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This work presents a stabilized formulation for phase‐field fracture of hyperelastic materials near the limit of incompressibility. At this limit, traditional mixed displacement and pressure formulations must satisfy the inf‐sup condition for solution stability. The mixed formulation coupled with the damage field can lead to an inhibition of crack opening as volumetric changes are severely penalized effectively creating a pressure‐bubble. To overcome this bottleneck, we utilize a mixed formulation with a perturbed Lagrangian formulation which enforces the incompressibility constraint in the undamaged material and reduces the pressure effect in the damaged material. A mesh‐dependent stabilization technique based on the residuals of the Euler–Lagrange equations multiplied with a differential operator acting on the weight space is used, allowing for linear interpolation of all field variables of the elastic subproblem. This formulation was validated with three examples at finite deformations: a plane‐stress pure‐shear test, a two‐dimensional geometry in plane‐stress, and a three‐dimensional notched sample. In the last example, we incorporate a hybrid formulation with an additive strain energy decomposition to account for different behaviors in tension and compression. The results show close agreement with analytical solutions for crack tip opening displacements and performs well at the limit of incompressibility.

 

Objective: Structural measurements after separation of cortical from trabecular bone are of interest to a wide variety of communities but are difficult to obtain because of the lack of accurate automated techniques. Methods: We d present a structure-based algorithm for separating cortical from trabecular bone in binarized images. Using the thickness of the cortex as a seed value, bone connected to the cortex within a spatially local threshold value is identified and separated from the remaining bone. The algorithm was tested on seven biological data sets from four species imaged using micro-computed tomography (μ-CT) and high-resolution peripheral quantitative computed tomography (HR-pQCT). Area and local thickness measurements were compared to images segmented manually. Results: The algorithm was approximately 11 times faster than manual measurements and the median error in cortical area was -4.47 ± 4.15%. The median error in cortical thickness was approximately 0.5 voxels for μ-CT data and less than 0.05 voxels for HR-pQCT images resulting in an overall difference of -28.1 ± 71.1 μm. Conclusion: A simple and readily implementable methodology has been developed that is repeatable, efficient, and requires few user inputs, providing an unbiased means of separating cortical from trabecular bone. Significance: Automating the segmentation of variably thick cortices will allow for the evaluation of large data sets in a time-efficient manner and allow for full-field analyses that have been previously limited to small regions of interest. The MATLAB code can be downloaded from https://github.com/TBL-UIUC/downloads.git.