2 Fundamental Mechanical Properties of Materials The goal of the following pages is to characterize materials in terms of some of the fundamental mechanical properties that were introduced in Chapter 1. A qualitative distinction between ductile,brittle,and elastic materials can be achieved in a relatively simple experiment us- ing the bend test,as shown in Figure 2.1.A long and compara- tively thin piece of the material to be tested is placed near its ends on two supports and loaded at the center.It is intuitively obvious that an elastic material such as wood can be bent to a much higher degree before breakage occurs than can a brittle material such as stone or glass.Moreover,elastic materials re- turn upon elastic deformation to their original configuration once the stress has been removed.On the other hand,ductile materi- als undergo a permanent change in shape above a certain thresh- old load.But even ductile materials eventually break once a large enough force has been applied. To quantitatively evaluate these properties,a more sophisti- cated device is routinely used by virtually all industrial and sci- entific labs.In the tensile tester,a rod-shaped or flat piece of the material under investigation is held between a fixed and a mov- able arm as shown in Figure 2.2.A force upon the test piece is exerted by slowly driving the movable cross-head away from the fixed arm.This causes a stress,o,on the sample,which is de- fined to be the force,F,per unit area,Ao,that is, (2.1) Ao Since the cross section changes during the tensile test,the ini-
2 The goal of the following pages is to characterize materials in terms of some of the fundamental mechanical properties that were introduced in Chapter 1. A qualitative distinction between ductile, brittle, and elastic materials can be achieved in a relatively simple experiment using the bend test, as shown in Figure 2.1. A long and comparatively thin piece of the material to be tested is placed near its ends on two supports and loaded at the center. It is intuitively obvious that an elastic material such as wood can be bent to a much higher degree before breakage occurs than can a brittle material such as stone or glass. Moreover, elastic materials return upon elastic deformation to their original configuration once the stress has been removed. On the other hand, ductile materials undergo a permanent change in shape above a certain threshold load. But even ductile materials eventually break once a large enough force has been applied. To quantitatively evaluate these properties, a more sophisticated device is routinely used by virtually all industrial and scientific labs. In the tensile tester, a rod-shaped or flat piece of the material under investigation is held between a fixed and a movable arm as shown in Figure 2.2. A force upon the test piece is exerted by slowly driving the movable cross-head away from the fixed arm. This causes a stress, , on the sample, which is defined to be the force, F, per unit area, A0, that is, A F 0 . (2.1) Since the cross section changes during the tensile test, the iniFundamental Mechanical Properties of Materials
2.Fundamental Mechanical Properties of Materials 13 FIGURE 2.1.Schematic representation of a bend test.Note that the convex surface is under tension and the concave surface is under compression.Both stresses are essen- tially parallel to the surface.The bend test is particularly used for brittle materials. tial unit area,Ao,is mostly used;see below.If the force is ap- plied parallel to the axis of a rod-shaped material,as in the ten- sile tester (that is,perpendicular to the faces Ao),then o is called a tensile stress.If the stress is applied parallel to the faces (as in Figure 2.3),it is termed shear stress,7. Many materials respond to stress by changing their dimen- sions.Under tensile stress,the rod becomes longer in the direc- tion of the applied force (and eventually narrower perpendicular to that axis).The change in longitudinal dimension in response to stress is called strain,e,that is: e=1-b-4 (2.2) where lo is the initial length of the rod and l is its final length. The absolute value of the ratio between the lateral strain (shrinkage)and the longitudinal strain (elongation)is called the Poisson ratio,v.Its maximum value is 0.5 (no net volume change).In reality,the Poisson ratio for metals and alloys is gen- erally between 0.27 and 0.35;in plastics (e.g.,nylon)it may be as large as 0.4;and for rubbers it is even 0.49,which is near the maximum possible value. Sample FIGURE 2.2.Schematic repre- 0 sentation of a tensile test equipment.The lower cross-bar is made to move downward and thus ex- tends a force,F,on the test piece whose cross-sectional area is Ao.The specimen to be tested is either threaded into the specimen holders or held by a vice grip
tial unit area, A0, is mostly used; see below. If the force is applied parallel to the axis of a rod-shaped material, as in the tensile tester (that is, perpendicular to the faces A0), then is called a tensile stress. If the stress is applied parallel to the faces (as in Figure 2.3), it is termed shear stress, . Many materials respond to stress by changing their dimensions. Under tensile stress, the rod becomes longer in the direction of the applied force (and eventually narrower perpendicular to that axis). The change in longitudinal dimension in response to stress is called strain, , that is: l l0 l0 l0 l , (2.2) where l0 is the initial length of the rod and l is its final length. The absolute value of the ratio between the lateral strain (shrinkage) and the longitudinal strain (elongation) is called the Poisson ratio, . Its maximum value is 0.5 (no net volume change). In reality, the Poisson ratio for metals and alloys is generally between 0.27 and 0.35; in plastics (e.g., nylon) it may be as large as 0.4; and for rubbers it is even 0.49, which is near the maximum possible value. 2 • Fundamental Mechanical Properties of Materials 13 10kg FIGURE 2.1. Schematic representation of a bend test. Note that the convex surface is under tension and the concave surface is under compression. Both stresses are essentially parallel to the surface. The bend test is particularly used for brittle materials. Sample F A0 FIGURE 2.2. Schematic representation of a tensile test equipment. The lower cross-bar is made to move downward and thus extends a force, F, on the test piece whose cross-sectional area is A0. The specimen to be tested is either threaded into the specimen holders or held by a vice grip.
14 2.Fundamental Mechanical Properties of Materials FIGURE 2.3.Distortion of a cube caused by shear stresses △a Txy and Tyx. Tyx The force is measured in newtons(1 N=1 kg m s-2)and the stress is given in N m-2 or pascal(Pa).(Engineers in the United States occasionally use the pounds per square inch(psi)instead, where 1 psi=6.895 X 103 Pa and 1 pound =4.448 N.See Ap- pendix II.)The strain is unitless,as can be seen from Eq.(2.2) and is usually given in percent of the original length. The result of a tensile test is commonly displayed in a stress-strain diagram as schematically depicted in Figure 2.4. Several important characteristics are immediately evident.Dur- ing the initial stress period,the elongation of the material re- sponds to o in a linear fashion;the rod reverts back to its orig- inal length upon relief of the load.This region is called the elastic range.Once the stress exceeds,however,a critical value, called the yield strength,oy,some of the deformation of the material becomes permanent.In other words,the yield point separates the elastic region from the plastic range of materials. Stress Tensile strength U T Yield strength Breaking strength y Plastic part Necking 复 △U FIGURE 2.4.Schematic rep- resentation of a Tension △e stress-strain diagram for a ductile material.For ac- Strain s tual values of oy and or, see Table 2.1 and Figure Compression 2.5
The force is measured in newtons (1 N 1 kg m s2) and the stress is given in N m2 or pascal (Pa). (Engineers in the United States occasionally use the pounds per square inch (psi) instead, where 1 psi 6.895 103 Pa and 1 pound 4.448 N. See Appendix II.) The strain is unitless, as can be seen from Eq. (2.2) and is usually given in percent of the original length. The result of a tensile test is commonly displayed in a stress–strain diagram as schematically depicted in Figure 2.4. Several important characteristics are immediately evident. During the initial stress period, the elongation of the material responds to in a linear fashion; the rod reverts back to its original length upon relief of the load. This region is called the elastic range. Once the stress exceeds, however, a critical value, called the yield strength, y, some of the deformation of the material becomes permanent. In other words, the yield point separates the elastic region from the plastic range of materials. 14 2 • Fundamental Mechanical Properties of Materials Plastic part Elastic part Stress Yield strength y Tensile strength T Breaking strength B Tension Compression Strain Necking FIGURE 2.4. Schematic representation of a stress–strain diagram for a ductile material. For actual values of y and T, see Table 2.1 and Figure 2.5. a xy yx a FIGURE 2.3. Distortion of a cube caused by shear stresses xy and yx.
2.Fundamental Mechanical Properties of Materials 15 Nylon WoodL PVC Ice Polyurethane Concrete W Pressure- foam a-Fe Stainless vessel Al Epoxy steel steel Ultrapure Au Diamond fcc metals Cast iron Ni SiO2 SiC Pb Cu Alkali ALO3 halides 10 100 1.,000 10,000 o,[MNm-2] FiGURE 2.5.Yield Polymers strengths of materials (given in meganewtons Ceramics per square meter or Metals megapascals;see Ap- Composites- pendixΠ). This is always important if one wants to know how large an ap- plied stress needs to be in order for plastic deformation of a workpiece to occur.On the other hand,the yield strength pro- vides the limit for how much a structural component can be stressed before unwanted permanent deformation takes place As an example,a screwdriver has to have a high yield strength; otherwise,it will deform upon application of a large twisting force.Characteristic values for the yield strength of different materials are given in Table 2.1 and Figure 2.5. The highest force (or stress)that a material can sustain is called the tensile strength,or(Figure 2.4).At this point,a localized decrease in the cross-sectional area starts to occur.The material is said to undergo necking,as shown in Figure 2.6.Because the cross section is now reduced,a smaller force is needed to con- tinue deformation until eventually the breaking strength,oB,is reached (Figure 2.4). The slope in the elastic part of the stress-strain diagram(Fig- ure 2.4)is defined to be the modulus of elasticity,E,(or Young's modulus): △d=E. (2.3) △e Equation(2.3)is generally referred to as Hooke's Law.For shear stress,T[see above and Figure 2.3],Hooke's law is appropriately written as: △=G, (2.4) △y Sometimes called ultimate tensile strength or ultimate tensile stress,ours
This is always important if one wants to know how large an applied stress needs to be in order for plastic deformation of a workpiece to occur. On the other hand, the yield strength provides the limit for how much a structural component can be stressed before unwanted permanent deformation takes place. As an example, a screwdriver has to have a high yield strength; otherwise, it will deform upon application of a large twisting force. Characteristic values for the yield strength of different materials are given in Table 2.1 and Figure 2.5. The highest force (or stress) that a material can sustain is called the tensile strength, 1 T (Figure 2.4). At this point, a localized decrease in the cross-sectional area starts to occur. The material is said to undergo necking, as shown in Figure 2.6. Because the cross section is now reduced, a smaller force is needed to continue deformation until eventually the breaking strength, B, is reached (Figure 2.4). The slope in the elastic part of the stress–strain diagram (Figure 2.4) is defined to be the modulus of elasticity, E, (or Young’s modulus): E. (2.3) Equation (2.3) is generally referred to as Hooke’s Law. For shear stress, [see above and Figure 2.3], Hooke’s law is appropriately written as: G, (2.4) 2 • Fundamental Mechanical Properties of Materials 15 FIGURE 2.5. Yield strengths of materials (given in meganewtons per square meter or megapascals; see Appendix II). 1Sometimes called ultimate tensile strength or ultimate tensile stress, UTS. Ultrapure fcc metals Polyurethane foam Wood Nylon PVC Ice Concrete –Fe Al Au Pb Ni Cu Cast iron Epoxy Stainless steel Alkali halides W SiO2 Pressurevessel steel Al2O3 SiC Diamond 1 10 100 1,000 10,000 y[MNm–2] Polymers Ceramics Metals Composites
16 2.Fundamental Mechanical Properties of Materials F Necking FIGURE 2.6.Necking of a test sample that was stressed in a tensile machine. where y is the shear strain Aa/a tan a =a and G is the shear modulus. The modulus of elasticity is a parameter that reveals how "stiff" a material is,that is,it expresses the resistance of a material to elastic bending or elastic elongation.Specifically,a material hav- ing a large modulus and,therefore,a large slope in the stress-strain diagram deforms very little upon application of even a high stress.This material is said to have a high stiffness.(For average values,see Table 2.1.)This is always important if one re- quires close tolerances,such as for bearings,to prevent friction. Stress-strain diagrams vary appreciably for different materials and conditions.As an example,brittle materials,such as glass, stone,or ceramics have no separate yield strength,tensile strength, or breaking strength.In other words,they possess essentially no plastic (ductile)region and,thus,break already before the yield strength is reached [Figure 2.7(a)].Brittle materials (e.g.,glass) are said to have a very low fracture toughness.As a consequence, tools (hammers,screwdrivers,etc.)should not be manufactured from brittle materials because they may break or cause injuries. Ductile materials (e.g.,many metals)on the other hand,with- stand a large amount of permanent deformation (strain)before they break,as seen in Figure 2.7(a).(Ductility is measured by the amount of permanent elongation or reduction in area,given in percent,that a material has withstood at the moment of fracture.) Many materials essentially display no well-defined yield strength in the stress-strain diagram;that is,the transition be- tween the elastic and plastic regions cannot be readily determined [Figure 2.7(b)].One therefore defines an offset yield strength at which a certain amount of permanent deformation(for example
where is the shear strain a/a tan and G is the shear modulus. The modulus of elasticity is a parameter that reveals how “stiff” a material is, that is, it expresses the resistance of a material to elastic bending or elastic elongation. Specifically, a material having a large modulus and, therefore, a large slope in the stress–strain diagram deforms very little upon application of even a high stress. This material is said to have a high stiffness. (For average values, see Table 2.1.) This is always important if one requires close tolerances, such as for bearings, to prevent friction. Stress–strain diagrams vary appreciably for different materials and conditions. As an example, brittle materials, such as glass, stone, or ceramics have no separate yield strength, tensile strength, or breaking strength. In other words, they possess essentially no plastic (ductile) region and, thus, break already before the yield strength is reached [Figure 2.7(a)]. Brittle materials (e.g., glass) are said to have a very low fracture toughness. As a consequence, tools (hammers, screwdrivers, etc.) should not be manufactured from brittle materials because they may break or cause injuries. Ductile materials (e.g., many metals) on the other hand, withstand a large amount of permanent deformation (strain) before they break, as seen in Figure 2.7(a). (Ductility is measured by the amount of permanent elongation or reduction in area, given in percent, that a material has withstood at the moment of fracture.) Many materials essentially display no well-defined yield strength in the stress–strain diagram; that is, the transition between the elastic and plastic regions cannot be readily determined [Figure 2.7(b)]. One therefore defines an offset yield strength at which a certain amount of permanent deformation (for example, 16 2 • Fundamental Mechanical Properties of Materials Necking F F FIGURE 2.6. Necking of a test sample that was stressed in a tensile machine
2.Fundamental Mechanical Properties of Materials 17 TABLE 2.1.Some mechanical properties of materials Modulus of Yield Tensile elasticity, strength, strength, Material E [GPa] oy [MPa] or [MPa] Diamond 1,000 50,000 same SiC 450 10,000 same W 406 1000 1510 Cast irons 170-190 230-1030 400-1200 Low carbon steel, 196 180-260 325-485 hot rolled Carbon steels.water- -200 260-1300 500-1800 quenched and tempered Fe 196 50 200 Cu 124 60 400 Si 107 10%Sn bronze 100 190 二 SiO2(silica glass) 94 7200 about the same Au 82 40 220 Al 69 40 200 Soda glass 69 3600 about the same Concrete 50 25* Wood to grain 9-16 33-50*;73-121+ Pb 14 11 14 Spider drag line 2.8-4.7 870-1420 Nylon 3 49-87 60-100 Wood〦to grain 0.6-1 5* 3-10*:2-8+ Rubbers 0.01-0.1 30 PVC 0.003-0.01 45 *compression;+tensile. Note:The data listed here are average values.See Chapter 3 for the di- rectionality of certain properties called anisotropy;see also Figure 2.4.) For glasses,see also Table 15.1. 0.2%)has occurred and which can be tolerated for a given ap- plication.A line parallel to the initial segment in the stress-strain curve is constructed at the distance e=0.2%.The intersect of this line with the stress-strain curve yields oo.2 [Figure 2.7(b)]. Some materials,such as rubber,deform elastically to a large extent,but cease to be linearly elastic after a strain of about 1%. Other materials(such as iron or low carbon steel)display a sharp yield point,as depicted in Figure 2.7(c).Specifically,as the stress is caused to increase to the upper yield point,no significant plas- tic deformation is encountered.From now on,however,the ma- terial will yield,concomitantly with a drop in the flow stress,(i.e., the stress at which a metal will flow)resulting in a lower yield point and plastic deformation at virtually constant stress [Figure 2.7(c)].The lower yield point is relatively well defined but fluc-
2 • Fundamental Mechanical Properties of Materials 17 TABLE 2.1. Some mechanical properties of materials Modulus of Yield Tensile elasticity, strength, strength, Material E [GPa] y [MPa] T [MPa] Diamond 1,000 50,000 same SiC 450 10,000 same W 406 1000 1510 Cast irons 170–190 230–1030 400–1200 Low carbon steel, 196 180–260 325–485 hot rolled Carbon steels, water- 200 260–1300 500–1800 quenched and tempered Fe 196 50 200 Cu 124 60 400 Si 107 — — 10% Sn bronze 100 190 — SiO2 (silica glass) 94 7200 about the same Au 82 40 220 Al 69 40 200 Soda glass 69 3600 about the same Concrete 50 25* — Wood to grain 9–16 — 33–50*; 73–121 Pb 14 11 14 Spider drag line 2.8–4.7 — 870–1420 Nylon 3 49–87 60–100 Wood to grain 0.6–1 5* 3–10*; 2–8 Rubbers 0.01–0.1 — 30 PVC 0.003–0.01 45 — *compression; tensile. Note: The data listed here are average values. See Chapter 3 for the directionality of certain properties called anisotropy; see also Figure 2.4.) For glasses, see also Table 15.1. 0.2%) has occurred and which can be tolerated for a given application. A line parallel to the initial segment in the stress–strain curve is constructed at the distance 0.2%. The intersect of this line with the stress–strain curve yields 0.2 [Figure 2.7(b)]. Some materials, such as rubber, deform elastically to a large extent, but cease to be linearly elastic after a strain of about 1%. Other materials (such as iron or low carbon steel) display a sharp yield point, as depicted in Figure 2.7(c). Specifically, as the stress is caused to increase to the upper yield point, no significant plastic deformation is encountered. From now on, however, the material will yield, concomitantly with a drop in the flow stress, (i.e., the stress at which a metal will flow) resulting in a lower yield point and plastic deformation at virtually constant stress [Figure 2.7(c)]. The lower yield point is relatively well defined but fluc-
18 2.Fundamental Mechanical Properties of Materials Brittle material (diamond) Necking 00.2 Ductile material (Cu) 0.2% (a) (b) Nonlinear elastic deformation Tensile Upper (viscoelasticity) strength yield point Yield strength Necking Plastic deformation Lower Linear elastic yield point deformation (c) (d) Low temperature High temperature 心 (e) FIGURE 2.7.Schematic representations of stress-strain diagrams for various materials and conditions:(a)brittle (diamond,ceramics,ther- moset polymers)versus ductile (metals,alloys)materials;(b)defini- tion of the offset yield strength;(c)upper and lower yield points ob- served,for example,in iron and low carbon steels;(d)thermoplastic polymer;and (e)variation with temperature
18 2 • Fundamental Mechanical Properties of Materials FIGURE 2.7. Schematic representations of stress–strain diagrams for various materials and conditions: (a) brittle (diamond, ceramics, thermoset polymers) versus ductile (metals, alloys) materials; (b) definition of the offset yield strength; (c) upper and lower yield points observed, for example, in iron and low carbon steels; (d) thermoplastic polymer; and (e) variation with temperature. 0.2% 0.2 (b) y T B (a) Brittle material (diamond) Ductile material (Cu) (e) Low temperature High temperature (c) (d) Upper yield point Lower yield point Nonlinear elastic deformation (viscoelasticity) Yield strength Necking Linear elastic deformation Plastic deformation Tensile strength Necking
2.Fundamental Mechanical Properties of Materials 19 tuates about a fixed stress level.Thus,the yield strength in these cases is defined as the average stress that is associated with the lower yield point.Upon further stressing,the material eventually hardens,which requires the familiar increase in load if additional deformation is desired.The deformation at the lower yield point starts at locations of stress concentrations and manifests itself as discrete bands of deformed material,called Ltiders bands,which may cause visible striations on the surface.The deformation oc- curs at the front of these spreading bands until the end of the lower yield point is reached. A few polymeric materials,such as nylon,initially display a linear and,subsequently,a nonlinear (viscoelastic)region in the stress-strain diagram [Figure 2.7(d)].Moreover,beyond the yield strength,a bathtub-shaped curve is obtained,as depicted in Fig- ure2.7(d). Stress-strain curves may vary for different temperatures [Fig- ure 2.7(e)].For example,the yield strength,as well as the tensile strength,and to a lesser degree also the elastic modulus,are of- ten smaller at elevated temperatures.In other words,a metal can be deformed permanently at high temperatures with less effort than at room temperature.This property is exploited by indus- trial rolling mills or by a blacksmith when he shapes red-hot metal items on his anvil.The process is called hot working. On the other hand,if metals,alloys,or some polymeric mate- rials are cold worked,that is,plastically deformed at ambient tem- peratures,eventually they become less ductile and thus harder and even brittle.This is depicted in Figure 2.8(a),in which a ma- terial is assumed to have been stressed beyond the yield strength. Upon releasing the stress,the material has been permanently de- formed to a certain degree.Restressing the same material [Fig- ure 2.8(b)]leads to a higher or and to less ductility.The plastic deformation steps can be repeated several times until eventually oy=or=oB.At this point the workpiece is brittle,similar to a ceramic.Any further attempt of deformation would lead to im- mediate breakage.The material is now work hardened(or strain hardened)to its limit.A coppersmith utilizes cold working(ham- mering)for shaping utensils from copper sheet metal.The strain hardened workpiece can gain renewed ductility,however,by heating it above the recrystallization temperature (which is ap- proximately 0.4 times the absolute melting temperature).For copper,the recrystallization temperature is about 200C. The degree of strengthening acquired through cold working is given by the strain hardening rate,which is proportional to the slope of the plastic region in a true stress-true strain curve.This needs some further explanation.The engineering stress and the
tuates about a fixed stress level. Thus, the yield strength in these cases is defined as the average stress that is associated with the lower yield point. Upon further stressing, the material eventually hardens, which requires the familiar increase in load if additional deformation is desired. The deformation at the lower yield point starts at locations of stress concentrations and manifests itself as discrete bands of deformed material, called Lüders bands, which may cause visible striations on the surface. The deformation occurs at the front of these spreading bands until the end of the lower yield point is reached. A few polymeric materials, such as nylon, initially display a linear and, subsequently, a nonlinear (viscoelastic) region in the stress–strain diagram [Figure 2.7(d)]. Moreover, beyond the yield strength, a bathtub-shaped curve is obtained, as depicted in Figure 2.7(d). Stress–strain curves may vary for different temperatures [Figure 2.7(e)]. For example, the yield strength, as well as the tensile strength, and to a lesser degree also the elastic modulus, are often smaller at elevated temperatures. In other words, a metal can be deformed permanently at high temperatures with less effort than at room temperature. This property is exploited by industrial rolling mills or by a blacksmith when he shapes red-hot metal items on his anvil. The process is called hot working. On the other hand, if metals, alloys, or some polymeric materials are cold worked, that is, plastically deformed at ambient temperatures, eventually they become less ductile and thus harder and even brittle. This is depicted in Figure 2.8(a), in which a material is assumed to have been stressed beyond the yield strength. Upon releasing the stress, the material has been permanently deformed to a certain degree. Restressing the same material [Figure 2.8(b)] leads to a higher T and to less ductility. The plastic deformation steps can be repeated several times until eventually y T B. At this point the workpiece is brittle, similar to a ceramic. Any further attempt of deformation would lead to immediate breakage. The material is now work hardened (or strain hardened) to its limit. A coppersmith utilizes cold working (hammering) for shaping utensils from copper sheet metal. The strain hardened workpiece can gain renewed ductility, however, by heating it above the recrystallization temperature (which is approximately 0.4 times the absolute melting temperature). For copper, the recrystallization temperature is about 200°C. The degree of strengthening acquired through cold working is given by the strain hardening rate, which is proportional to the slope of the plastic region in a true stress–true strain curve. This needs some further explanation. The engineering stress and the 2 • Fundamental Mechanical Properties of Materials 19
20 2.Fundamental Mechanical Properties of Materials o O' 02 0y1 y 0y0 E0 81 E1 E2 g" E2E3 permanent permanent permanent deformation deformation deformation (a) (b) (c) FIGURE 2.8.Increase of yield strength (and reduction of ductility)by re- peated plastic deformation.(a)Sample is moderately stressed until some plastic deformation has occurred,and then it is unloaded, which yields permanent deformation.(b)The sample is subsequently additionally permanently deformed.Note that the coordinate system has shifted after unloading from eo to e1.(c)Limit of plastic deforma- tion is reached after renewed stressing. engineering strain,as defined in Equations(2.1)and (2.2),are essentially sufficient for most practical purposes.However,as mentioned above,the cross-sectional area of a tensile test spec- imen decreases continuously,particularly during necking.The latter causes a decrease of o beyond the tensile strength.A true stress and true strain diagram takes the varying areas into con- sideration(Figure 2.9(a)).One defines the true stress as: 0= (2.5) where A;is now the instantaneous cross-sectional area that varies during deformation.The true strain is then: (2.6) ISee Problems 8 and 9
engineering strain, as defined in Equations (2.1) and (2.2), are essentially sufficient for most practical purposes. However, as mentioned above, the cross-sectional area of a tensile test specimen decreases continuously, particularly during necking. The latter causes a decrease of beyond the tensile strength. A true stress and true strain diagram takes the varying areas into consideration (Figure 2.9(a)). One defines the true stress as: t A F i , (2.5) where Ai is now the instantaneous cross-sectional area that varies during deformation. The true strain is then:1 t li l 0 d l l ln l l 0 i ln A A 0 i . (2.6) 20 2 • Fundamental Mechanical Properties of Materials 1See Problems 8 and 9. (a) (b) (c) 0 1 1 y2 y3 2 2 3 ' " y1 y1 ' ' " " y0 y2 permanent deformation permanent deformation permanent deformation FIGURE 2.8. Increase of yield strength (and reduction of ductility) by repeated plastic deformation. (a) Sample is moderately stressed until some plastic deformation has occurred, and then it is unloaded, which yields permanent deformation. (b) The sample is subsequently additionally permanently deformed. Note that the coordinate system has shifted after unloading from 0 to 1. (c) Limit of plastic deformation is reached after renewed stressing.
2.Fundamental Mechanical Properties of Materials 21 In o Plastic region Plastic region slope n<I Elastic region (b) Elastic region slope n=1 In er FIGURE 2.9.(a)True stress versus true strain diagram (compare to (a) Figure 2.4).(b)In ot versus In e diagram. In many cases,and before necking begins,one can approximate the true stress-true strain curve by the following empirical equation: :=K(e)", (2.7) where n is the strain hardening exponent (having values of less than unity)and K is another materials constant (called the strength coefficient)which usually amounts to several hundred MPa.Taking the (natural)logarithm of Eq.(2.7)yields: In o:=n In e+In K, (2.8) which reveals that the strain hardening exponent(or strain hard- ening rate),n,is the slope in the plastic portion of an In o versus In e diagram,see Figure 2.9(b). The tensile test and the resulting stress-strain diagrams have been shown above to provide a comprehensive insight into many of the mechanical properties of materials.For specialized appli- cations,however,a handful of further tests are commonly used. Some of them will be reviewed briefly below. The hardness test is nondestructive and fast.A small steel sphere (commonly 10 mm in diameter)is momentarily pressed into the surface of a test piece.The diameter of the indentation is then measured under the microscope,from which the Brinell hard- ness number(BHN)is calculated by taking the applied force and the size of the steel sphere into consideration.The BHN is directly proportional to the tensile strength.(The Rockwell hardness tester uses instead a diamond cone and measures the depth of the in- dentation under a known load whereas the Vickers and Knoop mi- crohardness techniques utilize diamond pyramids as indenters.) Materials,even when stressed below the yield strength,still may eventually break if a large number of tension and compres-
Elastic region Plastic region slope n=1 ln t ln t slope n<1 In many cases, and before necking begins, one can approximate the true stress–true strain curve by the following empirical equation: t K(t)n, (2.7) where n is the strain hardening exponent (having values of less than unity) and K is another materials constant (called the strength coefficient) which usually amounts to several hundred MPa. Taking the (natural) logarithm of Eq. (2.7) yields: ln t n ln t ln K, (2.8) which reveals that the strain hardening exponent (or strain hardening rate), n, is the slope in the plastic portion of an ln t versus ln t diagram, see Figure 2.9(b). The tensile test and the resulting stress–strain diagrams have been shown above to provide a comprehensive insight into many of the mechanical properties of materials. For specialized applications, however, a handful of further tests are commonly used. Some of them will be reviewed briefly below. The hardness test is nondestructive and fast. A small steel sphere (commonly 10 mm in diameter) is momentarily pressed into the surface of a test piece. The diameter of the indentation is then measured under the microscope, from which the Brinell hardness number (BHN) is calculated by taking the applied force and the size of the steel sphere into consideration. The BHN is directly proportional to the tensile strength. (The Rockwell hardness tester uses instead a diamond cone and measures the depth of the indentation under a known load whereas the Vickers and Knoop microhardness techniques utilize diamond pyramids as indenters.) Materials, even when stressed below the yield strength, still may eventually break if a large number of tension and compres- 2 • Fundamental Mechanical Properties of Materials 21 FIGURE 2.9. (a) True stress versus true strain diagram (compare to Figure 2.4). (b) ln t versus ln t diagram. y t t Plastic region Elastic region T (a) (b)
