I am trying to understand a claim made in many papers (Geertsma, 1957a; Rice, 1976) and textbooks (Jaeger and Cook; Zimmerman) related to mechanics of porous media.

Consider an isotropic linear elastic porous medium initially under an external hydrostatic normal stress, s, and an internal pore pressure p. Increase the two normal tractions (force/area) on the external and the internal surface of the matrix by the same amount dp. The claim is that the strains in the porous body upon application of this incremental normal stress will lead to isotropic local strains both in the matrix as well as in the pores.

There is no proof given; however the reasoning given is convincing, which says that since the matrix has dp additional normal stress it will lead to local volumetric strain in the matrix that is equal to dp/Km where Km is the matrix bulk modulus. The strain in any direction in the matrix will be dp/3Km due to the assumption of isotropy of the matrix. This strain configuration in the matrix will lead to a unique strain configuration in the pores which would require them to be equal to dp/3Km as well.

When I applied the above case to a thick spherical vessel, I don't think I will get equal strains in all directions, i.e. radial strains and tangential strains will be unequal.

  • $\begingroup$ Those authors are assuming (implicitly or explicitly) that the medium is uniform. A pressure vessel isn't uniform; it consists of a solid wall and the (typically fluid) contents. This nonuniformity leads to anisotropy; as you note; the in-plane wall stress differs from the out-of-plane wall stress, for example. $\endgroup$ – Chemomechanics May 30 '20 at 17:39

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