Review from Mechanical Properties of Materials

Transcription

1 Fall 24 Review from Mechanical Properties of Materials Concept of Stress and Strain Figure 1 shows the behavior of a clindrical bod subjected to tension and compression. Tension specimen etends along its ais and contracts on the orthogonal (circumferential) aes Compression specimen compresses along its ais and epands on the orthogonal (circumferential) aes z Figure 1: Specimen under tension and compression (Callister(2)) Aial stress (units of psi or MPa) is the applied aial load P divided b the original cross sectional area, A : = P σ (1) A Aial strain (dimensionless, in/in, m/m) relates the change in length of the specimen to the original length. It is calculated as follows: where l l l = = l is the original length is the aial strain (2) l l During a tension test the sample will elongate, therefore positive. l > l and the strain will be 9/24/24 1

2 Fall 24 Mechanical Properties of Material Certain mechanical properties of the material can be determined from a plot of stress against strain. Modulus of Elasticit In the linear elastic range of response, the stress is proportional to the strain as shown in Figure 2 below. Unload Stress Slope=E=modulus of elasticit Load The relationship is written as: Strain Figure 2: Initial portion of stress-strain curve σ = E Hooke s Law where E is known as the modulus of elasticit or Young s modulus. Tpical values of E for steel, aluminum and titanium are given in Table 1. Table 1: Tpical values of modulus of elasticit Modulus of elasticit, E Metal GPa ksi Steel 27 3 Aluminum 69 1 Titanium The largest value of the stress for which Hooke s law can be used for a given material is known as the proportional limit, P L, of that material. It marks the change from linear elastic behavior to non-linear elastic behavior (see Figure 3). 9/24/24 2

3 Fall 24 Yielding and ield strength The ield point marks the transition from essentiall elastic behavior (strain is recoverable upon unloading) to plastic behavior (some permanent deformation upon unloading). Some metals (including mild steel) ehibit the stress-strain relationship shown in Figure 3, part (a) where the ield point is clearl defined. In this figure, the ield point and the proportional limit coincide. The elastic to plastic transition is well defined b an upper ield point. However, for such materials the ield strength is taken as the average strength associated with the lower ield point. For metals ehibiting a stress-strain relationship similar to that shown in part (b), with no well defined ield point, the.2% offset strain is used to determine the ield strength. The ield strength is tpicall denoted as σ. Upper ield point and P L σ Y P L stress σ Y Lower ield point stress strain (a) Figure 3: Yield Point.2% offset strain (b) Plastic Behavior As a metal deforms past its ield point, plastic deformations will result. Plastic deformation is associated with slip in crstalline materials such as metals (ou will talk about this more in CIE 327). The process is not reversible. Tensile strength After ielding, the stress necessar to increase deformation increases to a maimum, shown as point M in Figure 4. Also shown in this figure are the states of deformation of a test specimen at various points along the stress-strain curve. The maimum stress is termed the tensile strength, tpicall denotes as σ. u 9/24/24 3

4 Fall 24 Breaking Strength Figure 4: Stress-Strain curve to failure (Callister (2)) Point F in Figure 4 corresponds to the breaking (or fracture) strength. Ductilit Ductilit is a measure of the plastic deformation capacit of a metal. Consider Figure 5 below: Figure 5: Brittle and ductile behavior (from Callister) A brittle (opposite of ductile) material is one with limited (or no) plastic deformation capacit. One metric for brittleness is a fracture strain of 5% or less. 9/24/24 4

5 Fall 24 A measure of ductilit can be obtained b calculating either % elongation or % reduction in area. The percent elongation is the percentage of plastic strain at fracture and is given b % EL l l f = (3) l 1 where l is the fracture length. When reporting %EL, the original gage length (often 2 in f or 5 mm) must also be reported. For commonl used steels, the specified minimum elongation is 21%. % reduction in area is calculated from the following equation: % AR A f = (4) A A 1 High values of %AR indicate necking of the specimen and ductile behavior. Tpical values for structural steel are 6 to 7%. Table 2 below gives values of ield strength, tensile strength and ductilit for mild steel, aluminum and titanium. Table 2: Tpical values of ield strength, tensile strength and ductilit Yield strength Tensile strength Ductilit %EL Metal MPa ksi MPA ksi (in 5 mm) Structural Steel % Aluminum (661) % Titanium % Strain Energ The work done b the load P as it is slowl applied to the specimen results in an increase in stored energ associated with deformation of the rod. This energ is referred to as the strain energ of the rod. Strain Energ = U Pd (4) = 1 which is equal to the area under the load deformation diagram between = and = 1 as shown in Figure 6 9/24/24 5

6 Fall 24 P P U=Area 1 d Figure 6: Strain Energ For a linear elastic material, 1 U = P 1 (5) 1 2 where P 1 is the value of the load corresponding to the deformation 1. The units of energ are Joule (N.m) in SI units or in.lb in U.S. Customar units. The strain energ is dependent on the length and cross-sectional area of the test specimen. Strain energ densit or strain energ per unit volume is independent of the dimensions of the specimen. Strain energ densit is actuall the area under the stressstrain curve. Resilience The strain energ densit at ield is known as the resilience modulus. Resilience is the capacit of a material to absorb energ elasticall upon loading. Mathematicall, this can be epressed as U r = σd The modulus of resilience represents the energ per unit volume that the material can absorb without ielding. Figure 7 shows that for a linear elastic material (6) U r 2 σ = σd =.5σ = (7) 2E 9/24/24 6

7 Fall 24 σ σ Y Y Modulus of resilience.2 Y Figure 7: modulus of resilience Units of resilience are joules per cubic meter or inch-pounds force per cubic inch. Resilient materials have a high ield strength and a low modulus of elasticit, properties useful for components such as springs. The capacit of a structure to withstand an impact load without being permanentl deformed also depends on its resilience, Toughness Qualitativel, the toughness is the strain energ densit (area under the stress-strain curve) up to the point of fracture (see Figure 8 ) and is given b: f σd (8) The units for toughness are the same as those for resilience (energ per unit volume). Similar to the calculation of resilience, the toughness of a material can be determined from numerical integration of the stress-strain data. For a material to be considered tough, it must possess significant ductilit and strength. Wh? 9/24/24 7

8 Fall 24 σ Modulus toughness of Rupture.2 f Figure 8: Toughness Fracture Fracture is the separation of a bod into two or more parts. Two modes of fracture are possible, ductile or brittle. The classification is based on whether the material can sustain substantial plastic deformation. Consider the three fractured specimens shown in Figure 9. Figure 9: Tpes of fracture (from Callister) Part (a) of the figure shows a highl ductile fracture in which the specimen necks down to a point. Part (b) of the figure shows a moderatel ductile fracture after some necking (epected from steel specimens) Part (c) of the figure shows brittle fracture without an plastic deformation. 9/24/24 8

9 Fall 24 The Figure 1 (Callister(2) shows the progression of fracture for a material which shows moderatel ductile failure. Five stages in the cup-and cone fracture are shown. Figure 1: Fracture of moderatel ductile material (a) Initial necking of the specimen (b) Small cavities or micro voids from in the interior of the cross section (c) Micro voids enlarge, come together, and coalesce to form an elliptical crack, which has its long ais perpendicular to the stress direction. (d) Crack continues to grow (e) Fracture is preceded b the rapid propagation of a crack around the outer perimeter of the neck and b shear deformation at an angle of 45 degrees to the tension ais (the angle at which the shear stresses are maimum). In contrast to the cup and cone fracture of a ductile material, a brittle material tends to fail along a horizontal plane where the normal stresses are a maimum. References Callister, W.D. Materials Science and Engineering, 5th ed., John Wile and Sons, New York, 2 9/24/24 9