# Boussinesq Elasticity Problem Derivation

• afallingbomb
In summary, the Boussinesq Elasticity Problem is a mathematical model used to describe the behavior of a linearly elastic solid subject to small and slowly varying loads. It is derived from the Navier's equations of motion and the Hooke's law, and takes into account the effects of both normal and tangential stresses on the solid. The resulting equations can be solved to determine the displacement and stress fields within the solid, providing valuable insights into its mechanical properties. The Boussinesq Elasticity Problem has applications in various fields such as geomechanics, soil mechanics, and structural analysis.
afallingbomb
Where can I find the derivation of the Boussinesq problem of elasticity (point load on an isotropic elastic infinite half-plane)? Besides the original 19th century paper, of course. Thanks!

afallingbomb said:
Where can I find the derivation of the Boussinesq problem of elasticity (point load on an isotropic elastic infinite half-plane)? Besides the original 19th century paper, of course. Thanks!

Go to:

http://en.wikipedia.org/wiki/Linear_elasticity

2/3rds of the way down you find,

Boussinesq-Cerruti solution - point force at the origin of an infinite isotropic half-space,

click on "show" to get,

"Boussinesq-Cerruti solution - point force at the origin of an infinite isotropic half-space

Another useful solution is that of a point force acting on the surface of an infinite half-space. It was derived by Boussinesq[4] and a derivation is given in Landau & Lifgarbagez.[3]:§8 In this case, the solution is again written as a Green's tensor which goes to zero at infinity, and the component of the stress tensor normal to the surface vanishes. This solution may be written in Cartesian coordinates as: ..."

## What is the Boussinesq Elasticity Problem?

The Boussinesq Elasticity Problem is a mathematical model that describes the deformation and stress distribution in an elastic material subject to a concentrated force or load. It was first proposed by French mathematician Joseph Valentin Boussinesq in the 19th century.

## What are the assumptions made in the derivation of the Boussinesq Elasticity Problem?

The derivation of the Boussinesq Elasticity Problem is based on several assumptions, including:

• The material is homogeneous and isotropic.
• The load is applied at a point on the surface of the material.
• The material is linearly elastic, meaning that the stress is directly proportional to the strain.
• The material is in a state of plane stress, meaning that the stress and strain only vary in two dimensions.
• The material is incompressible, meaning that the volume of the material remains constant under stress.

## What is the significance of the Boussinesq Elasticity Problem in engineering?

The Boussinesq Elasticity Problem is a fundamental model in the field of elasticity and is commonly used in engineering to analyze the stress and deformation of structures subject to concentrated loads. It has applications in various fields, including civil engineering, mechanical engineering, and geotechnical engineering.

## How is the Boussinesq Elasticity Problem solved?

The Boussinesq Elasticity Problem is typically solved using analytical methods, such as the method of superposition or the Fourier transform method. In some cases, numerical methods may also be used, such as the finite element method.

## What are the limitations of the Boussinesq Elasticity Problem?

While the Boussinesq Elasticity Problem is a useful model for analyzing elastic materials, it has some limitations. It assumes a linearly elastic material, which may not accurately describe the behavior of some materials under high loads. It also does not take into account the effects of shear stress, which may be significant in some cases. Additionally, the assumptions made in the derivation may not hold true for all materials and loading conditions.

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