Stress-Strain Relationship PDF

STRESS-STRAIN 3 NORMAL STRESS-STRAIN RELATIONSHIP If mild steel (low-carbon steel) specimens undergo tensile stress as OBJECTIVES:...

STRESS-STRAIN 3 NORMAL STRESS-STRAIN RELATIONSHIP If mild steel (low-carbon steel) specimens undergo tensile stress as OBJECTIVES: shown in the figure, we should obtain the graph. This could easily be done in laboratory experiments. The graph shows several changes in the stress-strain In engineering the deformation of a body is specified using the concepts relationship. of normal and shear strain. In this lesson we will define these quantities and show how they can be determined for various types of problem STRESS-STRAIN DIAGRAM INTRODUCTION The effects of stresses are deformation which is measured as strain ε. The study of strain will help us to solve more problems which cannot be done just by using the equations of equilibrium. Such problems were previously termed Statically Indeterminate. Strain measures the deformation by dividing the change in length by the original length. ε=δ/L Where δ = the deformation (m) L = the original length (m) Since δ and L both have dimensions of length, there is no dimension for ε. STRAIN ❖ PROPORTIONAL LIMIT Up to this point the relationship of stress to strain is linear, that is, stress is proportional to strain. This linear portion of the stress-strain diagram is probably the most important part of the curve and will be looked at in greater detail later. ❖ ELASTIC LIMIT As the name implies, when stress is removed before this point is reached, the material will return to its original length. Increasing the stress beyond this point will PERMANENTLY deform the material and it will never return to its original length. ENGS 24B MGSD  ❖ YIELD POINT YIELD POINT At this stage, the material will increase in length without any further increase in stress. This phenomenon is unique in mild steel. No other materials exhibit this yield point in their stress-strain phenomena is unique in mild steel. For such materials, the yield point is calculated by the offset method. The yield point for such materials will take the value of the stress at the point where a straight line drawn parallel to the tangent of the curve at the origin (zero σ, zero ε ) but with an offset of 0.002 strain. This is purely for ease of obtaining the yield point and is also used in normal practice. ❖ ULTIMATE STRESS OR STRENGTH This is taken as the highest stress that the material can take. ❖ RUPTURE STRESS OR STRENGTH NECKING This is the stress in the materials fails, that is, breaks, as shown in the figure. The actual rupture stress can never be accurately measured, since after the ultimate stress has been exceeded, the material becomes ‘plastic’ and ‘necking’ occurs drastically. This has the effect of decreasing the area of the cross section of the material. If the exact area at the point the material fails is taken, the rupture stress could be higher than the ultimate stress. However, the ultimate stress is normally taken as the highest stress of the material. NORMAL STRESS-STRAIN RELATIONSHIP PROPORTINAL LIMIT (Hooke’s Law) From the origin O to the point called proportional limit, the stress-strain curve is a straight line. This linear relation between elongation and the axial force causing was first noticed by Sir Robert Hooke in 1678 and is called Hooke’s Law that within the proportional limit, th stress is directly proportional to strain or σ∝ε or σ=kε The constant pf proportionality k is called the Modulus of Elasticity E or Young’s Modulus and is equal to the slope of the stress-strain diagram from O to P. Then σ=Eε ENGS 24B MGSD  DUCTILITY ISOTROPIC AND ANISOTROPICMATERIAL  The percent elongation is the ratio of the plastic elongation of a  When a material behaves in a similar manner regardless of the tensile specimen after ultimate failure within a set of gage marks to the direction of the loads, it is referred to as an isotropic material. original length between the gage marks. It is one measure of ductility. Example: glass, plastics and metals  A ductile material is one that can be stretched, formed, or drawn to a significant degree before fracture. A metal that exhibits a percent  Materials that exhibit different behavior and strength in compression elongation greater than 5.0% is ductile. than in tension is called anisotropic behavior.  A brittle material is one that fails suddenly under load with little or no Example: cast metals, some plastics, concrete, wood, and plastic deformation. A metal that exhibits a percent elongation of less composites than 5.0% is brittle. STIFFNESS  Stiffness of a material is a function of its modulus of elasticity, sometimes called Young’s modulus  The modulus of elasticity, E, is a measure of the stiffness of a material detertmined by the slope of the straight-line portion of the stress–strain curve. It is the ratio of the change of stress to the corresponding change in strain. POISSON’S RATIO Poisson’s ratio as the ratio of the amount of lateral strain to axial strain. ENGS 24B MGSD  FATIGUE STRENGTH OR ENDURANCE STRENGTH Parts subjected to repeated applications of loads or to stress conditions that vary with time over several thousands or millions of cycles fail because of the phenomenon of fatigue. Materials are tested under controlled cyclic loading to determine their ability to resist such repeated loads. The resulting data are reported as the fatigue strength, also called the endurance strength of the material. MODULUS OF RESILIENCE HARDNESS Modulus of resilience is the work done on a unit volume of material as the force is gradually increased from O to P, in N·m/m3. This may be calculated as the area under the stress-strain curve from the origin O to up to the elastic limit E. The resilience of the material is its ability to absorb energy without creating a permanent distortion. The resistance of a material to indentation by a penetrator is an indication of its hardness. MODULUS OF TOUGHNESS Several types of devices, procedures, and penetrators measure hardness; the Brinell hardness tester and the Rockwell hardness tester Modulus of toughness is the work done on a unit volume of material as are most frequently used for machine elements. the force is gradually increased from O to R, in N·m/m 3. This may be calculated as the area under the entire stress-strain curve (from O to R). The toughness of a For steels, the Brinell hardness tester employs a hardened steel ball material is its ability to absorb energy without causing it to break. 10 mm in diameter as the penetrator under a load of 3000 kg force. The load causes a permanent indentation in the test material, and the Working Stress, Allowable Stress, and Factor of Safety diameter of the indentation is related to the Brinell hardness number, which is abbreviated HB. Working stress is defined as the actual stress of a material under a given loading. The maximum safe stress that a material can carry is termed as the The Rockwell hardness tester uses a hardened steel ball with a 1.588 allowable stress. The allowable stress should be limited to values not exceeding mm diameter under a load of 100 kg force for softer metals, and the the proportional limit. However, since proportional limit is difficult to determine resulting hardness is listed as Rockwell B, RB, or HRB. For harder metals, accurately, the allowable stress is taken as either the yield point or ultimate such as heat-treated alloy steels, the Rockwell C scale is used. A load of strength divided by a factor of safety. The ratio of this strength (ultimate or yield 150 kg force is placed on a diamond penetrator (a brale penetrator) made strength) to allowable strength is called the factor of safety. in a sphero-conical shape. Rockwell C hardness is sometimes referred to as RC or HRC. Many other Rockwell scales are used. ENGS 24B MGSD PROBLEM The following data were recorded during the tensile test of a 14-mm-diameter mild steel rod. The gage length was 50 mm. Elongation Elongation Load (N) Load (N) (mm) (mm) 0 0 46 200 1.25 6 310 0.010 52 400 2.50 12 600 0.020 58 500 4.50 18 800 0.030 68 000 7.50 25 100 0.040 59 000 12.5 31 300 0.050 67 800 15.5 37 900 0.060 65 000 20.0 40 100 0.163 65 500 Fracture 41 600 0.433 Plot the stress-strain diagram and determine the following mechanical properties: (a) proportional limits; (b) modulus of elasticity; (c) yield point; (d) ultimate strength; and (e) rupture strength. ENGS 24B MGSD