MECHANICAL BEHAVIOR OF MATERIALS

MECHANICAL PROPERTIES

Mechanical behaviour of a material, i.e. its response or deformation to an applied external load in terms of strength, hardness, toughness, and ductility, is important for a structural or design engineer. Different types of loads as steady load, impact load, loads applied over long duration, cyclic loads, etc. are applied on standard specimens to study various mechanical properties.

Figure 1 Schematic representation of the tensile testing machine. Specimen is elongated by the movable fixture; the magnitude of the applied force and the amount of extension are measured by the load cell and extensometer, respectively.

movable fixture

specimen

extensometer

load cell

TENSILE TESTING

Figure 2 Gauge length of a standard sample

lo = gauge length

F F

Ao

radius:

dia: dia:

A simple tensile test is used to determine yield strength, ultimate strength, ductility, Young’s modulus of material. Figure 2 shows the gauge length of a

standard sample, l0 area of cross-section is A0, subjected to tensile load F, then engineering stress,

0AF

and engineering strain,

If the graph between force and change in length is linear as shown in Figure 3, the slope of the curve between F versus dl gives Young’s modulus

Figure 3 Force extension curve

As the bar elongates, there is increase in length but there is decrease in diameter of the bar. If the axial strain, i.e. δl/l, is positive, then lateral strain

δD/D is negative. The ratio of lateral strain to longitudinal or axial strain is known as Poisson’s ratio.

Figure 4 Typical stress – strain curve of a ductile specimen

Str

ess

Strain

Strain

Str

ess

Figure 5 Stress – strain curves for ductile and brittle materials

Strain

Str

ess

ductile

brittle

For the non-linear behaviour, either the tangent modulus or secant modulus is normally used. Figure 6 shows a non-linear graph between force and change in length. At point A, secant modulus, Esec = slope of line OA

tangent modulus, Etan = slope of tangent TT at A

Figure 6 Force extension curve

On atomic scale, Young’s modulus is defined as

where x is the distance between two atoms, and x0 is the interatomic distance for equilibrium, Figure 7.

Figure 7 Net force vs interatomic distance graph

Theoretical value of Young’s modulus on atomic scale is about 100 times the value of Young’s modulus of actual solid material.

If a rectangular block fixed at one face is subjected to shear stress t on the opposite face, shear strain g is developed. If variation of g is linear with the variation of shear stress, t, Figure 8, then

Figure 8 Shear stress and shear strain

Shear stress vs

Shear strain

g

g

Modulus of rigidity,

g

tG

Elastic constants E, G, and n for engineering materials are given in the Table.

Elastic Limit Point P corresponds to the elastic limit of the material where micro sclae plastic deformation starts.

Figure 9 Elastic limit on the stress-strain curve

Strain

Str

ess

Str

ess

Strain

Lower yield

point

upper yield

point

elastic plastic

y

y

Yield Strength Yielding is characterized by initial departure from linearity of stress-strain curve. Deformation in any member after yielding has begun is permanent and many structures are designed only for the elastic deformation. Therefore, it is very important to know the yield point, where there is the onset of yielding. Figure 10 shows stress-strain curve for a ductile material, in which σ – ε curve has departed from linearity at point A. This point is known as yield point and the strength of the material at this point is known as yield strength.

Figure 10 Stress-strain curve

Yield Strength For some materials, yield point is not clearly observed, where there is non-linear elastic region. The usual practice is to define the yield strength as a stress to produce some specified amount of strain, i.e. 0.002 or 0.2% strain as shown in Figure 11. A tangent is drawn at origin O on the non-linear σ – ε curve. Taking OB = 0.002 strain, a line BA is drawn parallel to tangent at O. Then stress at A (AB intersecting the stress-strain curve at A) is defined as 0.2%, Proof Stress.

Figure 11 Proof stress

For some materials, elastic–plastic deformation is well defined and occurs abruptly at yield point. The material exhibits two yield points, i.e. upper and lower yield points. At the upper yield point, plastic deformation is initiated with an actual decrease in internal resistance of the material as in the case of mild steel, a medium carbon steel. This steel contains atmosphere of carbon and nitrogen atoms as impurities. Carbon and nitrogen atoms are smaller than iron atoms. During the plastic deformation, there is the movement of dislocations on slip planes and these dislocations glide over easily on carbon and nitrogen atoms and the resistance of the material is decreased but when these dislocations reach grain boundaries, the resistance of the material is increased because grain boundaries are harder than internal structure. As a result, there is increase in stress from lower yield point as shown in Figure 21.

Figure 12 Stress-strain curve for mild steel

After upper yield point, continued deformation fluctuates slightly about some constant stress termed as lower yield point and subsequently rises with increasing strain, so the lower yield point is also known as point of strain hardening. At the maximum load, point D, a necking takes place in the material, and further extension takes places in the material in the vicinity of the neck. The material becomes plastically unstable, developing a neck. Dislocations in the material are responsible for neck formation. Stress at the maximum load is termed as ultimate tensile strength, i.e.,

Specimen breaks making cup and cone type fracture, a typical behaviour of mild steel (Figure 13).

Figure 13 Cup-cone fracture

Ultimate Tensile Strength It is the maximum value of the stress on the engineering stress – strain curve.

Figure 14 Various stages of the stress – strain curve as the specimen elongates and undergoes necking. Point M corresponds to the ultimate tensile strength (UTS)

Strain

Str

ess

uts

As the tensile deformation continues in the sample, the area of cross-section continuously decreases with deformation and true stress in the material is more than the engineering stress, in which original area of cross-section A0 is taken into account. Area under the load-extension curve gives the energy absorbed by sample up to fracture-indicating toughness of the material. If we assume constant volume during plastic deformation, then

At Lt = A0 L0

or P At Lt = P A0 L0

where At and Lt are the true area and true length at same load P; A0 and L0 are the initial area of the cross-section and length

So,

or true stress,

σt

=

where δL is extension

= σ(1 + ε), where ε is engineering strain.

Similarly true strain is defined as the change in length at a particular stage divided by the length of sample up to that stage.

True strain,

Strain

Str

ess

engineering

true

modified

Comparison of engineering and true stress-strain curves. Point M where necking starts on the engineering stress strain curve corresponds to point M' on the true curve.The modified

true stress-strain curve takes into consideration the complex (multiaxial) stress state at the necking point.

Figure 15 Engineering and true stress-strain curves

Ductility Ductility of the material is defined by the percentage elongation in the specimen up to fracture

For common engineering materials, yield strength, ultimate tensile strength, and percentage elongation are given in the Table:

Mechanical properties

Resilience Resilience is the ability of a material to absorb energy when it is deformed elastically and release that energy upon unloading.

Figure 16 Graphical representation of determination of the modulus of resilience of a material from the engineering stress strain curve

Definition of modulus of resilience

Resilience for linear

elastic behavior

Relation of resilience

with Hooke’s law

forfor linear elastic

behavior

Toughness

Toughness is a term related to mechanics which is is used in a number of occasions. Firstly, more specificially fracture toughness, is a property which shows the resistance of the materials against fracture in the presence of a crack (or a defect causing stress concentration).

Secondly, toughness can be defined as the capability of a material to absorb energy prior to fracture and amount of plastic deformation. Therefore toughness will be the optimum combination of stress and strain.

FAILURE ANALYSIS

Failure of any engineering material is an undesirable event resulting in loss of human lives, materials, and disruption in services. We will discuss about different types of fractures.

Simple fracture is the separation of a body into two or more pieces in response to imposed loads on body. The applied stress due to external loads may be tensile, compressive, shear, or torsional shear.

Let us first consider the effect of uniaxial tensile loads, due to which two fracture modes are possible, i.e. (a) ductile fracture with plastic deformation and (b) brittle fracture without any plastic deformation.

Figure 17 (a) shows a highly ductile fracture, in which necking takes place to a point, as in the case of loading of a lead wire or a pure gold wire resulting in hundred percent reduction in area.

Figure 17 Ductile type of fracture

Figure 17 (b) shows a moderately ductile fracture, after some necking as in the case of mild steel, under tensile load.

Figure 17 (c) shows a brittle fracture, without any plastic deformation as in the case of grey cast iron under tensile load.

Ductility and brittleness are relative terms and ductility may be quantified in terms of (i) percentage elongation and (ii) percentage reduction in area. Ductility is a function of state of stress, strain rate, and temperature of the material.

Figure 18 Different types of fractures

Any fracture process involves two stages, i.e. crack formation and crack propagation, in response to an imposed stress and the mode of fracture is highly dependent on the mechanism of crack propagation. Ductile fracture is characterized by extensive plastic deformation in the vicinity of an advancing crack and due to plastic deformation, the tip of the crack gets blunted resulting in stable crack and the fracture process proceeds slowly as the crack length is extended with an increase in applied stress. But in brittle fracture, cracks may spread extremely rapidly with very little accompanying deformation. Such cracks are unstable and crack propagation once started will continue spontaneously with any increase in applied stress.

Ductile fracture is always preferred over brittle fracture because of the following reasons:

1.Brittle fracture occurs suddenly and catastrophically without any warning—a highly undesirable feature. 2.In ductile fracture, the presence of plastic deformation gives warning and preventive measures can be taken. 3.More strain energy is required to induce ductile fracture. 4.Most metals and alloys are ductile. 5.Ceramics are remarkably brittle. 6.Polymers exhibit both types of fractures.

For common ductile materials, fracture is preceded by only a moderate amount of necking and fracture occurs in following stages:

1. First of all necking begins. 2. Small cavities or microvoids formed in the interior of cross-section. 3. With continued deformation, microvoids enlarge and coalesce to form an elliptical crack.

Figure 18 Different types of fractures

4. Rapid propagation of crack around the outer perimeter of the neck. 5. Shear deformation of an approx 45° angle with the tensile stress axis. 6. A cup and cone type fracture occurs as shown in Figure 17 (b).

The central interior region of the fractured surface has an irregular and fibrous appearance, which is indicative of plastic deformation.

Brittle Fracture

In brittle fracture, the direction of a crack motion is very nearly perpendicular to the direction of applied tensile stress, resulting in a relatively flat surface of fracture. Any sign of gross plastic deformation is absent. Brittle fracture surfaces contain lines or ridges radiating from the origin of the crack in a form like pattern. Brittle fracture in amorphous materials such as ceramic glasses yields a shiny and smooth surface. For most brittle crystalline materials fracture is transgranular, i.e. fracture passes through grains. In some alloys, crack propagation is along grain boundaries and fracture is known as intergranular.

Ductile Fracture vs. Brittle Fracture

Ductile Fracture vs. Brittle Fracture