Geometry, mechanics, and Eschelby inclusions
The fractures in chert nodules in this exposure of Pennsylvanian Bird Springs Formation, in the Arrow Canyon Wilderness Area of Southern Nevada, show a contrast in occurrence compared to those in the limestone matrix. The difference is so strong that its explanation is not obvious.
The geometry of the 2-meter-wide exposure in this image is a 3-dimensional corner so I’ll describe it in case it is not evident. The large, planar surface that occupies most of the center of this photograph is a joint surface – let’s call it set-N because it strikes north – that is normal to the joint on the left side of the photo, which is a member of set-E (for its easterly strike). This photo looks obliquely upward such that the uniformly tan surface in the center top of the image is a bedding surface. The strata dip moderately towards east, directly away from the photographer. Finally, both joint sets are orthogonal to bedding.
The joints in the limestone are spaced about 1-2 meters on average. Consider the distance from the set-E joint mentioned above to the next set-E joint, which can be seen crossing the tan bedding surface. This one does not crosscut the bed containing the central set-N joint. The next set-E joint that crosscuts the bed in the center is located at the right-hand margin of the photo. The average spacing of the set-N joints is similarly 1-2 meters.
Now zoom in and look at the chert nodule the pencil is pointing at. Clicking the image once or twice should activate zooming. The nodule contains a joint system with spacing on the order of 1-2 centimeters. Their surfaces are somewhat irregular, but they are oriented approximately parallel to the set-N and set-E joints. Both the larger joints and these in chert are apparently opening-mode cracks.
The joints in chert have the same orientation as those in the limestone matrix, including the fact they are generally normal to bedding, but are more closely spaced – by a factor of 100. Although I stated the joints are all normal to bedding, the chert joints are actually not normal to the boundary of the nodules. See, for example, how the left margin of the nodule by the pencil steepens relative to bedding, but the joints in this part of the nodule are apparently unaffected. Why don’t the fractures treat this margin as a mechanical bedding surface? Based on the change in their frequency of occurrence at this interface it does seem that the surface has some mechanical significance.
A considerable amount of intellectual energy has been invested in modeling of inclusions of one material in a matrix of different mechanical property, under the influence of a remote anisotropic stress. Such inclusions, generally of ellipsoidal shape, have been modeled analytically and numerically in many academic papers, usually concentrating on the perturbation of stress or strain in the matrix. These are referred to as Eschelby inclusions, in honor of one investigator who modeled such inclusions. If you decide to investigate the model results prepare yourself for a bit of mathematics.
I think we can consider the chert joints to be a response to a brittle-elastic strain field, presumably reflecting extension normal to the joint surfaces. Yet their geometry is not strongly controlled by the orientation of the chert/limestone contact. Rather, they are parallel to the large, widely spaced joints in the matrix. This hints that the fractures in chert and limestone layers formed under the influence of the same far-field strain. The theoretical Eschelby analyses I perused do not predict this behavior; they would show a strong control on fractures (strain) as the chert/limestone mechanical boundary is approached, affecting both their geometry and density.
Stiffness likely plays a role in the contrast of fracture density between the chert and limestone. A stiffer, more-brittle rock should show a higher density of fractures than a less brittle rock. A quick review for Young’s modulus of chert and of limestone (not the formation pictured, but rather measurements from various publications about other chert and limestone) suggests chert has a higher modulus, but only about 25%-50% higher. Can that alone account for a 100x increase in fracture abundance?
Perhaps the relatively thin layers defined by the nodules result in fractures much more closely spaced than fractures formed in response to the larger-scale mechanical layering of the limestone. We’ve seen the layer-thickness/joint-spacing relationship before (#1, #11). Perhaps coupling this layer-thickness effect with the modulus stiffness effect can account for the closely spaced fractures in the chert. Do you have any alternative ideas?