2.Types of deformation
3.Concepts of stress and strain
true stress and strain
4.Stress-Strain curves for tension
Elastic vs non-elastic materials
6. Stress-Strain behaviour for different materials
7. Elastic constants
Many materials, when in service, are subjected to forces or loads; examples include the aluminum alloy from which an airplane wing is constructed and the steel in an automobile axle. In such situations it is necessary to know the characteristics of the material and to design the member from which it is made such that any resulting deformation will not be excessive and fracture will not occur.
Most of the materials used in engineering are metallic in nature. The prime reason simply is the versatile nature of their properties those spread over a very broad range compared with other kinds of materials. Many engineering materials are subjected to forces both during processing/fabrication and in service.
Effect of force on the body
When a force is applied on a solid material, it may result in translation, rotation, or deformation of that material. Aspects of material translation and rotation are dealt by engineering dynamics. We restrict ourselves here to the subject of material deformation under forces.
Effect of force on the body
Solid material are defined such that change in their volume under applied forces in very small, thus deformation is used as synonymous to distortion.
2. Types of deformation
Material deformation can be permanent or temporary. Permanent deformation is irreversible i.e. stays even after removal of the applied forces, while the temporary deformation disappears after removal of the applied forces. It is also a function of time.
3. Concepts of stress and strain
- Forces applied act on a surface of the material, and thus the force intensity, force per unit area, is used in analysis.
- Analogous to this, deformation is characterized by percentage —change in length per unit length in three distinct directions.
There are three principal ways in which a load may be applied: namely, tension, compression, and shear. In engineering practice many loads are torsional rather than pure shear.
The mechanical properties of materials are ascertained by performing carefully designed laboratory experiments that replicate as nearly as possible the service conditions.
- Consider a uniform bar of cross sectional area Ao subjected to an axial tensile force F.
- The stress at any section normal to the line of action of the tensile force F is specifically called tensile stress.
- Since internal resistance R at some plane X-X is equal to the applied force F, we have,
σ= (internal resistance at x-x)/(resisting area at x-x)
σ = R/A = F/Ao.
- Due to the tensile stress there is an increase in the length of the body and decrease in the cross section area of the body.
- Tensile stress is a type of normal stress, so it acts at 90 degree to the area.
- The strain which is induced due to tensile stress is called tensile strain. It is equals to the ratio of increase in the length to the original length.
The tensile testing machine is designed to elongate the specimen at a constant rate, and to continuously and simultaneously measure the instantaneous applied load (with a load cell) and the resulting elongations (using an extensometer). A stress–strain test typically takes several minutes to perform and is destructive; that is, the test specimen is permanently deformed and usually fractured.
The load–deformation characteristics are dependent on the specimen size.
- For example, it will require twice the load to produce the same elongation if the cross-sectional area of the specimen is doubled.
- To minimize these geometrical factors, load and elongation are normalized to the respective parameters of engineering stress and engineering strain.
Engineering stress s is defined by the relationship
σ = F/Ao
Where F = instantaneous force, Ao = original cross-sectional area
Because of the applied force on the specimen, corresponding deformation happens in the specimen is called as strain and in this case of tensile force, there is an elongation in the body and engineering strain is defined according as below
ε = (l-lo) / lo = l/lo
- Where lo = initial length of the specimen before the application of the force
l = final length of the specimen
- Engineering strain (subsequently called just strain) is unitless.
- Sometimes strain is also expressed as a percentage, in which the strain value is multiplied by 100.
True stress and strain
The stress, as computed from engineering stress is on the basis of the original cross-sectional area, before any deformation, and does not take into account this reduction in area at the neck.
- Sometimes it is more meaningful to use a true stress–true strain scheme.
Where σT = true stress, Ai = instantaneous area cross-section
True strain is given as
εT = ln(li/lo)
Where li= instantaneous length, lo= initial length
- Relations between true and engineering stress- strains are
σT=σ(1+ε) and εT=ln(1+ε)
What is the procedure to follow above tension test?
- It is imperative that there be some consistency in the manner in which tests are conducted, and in the interpretation of their results.
- This consistency is accomplished by using standardized testing techniques.
- In the United States the most active organization is the American Society for Testing and Materials (ASTM).
4. Stress-Strain curves for tension
Material: Mild steels(Ductile material)
stress-strain curve fro mild steel
Proportional Limit (Hooke’s Law)
- From the origin O to the point ‘P’ is called proportional limit
- The stress-strain curve is a straight line.
- This linear relation between elongation and the axial force causing was related by Hooke’s Law that within the proportional limit, the stress is directly proportional to strain or
σ ∝ ε or σ= E ε
Modulus of Elasticity
The constant of proportionality E is called the Modulus of Elasticity E or Young’s Modulus.
- This is equal to the slope of the stress-strain diagram from O to P.
- The elastic limit is the limit beyond which the material will no longer go back to its original shape when the load is removed
- It is the maximum stress that may be developed such that there is no permanent or residual deformation when the load is entirely removed.
- It is up to point ‘E’ in the graph
Elastic and Plastic Ranges
- The region in stress-strain diagram from O to P is called the elastic range.
- The region from P to F is called the plastic range.
Yield point is the point at which the material will have an appreciable elongation or yielding without any increase in load.
- Beyond the elastic limit plastic deformation occurs and strains are not totally recoverable.
- There will be thus permanent deformation or permanent set when load is removed
- These two points Y1 and Y2 are termed as upper and lower yield points respectively. The stress at the yield point is called the yield strength.
When to use offset method to find out yield point?
- In the most of the cases, yield point is so near the proportional limit and the two may be taken as one.
- Materials like heat treated steels and cold drawn steels do not posses a well defined yield points.
- In order to find the yield point or yield strength in such cases, an offset method is applied.
In this method a line is drawn parallel to the straight line portion of initial stress diagram by off setting this by an amount equal to 0.2% of the strain as shown below and this happens especially for the low carbon steel.
Why two yield points in mild steels?
Two yield points
This yield point phenomenon is associated with interstitial and substitutional impurities in case of mild steel.
- Interstitial carbon and nitrogen are much larger when compared to the void they occupy in ferrite.
- This will create Cottrell atmosphere around the impurities to reduce their distortion energy.
- So more stress is needed to overcome this atmosphere resulting in upper yield point(Y1) as in the figure.
After this point, dislocation need less stress to slide along slip planes which is called as lower yield point(Y2).
- Following the elastic deformation, material undergoes plastic deformation.
- Microscopically, it involves breaking atomic bonds, moving atoms, then restoration of bonds.
- Crystalline solids deform by processes
- slip and twinning in particular directions.
- Amorphous solids deform by
- viscous flow mechanism without any directionality.
Because of the complexity involved, theory of plasticity neglects the following effects:
- Anelastic strain, which is time dependent recoverable strain.
- Hysteresis behavior resulting from loading and unloading of material.
- Bauschinger effect – dependence of yield stress on loading path and direction.
Perfectly plastic region
It is the region on the curve after yield point(Y2) where strain occurs without any increase in stress. This can be considered as a point B which is very nearer to the yield point
- It is the strengthening of a metal by plastic deformation.
- This is the region from point B to U, where strain increases in a faster rate.
- Material in this region undergoes change in atomic and crystalline structure.
- This portion is not used for structural design
The maximum ordinate in the stress-strain diagram is the ultimate strength or tensile strength.In this case it is at point ‘U’ along the y-axis.
Rupture strength is the strength of the material at rupture. This is also known as the breaking strength identified as point ‘F’ on the curve.
- This rupture point is not there in brittle materials and rupture occurs at ultimate point it self, all of a sudden.
Working Stress, Allowable Stress, and Factor of Safety
Working stress: is defined as the actual stress of a material under a given loading.
Allowable stress:The maximum safe stress that a material can carry is termed as the allowable stress. The allowable stress should be limited to values not exceeding the proportional limit.
Factor of safety:However, since proportional limit is difficult to determine accurately, the allowable stress is taken as either the yield point or ultimate strength divided by a factor of safety.
The ratio of this strength (ultimate or yield strength) to allowable strength is called the factor of safety.
5. Important properties
- If the material is isotropic then its mechanical and thermal properties are the same in all directions.
- Isotropic materials can have a homogeneous or non-homogeneous microscopic structures.
- Eg: steel demonstrates isotropic behavior, although its microscopic structure is non-homogeneous.
Anisotropic or Orthotropic Materials
- In these the mechanical or thermal properties are unique and independent in three mutually perpendicular directions.
- Eg: wood, many crystals, and rolled metals.
- A material is homogenous if it has the same composition through out body.
- Hence the elastic properties are the same at every point in the body.
- However, the properties need not to be the same in all the direction for the material to be homogenous.
Isotropic materials have the same elastic properties in all the directions. Therefore, the material must be both homogenous and isotropic in order to have the lateral strains to be same at every point in a particular component.
- It is a measure of the degree of plastic deformation that has been sustained at fracture.
- A material that experiences very little or no plastic deformation upon fracture is termed brittle.
Ductility may be expressed quantitatively as either percent elongation or percent reduction in area.
A brittle material is one which exhibits a relatively small extensions or deformations to fracture, so that the partially plastic region of the tensile test graph is much reduced.
This type of graph is shown by the cast iron or steels with high carbon contents or concrete.
- It is the ability of a solid to bend or be hammered into other shapes without breaking.
- Examples of malleable metals are gold, iron, aluminum, copper, silver, and lead. Gold and silver are highly malleable.
- It is the capacity of a material to absorb energy when it is deformed elastically and then, upon unloading, to have this energy recovered.
- The associated property is the modulus of resilience, which is the strain energy per unit volume required to stress a material from an unloaded state up to the point of yielding
Modulus of Resilience
- Modulus of resilience is the work done on a unit volume of material as the force is gradually increased from O to P, in N·m/m3.
- This may be calculated as the area under the stress-strain curve from the origin O to up to the elastic limit E.
The resilience of the material is its ability to absorb energy without creating a permanent distortion.
- It is the area under the stress-strain curve up to the point of fracture.
- For a material to be tough, it must display both strength and ductility.
- Ductile materials are tougher than brittle ones.
Which material has more toughness?
comparison of toughness
- The stress–strain curves are plotted for both material types.
- Even though the brittle material has higher yield and tensile strengths, it has a lower toughness than the ductile one, by virtue of lack of ductility
Modulus of Toughness
- Modulus of toughness is the work done on a unit volume of material as the force is gradually increased from O to F.
- This may be calculated as the area under the entire stress-strain curve.
- The toughness of a material is its ability to absorb energy without causing it to break.
- It is a measure of a material’s resistance to localized plastic deformation (e.g., a small dent or a scratch).
Tests to measure hardness
- The Rockwell tests constitute the most common method used to measure hardness because it is very simple to perform and require no special skills.
- Other well known test is In Brinell test, where a spherical indenter is forced into the surface of the metal to be tested.
Correlation Between Hardness and Tensile Strength
- Both tensile strength and hardness are indicators of a metal’s resistance to plastic deformation.
- Consequently, they are roughly proportional
- the strength of a material is its ability to withstand an applied load without failure or plastic deformation.
- The field of strength of materials deals with forces and deformations that result from their acting on a material.
These forces can be produced by any type stresses induced on the materials.
- The impact strength describes the ability of a material to absorb shock and impact energy without breaking.
- The impact strength is calculated as the ratio of impact absorption to test specimen cross-section.
- This is dependent upon temperature and the shape of the test specimen.
|elastic deformation is time independent
||time-dependent elastic behaviour
|upon release of the load the strain is totally recovered
|| Here also strain will be totally recoverable but not instantaneously
||For metals the anelastic component is neglected. But for polymeric materials its magnitude is significant, which is called as viscoelastic behavior
Elastic vs non-elastic materials
||Non linear elastic
|Obeys Hooke’s law
||Doesn’t obey Hooke’s law
|Has constant young’s modulus
||No constant young’s modulus
|Possesses yield point
||Possesses proof strain
Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Synthetic polymers, wood, and human tissue, as well as metals at high temperature, display significant viscoelastic effects.
Uses of Viscoelastic materials:
- These are used for isolating vibration, dampening noise, and absorbing shock.
- They give off the energy absorbed as heat.
The majority of engineering failures are due to fatigue. It is defined as the tendency of a material to fracture by means of progressive brittle cracking under repeated alternating or cyclic stresses of an intensity considerably below the normal strength.
- The number of cycles required to cause fatigue failure at a particular peak stress is generally quite large, but it decreases as the stress is increased.
- In case of some mild steels, cyclic stresses can be continued indefinitely provided the peak stress(fatigue strength) is below the endurance limit value.
Endurance limit is where the amplitude (or range) of cyclic stress that can be applied to the material without causing fatigue failure.
Factors affecting fatigue
- Frequency of loading
- Loading condition
- Stress concentration
The fatigue failure is due to progressive propagation of flaws in steel under cyclic loading. This is partially enhanced by the stress concentration at the tip of such flaw or crack.
Three stages of fatigue failure:
- crack initiation in the areas of stress concentration (near stress raisers)
- incremental crack propagation
- final rapid crack propagation after crack reaches critical size
Fatigue failure examples:
Any part subjected to repeated stresses will fail by fatigue like crack in turbine blades, breaking wire, etc..
How to increase fatigue life of materials?
Fatigue failure mainly starts from the defects on the surface. So this can be achieved by modifying the surfaces of materials through either alloying additions, mechanical deformation, or microstructural control.
- It is a method of cold working in which compressive stresses are induced in the exposed surface layers of metallic parts by the impingement of a stream of shots directed at the metal surface at high velocity under controlled conditions.
- The shot peening can be applied to various materials and their weldment like steels cast steels, cast iron, Cu alloys, Al alloys.
- It is a process that utilizes cold working and non-uniform plastic strain to control surface hardness and residual stress to result in parts with improved fatigue resistance.
- Deep rolling also can be used to improve surface finish and radial profiles in fillets.
- Crankshafts, axle shafts, and fasteners have been the primary areas of interest for the application of deep rolling
Many other techniques are in use to improve surface hardness to avoid failure by fatigue
- Creep may be defined as a time-dependent deformation at elevated temperature and constant stress.
- At elevated temperatures and stresses, much less than the high-temperature yield stress, metals undergo permanent plastic deformation called creep.
Creep failure examples:
The end of useful service life of the high-temperature components in a boiler (the superheater and reheater tubes and headers, for example) is usually a failure by a creep or stress-rupture mechanism. Other temperature applications like oil refineries and steam turbines may also fail by creep.
Creep is generally minimized in materials with:
- High melting temperature
- High elastic modulus
- Large grain sizes (inhibits grain boundary sliding)
The following materials are especially resilient to creep:
- Stainless steels
- Refractory metals (containing elements of high melting point, like Nb, Mo, W, Ta)
- “Superalloys” (Co, Ni based: solid solution hardening and secondary phases)
Failure in materials
|Failure happens after some deformation
||Failure happens without or with little deformation
|There is reduction in cross sectional area
||Negligible reduction in the area cross section
|Fracture occurs after necking
||Fracture happens instantaneously and all of a sudden
6. Stress-Strain behaviour for different materials
7. Elastic constants
An elastic modulus is a number that measures an object or substance’s resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is defined as the slope of its stress–strain curve in the elastic deformation region.
E= Direct stress/Linear strain =σ/ε
The shear modulus is the elastic modulus we use for the deformation which takes place when a force is applied parallel to one face of the object while the opposite face is held fixed by another equal force.
- The measurement of distortion is done by shear strain and in pure shear, only body distortion happens but its volume remains same.
- If the shear strain is φ then the linear strain in the diagonal of the specimen is ε =φ/2.
Shear modulus G= shear stress/shear strain = τ/φ
K= Direct stress/ Volumetric strain = σ/εv
Compressive tests are used when a material’s behavior under large and permanent (i.e., plastic) strains is desired, as in manufacturing applications, or when the material is brittle in tension.
1.What is temperature stress and how it is different from others?
Ans: This is stress caused by a change in the thermal state of a body upon heating, cooling, or prolonged exposure to elevated or low temperatures. Stress is not produced by simple variation in temperature without obstruction to the deformation of the body, caused by this temperature change.
- William D. Callister, Jr.David G. Rethwisch: Materials Science and Engineering: An Introduction, Wiley publication, 2014
- NPTEL material science material by Satish Vasu Kailas (IISc)