This document discusses mechanics of fibrous structures and materials. It begins with listing references used for the lectures. It then defines mechanics of materials as dealing with how solid bodies behave under various types of loading. The main goals are to determine stresses, strains, and displacements in structures due to loads. It discusses the importance of understanding mechanics of materials, as the mechanical behavior of materials is essential for safe design of structures like buildings and machines. For textiles specifically, the mechanical behavior is important for processing and end product performance. The document then covers fundamental concepts like stress, strain, tensile and compressive stresses and strains. It discusses analyzing material behavior through tensile testing and plotting stress-strain curves, which provide

1. Mechanics of Fibrous Structure
(TM-3113)
M Irfan
Department of Textile Engineering
National Textile University Faisalabad

2. References
The lectures are based on following reference books
• J.W.S. Hearle. " Structural Mechanics of Fibres Yarns and Fabrics"
• (2004)
• A.E. Bogdanovich, C. M. Pastore. "Mechanics of Textile and
Laminated Composites" (1996)
• B. Strong. "Plastics (Materials & Processing) (2001)
• Ferdinand, P. Beer, E. Russell Johnston Jr., John, T. Dewolf.
"Mechanics of Materials" (2004)
• Jinlian, Hu. "Structure and Mechanics of Woven Fabrics" (2004)
• Thormen H Courty. "Mechanical Behavior of Materials" (2005)
• E.J.Hearn. "Mechanics of Materials" (2001)
• P. Schwartz " Structure & Mechanics of Textile Assemblies" (2003)
• W.E.Morton. " Physical properties of Textile Fibres" (2002)
• W.A. Hanton. " Mechanics for Textile Students" (2007)

3. Mechanics of Materials
• “Mechanics of materials is a branch of applied mechanics that deals
with the behavior of solid bodies subjected to various types of loading”.
• The main objective of the mechanics of materials is to determine
stresses, strains, and displacements in structures and their components
due to the loads acting on them.
• The complete picture of mechanical behavior of materials (or different
structures made of materials) can be obtained if we can find these
quantities for load values up to the point of failure.

4. Importance of Mechanics of Materials
• Mechanics of materials is an important subject in many engineering
disciplines as understanding mechanical behavior of materials under
various forms of loading is essential for safe and secure design of any kind
of structure.
– buildings, bridges, machines, airplanes, aerospace structures etc.
• Mechanics of materials deals with mechanical behavior of real bodies as it
determines stresses and strains within bodies of finite dimensions that
deform under load.

5. Importance of Mechanics of Materials
Failure may happen due to wrong
material selection and design,
manufacturing and design faults

6. Importance in textiles
• Mechanical behavior of textile materials and structures, i.e.
response of textile materials and/or structures to applied
forces and deformations, is very important
– Processing in downstream processes
– Performance of end product

7. Fundamental concepts: stress and strain
• The most fundamental concepts in mechanics of materials are stress and strain
• Stress: Stress can be defined as force per unit area of a material. It is denoted by
Greek letter σ (sigma)
σ = 𝑃/𝐴
• SI unites of stress are N/m2 or Pascal

8. Stretch
Restoration force

9. Example:
Take a prismatic bar (a prismatic bar is a bar with uniform cross
sectional area) which is under axial load (a force directed along the
axis of the bar)

10. Fundamental concepts: stress and strain
• Suppose the axial force P is the only force acting
at the ends of the bar
• the original length of the bar is denoted by the
letter L, and the increase in length due to the
loads is denoted by the Greek letter δ (delta)
• mn is the cross section perpendicular to the
longitudinal axis of the bar
• Under the load, the bar is under continuously
distributed stresses acting on the cross section
mn with axial force P as the resultant of those
stresses
• The resultant P is equal to the magnitude of the
stress times the cross-sectional area A of the bar,
i.e P= σA
• Thus we get the relation σ = 𝑃/𝐴

11. • Normal Stresses: in case the stresses act in a direction
perpendicular (normal) to the cross section of the bar, these
are called normal stresses.
• When the bar is stretched by the force P, the stress is called
tensile stress
• If the bar is compressed, the stress is known as compressive
stress
• Tension force will tend to lengthen the bar, while compressive
stress will tend to shorten the bar
• tensile stresses are considered positive and compressive
stresses as negative
• Applied load can be static or dynamic
Fundamental concepts: stress and strain

12. • Strain:
– Change in the dimensions of an object occurs under the influence of an external
load, the ratio of this change to the original length of the object is known as
strain
– Change in dimension means becoming longer under tension or shorter when in
compression
– In the previous example of the bar, δ represents the elongation of the bar under
tension, then the strain (represented by Greek word ε, epsilon) is given by the
equation ε = δ/𝐿
Fundamental concepts: stress and strain

13. Fundamental concepts: stress and strain
• Strain:
– When the bar is in tension, the strain is called a tensile strain that tends
to elongate the bar
– When the bar is in compression, the strain is called compressive strain
that tends to shorten the bar
– The strain associated with normal stresses is known as normal strain
– Since strain is ratio between two lengths, it is dimensionless and has no
units
– Strain can be expressed as a percent
– Tensile strain is usually taken as positive and compressive strain as
negative.

14. How to analyze mechanical behavior of
materials
• The usual way to analyze how materials behave
when they are subjected to different loads is to
do tensile testing on tensile test machine
• Sample is gripped between jaws of the machine
and load is applied. The deformation under
load is recorded
• For comparison of different results, the test
method (specimen size, load application
method) should be standardized
• One of the major standards are ASTM
standards (American Society for Testing and
Materials)
• Compression tests are also carried. For
example, concrete is mostly tested under
compression

15. Stress-strain curves
• The results obtained from the testing machine (load applied and resulting
deformation) are converted to stresses and strains.
• The load can be applied under various modes, for example static or dynamic
load
– Static load: non fluctuating load, that is the rate of loading is not of interest,
usually caused by gravity
– Dynamic load: alternating load, can be in cyclic manner, the magnitude and
sign of the load changes with time

16. Types of loading

17. Stress-strain curves
In the example of a test bar, the axial stress in the bar is calculated by dividing
the axial load P by the cross-sectional area A
The obtained stress is called nominal stress if initial area of the bar is used in
the calculation
The stress is called true stress if it is calculated by using the actual area of the
bar at the cross section where failure occurred

18. Stress-strain curves
• The average axial strain ε in the test specimen is found by
dividing the measured elongation δ between the gauge
marks by the gauge length L (δ /L)
• The length of the specimen between the gauge marks
(between the machine jaws) is known as gauge length L
• If the initial gauge length (length of the specimen before
test) is used in the calculation, the obtained strain is known
as nominal strain
• Since the distance between the gauge marks increases as the
tensile load is applied, we can calculate the true strain (or
natural strain) at any value of the load by using the actual
distance between the gauge marks
• In tension, true strain is always smaller than nominal strain

19. Stress-strain curves
• After performing a tension or compression test and
determining the stress and strain at various magnitudes of
the load, we can plot a diagram of stress versus strain. Such a
stress-strain diagram is a characteristic of the particular
material being tested and conveys important information
about the mechanical properties and behavior of the
material
• Strains are plotted on the horizontal axis and stresses on the
vertical axis
• A typical stress-strain curve for mild steel is shown in the
next slide

20. Stress-strain curves
• Figure shows a stress-
strain curve when a steel
circular bar of uniform
cross-section is subjected
to a gradually increasing
tensile load until failure
occur
• When loaded, the change
in length is recorded by
extensometer and graph
of stress and strain is
produced

21. Stress-strain curves
• The diagram begins with a straight line from the origin O to
point A, which means that the relationship between stress and
strain in this initial region is not only linear but also
proportional
• Beyond point A, the proportionality between stress and strain
no longer exists; hence the stress at A is called the
proportional limit.
• The slope of the straight line from O to A is called the modulus
of elasticity. Because the slope has units of stress divided by
strain, modulus of elasticity has the same units as stress
• For a short period beyond this point the material may still be
elastic in the sense that deformations are completely
recovered when load is removed. The limiting point B for this
condition is termed the elastic limit, but for most practical
purposes the point A and B are considered coincident

22. • Hook’s Region
Elastic region

23. Stress-strain curves
• Beyond the elastic limit plastic deformation occurs and strains are
not totally recoverable thus leading to permanent deformation or
permanent set
• Point C is termed as upper yield point and point D is termed as lower
yield point
– upper yield point corresponds to the load reached just before yield starts
– lower yield point, which corresponds to the load required to maintain yield
• For some materials upper and lower yield points may coincide while
for some other materials yield point may not exist at all
• Beyond the yield point some increase in load is required to take the
strain to point E on the graph
• Between D and E the material is said to be in the elastic-plastic
state, some of the section remaining elastic and hence contributing
to recovery of the original dimensions if load is removed, the
remainder being plastic

24. Stress-strain curves
• Beyond E the cross-sectional area of the bar begins to
reduce rapidly over a relatively small length of the bar
and the bar is said to neck
• This necking takes place whilst the load reduces, and
fracture of the bar finally occurs at point F
• The nominal stress at failure, termed the maximum or
ultimate tensile stress, is given by the load at E divided
by the original cross-sectional area of the bar. This is also
known as the tensile strength of the material of the bar.
Owing to the large reduction in area produced by the
necking process the actual stress at fracture is often
greater than the above value. Since, however, designers
are interested in maximum loads which can be carried by
the complete cross-section, the stress at fracture is
seldom of any practical value

25. Stress-strain curves
• If the actual cross-sectional area at the narrow part of
the neck is used to calculate the stress, the true stress-
strain curve is obtained
• The total load the bar can carry does indeed diminish
after the ultimate stress is reached (as shown by curve
DE), but this reduction is due to the decrease in area of
the bar and not due to a loss in strength of the material
itself
• In reality, the material withstands an increase in true
stress up to failure
• Because most structures are expected to function at
stresses below the proportional limit, the conventional
stress-strain curve OABCDEF, which is based upon the
original cross-sectional area of the specimen and is easy
to determine, provides satisfactory information for use in
engineering design

26. Stress-strain curves
• After undergoing the large
strains that occur during
yielding , the steel begins to
strain harden
• During strain hardening, the
material undergoes changes in
its crystalline structure,
resulting in increased resistance
of the material to further
deformation. Elongation of the
test specimen in this region
requires an increase in the
tensile load, and therefore the
stress-strain diagram has a
positive slope in this region
Strain hardening

27. Stress-strain curves
Another example of typical stress-strain curve for steel
True stress
Not drawn to scale
Drawn to scale

28. Stress-strain curves
• The presence of a clearly defined yield point followed by
large plastic strains is an important characteristic of
materials like structural steel
• Metals such as structural steel that undergo large
permanent strains before failure are classified as ductile
• A desirable feature of ductile materials is that visible
distortions occur if the loads become too large, thus
providing an opportunity to take remedial action before an
actual fracture occurs
• Materials exhibiting ductile behavior are capable of
absorbing large amounts of strain energy prior to fracture

29. Stress-strain curves
• Some materials, for example
aluminum alloys, typically do not
have a clearly definable yield point
• they do have an initial linear region
with a recognizable proportional limit
• For such material that does not have
an obvious yield point and yet
undergoes large strains after the
proportional limit is exceeded, an
arbitrary yield stress may be
determined by the offset method
• it should be distinguished from a true
yield stress by referring to it as the
offset yield stress

30. Stress-strain curves
• Stress-strain curves for rubber:

31. Stress-strain curves
• Stress-strain curves for rubber:
– Rubber maintains a linear relationship between stress and strain up
to relatively large strains as compared to metals
– Beyond the proportional limit, the behavior depends upon the type
of rubber
– Some kinds of soft rubber will stretch enormously without failure,
reaching lengths several times their original lengths
– The material eventually offers increasing resistance to the load, and
the stress-strain curve turns markedly upward
– although rubber exhibits very large strains, it is not a ductile
material because the strains are not permanent. It is, of course, an
elastic material

32. Stress-strain curves
• The ductility of a material in tension can be characterized by
its elongation or by the reduction in area at the cross
section where fracture occurs. The percent elongation is
defined as follows:
𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝑒𝑙𝑜𝑛𝑔𝑎𝑡𝑖𝑜𝑛 = (𝐿1 − 𝐿𝑜)/𝐿𝑜
where Lo is the original gauge length and L1 is the distance
between the gauge marks at fracture
• The percent reduction in area measures the amount of
necking that occurs and is defined as follows:
𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝑟𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑖𝑛 𝑎𝑟𝑒𝑎 = (𝐴1 − 𝐴𝑜)/𝐴𝑜
in which Ao is the original cross-sectional area and A1 is the
final area at the fracture section

33. Modulus of elasticity: Hook’s law
• Most engineering structures are designed to undergo relatively
small deformations, involving only the straight-line portion of the
corresponding stress-strain diagram. For that initial portion of the
diagram, the stress σ is directly proportional to the strain ε and is
known as Hooks’ Law
𝛿 = 𝐸𝜀
The coefficient E is called the modulus of elasticity of the material
involved, or also Young's modulus
• modulus E is expressed in the same units as the stress namely in
pascals, and in psi if U.S. customary units are used
• The largest value of the stress for which Hooke's law can be used
for a given material is known as the proportional limit of that
material
• In the case of ductile materials possessing a well-defined yield
point, the proportional limit almost coincides with the yield point

34. Modulus of elasticity: Hook’s law
• large variations in the yield
strength, ultimate strength and
final strain (ductility) but same
modulus of elasticity and
stiffness

35. Modulus of elasticity: Hook’s law
• Some of the physical properties of structural metals, such as
strength, ductility, and corrosion resistance, can be greatly
affected by alloying, heat treatment, and the manufacturing
process used. For example, we note from the stress-strain
diagrams of pure iron and of three different grades of steel
(shown in next slide) that large variations in the yield
strength, ultimate strength and final strain (ductility) exist
among these four metals. All of them, however, possess the
same modulus of elasticity; in other words, their "stiffness,"
or ability to resist a deformation within the linear range, is
the same. Therefore, if a high-strength steel is substituted
for a lower-strength steel in a given structure, and if all
dimensions are kept the same, the structure will have an
increased load carrying capacity, but its stiffness will remain
unchanged

36. stress and strain curves of different fibers

37. stress and strain curves of different fibers