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Elastic shocks in relativistic rigid rods and balls.

ABSTRACT: We study the free boundary problem for the 'hard phase' material introduced by Christodoulou in (Christodoulou 1995 Arch. Ration. Mech. Anal. 130, 343-400), both for rods in (1?+?1)-dimensional Minkowski space-time and for spherically symmetric balls in (3?+?1)-dimensional Minkowski space-time. Unlike Christodoulou, we do not consider a 'soft phase', and so we regard this material as an elastic medium, capable of both compression and stretching. We prove that shocks must be null hypersurfaces, and derive the conditions to be satisfied at a free boundary. We solve the equations of motion of the rods explicitly, and we prove existence of solutions to the equations of motion of the spherically symmetric balls for an arbitrarily long (but finite) time, given initial conditions sufficiently close to those for the relaxed ball at rest. In both cases we find that the solutions contain shocks if and only if the pressure or its time derivative do not vanish at the free boundary initially. These shocks interact with the free boundary, causing it to lose regularity.

SUBMITTER: Costa JL

PROVIDER: S-EPMC6545059 | biostudies-literature | 2019 May

REPOSITORIES: biostudies-literature

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Elastic shocks in relativistic rigid rods and balls.

Costa João L JL Natário José J

Proceedings. Mathematical, physical, and engineering sciences 20190515 2225

We study the free boundary problem for the 'hard phase' material introduced by Christodoulou in (Christodoulou 1995 <i>Arch. Ration. Mech. Anal.</i> <b>130</b>, 343-400), both for rods in (1 + 1)-dimensional Minkowski space-time and for spherically symmetric balls in (3 + 1)-dimensional Minkowski space-time. Unlike Christodoulou, we do not consider a 'soft phase', and so we regard this material as an elastic medium, capable of both compression and stretching. We prove that shocks must be null hy ...[more]

PMID: 31236048

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Project description:The microstructure of trabecular bone is usually perceived as a collection of plate-like and rod-like trabeculae, which can be determined from the emerging high-resolution skeletal imaging modalities such as micro-computed tomography (?CT) or clinical high-resolution peripheral quantitative CT (HR-pQCT) using the individual trabecula segmentation (ITS) technique. It has been shown that the ITS-based plate and rod parameters are highly correlated with elastic modulus and yield strength of human trabecular bone. In the current study, plate-rod (PR) finite element (FE) models were constructed completely based on ITS-identified individual trabecular plates and rods. We hypothesized that PR FE can accurately and efficiently predict elastic modulus and yield strength of human trabecular bone. Human trabecular bone cores from proximal tibia (PT), femoral neck (FN) and greater trochanter (GT) were scanned by ?CT. Specimen-specific ITS-based PR FE models were generated for each ?CT image and corresponding voxel-based FE models were also generated in comparison. Both types of specimen-specific models were subjected to nonlinear FE analysis to predict the apparent elastic modulus and yield strength using the same trabecular bone tissue properties. Then, mechanical tests were performed to experimentally measure the apparent modulus and yield strength. Strong linear correlations for both elastic modulus (r(2) = 0.97) and yield strength (r(2) = 0.96) were found between the PR FE model predictions and experimental measures, suggesting that trabecular plate and rod morphology adequately captures three-dimensional (3D) microarchitecture of human trabecular bone. In addition, the PR FE model predictions in both elastic modulus and yield strength were highly correlated with the voxel-based FE models (r(2) = 0.99, r(2) = 0.98, respectively), resulted from the original 3D images without the PR segmentation. In conclusion, the ITS-based PR models predicted accurately both elastic modulus and yield strength determined experimentally across three distinct anatomic sites. Trabecular plates and rods accurately determine elastic modulus and yield strength of human trabecular bone.

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Elastic shocks in relativistic rigid rods and balls.

Publications

Elastic shocks in relativistic rigid rods and balls.

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