Stress strain relationship and elastic constants pdf to jpg

Mechanics of Elastic Solids - Lesson - TeachEngineering Elasticity. Constitutive Relation. The descriptions of stress and strain individually Elasticity. Linear Isotropic Elasticity: the Bulk Modulus. Hooke's Law or. Changes in Length—Tension and Compression: Elastic Modulus; Sideways Stress: Shear Modulus; Changes in Volume: Bulk Modulus. Young's modulus or Young modulus is a mechanical property that measures the stiffness of a solid material. It defines the relationship between stress (force per unit area) and strain In practice, Young's moduli are given in megapascals ( MPa or N/mm2) or .. "Ultrasonic Study of Osmium and Ruthenium" (PDF). Platinum.

Modulus of Elasticity is a representation of the stiffness of a material that behaves elastically. Modulus of elasticity is explained mathematically through the following equation: The standard unit for stress is Pascals and strain is unitless, so the unit for modulus of elasticity is also Pascals Pa.

Young's modulus - Wikipedia

Modulus of elasticity is how engineers characterize elastic material behavior. This is useful for knowing how materials behave, material selection for device design, and calculating stress in a material since it is easier to measure deformation than it is to determine the exact force on a material. The modulus of elasticity is similar to the stiffness, k, of springs.

In-Class Examples With the students, review the three examples included in the attached presentation file, and then have students individually complete the worksheet. Then continue presenting the material, below, followed by the homework handout. Measuring Modulus of Elasticity Modulus of elasticity and yield stress are two common material properties that can be calculated from performing tensile tests with a mechanical testing system.

Mechanical testing systems are designed so that engineers can clamp the selected material between two grips. The bottom grip remains stationary while the top grip moves up at a specified displacement rate. The testing system records the amount of force that it takes to stretch the material and the corresponding displacement of the grips. As described earlier, engineers measure the initial cross sectional area of a specimen and the initial length between the grips so that they can calculate stress from the force data and strain from the displacement data.

This data is used to create what engineers call stress-strain diagrams. So that engineers can select materials for their devices that meet the design requirements, many materials have been tested numerous times and their properties published in materials handbooks.

For material properties to be published so that all engineers can use them without testing the materials themselves, a set of standards was put into practice to ensure that testing is conducted in the same manner for all materials.

One important part of these standards is the geometry of the test specimen.

Young's modulus

If the test specimen is a rectangle, then often a higher stress state is experienced at the grips. This is due to the grips and cannot be avoided with this type of test. It results in the specimen breaking at the grips such that engineers cannot accurately calculate the stress that caused the material to break. To solve this problem, engineers developed a different geometry that eliminates this source of error: The dog bone shape is larger at the top and bottom where the grips attach to the specimen and the cross sectional area is smaller between the grips.

The smaller cross sectional area allows the stress to be concentrated in the center of the specimen so that the effects of the grips do not interfere with testing.

Using this specimen shape, engineers can be sure that the stress measured by the testing system is the stress state actually experienced by the material. Engineers use dog bone-shaped specimens for tensile testing and analyze the resulting stress-strain diagram for useful information about the material behavior. Briggs, ITL Program, College of Engineering, University of Colorado Boulder Figure 1 shows an example dog bone-shaped specimen and a typical stress-strain diagram for a ductile, elastic material such as steel.

Engineers gather much useful information from this diagram to learn about the behavior of a material, including its modulus of elasticity and yield stress.

Let's look more closely at this graph and explain the most important features that engineers use and record.

The elastic range is defined by the linear portion of the stress-strain curve. The slope of this line is defined by the modulus of elasticity. If a material is stretched only in this region and then the force is released, then the material follows the same line down while being unloaded. The material thus returns to its original dimensions.

Elasticity - Stress and Strain - Physics LibreTexts

Again, this is seen with a spring; when it is stretched and then released, it returns to its original configuration. The plastic range is the portion of the diagram to the right of the elastic region; this is the region of permanent deformation. If a material is stretched into this region, then it starts to permanently deform. When the force is released, the unloading curve is linear with the same slope as the elastic range, so it does not following its loading curve. The line never reaches the origin again and thus the material never returns to its original configuration. The amount of permanent deformation is defined by the intersection along the x-axis of the unloading curve. If you have ever stretched a spring too much perhaps while playing with a slinkyyou have experienced this plastic region and have observed permanent deformation.

No matter how hard you try, the spring will not go back to its original configuration. Yield stress is the minimum stress that causes permanent deformation. Since permanent deformation may define failure of a component in a device, engineers want to design devices so that no component is exposed to forces that produce the yield stress in its material. If this happens, it leads to failure; this is why engineers often use the yield stress value when designing systems.

Ultimate tensile stress is the maximum stress that a material can withstand; it is the maximum point on the diagram. Directional materials[ edit ] Young's modulus is not always the same in all orientations of a material.

Most metals and ceramics, along with many other materials, are isotropicand their mechanical properties are the same in all orientations. However, metals and ceramics can be treated with certain impurities, and metals can be mechanically worked to make their grain structures directional.

Lecture 30 - Elastic Stress Strain Relationship

These materials then become anisotropicand Young's modulus will change depending on the direction of the force vector. Anisotropy can be seen in many composites as well.

5.3: Elasticity - Stress and Strain

For example, carbon fiber has a much higher Young's modulus is much stiffer when force is loaded parallel to the fibers along the grain. Other such materials include wood and reinforced concrete. Engineers can use this directional phenomenon to their advantage in creating structures. For a generally anisotropic material there are 81 independent elastic constants. With additional stress symmetry the number of independent elastic constants reduces to Further, with strain symmetry this number reduces to A hyperelastic material with stress and strain symmetry has 21 independent elastic constants.

The material with 21 independent elastic constants is also called as anisotropic or aelotropic material. Further reduction with one plane of material symmetry gives 13 independent elastic constants.

These materials are known as monoclinic materials. Additional orthogonal plane of symmetry reduces the number of independent elastic constants to 9. These materials are known as orthotropic materials. Further, if a material has two orthogonal planes of symmetry then it is also symmetric about third mutually perpendicular plane.