Figure 1: General Plane Motion (Translation and Rotation)

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1 STRIN ND TH TRNSFORMTION OF STRIN INTRODUCTION - DFORMBL BODY MOTION ) Rigid Bod Motion T T Translation Rotation Figure : General Plane Motion (Translation and Rotation) Figure shows the general plane motion of a rigid bod consisting of a translation, where all points on the bod have the same displacement (T,T ), and a rotation, where all lines on the bod have the same rotation (R z ). In rigid bod motion, no line changes length nor is there a change in the angle between an two lines. ) Deformation and Strain C O Direct Strain v B u B' C O C C' ' Shear Strain Figure : Deformable Bod Motion (Strain) Figure shows the plane deformation of a bod. This motion involves a change of volume, direct (normal) strain, and a change of shape, shear strain. Direct strain represents a change in length of a line and shear strain represents a change in angle between lines. In Figure, direct strain is demonstrated b the change in length of line OB to OB and shear strain is demonstrated b the change in angle CO to C O. s strain displacements ma var from point to point (and line to line) strain is defined as a measure of relative displacement. e.g. In Figure the average direct strain is given b u/o in the direction and v/oc in the direction. The average shear strain is given b ( + C )/CO. 3) nerg When a deformable bod is acted upon b forces it moves. This motion ma consist of both rigid bod motion and deformation. Rigid bod motion is associated with kinetic energ and gravitational potential energ and deformation is associated with strain potential energ. Rz UTS Mechanical ngineering

2 Strength of ngineering Materials: Strain and the Transformation of Strain INFINITSIML STRIN ) Direct (Normal) Strain Figure 3 shows a line element in a bod and the change in length of this line represents direct strain. ngineering direct strain is defined as: change in length original length L L o In general, strain is not uniform over a finite length so engineering direct strain is defined b the etension of an elemental length. i.e. L L L0 If u and v are the displacements in the and directions, the engineering direct strain is defined as: ) Shear Strain du d dv...() d L o ²L L Figure 3: Direct Strain (a) (b) (c) (d) Figure 4: Shear Strain and Rotation Figure 4 shows deformations involving the rotation of lines in an element. Figure 4a and 4b show two eamples of shear strain but in Figure 4c, although there is rotation of a line, there is no shear action. There is thus a need to distinguish between the shear rotation of a line and the rigid bod rotation of a line. This is achieved b defining shear strain b the change in angle between two orthogonal lines as shown in Figure 4d. The angle is known as tensor shear strain and is known as engineering shear strain where: =... () Note that Figure 4a represents tensor shear strain = / with a clockwise rotation of = /. Figure 4b also represents tensor shear strain = / but with a counter-clockwise rotation of = /. Shear strain is positive when there is an etension of the diagonal with positive slope. The shear strain shown in Figure 4d is positive. UTS Mechanical ngineering

3 Strength of ngineering Materials: Strain and the Transformation of Strain 3 PRINCIPL STRINS When a bod deforms there will alwas be elements defined b orthogonal aes which do not undergo shear deformation. The aes defining these elements are the principal aes and the strains in these directions are the principal strains. It can be shown that the principal strains are the maimum and minimum direct strains. e.g. In Figure 5 the square aligned with the aes (BCD) is subjected to pure shear but the diagonals of this square do not rotate. Therefore the sides of a square aligned with the aes (abcd) do not rotate and the abcd square is subjected to normal strains but not shear strain. s shown below, it is possible to use these principal strains to obtain the strains in an other direction. Transformation of Strain In Figure 6, line OB on the element aligned with the (principal) aes etends and rotates to OB. For this line, the shear and normal strain in the directions are thus given b: BC OB CB ' OB In the directions, the (principal) strains are given b: B OD B' DB The displacement of OB can therefore be written in terms of the strains in both the and the directions: C c a unstra ined strained Figure 5: Pure Shear and Principal es O d b C B' D B B D B' i.e. vector BB' where: B OD OB cos B B' DB OB sin and: Figure 6: Transformation of Strain Leading to: CB' OB OB cos OB sin BC OB OB cos sin OB sin cos cos sin...(3) cos sin... (4) UTS Mechanical ngineering

4 Strength of ngineering Materials: Strain and the Transformation of Strain 4 Substituting gives: cos, sin cos, cos sin sin cos... (5) sin... (6) s with the transformation of stress, qns. (5) and (6) are the parametric equations of a circle known as Mohr s circle of strain. Y R = O X Figure 7: Mohr s Circle of Strain From Figure 7 it can be seen that the maimum shear strain occurs on planes which are ±45 from the principal planes and that the maimum shear strain is given b: ma... (7) On the planes of maimum shear both direct strains are ( + )/. Mohr s circle also shows that for all orthogonal planes, the sum of the direct strains is a constant:... (8) UTS Mechanical ngineering

5 Strength of ngineering Materials: Strain and the Transformation of Strain 5 HOOK S LW ND LSTIC CONSTNTS ) Young s Modulus and Modulus of Rigidit Consider a bod subjected to simple tension where: 3 0 From Hooke s law for plane strain (and a linear material) the principal strains are: 3 and the maimum shear strain is: 3 ma ma From Mohr s circle for stress, the maimum shear stress is given b: ma / / so we have: and hence: G ) Dilation and Bulk Modulus / G ma G...(9) Dilation (e) is the change of volume per unit volume and bulk modulus (K) is a volume stiffness defined b uniform pressure (p)/e. i.e. e vol Vol K p e When a bod is subjected to a general state of stress it can be shown that the dilation is given b: e 3 and in a state of uniform pressure, the principal stresses are: 3 p Using Hooke s law, the dilation for this state of stress is: Hence: e K p e 3p...(0) 3 Figure 8: Simple Tension p p Figure 9: Uniform Pressure p UTS Mechanical ngineering

6 Strength of ngineering Materials: Strain and the Transformation of Strain 6 LCTRICL RSISTNC STRIN GUGS The electrical resistance, R, of a length of wire is given b R = L/ where = resistivit, L = length and = cross sectional area. If the length of the wire changes there will be a change in resistance and this propert is used to measure strain. The most common strain gauges are manufactured from foil bonded to a non-conductive backing. The gauge is then bonded to the test material. s shown in Figure, gauges are tpicall formed b a number of "loops" which are elongated (or shortened) b the direct strain. Note that the area of the foil at the ends of each "loop" is greater than the nominal area. This reduces the gauge's transverse resistance thus reducing the gauge s sensitivit to transverse strain. In addition to strain, environmental Figure : Resistance Strain Gauge effects, especiall variations in temperature, will also change a gauge's resistance. Some gauges have thermal properties that match those of the test material and these gauges do not require an allowance to be made for temperature changes. However, there is usuall a limit on the range of operating temperatures and test materials for such gauges. These limitations can be overcome b the use of a "dumm" gauge. The dumm gauge is bonded to a sample of the same material as the test material and then subjected to the same environment, but not the same loads, as the "active" gauge. The change in resistance of a gauge is therefore given b: where: dr R dr R dr R T K dr R dr change in R due to strain developed b load R dr change in R due to environmental effects R T K = "gauge factor" = (direct) strain developed b the load Tpicall, the small change in a gauge s resistance is detected b a strain gauge amplifier which measures the potential difference across a "balanced" Wheatstone Bridge. T UTS Mechanical ngineering

7 Strength of ngineering Materials: Strain and the Transformation of Strain 7 The Wheatstone Bridge The voltage detected b the bridge is given b: R V R R 4 R R 4 When R = R R 4, = 0 and the bridge is said to be balanced. In the balanced condition: R R R R R 4 r (sa) and a small change in resistance, dr, will produce a small change in voltage, d, where: d V r r dr The Quarter Bridge dr d dr 4 R R R 4 For strain gauge applications it is common to make R the active gauge, R the dumm gauge and and R 4 standard resistors with = R 4 so that r =. This arrangement is known as a quarter bridge whereb: giving: dr R K dr R T ; d V K dr ( ) R meas 4 K d V dr dr R R T ; d dr 4 0 dr T R 0 0 T K 4... (0) The strain gauge amplifier indicates strain b measuring d with the factor of 4 included in the output. The amplifier also requires an input value for the gauge factor. For most foil gauges the gauge factor is approimatel equal to. If the amplifier gauge factor setting is different to the actual gauge factor, the true strain is given b: Half and Full Bridges true meas K setting K true... () In some applications, where = -, it is possible to use two active gauges (half bridge). In this configuration: dr R K dr R T ; d V K dr ( ) R and the true strain is given b: true dr K dr R R T ; d dr 4 0 K dr T R T 0 0 K meas R 4 K setting K true... () V Figure : Wheatstone Bridge UTS Mechanical ngineering

8 Strength of ngineering Materials: Strain and the Transformation of Strain 8 If = -, it is also possible to use four active gauges (full bridge) with dr and d indicating and dr and dr 4 indicating so that: and: dr R d K dr R d V 4 K dr R true meas 4 T ; dr R dr 4 R 4 K dr R T K dr R T K T K setting K true... (3) Strain Gauge Rosettes The complete state of strain at a point can be measured b an arrangement of three strain gauges. The most common arrangement is the rectangular rosette shown in Figure 3. This rosette measures three direct strains 0, 45 and 90. If the 0 gauge is taken to be aligned with the ais then the transformation of strain equation gives: 0 45 giving ssuming plane stress conditions ( 3 = 0) and appling Hooke s law, the strains are: Figure 3: The Rectangular Rosette ; ; G Rearranging in favour of stress gives: ; Substituting the measured values ields: ; G 0 90; 90 0 ; G (4) ' cos sin sin cos UTS Mechanical ngineering