
objects, using the index finger of their dominant hand. Nine of theparticipants tested the concave object and then the convex object after
a break of a few minutes. The other half tested the reference stimuli
in the reverse order. The comparative stimuli were randomly arrangedon the desk, not in the order of physical hardness. The participants
were instructed to push the center of the stimuli using a finger pad toinvestigate their softness. Sliding was not allowed. They then selected
a comparative stimulus whose material softness felt most similar tothat of the reference stimulus. There was no time limit set for the task;
however, most participants finished the task within 2–3 min, during
which they could touch the stimuli at any time. Participants wore
smoke glasses so that they could barely see the cylindrical stimuli
to touch. The participants conducted these matching tasks only once
for each reference stimulus.
3) Participants: Seventeen university students (mean and standard
deviation: 23.0 ± 1.3 years old, 3 females) participated in Experiment
2 after providing written informed consent. Twelve among them also
participated in Experiment 1.
4) Data analysis: We used the t-test to compare the Young’s
modulus of the reference stimuli (0.55 MPa) with the Young’s moduli
of flat objects that the participants selected as having subjectively
equal softness to the references. This comparison was separately
conducted for each of the two reference stimuli.

the spherical concave (Samples 10–12), convex (Samples 13–15), andflat (Sample 16) samples.
As shown in Fig. 3 (c), the softness ranks of all the samples
were not equal (χ2 = 129.1, df = 15, p = 3.16 × 10−20 )when they were tested by the fifth finger. Sample 8, which is aconcave shape corresponding to the first finger pad, was ranked
softest and judged statistically softer than two concave resembling
the fifth finger (Samples 1 and 2), two spherical concave (Samples
11 and 12), convex (Samples 13–15), and flat samples (Sample 16).
As shown in Fig. 4 (c), the ranks of the six sample groups were
not equal (χ2 = 146.4, df = 5, p = 7.83 × 10−30 ). The concaves
corresponding to the size of the first finger (Samples 7–9) were softer
than the concaves resembling the fifth finger (Samples 1–3), sphericalconcaves (Samples 10–12), convexes (Samples 13–15), and flat object(Sample 16).
B. Experiment 2: Matching tasks
Fig. 5 shows the mean and standard error values of the Young’s
moduli of the flat objects that the participants selected as having subjectively equal material softness to the concave and convex surfaces.
Note that the Young’s modulus of the rubber that was used for the
concave and convex surfaces is 0.55 MPa. As the left bar shows,the concave surface of 0.55 MPa was matched with a flat surface
with a rubber hardness of 0.23 ± 0.035 MPa (mean and standard
error). These subjectively equal Young’s moduli were significantly
smaller than that of the concave object, i.e., 0.55 MPa (t(16) = 9.06,
p = 1.07 × 10−7 , two-tailed t-test), which indicates that the concave
surface felt softer than a flat surface with an equal Young’s modulus.
In contrast, as the right bar shows, the softness of the convex object
was matched with that of a flat object of 1.68 ± 0.17 MPa, which isgreater than 0.55 MPa (t(16) = 6.65, p = 5.61 × 10−6 , two-tailed
t-test). This indicates that the convex surface felt harder than a flat
surface.

III. R ESULTS
A. Results of Experiment 1: Ranking tasks
Fig. 3 shows the results of the ranking tasks where 16 typesof samples were tested with the first, third, or fifth finger of theparticipant. The vertical axes show the softness ranks, and the samples
with lower ranks were judged to be comparatively softer. Similarly,
Fig. 4 shows the ranks after the same types of surfaces were merged,
irrespective of depth h.
As shown in Fig. 3 (a), the samples tested using the first finger
exhibited differences in their ranks (χ2 = 112.9, df = 15,
p = 4.47 × 10−17 ). Sample 8, which had a concave resembling
the first finger pad, was judged to be the softest, and its ranks weresignificantly lower than those of the concave samples resembling thefifth finger (Samples 1–3), the convex samples (Samples 14–16), andthe flat sample (Sample 16). Furthermore, as shown in Fig. 4 (a),the sample groups exhibited differences in their ranks (χ2 = 132.2,
df = 5, p = 8.20 × 10−27 ). The concave group (Samples 7–9) withsizes similar to that of the first finger pad exhibited the lowest ranks.
As shown in Fig. 3 (b), the softness ranks of the 16 samples
were not equal (χ2 = 120.4, df = 15, p = 1.58 × 10−18 ) whentested using the third finger. Sample 9, which is a concave surface
corresponding to the first finger pad, was judged to be the softest,
and it exhibited statistically lower ranks than the spherical concave(Samples 10–12), convex (Samples 13–15), and flat objects (Sample
16). As shown in Fig. 4 (b), the six sample groups also differed interms of their ranks (χ2 = 137.8, df = 5, p = 5.15×10−28 ), and thegroup corresponding to the first finger was judged to be the softest.
This group was judged to be statistically softer than the groups of

IV. D ISCUSSIONIn Experiment 1, we expected that the concave objects, the sizesof which were most similar to those of the finger pads, would be
judged as the softest by the participants. This is because the contact
area is magnified when the finger pad just fits into the concave,provided that the contact area is a major determinant of softness
perception. However, regardless of the fingers, i.e., first, third, andfifth fingers, the largest concaves, the sizes of which were similar tothat of the first finger, were judged to be the softest. We consider thisto still be inconclusive because the participants may have failed to
fit their finger pads into the appropriately matched concaves, which
may have produced results that we did not anticipate. Furthermore,individual differences in finger dimensions and shapes may haveaffected the results. Note that we adopted ellipsoidal and sphericalconcaves, and their dimensions were determined on the basis of mean
finger dimensions; however, the shapes and sizes of finger pads areindividually different. For more controlled studies, we need to prepare
tailor-made concaves for individuals and ensure that their finger pads
exactly fit into the concaves. However, the results of Experiment 1
indicate that concaves that are slightly larger than finger pads are
effective in causing perceived softness. The participants’ answers
regarding the spherical concaves fluctuated heavily, as shown in Figs.
3 and 4. This may be because of the shape mismatch between theindividual finger pads and concaves. When the finger pad is pushed
on a concave that is differently shaped or smaller than the finger
pad, the finger pad contacts the edge between the flat surface andthe concave, which may disperse the effect of concaves on softness

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4(c) Tested by fifth finger(b) Tested by third finger(a) Tested by first finger

∗∗

∗∗
∗
15

15

10

10

10

5

5

5

0

Convex Flat

Concave
Sample number

Convex Flat

1 2 3 4 5 6 7 8 9 10 11 1213 14 1516
Fifth
fingerSphere

Thumb

Third
fingerFifthfingerConcave
Sample number0

1 2 3 4 5 6 7 8 9 10 11 1213 14 1516

Sphere

Thumb

Third
fingerFifthfinger

1 2 3 4 5 6 7 8 9 10 11 12 13 14 1516

Sphere

15

0

∗

∗

Thumb

∗∗

Third
finger∗Rank

Perceptually softer Perceptually harder∗∗

Concave
Sample number

Convex Flat

Fig. 3: Box plots of the ranks of all samples. The left, middle, and right columns represent the results when touched by the thumb, third finger, andfifth finger, respectively. Samples with low/high ranks were judged soft/hard. Samples 1–3, 4–6, and 7–9 had ellipsoidal concaves resembling
the sizes of the fifth, third, and first fingers. Samples 10–12 had spherical concaves. Samples 13–15 had ellipsoidal convexes. Sample 16 had
a flat surface. * and ** indicate the statistical differences in the ranks between two samples at p < 0.05 and p < 0.01, respectively.
(b) Tested by third finger(c) Tested by fifth finger
∗∗

∗∗

∗∗
∗

∗

∗

15

15

15

10

10

10

5

5

5

0

Concave
Sample number

7—9 10—12 13—15 16

1—3

4—6

7—9 10—12 13—15 16

Thumb

Fifth
fingerThird
fingerThumbConvex Flat

Sphere4—6
Third
finger0
1—3
Fifth
fingerSphereThird
finger

7—9 10—12 13—15 16
Thumb

4—6

Fifth
finger0
1—3

Concave
Sample number

Convex Flat

Sphere∗Rank

Perceptually softer

Perceptually harder(a) Tested by first finger

Concave
Sample number

Convex Flat

Fig. 4: Box plots of the ranks of all sample groups. The left, middle, and right columns represent the results obtained when the samples were touchedby the thumb, third finger, and fifth finger, respectively. Samples were grouped based on the size of the concaves or the surface shapes. *and ** indicate the statistical differences in the ranks between two groups at p < 0.05 and p < 0.01, respectively.

perception. In contrast, the participants’ answers regarding the thumbsized concaves did not fluctuate as much as those regarding thespherical ones. The thumb-sized concaves were larger than or nearly
equal to the pads of all participants’ fingers, and stable effects on
softness perception may be expected.

The primary interest of this study is why finger-sized concaves
and convexes affect our softness perception of objects. It would
be reasonable to discuss the softness perception of elastic objects
from the viewpoint of stiffness and the elastic modulus [2]. We
discuss the relationship between the contact force, contact area, anddisplacement at contact based on the Hertz contact theory, which does
not accurately hold for finger pads; however, it is still meaningful tocomprehend the general trends [25]. According to the theory [26],when two spheres are in contact with a normal force F , the radius

a of the circular contact area is approximated as1
3F R∗ 3
a=
,
4 E∗

(1)

where R∗ is the equivalent radius determined by the radii of the two
spheres, R1 and R2 :
1
1
1
=
+
.
R∗
R1
R2

(2)

E ∗ is the equivalent elastic modulus determined by Young’s moduli(E1 and E2 ) and the Poisson ratios (ν1 and ν2 ) of the two materials:
11 − ν121 − ν22
=
+
.
∗
E
E1
E2

(3)

Let E1 , R1 , and ν1 denote the parameters of the human fingers,
referring to [23], [27], we set E1 = 0.1 MPa, R1 = 6.8 mm, andν1 = 0.48. For a concave object, R2 = 8.3 mm, and E2 = 0.55 MPa.

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Elastic modulus (MPa)3∗∗∗
1.68  0.17

2

5

TABLE III: Apparent Young’s moduli and spring constants of threeobjects with an actual Young’s modulus of 0.55 MPa.
Apparent Young’s moduli were computed from the forcearea curves, assuming that the stimuli were flat. Apparent
spring constants were defined within 0.5–1.5 N.

∗∗∗
1

0.23  0.035

Concave
Convex
Flat

0.55

Perceived Young’s modulus
0.018 MPa0.55 MPa

Perceived spring constant1369 N/m
634 N/m
774 N/m0Flat surface
Concave or
Flat surface
subjectively
convex surface
subjectively
equal to convex(Constant)equal to concave

Fig. 5: Mean and standard error values of the Young’s moduli of flatmaterials selected by the participants as subjectively equal tothe concave or convex object. Left: Young’s moduli subjectively
equal to the concave surface made of rubber (0.55 MPa). Center:
Young’s modulus of the rubber used for the concave and convex
surfaces. Right: Young’s moduli of the materials that were subjectively equal to the convex surface made of rubber (0.55 MPa).
*** indicates statistical significance level of p < 0.001.

Contact area ߨܽ ଶ (mm2)(a) Force-area curve
Flat (0.018 MPa)
150

Concave (0.55 MPa)100
Flat (0.23 MPa)
Flat (0.55 MPa)
Flat (1.68 MPa)
Convex (0.55 MPa)

50
0
0

1.0
0.5
Contact force (N)1.5

Displacement ߜ (mm)(b) Force-displacement curve

Convex (0.55 MPa)
Flat (0.23 MPa)
Flat (0.55 MPa)
Flat (1.68 MPa)

32

Concave (0.55 MPa)

1
0
0

1.0
0.5
Contact force (N)

1.5

Fig. 6: Curves simulated by Hertz contact theory. (a) Force-area curve,(b) force-displacement curve.

For a flat object, R2 is infinity. E2 is either 0.23 MPa, 0.55 MPa, or
1.68 MPa. For a convex object, R2 is −8.3 mm, and E2 = 0.55 MPa.
For the R1 and R2 values, we used the short axes of the third finger(6.8 mm) and thumb-sized concave and convex (8.3 mm). ν2 was
set to 0.48 considering the rubber material used in this study. Fig. 6(a) shows the curves between the force and contact area when thisspherical finger is in contact with concave, flat, and convex objects.
Fig. 6 (b) shows the curves between the force and displacement,from which the stiffness or spring constant can be calculated. The
displacement is the change in the distance between the centers of thetwo objects, computed as follows:
a2
.
(4)
R∗
Fig. 6 (a) shows that the contact area is smallest for the convex
object (blue curve), and its apparent hardness or elastic modulus isδ=

the largest. The contact area is largest for the concave object (black
curve), and its apparent hardness is the smallest. These curves arevery different from those of the flat object with the same hardness (redcurve), which indicates that the concave and convex shapes change
the force-area curves and apparent hardness. Based on the forcedisplacement curve shown in Fig. 6 (b), the convex object (blue)
appears less hard than the flat object of equal Young’s modulus(red). The apparent stiffness or spring constant of the convex object
is smaller than that of the flat object. Hence, a convex increases
the apparent Young’s modulus and decreases the apparent spring
constant. Furthermore, the concave decreases the apparent Young’s
modulus and increases the apparent spring constant. These two typesof quantities, i.e., spring constant and Young’s modulus, affect thesoftness judgment simultaneously; however, the role of Young’s
modulus may be more important [2] for the object hardness used in
the present study. The changes in the apparent Young’s moduli causedby convex and concave shapes may perceptually surpass those in theapparent spring constants.
Table III lists the nominal or apparent Young’s moduli and spring
constants of concave, convex, and flat objects of the same actual
Young’s modulus: 0.55 MPa. The apparent Young’s modulus of theconcave was determined using the least squares method such that theforce-area curve of the flat surface best fitted that of the concave. The
apparent spring constant was computed using the force-displacement
curve between 0.5–1.5 N. The apparent Young’s modulus of theconcave object is 0.018 MPa, although its nominal material hardness
is 0.55 MPa. This means that the force-area curve of the concave
object is close to that of a flat object of 0.018 MPa (dotted yellow
curve in Fig. 6 (a)). The apparent spring constant of the concave
object is 1369 N/m, which is greater than that of a flat object of
0.55 MPa. The apparent Young’s modulus of the convex object does
not exist because no flat object can achieve similar curves, even with
infinite hardness. The apparent spring constant of the convex object
is 634 N/m, which is smaller than that of the flat object.
According to the above analyses, in this setting, it was suggested
that the force-area curves or Young’s moduli more dominantly
affected the perceived softness than the force-displacement curves
or spring constants. Earlier studies also agreed that the contact area
or Young’s moduli are highly significant for softness perception [2],[8], [10], [11], [13]. In contrast, a study by Xu et al. [5] is inconsistent
with the present study. They investigated the softness perception ofplums that are largely spherical and similar to our convex stimuli. In
terms of the shape of the stimuli, [5] and the present study should becompared, whereas most of the earlier studies employed flat surfaces.
In [5], no evidence was found suggesting that contact areas wereused for judging the softness of plums. They concluded that theforce-displacement curve and the composite spring constant derived
from the curve were good predictors for the perceived softness.
Although the present study and [5] used spherical convexes as stimuli,
their conclusions are very different. We speculate that the irregularvariation of natural plums decreased the reliability of contact area
cues in the experiments of [5] and influenced the softness judgmentof the assessors in [5]. The differences in the conclusions about the

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role of contact areas between the present study and [5] indicate thatfurther studies are necessary to comprehend the softness perceptionof non-flat objects.
One concern of the present study may be in the experimental
methods used for the ranking (Experiment 1) and matching tasks(Experiment 2). In particular, the task in Experiment 1 suggested tothe participants that the stimuli should differ in terms of perceived
softness. The method of constant stimuli may provide more convincing conclusions in the scenario where the pair of a concave or convex
object and flat object was compared in terms of softness. Owing to therestrictions placed under the COVID-19 pandemic, we avoided themethod of constant stimuli because it requires a long task duration.
However, the effects of the convex and concave shapes observed inthis study are substantial. For the reference stimulus of ∼1 MPa, theWeber ratio was approximately 15% [2]. In our results, the softnessof a concave object with a Young’s modulus of 0.55 MPa was judged
to be close to that of a flat object of 0.23 MPa. This difference inYoung’s moduli is greater than that calculated using the Weber ratio,
and is considered meaningful.
We also note the potential cognitive biases of the surface shapes.
Round shapes are linked to feelings of softness in a synesthetic
manner [28]. The participants could have visually recognized therounded concaves or convexes before or after touching them, which
could have biased their judgment of softness. If so, the perceived
softness of both the concaves and convexes would be expected tobe similarly influenced, whereas opposite effects were found in theexperiments; the concaves felt softer and the convexes felt harder. It
is thus reasonable to conclude that the synesthetic biases of roundshapes were less significant than the effects of tactile interaction
between the finger pad and surface shapes.
V. C ONCLUSION
Physical hardness, such as the elastic moduli, and perceived
hardness are different. An earlier hypothesis that the finger contact
area is a major cue for softness perception suggests that macroscopic
surface shapes influence perceived softness. We tested this possibility
in Experiment 1 and compared the material softness of objects with
finger-sized concaves, convexes, and flat surfaces. The objects withconcaves, the sizes of which were similar to that of the average
first finger, were judged to be softer than the others. Furthermore,
the objects with convexes were judged to be harder than the others.
In Experiment 2, the material softness of the concave and convex
objects was matched with flat objects with different Young’s moduli.
The concave object of 0.55 MPa corresponded to a flat object of
0.23 MPa. The convex object of 0.55 MPa corresponded to a flat
object of 1.68 MPa. These experiments demonstrated that finger-sized
concaves and convexes influenced the softness experienced whenpushing these surfaces with the finger pads. The findings of thisstudy will assist in surface design and provide some insights into
the principles of softness perception. In future work, these findings
should be combined with detailed measurements of contact statuses.
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