1. Objectives Experimental determination of the elastic modulus E and Poisson's ratio v of an isotropic material. Experiment 4 Determining Elastic Constants of Isotropic Materials 2. Reference: Morrow & Kokernak, Statics and Strength of Materials, 7th Ed, pp. 240-243 & 255 3. Background Information To measure deformations, instruments called strain gauges are often used. Various types of strain gauges have been developed and applied: mechanical, optical, pneumatic, etc., but the most common are electric strain gauges. Strain gages are firmly glued to the outer surface of the sample. When the sample is deformed, the strain gages are also deformed, and as their electrical resistance changes to measure strain. Wire and foil strain gauges are widely used in material testing and monitoring. Strain-sensitive elements of wire strain gages are made in an accordion-shaped loop (to maximize length in a small area). Strain-sensitive elements of foil strain gages use a similar configuration but are made of thin foil. Foil strain gauges are generally considered more accurate than wire gauges for most applications. Structurally, strain gages (Figure 4.2) consist of a thin varnish, paper or metal substrate on which a strain-sensitive element is fixed with glue. The strain gauge is connected to the strain gauge using output conductors soldered to it. 3 Figure 4.1 - Strain Gauge Scheme: 1 - Substrate, 2 – Strain-Sensing Element, 3 – Output Conductors - For this experiment, two foil type strain gages are glued to the sample (Figure 4.2): one oriented in the longitudinal direction and one perpendicularly in the transverse direction. a A F y F Z X 2 Figure 4.2 - The Location of the Strain Gauges on the Rod: 1 - Rod, 2 - Transverse Strain Gauge, 3 – Longitudinal Strain Gauge The strain gauge is most often assembled using a bridge circuit. When using strain gages, the issue of temperature compensation is important since a change in temperature causes a change in the electrical resistance of the strain gage. Temperature compensation is carried out by introducing a compensation strain gage (R2) into the corresponding arm of the bridge circuit. The compensation strain gage is glued to a load-free bar of material of the same grade as the material of the test object and is placed in the same temperature conditions as the working strain gage. A schematic diagram of the strain gauge circuit is shown in Figure 4.4. R R 1 0 2 Rr R Figure 4.3 - Scheme of a Bridge Circuit for Measuring Strain Using Strain Gauges: R₁ - Resistance of the Working Strain Gage; R₂ – Resistance of the Compensation Strain Gage; R3, R4 -Resistances Built into the Device; S- Slide-wire Scale; G - Galvanometer (an instrument for detecting and measuring small electric currents) R The laboratory is equipped with a device for calibrating a strain gauge. This apparatus is used to setup the Digital Tensometer, using a known amount of strain caused by known loads (applied weights) acting on a cantilever beam of known dimensions (Figure 4.4). r 3 F Figure 4.4 - Device for Calibrating the Deformations Meter Scale Division: 1 – Beam, 2 – Strain Gage, 3 – Deflection Indicator Hooke's Law determines the proportionality between stresses and elastic strains: σ₂ = E.E., (4.1) where E is the Modulus of Elasticity or Young's Modulus. Hooke's law is valid up to stress opr, called the material Proportional Limit or Limit of Proportionality. Under axial loading of the rod, longitudinal deformation occurs, but also an associated transverse deformation take place. When the sample is loaded, the ratio between the transverse and longitudinal deformations is constant (up to the limit of proportionality of the material). The absolute value of the ratio of transverse to longitudinal strain is called the Poisson's Ratio or transverse strain coefficient: Calibration of Digital Tensometer V= E trans &long 1 E To determine the strain using a strain gauge, it is necessary to know the scale division value of the Digital Tensometer. The process of determining the value of dividing the scale of the device is called its calibration. Calibration is carried out using a calibration beam (Figure 4.5). This is a wedge-shaped cantilever beam with a constant thickness. The curvature of the elastic line of the beam can be determined by the following relationship: Mx р E.1. = (4.2) 2 (4.3) where p - the radius of curvature of the arc of the elastic line of the beam, Mx - the bending moment, E – Young's modulus, Ix – the moment of inertia of the section relative to the x axis. When calibrating, the strain &z can be determined using a mechanical deflection meter, a diagram of which is shown in Figure 4.5. The deflection meter allows you to measure the deflection arrow fof the calibration beam at the base a.