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