Constantes de matériau

Most engineering design today uses the theory of elasticity and several material constants to estimate the performance criteria of a structure.

In contrast to the production of cars, for example, where millions of identical copies are made, an extensive prototype testing is possible. Each civil engineering structure is unique, and its design vastly relies on an analytical modeling and different material constants.

The two most common material constants used in civil engineering design are the Modulus of Elasticity, which relates stress to strain, and Poisson's Ratio, which is the ratio of lateral to longitudinal strains.

In this video, we will measure stress and strain with equipment typically found in a construction materials laboratory, and use these quantities to determine the material constants of an aluminum bar.

The most common model used for analysis is linear elasticity, or Hooke's Law, which postulates that the applied force is directly proportional to deformation.

In engineering, stress is defined as the force per unit area, while strain is defined as the change in dimension when subjected to a force, divided by the original magnitude of that dimension. According to Hooke's Law, stress is proportional with strain, and the constant of proportionality is the constant of elasticity. If we can measure the force, the strain, and the original area, we can find E. This is the particular case of a unit directional load.

Let's look now at the general case where a piece of structure is subjected to 3D loads. Considering an X, Y, Z coordinate system, at any given point the solid body is subjected to three normal components and three sheer components of stress. Breaking the equations of equilibrium for forces and moments along all axes results in a series of equations for the normal strain and the sheer strain.

Six equations of this type, three for normal strains and three for sheer strains, are needed to establish the global deformations. These equations contain three constants, the Modulus of Elasticity (E), Poisson's Ratio (μ), and the Sheer Modulus (G). The Sheer Modulus is defined as the change in angular deformation given the sheer stress or surface traction. Poisson's Ratio is defined as the ratio of transversal to longitudinal strain. Since G can be expressed using E and μ, only two of the three constants need to be measured in order to define all three.

For the state of stress, represented in the X, Y, Z coordinate system, there is an equivalent representation on a new coordinate system of principal axes one, two, and three, where there are no sheer stresses. The normal stresses in this particular system are called Principal Normal Stresses. Among these, there is a minimum and respectively maximum principal stress acting on any plane. The state of stress and strain on a surface is determined if at least three independent strain measurements are made.

In the laboratory, a rosette strain gauge composed of three strain gauges aligned at 45 degrees to one another, is used to measure strain in three different directions. From here, the complete state of stress on a surface can be defined using Mohr's circle, to calculate maximum and minimum principal strains and the angle between the measured strains and the principal strains.

In this experiment, we will use a simple cantilever beam instrumented with strain gauges to illustrate the concepts of principal strains and stresses and measure the Young's Modulus and the Poisson's Ratio.

Obtain a regular aluminum bar, of dimensions 12 inches by 1 inch by 1/4 inch. An aluminum 6061-T6 or stronger is recommended.

Drill a hole at one end of the beam to serve as a loading point, and mark a location on the beam, about eight inches from the center of the hole, where the strain gauges will be installed. Measure area of the bar carefully, using calipers. Perform three replicates in three different locations, to obtain a good average of the dimensions. From these measurements, calculate the moment of inertia of the bar.

Next, obtain a rosette strain gauge with a sensing grid of approximately 1/4 inch long by 1/8 inch wide on each gauge. Note the calibration factor or gauge factor. Mark the location where the strain gauge will be installed. Then, degrease this area, obtain a very smooth surface by sanding with progressively finer grades of sandpaper, clean the surface with a neutralizer. Mix the epoxy components and install the strain gauges. Installation and glue curing procedures should follow manufacturer specifications.

Make sure to test the resistance of the gauges using an ohm meter and their current leakage to the specimen bar before proceeding. A micro-measurements 1300 strain gauge tester will be used herein for this purpose. Repeat these operations to install a single strain gauge longitudinally on the surface and directly below the strain gauge rosette.

Insert the specimen into a secure vice, that ensures the aluminum beam will behave as a cantilever beam. Now, connect the strain gauges to a recording device. Make sure that the wiring is correct as per the strain indicator instructions, and that you know which channel corresponds to each strain gauge.

Then, enter the appropriate gauge factors for each gauge in the indicator. If possible, calibrate the strain gauge outputs and the strains in the indicator. Make sure to record initial load and strains. Now, slowly apply seven increments of 0.5 kilograms of load at the beam tip. Pause at every step and allow measurements to stabilize before recording readings. Next, slowly apply eight decrements of 0.5 kilograms. Make sure to pause at every step and allow measurements to stabilize before recording readings.

Raw data shown in the table consists of the load step number, applied loads, the strain from the top rosette strain gauge, and the strain from the single bottom strain gauge. The initial and final load steps will not be used in the calculations, as the readings are small and will not produce accurate results.

Next, using the strain values from the top rosette strain gauge, compute the principal strains, the angle of inclination, and the Poisson's ratio as the ratio of the maximum to minimum principal strain. Plotting the maximum and minimal principal strains corresponds to plotting the longitudinal and transfer strains; and thus, the slope of this line corresponds to Poisson's ratio. The value obtained is very close to the generally accepted value of 0.3, and the R squared measure indicates very good linearity.

A good physical interpretation of the rosette strain gauge data can be gained from plotting the principal strains on a Mohr's circle. Note that the three measurements shown here for the case of the maximum load of 9.93 pounds corresponds to three points in the circles at 90 degrees to one another, starting at an angle of about 27.4 degrees, counter-clockwise from the X axis.

Next, from the load values we calculate the bending stresses. The Young's Modulus is given by the ratio of the stress of the maximum principal strain, which we had calculated in Table 2. Now, plot the stress versus strain, and compute the slope of this line, which corresponds to Young's Modulus. The value obtained is very close to the theoretical value of 10,000 KSI. Finally, draw the Mohr's circle for plane stress.

Material constants are used together with theoretical models to improve and optimize the design of many engineering products, from consumer goods to aircrafts and skyscrapers.

For waterproofing the facade of a brick building, the engineer must determine, among other factors, how much force the mortar between bricks can resist before it cracks. Different analytical models and material constants are employed in order to decide what type of mortar should be chosen for construction, based on the load that the facade will likely see.

In the design of a soda can, a manufacturer must minimize the thickness of the aluminum wall in order to decrease the costs. Before moving to the prototype phase, theoretical studies taking into account the material properties may be performed in order to optimize the can shape and dimensions.

You've just watched JoVE's Introduction to Material Constants. You should now understand the basics of the theory of elasticity. You should also know how to measure the Modulus of Elasticity and the Poisson's ratio, two fundamental material constants widely used for practical engineering applications.

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