# Stress and strain relationship equation

### Stress–strain curve - Wikipedia

Hooke's law is the proportionality gives the elastic stress-strain relationship. Holloman-Ludwig equation also gives a relationship between. Stress – Strain Relations: The Hook's law, states that within the elastic limits the stress is These equation expresses the relationship between stress and strain . A member is under axial loading when a force acts along its axis. The internal force is normal to the plane of the section and the corresponding.

The linear portion of the curve is the elastic region and the slope is the modulus of elasticity or Young's Modulus Young's Modulus is the ratio of the compressive stress to the longitudinal strain. Many ductile materials including some metals, polymers and ceramics exhibit a yield point. Plastic flow initiates at the upper yield point and continues at the lower one. At lower yield point, permanent deformation is heterogeneously distributed along the sample.

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

The deformation band which formed at the upper yield point will propagate along the gauge length at the lower yield point. The band occupies the whole of the gauge at the luders strain.

• Stress Strain Relation
• Stress-Strain Relationship

Beyond this point, work hardening commences. The appearance of the yield point is associated with pinning of dislocations in the system. Specifically, solid solution interacts with dislocations and acts as pin and prevent dislocation from moving. Therefore, the stress needed to initiate the movement will be large. As long as the dislocation escape from the pinning, stress needed to continue it is less.

After the yield point, the curve typically decreases slightly because of dislocations escaping from Cottrell atmospheres. As deformation continues, the stress increases on account of strain hardening until it reaches the ultimate tensile stress.

Until this point, the cross-sectional area decreases uniformly and randomly because of Poisson contractions. The actual fracture point is in the same vertical line as the visual fracture point.

However, beyond this point a neck forms where the local cross-sectional area becomes significantly smaller than the original. If the specimen is subjected to progressively increasing tensile force it reaches the ultimate tensile stress and then necking and elongation occur rapidly until fracture.

Strain is a unitless measure of how much an object gets bigger or smaller from an applied load. Shear strain occurs when the deformation of an object is response to a shear stress i. Mechanical Behavior of Materials Clearly, stress and strain are related. Stress and strain are related by a constitutive law, and we can determine their relationship experimentally by measuring how much stress is required to stretch a material.

This measurement can be done using a tensile test. In the simplest case, the more you pull on an object, the more it deforms, and for small values of strain this relationship is linear. This linear, elastic relationship between stress and strain is known as Hooke's Law. If you plot stress versus strain, for small strains this graph will be linear, and the slope of the line will be a property of the material known as Young's Elastic Modulus.

## Mechanics of Materials: Strain

This value can vary greatly from 1 kPa for Jello to GPa for steel. In this course, we will focus only on materials that are linear elastic i.

From Hooke's law and our definitions of stress and strain, we can easily get a simple relationship for the deformation of a material. Intuitively, this exam makes a bit of sense: If the structure changes shape, or material, or is loaded differently at various points, then we can split up these multiple loadings using the principle of superposition.

Generalized Hooke's Law In the last lesson, we began to learn about how stress and strain are related — through Hooke's law. But, up until this point we've only considered a very simplified version of Hooke's law: In this lesson, we're going to consider the generalized Hooke's law for homogenousisotropicand elastic materials being exposed to forces on more than one axis. First things first, even just pulling or pushing on most materials in one direction actually causes deformation in all three orthogonal directions.

Let's go back to that first illustration of strain. This time, we will account for the fact that pulling on an object axially causes it to compress laterally in the transverse directions: This property of a material is known as Poisson's ratio, and it is denoted by the Greek letter nu, and is defined as: Or, more mathematically, using the axial load shown in the above image, we can write this out as an equation: Since Poisson's ratio is a ratio of two strains, and strain is dimensionless, Poisson's ratio is also unitless.

### Stress And Strain - Relation of a Material | Stress Strain Curve

Poisson's ratio is a material property. Poisson's ratio can range from a value of -1 to 0. For most engineering materials, for example steel or aluminum have a Poisson's ratio around 0.

Incompressible simply means that any amount you compress it in one direction, it will expand the same amount in it's other directions — hence, its volume will not change.

Physically, this means that when you pull on the material in one direction it expands in all directions and vice versa: We can in turn relate this back to stress through Hooke's law. This is an important note: In reality, structures can be simultaneously loaded in multiple directions, causing stress in those directions.

## Stress–strain curve

A helpful way to understand this is to imagine a very tiny "cube" of material within an object. That cube can have stresses that are normal to each surface, like this: And, as we now known, stress in one direction causes strain in all three directions.

So now we incorporate this idea into Hooke's law, and write down equations for the strain in each direction as: These equations look harder than they really are: