MNG 213 Mechanical Behavior of Materials PDF

MENG 201 MNG 213 Module 1 Module 3 Review Mechanical Behavior of Materials Dr. MAHMOUD SHAABAN September 2024  Stress Preamble Preamble In this lecture we aim to ►Discuss the S...

MENG 201 MNG 213 Module 1 Module 3 Review Mechanical Behavior of Materials Dr. MAHMOUD SHAABAN September 2024  Stress Preamble Preamble In this lecture we aim to ►Discuss the Stress-Strain Behavior of Materials ►Analyze the material properties related to stress analysis ►Discuss the strain energy ►Analyze the Poisson ratio ►Understand the shear stress diagram ►Discuss fatigue and creep failures MNG 213 – Dr. Mahmoud Shaaban - 2024 2  Tension Tests 3 MNG 213 – Dr. Mahmoud Shaaban - 2024  Stress Tension Test Testing Material Properties ► The strength of a material depends on its ability to sustain a load without undue deformation or failure. ► This property is inherent in the material itself and must be determined by experiment. ► A tension test is used primarily to determine the relationship between the average normal stress and average normal strain in many engineering materials such as metals, ceramics, polymers, and composites MNG 213 – Dr. Mahmoud Shaaban - 2024 4  Stress Tension Test The Specimen ► A specimen is made into a “standard” shape and size. ► It has a constant circular cross section with enlarged ends, so that failure will not occur at the grips. ► Before testing, two small punch marks are placed along the specimen’s uniform length. ► Measurements are taken of both the specimen’s initial cross-sectional area, and the gauge-length distance between the punch marks. MNG 213 – Dr. Mahmoud Shaaban - 2024 5  Stress Tension Test Tension Test ► In order to apply an axial load with no bending of the specimen, the ends are usually seated into ball-and-socket joints. ► A testing machine is then used to stretch the specimen at a very slow, constant rate until it fails. ► The machine is designed to read the load required to maintain this uniform stretching. MNG 213 – Dr. Mahmoud Shaaban - 2024 6  Stress Tension Test Measuring Strain ► At frequent intervals, data is recorded of the applied load and elongation. ► The strain can be read manually from the pinch marks, or directly using an electrical-resistance strain gauge. ► By measuring the electrical resistance of the wire, the gauge may be calibrated to read values of normal strain directly MNG 213 – Dr. Mahmoud Shaaban - 2024 7  Stress Tension Test Measuring Strain ► Brittle materials, such as gray cast iron, exhibit a much higher resistance to axial compression ► For this case any cracks or imperfections in the specimen tend to close up, and as the load increases the material will generally bulge or become barrel shaped as the strains become larger. MNG 213 – Dr. Mahmoud Shaaban - 2024 8  Stress-Strain Diagram 9 MNG 213 – Dr. Mahmoud Shaaban - 2024  Stress Stress-Strain Diagram Stress-Strain Diagrams ► The load and deformation data are used to calculate various values of the stress and corresponding strain in the specimen ► A plot of the results produces a curve called the stress–strain diagram MNG 213 – Dr. Mahmoud Shaaban - 2024 10  Stress Stress-Strain Diagram Conventional σ-ε Diagram ► The corresponding values of σ-ε are plotted ► the vertical axis is the stress ► the horizontal axis is the strain ► Two stress–strain diagrams for a particular material will be quite similar, but will never be exactly the same. ► The results depend on the material’s composition, microscopic imperfections ► The way it is manufactured, the rate of loading, and the temperature during the time of the test. MNG 213 – Dr. Mahmoud Shaaban - 2024 11  Stress Stress-Strain Diagram Conventional σ-ε Diagram MNG 213 – Dr. Mahmoud Shaaban - 2024 12  Stress Stress-Strain Diagram Ductile Steel Diagram Yielding ► A slight increase in stress above the elastic limit will result in a breakdown of the material and cause it to deform permanently. ► This behavior is called yielding, and it is indicated by the rectangular dark orange region of the curve. ► The stress that causes yielding is called the yield stress or yield point, and the deformation that occurs is called plastic deformation. MNG 213 – Dr. Mahmoud Shaaban - 2024 13  Stress Stress-Strain Diagram Ductile Steel Diagram Yielding ► For low carbon steels or those that are hot rolled, the yield point is often distinguished by two values. ► The upper yield point occurs first, followed by a sudden decrease in load-carrying capacity to a lower yield point. ► Notice that once the yield point is reached, then the specimen will continue to elongate (strain) without any increase in load. ► When the material is in this state, it is often referred to as being perfectly plastic. MNG 213 – Dr. Mahmoud Shaaban - 2024 14  Stress Stress-Strain Diagram Ductile Steel Diagram Strain Hardening ► When yielding has ended, an increase in load can be supported by the specimen, ► This results in a curve that rises continuously but becomes flatter until it reaches a maximum stress referred to as the ultimate stress. ► The rise in the curve in this manner is called strain hardening. MNG 213 – Dr. Mahmoud Shaaban - 2024 15  Stress Stress-Strain Diagram Ductile Steel Diagram Necking ► Up to the ultimate stress, as the specimen elongates, its cross-sectional area will decrease. ► This decrease is fairly uniform over the specimen’s entire gauge length ► At the ultimate stress, the cross-sectional area will begin to decrease in a localized region of the specimen. ► As a result, a constriction or “neck” tends to form in this region as the specimen elongates further ► Here the stress–strain diagram tends to curve downward until the specimen breaks at the fracture stress MNG 213 – Dr. Mahmoud Shaaban - 2024 16  Stress Stress-Strain Diagram True Stress-Strain Diagram ► Uses the actual cross-sectional area and specimen length at the instant the load is measured. ► The values of stress and strain found from these measurements are called true stress and true strain, and a plot of their values is called the true stress–strain diagram. ► There is a large divergence within the necking region. ► Most engineering design is done so that the material supports a stress within the elastic range. MNG 213 – Dr. Mahmoud Shaaban - 2024 17  Behavior of Ductile and Brittle Materials 18 MNG 213 – Dr. Mahmoud Shaaban - 2024  Stress Ductile and Brittle Materials Ductile - Mild Steel ► Any material that can be subjected to large strains before it fractures is called a ductile material. ► Mild steel is a typical example. ► Engineers often choose ductile materials for design because these materials are capable of absorbing shock or energy, ► If they become overloaded, they will usually exhibit large deformation before failing. MNG 213 – Dr. Mahmoud Shaaban - 2024 19  Stress Ductile and Brittle Materials Ductile - Mild Steel MNG 213 – Dr. Mahmoud Shaaban - 2024 20  Stress Ductile and Brittle Materials Ductile - Mild Steel MNG 213 – Dr. Mahmoud Shaaban - 2024 21  Stress Ductile and Brittle Materials Ductile - Mild Steel ► Ductility is reported as a percent elongation at time of fracture ► It can also be reported as percent reduction in area at time of fracture MNG 213 – Dr. Mahmoud Shaaban - 2024 22  Stress Ductile and Brittle Materials Ductile - Aluminum MNG 213 – Dr. Mahmoud Shaaban - 2024 23  Stress Ductile and Brittle Materials Yield Strength ► In most metals, however, constant yielding will not occur beyond the elastic range. ► One metal for which this is the case is aluminum. ► Aluminum metal often does not have a well-defined yield point. ► It is standard practice to define a yield strength using a graphical procedure called the offset method. MNG 213 – Dr. Mahmoud Shaaban - 2024 24  Stress Ductile and Brittle Materials The offset Method ► A value of yield strain is chosen (typically 0.2% ) ► From this point on the axis, a line parallel to the initial straight-line portion of the stress–strain diagram is drawn. ► The point where this line intersects the curve defines the yield strength. ► From the graph, the yield strength of this Aluminum Alloy is 51 ksi = 352 MPa. MNG 213 – Dr. Mahmoud Shaaban - 2024 25  Stress Ductile and Brittle Materials Brittle Materials ► Materials that exhibit little or no yielding before failure are referred to as brittle materials. ► Fracture takes place initially at an imperfection or microscopic crack and then spread rapidly across the specimen, causing complete fracture. ► Since the appearance of initial cracks in a specimen is quite random, brittle materials do not have a well- defined tensile fracture stress. ► Instead the average fracture stress from a set of observed tests is generally reported. MNG 213 – Dr. Mahmoud Shaaban - 2024 26  Stress Ductile and Brittle Materials Brittle or Ductile? ► Most materials exhibit both ductile and brittle behavior. ► Steel has brittle behavior when it contains a high carbon content, and it is ductile when the carbon content is reduced. ► At low temperatures materials become harder and more brittle, and at high temperatures they become softer and more ductile. MNG 213 – Dr. Mahmoud Shaaban - 2024 27  Stress Ductile and Brittle Materials Brittle - Cast Iron MNG 213 – Dr. Mahmoud Shaaban - 2024 28  Stress Ductile and Brittle Materials Nonlinear Elastic - Rubber MNG 213 – Dr. Mahmoud Shaaban - 2024 29  Stress Ductile and Brittle Materials Concrete MNG 213 – Dr. Mahmoud Shaaban - 2024 30  Hooke’s Law 31 MNG 213 – Dr. Mahmoud Shaaban - 2024  Stress Hooke’s Law Hook’s Law ► The stress–strain diagrams for most engineering materials exhibit a linear relationship between stress and strain within the elastic region. ► An increase in stress causes a proportionate increase in strain. ► This fact was discovered by Robert Hooke in 1676 using springs. MNG 213 – Dr. Mahmoud Shaaban - 2024 32  Stress Hooke’s Law Young’s Modulus ► E represents the constant of proportionality, which is called the modulus of elasticity or Young’s modulus, named after Thomas Young. ► The modulus of elasticity represents the slope of the stress-strain line. ► Since strain is dimensionless, E will have the same units as stress MNG 213 – Dr. Mahmoud Shaaban - 2024 33  Stress Hooke’s Law Young’s Modulus ► The modulus of elasticity is a mechanical property that indicates the stiffness of a material. ► Materials that are very stiff, such as steel, have large values of E (~ 200 Gpa). ► Spongy materials such as vulcanized rubber may have low values (0.70 MPa]. MNG 213 – Dr. Mahmoud Shaaban - 2024 34  Stress Hooke’s Law Young’s Modulus ► Only use E in the elastic range. ► If the stress is greater than the proportional limit, the stress-strain relationship is not linear and Hooke’s law is not valid. MNG 213 – Dr. Mahmoud Shaaban - 2024 35  Stress Hooke’s Law Strain Hardening ► If a specimen of ductile material, such as steel, is loaded into the plastic region and then unloaded, elastic strain is recovered as the material returns to its equilibrium state. ► As a result the material is subjected to a permanent set. MNG 213 – Dr. Mahmoud Shaaban - 2024 36  Stress Hooke’s Law Strain Hardening ► If the load is reapplied, the atoms in the material will again be displaced until yielding occurs at or near the stress and the stress– strain diagram continues along the same path as before. ► This new stress–strain diagram has a higher yield point a consequence of strain hardening. ► The material now has a greater elastic region but less ductility. MNG 213 – Dr. Mahmoud Shaaban - 2024 37  Examples 38 MNG 213 – Dr. Mahmoud Shaaban - 2024  Stress Examples MNG 213 – Dr. Mahmoud Shaaban - 2024 39  Stress Examples MNG 213 – Dr. Mahmoud Shaaban - 2024 40  Stress Examples Example A tension test for a steel alloy results in the stress–strain diagram shown in Fig. 3– 18. Calculate the modulus of elasticity and the yield strength based on a 0.2% offset. Identify on the graph the ultimate stress and the fracture stress. MNG 213 – Dr. Mahmoud Shaaban - 2024 41  Stress Examples Example An aluminum rod shown in Fig. 3–20a has a circular cross section and is subjected to an axial load of 10 kN. Determine the approximate elongation of the rod when the load is applied. Take Eal = 70 GPa. MNG 213 – Dr. Mahmoud Shaaban - 2024 42  STRAIN ENERGY 43 MNG 213 – Dr. Mahmoud Shaaban - 2024  Stress Strain Energy Strain Energy ► As a material is deformed, it tends to store energy internally. ► Consider a volume element of material subjected to uniaxial stress MNG 213 – Dr. Mahmoud Shaaban - 2024 44  Stress Strain Energy Strain Energy ► The force increases from 0 to F, body deforms ► The work done on the element by the force is equal to the average force magnitude times the displacement MNG 213 – Dr. Mahmoud Shaaban - 2024 45  Stress Strain Energy Strain Energy ► The volume of the element is ► Therefore, the strain energy is ► The strain energy density is MNG 213 – Dr. Mahmoud Shaaban - 2024 46  Stress Strain Energy Elastic Strain Energy ► The strain energy density is ► And Hooke’s law states that ► Therefore, for elastic materials under uniaxial stress, MNG 213 – Dr. Mahmoud Shaaban - 2024 47  Stress Strain Energy Modulus of Resilience ► The strain energy density at the proportional limit is the modulus of resilience ► From the elastic region of the stress–strain diagram notice that is equivalent to the shaded triangular area under the diagram. ► Physically a material’s resilience represents the ability of the material to absorb energy without any permanent damage to the material. MNG 213 – Dr. Mahmoud Shaaban - 2024 48  Stress Strain Energy Modulus of Toughness ► The modulus of toughness represents the entire area under the stress–strain diagram ► It indicates the strain-energy density of the material just before it fractures. ► This property becomes important when designing members that may be accidentally overloaded. ► Alloying metals can also change their resilience and toughness. MNG 213 – Dr. Mahmoud Shaaban - 2024 49  POISSON’S RATIO 50 MNG 213 – Dr. Mahmoud Shaaban - 2024  Stress Poisson’s Ratio Poisson’s Ratio ► When a deformable body is subjected to an axial tensile force, ► it elongate ► It contracts laterally. ► Likewise, a compressive force causes a body to contract in the direction of the force and yet its sides expand laterally. MNG 213 – Dr. Mahmoud Shaaban - 2024 51  Stress Poisson’s Ratio Poisson’s Ratio ► Consider a bar having an original radius r and length L and subjected to the tensile force P. ► This force elongates the bar by an amount δ, and its radius contracts by an amount δ’ MNG 213 – Dr. Mahmoud Shaaban - 2024 52  Stress Poisson’s Ratio Poisson’s Ratio ► Within the elastic range, the ratio of these strains is a constant. ► This constant is referred to as Poisson’s ratio, ν (nu) ► It is a dimensionless value that is unique for a homogeneous and isotropic material. ► The value is always positive. A negative sign is used as long. and lat. strains are opposite MNG 213 – Dr. Mahmoud Shaaban - 2024 53  Stress Poisson’s Ratio Poisson’s Ratio ► Poisson’s ratio is a dimensionless quantity, and for most non porous solids it has a value that is generally between 1/4 and 1/3. ► For an “ideal material” having no lateral deformation when it is stretched or compressed Poisson’s ratio will be 0. ► The maximum possible value for Poisson’s ratio is 0.5. MNG 213 – Dr. Mahmoud Shaaban - 2024 54  Stress Poisson’s Ratio Example ► If an axial force of is applied to the bar, determine the change in its length and the change in the dimensions of its cross section after applying the load. The material behaves elastically. Poisson ratio is 0.3. MNG 213 – Dr. Mahmoud Shaaban - 2024 55  SHEAR STRESS DIAGRAM 56 MNG 213 – Dr. Mahmoud Shaaban - 2024  Stress Shear Diagrams Shear Test ► The behavior of a material subjected to pure shear can be studied in a laboratory using specimens in the shape of thin tubes and subjecting them to a torsional loading. ► Measurements are made of the applied torque and the resulting angle of twist, the data can be used to plot the shear stress–strain diagram. MNG 213 – Dr. Mahmoud Shaaban - 2024 57  Stress Shear Diagrams Shear Diagram MNG 213 – Dr. Mahmoud Shaaban - 2024 58  Stress Shear Diagrams Modulus of Rigidity ► Hooke’s law for shear can be written as ► Here G is called the shear modulus of elasticity or the modulus of rigidity ► Its value represents the slope of the line on the diagram, MNG 213 – Dr. Mahmoud Shaaban - 2024 59  Creep and Fatigue 60 MNG 213 – Dr. Mahmoud Shaaban - 2024  Stress Shear Diagrams Long-Term Stress Effects ► Stresses result in a failure differently if the load is applied ► for very long periods or ► For a high number of cycles of application and removal ► At elevated temperature ► These conditions need a special treatment in design MNG 213 – Dr. Mahmoud Shaaban - 2024 61  Stress Shear Diagrams Creep ► When a material has to support a load for a very long period of time, it may continue to deform until a sudden fracture occurs ► both stress and/or temperature play a significant role in the rate of creep. MNG 213 – Dr. Mahmoud Shaaban - 2024 62  Stress Shear Diagrams Creep ► When creep becomes important, a member is usually designed to resist a specified creep strain for a given period of time. ► The creep strength represents the highest stress the material can withstand during a specified time without exceeding an allowable creep strain. ► For design, a given temperature, duration of loading, and allowable creep strain must all be specified. ► For example, a creep strain of 0.1% per year has been suggested for steel in bolts and piping. MNG 213 – Dr. Mahmoud Shaaban - 2024 63  Stress Shear Diagrams Fatigue ► When a metal is subjected to repeated cycles of stress or strain, it causes its structure to fracture. ► This behavior is called fatigue, and it is usually responsible for a large percentage of failures in connecting rods and crankshafts of engines; steam or gas turbine blades; connections or supports for bridges, railroad wheels, and axles; and other parts subjected to cyclic loading. ► Fatigue fracture will occur at a stress that is less than the material’s yield stress. MNG 213 – Dr. Mahmoud Shaaban - 2024 64  Stress Shear Diagrams Fatigue ► It is necessary to determine a limit below which no evidence of failure can be detected after applying a load for a specified number of cycles. ► This limiting stress is called the endurance or fatigue limit. ► The fatigue stress limit S versus the number of cycles-to-failure N is called an S–N diagram or stress–cycle diagram. ► Most often, the values of N are plotted on a logarithmic scale since they are generally quite large. MNG 213 – Dr. Mahmoud Shaaban - 2024 65  Stress Shear Diagrams S-N Diagram MNG 213 – Dr. Mahmoud Shaaban - 2024 66  See you next time! MNG 213 – Dr. Mahmoud Shaaban - 2024 67