1) Materials deform when stressed, returning to original shape within the elastic limit. Beyond this, deformation is permanent.
2) Hooke's law describes the linear relationship between stress and strain within the elastic limit. The slope is Young's modulus, a measure of stiffness.
3) Poisson's ratio defines the lateral contraction that occurs when a material is stretched. Most materials contract laterally to some degree.
1. When a force is applied to a body, it causes the body to deform or change shape. This deformation is called strain. Direct stress is calculated as the applied force divided by the cross-sectional area.
2. Materials deform both elastically and plastically when stressed. Elastic deformation is reversible but plastic deformation causes a permanent change in shape. Hooke's law describes the linear elastic behavior of many materials, where stress is directly proportional to strain up to the elastic limit.
3. Thermal expansion and contraction can induce stress in materials as temperature changes unless deformation is unconstrained. The total strain is the sum of strain due to stress and strain due to temperature changes.
This document provides an overview of fundamental mechanical engineering concepts including stress, strain, Hooke's law, stress-strain diagrams, elastic constants, and mechanical properties. It defines stress as force per unit area and strain as the deformation of a material from stress. Hooke's law states that stress is directly proportional to strain within the elastic limit. Stress-strain diagrams are presented for ductile and brittle materials. Key elastic constants like Young's modulus, shear modulus, and Poisson's ratio are defined along with their relationships. Mechanical properties of materials like elasticity, plasticity, ductility, strength, brittleness, toughness, hardness, and stiffness are also summarized.
This document provides an overview of fundamental mechanical engineering concepts including stress, strain, Hooke's law, stress-strain diagrams, and elastic properties of materials. Key points include:
- Stress is defined as force per unit area. Normal stress acts perpendicular to the area while shear stress acts tangentially.
- Strain is the deformation from applied stress. Tensile and compressive strains refer to changes in length while shear and volumetric strains refer to other types of deformations.
- Hooke's law states that stress is directly proportional to strain within the elastic limit. The modulus of elasticity is the constant of proportionality.
- Stress-strain diagrams graphically show the relationship between stress and strain
Loads can be tensile (pulling) or compressive (pushing) forces. Common types of loads include dead loads from structural weight, live loads from moving objects, impact loads from vibrations, and cyclic loads from repeated forces. When loads are applied, they cause stress in materials. Stress is the internal resisting force per unit area. Stresses can be tensile (pulling), compressive (pushing), or shear (tangential). Corresponding strains are the changes in dimensions from stresses. Hooke's law states that within the elastic limit, stress is proportional to strain by a constant modulus of elasticity.
The document discusses stress and strain in engineering structures. It defines load, stress, strain and different types of each. Stress is the internal resisting force per unit area within a loaded component. Strain is the ratio of dimensional change to original dimension of a loaded body. Loads can be tensile, compressive or shear. Hooke's law states stress is proportional to strain within the elastic limit. The elastic modulus defines this proportionality. A tensile test measures the stress-strain curve, identifying elastic limit and other failure points. Multi-axial stress-strain relationships follow Poisson's ratio definitions.
This document provides information on stress, strain, elasticity, Hooke's law, and other fundamental concepts in strength of materials. Some key points:
- Stress is defined as the internal resisting force per unit area within a material when subjected to external forces. It is proportional to applied load and inversely proportional to cross-sectional area.
- Strain is the ratio of deformation to original dimension of a material. There are different types including tensile, compressive, and shear strains.
- Hooke's law states that within the elastic limit, stress is proportional to strain. The proportionality constant is known as modulus of elasticity.
- Materials behave elastically and return to their original shape when
The document discusses stresses and strains, including:
1) It defines stress-strain diagrams, which plot stress versus strain, and describes the different regions including the elastic region, yield point, plastic region, and fracture point.
2) It explains concepts such as Hooke's law, elastic limit, yield strength, tensile strength, and strain hardening.
3) It discusses modulus of elasticity (Young's modulus), which is a measure of a material's stiffness, and Poisson's ratio, which relates lateral and linear strains.
1. When a force is applied to a body, it causes the body to deform or change shape. This deformation is called strain. Direct stress is calculated as the applied force divided by the cross-sectional area.
2. Materials deform both elastically and plastically when stressed. Elastic deformation is reversible but plastic deformation causes a permanent change in shape. Hooke's law describes the linear elastic behavior of many materials, where stress is directly proportional to strain up to the elastic limit.
3. Thermal expansion and contraction can induce stress in materials as temperature changes unless deformation is unconstrained. The total strain is the sum of strain due to stress and strain due to temperature changes.
This document provides an overview of fundamental mechanical engineering concepts including stress, strain, Hooke's law, stress-strain diagrams, elastic constants, and mechanical properties. It defines stress as force per unit area and strain as the deformation of a material from stress. Hooke's law states that stress is directly proportional to strain within the elastic limit. Stress-strain diagrams are presented for ductile and brittle materials. Key elastic constants like Young's modulus, shear modulus, and Poisson's ratio are defined along with their relationships. Mechanical properties of materials like elasticity, plasticity, ductility, strength, brittleness, toughness, hardness, and stiffness are also summarized.
This document provides an overview of fundamental mechanical engineering concepts including stress, strain, Hooke's law, stress-strain diagrams, and elastic properties of materials. Key points include:
- Stress is defined as force per unit area. Normal stress acts perpendicular to the area while shear stress acts tangentially.
- Strain is the deformation from applied stress. Tensile and compressive strains refer to changes in length while shear and volumetric strains refer to other types of deformations.
- Hooke's law states that stress is directly proportional to strain within the elastic limit. The modulus of elasticity is the constant of proportionality.
- Stress-strain diagrams graphically show the relationship between stress and strain
Loads can be tensile (pulling) or compressive (pushing) forces. Common types of loads include dead loads from structural weight, live loads from moving objects, impact loads from vibrations, and cyclic loads from repeated forces. When loads are applied, they cause stress in materials. Stress is the internal resisting force per unit area. Stresses can be tensile (pulling), compressive (pushing), or shear (tangential). Corresponding strains are the changes in dimensions from stresses. Hooke's law states that within the elastic limit, stress is proportional to strain by a constant modulus of elasticity.
The document discusses stress and strain in engineering structures. It defines load, stress, strain and different types of each. Stress is the internal resisting force per unit area within a loaded component. Strain is the ratio of dimensional change to original dimension of a loaded body. Loads can be tensile, compressive or shear. Hooke's law states stress is proportional to strain within the elastic limit. The elastic modulus defines this proportionality. A tensile test measures the stress-strain curve, identifying elastic limit and other failure points. Multi-axial stress-strain relationships follow Poisson's ratio definitions.
This document provides information on stress, strain, elasticity, Hooke's law, and other fundamental concepts in strength of materials. Some key points:
- Stress is defined as the internal resisting force per unit area within a material when subjected to external forces. It is proportional to applied load and inversely proportional to cross-sectional area.
- Strain is the ratio of deformation to original dimension of a material. There are different types including tensile, compressive, and shear strains.
- Hooke's law states that within the elastic limit, stress is proportional to strain. The proportionality constant is known as modulus of elasticity.
- Materials behave elastically and return to their original shape when
The document discusses stresses and strains, including:
1) It defines stress-strain diagrams, which plot stress versus strain, and describes the different regions including the elastic region, yield point, plastic region, and fracture point.
2) It explains concepts such as Hooke's law, elastic limit, yield strength, tensile strength, and strain hardening.
3) It discusses modulus of elasticity (Young's modulus), which is a measure of a material's stiffness, and Poisson's ratio, which relates lateral and linear strains.
1. The document defines key terms related to loads, stresses, strains, and elastic behavior of materials. It describes different types of loads, stresses, strains and their relationships based on Hooke's law.
2. Formulas are provided for calculating tensile stress, compressive stress, shear stress, elastic modulus, and deformation of tapered and composite bars.
3. The principles of superposition and self-weight induced stresses in cantilever beams are also summarized.
Ekeeda is an online portal which creates and provides exclusive content for all branches engineering.To have more updates you can goto www.ekeeda.com..or you can contact on 8433429809...
1-Machine design - Stresses in Machine Members (2) - Copy.pptxssuser2e7793
Types of stresses include tensile, compressive, shear, torsional, and bearing. Stresses are caused by external forces and loads acting on a body. Stress is equal to force divided by cross-sectional area. Strain is the deformation or change in length caused by stresses. Hooke's law states stress is proportional to strain. Shear stress is caused by tangential forces across a section and shear strain is the resulting angular deformation. Torsional shear stress results from opposing torque or twisting moments.
This document provides an overview of simple stress and strain concepts. It defines load, stress, strain, Hooke's law, elastic moduli, stress-strain curves, and analyses of bars with varying cross-sections, tapered sections, and composite sections. Key points include: load is defined as external forces on a structure; stress is the internal resisting force per unit area; strain is the ratio of dimensional change to original dimension; Hooke's law states stress is proportional to strain within the elastic limit; and stress-strain curves from tensile tests show elastic and plastic deformation regions.
This document provides an overview of basic concepts in strength of materials, including stress, strain, and different types of stresses. It defines stress as the internal force of resistance per unit area offered by a body against deformation. Stress is calculated as force divided by area. Normal stress acts perpendicular to the cross-sectional area and can be tensile or compressive. Shear stress refers to a cutting action and is represented by the symbol tau. The document also defines strain as the change in length divided by the original length, represented by epsilon. It provides stress elements and equations to calculate direct normal and shear stresses.
mechanics of materials ,strength of materials, bar ,Tensile stress, compressive stress, shear stress, normal stress,Tensile strain, compressive strain, shear strain
The document discusses stress and strain in materials. It introduces the key concepts of normal stress, shear stress, bearing stress, and thermal stress. Normal stress acts perpendicular to a cross-section, shear stress acts tangentially, and bearing stress occurs at contact points. The relationships between stress, strain, elastic modulus, and Poisson's ratio are explained. Methods for calculating stress and strain in axial loading, torsion, bending and combined loading are presented through examples. The stress-strain diagram is discussed to show material properties like yield strength and ductility.
This document provides an overview of the syllabus and objectives for the course CE8395 Strength of materials for Mechanical Engineers. It outlines the 5 units that will be covered: 1) Stress, Strain and Deformation of Solids, 2) Transverse Loading on Beams and Stresses in Beam, 3) Torsion, 4) Deflection of Beams, and 5) Thin Cylinders, Spheres and Thick Cylinders. Key concepts that will be studied include stresses, strains, principal stresses, shear force and bending moment in beams, torsion, deflections, and stresses in thin shells and cylinders. The document also provides two mark questions and answers related to stress, strain, elastic properties
1. The document discusses various types of mechanical loading and stresses including tensile, compressive, shear, bending, and torsional stresses.
2. It describes different types of strains and properties of materials like elasticity, plasticity, ductility. Hooke's law and relationships between stress and strain are explained.
3. Methods for analyzing stresses in machine components subjected to combinations of loads are presented, including principal stresses, Mohr's circle, and thermal stresses. Bending stresses and shear stresses are analyzed for beams under different support conditions.
This document is a presentation on stress and strain analysis given by Mr. Oduor Wafulah. It defines stress and strain, discusses related terminology, and outlines the different types of stress and strain. It also covers Hooke's law, which states that stress is proportional to strain, and stress-strain diagrams. Factors like elasticity, elastic limits, and modulus of elasticity are examined in relation to the stress-strain relationship. Beams theory and the theories of Timoshenko and torsion are also briefly introduced.
mechanics of materials presentation - vtuSuryaRS10
This document discusses concepts related to mechanics of materials including stress, elastic constants, thermal stresses, and the relationships between them. It defines stress as the intensity of internally distributed forces that resist external forces. It explains the four elastic constants - Young's modulus, shear modulus, bulk modulus, and Poisson's ratio. Thermal stresses that develop due to restraint against thermal expansion when temperature changes are also discussed. Complete restraint causes compressive thermal stresses proportional to the temperature change and coefficient of thermal expansion.
This document discusses key concepts in strength of materials and engineering basics. It defines stress as the force per unit area on a material, and strain as the deformation or change in shape of a material under stress. The document outlines different types of stresses like tensile, compressive, and shear stress and the corresponding strains. It also discusses stress-strain curves and elastic properties like Young's modulus and Poisson's ratio. Finally, it covers topics like types of beams, loads, mechanical properties and more.
This document discusses key concepts in strength of materials and engineering basics. It defines stress as the force per unit area on a material, and strain as the deformation or change in shape of a material under stress. The document outlines different types of stresses like tensile, compressive, and shear stress and the corresponding strains. It also discusses stress-strain curves and elastic properties like Young's modulus and Poisson's ratio. Finally, it covers types of beams, loads, and mechanical properties of materials.
The document discusses the behavior of materials under stress and strain. It defines stress as the internal resistance of a material to external loads, and strain as the deformation or change in shape of a material under stress. The key types of stress are tensile, compressive, and shear stress. Hooke's law states that stress is proportional to strain within the material's elastic limit, after which plastic deformation occurs. The elastic modulus, shear modulus, and bulk modulus describe a material's response to different types of stress.
This document discusses linear and non-linear elasticity concepts relevant to rock mechanics. It defines key terms like stress, strain, elastic moduli, and principal stresses/strains. It describes how stress and strain relate for isotropic materials using Hooke's law and elastic constants. It also covers the stress tensor, Mohr's circle, strain energy, and the differences between linear, perfectly elastic, elastic with hysteresis, and permanently deforming non-linear elastic models.
This document provides an overview of strength of materials and introduces key concepts. It discusses stress and strain, ductile and brittle materials, and stress-strain diagrams. Stress is defined as the internal resisting force per unit area acting on a material. Strain is the ratio of change in dimension to the original dimension when a body is subjected to external force. Ductile materials show deformation under stress, while brittle materials do not. The stress-strain diagram shows the relationship between stress and strain for ductile and brittle materials.
This document provides an introduction and overview of mechanics of materials. It defines key terms like stress, strain, normal stress, shear stress, factor of safety, and allowable stress. It also gives examples of calculating stresses in structural members subjected to various loads. The document is an introductory reading for a mechanics of materials course that will analyze the relationship between external forces and internal stresses and strains in structural elements.
The document provides an introduction to mechanics of deformable solids. It defines stress as force per unit area and distinguishes between normal and shear stresses. Normal stresses are stresses acting perpendicular to a surface, and can be tensile or compressive. Shear stresses act parallel to a surface. The general state of stress at a point involves six independent stress components - normal stresses on three perpendicular planes and shear stresses on those planes. Notation for stresses depends on the coordinate system used.
International Conference on NLP, Artificial Intelligence, Machine Learning an...gerogepatton
International Conference on NLP, Artificial Intelligence, Machine Learning and Applications (NLAIM 2024) offers a premier global platform for exchanging insights and findings in the theory, methodology, and applications of NLP, Artificial Intelligence, Machine Learning, and their applications. The conference seeks substantial contributions across all key domains of NLP, Artificial Intelligence, Machine Learning, and their practical applications, aiming to foster both theoretical advancements and real-world implementations. With a focus on facilitating collaboration between researchers and practitioners from academia and industry, the conference serves as a nexus for sharing the latest developments in the field.
1. The document defines key terms related to loads, stresses, strains, and elastic behavior of materials. It describes different types of loads, stresses, strains and their relationships based on Hooke's law.
2. Formulas are provided for calculating tensile stress, compressive stress, shear stress, elastic modulus, and deformation of tapered and composite bars.
3. The principles of superposition and self-weight induced stresses in cantilever beams are also summarized.
Ekeeda is an online portal which creates and provides exclusive content for all branches engineering.To have more updates you can goto www.ekeeda.com..or you can contact on 8433429809...
1-Machine design - Stresses in Machine Members (2) - Copy.pptxssuser2e7793
Types of stresses include tensile, compressive, shear, torsional, and bearing. Stresses are caused by external forces and loads acting on a body. Stress is equal to force divided by cross-sectional area. Strain is the deformation or change in length caused by stresses. Hooke's law states stress is proportional to strain. Shear stress is caused by tangential forces across a section and shear strain is the resulting angular deformation. Torsional shear stress results from opposing torque or twisting moments.
This document provides an overview of simple stress and strain concepts. It defines load, stress, strain, Hooke's law, elastic moduli, stress-strain curves, and analyses of bars with varying cross-sections, tapered sections, and composite sections. Key points include: load is defined as external forces on a structure; stress is the internal resisting force per unit area; strain is the ratio of dimensional change to original dimension; Hooke's law states stress is proportional to strain within the elastic limit; and stress-strain curves from tensile tests show elastic and plastic deformation regions.
This document provides an overview of basic concepts in strength of materials, including stress, strain, and different types of stresses. It defines stress as the internal force of resistance per unit area offered by a body against deformation. Stress is calculated as force divided by area. Normal stress acts perpendicular to the cross-sectional area and can be tensile or compressive. Shear stress refers to a cutting action and is represented by the symbol tau. The document also defines strain as the change in length divided by the original length, represented by epsilon. It provides stress elements and equations to calculate direct normal and shear stresses.
mechanics of materials ,strength of materials, bar ,Tensile stress, compressive stress, shear stress, normal stress,Tensile strain, compressive strain, shear strain
The document discusses stress and strain in materials. It introduces the key concepts of normal stress, shear stress, bearing stress, and thermal stress. Normal stress acts perpendicular to a cross-section, shear stress acts tangentially, and bearing stress occurs at contact points. The relationships between stress, strain, elastic modulus, and Poisson's ratio are explained. Methods for calculating stress and strain in axial loading, torsion, bending and combined loading are presented through examples. The stress-strain diagram is discussed to show material properties like yield strength and ductility.
This document provides an overview of the syllabus and objectives for the course CE8395 Strength of materials for Mechanical Engineers. It outlines the 5 units that will be covered: 1) Stress, Strain and Deformation of Solids, 2) Transverse Loading on Beams and Stresses in Beam, 3) Torsion, 4) Deflection of Beams, and 5) Thin Cylinders, Spheres and Thick Cylinders. Key concepts that will be studied include stresses, strains, principal stresses, shear force and bending moment in beams, torsion, deflections, and stresses in thin shells and cylinders. The document also provides two mark questions and answers related to stress, strain, elastic properties
1. The document discusses various types of mechanical loading and stresses including tensile, compressive, shear, bending, and torsional stresses.
2. It describes different types of strains and properties of materials like elasticity, plasticity, ductility. Hooke's law and relationships between stress and strain are explained.
3. Methods for analyzing stresses in machine components subjected to combinations of loads are presented, including principal stresses, Mohr's circle, and thermal stresses. Bending stresses and shear stresses are analyzed for beams under different support conditions.
This document is a presentation on stress and strain analysis given by Mr. Oduor Wafulah. It defines stress and strain, discusses related terminology, and outlines the different types of stress and strain. It also covers Hooke's law, which states that stress is proportional to strain, and stress-strain diagrams. Factors like elasticity, elastic limits, and modulus of elasticity are examined in relation to the stress-strain relationship. Beams theory and the theories of Timoshenko and torsion are also briefly introduced.
mechanics of materials presentation - vtuSuryaRS10
This document discusses concepts related to mechanics of materials including stress, elastic constants, thermal stresses, and the relationships between them. It defines stress as the intensity of internally distributed forces that resist external forces. It explains the four elastic constants - Young's modulus, shear modulus, bulk modulus, and Poisson's ratio. Thermal stresses that develop due to restraint against thermal expansion when temperature changes are also discussed. Complete restraint causes compressive thermal stresses proportional to the temperature change and coefficient of thermal expansion.
This document discusses key concepts in strength of materials and engineering basics. It defines stress as the force per unit area on a material, and strain as the deformation or change in shape of a material under stress. The document outlines different types of stresses like tensile, compressive, and shear stress and the corresponding strains. It also discusses stress-strain curves and elastic properties like Young's modulus and Poisson's ratio. Finally, it covers topics like types of beams, loads, mechanical properties and more.
This document discusses key concepts in strength of materials and engineering basics. It defines stress as the force per unit area on a material, and strain as the deformation or change in shape of a material under stress. The document outlines different types of stresses like tensile, compressive, and shear stress and the corresponding strains. It also discusses stress-strain curves and elastic properties like Young's modulus and Poisson's ratio. Finally, it covers types of beams, loads, and mechanical properties of materials.
The document discusses the behavior of materials under stress and strain. It defines stress as the internal resistance of a material to external loads, and strain as the deformation or change in shape of a material under stress. The key types of stress are tensile, compressive, and shear stress. Hooke's law states that stress is proportional to strain within the material's elastic limit, after which plastic deformation occurs. The elastic modulus, shear modulus, and bulk modulus describe a material's response to different types of stress.
This document discusses linear and non-linear elasticity concepts relevant to rock mechanics. It defines key terms like stress, strain, elastic moduli, and principal stresses/strains. It describes how stress and strain relate for isotropic materials using Hooke's law and elastic constants. It also covers the stress tensor, Mohr's circle, strain energy, and the differences between linear, perfectly elastic, elastic with hysteresis, and permanently deforming non-linear elastic models.
This document provides an overview of strength of materials and introduces key concepts. It discusses stress and strain, ductile and brittle materials, and stress-strain diagrams. Stress is defined as the internal resisting force per unit area acting on a material. Strain is the ratio of change in dimension to the original dimension when a body is subjected to external force. Ductile materials show deformation under stress, while brittle materials do not. The stress-strain diagram shows the relationship between stress and strain for ductile and brittle materials.
This document provides an introduction and overview of mechanics of materials. It defines key terms like stress, strain, normal stress, shear stress, factor of safety, and allowable stress. It also gives examples of calculating stresses in structural members subjected to various loads. The document is an introductory reading for a mechanics of materials course that will analyze the relationship between external forces and internal stresses and strains in structural elements.
The document provides an introduction to mechanics of deformable solids. It defines stress as force per unit area and distinguishes between normal and shear stresses. Normal stresses are stresses acting perpendicular to a surface, and can be tensile or compressive. Shear stresses act parallel to a surface. The general state of stress at a point involves six independent stress components - normal stresses on three perpendicular planes and shear stresses on those planes. Notation for stresses depends on the coordinate system used.
International Conference on NLP, Artificial Intelligence, Machine Learning an...gerogepatton
International Conference on NLP, Artificial Intelligence, Machine Learning and Applications (NLAIM 2024) offers a premier global platform for exchanging insights and findings in the theory, methodology, and applications of NLP, Artificial Intelligence, Machine Learning, and their applications. The conference seeks substantial contributions across all key domains of NLP, Artificial Intelligence, Machine Learning, and their practical applications, aiming to foster both theoretical advancements and real-world implementations. With a focus on facilitating collaboration between researchers and practitioners from academia and industry, the conference serves as a nexus for sharing the latest developments in the field.
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECTjpsjournal1
The rivalry between prominent international actors for dominance over Central Asia's hydrocarbon
reserves and the ancient silk trade route, along with China's diplomatic endeavours in the area, has been
referred to as the "New Great Game." This research centres on the power struggle, considering
geopolitical, geostrategic, and geoeconomic variables. Topics including trade, political hegemony, oil
politics, and conventional and nontraditional security are all explored and explained by the researcher.
Using Mackinder's Heartland, Spykman Rimland, and Hegemonic Stability theories, examines China's role
in Central Asia. This study adheres to the empirical epistemological method and has taken care of
objectivity. This study analyze primary and secondary research documents critically to elaborate role of
china’s geo economic outreach in central Asian countries and its future prospect. China is thriving in trade,
pipeline politics, and winning states, according to this study, thanks to important instruments like the
Shanghai Cooperation Organisation and the Belt and Road Economic Initiative. According to this study,
China is seeing significant success in commerce, pipeline politics, and gaining influence on other
governments. This success may be attributed to the effective utilisation of key tools such as the Shanghai
Cooperation Organisation and the Belt and Road Economic Initiative.
Harnessing WebAssembly for Real-time Stateless Streaming PipelinesChristina Lin
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
Introduction- e - waste – definition - sources of e-waste– hazardous substances in e-waste - effects of e-waste on environment and human health- need for e-waste management– e-waste handling rules - waste minimization techniques for managing e-waste – recycling of e-waste - disposal treatment methods of e- waste – mechanism of extraction of precious metal from leaching solution-global Scenario of E-waste – E-waste in India- case studies.
6th International Conference on Machine Learning & Applications (CMLA 2024)ClaraZara1
6th International Conference on Machine Learning & Applications (CMLA 2024) will provide an excellent international forum for sharing knowledge and results in theory, methodology and applications of on Machine Learning & Applications.
2. Concept of elasticity and plasticity
Strength of Material: When an external force acts on a body, the body tend to
undergo some deformation. Due to the cohesion between the molecules, the
body resists deformation. The resistance by which material of the body
opposes the deformation is known as strength of materials.
Elasticity: Property of material by which it returns to its original shape and
size after removing the applied load, is called elasticity. And the material
itself is said to be elastic.
Plasticity: Characteristics of material by which it undergoes inelastic strains
(Permanent Deformation) beyond the elastic limit, known as plasticity. This
property is useful for pressing and forging.
3.
4. Direct or Normal Stress
When a force is transmitted through a body, the body tends to change its
shape or deform. The body is said to be strained
Direct Stress =
Applied Force (F)
Cross Sectional Area (A)
Units: Usually N/m2 (Pa), N/mm2 , MN/m2 , GN/m2 or N/cm2
Note: 1 N/mm2 = 1 MN/m2 = 1 MPa
5. Unit of Stress
1
𝑁
𝑚2 = 1
𝑁
1000 𝑚𝑚 2 = 1
𝑁
106𝑚𝑚2 = 10−6 𝑁
𝑚𝑚2
1
𝑁
𝑚𝑚2 = 106 𝑁
𝑚2
{ Also, 1
𝑁
𝑚2 = Pascal = 1 Pa, and 106
is Mega (M)}
∴ 1
𝑁
𝑚𝑚2 = 1 MPa
6. Unit of Force
In MKS and SI units, the fundamental units are:
Metre – Length
Kilogram – mass
Second – time.
From the Newtons second law of motion,
Force ∝ rate of change of momentum.
∝ rate of change of (mass × velocity)
∝ mass × rate of change of velocity
∝ mass × acceleration.
Force = mass × acceleration.
Unit – Kg 𝑚 𝑠2
= 9.81 N.
Difference is only when selecting the unit of force.
MKS - 𝑘𝑔 𝑚 𝑠2
or Kg - wt.
SI – Newton (N)
7. Direct or Normal Stress
Direct stress may be tensile or compressive and result from forces acting
perpendicular to the plane of the cross-section
8. Direct or Normal Strain
When loads are applied to a body, some deformation will occur resulting in a
change in dimension.
Consider a bar, subjected to axial tensile loading force, F. If the bar extension
is dl and its original length (before loading) is l, then tensile strain is:
Direct Strain ( ) =
Change in Length
Original Length
i.e. = dl/l
9. Direct or Normal Strain
As strain is a ratio of lengths, it is dimensionless.
Similarly, for compression by amount, dl: Compressive strain = - dl/L
Note: Strain is positive for an increase in dimension and negative for a
reduction in dimension.
10. Shear Stress and
Shear Strain
Shear stresses are
produced by equal and
opposite parallel
forces not in line.
The forces tend to
make one part of the
material slide over the
other part.
Shear stress is
tangential to the area
over which it acts.
11. STRAIN
It is defined as deformation per unit length
it is the ratio of change in length to original length
Tensile strain (ε) =
increase in length
Original length L
= δ / L (+ Ve)
Compressive strain (ε) =
decrease in length
Original length L
= δ / L(- Ve)
12. Ultimate Strength
The strength of a material is a measure of the stress that it can take when in
use. The ultimate strength is the measured stress at failure, but this is not
normally used for design because safety factors are required. The normal way
to define a safety factor is :
safety factor =
stress at failure
stress when loaded
or
𝐶𝑎𝑝𝑎𝑐𝑖𝑡𝑦
𝐷𝑒𝑚𝑎𝑛𝑑
=
Ultimate stress
Permissible stress
,
If
𝐶𝑎𝑝𝑎𝑐𝑖𝑡𝑦
𝐷𝑒𝑚𝑎𝑛𝑑
≥ 1, Safe.
13. Strain
We must also define strain. In engineering this is not a measure of force but is
a measure of the deformation produced by the influence of stress. For tensile
and compressive loads:
Strain is dimensionless, i.e. it is not measured in metres, kilograms etc. strain
(ε) =
increase in length x
original length L
For shear loads the strain is defined as the angle γ This is measured in radians
shear strain (γ)=
shear displacement x
width L
14. Shear Stress and
Shear Strain
Shear strain is the distortion
produced by shear stress on an
element or rectangular block as
above. The shear strain,
(gamma) is given as
𝛾 =
𝑇𝑟𝑎𝑛𝑠𝑣𝑒𝑟𝑠𝑎𝑙 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝐴𝐷
𝛾 =
𝐷𝐷′
ℎ
=
𝑑𝑙
ℎ
= 𝑡𝑎𝑛𝜑 A B
C
D C’
D’
P
L
h
𝜑 𝜑
𝑑𝑙
Resistance (R)
𝑺𝒉𝒆𝒂𝒓 𝑭𝒐𝒓𝒄𝒆, 𝝉 =
𝑺𝒉𝒆𝒂𝒓 𝑹𝒆𝒔𝒊𝒔𝒕𝒂𝒏𝒄𝒆 (𝑹)
𝑺𝒉𝒆𝒂𝒓 𝑨𝒓𝒆𝒂 (𝑨)
=
𝑹
𝑨
=
𝑷
𝑳×𝟏
15. Shear Stress and Shear Strain
For small φ, γ = φ
Shear strain then becomes the change in the right angle.
It is dimensionless and is measured in radians.
17. Modulus of Elasticity
If the strain is "elastic" Hooke's law may be used to define
Youngs Modulus E =
Stress
= Strain
=
W
x
L
A
Young's modulus is also called the modulus of elasticity or stiffness and is a
measure of how much strain occurs due to a given stress. Because strain is
dimensionless Young's modulus has the units of stress or pressure
18. Volumetric Strain
Hydrostatic stress refers to tensile or compressive stress in all dimensions
within or external to a body
Hydrostatic stress results in change in volume of the material.
Consider a cube with sides x, y, z. Let dx, dy, and dz represent increase in
length in all directions.
i.e. new volume = (x + dx) (y + dy) (z + dz)
19. Volumetric Strain
Neglecting products of small quantities:
New volume = x y z + z y dx + x z dy + x y dz
Original volume = x y z
= z y dx + x z dy + x y dz
Volumetric strain=
z y dx + x z dy + x y dz
x y z
ε v = dx/x + dy/y + dz/z
ε v =εx +ε y +ε z
20. Elasticity and Hooke’s Law
All solid materials deform when they are stressed, and as stress is increased,
deformation also increases.
If a material returns to its original size and shape on removal of load causing
deformation, it is said to be elastic.
If the stress is steadily increased, a point is reached when, after the removal
of load, not all the induced strain is removed.
This is called the elastic limit.
21. Hooke’s Law
Stress and strain has linear
relationship between O to A.
i.e., 𝜎 𝛼 𝜀
Hook's law applies only between
O and A, which is the
proportionality limit.
Not proportional beyond point
A.
Slope of the line OA represents
Youngs modulus (E)
Higher slope means stiffer and
stronger material.
It is a measure of the stiffness
of a material.
𝐸 =
𝜎
𝜀
Slope
O
22. O – origin
A- Proportionality limit
B – Elastic limit
C – Upper yield point
D – Lower yield point
E – Ultimate tensile strength
F – Fracture point
O
23. Hooke’s Law
States that providing the limit of proportionality of a material is not
exceeded, the stress is directly proportional to the strain produced.
If a graph of stress and strain is plotted as load is gradually applied, the first
portion of the graph will be a straight line.
The slope of this line is the constant of proportionality called modulus of
Elasticity, E or Young’s Modulus.
It is a measure of the stiffness of a material.
Modulus of Elasticity,=
Direct stress
Direct strain
E=
σ
ε
24. Stress strain relation in 2 D Cases.
In (1D) stress system, 𝜎 ∝ 𝜀 and
𝜎
𝜀
= E.
𝜎 = Normal stress, 𝜀 = Strain and E = Youngs modulus.
2D stress system
Longitudinal strain =
𝛿𝐿
𝐿
Lateral strain =
𝛿𝑏
𝑏
Note: If longitudinal strain is tensile, lateral stain will be
compressive. If longitudinal stain is compressive, lateral strain
will be tensile.
25. Stress strain relation in 2 D Cases.
Poisson’s Ratio (𝜇) =
𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑠𝑡𝑎𝑟𝑖𝑛
𝑙𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛
(or) Lateral strain = - 𝜇 × 𝑙𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛. (As
lateral strain is opposite in sign to longitudinal
strain.)
Determines how a body responds to a applied
stresses.
27. Stress strain relation in 2 D Cases.
Let 𝜎 1 = Normal stress in x- direction and
𝜎 2 = Normal stress in y – direction.
Strain produced by 𝜎 1
Longitudinal strain produced by 𝜎 1 will be equal
𝜎 1
𝐸
. (x axis)
Lateral strain produced by 𝜎 1 will be equal to −𝜇
𝜎 1
𝐸
. (y axis)
Strain produced by 𝜎 2
Longitudinal strain produced by 𝜎 2 will be equal
𝜎 2
𝐸
. (y axis)
Lateral strain produced by 𝜎 2 will be equal to −𝜇
𝜎 2
𝐸
. (x axis)
Total Strain
Let, 𝑒1the total stain in x direction =
𝜎 1
𝐸
−𝜇
𝜎 2
𝐸
𝑒2the total stain in y direction =
𝜎 2
𝐸
−𝜇
𝜎 1
𝐸
A B
C
D
𝜎 1
𝜎 1
𝜎 2
𝜎 2
28. Stress strain relation in 3 D Cases.
Strain produced by 𝜎 1
The stress 𝜎 1 will produce strain in the direction of x and also in the
direction y and z.
Longitudinal Strain in the direction of x =
𝜎 1
𝐸
(x axis)
Lateral strain in y direction= −𝜇
𝜎 1
𝐸
. (y axis)
Lateral strain in z direction = −𝜇
𝜎 1
𝐸
. (z axis)
Strain produced by 𝜎 2
Longitudinal Strain in the direction of y =
𝜎 2
𝐸
(y axis)
Lateral strain in x direction = −𝜇
𝜎 2
𝐸
(x axis)
Lateral strain in z direction = −𝜇
𝜎 2
𝐸
(z axis)
x
y
z
𝜎 1
𝜎 2
𝜎 3
29. Stress strain relation in 3 D Cases.
Strain produced by 𝜎 3
Longitudinal strain in the direction of z =
𝜎 3
𝐸
(z axis)
Lateral strain in x direction = −𝜇
𝜎 3
𝐸
(x axis)
Lateral strain in y direction = −𝜇
𝜎 3
𝐸
(y axis)
Total Strain (let 𝑒1, 𝑒2, and 𝑒3 be the total strain in x, y, and z direction respectively)
𝑒1 =
𝜎 1
𝐸
−𝜇
𝜎 2
𝐸
−𝜇
𝜎 3
𝐸
(𝑡𝑜𝑡𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛 𝑖𝑛 𝑥 𝑎𝑥𝑖𝑠).
𝑒2 =
𝜎 2
𝐸
−𝜇
𝜎 1
𝐸
−𝜇
𝜎 3
𝐸
(𝑡𝑜𝑡𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛 𝑖𝑛 𝑦 𝑎𝑥𝑖𝑠).
𝑒3 =
𝜎 3
𝐸
−𝜇
𝜎 1
𝐸
−𝜇
𝜎 2
𝐸
(𝑡𝑜𝑡𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛 𝑖𝑛 𝑧 𝑎𝑥𝑖𝑠).
The above 3 equations give stress and strain relationship for 3 orthogonal normal stress system.