Strain Energy MCQ Quiz - Objective Question with Answer for Strain Energy - Download Free PDF

Concept:

Proof Resilience is the strain energy stored per unit volume when a material is deformed up to its elastic limit. It depends on the square of the elastic strength (proof stress).

Given:

• Initial elastic strength = \(\sigma_e\)
• Elastic strength increases by 3 times → New elastic strength = \(3\sigma_e\)
• Young's modulus (E) remains constant

Step 1: Recall Proof Resilience formula

\( U = \frac{\sigma_e^2}{2E} \)

Step 2: Calculate new Proof Resilience

With new elastic strength \(3\sigma_e\):

\( U_{\text{new}} = \frac{(3\sigma_e)^2}{2E} = \frac{9\sigma_e^2}{2E} = 9U \)

Conclusion:

When elastic strength increases 3 times, Proof Resilience increases by 9 times because it is proportional to the square of the elastic strength.

Explanation:

Understanding the Toughness of a Material

Definition: Toughness is a measure of a material's ability to absorb energy and plastically deform without fracturing. It is an important property that indicates how much energy a material can absorb before it fails. Toughness is essential in applications where materials are subjected to sudden impacts or stresses, as it determines how well a material can withstand such forces without breaking.

Stress-Strain Curve: To understand toughness, it is crucial to analyze the stress-strain curve of a material. The stress-strain curve is a graphical representation that shows the relationship between the stress applied to a material and the resulting strain (deformation) it experiences. The curve typically has distinct regions that represent different stages of deformation: the elastic region, the yield point, the plastic region, and the fracture point.

Area Under the Curve: The toughness of a material is represented by the total area under the stress-strain curve up to the point of fracture. This area encompasses both the elastic and plastic regions of deformation, indicating the total energy absorbed by the material before it breaks. The larger the area under the curve, the tougher the material is.

Elastic Region: In the initial part of the stress-strain curve, the material deforms elastically. In this region, the material returns to its original shape upon the removal of the applied stress. The elastic region is characterized by a linear relationship between stress and strain, and the area under this part of the curve represents the elastic energy stored in the material.

Plastic Region: Beyond the yield point, the material enters the plastic region, where it undergoes permanent deformation. In this region, the material does not return to its original shape after the stress is removed. The area under the plastic region of the curve represents the energy absorbed by the material as it deforms plastically.

Importance of Total Area: The total area under the stress-strain curve, which includes both the elastic and plastic regions, is a comprehensive measure of toughness. It accounts for the energy absorbed during both reversible (elastic) and irreversible (plastic) deformation. Therefore, the correct representation of toughness is the total area under the curve, as it reflects the material's ability to absorb energy up to the point of fracture.

Analysis of Other Options:

Option 2: Area of Plastic Region

While the area of the plastic region does contribute to the toughness of a material, it does not represent the total energy absorbed by the material. The plastic region only accounts for the energy absorbed during permanent deformation, ignoring the energy stored elastically before yielding. Therefore, considering only the plastic region provides an incomplete measure of toughness.

Option 3: Area of Elastic Region

The area of the elastic region represents the energy stored in the material during reversible deformation. However, this is only a part of the total energy absorbed by the material before failure. Toughness encompasses both elastic and plastic deformation, so focusing solely on the elastic region underestimates the material's true toughness.

Option 4: Slope of the Elastic Region

The slope of the elastic region, known as the modulus of elasticity or Young's modulus, indicates the material's stiffness. It does not provide information about the total energy absorption capacity or toughness. While stiffness is an important mechanical property, it is not a measure of toughness.

Explanation:

The work done by the load in stretching the bar is known as Strain Energy.

Strain Energy:

Strain energy is the energy stored in a body due to deformation. When a bar or any other structural member is subjected to a load, it deforms and this deformation causes the internal energy of the material to increase. This increase in internal energy due to the load-induced deformation is called strain energy. In simpler terms, strain energy is the work done by the load in stretching or compressing the bar. Mathematically, it is given by the area under the load-deformation curve.

Detailed Explanation:

When a material is subjected to external forces, it deforms, and this deformation leads to the development of internal stresses and strains within the material. The energy required to cause this deformation is stored in the material as strain energy. The concept of strain energy is fundamental in the field of mechanics of materials and structural engineering.

For a linear elastic material, the relationship between stress and strain is linear, and the strain energy (U) can be calculated using the following formula:

U = 0.5 × σ × ε × V

where:

• σ is the stress applied to the material.
• ε is the strain experienced by the material.
• V is the volume of the material.

This equation shows that the strain energy is proportional to the product of stress and strain, and the volume of the material. In a uniaxial loading scenario, such as stretching a bar, the strain energy can also be expressed in terms of the load (P) and the deformation (ΔL) as:

U = 0.5 × P × ΔL

Here, P is the load applied to the bar, and ΔL is the change in length of the bar due to the load. The factor of 0.5 comes from the fact that the load-deformation relationship is linear, and the work done is represented by the area of the triangle under the load-deformation curve.

Strain energy is an important concept because it helps engineers understand how materials and structures will behave under different loading conditions. It is used in the design and analysis of structures to ensure that they can safely withstand the loads they will encounter during their service life.

Analysis of Other Options:

1. Potential Energy: Potential energy is the energy possessed by an object due to its position or configuration. For example, an object at a height above the ground has gravitational potential energy. While strain energy is a form of potential energy (specifically elastic potential energy), the term "potential energy" is more general and not specific to the context of stretching a bar.

2. Kinetic Energy: Kinetic energy is the energy possessed by an object due to its motion. It is given by the equation KE = 0.5 × m × v², where m is the mass of the object and v is its velocity. Kinetic energy is not relevant to the context of stretching a bar, as it pertains to motion rather than deformation.

3. Dislocation Energy: Dislocation energy refers to the energy associated with dislocations in a crystal structure. Dislocations are defects in the crystal lattice that play a significant role in the plastic deformation of materials. While dislocation energy is related to the mechanical behavior of materials, it is not the same as the strain energy stored due to stretching a bar.

In conclusion, the correct option is Strain Energy because it specifically refers to the energy stored in a material due to deformation under an applied load. This concept is fundamental in understanding the behavior of materials and structures under various loading conditions.

Explanation:

Proof Resilience

• Proof resilience is defined as the maximum amount of energy that can be stored in a material up to the elastic limit. This energy is stored in the form of strain energy, which is the energy absorbed by the material when it is deformed elastically.

Elastic Limit:

• The elastic limit is the maximum stress that a material can withstand without undergoing permanent deformation. When the material is loaded within its elastic limit, it returns to its original shape after the load is removed.

Calculation of Proof Resilience: The proof resilience can be calculated using the area under the stress-strain curve up to the elastic limit. Mathematically, it can be expressed as:

Proof Resilience = \(\frac{1}{2}\) × Stress × Strain

Important Points:

• Proof resilience is a measure of the energy absorption capacity of a material within the elastic limit.
• It is an important parameter for materials that are subjected to cyclic loading, as it indicates the material's ability to absorb energy without undergoing permanent deformation.
• The unit of proof resilience is typically Joules (J) or any other unit of energy.

Concept:

Area under the Stress-Strain Curve:

• The area under the stress-strain curve represents the toughness of the material. Toughness is defined as the ability of the material to absorb energy up to failure. It is an indication of the amount of energy per unit volume that a material can absorb before rupturing.

Modulus of toughness:

• Modulus of toughness is the ability to absorb energy up to fracture. From the stress-strain diagram, the area under the complete curve gives the measure of modules of toughness.

Modulus of resilience:

• Modulus of resilience is defined as proof resilience per unit volume. It is the area under the stress-strain curve up to the elastic limit.

Resilience:

• Resilience is defined as the capacity of a strained body for doing work on the removal of the straining force.

Proof resilience:

• Proof resilience is defined as the maximum strain energy stored in a body. So, it is the quantity of strain energy stored in a body when strained up to the elastic limit (ability to store or absorb energy without permanent deformation).

Explanation:-

Resilience

• The total strain energy stored in a body is commonly known as resilience. Whenever the straining force is removed from the strained body, the body is capable of doing work. Hence resilience is also defined as the capacity of a strained body for doing work on the removal of the straining force.
• It is the property of materials to absorb energy and to resist shock and impact loads.
• It is measured by the amount of energy absorbed per unit volume within elastic limit this property is essential for spring materials.
• The resilience of material should be considered when it is subjected to shock loading.

Proof resilience

• The maximum strain energy, stored in a body, is known as proof of resilience. The strain energy stored in the body will be maximum when the body is stressed upto the elastic limit. Hence the proof resilience is the quantity of strain energy stored in a body when strained up to the elastic limit.
• It is defined as the maximum strain energy stored in a body.
• So, it is the quantity of strain energy stored in a body when strained up to the elastic limit (ability to store or absorb energy without permanent deformation).

Modulus of resilience

• It is defined as proof resilience per unit volume.
• It is the area under the stress-strain curve up to the elastic limit.

Toughness:

• Toughness is defined as the ability of the material to absorb energy before fracture takes place.
• This property is essential for machine components that are required to withstand impact loads.
• Tough materials have the ability to bend, twist or stretch before failure takes place.
• Toughness is measured by a quantity called modulus of toughness. Modulus of toughness is the total area under the stress-strain curve in a tension test.
• Toughness is measured by Izod and Charpy impact testing machines.
• When a material is heated it becomes ductile or simply soft and thus less stress is required to deform the material and the stress-strain curve will shift down and the area under the curve decreases thus toughness decreases.
• Toughness decreases as temperature increases.

Hardness:

• Hardness is a measure of the resistance to localized plastic deformation induced by either mechanical indentation or abrasion. 
• Hardness Testing measures a material’s strength by determining resistance to penetration.
• There are various hardness test methods, including Rockwell, Brinell, Vickers, Knoop and Shore Durometer testing.

When a material is subjected to repeated stresses, it fails at stresses below the yield point stresses. Such type of failure of a material is known as fatigue.

The slow and continuous elongation of a material with time at constant stress and high temperature below the elastic limit is called creep.

The elastic strain energy stored in a member of length s (it may be curved or straight) due to axial force, bending moment, shear force and torsion are summarized below:

The strain energy of a beam

• Depends on the shear force in the beam
• Depends on the bending moment in the beam
• It is different than the potential energy

Explanation:

Castigliano’s first theorem-

• For linearly elastic structures, where external forces only cause deformations, the complementary energy is equal to the strain energy.
• For such structures, Castigliano’s first theorem may be stated as the first partial derivative of the strain energy of the structure with respect to any particular displacement gives the force causing the deflection at that point
• This first theorem is applicable to linearly or nonlinearly elastic structures in which the temperature is constant and the supports are unyielding.

\(\begin{array}{l} \frac{{\partial U}}{{\partial \Delta }} = P\\ \frac{{\partial U}}{{\partial \theta }} = M \end{array}\)

Castigliano’s second theorem-

• The first partial derivative of the total internal energy in a structure with respect to the force applied at any point is equal to the deflection at the point of application of that force in the direction of its line of action.
• The second theorem of Castigliano is applicable to linearly elastic (Hookean material) structures with constant temperature and unyielding supports.

\(\begin{array}{l} \frac{{\partial U}}{{\partial P}} = \Delta \\ \frac{{\partial U}}{{\partial M}} = \theta \end{array}\)

Explanation:-

Resilience

• It is the property of materials to absorb energy and to resist shock and impact loads.
• It is measured by the amount of energy absorbed per unit volume within elastic limit this property is essential for spring materials.
• The resilience of material should be considered when it is subjected to shock loading.

Proof resilience

• It is defined as the maximum strain energy stored in a body.
• So, it is the quantity of strain energy stored in a body when strained up to the elastic limit (ability to store or absorb energy without permanent deformation).

Modulus of resilience

• It is defined as proof resilience per unit volume.
• It is the area under the stress-strain curve up to the elastic limit.

When a material is subjected to repeated stresses, it fails at stresses below the yield point stresses. Such type of failure of a material is known as fatigue.

The slow and continuous elongation of a material with time at constant stress and high temperature below the elastic limit is called creep.

Additional Information

Thermal Stress – Stress caused due to the change in temperature

Fatigue – Fatigue occurs when a structure is subjected to cyclic loading.

Wear and tear – This is a deterioration or damage happen to something naturally has over time through normal day to day use.

Modulus of toughness

• It is the ability to absorb energy up to fracture.
• From the stress-strain diagram, the area under the complete curve gives the measure of modules of toughness.

Concept:

In gradual loading, the loading starts from zero and increases gradually till the body is fully loaded, while in sudden loading, the load is suddenly applied on the body.

\({σ _{gradual}} = \frac{P}{A}\)

\({σ _{sudden}} = \frac{{2P}}{A}\)

\({\rm{Strain\;energy}}\left( {\rm{U}} \right) = \frac{{{{\rm{σ }}^2}{\rm{V}}}}{{2{\rm{E}}}}\)

Calculation:

Given:

σ_gradual = σ, σ_sudden = 2σ.

\({\rm{Strain\;energy}}\left( {\rm{U}} \right) = \frac{{{{\rm{σ }}^2}{\rm{V}}}}{{2{\rm{E}}}}\)

\({{\rm{U}}_{{\rm{gradual}}}} = \frac{{{{\rm{σ }}^2}{\rm{V}}}}{{2{\rm{E}}}}{\rm{\;}},\)

\({{\rm{U}}_{{\rm{suddenly}}}} = \frac{{{{\left( {2{\rm{σ }}} \right)}^2}{\rm{V}}}}{{2{\rm{E}}}} = \frac{{4{{\rm{σ }}^2}{\rm{V}}}}{{2{\rm{E}}}}\)

\(\frac{{{{\rm{U}}_{{\rm{sudden}}}}}}{{{{\rm{U}}_{{\rm{gradual}}}}}} = \frac{{4{{\rm{σ }}^2}{\rm{V}}}}{{2{\rm{E}}}} \times \frac{{2{\rm{E}}}}{{{{\rm{σ }}^2}{\rm{V}}}} = 4\)

The strain energy stored in a body due to suddenly applied load compared to when it is applied gradually is four times.

Concept:

• The energy which is absorbed in the body due to the straining effect is known as strain energy (U). 

​\({\bf{U}} = \frac{1}{2} \times {\bf{\sigma }} ~\times~{\epsilon} = \frac{{{{\bf{\sigma }}^2}}}{{2{\bf{E}}}} \times {\bf{vol}}\)

• Whenever the straining force is removed from the strained body, the body is capable of doing work. Hence the resilience is also defined as the capacity of a strained body for doing work on the removal of the straining force
• The strain energy stored in the body will be maximum when the body is stressed up to the elastic limit
• Resilience is also defined as the energy absorbed by a component within an elastic region. 
• The proof resilience is the maximum quantity of strain energy stored in a body when strained up to the elastic limit
• Modulus of resilience is defined as proof resilience of a material per unit volume
• Impact energy is defined as the energy absorbed by the component just before its fracture. It is also called as toughness. 

Concept:

Strain energy stored in prismatic bar due to axial load is:

\(U = \frac{{{P^2}L}}{{2AE}}\)

Calculation:

\({U_1}\left( {{P_1}} \right) = \frac{{P_1^2L}}{{2AE}}\)

\({U_2}\left( {{P_2}} \right) = \frac{{P_2^2L}}{{2AE}}\)

\(U\left( {{P_1} + {P_2}} \right) = \frac{{{{\left( {{P_1} + {P_2}} \right)}^2}L}}{{2AE}} = \frac{{\left( {P_1^2 + P_2^2 + 2{P_1}{P_2}} \right)L}}{{2AE}}\)

\(U\left( {{P_1} + {P_2}} \right) = \frac{{P_1^2L}}{{2AE}} + \frac{{P_2^2L}}{{2AE}} + \frac{{2{P_1}{P_2}L}}{{2AE}} = {U_1} + {U_2} + \frac{{{P_1}{P_2}L}}{{AE}}\)

Concept:

The strain energy is the energy stored in a body due to its elastic deformation.

The most general formula for strain energy is:

U = \(\frac{1}{2}\) x F x δ 

Where, F is the applied force and δ is deformation

However, when stress is proportional to strain, ϵ, the strain energy formula is:

U = \(\frac{1}{2}\) x V x σ x ϵ 

Where, V is the volume of the material.

Based on the above formulas, the strain energy in the material due to different types of forces/moments is given as:

1. Strain energy due to bending is:

U = ∫ \(\frac{M^2}{2EI}\)dx

2. Strain energy due to torsion is:

U = ∫ \(\frac{T^2L}{2GJ}\)dx

Concept:

The two theorems of Castigliano's for structural analysis are as follows:

Castigliano’s 1^st theorem: The partial derivative of total strain energy of the system with respect to any particular deflection at a point is equal to the force applied at that point in the same direction as that of the deflection.

\(\frac{{\partial {\rm{u}}}}{{\partial {\rm{\delta }}}} = {\rm{w}}\)

Castigliano’s 2^nd theorem: The partial derivative of strain energy of the system with respect to load at any point is equal to deflection at that point.

\(\frac{{\partial {\rm{u}}}}{{\partial {\rm{w}}}} = {\rm{\delta }}\)

Also, the partial derivative of strain energy of the system with respect to couple at any point is equal to slope at that point.

\(\frac{{\partial {\rm{u}}}}{{\partial {\rm{M}}}} = {\rm{\theta }}\)

Where,

u is the strain energy of the system.

Note:

These theorems are valid in both the beams and truss, but in truss strain energy is only due to axial loads.

Concept:

Strain energy in pure shearing:

• Strain Energy is the energy stored by the body due to deformation.
• In the figure below if the material is subjected to Shearing force P, the face BC will move and produce a shear strain. 

Strain energy U = Work done = Force x displacement = \(\frac{1}{2}\times{{P}} \times {{C}}{{{C}}_1} = \frac{1}{2}\times{{P}} \times {{CD}} \times \phi {\rm{\;}}\)
\(P = \tau \times BC \)

\(\phi = \frac{\tau}{C} \)

where,

\(\tau \) = shear stress, ϕ = shear strain, C = Modulus of Rigidity

V = volume of block = BC x CD x unit depth

Considering unit depth to the diagram, we have

\({{U}} = \frac{1}{2}\times{\rm{P}} \times {\rm{CD}} \times \phi {\rm{\;}} = \frac{1}{2}\times{\rm{\tau }} \times {\rm{BC}} \times {\rm{CD}} \times \frac{{\rm{\tau }}}{{\rm{C}}}\; = \frac{{{{\bf{\tau }}^2}}}{{2{{C}}}}\; \times {{V}}\)

Calculation:

Given:

E = Young’s modulus, v = Poisson’s ratio, homogenous and isotropic material

E = 2G (1+ ν) ⇒ \(2{{G}} = \frac{{{E}}}{{1 + {{ν }}}}\)

Strain energy per unit volume \(= \frac{{{{\bf{\tau }}^2}}}{{2G}} = \frac{{{\tau ^2}}}{E}\left( {1 + ν } \right)\)

Key Points

Relation between Young's modulus (E), Bulk modulus (K) ,and Rigidity modulus (G):

E = 2G (1 + ν)

E = 3K (1 - 2ν)

\(E = \frac {9KG}{3K + G}\)​