Chapter 3: Materials
Deformability and strength of each element of the living body is supporting a life
activity.

In this chapter, we learn the basics of materials engineering, while thinking

about how to analyze the various parts of the body as material.

3.1 Deformation
3.1.1 Classification of deformation
If you hang a weight on a coil spring, the coil spring stretches.
increases, as the weight increases.
mass of the weight.

The stretch

The increase of the length is proportional to the

If the weight is removed, the length of the spring returns to its

original value (Fig. 3.1).

Fig. 3.1: Spring.

© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, JapanAn object deforms, when it receives forces.

The deformation is classified into

extension, compression, bending, and torsion (Fig. 3.2).

The direction of loading isclassified into extension, compression, bending, and twisting.

To compare thedeformability, the deformation according to the internal pressure is measured in a pipewith a cavity, such as a blood vessel.

Fig. 3.2: Mode of deformation.

3.1.2 Cutting and fixation of specimen
An organism is composed of various elements.
artery consists of multiple layers of tissue.

For example, a vessel wall of an

A layer of endothelial cells (endothelium)

covers inner wall of a layer of smooth muscle. Each part of the biological tissue has© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japanorientation: a cell, a group of cells, an extracellular matrix, and a group of extracellularmatrices (Fig. 3.3).

Fig. 3.3: Orientation of cells and matrices.

A part of objective material is cut for material testing.
handled as a composite material with an orientation.

A part of a living body is

Apart from a non-directional

uniform material, the structure and the direction of elements should be selected for a
specimen.
The direction of load on the material is also selected: tangential, radial, or
longitudinal direction in a tubular material (e.g. blood vessel wall) (Fig. 3.4).

The

direction should be defined according to the direction of the structure of biologicalmaterials: the cell, the cell-extracellular matrix or the tissue.

It is important to cut out

the specimen into a shape suitable for a direction to apply the load.

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Fig. 3.4: Direction of forces at tube wall.

Several biological materials have large compliance.
necessary on the chucking of the specimen.

The special attention is

The strong fixing force makes the local

deformation of the fixed part, where destruction might occur.

On the contrary, theweak fixation produces slip at fixation point.
The transmission of the force to the specimen should be examined carefully.
many cases, cells are cultured on the scaffold.

In

When the scaffold deforms, thedeformation of the cell depends on the adhesion between the cell and the scaffold (Fig.
3.5).

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Fig. 3.5: Transmission of force.

If the cell adheres to the scaffold with every area, the deformation of the cellsynchronizes with that of the scaffold. When the cell does not adhere to the scaffold
with several areas, the deformation of the cell does not synchronize with that of thescaffold.

The deformation depends on the combination of the adhesion portion.

cell may be sheared, even if the scaffold is pulled.

The

When the slip occurs at theadhesion part, the deformation of the scaffold is not transmitted to the cell (Fig. 3.5(a)).
On the other hand, the shear stress can be transmitted to the cell in the fluid (Fig.
3.5(b)) [14].

By the shear stress in the fluid on the wall, the cell adhered on the wall

deforms, migrates and peels off (see 4.2.5).
In a material of large compliance, the direction of deformation should be flexible.
The equilibrium point is explored, while the force is applied (see 3.1.3).

© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, JapanA device with a large area of contact surface disperses the fixation pressure.

Thepneumatic pressure at the fixation device can control the fixation pressure (Fig. 3.6(a)).
The film-like or linear specimen is rolled around a cylindrical fixation device.

The

frictional force between the specimen and the cylinder disperses fixation force to avoid
local concentration of fixation stress (Fig. 3.6(b)).

The larger cross-sectional area offixation part of the specimen is effective to increase the fixation force (Fig. 3.6(c)).

Fig. 3.6: Fixation.

3.1.3 Setting of origin and range of measurement
Because of the various state of each element in the composite specimen, the originof the tensile test should carefully be determined.

An initial load should be added at© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

the origin, in order to apply load on every element in the specimen.
The ligament shows large deformation, when the load is small.
increase of deformation decreases, as the increase of the load.

The rate of

The elastic modulus of

the whole material might increase, as each element in the ligaments aligns to thedirection of the load (Fig 3.7).

Fig. 3.7: Origin.

To escape from concentration of the deformation adjacent to the fixing portion, thepart for measuring the deformation of the specimen is set at the distant portion from the© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japanfixed portion. At the measurement part, the uniformity of the deformability of thematerial had better been achieved: the uniform cross-sectional area of the specimen.
The mark indicating the measurement position (Fig. 3.6(c)) is effective.

3.1.4 Stress-strain diagram
When a load is applied to the material, the deformation progresses with increase of
the load. The thin material is easy to deform than a thick material.

Instead of theforce, the force per cross-sectional area is used to compare the deformability of thematerial.

The force per cross-sectional area is called the “stress”.

The unit of thestress is Pa (Pascal), which is the same as that of the pressure.
The pressure is defined as the balanced force per unit area in the free flowing fluid;
such as gas or liquid. The pressure is perpendicularly loaded on every plane.

The

stress, on the other hand, is defined as the force per unit area, and has a direction at anarbitrary plane (see 4.1.1). The stress (compression and tensile) perpendicular to theplane is called the “normal stress”.

The stress parallel to the plane is called the“shear stress” (Fig. 3.8).

Fig. 3.8: Stress.

The larger material generates the larger deformation.

The ratio of the deformed

length Δx against the original length x is called “strain” ε.

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ε＝Δx／x(3.1)

The strain is the ratio of the “length” and “length”.
The term “strain” is conventionally used like a unit.

It is a dimensionless number.
In the deformation of the solid

metal, “micro-strain” is used as 10－6.
A material is not uniformly deformed in all directions.

When the material isdeformed in one direction, the deformation occurs also in the vertical direction.
ratio is expressed by “Poisson's ratio”.

The

When the material is pulled in one direction,

the ratio of “elongation at the parallel direction of the force application (strain Δx)” to“contraction at the vertical direction (strain Δy)” is Poisson's ratio κ.
κ＝Δy／Δx(3.2)

Δy =－Δx in the uniform extension to every direction, which has the Poisson's ratioof κ =－1.
Consider about the following deformation: “the quadrangular prism with the heightx at the side y of the bottom square”, while maintaining its volume, deforms to “thequadrangular prism with the height x + Δx at the side y－Δy of the bottom square” (Fig.
3.9).

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Fig. 3.9: Poisson’s ratio.
(x＋Δx)(y－Δy)2＝xy2(3.3)

If you expand the left-hand side of the equation,(x＋Δx)(y2－2 y Δy＋Δy2)＝x y2＋y2Δx－2 x y Δy－2 y Δx Δy＋x Δy2＋Δx Δy2
＝x y2

(3.4)

When Δx and Δy are minute amounts, the square term of Δx and Δy can bewithdrawn.

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y2 Δx－2 x y Δy＝0

(3.5)

Since x = y in the cube,x2 Δx－2 x2 Δy＝0

(3.6)

Δy／Δx＝1／2(3.7)

Poisson's ratio is 0.5, when the cube is deformed into a rectangular with maintaining
its volume (Eq. 3.7).

Poisson's ratio is usually smaller than 0.5 in the solid, because

the volume increases slightly with the elongation.
In the solid, the tension in one direction usually causes the compressive strain in theperpendicular direction, and the shear strain in the oblique direction (Fig. 3.10).

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Fig. 3.10: Strains in tension.

A metal resistor, known as a strain gauge, is frequently used to measure the strain.
The strain gauges has a direction at the resistor.

Attention should be payed to thedirection of the strain and the strain gauge.
The resistance of the strain gauge changes according to the strain.

In other words,

the electrical resistance changes according to the length and cross-sectional area of themetal resistor (Fig. 3.11). When the wire of resistance is pulled, both the increase of
the length and the decrease of the cross-sectional area of the wire increases the electricalresistance. The change of the electrical resistance is detected by the bridge circuit (Fig.

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2.11).
The graph representing the relationship between the strain (the horizontal axis) andthe stress (the vertical axis) is called stress-strain diagram (Fig. 3.12).

Thecross-sectional area of the specimen changes with the deformation.

Fig. 3.11: Strain gauge.

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Fig. 3.12: Stress-strain diagram.

For example, when the length extends from x0 to x1 by a load F1, the cross-sectional
area of the vertical direction decreases from A0 to A1.

The value τ1 calculated by

dividing the load F1 by the cross-sectional area A1 at the time of deformation is called
the true stress.

The value τ0, on the other hand, calculated by dividing the load F1 by

the cross-sectional area A0 before deformation, is called nominal stress (Fig. 3.13).

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Fig. 3.13: True stress.
τ1＝F1／A1(3.8)

τ0＝F1／A0(3.9)

3.1.5 Elastic region and plastic region
The reversible deformation to its original shape after removal of the force is called
the “elastic deformation”.
modulus E (Fig. 3.12).
stress.

The ratio of “stress τ” and “strain ε” (τ / ε) is called elastic

The unit of the elastic modulus is Pa, which is the same unit as

The stress is proportional to the strain at the idealized Hookean elastic

material.
The reversible deformation region, where the shape returns to its original one after
exclusion of the load, is called the elastic region.

In the region, the relationship© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japanbetween the strain and the load is one-to-one correspondence.
independent of the order of the load.

Elastic deformation is

In other words, it does not have the hysteresis

effect.
In the elastic deformation model, the deformation rate does not affect the
deformation.

It is instantly deformed, when a load is applied.

In the viscous flow

model (see 4.1.2), on the other hand, the deformation depends on the deformation rate.
Beyond the elastic region, the deformation remains after exclusion of the load.
The deformation is called plastic deformation, and the region is called the plastic
region.

The boundary between the elastic region and the plastic region is called yield

point (Fig. 3.12).
The composite material may have several boundaries like a yield point. Most of thebiological tissue has the hysteresis effect at deformation.

The attention should be

payed to the order of the deformation process at the biological tissue.

3.1.6 Sphere
At the hemisphere of radius r, “the product of the pressure difference ΔP (between
the inner pressure and outer pressure across the sphere) and the area πr2 of thecross-sectional circle of the sphere” balances with “the product of the circumference
length 2πr of the cross-sectional circle of the sphere and the surface tension per unit
length γ of circumference of the sphere” (Fig. 3.14).

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Fig. 3.14: Balance of forces in hemisphere.
ΔP π r2＝2π r γ(3.10)

By transforming the equation (3.10),
ΔP＝2γ／r(3.11)The equation (3.10) is the expression of Young-Laplace at the droplet of the sphere.
Unit of the surface tension γ is N m－1.

When an erythrocyte is aspirated by thecapillary, the tension γ of the erythrocyte membrane can be estimated by the above
equation with the radiuses of curvature r1 and r2 and with the ambient pressures P1 andP3 (Fig. 3.15).

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Fig. 3.15: Tensile force at membrane of erythrocyte.

To measure the deformation of the suspended particle in the liquid, the counterrotating rheoscope (see 4.2.4) is available.

The deformation can be quantified by theratio of the major axis and minor axis at the spheroid deformed from the sphere in theshear field (see 6.1.1).

3.1.7 Bending
The bending test is applied to measure the bending deformation of the material (Fig.
3.16). The graph, which displays the distribution of shearing force at the material, iscalled shearing force diagram.

In the shear force diagram, the shear force of“downward on the left side of the cross section and upward on the right side of the crosssection (counterclockwise)" is defined as positive.

The graph, which displays the

distribution of the bending moment at the material is called the bending moment
diagram.

In the bending moment diagram, the upward moment is defined as positive.

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Fig. 3.16: Bending.

In Fig. 3.16, consider the shear force and the bending moment in the cross-section.
The force F is applied to the vertical direction at the center of the straight horizontal
beam of length a. The force applied at a point is called concentrated load.

The

forces applied with distribution, on the other hand, is called distributed load: thegravity due to the mass of the beam itself is taken into account.
In Fig. 3.17 (a), the beam is supported by both ends of the simple support. The
moment is not generated at the simple support, since the rotation is not fixed at the ends.
At the rigid support, on the other hand, the moment M is generated, since the end
cannot rotate (Fig. 3.17 (b)).

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Fig. 3.17: Simple & rigid support.

The x-axis is defined along the beam (the length of a) from the left end to the rightin Fig. 3.16(a).

The upward force is defined as positive.

Eq. 3.12 describes thebalance of the forces F.
F／2＋F／2－F ＝0(3.12)

The counter-clockwise moment is defined as positive in Fig. 3.16.

Eq. 3.13

describes the balance of moment M around the left edge of the beam.
－(a／2)F＋a(F／2)＝0(3.13)In Fig. 3.16 (b), the force F/2 downward at the position of x is necessary for thebalance of the force at the parts of the beam: from the left end to the position x (0 < x <© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japana/2).

The force F/2 upward at the position of x, on the other hand, is necessary for thebalance of the force at the remaining right part of the beam: from the position x to theright end.

Thus, shearing forces are generated in the cross-section at the position x of

the beam: downward left and upward right (counterclockwise).
In Fig. 3.16 (b), the counterclockwise moment M at the position of x is necessary forthe balance of the moment at the parts of the beam: from the left end to the position x (0< x < a/2).

M －x (F／2)＝0(3.14)

The clockwise moment M at the position of x, on the other hand, is necessary for thebalance of the moment at the remaining right part of the beam: from the position x to theright end.

Thus, the upward moment is generated in the cross-section at the position xof the beam.
At the three-point bending test in Fig. 3.18, estimate the bending moment andshearing force in the cross-section of the beam.

A downward force F1 is verticallyapplied at the center of the horizontal beam of length a.

At the simple support, eachend of the beam is supported by a vertical upward force F1/2, respectively.
half of the cross-section of the beam, the shear force +F1/2 is applied.
of the cross-section of the beam, － F1/2 is applied.

At the left

At the right half

The bending moment is(a/2)(F1/2) at the center of the beam, and decreases linearly toward zero at both ends.

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Fig. 3.18 Three-point bending test.

At the four-point-bending test in Fig. 3.19, the bending moment and the shear
force in the cross-section are calculated as follows.

The forces F1 are appliedvertically downward at the two points (a/3 and 2a/3 from the left) of the horizontal
beam of the length a, respectively.

The both sides of the simple support are supportedby the vertical upward forces F1, respectively.

At the cross-section in the left part ofone third of the beams, the shear force + F1 is applied.

At the cross-section in the rightpart of one third of the beams, the shear force －F1 is applied.

Bending moment is(a/3) F1 at the central part of the beam between two points of application of the forces,and decreases linearly toward zero at both ends.

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Fig. 3.19: Four-point bending test.

The 4-point bending has both the constant of the moment and zero of the shearing
force between two points of application of the forces.
The bending deformation depends on the shape of the cross section of the beam,
because the larger strain generates at the point farther from the center line (Fig. 3.20).
The value calculated by integrating y2 with respect to the very small area dA at the
distance y from the z-axis is called the moment of inertia of area about the z-axis Iz.

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Fig. 3.20: Center line and strain.
Iz＝∫y2 dA(3.15)The moment of inertia of area represents the difficulty of bending deformation of
the material.

3.2 Properties and Destruction of Material
3.2.1 Fatigue fracture
A material destroys, when it can no longer maintain the continuity as thedeformation proceeds.
“strength”.

The maximum strength before the fracture is referred to the

In some material, maximum distortion is used for the index of thestrength before the destruction (Fig. 3.21).

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Fig. 3.21: Yield and fracture.

The deformation or the load, which does not lead to the fracture, leads to fracture by
repetition in some cases.

This fracture is called fatigue fracture.

The applied wayof cyclic loading can be classified into three types: extension, compression, andextension and compression (Fig. 3.22).

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Fig. 3.22: Repetitive load.

The stripe pattern of the tracings is observed at the fracture surface of the fatigue
fracture.

The pattern is called striation.

The unevenness pattern such as after tearing

is observed, on the other hand, at the instantly fractured surface.
dimple (Fig. 3.23).

The pattern is called

Observation of these patterns of the fracture surface is effective todetermine the fracture mode of the metal material.

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Fig. 3.23 Fracture surface.

The stress amplitude and the number of cycles are parameters at the fatigue
fracture (Fig. 3.24). An erythrocyte has a large deformability, and is hard to break.
The fatigue fracture of the erythrocyte occurs under the repeated deformation [15].

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Fig. 3.24: Stress amplitude vs. number of cycles.

The biconcave of an erythrocyte deforms into an ellipsoid in response to shear
stress in a shear flow (see 6.1.1).

During the tank tread motion of the membrane, themembrane repeats deformation between the tensile and the compression according to
the change of the curvature. When the shear rate increases, the speed of the tank tread
motion increases.

At the fatigue fracture of erythrocyte in the shear fluid, the productof “the shear rate” and “the exposure time to the shear field” corresponds to the numberof repetitions of fatigue failure of the material (Fig. 3.25) [15].

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Fig. 3.25: Erythrocyte fatigue in flow.

In the erythrocyte, the fatigue failure at the membrane makes vibrating deformationof the erythrocyte in the shear flow, the contents of the erythrocytes come out to theoutside (Fig. 3.26) [47].

The phenomenon, that the hemoglobin comes out to theoutside of the erythrocyte, is called hemolysis.

Hemoglobin is a protein, whichtransports oxygen.

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Fig. 3.26: Erythrocyte destruction.

Hemoglobin outside of the red blood cells adsorbs to the tissue, or is discharged tothe outside the organization. The free hemoglobin cannot keep circulating in the blood
circulation circuit, and the oxygen-carrying function is lost.

Moreover, the erythrocyte

with poor deformability is captured in the microcirculation system, and cannot keep
circulating in the blood circulation circuit. The destruction of erythrocytes can befunctionally evaluated by the ratio of hemoglobin released to the outside of erythrocytes(hemolysis ratio) [15].

3.2.2 Crystal and lattice defectA substance consisted of regular atomic arrangement is called a crystal.
crystal is hard to be destroyed because of the strong interatomic force.

The

A substance, of

which the entire material is made of a single crystal, is called monocrystal.
The typical regular arrangements are the face-centered cubic lattice and© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japanclose-packed hexagonal lattice.

The face-centered cubic lattice consists of the layersof the repeated pattern of ABCABCABC.

The close-packed hexagonal lattice consistsof the layers of the repeated pattern of ABABAB (Fig. 3.27).

Fig. 3.27: Close-packed lattice.

The regularity of coordination of the atoms in the three-dimensional space defects atthe surface, which has the open side of the neighbor (Fig. 3.28).

The defect relates tothe fracture initiated from the surface.

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Fig. 3.28: Surface.

The material is usually an aggregation of single crystals, and is called polycrystal.
The crystalline part in the polycrystal is called the crystal grain (Fig. 3.29).

Thebinding force at the interface between the crystal grains is weaker than that in the crystal
grain.

The fracture occurs more frequently at the interface between the crystal grains

than at the inside of the crystal grain.

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Fig. 3.29: Poly-crystal.

In crystals, the displacement of atoms from the regular position is called the latticedefect (Fig. 3.30).
vacancy.

The defect, where the atomic arrangement is missing, is called the

The defect, where the position is shifted, is called the dislocation.

The

plastic deformation can be considered transfer phenomena of the lattice defects.

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Fig. 3.30: Lattice defect.

Micro cracks generates at the point of low strength in the material: the interface
between the crystal grains, and the lattice defect.

The fracture starts at the crack.

The

distribution of cracks in the material should be examined to estimate the destruction of
the material.
The crack inside of the material can be measured by the reflection of the soundwave (non-destructive inspection).

The internal structure is measured by theultrasound, X-rays, and MRI (magnetic resonance imaging) in vivo.

3.2.3 Stress concentration
When the loaded force and the deformation at the material are not uniform in thematerial, the distribution of stress is generated.

A phenomenon of the partial increaseof the stress is called the stress concentration.

The stress concentration in thematerial is often generated, where the shape or the property of the material is
discontinuous.

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The stress concentration occurs at the surface of the small curvature.

The smaller

the curvature of the tip of the crack, the faster the crack progresses.

Increase the radiusof curvature of the crack tip decelerates the progress of cracks.

A notch with theadjusted radius of curvature of the tip is provided at the observation position in the
breaking strength test of materials (Fig. 3.31).

Fig. 3.31: Stress concentration.

The long axis of the cultured cell make orientation, when the repeated extension andcontraction is applied to the scaffold.

The shear stress by the flow affects theorientation and differentiation of cells. The parallel and perpendicular orientations ofcells to the flow direction are observed (Fig. 3.32) [16].

The orientation might beadaptation of the cells to minimize the internal mechanical stress.

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Fig. 3.32(a): Orientation of endothelial cells.

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Fig. 3.32(b): Orientation of C2C12.

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Fig. 3.33: Electrodes.

By the flow stimulation, myoblasts differentiate and fuse to myotubes.

Myotubes

contracts in synchronization with the pulse, when they are stimulated by electrical pulse
with the electrode immersed in the culture medium (Fig. 3.33).

3.2.4 Composite material and environment
A material, which consists of various elements, is called composite material.

The

property of the composite material depends not only on the individual element, but also
on the interaction between elements.
The biological tissue consists of cells and the extracellular matrix.

The© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japanextracellular matrix has a large contribution to the deformation of the tissue.
The cell membrane is the partition between internal and external sides of the cell.
The cell membrane consists of lipid bilayers supported by protein molecules.

The

membrane can be resealed after the fracture, even if the contents are released to theoutside.

The erythrocyte without contents is called ghost.

Without cutting out as a specimen, it is ideal for measurement to apply force to thetargeted tissue in situ.

A laryngoscope is a tool, which is used to assist passing thetube for the artificial respirator through the epiglottis.

The pressure on the epiglottis,

on the tongue, and on the cutting teeth applied by the laryngoscope can be measured
with the pressure-sensitive paper pasted on the blade of the laryngoscope (Fig. 3.34)[17].

The limit pressure can be determined, to prevent damaging the tissue during

intubation.

Fig. 3.34: Force applied on laryngoscope.

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It is not usually easy to use a part of a living body or a whole living body for themeasurement.

The material, which imitates property of the living body, is called

phantom.
Several kinds of phantoms are available for the properties of a living body:mechanical property, electrical property, magnetic property, and thermal property.

Thebottle containing a copper sulfate solution is applied to test the homogeneity of the MRI
image (Fig. 3.35) [18].

The agar is applied to measure the permeability (Fig. 3.36)[19].

Fig. 3.35: Phantom.

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Fig. 3.36: Penetration of phenol-red into agar.

The property of the material is changed by the environment.

The environment invivo is as follows. The electrolyte solution containing polymers is filled in vivo.

Thebody temperature is maintained, the carbon dioxide partial pressure is high, pH is not
7.0 (neutral).

Several phenomena occur at the surface of the material in vivo:

molecular adsorptions, and chemical reactions.

By contact with the blood, several

biological reactions occur at the surface: platelet adhesion, and the formation of clots(see 6.2.4).
Several materials are applied for implantation: stainless steel, titanium alloys, andceramics.

Stainless steel SUS316L used for the artificial joint includes C 0.03%, Cr

17%, Ni 12%, and Mo % to improve the corrosion resistance.

Questions

Q 3.1: List points to be devised, when you get and supply a test piece of the biological© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japantissue for a tensile test.

Q 3.2: Estimate the number of repetitions of the opening and closing of the leaflets of
the artificial valve in 10 years to think about the durability of the pivot.

Q 3.3: Find the filling rate, when rigid spheres are closely stacked without a gap in asequence of the face-centered cubic lattice (Fig. 3.37).

Q 3.4: Describe ingenuity in the fracture test of materials, and features of the fracturesurface.

Q. 3.3 Fig. 3.37: Face-centered cubic unit lattice.

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