science.sciencemag.org/content/369/6504/689/suppl/DC1

Supplementary Materials for

How hair deforms steel

Gianluca Roscioli, Seyedeh Mohadeseh Taheri-Mousavi, Cemal Cem Tasan*

*Corresponding author. Email: tasan@mit.edu

Published 7 August 2020, Science 369, 689 (2020) DOI: 10.1126/science.aba9490

This PDF file includes:

Materials and Methods Figs. S1 to S10 Captions for Movies S1 to S5 References

Other Supplementary Material for this manuscript includes the following: (available at science.sciencemag.org/content/369/6504/689/suppl/DC1)

Movies S1 to S5

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Materials and Methods

Blade cross-section and general sample preparation

Samples for interrupted tests were used and imaged in their commercial settings, housed in their

own plastic handles. In order to use the blades in in-situ SEM hair-cutting experiments and cross-

sectional imaging, they were removed from the plastic handles using a low-speed diamond saw, 5

cutting in a direction perpendicular to the blades and starting from the opposite side of the sharp

edge in order to avoid large deformations in the region of interest. Samples for cross-sectional

imaging for ECCI and EBSD were then cut using wire Electrical Discharge Machining (EDM)

and cold-mounted to avoid unwanted heat treatments. The samples were mechanically ground

using silicon carbide abrasive paper down to P4000 and polished with diamond suspensions of 9, 10

3 and 1 µm diameter before the final polishing step using silica particles with a diameter of 40 nm.

Further characterization of blades’ steel and human hair

Scanning electron microscope images of commercial razor blades cross-sections and energy

dispersive x-ray spectroscopy results (fig. S1) reveal similar composition and lath martensite 15

microstructure among the steels used, in agreement with commercial data (52). Analyzing more in

detail Sample 1 in fig. S1, image analysis revealed a carbide area fraction of 5.84%, with an

average carbide area of 0.42 +/- 0.26 (SD) µm2, corresponding to an average diameter of 0.73 µm

when considering a circular cross-section. In addition, EBSD analyses revealed an area fraction of

retained austenite of 4.75%. The parent austenite grain size was calculated using orientation 20

relationships to the present lath martensite, using the ARPGE software (53), resulting in an average

value of 4.93 µm, with a maximum of 10.42 µm. These small values are in agreement with the

short austenization step of these steels prior to rapid quenching (52).

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Human hairs were characterized through SEM imaging and pico-indentation hardness tests. The

hardness tests were performed on the outer cuticle layer, along a radial direction perpendicular to

the hair axis. The tests resulted in an average hardness of 170 +/- 23 (SD) MPa, in agreement with

literature data (39).

5

In-situ hair-cutting experiments in SEM

Two different sets of in-situ hair-cutting experiments have been performed inside an SEM, with

two separate micro-mechanical setups.

The first setup used a Deben Gatan micro-tensile deformation stage in compression mode, with

customized clamps manufactured from stainless steel using a wire EDM. The clamp geometry 10

accounts for a 21° tilt of the blade with respect to the cutting direction, and a cutting depth of ~100

µm from the base of the hair, mimicking cartridge design. Figure S5 shows details of the clamps;

Figs. 2A and 2B show the final setup on the micro-tensile stage. This setup enables cutting of a

single hair, multiple aligned hairs, or even a ‘forest’ of hairs randomly or carefully distributed.

Cutting experiments were performed with displacement rates of 0.4 mm/min and 0.2 mm/min. 15

Blade chipping was obtained in both cases, although a smaller rate was preferred for video

capturing. Although it is possible to detect a force when cutting multiple hairs and using a low-

load cell (150N), the force to cut a single hair is lost in background noise.

The second set of experiments involved the use of a Hysitron PI 88 Picoindenter setup. A

customized tip was machined to host the hair, meanwhile the blade was mounted on a rotary-tilting 20

stage in order to replicate blade tilting to 21° (see Figs. 2D and 2E). The experiments were

performed in nanoindentation/compression mode, using a high-load transducer (500 mN, 150 µm).

The tests were performed in displacement-control, setting the maximum displacement equal to the

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maximum reachable from the setup, i.e. 150 µm. In order to cut large diameter hairs, the tests were

conducted through two consecutive cuts, and the results were later combined. It is important to

note that indentation experiments using this setup enable measurement of the normal cutting force

with respect to blade displacement, and not indentation depth in the hair. Correlation of the position

of the blade with respect to hair depth has been performed through video-frames analysis. 5

Analytical model – Mixed Mode II and III on a bi-material semi-infinite plate with interfacial

crack

The general definition of the energy release rate for an interfacial crack propagating along a bi-

material interface under mixed mode I, II and III can be written as (54–56) 10

𝐽𝐽 = 1cosh(𝜋𝜋𝜋𝜋)2

|𝐾𝐾|2

𝐸𝐸∗+ 𝐾𝐾𝐼𝐼𝐼𝐼𝐼𝐼

2

2𝜇𝜇∗ (1)

where, in the case of plane stress, 2𝐸𝐸∗

= 1𝐸𝐸1

+ 1𝐸𝐸2

and 2𝜇𝜇∗

= 1𝜇𝜇1

+ 1𝜇𝜇2

.

The parameter 𝜀𝜀 is defined as 𝜀𝜀 = 12𝜋𝜋

ln �𝑘𝑘1 𝜇𝜇1⁄ +1 𝜇𝜇2⁄𝑘𝑘2 𝜇𝜇2⁄ +1 𝜇𝜇1⁄ �, where 𝑘𝑘𝑖𝑖 = (3 − 𝜈𝜈𝑖𝑖)/(1 + 𝜈𝜈𝑖𝑖). If the two

materials are the same, 𝜀𝜀 = 0 and therefore cosh(𝜋𝜋𝜀𝜀) = 1. 𝐸𝐸, 𝜇𝜇, and 𝜈𝜈 are the Young’s modulus,

the shear modulus and the Poisson’s ratio, respectively; while 1 and 2 differentiates the two 15

materials on opposite sides of the crack.

The first term in the Eq. 1 contains the complex stress intensity factor 𝐾𝐾, given by the mixed

contribution of mode I and II for a notch at the interface of two different materials. Although

Erdogan’s definition of this complex stress intensity factor (57) implies the same physical units as

for the homogeneous case (58) (𝑀𝑀𝑀𝑀𝑀𝑀 √𝑚𝑚, when using SI units), we use the definition from Rice 20

and Sih (59, 60) for simplicity:

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𝐾𝐾 = �𝜎𝜎𝑦𝑦𝑦𝑦∞ + 𝑖𝑖𝜎𝜎𝑥𝑥𝑦𝑦∞ �(1 + 2𝑖𝑖𝜀𝜀)𝐿𝐿−𝑖𝑖𝜋𝜋�𝜋𝜋𝐿𝐿2

where L is a characteristic length of the system for which we use 𝐿𝐿 = 2𝑀𝑀, a being the length of the

crack.

Considering no contribution from the opening mode (𝜎𝜎𝑦𝑦𝑦𝑦∞ = 0), and that 𝐾𝐾𝐼𝐼𝐼𝐼𝐼𝐼 = 𝜏𝜏3√𝜋𝜋𝑀𝑀, the energy

release rate (J) of a notch under a mixed mode II and III loading is 5

𝐽𝐽 = 𝑎𝑎𝜋𝜋(1+4𝜋𝜋2)(𝜇𝜇1+𝑘𝑘1𝜇𝜇2+𝜇𝜇2+𝑘𝑘2𝜇𝜇1)𝜏𝜏22sech (𝜋𝜋𝜋𝜋)2

16𝜇𝜇1𝜇𝜇2+ 𝑎𝑎𝜋𝜋

4� 1𝜇𝜇1

+ 1𝜇𝜇2� 𝜏𝜏32 . (2)

In order to analyze the effect of the stress character (varying from pure mode II to pure mode III)

on the energy release rate, we fixed the magnitude of the total stress

𝜏𝜏𝑡𝑡𝑡𝑡𝑡𝑡 = �𝜏𝜏22 + 𝜏𝜏32

and rewrote Eq. 2 as a function of the total stress angle 10

𝛼𝛼 = atan �𝜏𝜏3𝜏𝜏2� .

It is easy to show that this can be achieved through the following substitutions:

𝜏𝜏2 = � 𝜏𝜏𝑡𝑡𝑡𝑡𝑡𝑡2

1 + tan2 𝛼𝛼

𝜏𝜏3 = tan𝛼𝛼 �𝜏𝜏𝑡𝑡𝑡𝑡𝑡𝑡2

1 + tan2 𝛼𝛼

The resulting equation has been used to generate Fig. 3D, where it is possible to see how the energy 15

release rate is affected by materials parameters. In order to later compare the results with the

simulations run on a wedge geometry, the total stress magnitude used was 50 MPa and the crack

length 3 µm.

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FEM simulations of the blade containing a sequence of notches

A continuum model of a three-dimensional blade was constructed in ABAQUS (61) for our FE

simulations. As shown in Fig. 3A, the geometry of the model follows the geometry obtained from

the cross-section images of the blades. The martensitic steel used in the blade was modeled as a

homogeneous isotropic elasto-plastic material, having a Young’s modulus of 200 GPa, a Poisson’s 5

ratio of 0.3, and a flow stress curve as provided in Fig. S7 obtained from tensile experiments.

The imposed stress of the hair on the blade has been modelled as a static surface traction acting

evenly only on one side of the blade, applied in a ramp load, since when the cut is not orthogonal

the other side of the blade is not in contact. The magnitude was set to 50 MPa, corresponding to a

maximum force of ~0.2 N (most critical value from the single hair cutting experiments of Figs. 10

2D-F and S8) over a 100-µm-diameter semi-circular area corresponding to half of the cross-section

of a hair. Several simulations have been carried out, corresponding to different surface traction

directions (force angles γ in Fig. 3A).

The roughness along the sharp edge has been represented as the extreme condition, i.e. a sequence

of notches with characteristic size (1 µm wide, 3 µm long) derived from blade profile images (see, 15

for instance, Fig. 1C). Referring to Fig. 3A, fixed boundary conditions were applied both to the

bottom plane and to the right cross-sectional one. Symmetry boundary conditions were applied to

the left cross-sectional plane, thus to simulate only half of the model being deformed by the hair.

The size of blade portion simulated consisted in a triangular cross-section 300 µm tall with a 17°

vertex angle, 150 µm deep. A non-constant mesh containing 1,400,000 elements was used 20

throughout the model, coarse towards the bottom-right side (furthest point from the hair) and fine

towards the region containing the notches. A constant mesh size was used in the notched region,

set equal to 100 nm such that the plastically deformed regions resulting from the simulations at

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higher force angles contained at least 10 elements. A mesh convergence study has been conducted

using models with a number of elements approximately equal to 150,000, 250,000, 460,000, and

1,400,000. Although convergence was already reached for the case with 460,000 elements, all the

simulations have been run using the finer mesh in order to increase accuracy around the notches.

The resulting von Mises stress distribution on the cross-section of the blade are reported in Fig. 5

3B for two particular force angles and in movie S5 for all of them. Following the schematics in

Fig. 3A, the first four notches from the left in Fig. 3B are in contact with the hair, the fifth notch

is in contact with the hair edge (brown dashed line), and the other notches are not contacted by the

hair. The stress value at the tip of selected notches – inside, at the edge of, and outside the hair –

have been plotted as a function of the force angle in Fig. 3C. 10

FEM calculation of the J-integral for a crack propagating from a notch perpendicular to the

sharp edge

A three-dimensional sample (Fig. 3E) was constructed in ABAQUS (61) once more, with loading

conditions and geometry similar to the previous simulations. Only the critical notch with the 15

highest stress intensity factor was considered in our model, i.e. the one in proximity of the edge of

the hair. Fixed boundary conditions were applied to both cross-sectional planes and to the bottom

plane. Figure 3E is a simplified schematic aimed at clarifying the parameters of these simulations,

and it differs from the real simulated model in the region where the surface traction is applied: in

the real simulated model, the notch is in the middle of the blade and the surface traction acts on a 20

semi-circular area with a diameter of 100 µm right next to it, representing half of the cross-section

of the hair. This surface traction has been applied along different directions (different angles γ

referring to Fig. 3E), with 10° increments from 20° to 60°. The energy release rate has been

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evaluated at the tip of the notch along different crack propagation directions (different angles θ

referring to Fig. 3E) ranging between 0° (vertical) and 90° (horizontal), with 10° increments.

In this series of simulations, the effect of material’s heterogeneity at the tip of the notch on the

energy release rate was systematically studied by changing the two materials on each side of the

notch. Each side has been modelled as an elastic material with a Poisson’s ratio of 0.3 and a 5

Young’s modulus that could assume two possible values, 170 GPa or 200 GPa, corresponding to

a compliant (C) and a stiff (S) phase, respectively. These values were chosen following the data

obtained from continuous stiffness measurements reported in Fig. S9. Parametric simulations were

carried out for the four possible materials combinations (C-C, S-S, C-S, S-C), for the different

surface traction directions and for the different crack propagation angles. The J-integral values 10

were compared with the values obtained from a model having an average Young’s modulus of 185

GPa.

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Supplementary Figures

Fig. S1. Comparison of commercial razor blades.

Energy dispersive X-ray spectroscopy (EDS) results are shown for several commercial razor blades brands 5

and grades (i.e. razors with 2 or 5 blades). SEM electron backscattered images are shown for each sample,

correlated to the EDS results through the image border color. It is possible to notice that all the razor blades

have a carbide-rich lath martensitic microstructure, and show similar composition from point analyses. Tip

bending in some of the images is an artifact due to blade handling during cartridge removal and sample

polishing. In order to systematically compare the results, the blades used throughout the rest of the paper 10

belong to Sample 1.

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Fig. S2. Blade microstructure.

(A) SEM back-scattered electron image of the blade cross-section near the tip. (B to E) EBSD images

representing an image quality map with carbides highlighted in blue (B), an inverse pole figure map (C), a

Kernel Average Misorientation map (D), and a grain boundary map (E). No preferential orientation appears 5

from the inverse pole figure map (C), but a highly deformed layer (D) from the honing process is present

on both sides of the sharp edge, merging at the tip in a region with higher grain boundary density (E). (F)

Line-profiles from (E) of the Image Quality value along a direction that includes the tip (line 1) and one

perpendicular to it (line 2) to highlight the deformed regions in the blade (tip and sides).

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Fig. S3. Interrupted shaving test.

(A to D) Scanning electron images taken at 20kV, using an Everhart-Thornley detector, showing the

evolution of a portion of the blade at its pristine conditions and after 1, 5 and 10 shaving. The halo around

the sharp edge is formed by the high-voltage imaging of the hard-coatings layer. (E) Outcome of an edge-5

detecting Python algorithm, showing the four edges from top to bottom, translated of an arbitrary constant

amount with respect to each other. The algorithm has been used to determine the wear of the blades from

the profile images in units of volume per unit length, resulting in 12 nm3/nm after 1 and 5 shaves (B and C,

respectively) and 13 nm3/nm after 10 shaves (D). Considering the noise present in the images due to

contamination-induced charging, the results obtained can be considered within the resolution limits of this 10

method.

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Fig. S4. Blade chipping details

Top view of a blade with two chipped regions after an in-situ SEM cutting experiment using Velcro. The

region on the left shows full chipping, meanwhile the one on the right is undergoing failure. The insets

show the ductile fracture surface in the martensitic stainless steel the substrate is made of (left), and the 5

brittle fracture occurring in the hard-coatings layer due to substrate out-of-plane bending (right).

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Fig. S5. Customized clamps for in-situ SEM shaving experiment.

Details of the clamps used to adapt a Deben Gatan micro-tensile stage, used in compression mode, for in-

situ SEM shaving experiments. The blade on the clamp is tilted at approximately 21 degrees with respect

to the travelling direction to replicate its state on the cartridge. Analogously, the hairs are fixed using silver 5

paint to replicate their state in a follicle: almost fixed at one side but able to bend and to be ‘pulled out’.

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Fig. S6. Sequence of in-situ Scanning Electron Microscope shaving experiments.

(A to D) Series of four consecutive in-situ SEM shaving experiments performed using the same portion of

a blade. The image at the top of each sequence is before the cut, the one at the bottom is right after it. The

point of view is shown in the schematic top-left inset in (A). The first three cuts did not produce any damage 5

in the blade (inset in C showing a non-chipped blade), meanwhile the fourth cut produced a chip (inset in

D). The four white arrows in the two insets point at the same features of the blade to help identify the

position of the chip. The two scale bars shown correspond to 20 µm. The sequence of videos of these

experiments is reported in movie S3, together with two more cuts that produced one additional chip each.

10

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Fig. S7. Blade’s martensitic stainless steel mechanical properties.

The first two plots show the engineering and true stress-strain curves, respectively, for the martensitic

stainless steel the blades are made of. The third plot represents the effective true stress as a function of the

effective plastic true strain. This data have been used as input in Abaqus for the elasto-plastic simulations. 5

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Fig. S8. Single-hair cutting-force measurements for different hairs.

Plots of the hair cutting force as a function of blade displacement for three hairs having diameters of 90

µm, 120 µm and 200 µm. The initial value of the load is not zero due to a pre-approach step, during which

the blade and the hair gets in contact. It must be noted that, due to hair bending and dragging during the 5

initial stages of cutting, the “blade displacement” plotted on the x-axis does not correspond to the cutting

depth. This is also the reason why the final value of the displacement of the blade does not match the

diameter of the hair.

10

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Fig. S9. Continuous stiffness measurements results.

Results of continuous-stiffness-measurements mode nanoindentation experiments on the carbide-rich lath

martensitic stainless steel. The two plots show how the values for the hardness (left) and storage modulus

(right) vary with indentation depth for 100 different indents. 5

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Fig. S10. In-situ SEM shaving experiment with half milled blade.

(A) Sideview of a blade partially half-milled using a Focused Ion Beam (point of view in the inset) in order

to be able to look at its cross-section and determine microstructural effects on deformation localization. (B)

Same portion of the blade after having cut several hairs during an in-situ SEM shaving experiment. It is 5

possible to distinguish two features in (B): (i) a chip being created on the left half, and (ii) a post-mortem

image of the blade once the chip has formed. During the chipping formation stage on the left side, it is

possible to notice few cracks starting from the sharp edge, and deformation bands that connect the several

carbides with each other, demonstrating that failure occurs mainly at the weak carbide-martensite matrix

interfaces. The inset at the bottom left in (B) highlights how a crack (yellow arrows) is initiated at the edge 10

when a hair hits the compliant phase separated from a hard carbide (highlighted by red arrows) by an

asperity. The inset on the bottom right of (B), instead, shows a detail of carbide cracking and failure at the

carbide-matrix interface occurring during out-of-plane bending. Finally, on the right half of (B) it is possible

to look at a ‘step forward’ in time: a chip has formed in the blade due to plastic deformation and failure,

meanwhile the hard coatings show a brittle crack. 15

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Movie S1. Sequence of interrupted shaving test.

Animated sequence of low-voltage scanning electron images taken using an Everhart-Thornley detector,

that shows the evolution of a portion of the blade in its pristine conditions and after 1, 5 and 10 shavings.

Despite the contamination accumulated during the ex-situ shaving, it is evident that the edge morphology

does not change significantly between the several shavings. 5

Movie S2. In-situ SEM single hair shaving experiment.

In-situ SEM video of a shaving experiment performed on a single hair being cut against the grain. The hair

bends backwards, changing the stress state on the blade during the cut as well as the interaction volume as 10

the blade penetrates through it. The final outcome consists in two separate chips on the blade.

Movie S3. In-situ SEM sequence of several hair shaving experiments.

Video showing six consecutive in-situ SEM shaving experiments performed with the same blade. The hairs 15

have been aligned such that consecutive cuts are performed by the same portion of the blade. The first three

cuts do not cause any chips on the blade, meanwhile the latter three produce one chip each. The dark region

at the bottom is a marker used on the blade to align the hairs among consecutive cuts.

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Movie S4. In-situ SEM single-hair cutting-force measurement.

Video of the experimental measurement of the force needed to cut a single hair, performed inside a scanning

electron microscope using a Picoindenter setup (inset at the top-center). A plot of the load evolution as a

function of the blade displacement is shown alongside the video captured from the SEM.

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Movie S5. Stress distribution in the blade at different cutting angles.

Animated sequence of the results of the first set of 3D simulations. The images show a color coded von

Mises stress distribution along the lengthwise cross-section of the blade at different angles of application

of the force. The left edge is the plane where symmetry boundary conditions have been applied

(corresponding to the center of the hair). The first four notches from left are in contact with the hair, the 5

fifth is in contact with the hair edge, the sixth is not in contact with the hair. The other four notches outside

the hair have been omitted from these pictures due to low stress intensification.

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