# Homework help: anisotropic elasticity equations

• santamartina
In summary, the conversation discusses the topic of anisotropic elasticity and the use of Lekhnitskii's book for reference. The speaker is having trouble reconciling the general equations of elasticity for an orthotropic body with their own calculations. They are seeking help in identifying their mistake and suggest using the continuity equation to eliminate any cross derivatives.

## Homework Statement

Hello, I'm studying anisotropic elasticity. One of the books I'm using is Lekhnitskii's. The book presents the general equations of the theory of elasticity for an orthotropic body as follows:

## The Attempt at a Solution

However, when I combine the strain-displacement equations and the Hook's Law and then replace them into the equilibrium equations I cannot get the same expression. My reasoning looks like this (for example, the first equation of the system):

A11.Uxx + A12.Vyx + A13.Wzx + A66.(Uyy + Vyx) + A55.(Wzx + Uzz) = 0

Obviously, I'm doing something wrong. What is my mistake?

Many thanks for your kind help.

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Maybe they are able to eliminate the cross derivatives by combining the equations with the continuity equation.

## 1. What is anisotropic elasticity?

Anisotropic elasticity is a branch of mechanics that studies the behavior of materials that have different mechanical properties in different directions. This means that the material's response to external forces is not the same in all directions.

## 2. What are the equations used in anisotropic elasticity?

The equations used in anisotropic elasticity are known as the anisotropic elasticity equations. These equations describe the relationship between stress and strain in a material that exhibits anisotropic behavior. They are typically represented by matrices and can be solved using various numerical methods.

## 3. How is anisotropic elasticity different from isotropic elasticity?

In isotropic elasticity, the material has the same mechanical properties in all directions, while in anisotropic elasticity, the properties vary in different directions. This means that the equations used to describe the behavior of these materials are different, and anisotropic materials require more complex mathematical models to accurately predict their behavior.

## 4. What are some real-world applications of anisotropic elasticity?

Anisotropic elasticity is used in various industries, including aerospace, automotive, and engineering, to design and analyze materials with directional properties. It is also used in the development of composites, such as carbon fiber, which have anisotropic behavior and are widely used in high-performance applications.

## 5. How can anisotropic elasticity equations be solved?

Anisotropic elasticity equations can be solved using various numerical methods, such as the finite element method, finite difference method, and boundary element method. These methods involve breaking down the equations into smaller, more manageable parts and using computational techniques to solve them. Advanced software and computer simulations are often used to solve anisotropic elasticity problems.