# Manipulating 2nd Order Tensor Expression: Isolating Q

• afallingbomb
In summary, the conversation discusses the notation and concept of the double dot product or double contraction between second order tensors. The question is how to manipulate the expression A=Q:B to isolate Q on one side of the equation, and the answer suggests finding a general solution for Q with arbitrary coefficients. The conversation also touches on the physical motivation behind this problem and the fact that in the most general, anisotropic case, there is no unique answer.
afallingbomb
Given the expression

$$\textbf{A} = \textbf{Q} : \textbf{B}$$

where A and B are second order tensors of rank 2 and Q is a second order tensor of rank 4.

How can I manipulate this expression to calculate for Q given A and B? In other words, isolate Q on one side of the equation. Thank you very much!

The notation you are using

A=Q:B

is not one of the many standard textbooks. Perhaps you can explain it or provide a reference to some easily available text.

Last edited by a moderator:
The double dot product between a fourth order tensor and a second order tensor is a second order tensor (now I'm referring to order as the number of subscripts... rank/order are used interchangeably in the literature). I can work it out using dyadics, but I'm not sure how to move around terms in the equation to isolate Q. There are many products to choose from and I'm not very comfortable with the rules, especially using an inverted second order tensor and perhaps (single or double?) contracting it on the left.

Alright, make it simpler. Let A,B be vectors and Q a second order tensor. So what you have then is an equation of the form

a = Qb

where a and b are vectors. Can you calculate nxn coefficients form n equations? You can't. But you can try to find a general solution for Q, with arbitrary coefficients. Is that what you mean? If so, try to do it for matrices and vectors - you will see what kind of algebra is needed.

The idea is: you have an inhomogeneous linear equation (for Q). Therefore a general solution is a sum of a particular solution of this equation and a general solution of the homegeneous one.

arkajad, thank you very much. You made me see that my problem was not well posed.

The physical motivation of this problem stemmed from my curiosity on determining a unique material property (stiffness or compliance) from a specified stress and strain state at a material point. In the most general, anisotropic case, there is no unique answer to this problem.

You are welcome!

## 1. What is a 2nd order tensor?

A 2nd order tensor is a mathematical object used to represent physical quantities that have both magnitude and direction. It is a multidimensional array that can be manipulated using mathematical operations.

## 2. How can I isolate Q from a 2nd order tensor expression?

To isolate Q from a 2nd order tensor expression, you can use the properties of transpose and inverse operations. By taking the transpose of the expression, you can swap the rows and columns, and then use the inverse operation to isolate Q on one side of the equation.

## 3. What are the practical applications of manipulating 2nd order tensor expressions?

Manipulating 2nd order tensor expressions is essential in various fields of science and engineering, such as mechanics, fluid dynamics, and electromagnetism. It allows us to solve complex problems involving physical quantities and analyze the behavior of materials under different conditions.

## 4. Are there any specific rules to follow when manipulating 2nd order tensor expressions?

Yes, there are specific rules that must be followed when manipulating 2nd order tensor expressions. These include the commutative, associative, and distributive properties, as well as the properties of transpose and inverse operations.

## 5. Can I use computer software to manipulate 2nd order tensor expressions?

Yes, there are various software programs and libraries available that can perform operations on 2nd order tensor expressions. Some popular options include MATLAB, Python's NumPy library, and Mathematica. These programs have built-in functions and tools for manipulating tensors and performing calculations.

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