Studying from Love's Treatise on elasticity im having some difficulties.(adsbygoogle = window.adsbygoogle || []).push({});

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

[tex](\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[/tex]

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

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

[tex](\lambda + \mu )\nabla^2\phi + \rho \Phi=0[/tex]

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

The scalar component of the force [tex] (X,Y,Z) [/tex] can be written :

[tex] \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'[/tex]

Where [tex]X',Y',Z'[/tex] 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 [tex]\Phi[/tex] via the aforementioned integral.

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# Transmission of force(elasticity)

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