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# Stress Strain In Elastic And Plastic Deformation Pdf

By  Leo M.

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Published: 22.05.2021

We referred to the proportionality constant between stress and strain as the elastic modulus.

## 12.6: Elasticity and Plasticity

We referred to the proportionality constant between stress and strain as the elastic modulus. But why do we call it that?

What does it mean for an object to be elastic and how do we describe its behavior? Elasticity is the tendency of solid objects and materials to return to their original shape after the external forces load causing a deformation are removed.

An object is elastic when it comes back to its original size and shape when the load is no longer present. Physical reasons for elastic behavior vary among materials and depend on the microscopic structure of the material. For example, the elasticity of polymers and rubbers is caused by stretching polymer chains under an applied force. In contrast, the elasticity of metals is caused by resizing and reshaping the crystalline cells of the lattices which are the material structures of metals under the action of externally applied forces.

The two parameters that determine the elasticity of a material are its elastic modulus and its elastic limit. A high elastic modulus is typical for materials that are hard to deform; in other words, materials that require a high load to achieve a significant strain.

An example is a steel band. A low elastic modulus is typical for materials that are easily deformed under a load; for example, a rubber band. If the stress under a load becomes too high, then when the load is removed, the material no longer comes back to its original shape and size, but relaxes to a different shape and size: The material becomes permanently deformed. The elastic limit is the stress value beyond which the material no longer behaves elastically but becomes permanently deformed.

Our perception of an elastic material depends on both its elastic limit and its elastic modulus. For example, all rubbers are characterized by a low elastic modulus and a high elastic limit; hence, it is easy to stretch them and the stretch is noticeably large. Among materials with identical elastic limits, the most elastic is the one with the lowest elastic modulus. When the load increases from zero, the resulting stress is in direct proportion to strain in the way given by Equation Conversely, the response force from the spring to an applied stretch is directly proportional to the stretch.

In the same way, the deformation of a material under a load is directly proportional to the load, and, conversely, the resulting stress is directly proportional to strain. The linearity limit or the proportionality limit is the largest stress value beyond which stress is no longer proportional to strain. Beyond the linearity limit, the relation between stress and strain is no longer linear. When stress becomes larger than the linearity limit but still within the elasticity limit, behavior is still elastic, but the relation between stress and strain becomes nonlinear.

For stresses beyond the elastic limit, a material exhibits plastic behavior. This means the material deforms irreversibly and does not return to its original shape and size, even when the load is removed. When stress is gradually increased beyond the elastic limit, the material undergoes plastic deformation. Rubber-like materials show an increase in stress with the increasing strain, which means they become more difficult to stretch and, eventually, they reach a fracture point where they break.

Ductile materials such as metals show a gradual decrease in stress with the increasing strain, which means they become easier to deform as stress-strain values approach the breaking point.

Microscopic mechanisms responsible for plasticity of materials are different for different materials. We can graph the relationship between stress and strain on a stress-strain diagram. Each material has its own characteristic strain-stress curve. In this figure, strain is a fractional elongation not drawn to scale.

When the load is gradually increased, the linear behavior red line that starts at the no-load point the origin ends at the linearity limit at point H. For further load increases beyond point H, the stress-strain relation is nonlinear but still elastic.

In the figure, this nonlinear region is seen between points H and E. Ever larger loads take the stress to the elasticity limit E, where elastic behavior ends and plastic deformation begins. Beyond the elasticity limit, when the load is removed, for example at P, the material relaxes to a new shape and size along the green line. This is to say that the material becomes permanently deformed and does not come back to its initial shape and size when stress becomes zero.

The material undergoes plastic deformation for loads large enough to cause stress to go beyond the elasticity limit at E.

The material continues to be plastically deformed until the stress reaches the fracture point breaking point. Beyond the fracture point, we no longer have one sample of material, so the diagram ends at the fracture point. For the completeness of this qualitative description, it should be said that the linear, elastic, and plasticity limits denote a range of values rather than one sharp point.

The value of stress at the fracture point is called breaking stress or ultimate stress. Materials with similar elastic properties, such as two metals, may have very different breaking stresses. For example, ultimate stress for aluminum is 2. We can make a quick estimate, based on Equation Samuel J.

Learning Objectives Explain the limit where a deformation of material is elastic Describe the range where materials show plastic behavior Analyze elasticity and plasticity on a stress-strain diagram. Contributors and Attributions Samuel J.

## Representative Stress-Strain Curve by Spherical Indentation on Elastic-Plastic Materials

Chao Chang, M. Garrido, J. Tensile stress-strain curve of metallic materials can be determined by the representative stress-strain curve from the spherical indentation. Experiment tests were performed to evaluate these constraint factors. The instrumented indentation technique consists of applying load to the sample by means of an indenter of known geometry, while the applied load and the penetration depth of the indenter are recorded simultaneously during a loading and unloading cycle.

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Functionally graded material shafts are the main part of many modern rotary machines such as turbines and electric motors. The purpose of this study is to present an analytical solution of the elastic-plastic deformation of functionally graded material hollow rotor under a high centrifugal effect and finally determine the maximum allowed angular velocity of a hollow functionally graded material rotating shaft. Introducing non-dimensional parameters, the equilibrium equation has been analytically solved. It is shown that material variation has a considerable effect on the stress and strain components and radial displacement. Considering variable density and yield stress causes yielding onset from inner, outer, or simultaneously from both inner and outer rotor shaft radius in contrast to earlier researches that modulus of elasticity was the only variable.

A model of a rigid body is an idealized example of an object that does not deform under the actions of external forces. It is very useful when analyzing mechanical systems—and many physical objects are indeed rigid to a great extent. The extent to which an object can be perceived as rigid depends on the physical properties of the material from which it is made. For example, a ping-pong ball made of plastic is brittle, and a tennis ball made of rubber is elastic when acted upon by squashing forces.

Stress is plotted on the Y-Axis and Strain is plotted on the X-axis. In material science and mechanical engineering, the stress-strain curve is widely used to understand the strength, deformation, and failure criteria of any material. In this article, we will explore details about the stress-strain curve. In that instrument, a force on the standard specimen is increased till its failure and a plotter keeps recording the stress and strain. The Yield Strength of a material is the maximum stress after which the elongation becomes plastic and permanent deformation starts. Once the yield strength of a material is reached, large deformation occurs with very little increase in the applied load. The material will regain its shape once the stress is removed if the yield point is not reached.

Elasticity is a measure of how much an object deforms strain when a given stress force is applied. If a bulldozer pushes a car into a wall, the car will not move once it hits the wall, but it will noticeably change shape. A change in shape due to the application of a force is a deformation. Even very small forces are known to cause some deformation. For small deformations, two important characteristics are observed. First, the object returns to its original shape when the force is removed—that is, the deformation is elastic for small deformations. Elasticity is a measure of how difficult it is to stretch an object.

Chapter 6. Mechanical Properties of Metals. Mechanical Properties refers to the behavior of material when external forces are applied. Stress and strain ⇒.

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