ce 221: mechanics of solids i chapter 3: · pdf filechapter 3: mechanical properties of...
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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
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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
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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 micro-mechanics
• 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
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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 cross-sectional area A0 and gauge length L0
• Strain gauges are placed at the middle section of the specimen
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The Tension and Compression Test • A specimen is then placed in a machine and stretched at a very
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
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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
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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
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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
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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 linear-
elastic • 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
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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
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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
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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 stress-
strain 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
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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%
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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
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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
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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
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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
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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
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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
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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
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Strain Energy • During deformation, a material store energy internally
throughout its volume • This is called strain energy
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Strain Energy • The strain energy per unit volume or strain-energy
density
• For a linear elastic material, Hooke’s law applies, hence
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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
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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
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Example
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Example
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Example
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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
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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
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Example
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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.
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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
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The Shear Stress-Strain Diagram • Material constant can be related as
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Example
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Example
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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
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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
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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
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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.
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