Elastic Strain Energy due to Gradual Loading.
Elastic Strain Energy due to Sudden Loading.
Elastic Strain energy due to impact loading.
Elastic Strain Energy due to Principal Stresses.
Energy of Dilation And Distortion.
This presentation gives an introduction to mechanical vibration or Theory of Vibration for BE courses. Presentation is prepared as per the syllabus of VTU.For any suggestions and criticisms please mail to: hareeshang@gmail.com or visit:ww.hareeshang.wikifoundry.com.
Thanks for watching this presentation.
Hareesha N G
Static force analysis, Unit-1 of Dynamics of machines of VTU Syllabus compiled by Hareesha N Gowda, Asst. Prof, Dayananda Sagar College of Engg, Blore. Please write to hareeshang@gmail.com for suggestions and criticisms.
This presentation gives an introduction to mechanical vibration or Theory of Vibration for BE courses. Presentation is prepared as per the syllabus of VTU.For any suggestions and criticisms please mail to: hareeshang@gmail.com or visit:ww.hareeshang.wikifoundry.com.
Thanks for watching this presentation.
Hareesha N G
Static force analysis, Unit-1 of Dynamics of machines of VTU Syllabus compiled by Hareesha N Gowda, Asst. Prof, Dayananda Sagar College of Engg, Blore. Please write to hareeshang@gmail.com for suggestions and criticisms.
This document gives the class notes of Unit 6: Bending and shear Stresses in beams. Subject: Mechanics of materials.
Syllabus contest is as per VTU, Belagavi, India.
Notes Compiled By: Hareesha N Gowda, Assistant Professor, DSCE, Bengaluru-78.
PPT describes the engine performance parameters of the I.C. engine.
Engine performance is an indication of the degree of success of the engine performs its assigned task, i.e. the conversion of the chemical energy contained in the fuel into the useful mechanical work. The engine performance is indicated by the term efficiency, η. Five important engine efficiencies and other related engine performance parameters are:
Power
Indicated Thermal Efficiency (ηith)
Brake Thermal Efficiency (ηbth)
Mechanical Efficiency (ηm)
Volumetric Efficiency (ηv)
Relative Efficiency or Efficiency Ratio (ηrel)
Mean Effective Pressure (Pm)
Specific Fuel Consumption (sfc)
Fuel-Air or Air-Fuel Ratio (F/A or A/F)
Calorific Value (CV)
Power:-
The main purpose of running an engine is to obtain mechanical power.
Brake Power (B.P.)
The power developed by an Engine at the output shaft is called the brake power.
Brake Power= Brake Workdone/Time
B.P.=BWD/sec.
Indicated power (I.P.)
The total power developed by Combustion of fuel in the combustion chamber is called indicated power.
Indicated Power= Indicated Workdone/Time
I.P.=IWD/sec.
Frictional Power (F.P.)
The difference between I.P. and B.P. is called frictional power (f.p.).
FP = IP – BP
Thermal Efficiency (ηth)
Thermal efficiency is the ratio of Power to energy supplied by the fuel.
ηth= Power/ Energy
In I.C. Engine, thermal efficiency can be classified into two categories i.e.
Indicated Thermal Efficiency (ηith)
Indicated thermal efficiency is the ratio of indicated power to the heat supplied or added.
ηith= IP/Qs
2. Brake Thermal Efficiency (ηith)
Brake Thermal Efficiency is the ratio of brake power to the heat supplied or added.
ηbth= BP/Qs
Volumetric Efficiency (ηv)
This is one of the most important parameters which decide the performance of four-stroke engines. Four stoke engines have distinct suction stoke, volumetric efficiency indicates the breathing ability of the engine.
Volumetric efficiency is defined as the ratio of actual flow rate of air into the intake system to rate at which the volume is displaced by the system.
ηv= (푚 ̇"a/a" )/(푉푑푖푠푝푎푐푒푑 푋 푁/2)
"a"= Inlet density is taken atmospheric air density
N= Number of the cylinder in use
The various forces acts on the reciprocating parts of an engine.
The resultant of all the forces acting on the body of the engine due to inertia forces only is known as unbalanced force or shaking force.
This document gives the class notes of Unit 6: Bending and shear Stresses in beams. Subject: Mechanics of materials.
Syllabus contest is as per VTU, Belagavi, India.
Notes Compiled By: Hareesha N Gowda, Assistant Professor, DSCE, Bengaluru-78.
PPT describes the engine performance parameters of the I.C. engine.
Engine performance is an indication of the degree of success of the engine performs its assigned task, i.e. the conversion of the chemical energy contained in the fuel into the useful mechanical work. The engine performance is indicated by the term efficiency, η. Five important engine efficiencies and other related engine performance parameters are:
Power
Indicated Thermal Efficiency (ηith)
Brake Thermal Efficiency (ηbth)
Mechanical Efficiency (ηm)
Volumetric Efficiency (ηv)
Relative Efficiency or Efficiency Ratio (ηrel)
Mean Effective Pressure (Pm)
Specific Fuel Consumption (sfc)
Fuel-Air or Air-Fuel Ratio (F/A or A/F)
Calorific Value (CV)
Power:-
The main purpose of running an engine is to obtain mechanical power.
Brake Power (B.P.)
The power developed by an Engine at the output shaft is called the brake power.
Brake Power= Brake Workdone/Time
B.P.=BWD/sec.
Indicated power (I.P.)
The total power developed by Combustion of fuel in the combustion chamber is called indicated power.
Indicated Power= Indicated Workdone/Time
I.P.=IWD/sec.
Frictional Power (F.P.)
The difference between I.P. and B.P. is called frictional power (f.p.).
FP = IP – BP
Thermal Efficiency (ηth)
Thermal efficiency is the ratio of Power to energy supplied by the fuel.
ηth= Power/ Energy
In I.C. Engine, thermal efficiency can be classified into two categories i.e.
Indicated Thermal Efficiency (ηith)
Indicated thermal efficiency is the ratio of indicated power to the heat supplied or added.
ηith= IP/Qs
2. Brake Thermal Efficiency (ηith)
Brake Thermal Efficiency is the ratio of brake power to the heat supplied or added.
ηbth= BP/Qs
Volumetric Efficiency (ηv)
This is one of the most important parameters which decide the performance of four-stroke engines. Four stoke engines have distinct suction stoke, volumetric efficiency indicates the breathing ability of the engine.
Volumetric efficiency is defined as the ratio of actual flow rate of air into the intake system to rate at which the volume is displaced by the system.
ηv= (푚 ̇"a/a" )/(푉푑푖푠푝푎푐푒푑 푋 푁/2)
"a"= Inlet density is taken atmospheric air density
N= Number of the cylinder in use
The various forces acts on the reciprocating parts of an engine.
The resultant of all the forces acting on the body of the engine due to inertia forces only is known as unbalanced force or shaking force.
CONTENT:
1. Elastic strain energy
2. Strain energy due to gradual loading
3. Strain energy due to sudden loading
4. Strain energy due to impact loading
5. Strain energy due to shock loading
6. Strain energy due to shear loading
7. Strain energy due to bending (flexure)
8. Strain energy due to torsion
9. Examples
When a body is subjected to gradual, sudden or impact load, the body deforms and work is done upon it. If the elastic limit is not exceed, this work is stored in the body. This work done or energy stored in the body is called strain energy.
When a body is subjected to gradual, sudden or impact load, the body deforms and work is done upon it. If the elastic limit is not exceed, this work is stored in the body. This work done or energy stored in the body is called strain energy.
Strain energy is a type of potential energy that is stored in a structural member as a result of elastic deformation. The external work done on such a member when it is deformed from its unstressed state is transformed into (and considered equal to the strain energy stored in it.
This PPT contain the basic topic about the strength of the material. Such as stress, strain, energy, principle of super position and various other topic of solid mechanics.
Stress is a very significant factor the formation of the structures. These structures either can be formed by natural processes or by manmade processes. In this chapter, we will discuss the basics of stress, stress states, signs, Mohr’s Stress Diagram.
introduction
Classification Of Aggregates, Good Qualities of an Ideal Aggregate: ,Tests on Aggregate:- , Specıfıc gravıty of Aggregate. , Flakiness & Elongation Index , Fineness Modulus (f.m):
It is refers to the downward sliding of huge quantities of land mass
Downward movement of slope forming material composed of rocks and soil or combination of all these material along surfaces of separation by FALLING, SLIDING AND FLOWING either sudden or slow from one place to another place.
All given topics covered with animations how to solve problem of E.G.
1. Scales
2. Engineering Curves - I
3. Engineering Curves - II
4. Loci of Points
5. Orthographic Projections - Basics
6. Conversion of Pictorial View into Orthographic Views
7. Projections of Points and Lines
8. Projection of Planes
9. Projection of Solids
10. Sections & Development
11. Intersection of Surfaces
12. Isometric Projections
13. Exercises
14. Solutions – Applications of Lines
Introduction
Elements of Flexible Manufacturing System
Objective of Flexible Manufacturing System
Classification of Flexible Manufacturing System
Flexible Manufacturing System layout
Advantages & Limitation of Flexible Manufacturing System
Application of Flexible Manufacturing System
Manufacturing Flexibility
Classification of manufacturing process
Classification based on Function of Process
Classification based on quantity of production
Selection of manufacturing process
Casting Process
Joining Processes
Machining Processes
Surface Finishing Processes
Classification of manufacturing processAkhtar Kamal
Classification of manufacturing process...
Process for changing Physical properties of work piece.
Casting process
Primary metal working processes.
Shearing and Forming processes.
Joining processes.
Machining processes.
Surface finishing processes.
system of algebraic equation by Iteration methodAkhtar Kamal
solve the system of algebraic equation by Iteration method
classification of Iteration method:-
(1) Jacobi's method
(2) Gauss-Seidel method
each problem
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
For more technical information, visit our website https://intellaparts.com
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
CW RADAR, FMCW RADAR, FMCW ALTIMETER, AND THEIR PARAMETERSveerababupersonal22
It consists of cw radar and fmcw radar ,range measurement,if amplifier and fmcw altimeterThe CW radar operates using continuous wave transmission, while the FMCW radar employs frequency-modulated continuous wave technology. Range measurement is a crucial aspect of radar systems, providing information about the distance to a target. The IF amplifier plays a key role in signal processing, amplifying intermediate frequency signals for further analysis. The FMCW altimeter utilizes frequency-modulated continuous wave technology to accurately measure altitude above a reference point.
Understanding Inductive Bias in Machine LearningSUTEJAS
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
2. INTRODUCTION
1) Elastic Strain Energy due to Gradual
Loading.
2) Elastic Strain Energy due to Sudden Loading.
3) Elastic Strain energy due to impact loading.
4) Elastic Strain Energy due to Principal
Stresses.
5) Energy of Dilation And Distortion.
Akhtar Kamal
3. What is strain energy?
When the body is subjected to gradual, sudden or impact load, the
body deforms, and work done upon it.
The material behave like a perfect spring and oscillates about its
mean position.
If the elastic limit is not exceeded, this work is stored in the body.
This work done or energy stored in the body is called strain energy.
Akhtar Kamal
4. Some Important Definition And Question
(1)Resilience:
Total strain energy stored in body is called resilience. It is denoted as ‘’
𝒖 =
𝝈 𝟐
𝟐𝑬
× 𝑽
Where… 𝝈 = 𝒔𝒕𝒓𝒆𝒔𝒔
𝑽 = 𝑽𝒐𝒍𝒖𝒎𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒃𝒐𝒅𝒚
(2)Proof Resilience:
Maximum strain energy which can be stored in a body at elastic limit is called proof
resilience. It is denoted as ‘𝒖 𝒑’
𝒖 𝒑 =
𝝈 𝑬
𝟐
𝟐𝑬
× 𝑽
Where… 𝝈 𝑬 = 𝒔𝒕𝒓𝒆𝒔𝒔 at elastic limit
(3)Modulus of resilience:
Maximum strain energy which can be stored in a body per unit volume, at elastic limit is
called Modulus of resilience. It is denoted as ‘𝒖 𝒎’
𝒖 𝒎 =
𝝈 𝑬
𝟐
𝟐𝑬
Akhtar Kamal
5. Strain Energy Due to Gradual Loading
Considr a bar of length is 𝑙 and uniform section
area 𝐴, subjected to gradual load 𝑃.
Akhtar Kamal
6. Stress Due to Gradual Load
Since the load is applied gradually,(i.e. it increases from 0 to P),
average load is considered.
Work done on the bar = Area of the load – Deformation
diagram.
=
𝟏
𝟐
× 𝑷 × 𝜹𝒍 … . . (𝟏)
Work stored in the bar = Area of the resistance – Deformation
diagram.
=
𝟏
𝟐
× 𝑹 × 𝜹𝒍
=
𝟏
𝟐
× (𝝈 ∙ 𝑨) × 𝜹𝒍 … . . (𝟐)
Work done = Work stored
𝟏
𝟐
× 𝑷 × 𝜹𝒍 =
𝟏
𝟐
× (𝝈 ∙ 𝑨) × 𝜹𝒍
𝝈 =
𝑷
𝑨
… . . 𝐒𝐭𝐫𝐞𝐬𝐬 𝐝𝐮𝐞 𝐭𝐨 𝐠𝐫𝐚𝐝𝐮𝐚𝐥 𝐥𝐨𝐚𝐝.Akhtar Kamal
8. Elastic Strain Energy due to Sudden Loading
When the load is applied suddenly the value of
the load is P throughout the deformation.
But, Resistance R increases from 0 to R.
Work done on the bar= 𝑃 × 𝛿𝑙 … . . 1
Work store in the bar=
1
2
× 𝑅 × 𝛿𝑙
=
1
2
× 𝜎 × 𝐴 × 𝛿𝑙 … . . (2)
Akhtar Kamal
9. Work done = work store
𝑷 × 𝜹𝒍 =
𝟏
𝟐
× 𝝈 × 𝑨 × 𝜹𝒍
𝝈 =
𝟐𝑷
𝑨
Hence, the maximum stress intensity due to a suddenly applied load twice
the stress intensity produced by the load of the same magnitude applied
gradually.
Akhtar Kamal
18. Energy Of Dilation and Distortion
Total strain energy given by equation (1) of article 1.5 can
be separated into the following two strain energies .
a) Strain energy of dilatation (dilation) or volume metric
strain energy (strain energy of uniform compression
or tension).
b) Strain Energy of distortion(shear strain energy)
To accomplish this , Let the principal strains be 𝜺 𝟏, 𝜺 𝟐
𝒂𝒏𝒅 𝜺 𝟑,in the deration of principal stresses 𝝈 𝟏, 𝝈 𝟐 and 𝝈 𝟑
respectively.
Akhtar Kamal
20. Energy Of Dilation and Distortion
From the above discussion, following conclusions can be
made.
(a) If 𝝈 𝟏 = 𝝈 𝟐 = 𝝈 𝟑
𝜺 𝟏= 𝜺 𝟐 = 𝜺 𝟑,
This means that there is no distortion (so that no shearing
stresses and shearing strains will be present anywhere in the
block) but only volumetric change(dilation) occurs.
(b) If 𝝈 𝟏 + 𝝈 𝟐 + 𝝈 𝟑 = 𝟎, 𝜺 𝒗 = 𝟎
This means that if the sum of three principal stress is zero,
there is no volumetric change(dilation), but only the
distortion occurs.
The above to conclusion can be used to break the given three
principal stresses into two sets of principal stresses such that
one set produces dilation (volumetric change) only, while
the other produces distortion (shear stresses) only.
Akhtar Kamal
21. Consider a small block of length δℓ, width δb and
height δh subjected to three principal stresses σ1,
σ2 and σ3 as shown in figure
σ1 = Principal stress on face of area (δb × δh)
σ2 = Principal stress on face of area (δℓ × δh)
σ3 = Principal stress on face of area (δℓ × δb)
μ = Poisson’s ratio for the material.
Akhtar Kamal
23. .˙., Extention of the block in the direction of σ1
δℓı =εı · δı
δℓı =
1
𝐸
[σ1 – μ (σ2+σ3 )] δℓ
Akhtar Kamal
24. .˙. Strain energy due to σ1
=
1
2
(Load due to σ1 in the direction of σ1) × δℓ1
=
1
2
[σ1.δb.δh] x
1
𝐸
[σ1‒ μ (σ2+ σ3)] δℓ
=
1
2𝐸
[σ1² ‒ μ (σ1 σ2 +σ1 σ3)] (δb.δh.δℓ)]
=
1
2𝐸
[σ1² ‒ μ (σ1 σ2 +σ1 σ3)] δV
Where,
δV= volume of block
= δb.δh.δl
Akhtar Kamal
25. Similarly,
Strain energy due to σ2
=
1
2𝐸
[σ2² ‒ μ(σ2σ1 + σ2 σ3)]δV
Strain Energy due to σ3
=
1
2𝐸
[σ3² ‒ μ σ3 σ1 + σ3 σ2)]δV
Akhtar Kamal
26. .˙. δu = Total Strain energy for volume δV
= Sum of strain energies due to σ1,σ2 and σ3
=
1
2𝐸
[σ1² ‒ μ(σ1 σ2 +σ1 σ3)]δV
+
1
2𝐸
[σ2 ² ‒ μ(σ2 σ1 +σ2 σ3)]δV
+
1
2𝐸
[σ3 ² ‒ μ(σ3 σ1 + σ3 σ2)]δV
.˙. δu =
1
2𝐸
[σ1²+ σ2 ²+ σ3 ²- 2μ(σ1 σ2 + σ2 σ3 + σ3 σ1)] δV
Akhtar Kamal
27. Thus for a body of Volume V Subjected to the
principal Stresses σ1,σ2 and σ3, total strain energy
is given by,
u=
𝟏
𝟐𝑬
[σ1²+ σ2 ²+ σ3 ² ‒ 2μ(σ1 σ2 + σ2 σ3 + σ3 σ1)] V
Sign for Principal Stresses,
Tension = + ve
Compression = -ve
Akhtar Kamal
28. The expression for the strain energy for the simple
cases of stresses can be easily deducted from the
general equation (1) for the strain energy.
Akhtar Kamal
31. Let τ be simple shear in volume V
Then the principle stress will be,
σ1= τ , σ2 = ‒ τ , σ3 =0
Substituting these values in (1) we get,
u=
1
2𝐸
[τ ²(‒ τ )²+0 ‒ 2μ(τ)(‒ τ)] V
=
1
2𝐸
[2 τ ²+ 2 μ τ ²]V
=
𝜏 ²
𝐸
(1+ μ) V
But,E =2G (1+ μ) G = Modulus of rigidity
Therefore
1+ μ
𝐸
=
1
2𝐺
u=
τ
𝟐𝑮
V
Strain energy per unit volume = u =
τ ²
𝟐𝑮
Akhtar Kamal
32. Let p= hydrostatic tension or hydrostatic pressure
.˙. Either σ1 =p , σ2 =p and σ3=p
Or σ1= -p , σ2 = -p and σ3 = -p
Substituting any one in equation (1) we get,
u=
1
2𝐸
[ p² + p² + p² ‒ 2μ( p.p + p.p + p.p)] V
=
1
2𝐸
[3p² ‒ 2μ(3p²)]V
=
3p²
2𝐸
(1- 2μ)V
but E = 3k(1- 2μ)
.˙.
(1− 2μ)
𝐸
=
1
3𝑘
k = Bulk modulus
.˙. u =
p²
𝟐𝒌
V
Akhtar Kamal