CE 221: MECHANICS OF SOLIDS I
CHAPTER 3: MECHANICAL
PROPERTIES OF MATERIALS
By
Dr. Krisada Chaiyasarn
Department of Civil Engineering,
Faculty of Engineering
Thammasat university
Outline
• Tension and compression test
• Stress-strain diagram
• Stress-strain behaviour of ductile and brittle materials
• Hooke’s law
• Strain energy
• Poisson’s ratio
• Shear stress-strain diagram
The Tension and Compression Test
• The strength of a material depends on its ability to sustain a
load without undue deformation or failure
• This property is inherent, and can be determined by
experiment, otherwise, we will need to study micromechanics
• The tension and compression test is used to determine the
relationship between the average normal stress and
average normal strain in engineering materials, e.g. metals,
ceramics, polymers and composites
The Tension and Compression Test
• A specimen of the material is made into
a standard shape and size
• Circular cross-section with enlarged
ends to ensure failure not occur at the
grips
• Two punch marks with a constant crosssectional area A0 and gauge length L0
• Strain gauges are placed at the middle
section of the specimen
The Tension and Compression Test
• A specimen is then placed in a machine and stretched at a very
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slow constant rate until it fails
The load P is recorded,
The elongation δ = L – L0 between the punch marks will be
measured using extensometer
δ is used to calculate the average normal strain
Or the strain gauge is used directly to measure strain
The electrical wire is experiencing the same strain and causes the
resistance in electrical wire to change, hence the resistance in the
wire can be converted to strain
The Stress-Strain Diagram
• Normally, specimen may not be made into specific size,
hence the stress-strain diagram is reported instead to
study the material properties of a specimen
Conventional Stress-Strain Diagram
• Nominal or engineering stress assumes the stress is
constant over the cross section and throughout the gauge
length
• Hence, for the nominal stress, the applied load P is
divided by the specimen’s original cross-sectional area A0
• Likewise, nominal or engineering strain, the elongation δ
is divided by the original gauge length L0
The Conventional Stress-Strain Diagram
• The conventional stress-strain diagram is to plot the
corresponding values of σ and ε
• The diagram of a particular material will be similar but not
identical due to
• Slight material’s composition
• Microscopic imperfections
• The way it is manufactured
• The rate of loading
• The temperature
The stress-strain diagram - steel
• Elastic Behaviour
• The curve is a straight line
throughout the region
• Stress is proportional to strain
• The material is said to be linearelastic
• The upper stress is called the
proportional limit σpl
• After this point, the curve will bend
and continue to elastic limit σY
• If the load is removed, the specimen
will return to its original shape
• For steel σpl and σY is very similar,
and hard to detect
The stress-strain diagram - steel
• Yielding
• The material will break down and
cause it to deform permanently
• The stress at this point is called
yield stress or yield point σY
• The deformation is called plastic
deformation
• For carbon steel, the upper yield
point occurs first, then a decrease
in load-carrying capacity to a lower
yield point
• At yield point, the specimen
continues to elongate without
increase in load, this is called
perfectly plastic
The stress-strain diagram - steel
• Strain hardening
• An increase in load can be seen
• The load rises until it reaches a
maximum stress called ultimate
stress σu
• Necking
• The specimen continues to
elongates but the cross-sectional
area starts to decrease
• The decrease is uniform over the
gauge length
• The neck will form and the
specimen continues to elongate
until it breaks at the fracture stress,
σf
True Stress-Strain Diagram
• Actual cross-sectional area is used and instant load is
measured
• This produces actual true stress-strain diagram
• When the strain is small, the conventional and true stressstrain diagram coincide
• The differences is during the strain-hardening range
• The large divergence is seen within the necking region,
the specimen support a decreasing load.
• But the material actually sustains increasing stress until
failure
Engineering Design
• Normally, most engineering design is done within the
elastic range.
• This range, the strain is very small, hence the error using
the true and conventional values is very small, about 0.1%
Stress-Strain Behaviour of Ductile and
Brittle Materials
• Any material that can be subjected to large strains before it
fractures is called a ductile material.
• Example, mild steel
• The percentage elongation is the specimen’s fracture strain
expressed as a percent.
• The percent reduction in area can also be used to specify
ductility
• About 38% for a mild steel for percentage elongation and
60%for percentage reduction in area
Ductile material
• Yielding occurs at constant
stress
• Most metals do not exhibit
constant yielding, and yield point
is not easy to define.
• Normally, a yield strength is
define using an offset method,
where a 0.2% strain is offset,
and a parallel is drawn to define
a yield strength
• 1 ksi = 6.89 MPA
• E.g. brass, molybdenum, zinc,
aluminium
Ductile material
• Yield strength is not a physical
property, but it is a stress that
causes permanent strain
• Here, we assume yield strength,
yield point, elastic limit,
proportional limit all coincide
• Except rubber, which nonlinear
elastic behavior
• Wood is moderately ductile,
varies from species to species
• Wood is directional material
Brittle Materials
• Material exhibit little or no
yielding before failure
• Example, gray cast iron, concrete
• Can withstand much higher
compressive stress
• Cracks and imperfections tend to
close up and bulge out
• For concrete, compressive stress
is 12.5 times greater than tensile
strength
Hooke’s Law
• Most engineering materials exhibit a linear relationship
between stress and strain within the elastic range.
• Robert Hooke discover the law in 1676, and created
Hooke’s law
• E is called modulus of elasticity or Young’s Modulus,
named after Thomas Young
• E is the slope of initial straight-line of the stress-strain
diagram, up to the proportional limit
• E has the same unit as σ
Hooke’s Law
• For steel alloy, from soft
steel to hardest steel, E is
about 200 Gpa
• E can only be used in
material with linear elastic
behaviour
• If the stress is greater
than the proportional limit,
the stress-strain diagram
is not a straight, so E is
no longer valid
Strain Hardening
• If a specimen of ductile material is loaded to the plastic range, then unloaded,
the elastic strain is recovered, but the plastic strain remains.
• Hence the material is subjected to a permanent set.
• When the material is loaded again, it still continue along the elastic line, but
the yield point will be higher.
• It then has greater elastic range, but less plastic region
Strain Energy
• During deformation, a material store energy internally
throughout its volume
• This is called strain energy
Strain Energy
• The strain energy per unit volume or strain-energy
density
• For a linear elastic material, Hooke’s law applies, hence
Modulus of Resilience
• When the stress σ reaches the proportional limit, the strain-energy
density is referred to as the modulus of resilence
• It’s the shaded triangular area under the diagram.
• It is the physical property of a material indicating the ability of the
material to absorb energy without any permanent damage to the
material
Modulus of Toughness
• This quantity in the entire area under the stress-strain diagram.
• It indicates the strain energy density of the material just before it
fractures
• This is an important properties when designing a member that may be
overloaded.
• For steel, by changing the carbon in steel, the diagram will change,
hence the modulus of resilience and toughness will change
Example
Example
Example
Poisson’s Ratio
• When deforming a body, object elongate and contract in
more than one direction
• Example when a rubber is subjected to a compressive
stress, the block contract, but the radius or lateral strain
increase
• S.D. Poisson discover the ratio of elongation and lateral
strain is constant within the elastic range.
• Hence Poisson’s ratio, for an isotropic and
homogeneous material
Poisson’s Ratio
• The negative sign indicate longitudinal elongation and
lateral contraction and vice versa
• Only axial force cause these strain
• Poisson’s ratio has no unit
• For ‘ideal material’, no lateral deformation when stretched
or compressed, Poisson’s ratio will be 0
• Poisson’s ratio has the value 0 ≤ ν ≤ 0.5
Example
The Shear Stress-Strain Diagram
• When a small element is subjected to pure shear, equal
shear stresses are developed directed toward or away on
the corner’s element.
• For a homogeneous and isotropic material, the shear
stress will deform an element uniformly
• Pure shear is studied when a specimen is subjected to
torsion, and a shear stress-strain diagram can be
obtained.
The Shear Stress-Strain Diagram
• The material will exhibit linear-elastic
behaviour and it will have a
proportional limit, τpl, and it will then
reach an ultimate shear stress τu,
and then lose its shear strength and
reach fracture stress, τf
• Hooke’s Law applied for linear-elastic
material
• G is the shear modulus of elasticity
or the modulus of rigidity
• G has the same unit as τ
The Shear Stress-Strain Diagram
• Material constant can be related as
Example
Example
Creep
• When a material has to support a load for a very long
period of time, the permanent deformation is known as
creep
• Creep is time dependent permanent deformation
• For metal and ceramics, creep occurs when members are
subjected to high temperature
• Stress and/or temperature is a major cause of creep
• A member is designed to resist creep strain for a specified
time period, called creep strength
• A simple test is to test several specimens at a constant
temperature, with different axial stress, then measure the
time needed to produce allowable strain, a curve of stress
over time can then be plotted
Creep
• Creep strength will decrease for higher temperature or
higher stress
• Usually a factor of safety is applied to allow for creep, as
creep can be difficult to determine
Fatigue
• When a metal is subjected to repeated cycles of stress, it
causes the structure to break.
• Usually occurs in connecting rods, crankshafts, any part
with cyclic loading
• Fracture will occur at less than material’s yield stress
• Usually causes due to imperfections, when localized
stress is much greater than average stress, can cause
cracks, ductile material behaves like brittle
• Endurance or fatigue limit is the limiting stress when
applying a load for a specified number of cycles
• The S-N diagram or stress-cycle diagram is plotted to
determine endurance, S is stress, N is number of cycles
to failure
Fatigue
• For steel, the endurance is when the stress becomes
horizontal, from the graph it is 27 ksi or 186 Mpa
• For aluminum is not well-defined, we take the stress at the
a limit of 500 million cycles, any stress below this, the
fatigue in infinite.