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 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

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

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

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 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

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 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

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.