Transmission of force(elasticity)

[mediocre]

Studying from Love's Treatise on elasticity I am having some difficulties.

For an isotropic body under the influence of a force within a finite volume $$T$$ equiations of equillibrium via Hooke's law are given by :

$$(\lambda + \mu )(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z})\Delta + \mu\nabla^2(u,v,w) + \rho (X,Y,Z) = 0$$

Displacement vector field $$(u,v,w)$$ and applied force $$(X,Y,Z)$$ can be made, respectively,via Helmholtz decomposition into a curl free scalar field $$\phi,\Phi$$ and a div free vector field $$(F,G,H) , (L,N,M)$$.

Plugging these relations into the equilibrium equation yields 4 equations,of which one that combines the scalar fields is:

$$(\lambda + \mu )\nabla^2\phi + \rho \Phi=0$$

Now comes the difficult part (bear with me :) ):

The scalar component of the force $$(X,Y,Z)$$ can be written :

$$\Phi=-\frac{1}{4\pi}\int\int\int(X'\frac{\partial r^-^1}{\partial x}+Y'\frac{\partial r^-^1}{\partial y}+Z'\frac{\partial r^-^1}{\partial z})dx'dy'dz'$$

Where $$X',Y',Z'$$ denote values of force within T and r is the distance from x,y,z.

My question is why?I don't understand the way why we can write the $$\Phi$$ via the aforementioned integral.

 

[Studiot]

Isn't this just the compatibility requirement he deals with in articles 16 and 17 on pages 47 - 49?

It would be useful to know where you are coming from on this. I take my hat off to anyone brave enough to attempt Love.

 

[Andy Resnick]

Studying from Love's Treatise on elasticity I am having some difficulties.

For an isotropic body under the influence of a force within a finite volume $$T$$ equiations of equillibrium via Hooke's law are given by :

$$(\lambda + \mu )(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z})\Delta + \mu\nabla^2(u,v,w) + \rho (X,Y,Z) = 0$$

Displacement vector field $$(u,v,w)$$ and applied force $$(X,Y,Z)$$ can be made, respectively,via Helmholtz decomposition into a curl free scalar field $$\phi,\Phi$$ and a div free vector field $$(F,G,H) , (L,N,M)$$.

Plugging these relations into the equilibrium equation yields 4 equations,of which one that combines the scalar fields is:

$$(\lambda + \mu )\nabla^2\phi + \rho \Phi=0$$

Now comes the difficult part (bear with me :) ):

The scalar component of the force $$(X,Y,Z)$$ can be written :

$$\Phi=-\frac{1}{4\pi}\int\int\int(X'\frac{\partial r^-^1}{\partial x}+Y'\frac{\partial r^-^1}{\partial y}+Z'\frac{\partial r^-^1}{\partial z})dx'dy'dz'$$

Where $$X',Y',Z'$$ denote values of force within T and r is the distance from x,y,z.

My question is why?I don't understand the way why we can write the $$\Phi$$ via the aforementioned integral.

I don't exactly understand what you are asking. Looking at the relevant section (130), all that happens (I think) is a formal evaluation of the general solution.

That said, the notation is archaic and difficult to follow. Can you be a little more specific?

 

[Studiot]

Mediocre's first equation was derived and appears at the bottom of p133 (section 91)
The equation that the query was about was derived and appears on p47 -49 as I said before.

Yes I see that Love does further work on this in section 130.

Unlike some authors, Love does not generally refer to the source of something he has already proved earlier in the treatise. This can cause difficulties when dipping into the book.

 

[mediocre]

I'm sorry if i wasn't clear enough,but my problem was the general solution to the equation of the scalar potential.I can see the analogy with,for an example,the problem of electrostatics:

$$\nabla\vec{E}=4\pi\rho$$

But i wanted to learn the mathematics behind it,which is impossible because,as studiot said, there are practically no cited references in his book.
If you could perhaps recommend me a book on theory of potential i would be very grateful.

 

[Studiot]

Yes you really need a good grasp of classical applied maths to tackle Love for he takes no prisoners.

I usually recommend the following a 'light reading'

Div, Grad, Curl and All That by H M Schey

This is a marvellous introduction to the subject. Schey develops all the main suspects without making the math seem difficult. However it is written largely from an electrical point of view. It starts from a mathematical view and develops the physics.

Engineering Field theory by A J BadenFuller

This starts from the physics and develops the necessary mathematics to suit. It covers all the major branches of classical field theory, not just electromagnetism.

If you want to stick with continuum mechanics I will have to look to see what is available, but I think even Timoshenko assumes much mathematical proficiency in his
Theory of Elasticity, co authored with J N Goodier.

There is also a good Russian book that I will have to dig out.

You should look up (Airy) Stress functions.

The following books have good sections on your topic

The mechanics of Fracture and Fatigue by Parker
Advanced Mechanics of Materials by Boresi, Schmidt and Sidebottom
Mechanics of solid Materials Lemaitre and Chaboche (in French), English translation by Shrivastava

 

[mediocre]

Thank you very much.At least I've got some pointers as to where i should be looking...