Kathleen A. Issen

Conditions for compaction bands in porous rock

Reexamination of the results of Rudnicki and Rice for shear localization reveals that solutions for compaction bands are possible in a range of parameters typical of porous rock. Compaction bands are narrow planar zones of localized compressive deformation perpendicular to the maximum compressive stress, which have been observed in high-porosity rocks in the laboratory and field. Solutions for compaction bands, as an alternative to homogenous deformation, are possible when the inelastic volume deformation is compactive and is associated with stress states on a yield surface “cap.” The cap implies that the shear stress required for further inelastic deformation decreases with increasing compressive mean stress. While the expressions for the critical hardening modulus for compaction and shear bands differ, in both cases, deviations from normality promote band formation. Inelastic compaction deformation associated with mean stress (suggested by Aydin and Johnson) promotes localization by decreasing the magnitude of the critical hardening modulus. Axisymmetric compression is the most favorable deviatoric stress state for formation of compaction bands. Predictions for compaction bands suggest that they could form on the “shelf” typically observed in axisymmetric compression stress strain curves of porous rock at high confining stress. Either shear or compaction bands may occur depending on the stress path and confining stress. If the increase in local density and decrease in grain size associated with compaction band formation result in strengthening rather than weakening of the band material, formation of a compaction band may not preclude later formation of a shear band.

The influence of constitutive models on localization conditions for porous rock

Alternative constitutive models are employed to examine localization conditions for high porosity sandstone. Two deformation types are considered: shear bands and compaction bands (planar zones of pure compressional deformation, perpendicular to maximum compression). The proposed two-yield surface model, motivated in part by observations of two microstructural damage mechanisms, consists of a shear yield surface (dilatant, frictional mechanism), combined with a cap (compactant mechanism). Results from this model are consistent with limited experimental observations: both band types are permitted for probable values of key material parameters. Conversely, the single-yield surface model, previously employed for sandstone, cannot predict the observed compaction bands.

Creating Physiologically Realistic Vertebral Fractures in a Cervine Model

Approximately 50% of women and 25% of men will have an osteoporosis-related fracture after the age of 50, yet the micromechanical origin of these fractures remains unclear. Preventing these fractures requires an understanding of compression fracture formation in vertebral cancellous bone. The immediate research goal was to create clinically relevant (midvertebral body and endplate) fractures in three-vertebrae motion segments subject to physiologically realistic compressional loading conditions. Six three-vertebrae motion segments (five cervine, one cadaver) were potted to ensure physiologic alignment with the compressive load. A 3D microcomputed tomography (microCT) image of each motion segment was generated. The motion segments were then preconditioned and monotonically compressed until failure, as identified by a notable load drop (48-66% of peak load in this study). A second microCT image was then generated. These three-dimensional images of the cancellous bone structure were inspected after loading to qualitatively identify fracture location and type. The microCT images show that the trabeculae in the cervine specimens are oriented similarly to those in the cadaver specimen. In the cervine specimens, the peak load prior to failure is highest for the L4-L6 motion segment, and decreases for each cranially adjacent motion segment. Three motion segments formed endplate fractures and three formed midvertebral body fractures; these two fracture types correspond to clinically observed fracture modes. Examination of normalized-load versus normalized-displacement curves suggests that the size (e.g., cross-sectional area) of a vertebra is not the only factor in the mechanical response in healthy vertebral specimens. Furthermore, these normalized-load versus normalized-displacement data appear to be grouped by the fracture type. Taken together, these results show that (1) the loading protocol creates fractures that appear physiologically realistic in vertebrae, (2) cervine vertebrae fracture similarly to the cadaver specimen under these loading conditions, and (3) that the prefracture load response may predict the impending fracture mode under the loading conditions used in this study.

Characterization of Fatigue Fractures in Closed-Cell Aluminum Foam Using x-ray Micro-Computed Tomography

A post-mortem study of Alporas closed-cell aluminum foam specimens previously failed under strain-controlled fully reversed tension-compression fatigue was conducted using x-ray micro-computed tomography (μCT). Volumetric renders of the 3D structure of the material were produced. Fractures were identified and marked throughout voxel-based images of the specimens. This produced a 3D plot of fracture locations. At high strain amplitudes (0.175-0.5%), fractures formed an interconnected planar zone oriented approximately perpendicular to the loading axis; typically, the angle of the plane differed from that of a tension failure. Conversely, at low strain amplitudes (0.05-0.1%), short fractures have been formed diffusely within the specimen. In both cases, observed fractures were tortuous. Our previous work with surface strain mapping via digital image correlation (DIC) suggested that for all strain amplitudes, a crack, evidenced by a zone of high extensile strain, was formed and propagated through the material. This result was confirmed at high strain amplitudes, but not at low strain amplitudes. The discrepancy is attributed to three potential causes. Using DIC, short cracks cannot be accurately resolved with relatively coarse light intensity patterns. DIC images indicate fractures under load, while μCT imaging was conducted under zero load. Finally, the localized extension seen in DIC images could be attributed to strain with no resultant fractures.

Strain localization conditions in porous rock using a two-yield surface constitutive model

Abstract This work examines theoretical conditions for localized deformation in porous rock, with emphasis on two recently identified deformation structures: compaction bands and dilation bands. Field and laboratory observations report that compaction/dilation bands consist of pure compressional/dilational deformation, which form perpendicular to maximum/minimum compression. A bifurcation approach is employed, with a two-yield surface constitutive model, to develop localization conditions under axisymmetric stress states for different stress paths. The first yield surface corresponds to a dilatant, frictional-damage mechanism (brittle regime), while the yield surface cap corresponds to a compactant mechanism (ductile regime). In the transitional regime, where both mechanisms are active, this model successfully predicts compaction bands and shear bands observed in axisymmetric compression tests. Due to discontinuities in the predicted band angle for probable material parameter values, this model may explain the lack of low angle compacting shear band observations in experiments. The two-yield surface model may also be applicable for a non-traditional axisymmetric extension stress path: increasing confining pressure with constant axial compression. Conditions for dilation band formation for this stress path are significantly less restrictive than corresponding compaction band conditions, suggesting that dilation bands could be a common deformation mode for high porosity sandstone.

Third Invariant Dependent Single Yield Surface Model and Localization Conditions for High Porosity Sandstone

High porosity sandstones are observed to fail by the formation of localized bands in field and laboratory settings. Compaction bands form perpendicular to the direction of maximum compression, with pure compactant strain, while shear bands form at an angle to the direction of maximum compression, with shear strain accompanied by either compactant or dilatant strain normal to the band. Recent experimental evidence indicates that mechanical behavior of some high porosity sandstones depends on the third invariant of deviatoric stress (J 3). In this work, a J 3 dependent single yield surface constitutive model was developed, and band orientation predictions determined using the Rudnicki and Rice bifurcation theory. While in the laboratory, high porosity sandstones are typically tested under axisymmetric compression (ASC; intermediate principal stress equal to the least compressive principal stress), stress states in the field are often non-axisymmetric. Therefore, localization conditions were determined under P 1, a stress state perturbed from ASC. For ASC, the localization conditions resulting from the J 3 dependent model are identical to those from the J 3 independent model. The most favorable conditions for compaction band formation occur under ASC, while under P 1, localization conditions favor shear band formation. Mild to moderate J 3 dependence favors formation of shear bands over compaction bands, and a strong J 3 dependence prohibits localized deformation band formation. These results provide one possible expla-nation for relatively few field observations of compaction bands versus more commonly observed shear bands.

Failure of Castlegate Sandstone Under True Triaxial Loading

A test series designed to investigate and quantify the effect of the intermediate principal stress on the failure of Castlegate sandstone was completed. Using parallelepiped specimens and a true triaxial testing system, constant mean stress tests were conducted. Stress states ranged from axisymmetric compression to axisymmetric extension. Results suggest a possible failure dependence on the third invariant of deviatoric stress at lower mean stresses.

Simulation of bone aging and virtual reality visualization of cancellous bone structure

This paper describes a bone modeling effort and virtual reality visualization aimed towards characterizing and predicting human vertebral compression fractures with emphasis on quantifying osteoporosis and aging. A 3D model of the trabecular bone structure is constructed from a stack of muCT images of a human vertebra. Parameters for quantifying the bone are calculated for this structure from the 3D model and validated by comparing the results with values calculated from standard stereological methods. The resulting structure is then aged using biologically based computational algorithms. These structures are then visualized in a virtual reality environment enabling detailed examination of the regional microstructures to assess the potential for compression fracture formation.

Ring apophysis fractures induced by low-load low-angle repetitive flexion in an ex-vivo cervine model

Ring apophysis fractures of the spine occur in physically-active adolescents causing low back pain and the potential for chronic pain. Many of these fractures occur without memorable trauma, suggesting that the fractures occur during everyday movements and activities. The benign nature of this poorly understood potential mechanism of injury hampers appropriate diagnosis and early treatment. The purpose of this study was to establish an ex-vivo model of ring apophysis fracture and demonstrate that these fractures can be initiated by repetitive non-traumatic loading. Six 5-vertebra cervine lumbar (L1-L5) motion segments were cyclically loaded in low-angle low-load flexion (to 15° flexion, with peak load of 230±50N), a representative movement component of daily activities for both human and deer lumbar spines. Pinned end conditions replicated physiologically realistic loading. Ring apophysis fractures were created under low-load low-angle conditions in healthy vertebrae of similar bone mineral density and a similar degree of skeletal maturity to adolescent humans. All specimens developed ring apophysis fractures, some as early as 1400 cycles. The load-displacement data, and hysteresis loops during the cyclic loading, suggest that the fractures occurred gradually, i.e., without trauma. The ease at which these fractures were created suggests that ring apophysis fractures may be more prevalent than current diagnosis rates. Therefore, clinically, healthcare providers should include the potential for ring apophysis fracture in the differential diagnosis of all physically-active adolescents who present with back pain.

Strain localization conditions under true triaxial stress states.

This work uses a bifurcation approach to develop theoretical predictions for deformation band formation for a suite of true triaxial tests on Castlegate sandstone. In particular, the influence of the intermediate principal stress on strain localization is examined. Using common simplifying assumptions (localization occurs at peak stress, and the failure surface is similar to the yield surface), theoretical predictions captured the overall trends observed experimentally. However, agreement between predicted and observed band orientations for individual specimens was varied. This highlights the importance of detailed data analyses to accurately determine key material parameter values at the inception of localization.