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Design for Strength and Endurance – Chapter 4

_____________________________________________________________________________________ Static Material Properties - 77 - C.F. Zo0rowski 2002

Chapter 4 Static Material Properties

Screen Titles

Common Mechanical Tests Tensile Test Elastic Behavior Moduli & Poisson’s Ratio Plastic Behavior Nominal Stress Strain Diagram Necking Behavior True Strain True Stress True Stress Strain Diagram Offset Yield Stress Typical Tensile Properties Plain Carbon Steel Hardness Tests Brinell Hardness Test Rockwell Hardness Test Hardness Scale Comparison Review Exercise Off Line Exercises


_____________________________________________________________________________________ Static Material Properties - 78 - C.F. Zo0rowski 2002


Static Material Properties - - C.F. Zorowski 2002 79

1. Title page Chapter 4 deals with the subject of the static tensile strength and ductility properties of both ferrous and non-ferrous metals. The topics covered include: tensile testing, elastic and plastic behavior, comparative modulus properties, yield, ultimate and fracture stress, work hardening, elongation and area reduction at fracture, ductility, necking phenomena, true stress and strain, off set yield stress, typical metal tensile properties, effect of carbon content on steel tensile properties, hardness testing, comparison of Brinell and Rockwell hardness scales and their relationship to tensile strength. Sample exercises that demonstrate the application of the subject content are also included.

2. Page Index Listed on this page are all the individual pages in Chapter 4 with the exception of the sample exercises. Each title is hyperlinked to its specific page and can be accessed by clicking on the title. It is suggested that the reader first proceed through all pages sequentially. Clicking on the text button at the bottom of the page provides a pop up window with the text for that page. The text page is closed by clicking on the x in the top right corner of the frame. Clicking on the index button returns the presentation to the page index of chapter 4



3. Common Mechanical Tests. The three most commonly used mechanical tests for determining the static strength properties of both metals and non-metals are the tensile test, the compression test and hardness testing. Standard strength and ductility characteristics used particularly in the design of components that undergo elongation and shear loading come principally from tensile testing in which cylindrical test specimens are elongated to fracture. Compression testing is used principally for the testing of brittle materials like cast iron, concrete and stone or other such materials that are customarily use to carry compressive rather than tensile loading. This topic together with compressive properties will not be covered in this chapter. Hardness testing is a means of determining strength characteristics non-destructively by measuring the material surface indentation properties produced by standard indenters under specific magnitudes and forms of loading.

4. Tensile Test In a standard tensile test a cylindrical specimen of specified standard length is elongated slowly until fracture of the specimen takes place. The elongation of the specimen places the material under an axial tensile load P. Dividing this load by the original cross sectional area of the specimen results in an axial tensile stress sigma. This is referred to as the engineering stress that the specimen experiences under the elongation delta L. The state of elongation is defined in terms of the ratio of delta L to L which is defined as the engineering strain, epsilon. As epsilon increases there will be a corresponding increase in sigma.



5. Elastic Behavior The generic behavior of a ductile material subjected to a tensile test will now be discussed in terms the various characteristics of its elongation and load behavior. As elongation proceeds producing an increase in the applied value of strain it is observed that for small values of strain the resulting internal tensile stress will increase in direct proportion to the strain. These observations and the measurement of the magnitudes of both the engineering stress and strain are made on special machines designed specifically for conducting this type of test. This linear behavior of the material as depicted on the graphic follows Hooke’s law and is referred to as elastic behavior of the material. The slope of the linear portion of the stress strain curve is the modulus of elasticity, E, of the material. This property is also referred to as Young’s modulus or simply the modulus of the material. Its units are the same as stress since the units of strain are dimensionless.

6. Common Moduli & Poisson’s Ratio Listed in the table are actual measured average values of the elastic modulus E for a variety of ferrous and non-ferrous materials along with corresponding shear modulus G and Poisson’s ratio values. Note that the order of magnitude of the modulus E is ten to the sixth pounds per inch. This is indicative that the strains associated with elastic behavior are indeed very small. As an example a stress of thirty thousand psi in steel that behaves elastically corresponds to an associated strain of only one times ten to the minus three inches per inch or a thousandth of an inch per inch of length. Also note that the ratio of E for steel compared to lead is only a factor of about six greater. The corresponding values of the shear modulus G are also of magnitude of ten to the sixth pounds per inch. This is not surprising in that in chapter three it was shown theoretically that G should be between one third and one half E. Also note that all of the Poisson’s ratio values lie between the theoretical values of zero and one half and seem to cluster around 0.3 to 0.4.



7. Exercise - 1 The purpose of this exercise is to check the elastic property relationship developed theoretically in chapter 2. Using the relationship from Chapter 2 between the shear modulus, G, elastic modulus E and Poisson’s ratio apply the E and Nu values from the previous page to calculate G for copper and carbon steel. How do they compare with the average measured values given in the table? When you have completed your analysis click on the solution button to check your results or go on to the next page.

(Solution on Page 93)

8. Plastic Behavior The response of the material specimen to the tension test now enters a second phase of behavior. At some value of applied strain the material will deviate from the straight-line elastic behavior and the strain will begin to increase more rapidly than the induced stress. The stress level at which this deviation from linear behavior takes place is designated the yield stress in tension, sigma y. This second phase of response beyond the yield stress is called plastic behavior indicating some form of permanent unrecoverable deformation is taking place. If at some stress level greater than the yield stress the load is removed from the specimen the stress strain curve will unload along a line parallel to the initial elastic behavior until the load and stress goes to zero. At this point there is an unrecoverable residual strain in the specimen producing a permanent set which can be observed as an increase in length over the original length. If the specimen is reloaded the stress will increase along the unloading curve until it reaches the stress level from which it was unloaded. It will then continue to behave plastically again under additional elongation. Thus the yield stress of the material has been increased beyond what it was originally. This phenomenon is referred to as work hardening and represents a mechanism for increasing the yield stress of the material. This is what occurs when wire is drawn down to a smaller diameter through a die.



9. Nominal Stress Strain Diagram. Plastic behavior continues in the specimen with increased elongation until a third phase of extension behavior is observed. The induced stress continues to increase but with significant increases in the applied strain until the stress reaches a maximum value referred to as the ultimate stress or the tensile strength of the material, sigma u. At this point the value of the load and engineering stress begins to decrease and continues to do so until sufficient elongation takes place to produce physical fracture of the specimen into two separate pieces. In this third phase the deformation behavior of the specimen changes radically as the diameter of the specimen begins to suddenly reduce at a localized position along the length of the specimen. This phenomenon is referred to as necking and is very observable in a material like low carbon steel. The extent of the total strain at which fracture takes place is a measure of the ductility of the material. Ductility can be thought of as the degree of plastic behavior the material can undergo before failure takes place. The greater the fracture strain the more ductile the material is or the greater the amount of plastic behavior and deformation it can withstand before fracture occurs. is gamma xy times the quantity cosine squared theta minus sine squared theta.

10. Necking Behavior As the tensile specimen is stretched first through the elastic and the initial plastic range of behavior it elongates uniformly along its length while its diameter and subsequently the cross sectional area also decrease uniformly along its length as represented by the graphic on the left. When the ultimate stress is reached the deformation behavior undergoes a radical change. The rate at which work is being done on the material can no longer be distributed uniformly at a fast enough rate throughout the entire specimen. It becomes locally concentrated which results in the diameter of the bar decreasing more rapidly at that location than anywhere else along its length. This can be physically observed as a local necking down of the bar as indicated in the middle graphic. Further extension of the specimen results in a decrease in its load carrying capacity with a continuing decrease in the necked down diameter until fracture takes place and the specimen separates into two parts as depicted in the third graphic at the fracture stress. The behavior of the material beyond the ultimate stress level is really a state of instability. Materials that exhibit high degrees of ductility are more apt to undergo this necking process. It is very apparent in low carbon steels.



11. Exercise - 2 The purpose of this exercise is to apply to a physical problem the concepts and observations of nominal stress strain behavior for a ductile material undergoing tensile elongation and the effects of work hardening on the yield stress of the material. When you have completed your solution you can click on the solution button to check your results or go on to the next page.

(Solution on Page 93 and 94)

12. True Strain The definition of engineering strain as the change in length divided by the original length is sufficiently accurate for very small levels of strain but is not a true measure of strain under elongation into the plastic behavior of the material. A more accurate description of the strain is the incremental change in length dy associated with that specific length y integrated from the original length Lo to the length L at which the strain is desired. Carrying out this integration results in the true strain being given by the natural logarithm of the ratio of L to Lo. However if L is expressed as delta L plus L then this logarithmic term can be written as the log of the quantity one plus delta L over L. This second term is just the engineering strain. Since epsilon will always be smaller than one the natural log of the term one plus epsilon can be expressed as the power series epsilon times the quantity one minus one half epsilon plus one-third epsilon squared plus etc. Again considering that epsilon is a small quantity the higher order terms of the power series can be neglected giving finally an approximation for the true strain as the engineering strain times the quantity one minus the engineering strain dived by two. Thus it is seen that the true strain of the tensile specimen will always be somewhat smaller than the engineering strain but not by a great deal.



13. True Stress The definition of engineering stress is the applied load P divided by the original cross sectional area of the tensile specimen A. As the specimen elongates the cross sectional area will decrease, therefore the true tensile stress in the material is more accurately given by the applied load P divided by the actual cross sectional area Ao at that load value P. Thus the true stress can be expressed in terms of the engineering stress simply as the engineering stress multiplied by the original area A divided by the actual area at load P which is Ao. Since the actual cross sectional area under load will always be less than the original cross sectional area the true stress will always be greater than the engineering stress. In the elastic range of the material the reduced cross sectional area can be expressed in terms of the Poisson’s ratio effect. Again making use of a power series expansion and neglecting higher order strain terms the true stress is given approximately by the engineering stress times the quantity 1 plus two times the Poisson’s ratio, nu. In the plastic region but below sigma ultimate and assuming a constant volume process the true stress can be expressed as the engineering stress times the quantity one plus the engineering strain. This makes it somewhat higher than in the elastic region. After necking takes place it is difficult to express the true stress analytically in terms of the engineering stress. However, at fracture it is not uncommon to observe in very ductile materials that the cross sectional area can be reduced to 40 % of its original value which results in a true stress that can be more than double the fracture stress measured in engineering terms

14. True Stress Strain Diagram A generic true stress strain diagram for the tensile behavior of a ductile material is shown in the graphic compared to the engineering stress strain diagram for the same material. Up to the value of the ultimate stress it is observed that the true stress is somewhat higher than the engineering stress but that it follows the same form of behavior relative to the strain. Beyond the ultimate stress the true stress curve departs radically from the engineering stress curve. This is the consequence of the necking behavior, which decreases the minimum cross sectional area of the specimen dramatically. Consequently the true stress in this last deformation phase it actually observed to increase even though the load on the specimen is decreasing. As indicated on the previous page the true fracture stress can in some instances be greater than double the fracture stress in engineering terms. One might conclude from this that in a physical design application a material could be expected to carry stress loads higher than the ultimate stress defined by the peak on the engineering stress strain curve. This is an incorrect conclusion since the behavior beyond the ultimate stress is unstable. Remember that the load begins decreasing after this point even though the strain increases. Hence the ultimate stress represents a true realistic maximum design value for the material and is therefore referred to as the tensile strength of the material.



15. Offset Yield Stress All engineering materials do not exhibit a well-defined yield stress as represented on the previous page. This is particularly true of non-ferrous materials like copper, aluminum, magnesium and their alloys. Hence, an arbitrary standard procedure has been established to define a yield stress for such materials. The process for doing this is to draw a line parallel to the initial slope of the stress strain curve of the material at some prescribed value of strain on the epsilon axis as shown in the figure. The value of stress associated with where this parallel line intersects the stress strain curve is defined as the off set yield stress for that material. In most instances the value used for the magnitude of the off set stain is 0.2 %. Within this range beginning at zero strain the material is considered to behave elastically for design considerations.

16. Typical Tensile Properties The three most common tensile properties published for materials are the yield stress, the ultimate stress or tensile strength and the percent elongation associated with a specified gage length used in the test. The table presented here lists approximate ranges of these property values for a number of different materials. The ranges are a consequence of alloying and possible work hardening. Note that the stresses are given in the units of kpsi so that the values in the table must be multiplied by 1000 to obtain psi. Cast iron is not normally considered an effective tensile material for practical applications hence information is scarce. It should be observed that as the yield stress and tensile strength increase the ductility of carbon steel and stainless decreases significantly as indicated by the decrease in percent elongation. It is also observed that the highest yield and tensile strengths are achieved in the ferrous materials with a particularly wide range for stainless steel. Material property values that account for specific alloying and processing should be obtained from a reliable test source before use in design applications.



17. Plain Carbon Steel The effects of alloying and processing on material properties can be dramatic and are well demonstrated by considering plain carbon steel. The chart on this page shows how the yield point, tensile strength and ductility change with an increase in the percent carbon content. By increasing the carbon content from 0.1 % to 0.5 % the yield stress and ultimate stress (tensile strength) of hot rolled (HR) steel are both doubled in value. However this improvement in strength come with a significant decrease in ductility from 27 % elongation to 17% elongation. Even more dramatic changes occur from the hot rolled steels to cold drawn steels (CD) of similar alloy composition. Here it is observed that the work hardening effect dramatically improves the yield stress with less but still positive increase in the tensile strength also. For 0.5 % carbon the yield increases from 50 kpsi to 83 psi while the tensile strength improves from 89kpsi to 100 kpsi. However, the effect of cold work processing decreases the ductility for the same composition steel from 17 % to about 10% over a 2 in gage length. Not all material undergo as dramatic changes as carbon steel but it is important to know what the specific composition and properties of a given material are for a given design application.

18. Hardness Testing In the introduction to this chapter hardness testing was cited as one of the more common static material properties tests employed. Hardness is generally referred to as the resistance of the material to some form of surface indentation. These tests are useful from several points of view. To begin with the more common types of test can be related to one another quantitatively and correlated to the tensile strength of the material. Secondly they are non destructive and virtually have no effect on the strength of finished mechanical parts and as such can be used to control the quality and uniformity of their manufacture. The two most common forms of hardness testing that will be discussed here are the Brinell and Rockwell tests.



19. Brinell Hardness Test The Brinell hardness test makes use of a loading device that indents the surface of the material with a 10 mm diameter ball with a load of 3000 kg for ferrous materials. For non-ferrous materials the indenting load is reduced to 500 kg. The load is removed and the diameter of the indentation is then measured with a microscope. The pressure in kg per mm squared is then determined from the surface area of the impression. This pressure is designated the Brinell Hardness number or Bhn. For steels this number is generally range from 100 to 500. The formula for determining the Brinell hardness number in terms of the load in kg, the indenter ball diameter in mm and the indention diameter in mm is as given at the bottom of the page.

20. Rockwell Hardness Test. Rockwell harness testing also indents the surface of the material but its process and measurements are quite different from the Brinell test. In the Rockwell test the surface is first indented under a standard load. The load is then increased a specified amount and reduced back to the initial load and the increase in the depth of the indentation between the two loads is measured. This increment of additional indentation provides the Rockwell harness number. The initial load applied is 10 kg. The additional load may vary. Several scales of Rockwell hardness exist. The Rockwell C scale uses a sphero-conical indenter with a maximum load of 150 kg. The Rockwell B scale uses a 1/16 in. diameter ball with a maximum load of 100 kg. Typical numbers for medium carbon steel are Rockwell C-20 or B-100 while for very hard steel it would be Rockwell C-100.



21. Hardness Scale Comparisons The table on this page lists corresponding values of Brinell hardness number, Rockwell C scale and B scale numbers together with tensile strength in kpsi for steel. Note that Rockwell C scale measurements are discontinued at 15 because lower values are difficult to obtain and would be inaccurate. Similarly Rockwell B values are discontinued above about 100 for the same reason. Also observe that the Brinell numbers range from about 100 to 500 for steel as indicated earlier. Hardness comparison scales between Brinell and Rockwell numbers are available in a variety of sources such as the Machinist’s Handbook. Good sources for the correlation of these hardness numbers to tensile strength for other materials are much more difficult to come by. This simply indicates that such a correlation may need to be determined for a specific material design application by conducting both the tensile tests and hardness test on that specific material.

22. Exercise -3 This exercise deals with establishing a correlation between Brinell hardness numbers and tensile strength for steel from the table on the previous page. The hint is provided to help guide the solution process. When you have completed the development of this relationship check your result by clicking on the solution button and then proceed on to the next page.

(Solution on Pages 94 and 95)



23. Review Exercises This review exercise consists of a series of statements taken from the subject content of the chapter that are either true or false. Indicate your response by clicking on either the true or false button at the end of the statement. An immediate visual feedback will be provided. To remove the feedback and go on to the next statement hit the tab key or click the mouse. Clicking on the hot word, in red, in the statement will popup the page on which the subject of the statement is covered. After completing all questions go on to the next page

24. Off Line Exercises This off line exercise consists of two practical problems that require using material covered in this and earlier chapters. The first requires an understanding of a ductile material undergoing tensile test behavior and the difference and significance of true stress and strain as related to engineering stress and strain. The second problem deals with determining both Brinell and Rockwell hardness numbers given the results of a specific test. After reviewing these two problem statements as long as necessary and if you are through with this chapter click on the main menu button to exit or go to another chapter.

(Solution in Appendix)


_____________________________________________________________________________________Two Dimensional Stresses - 91 - C.F. Zorowski 2002

Chapter 4 Static Material Properties

Problem Solutions

Screen Titles

Calculation of Shear Modulus Mode of Deformation Elongation due to Applied Load Hardness Exercise Hardness Exercise (cont.)


_____________________________________________________________________________________Static Material Properties - 92 - C.F. Zorowski 2002



1. Calculation of Shear Modulus The modulus E for Copper from the table is 17.2 times ten to the sixth psi while its Poisson’s ration is 0.325. Substituting these values into the theoretical relationship for the shear modulus gives a value of 6.49 times ten to the sixth. This is exactly the value for G as given in the table. The modulus E for carbon steel from the table is 30 times ten to the sixth psi and its Poisson’s ratio is given as o.292. Substituting these values into the theoretical relationship for the shear modulus gives a calculated value of 11.6 times ten to the sixth psi. This compares with 11.58 times ten to the sixth given in the table. Thus the calculated shear modulus numbers are essentially the same as given in the property table. Click on the return button to return to the next page in chapter four.

2. Mode of Deformation Determine the stress by dividing the load by the cross sectional area of the bar. With the dimensions of ¼ in. by 5/8 in. the cross sectional area becomes 0.156 inches squared. Thus the stress is 4800 lbs. divided by 0.156 inches squared giving 30,770 psi. Since the stress level is greater than the yield stress the material has gone beyond its elastic range. Also since the calculated stress is below the ultimate stress the material is not yet loaded sufficiently high enough to cause it to begin necking. Thus it is appropriate to conclude that the material is in its initial plastic deformation mode.



3. Elongation due to Applied Load When the load is removed and reapplied the material will be work hardened and the yield stress will be increased to 30,700 psi. Thus the deformation of the reapplied load will deform the material elastically and the modulus E can be used to calculate the magnitude of the resulting strain as the ratio of the stress to the modulus. Substituting the values of these two parameters gives a strain of 1.025 times ten to the minus three inches per inch. Multiplying the strain by the length of the bar, 50 inches, gives a total elongation of .051 inches. Click on the return button to go back to the next page in chapter four.

4. Hardness Exercise Applying the hint assume that the tensile strength is a constant times the Brinell hardness number. Use the values as given in the table to determine the constant C for all combinations of Brinell hardness number and tensile strength. These values for C are presented in a table on the next page. From the 20th Edition of Machinery Handbook the value of the constant C is given as 515 for Bhn values less than 175 and 495 for Bhn values greater than 175.



5. Hardness Exercise (cont.) The table lists the Brinell hardness number in the first column, the tensile strength in the second column and the corresponding value of the constant C as the ration of tensile strength to Brinell hardness number in the third column. It is observed that the values of C are close to 500. The percentage variation between the calculated values of C and the values of C as given in Machinery Handbook are listed in column four. It is observed that the C values as given in Machinery Handbook predict the tensile strength to less that 2.5% error in all cases. Hence the assumption of a proportional relationship appears to be quite appropriate. Click on the return button to go back to the next page in chapter four.

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