Nature of displacement and the deformation tensor

The displacement is the vector difference between P and P'. The deformation gradient tensor is the derivative of the displacement. In summary, the deformation tensor is the derivative of displacement because it maps the change in position from P to P' to the change in position from Q to Q', making it a more accurate representation of the deformation. This also allows for the calculation of strain, which is the change in length between two points, such as PQ and P'Q'. The choice of using the derivative instead of the displacement itself is based on scientific reasoning and provides a more precise measure of the deformation.
  • #1
mohammed El-Kady
32
2
If we have two points P and Q in undeformed material and after deformation they become P' and Q'. The deformation tensor is the derivative of the displacement. What is the displacement? vector PP'? or the change from PQ to P'Q'?
is the second question is the strain "change in length".
Why the deformation is the derivative of displacement, why not the displacement itself? does it has scientific reason or just choice?
 
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  • #2
mohammed El-Kady said:
If we have two points P and Q in undeformed material and after deformation they become P' and Q'. The deformation tensor is the derivative of the displacement. What is the displacement? vector PP'? or the change from PQ to P'Q'?
is the second question is the strain "change in length".
Why the deformation is the derivative of displacement, why not the displacement itself? does it has scientific reason or just choice?
The deformation gradient tensor maps the differential vector between P and Q into the differential vector between P'and Q'.
 

1. What is the nature of displacement in relation to the deformation tensor?

The nature of displacement refers to the change in position or location of a point or object in a system. The deformation tensor is a mathematical representation of this displacement, showing how the shape and size of an object changes as it is subjected to external forces.

2. How is the deformation tensor calculated?

The deformation tensor is calculated by taking the partial derivatives of the displacement vector field with respect to the coordinates of the system. This results in a matrix of values that describes the amount and direction of deformation at each point in the system.

3. What is the significance of the deformation tensor in mechanics and engineering?

The deformation tensor is a crucial tool in mechanics and engineering as it allows for the analysis and prediction of how a material will behave under external forces. It is used in the study of elasticity, plasticity, and other material properties to design and optimize structures and machines.

4. Can the deformation tensor be used to model non-linear deformations?

Yes, the deformation tensor can be used to model non-linear deformations, such as in the case of large deformations or complex material behaviors. This is done by incorporating higher-order terms and non-linear equations into the calculation of the tensor.

5. How does the deformation tensor relate to strain and stress?

The deformation tensor is directly related to strain and stress, as it provides the mathematical framework for calculating these quantities. Strain is a measure of the amount of deformation in a material, while stress is a measure of the internal forces that cause this deformation. The deformation tensor helps to connect these two quantities and understand their relationship in a given system.

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