Lab 3: Stress, strain, stiffness, and time-dependent properties of ...

Lab 3: Stress, strain, stiffness, and time-dependent 

properties of biomaterials 

Bio427 — Biomechanics 

This lab explores experimental methods for quantifying the mechanical 

properties of biological materials (recall that material properties 

are independent of structural size and shape). In addition we 

will take our first foray into measurements of the dynamic mechanical 

characteristics of natural systems. We will be using both plant and 

animal material in this lab. 

Goals 

• Learn how to measure force and displacement so that you can 

construct a stress-strain curve 

• Understand data acquisition using a computer and an Arduino 

• Examine the time-dependent properties of biological materials 

• Explore the concept of anisotropy 

• Explore resonance and damping in natural systems 

Conceptual Basis 

Last week we explored the concepts of stress (σ), strain (ɛ ), stiffness 

(E, Young’s modulus) and strength (σ max ). These are all material 

properties and represent a biomechanical ways to characterize materials 

living systems. These physical quantities are summarized here: 

Figure 1: Stress-strain relationship 

for ligament: from Lee and Hyman, 

Modeling of failure mode in knee 

ligaments depending on the strain rate. 

BMC Musculoskeletal Disord. 2002 doi: 

10.1186/1471-2474-3-3 

σ = F A 

(1) 

ɛ = ∆L 

L o 

(2) 

E = dσ 

(3) 

dɛ 

where F is the applied force, A is the cross-sectional area over which 

that force operates, ∆L is the change in length in response to that 

force, and L o is the initial length of the specimen. Young’s modulus 

(E) is the slope of the stress-strain relationship which is constant for 

Hookean materials (that is, they obey Hooke’s Law). However, as we 

may learn in this lab, many biological materials are non-Hookean 

and the relationship between stress and strain is nonlinear. Moreover, 

biomaterials may also show anisotropic behavior – that is, the relationship 

between the stress and strain depends on the direction in

lab 3: stress, strain, stiffness, and time-dependent properties of biomaterials 2 

which the load is applied. For example, any biomaterial containing 

fibers with some average axial orientation will show some level of 

anisotropy. Examples of biomaterials with containing fibers include 

skin, intestine, muscle epimysium, wood, plant stems and many others. 

In addition to nonlinear and anisotropic behaviors, many biomaterials 

are viscoelastic (as we have discussed in lecture), showing a 

time-dependence in their their response to loads. There are a host of 

methods available for quantifying time-dependent properties including 

creep tests (constant force: isotonic loads), stress relaxation tests 

(constant length: isometric loads) as well as cyclic dynamic loads. 

One fascinating application of dynamic loads is called a free vibration 

test (colloquially called a "kaboing-wonga-wonga" test) in 

which you impulsively load a structure and observe the oscillations 

that follow that load. Those oscillations have a particular frequency 

for that structure (called the natural frequency) and, in the presence 

of any visco-elasticity, they damp out in time. The natural frequency 

for a given structure will depend on the stiffness and density of the 

material from which it is made: stiffer materials increase the natural 

frequency, but denser materials decrease the natural frequency. 

For a cylindrical beam of uniform section, there is a simple relationship 

that relates the natural frequency to the geometry of that beam 

and the stiffness and density of the material of which that beam is 

composed: 

ω n = 1.7 

√ 

ER 2 

ρL 4 (4) 

where E is Young’s modulus for the material from which the cylinder 

is made, R is the beam radius, L is the beam length and ρ is the 

density of the beam. For nearly all biological materials, we take the 

density to be 1000 kg m −3 , which is the approximate density of water. 

Thus, if you knew the length and radius of the beam and its density, 

you could observe its natural frequency and compute the stiffness of 

the material that constitutes the beam! 

Figure 2: On November 7, 1940 at 

11:00 am the Tacoma Narrows bridge 

was excited by wind vortices and 

establish large amplitude vibrations. 

This vibrating beam failed and fell into 

the sound. Though no lives were lost, it 

was a narrow escape 

Measuring forces 

In the lab exercises you have done so far, you have used measurements 

of length and time together with mathematical models to 

investigate biomechanical performance of humans in jumping and 

walking. In many cases, however, it is not possible to directly obtain 

the data we want, and in these cases we rely on transducers to 

convert signals from one type of energy to another. 1 If we wish to 

measure changes in force, which is a signal stored as mechanical 

1 

A common example of a transducer is 

the microphone in your cell phone: this 

is an electroacoustic transducer which 

converts sound into an electrical signal. 

Your voice produces propagating 

changes in air pressure, which cause the 

motion of a coil within a magnetic field 

inside the microphone. This results in 

a potential difference across the conductor, 

which constitutes an electrical 

signal which may be transmitted.


energy, we must convert this mechanical energy to electrical energy 

using a force transducer. These devices output a changing voltage 

proportional to the force that is applied to them. One of the most 

common elements used in force transducers is the foil strain gauge, 

which is what you will be using in this lab: these devices, as you will 

see if you look closely at one, are simply a very thin etched metal foil 

that essentially forms a very long, very thin wire that doubles back 

on itself many times – see Figure 3. 

A force transducer may be constructed using strain gauges by 

bonding the gauges to the surface of a linearly elastic (Hookean) 

beam. As the beam is deformed, the strain gauge is stretched very 

slightly. You may recall from physics that the resistance of a wire is 

given by 

R = ρ l (5) 

A 

where R is the resistance of the wire in ohms (Ω), ρ is the electrical 

resistivity of the wire material in ohm-meters (Ω m) 2 , l is the length 

of the wire, and A is the cross sectional area of the wire. So, as the 

strain gauge is stretched, its l increases, thereby increasing its resistance. 

This change in resistance can be measured using a Wheatstone 

bridge, which is a standard device used to measure unknown resistance 

which may be familiar from physics. 

By hanging known masses from the Hookean beam to which 

the gauge is bonded, we may calibrate the force transducer – by 

measuring the voltage output of the transducer when various known 

forces are applied to it, we can obtain a calibration curve which will 

then allow us to measure unknown forces by reading the voltage 

output from the transducer. 

Data acquisition 

Once we have a tranducer capable of converting a force signal to an 

electrical signal, we next need to take that electrical signal and record 

it for later analysis. In this lab course, we will use small, low-cost 

microcontroller boards called Arduinos to measure data from our 

sensors and relay this information to our laptops. 

Arduinos are extraordinarily versatile devices with myriad applications, 

as you will see as this lab course progresses. In this particular 

lab, we have configured the Arduino boards to read in a single 

analog voltage on one of its analog input channels, convert this analog 

signal to a digital representation of that signal 3 , and then relay a 

message containing this digital representation to the laptop via USB 

serial. We can then configure the laptop to listen for these messages 

and store these as data. 

Figure 3: A typical foil strain gauge 

(Source: Wikimedia Commons) 

2 

Note that ρ here is not the density of 

the material, it’s the resistivity. They 

somewhat annoyingly share the same 

symbol. 

Figure 4: Schematic for a wheatstone 

bridge. R 1 , R 2 and R 3 are all known 

resistors, but R X is unknown. In a 

single-gauge strain gauge application, 

R 1−3 are precision resistors, and R X is a 

strain gauge. 

3 

This is roughly analogous to recording 

sound on a computer through a 

microphone and sound card.


Methods 

There are two parts to this lab: stress-strain relationships and dynamic 

vibration tests of whole structures. In the former you will use 

force measurement and data acquisition to plot the stress strain relationship 

for some biomaterials and use that relationship to examine 

the stiffness and the extent to which that material has an anisotropic 

behavior. In the latter part, you will use free vibration tests to extract 

the stiffness of the material that constitutes a plant stem. 

It’s a stretch 

For this part of the lab you will use cyanoacrylate glue (superglue) 

to mount a piece of animal tissue on a force transducer. The animal 

tissue we will be using is pig intestine, which is a wonderfully thin 

and stretchy material typically used by humans as a sausage casing. 

We will measure the stress-strain relationship of pig intestine by 

cutting a small strip of the material and stretching it by applying a 

force to one end. This force will create stress within the material and 

cause it to elongate (strain). As you modulate the amount of stress 

applied to the material, you will measure its deformation and thereby 

derive its stress-strain curve. 

Preparing your sample The pig intestine is obtained from a local 

butcher and arrives sleeved around a strip of plastic. The material is 

difficult to work with once it is removed from the plastic core sheet, 

so you will want to do as much preparation of the sample as possible 

before cutting away the plastic. 

First, determine whether you want to test the material along the 

axial or circumferential direction – you will test the gut in both directions 

for this lab, so the order doesn’t matter. Next, find a clean 

section of the intestine and glue two pieces of stiff plastic to the material 

with a ∼ 2 cm wide section of material between them. Try not 

to glue your fingers to the intestine, or to each other. Once you have 

glued the plastic tabs to the gut, cut away material so that you are left 

with a thin strip of material glued at either end to the tabs. 

Measuring stress and strain In this section, you will obtain a stressstrain 

relationship for your sample by stretching your sample with 

known weights. Begin by ensuring that you have an Arduino board 

connected via USB cable to your laptop, and start the yodaq application. 

When you hit the start button on the GUI, you will see a live 

display of the voltage output from the force transducer. Hang your 

sample from the force transducer using a thin loop of cotton thread,


and adjust the offset knob on the transducer control box so that the 

output voltage from the transducer is 0 V. 

Next, you will hang weights from the free end of the sample to 

apply a stress. Our weights in this case are 1/4-20 nuts, which each 

weigh 12.6 g, and you will suspend these from your sample using a 

hooked paperclip weighing 0.2 g. 

As you add more mass to the free end of the sample, measure 

the extension of the sample with calipers or a ruler. Plot your results 

on the worksheet for both an axially-loaded sample and a 

circumferentially-loaded sample. 

Free vibration 

For this part of the lab we will be measuring the natural frequency 

of vibration for a biomaterial (a stick), and use these data to estimate 

the stiffness of bamboo. 

Recall from Eqn 4 that the natural frequency of a vibrating cylindrical 

beam is determined partly by its stiffness, as well as its density 

and shape. Therefore, if we can measure the natural frequency of 

vibration for a structure where we know the dimensions of the structure 

and the density of the material from which it is composed, we 

can estimate the stiffness of the material. 

We will measure the natural vibration frequency of these beams 

by applying a foil strain gauge (see Section ) directly to the surface of 

the beam, loading that beam and then releasing that load quickly (the 

kaboing-wonga-wonga test). As the beam vibrates, the strain gauge 

will be cyclically stretched and compressed, and we will be able to 

see this variation in strain as a function of time. 

Begin by ensuring that you have an Arduino board connected via 

USB to your laptop, and that the Arduino is supplying power to the 

Industrologic SGAU strain gauge amplifier. The strain gauge will 

form the fourth arm of a Wheatstone bridge, where the other three 

arms are precision 350 Ω resistors. 

Start yodaq again, and you should see an output voltage from the 

strain gauge amplifier that varies as strain is applied to the gauge. 

Adjust the VRO trimpot on the strain gauge amplifier to bring the 

output voltage up to the mid-point of the Arduino’s range. 

Apply a force to the end of the beam, and release this force to 

allow the beam to vibrate freely. You should see the output voltage 

from the strain gauge oscillate accordingly. Hit the “Stop” button 

after one such oscillatory period, and measure the time between 

successive peaks of the oscillation by clicking on the plot area. This 

is the period T of the natural vibration, and from this value you can 

Strain (%) 

Strain (%) 

5 

4 

3 

2 

1 

0 

1 

2 

3 

4 

0 2 4 6 8 10 

4 

3 

2 

1 

0 

1 

2 

3 

0 2 4 6 8 10 

Time (s) 

Figure 5: Kaboing-wonga-wonga tests 

for two otherwise identical cylindrical 

beams of differing length. A single 

period of oscillation is highlighted for 

each beam.


determine the natural frequency: 

ω n = 2π T 

Use this measured value along with Eqn 4 to estimate the Young’s 

modulus of the beam. 

Next, try and clamp the beam at a different point, so that its effective 

length is increased. Does the natural frequency change? Do 

you still get the same value for the Young’s modulus? If not, can you 

speculate as to why?

Lab 3: Stress, strain, stiffness, and time-dependent properties 

Note: Hand in one worksheet per lab group 

Lab Section: 

Name 1: Name 2: 

Stress (σ) 

Strain (ɛ) 

In the graph above, plot the stress strain curve for your samples. Use the same graph for the stress strain 

curves for both axial and circumferential loading. Remember to label your axis ticks! 

Stress-strain data 

1 Sample length (m) 

2 Sample thickness (m) 

3 Sample width (m) 

4 Cross-sectional area (m 2 ) 

Longitudinal 

Circumferential 

Free vibration data 

1 Sample length (m) 

2 Sample radius (m) 

3 Natural frequency (Hz) 

4 Young’s modulus 

Length 1 Length 2 

Questions 

1. What evidence do you have that intestinal lining is anisotropic? What do you think the function of 

anisotropy is in the function of an intestine? 

2. If the beam you used in the free vibration test was anisotropic, how would its natural frequency vary 

with the direction of loading?

Lab 3: Stress, strain, stiffness, and time-dependent properties of ...

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