Engineering properties of Materials, Study notes of Mechanics

Download Engineering properties of Materials and more Study notes Mechanics in PDF only on Docsity! MechaniCalc Mechanical Properties of Materials Database v The mechanical properties of a material affect how it behaves as it is loaded. The elastic modulus of the material affects how much it deflects under a load, and the strength of the material determines the stresses that it can withstand before it fails. The ductility of a material also plays a significant role in determining when a material will break as it is loaded beyond its elastic limit. Because every mechanical system is subjected to loads during operation, it is important to understand how the materials that make up those mechanical systems behave. This page describes the mechanical properties of materials that are relevant to the design and analysis of mechanical systems. Contents ¢ Stress and Strain © Stress-Strain Curve © True Stress and Strain ¢ Hooke's Law o Hooke's Law in Shear ¢ Poisson's Ratio e Strain Hardening ¢ Elastic and Plastic Strain e Ductility ¢ Ductile and Brittle Materials e Strain Energy © Strain Energy Density © Modulus of Resilience ° Modulus of Toughness e Stress-Strain Curve Approximation o Ramberg-Osgood Equation © Determining Constants for Ramberg-Osgood ¢ References Related Pages: + Engineering Materials + Strength of Materials + Materials Database MechaniCalc = The relationship between stress and strain in a material is determined by subjecting a material specimen to a tension or compression test. In this test, a steadily increasing axial force is applied to a test specimen, and the deflection is measured as the load is increased. These values can be plotted as a load-deflection curve. The deflection in the test specimen is dependent on both the material's elastic modulus as well as the geometry of the specimen (area and length). Since we are interested material behavior without regard to geometry, it is useful to generalize the data to remove the effect of geometry. This is done by converting the load values to stress values and converting the deflection values to strain values: Stress: c= P : A Strain: e= Lolo = So Lg Lo In the equation for stress, P is the load and Ap is the original cross-sectional area of the test specimen. In the equation for strain, L is the current length of the specimen and Lg is the original length. Stress-Strain Curve The values of stress and strain determined from the tensile test can be plotted as a stress-strain curve, as shown below: Stress-Strain Diagram Stress, 0 Strain, € Stress-Strain Curve Calculator Check out our Stress-Strain Curve calculator based on the methodology described here. There are several points of interest in the diagram above: e P: This is the proportionality limit, which represents the maximum value of stress at which the stress-strain curve is linear. MechaniCalc where T is the value of shear stress, ) is the value of shear strain, and G is the shear modulus of elasticity. The elastic modulus and the shear modulus are related by: E C= 2(1+v) where V is Poisson's ratio. More information on Hooke's law can be found here. Poisson's Ratio As load is applied to a material, the material elongates and the cross-sectional area is reduced. This reduction in cross-sectional area is called /ateral strain, and it is related to the axial strain by Poisson's ratio, V. For a circular specimen this reduction in area is realized as a reduction in diameter, and the Poisson's ratio is calculated as: lateral _ AD/Dy Enmial 5/Io Poisson's ratio only applies within the elastic region of the stress-strain curve, and it is typically about 0.3 for most metals. The theoretical maximum limit of Poisson's ratio is 0.5. Need Structural Calculators? We have a number of structural calculators to choose from. Here are just a few: * Beam Calculator © Bolted Joint Calculator ¢ Bolt Pattern Force Distribution e Lug Calculator e Column Buckling Calculator e Fatigue Crack Growth Calculator Von Misos Stress Unit a 40 aan inne Strain Hardening MechaniCalc = below, the strength of the material can be seen to increase between the yield point Y and the ultimate strength at point U. This increase in strength is the result of strain hardening. The ductile material in the figure below is still able to support load even after the ultimate strength is reached. However, after the ultimate strength at point U, the increase in strength due to strain hardening is outpaced by the reduction in load-carrying ability due to the decrease in cross sectional area. Between the ultimate strength at point U and the fracture point F, the engineering strength of the material decreases and necking occurs. In the stress-strain curve for the brittle material below, a very small region of strain hardening is shown between the yield point Y and the ultimate strength U. Note however that a brittle material may not actually exhibit any yielding behavior or strain hardening at all — in this case, the material would fail on the linear portion of the curve. This is more common in materials such as ceramics or concrete. Ductile Material Brittle Material a a U Y F vy UF or or ~~’ € elasticregion strainhardening necking elastic strain region hardening Because the strain hardening region occurs between the yield point and the ultimate point, the ratio of the ultimate strength to the yield strength is sometimes used as a measure of the degree of strain hardening in a material. This ratio is the strain hardening ratio: strain hardening ratio = S,y / Sty According to Dowling, typical values of strain hardening ratio in metals range from approximately 1.2 to 14. If a material is loaded beyond the elastic limit, it will undergo permanent deformation. After unloading the material, the elastic strain will be recovered (return to zero) but the plastic strain will remain. The figure below shows the stress-strain curve of a material that was loaded beyond the yield point, Y. The first time the material was loaded, the stress and strain followed the curve O-Y-Y', and then the load was removed once the stress reached the point Y'. Since the material was loaded beyond the elastic limit, only the elastic portion of the strain is recovered -- there is some permanent strain now in the material. If the material were to be loaded again, it would follow line O'-Y'-F, where 0'-Y' is the previous unloading line. The point Y' is the new yield point. Note that the line O'-Y' is linear with a slope equal to the elastic modulus, and the point Y' has a higher stress value than point Y. Therefore, the material now has a higher yield point than it had previously, which is a result of strain hardening that occurred by loading the material beyond the elastic limit. MechaniCalc plastic strain elastic strain (permanent) (recovered) By strain hardening the material, it now has a larger elastic region and a higher yield stress, but its ductility has been reduced (the strain between points Y-F is less than the strain between points Y-F). Elastic and Plastic Strain Up to the elastic limit, the strain in the material is also elastic and will be recovered when the load is removed so that the material returns to its original length. However, if the material is loaded beyond the elastic limit, then there will be permanent deformation in the material, which is also referred to as plastic strain. Elasticand Plastic Strain elastic limit go fc 2 = 3 plastic strain (permanent set) elastic strain & ee In the figure above, both elastic and plastic strains exist in the material. If the load is removed at the indicated point (0, €), the stress and strain in the material will follow the unloading line as shown. The elastic strain and plastic strain are indicated in the figure, and are calculated as: Elastic Strain: Eo = O/E MechaniCalc Strain Energy When force is applied to a material, the material deforms and stores potential energy, just like a spring. The strain energy (i.e. the amount of potential energy stored due to the deformation) is equal to the work expended in deforming the material. The total strain energy corresponds to the area under the load deflection curve, and has units of in-Ibf in US Customary units and N-m in SI units. The elastic strain energy can be recovered, so if the deformation remains within the elastic limit, then all of the strain energy can be recovered. elastic limit strain energy deflection from applied force Strain energy is calculated as: Load-Deflection Curve Deflection General Form: U = Work = ['F dL (area under load-deflection curve) (area under load-deflection curve) Within Elastic Limit: (spring potential energy) MechaniCalc = potential energy stored in a spring. Both equations give the same result, they are just derived somewhat differently. Strain Energy Density It is sometimes more convenient to work with strain energy density, which is the strain energy per unit volume. This is equal to the area under the stress-strain diagram: Ean u=[ ade 0 where the limits in the integral above are from a strain of 0 to Eapp’ which is the strain existing in the loaded material. Note that the units of strain energy density are psi in US Customary units and Pa in SI units. Modulus of Resilience The modulus of resilience is the amount of strain energy per unit volume (i.e. strain energy density) that a material can absorb without permanent deformation resulting. The modulus of resilience is calculated as the area under the stress-strain curve up to the elastic limit. However, since the elastic limit and the yield point are typically very close, the resilience can be approximated as the area under the stress-strain curve up to the yield point. Since the stress-strain curve is very nearly linear up to the elastic limit, this area is triangular. Modulus of Resilience F_—~ yield point ~™ elastic limit modulus of resilience, u, ak € The modulus of resilience is calculated as: Eat ey Up = [ ode= [ ade general form 0 0 2 2 Up = 2 nea = Fel oe Si triangular form 2 2E 2E where Og) and €g] are the stress and strain at the elastic limit, Sty is the tensile yield strength, and E is the elastic modulus. MechaniCalc Modulus of Toughness The modulus of toughness is the amount of strain energy per unit volume (i.e. strain energy density) that a material can absorb just before it fractures. The modulus of toughness is calculated as the area under the stress-strain curve up to the fracture point. An accurate calculation of the total area under the stress-strain curve to determine the modulus of toughness is somewhat involved. However, a rough approximation can be made by dividing the stress- strain curve into a triangular section and a rectangular section, as seen in the figure below. The height of the sections is equal to the average of the yield strength and the ultimate strength. The modulus of toughness can be approximated as: mw Sty + Stu . i ut 2 fu 3 fu 7 Sty + Stu _ Sty + Siu an * 2 fu 2 2E where Sty is the tensile yield strength, S;y is the tensile ultimate strength, &y is the strain at yield, Ey is the ultimate strain (total strain at failure), and E is the elastic modulus. Modulus of Toughness Approximation & & £ A better calculation of the modulus of toughness could be made by using the Ramberg-Osgood equation to approximate the stress-strain curve, and then integrating the area under the curve. It should be noted how greatly the area under the plastic region of the stress-strain curve (i.e. the rectangular portion) contributes to the toughness of the material. Since a ductile material can withstand much more plastic strain than a brittle material, a ductile material will therefore have a higher modulus of toughness than a brittle material with the same yield strength. Even though structures are typically designed to keep stresses within the elastic region, a ductile material with a higher modulus of toughness is better suited to applications in which an accidental overload may occur. Note that the units of the modulus of toughness are the same as the units of strain energy density, which are psi in US Customary units and Pa in SI units. MechaniCalc = to place the yield point within the plastic region of the curve. From the table above, it can be seen that the yield point and ultimate point within the plastic region are given by: + Yield Point: (Spy, 0.002) * Ultimate Point: (Stu, Ep) From the two points in the plastic region of the curve, the constants N and H for the Ramberg-Osgood equation can be calculated. The strain hardening exponent, N0, is calculated as: (Note 1) 1, — J28lSi/ Sw) log(e¢/0.002) The value for H is calculated using the yield point, (Sty, 0.002), as the point of reference, although either point would do: Diy 0.002" Now that the constants N and H have been determined, the equation for the total strain as a function of stress is known: oc o\lin $+) E H The equation above can be simplified by substituting the expression for H. The final equation for total strain as a function of stress is: A= 1fn a o e= E +-0.002( =.) Notes Note 1: Strain Hardening Exponent in Ramberg-Osgood Equation The strain hardening exponent, denoted by 17, should not be confused with the Ramberg-Osgood parameter, which is also denoted by I. The two parameters are reciprocals of one another, which only adds to the confusion. We use the strain hardening exponent in the Ramberg-Osgood equation rather than the Ramberg-Osgood parameter. The reason that we use the strain hardening exponent is that it is a general material property that is useful outside the context of the Ramberg-Osgood equation. Mailing List Subscribe to receive occasional updates on the latest improvements: Email Address Subscribe MechaniCalc References General References: 1. Budynas-Nisbett, "Shigley's Mechanical Engineering Design," 8th Ed. 2. Dowling, Norman E., "Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture, and Fatigue," 3rd Ed. 3. Gere, James M., "Mechanics of Materials," 6th Ed. 4. Hibbeler, Russell C., "Mechanics of Materials," 10th Ed. 5. Lindeburg, Michael R., "Mechanical Engineering Reference Manual for the PE Exam," 13th Ed. Specifications and Standards: 1. ASTM E8, "Standard Test Methods for Tension Testing of Metallic Materials,’ American Society for Testing and Materials, 2011. Shigley's Mechanical... $549.97 Shop now [ee Mechanical Behavior of... $67.43 Shop now MechaniCalc © 2014-2021 MechaniCalc, Inc. Terms & Conditions Privacy Policy Mechanics of Materials $158.06 Shop now Mechanics of Materials $210.49 Lf Shop now on PPI Mechanical Engineering... $100.39 Lf Shop now