# Strains and change in temperature

• Dell
In summary: The x-axis will stay the same point, and the y and z axes will move due to the strain caused by the change in temperature.
Dell
i know that the strains in a beam with no external forces on it, under a change in temperature will be

ε=αΔΤ

that i know is true for the strain on the axis of the length of the beam, but what about the height and width, if the length axis is x, what will the strains on the y and z axes be? are they 0?? common sense tells me that is the material is isotropic it must act the same in all directions therefore εxx=εyy=εzz=αΔΤ (presuming there is nothing limiting these changes in dimentions)

for the more specific case of a statically indetermined beam which is supported at both ends and cannot change its length, i know that now εxx=0 but what about yy and zz? are they now αΔΤ - σxxν/E? (according to hookes law)

if i draw a circle on a beam like this, where will it move to after deformation? on the x-axis will it stay the same point, and the y and z axis will it move or will it just get bigger (and become an ellipsoid)

Dell said:
i know that the strains in a beam with no external forces on it, under a change in temperature will be

ε=αΔΤ

that i know is true for the strain on the axis of the length of the beam, but what about the height and width, if the length axis is x, what will the strains on the y and z axes be? are they 0?? common sense tells me that is the material is isotropic it must act the same in all directions therefore εxx=εyy=εzz=αΔΤ (presuming there is nothing limiting these changes in dimentions)

Agreed.

Dell said:
for the more specific case of a statically indetermined beam which is supported at both ends and cannot change its length, i know that now εxx=0 but what about yy and zz? are they now αΔΤ - σxxν/E? (according to hookes law)

Yes, exactly.

Dell said:
if i draw a circle on a beam like this, where will it move to after deformation? on the x-axis will it stay the same point, and the y and z axis will it move or will it just get bigger (and become an ellipsoid)

Yes, it will become an ellipse.

I can confirm that your understanding of strains and change in temperature is correct. The strain on the length axis of a beam, assuming no external forces, can be calculated using the equation ε=αΔΤ, where α is the coefficient of thermal expansion and ΔΤ is the change in temperature. This holds true for isotropic materials, where the strain is equal in all directions and can be represented by εxx=εyy=εzz=αΔΤ.

In the case of a statically indeterminate beam that is supported at both ends and cannot change its length, the strain on the length axis (εxx) will be 0, as you mentioned. However, the strains on the y and z axes (εyy and εzz) will not be equal to αΔΤ, as there is a constraint on the length of the beam. Instead, these strains can be calculated using Hooke's law, which takes into account the stress (σxx), Poisson's ratio (ν), and Young's modulus (E). Therefore, the strains on the y and z axes would be εyy=εzz=αΔΤ - σxxν/E.

To answer your question about the movement of a circle drawn on a beam after deformation, it would depend on the direction of the applied force and the constraints on the beam. If the beam is free to deform in all directions, the circle would likely become an ellipsoid. However, if there are constraints on certain axes, the circle may not deform equally in all directions.

In conclusion, your understanding of strains and change in temperature is correct, and your reasoning for the strains on the y and z axes is also accurate. Keep questioning and exploring different scenarios to deepen your understanding of this concept.

## What is a strain and how does it relate to temperature?

A strain is a measure of the physical deformation that occurs in a material when it is subjected to stress. Temperature changes can cause a material to expand or contract, creating strain. This strain can have a significant impact on the material's properties and performance.

## How do different materials respond to changes in temperature?

Different materials have different coefficients of thermal expansion, which determines how much they will expand or contract when the temperature changes. Some materials, like metals, have a high coefficient of thermal expansion, while others, like ceramics, have a low coefficient of thermal expansion.

## What is thermal stress and how does it affect materials?

Thermal stress is the stress that occurs in a material due to a difference in temperature between different parts of the material. This can cause the material to expand or contract unevenly, leading to strain and potential damage. Understanding the thermal stress a material may experience is important in designing structures or products that will be exposed to temperature changes.

## How do engineers and scientists measure strain and changes in temperature?

There are various methods for measuring strain and changes in temperature, including using strain gauges, thermocouples, and thermal imaging. These tools allow scientists and engineers to accurately measure and monitor the effects of temperature changes on materials.

## What role does strain and temperature play in material failure?

Strain and temperature can both play significant roles in material failure. Excessive strain or thermal stress can cause a material to deform or break, leading to failure. Understanding how a material will respond to strain and temperature changes is crucial in preventing failure and ensuring the safety and reliability of structures and products.

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