,
the Poisson's ratio
of an isotropic solid, and ignoring the effect of gravity,
the vector equation to be solved is [70]
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The linear elastic properties of porous media can be computed by
finite difference or a finite element methods applied to digital images.
Expressed in terms
of the elastic vector displacement, ,
the Poisson's ratio
of an isotropic solid, and ignoring the effect of gravity,
the vector equation to be solved is [70]
For a two-phase image, solid and pores, where the solid has a uniform elastic moduli tensor and both elastic moduli are zero in the pore space, a finite difference approach can be used [71]. The boundary condition of zero normal force at a solid-pore boundary is automatically satisfied in the finite difference formulation [71]. When there are two or more kinds of solid material, or when the pore space is filled with an incompressible fluid, it is difficult to incorporate, into a finite difference formulation, the boundary conditions of continuity of elastic displacement and normal stress at boundaries between different elastic moduli regions. It is easier to use a finite element formulation, which makes use of the variational principle that the correct displacement solution minimizes the elastic energy under an applied strain [57,72]. The finite difference method would use a grid just like that shown in Fig. 6, with elastic displacements determined at the nodes in the pixel centers, while the finite element method would use a grid like that in Fig. 7, with elastic displacements determined at the pixel corners. In the displacement formulation [73] of the finite element method, continuity of displacement is satisfied automatically, but continuity of normal stresses is only approximate.
Figure 9 shows the component
σxx of the computed stress tensor
throughout the 22% porosity microstructure shown in Fig.
4,
where the solid (white) phase was fully connected with a Poisson
ratio of 0.2 and a Youngs's modulus of 1.0 in arbitrary units.
A horizontal strain (
xx)
of 0.01 has been applied across the sample.
Fig. 9 was obtained using
a finite element method [57].
The brighter
the gray scale, the higher the stress. The pores are shown in black, and the
compressed regions are shown in an uniform dark gray.
Because of the randomness of the porous material,
even though the average strain is tensile, there will still be regions of
compressive stress.
Notice that the areas of compressive stress are
always near a pore. On the other hand, the areas of high tensile stress
are almost
always at the bottom or top of a pore, due to the stress concentration effects
of a cavity in a tensile strain field [74].
Figure 10 shows the corresponding stress histogram.
The area under the histogram has been adjusted to be one, rather than
1 - porosity, because the zero stresses in the (empty) pores
have been ignored.
The effective Young's modulus of this porous material was about 1/3.
The effective moduli is easily determined by computing the average stress
tensor and then extracting the effective moduli using the applied strain
and well-known composite theory [45,46].
Ref. [75] describes a successful comparison with experiment
using the finite element elastic technique to compute the effective elastic
and shrinkage properties of porous VycorTM
glass.
Figure 9: Showing the horizontal tensile stress (σxx) distribution for the 22% porosity microstructure shown in Fig. 4, where the solid phase (white in Fig. 4) has a Poisson's ratio of 0.2. The brightness is proportional to the tensile stress magnitude, with pores shown in black. Areas of compressive stress are in dark gray.
Figure 10: Showing a stress histogram for the computed stress fields shown in Fig. 9. The applied strain was 0.01, and Young's modulus of the solid phase was 1.0, in arbitrary units.