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Mechanic of the fibrous Structure

- 1. Mechanics of Fibrous Structure (TM-3113) M Irfan Department of Textile Engineering National Textile University Faisalabad
- 2. References The lectures are based on following reference books • J.W.S. Hearle. " Structural Mechanics of Fibres Yarns and Fabrics" • (2004) • A.E. Bogdanovich, C. M. Pastore. "Mechanics of Textile and Laminated Composites" (1996) • B. Strong. "Plastics (Materials & Processing) (2001) • Ferdinand, P. Beer, E. Russell Johnston Jr., John, T. Dewolf. "Mechanics of Materials" (2004) • Jinlian, Hu. "Structure and Mechanics of Woven Fabrics" (2004) • Thormen H Courty. "Mechanical Behavior of Materials" (2005) • E.J.Hearn. "Mechanics of Materials" (2001) • P. Schwartz " Structure & Mechanics of Textile Assemblies" (2003) • W.E.Morton. " Physical properties of Textile Fibres" (2002) • W.A. Hanton. " Mechanics for Textile Students" (2007)
- 3. Mechanics of Materials • “Mechanics of materials is a branch of applied mechanics that deals with the behavior of solid bodies subjected to various types of loading”. • The main objective of the mechanics of materials is to determine stresses, strains, and displacements in structures and their components due to the loads acting on them. • The complete picture of mechanical behavior of materials (or different structures made of materials) can be obtained if we can find these quantities for load values up to the point of failure.
- 4. Importance of Mechanics of Materials • Mechanics of materials is an important subject in many engineering disciplines as understanding mechanical behavior of materials under various forms of loading is essential for safe and secure design of any kind of structure. – buildings, bridges, machines, airplanes, aerospace structures etc. • Mechanics of materials deals with mechanical behavior of real bodies as it determines stresses and strains within bodies of finite dimensions that deform under load.
- 5. Importance of Mechanics of Materials Failure may happen due to wrong material selection and design, manufacturing and design faults
- 6. Importance in textiles • Mechanical behavior of textile materials and structures, i.e. response of textile materials and/or structures to applied forces and deformations, is very important – Processing in downstream processes – Performance of end product
- 7. Fundamental concepts: stress and strain • The most fundamental concepts in mechanics of materials are stress and strain • Stress: Stress can be defined as force per unit area of a material. It is denoted by Greek letter σ (sigma) σ = 𝑃/𝐴 • SI unites of stress are N/m2 or Pascal
- 9. Example: Take a prismatic bar (a prismatic bar is a bar with uniform cross sectional area) which is under axial load (a force directed along the axis of the bar)
- 10. Fundamental concepts: stress and strain • Suppose the axial force P is the only force acting at the ends of the bar • the original length of the bar is denoted by the letter L, and the increase in length due to the loads is denoted by the Greek letter δ (delta) • mn is the cross section perpendicular to the longitudinal axis of the bar • Under the load, the bar is under continuously distributed stresses acting on the cross section mn with axial force P as the resultant of those stresses • The resultant P is equal to the magnitude of the stress times the cross-sectional area A of the bar, i.e P= σA • Thus we get the relation σ = 𝑃/𝐴
- 11. • Normal Stresses: in case the stresses act in a direction perpendicular (normal) to the cross section of the bar, these are called normal stresses. • When the bar is stretched by the force P, the stress is called tensile stress • If the bar is compressed, the stress is known as compressive stress • Tension force will tend to lengthen the bar, while compressive stress will tend to shorten the bar • tensile stresses are considered positive and compressive stresses as negative • Applied load can be static or dynamic Fundamental concepts: stress and strain
- 12. • Strain: – Change in the dimensions of an object occurs under the influence of an external load, the ratio of this change to the original length of the object is known as strain – Change in dimension means becoming longer under tension or shorter when in compression – In the previous example of the bar, δ represents the elongation of the bar under tension, then the strain (represented by Greek word ε, epsilon) is given by the equation ε = δ/𝐿 Fundamental concepts: stress and strain
- 13. Fundamental concepts: stress and strain • Strain: – When the bar is in tension, the strain is called a tensile strain that tends to elongate the bar – When the bar is in compression, the strain is called compressive strain that tends to shorten the bar – The strain associated with normal stresses is known as normal strain – Since strain is ratio between two lengths, it is dimensionless and has no units – Strain can be expressed as a percent – Tensile strain is usually taken as positive and compressive strain as negative.
- 14. How to analyze mechanical behavior of materials • The usual way to analyze how materials behave when they are subjected to different loads is to do tensile testing on tensile test machine • Sample is gripped between jaws of the machine and load is applied. The deformation under load is recorded • For comparison of different results, the test method (specimen size, load application method) should be standardized • One of the major standards are ASTM standards (American Society for Testing and Materials) • Compression tests are also carried. For example, concrete is mostly tested under compression
- 15. Stress-strain curves • The results obtained from the testing machine (load applied and resulting deformation) are converted to stresses and strains. • The load can be applied under various modes, for example static or dynamic load – Static load: non fluctuating load, that is the rate of loading is not of interest, usually caused by gravity – Dynamic load: alternating load, can be in cyclic manner, the magnitude and sign of the load changes with time
- 16. Types of loading
- 17. Stress-strain curves In the example of a test bar, the axial stress in the bar is calculated by dividing the axial load P by the cross-sectional area A The obtained stress is called nominal stress if initial area of the bar is used in the calculation The stress is called true stress if it is calculated by using the actual area of the bar at the cross section where failure occurred
- 18. Stress-strain curves • The average axial strain ε in the test specimen is found by dividing the measured elongation δ between the gauge marks by the gauge length L (δ /L) • The length of the specimen between the gauge marks (between the machine jaws) is known as gauge length L • If the initial gauge length (length of the specimen before test) is used in the calculation, the obtained strain is known as nominal strain • Since the distance between the gauge marks increases as the tensile load is applied, we can calculate the true strain (or natural strain) at any value of the load by using the actual distance between the gauge marks • In tension, true strain is always smaller than nominal strain
- 19. Stress-strain curves • After performing a tension or compression test and determining the stress and strain at various magnitudes of the load, we can plot a diagram of stress versus strain. Such a stress-strain diagram is a characteristic of the particular material being tested and conveys important information about the mechanical properties and behavior of the material • Strains are plotted on the horizontal axis and stresses on the vertical axis • A typical stress-strain curve for mild steel is shown in the next slide
- 20. Stress-strain curves • Figure shows a stress- strain curve when a steel circular bar of uniform cross-section is subjected to a gradually increasing tensile load until failure occur • When loaded, the change in length is recorded by extensometer and graph of stress and strain is produced
- 21. Stress-strain curves • The diagram begins with a straight line from the origin O to point A, which means that the relationship between stress and strain in this initial region is not only linear but also proportional • Beyond point A, the proportionality between stress and strain no longer exists; hence the stress at A is called the proportional limit. • The slope of the straight line from O to A is called the modulus of elasticity. Because the slope has units of stress divided by strain, modulus of elasticity has the same units as stress • For a short period beyond this point the material may still be elastic in the sense that deformations are completely recovered when load is removed. The limiting point B for this condition is termed the elastic limit, but for most practical purposes the point A and B are considered coincident
- 22. • Hook’s Region Elastic region
- 23. Stress-strain curves • Beyond the elastic limit plastic deformation occurs and strains are not totally recoverable thus leading to permanent deformation or permanent set • Point C is termed as upper yield point and point D is termed as lower yield point – upper yield point corresponds to the load reached just before yield starts – lower yield point, which corresponds to the load required to maintain yield • For some materials upper and lower yield points may coincide while for some other materials yield point may not exist at all • Beyond the yield point some increase in load is required to take the strain to point E on the graph • Between D and E the material is said to be in the elastic-plastic state, some of the section remaining elastic and hence contributing to recovery of the original dimensions if load is removed, the remainder being plastic
- 24. Stress-strain curves • Beyond E the cross-sectional area of the bar begins to reduce rapidly over a relatively small length of the bar and the bar is said to neck • This necking takes place whilst the load reduces, and fracture of the bar finally occurs at point F • The nominal stress at failure, termed the maximum or ultimate tensile stress, is given by the load at E divided by the original cross-sectional area of the bar. This is also known as the tensile strength of the material of the bar. Owing to the large reduction in area produced by the necking process the actual stress at fracture is often greater than the above value. Since, however, designers are interested in maximum loads which can be carried by the complete cross-section, the stress at fracture is seldom of any practical value
- 25. Stress-strain curves • If the actual cross-sectional area at the narrow part of the neck is used to calculate the stress, the true stress- strain curve is obtained • The total load the bar can carry does indeed diminish after the ultimate stress is reached (as shown by curve DE), but this reduction is due to the decrease in area of the bar and not due to a loss in strength of the material itself • In reality, the material withstands an increase in true stress up to failure • Because most structures are expected to function at stresses below the proportional limit, the conventional stress-strain curve OABCDEF, which is based upon the original cross-sectional area of the specimen and is easy to determine, provides satisfactory information for use in engineering design
- 26. Stress-strain curves • After undergoing the large strains that occur during yielding , the steel begins to strain harden • During strain hardening, the material undergoes changes in its crystalline structure, resulting in increased resistance of the material to further deformation. Elongation of the test specimen in this region requires an increase in the tensile load, and therefore the stress-strain diagram has a positive slope in this region Strain hardening
- 27. Stress-strain curves Another example of typical stress-strain curve for steel True stress Not drawn to scale Drawn to scale
- 28. Stress-strain curves • The presence of a clearly defined yield point followed by large plastic strains is an important characteristic of materials like structural steel • Metals such as structural steel that undergo large permanent strains before failure are classified as ductile • A desirable feature of ductile materials is that visible distortions occur if the loads become too large, thus providing an opportunity to take remedial action before an actual fracture occurs • Materials exhibiting ductile behavior are capable of absorbing large amounts of strain energy prior to fracture
- 29. Stress-strain curves • Some materials, for example aluminum alloys, typically do not have a clearly definable yield point • they do have an initial linear region with a recognizable proportional limit • For such material that does not have an obvious yield point and yet undergoes large strains after the proportional limit is exceeded, an arbitrary yield stress may be determined by the offset method • it should be distinguished from a true yield stress by referring to it as the offset yield stress
- 30. Stress-strain curves • Stress-strain curves for rubber:
- 31. Stress-strain curves • Stress-strain curves for rubber: – Rubber maintains a linear relationship between stress and strain up to relatively large strains as compared to metals – Beyond the proportional limit, the behavior depends upon the type of rubber – Some kinds of soft rubber will stretch enormously without failure, reaching lengths several times their original lengths – The material eventually offers increasing resistance to the load, and the stress-strain curve turns markedly upward – although rubber exhibits very large strains, it is not a ductile material because the strains are not permanent. It is, of course, an elastic material
- 32. Stress-strain curves • The ductility of a material in tension can be characterized by its elongation or by the reduction in area at the cross section where fracture occurs. The percent elongation is defined as follows: 𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝑒𝑙𝑜𝑛𝑔𝑎𝑡𝑖𝑜𝑛 = (𝐿1 − 𝐿𝑜)/𝐿𝑜 where Lo is the original gauge length and L1 is the distance between the gauge marks at fracture • The percent reduction in area measures the amount of necking that occurs and is defined as follows: 𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝑟𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑖𝑛 𝑎𝑟𝑒𝑎 = (𝐴1 − 𝐴𝑜)/𝐴𝑜 in which Ao is the original cross-sectional area and A1 is the final area at the fracture section
- 33. Modulus of elasticity: Hook’s law • Most engineering structures are designed to undergo relatively small deformations, involving only the straight-line portion of the corresponding stress-strain diagram. For that initial portion of the diagram, the stress σ is directly proportional to the strain ε and is known as Hooks’ Law 𝛿 = 𝐸𝜀 The coefficient E is called the modulus of elasticity of the material involved, or also Young's modulus • modulus E is expressed in the same units as the stress namely in pascals, and in psi if U.S. customary units are used • The largest value of the stress for which Hooke's law can be used for a given material is known as the proportional limit of that material • In the case of ductile materials possessing a well-defined yield point, the proportional limit almost coincides with the yield point
- 34. Modulus of elasticity: Hook’s law • large variations in the yield strength, ultimate strength and final strain (ductility) but same modulus of elasticity and stiffness
- 35. Modulus of elasticity: Hook’s law • Some of the physical properties of structural metals, such as strength, ductility, and corrosion resistance, can be greatly affected by alloying, heat treatment, and the manufacturing process used. For example, we note from the stress-strain diagrams of pure iron and of three different grades of steel (shown in next slide) that large variations in the yield strength, ultimate strength and final strain (ductility) exist among these four metals. All of them, however, possess the same modulus of elasticity; in other words, their "stiffness," or ability to resist a deformation within the linear range, is the same. Therefore, if a high-strength steel is substituted for a lower-strength steel in a given structure, and if all dimensions are kept the same, the structure will have an increased load carrying capacity, but its stiffness will remain unchanged
- 36. stress and strain curves of different fibers
- 37. stress and strain curves of different fibers