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Relation between E, G, K & v

​​​"Stress in a material induces strain and vice versa"


In a tensile test for characterisation of material property, as the force is applied, initially a linear relation/straight line (OA) is obtained between force and elongation or stress v/s strain. The end of this linear region is called the proportional limit. For some metals, like mild steel, the stress may then decrease slightly (the region AB), before increasing once again. The largest stress (point D on the curve) is called the ultimate stress. The stress at breaking point E is called fracture or rupture stress. 

If the specimen is loaded up to any point along line OA and then start unloading, the specimen will retrace the stress–strain curve and return to point O. In this elastic region, the material regains its original shape when the applied force is removed.

If we start unloading only after reaching point C, however, then we will come down the straight line FC, which will be parallel to line OA. At point F, the stress is zero, but the strain is nonzero. C thus lies in the plastic region of the stress-strain curve, in which the material is deformed permanently, and the permanent strain at point F is the plastic strain. The region in which the material deforms permanently is called plastic region. The total strain at point C is sum the plastic strain (OF) and an additional elastic strain (FG)

The point demarcating the elastic from the plastic region is called the yield point. The stress at yield point is called the yield stress. In practice, the yield point may lie anywhere in the region AB, although for most metals it is close to the propor- tional limit. For many materials it may not even be clearly defined. For such materials, we mark a prescribed value of offset strain recommended by ASTM A 370 to get point H  Starting from H we draw a line (HI) parallel to the linear part (OA) of the stress–strain curve. Offset yield stress would correspond to a plastic strain at point I. Usually the offset strain is given as a percentage. A strain of 0.2% equals ε = 0.002.

Elastic and linear are two distinct material descriptions.  Soft rubber is thus elastic but nonlinear material. 

Prinicipal Mechanical Properties of Materials

Strength: It is the resistance offered by a material when subjected to external loading. So, stronger the material the greater the load it can withstand. Depending upon the type of load applied the strength can be tensile, compressive, shear or torsional. The maximum stress that any material will withstand before destruction is called its ultimate strength 

Elasticity: Elasticity of a material is its power of coming back to its original position after deformation when the stress or load is removed. Elasticity is a tensile property of its material. The greatest stress that a material can endure without taking up some permanent set is called elastic limit 

Stiffness (Rigidity): The resistance of a material to deflection is called stiffness or rigidity. Steel is stiffer or more rigid than aluminium. Stiffness is measured by Young’s modulus E. The higher the value of the Young’s modulus, the stiffer the material. E is the ratio of stress over strain.

Plasticity: The plasticity of a material is its ability to undergo some degree of permanent deformation without failure. Plastic deformation will take place only after the elastic range has been exceeded Plasticity is used in several mechanical processes like forming, shaping, extruding and many other hot and cold working processes. In general, plasticity increases with increasing temperature and is a favourable property of material for secondary forming processes. Due to this properties various metal can be transformed into different products of required shape and size. This conversion into desired shape and size is effected either by the application of pressure, heat or both.

Ductility: Ductility of a material enables it to draw out into thin wire on application of the load. Mild steel is a ductile material. The wires of gold, silver, copper, aluminium, etc. are drawn by extrusion or by pulling through a hole in a die due to the ductile property. The ductility decreases with increase of temperature. The per cent elongation and the reduction in area in tension is often used as empirical measures of ductility.

Malleability: Malleability of a material is its ability to be flattened into thin sheets without cracking by hot or cold working. Aluminium, copper, tin, lead, steel, etc. are malleable metals. Lead can be readily rolled and hammered into thin sheets but can not be drawn into wire. Ductility is a tensile property, whereas malleability is a compressive property. Malleability increases with increase of temperature.

Brittleness: The brittleness of a material is the property of breaking without much permanent distortion. There are many materials, which break or fail before much deformation take place. Such materials are brittle e.g., glass, cast iron. Therefore, a non-ductile material is said to be a brittle material. Usually the tensile strength of brittle materials is only a fraction of their compressive strength. A brittle material should not be considered as lacking in strength. It only shows the lack of plasticity. On stress-strain diagram, these materials don’t have yield point and value of E is small.

Toughness: The toughness of a material is its ability to withstand both plastic and elastic deformations. It is a highly desirable quality for structural and machine parts to withstand shock and vibration. Manganese steel, wrought iron, mild steels are tough materials. Example:  If a load is suddenly applied to a piece of mild steel and then to a piece of glass the mild steel will absorb much more energy before failure occurs. Thus, mild steel is said to be much tougher than a glass.
oughness is a measure of the amount of energy a material can absorb before actual fracture or failure takes place. “The work or energy a material absorbs is called modulus of toughness” Toughness is also resistance to shock loading. It is measured by a special test on Impact Testing Machine.

Hardness: Hardness is closely related to strength. It is the ability of a material to resist scratching, abrasion, indentation, or penetration. It is directly proportional to tensile strength and is measured on special hardness testing machines by measuring the resistance of the material against penetration of an indentor of special shape and material under a given load. The different scales of hardness are Brinell hardness, Rockwell hardness, Vicker’s hardness, etc. Hardness of a metal does not directly relate to the hardenability of the metal. Hardenability is indicative of the degree of hardness that the metal can acquire through the hardening process. i.e., heating or quenching.

Hardenability: Hardenability is the degree of hardness that can be imparted to metal by process of hardening. A metal capable of being hardened throughout its structure is said to have high hardenability. The material is heated above a certain temperature and then suddenly quenched in a cold oil or water bath.

Impact Strength: It can be defined as the resistance of the material to fracture under impact loading, i.e., under quickly applied dynamic loads.Two standard tests are normally used to determine this property. (1) The IZOD impact test. (2) The CHARPY test.

Resilience: Resilience is the capacity of material to absorb energy elastically. On removal of the load, the energy stored is released as in a spring. The maximum energy which can be stored in a body up to elastic limit is called the proof resilience. The quantity gives capacity of the material to bear shocks and vibrations. The strain energy stored in a material of unit volume gives proof resilience and is measured by work stretching. 

Hooke’s law:  "Within Elastic limit Strain is proportional to Stress" i.e.  

OR                                           i.e.

where E is called modulus of elasticity or Young’s modulus. It represents the slope of the straight line in a stress–strain curve 

Bulk Modulus Of Elasticity (K)

Bulk modulus defines the ability of a material to resist deformation in terms of volume change, when subject to compression under pressure. It is given by the ratio of pressure applied to the corresponding relative decrease in volume of the material. Bulk modulus gives us a measure of how incompressible a solid is, as in, the more the value of K for a material, higher is its nature to be incompressible. 

The energy stored in a body due to deformation is the Strain Energy, and the strain energy per unit volume is the strain energy density.

The strain energy density at the yield point is called modulus of resilience This property is a measure of the recoverable (elastic) energy per unit volume that can be stored in a material. Since a spring is designed to operate in the elastic range, the higher the modulus of resilience, the more energy it can store.

The strain energy density at rupture is called modulus of toughness. This property is a measure of the energy per unit volume that can be absorbed by a material without breaking and is important in resistance to cracks and crack propagation. Whereas a strong material has high ultimate stress, a tough material has large area under the stress–strain curve. Strain energy density, complementary strain energy density, modulus of resilience, and modulus of toughness all have units of energy per unit volume. 

Shear Modulus or Modulus of Rigidity (G)
Within the proportional limit of a material, shear modulus G is defined as the ratio of the shear stress t to the shear strain. The shear modulus can be measured from a torsion test, or alternatively, it can be calculated from the Young's modulus and Poisson's ratio of a material using a closed-form elasticity relationship. 

Poisson's Ratio

Elongation of a member under a tensile load is accompanied by a corresponding lateral contraction. The deformation for a small portion of the tension coupon in an exaggerated scale is shown here. The section elongates, inducing tensile strain along the direction of load. It also contracts along the transverse direction, inducing a compressive strain –. Poisson's ratio is defined as the negative ratio of the lateral, or transverse, strain to the longitudinal strain.

Poisson's ratio is a positive fraction because the two strain components have opposite signs. It is a dimensionless quantity and has an approximate value of 0.3 for steel and aluminum. 

​Poisson’s ratio is a dimensionless quantity that has a value between 0 and 0.5 for most materials, although some composite materials can have negative values for ν . The theoretical range for Poisson’s ration is –1 ≤ ν ≤ -1/2  


Fig. 1- A typical Stress-Strain curve for metal

(a)Rubber                                     (b)Glass


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