# Simple stress and strain relationship pdf

### Stress Strain Curve Explanation | Stages | Mild Steel | Engineering Intro

representing the variation of the stress-strain behavior with temperature. A simple two-stage procedure is used for fitting the equations to the experimental data. Slope of stress strain plot (which is proportional to the elastic modulus) simple torsion test. M. M. □ Special relations for isotropic materials: 2(1 + ν). E. G. Stress – Strain Relations: The Hook's law, states that within the elastic limits the describe the entire stress – strain curve with simple mathematical expression.

The normal strain of a body is generally expressed as the ratio of total displacement to the original length.

**Stress & Strain Curve of ductile material in tension - GATE Lectures - ME, CE**

It is of two types: Shear strain Note 1: The angle is radians, not degrees. The volume of the solid is not changed by shear strain. Stress-Strain Relationship The stress-strain diagram is shown in the figure.

In brittle materials, there is no appreciable change in the rate of strain. There is no yield point and no necking takes place. In figure athe specimen is loaded only upto point A, when load is gradually removed the curve follows the same path AO and strain completely disappears. Such a behaviour is known as the elastic behaviour.

In figure bthe specimen is loaded upto point B beyond the elastic limit E. When the specimen is gradually loaded the curve follows path BC, resulting in a residual strain OC or permanent strain. Comparison of engineering stress and the true stress-strain curves shown below: The true stress-strain curve is also known as the flow curve.

True stress-strain curve gives a true indication of deformation characteristics because it is based on the instantaneous dimension of the specimen.

In engineering stress-strain curve, stress drops down after necking since it is based on the original area. In true stress-strain curve, the stress however increases after necking since the cross sectional area of the specimen decreases rapidly after necking.

The flow curve of many metals in the region of uniform plastic deformation can be expressed by the simple power law. The coefficient G is called the shear modulus of elasticity or modulus of rigidity. Properties of Materials Some properties of materials which judge the strength of materials are given below: Elasticity is the property by virtue of which a material is deformed under the load and is enabled to return to its original dimension when the load is removed.

Plasticity is the converse of elasticity. A material in the plastic state is permanently deformed by the application of load and it has no tendency to recover.

The characteristic of the material by which it undergoes inelastic strains beyond those at the elastic limit is known as plasticity. Ductility is the characteristic which permits a material to be drawn out longitudinally to a reduced section, under the action of a tensile force large deformation.

## Stress–strain analysis

Brittleness implies the lack of ductility. Instead, one assumes that the stresses are related to strain of the material by known constitutive equations. By Newton's laws of motionany external forces that act on a system must be balanced by internal reaction forces, [1] or cause the particles in the affected part to accelerate.

In a solid object, all particles must move substantially in concert in order to maintain the object's overall shape. It follows that any force applied to one part of a solid object must give rise to internal reaction forces that propagate from particle to particle throughout an extended part of the system.

With very rare exceptions such as ferromagnetic materials or planet-scale bodiesinternal forces are due to very short range intermolecular interactions, and are therefore manifested as surface contact forces between adjacent particles — that is, as stress. In principle, that means determining, implicitly or explicitly, the Cauchy stress tensor at every point. The external forces may be body forces such as gravity or magnetic attractionthat act throughout the volume of a material; [3] or concentrated loads such as friction between an axle and a bearingor the weight of a train wheel on a railthat are imagined to act over a two-dimensional area, or along a line, or at single point.

The same net external force will have a different effect on the local stress depending on whether it is concentrated or spread out. Types of structures[ edit ] In civil engineering applications, one typically considers structures to be in static equilibrium: In mechanical and aerospace engineering, however, stress analysis must often be performed on parts that are far from equilibrium, such as vibrating plates or rapidly spinning wheels and axles.

In those cases, the equations of motion must include terms that account for the acceleration of the particles. In structural design applications, one usually tries to ensure the stresses are everywhere well below the yield strength of the material. In the case of dynamic loads, the material fatigue must also be taken into account.

However, these concerns lie outside the scope of stress analysis proper, being covered in materials science under the names strength of materialsfatigue analysis, stress corrosion, creep modeling, and other. Experimental methods[ edit ] Stress analysis can be performed experimentally by applying forces to a test element or structure and then determining the resulting stress using sensors. In this case the process would more properly be known as testing destructive or non-destructive. Experimental methods may be used in cases where mathematical approaches are cumbersome or inaccurate.

Special equipment appropriate to the experimental method is used to apply the static or dynamic loading. There are a number of experimental methods which may be used: Tensile testing is a fundamental materials science test in which a sample is subjected to uniaxial tension until failure.

- Stress Strain Curve Explanation
- Stress and Strain Study Notes for Mechanical Engineering

The results from the test are commonly used to select a material for an application, for quality controlor to predict how a material will react under other types of forces. Properties that are directly measured via a tensile test are the ultimate tensile strengthmaximum elongation and reduction in cross-section area.

From these measurements, properties such as Young's modulusPoisson's ratioyield strengthand the strain-hardening characteristics of the sample can be determined.

Strain gauges can be used to experimentally determine the deformation of a physical part. A commonly used type of strain gauge is a thin flat resistor that is affixed to the surface of a part, and which measures the strain in a given direction.

### NPTEL :: Mechanical Engineering - Strength of Materials

From the measurement of strain on a surface in three directions the stress state that developed in the part can be calculated. Neutron diffraction is a technique that can be used to determine the subsurface strain in a part.

Stress in plastic protractor causes birefringence. The photoelastic method relies on the fact that some materials exhibit birefringence on the application of stress, and the magnitude of the refractive indices at each point in the material is directly related to the state of stress at that point. The stresses in a structure can be determined by making a model of the structure from such a photoelastic material. Dynamic mechanical analysis DMA is a technique used to study and characterize viscoelastic materials, particularly polymers.

The viscoelastic property of a polymer is studied by dynamic mechanical analysis where a sinusoidal force stress is applied to a material and the resulting displacement strain is measured. For a perfectly elastic solid, the resulting strains and the stresses will be perfectly in phase.

For a purely viscous fluid, there will be a 90 degree phase lag of strain with respect to stress. Viscoelastic polymers have the characteristics in between where some phase lag will occur during DMA tests. Mathematical methods[ edit ] While experimental techniques are widely used, most stress analysis is done by mathematical methods, especially during design. Differential formulation[ edit ] The basic stress analysis problem can be formulated by Euler's equations of motion for continuous bodies which are consequences of Newton's laws for conservation of linear momentum and angular momentum and the Euler-Cauchy stress principletogether with the appropriate constitutive equations.

These laws yield a system of partial differential equations that relate the stress tensor field to the strain tensor field as unknown functions to be determined. Solving for either then allows one to solve for the other through another set of equations called constitutive equations. Both the stress and strain tensor fields will normally be continuous within each part of the system and that part can be regarded as a continuous medium with smoothly varying constitutive equations.

The external body forces will appear as the independent "right-hand side" term in the differential equations, while the concentrated forces appear as boundary conditions. An external applied surface force, such as ambient pressure or friction, can be incorporated as an imposed value of the stress tensor across that surface. External forces that are specified as line loads such as traction or point loads such as the weight of a person standing on a roof introduce singularities in the stress field, and may be introduced by assuming that they are spread over small volume or surface area.

The basic stress analysis problem is therefore a boundary-value problem. Elastic and linear cases[ edit ] A system is said to be elastic if any deformations caused by applied forces will spontaneously and completely disappear once the applied forces are removed. The calculation of the stresses stress analysis that develop within such systems is based on the theory of elasticity and infinitesimal strain theory.

When the applied loads cause permanent deformation, one must use more complicated constitutive equations, that can account for the physical processes involved plastic flowfracturephase changeetc.

That is, the deformations caused by internal stresses are linearly related to the applied loads. In this case the differential equations that define the stress tensor are also linear. Linear equations are much better understood than non-linear ones; for one thing, their solution the calculation of stress at any desired point within the structure will also be a linear function of the applied forces.

For small enough applied loads, even non-linear systems can usually be assumed to be linear. Built-in stress preloaded [ edit ] Example of a Hyperstatic Stress Field.