Force Balance Law Research Articles

On the application of a lattice method to configurational and fracture mechanics

This study elucidates the concept of material forces at the tip of an evolving crack in brittle homogeneous materials by using a numerical lattice approach. A 2D lattice with an erosion algorithm is employed in the context of LEFM to investigate two well-known classic fracture problems in Mode I, i.e., a center-cracked rectangular panel with finite dimensions and a three-point bending beam with a notch. Validated by the well-developed analytical solutions for these two configurations, the lattice is used to derive crack tip driving forces in the direction of crack propagation based on the change of global stiffness matrix of the mesh before and after crack growth without any stress calculations, and to obtain the material forces opposing the tip motion utilizing the Eshelby-stress tensor and local force balance law in cracked bodies. Comparing these two distinct approaches, it is observed that the discrete material forces at the crack tip are closely equal to the tip driving forces, but with different signs, confirming that the lattice approach approximates the values of crack tip material forces using Eshelby-stress distributions. Moreover, satisfying C1 continuity by including rotational displacements for the Euler–Bernoulli based frame struts, there is no need for the lattice model to update interior computational point positions in the mesh to eliminate spurious material forces away from the tip, a complication observed in Finite Element formulations. The discrete material forces are then characterized for the two problems by only three material parameters, i.e., Young's modulus, uniaxial tensile strength, and the configurational characteristic width or crack band. After conducting a regression analysis on the material forces with respect to different tensile strength values, a parabolic relationship emerges between the material forces and tensile strengths. This feature is employed to extract the crack band width values by the lattice approach, which are explicitly used to obtain the crack tip material forces of the two classic problems. Being simple in terms of constitutive formulation, failure criterion, and capable of obtaining the configurational characteristic width or crack band for different boundary value problems for brittle homogeneous isotropic materials, the numerical lattice formulation sheds light on the important concept of material force in Configurational and Fracture Mechanics.

On Energy Release Rates and Configurational Forces for Interfacial Propagating Cracks: A Lattice Approach With a Brittle Erosion Technique

A numerical 2D lattice approach with an erosion algorithm is employed to analyze bimaterial interface fracture quantities in brittle heterogeneous materials in the context of linear elastic fracture mechanics (LEFM). The concept of configurational force is elucidated and the importance of nodal configurational changes in a mesh where no stress–strain analyses are needed is investigated. Three fracture problems, i.e., an infinite panel with a bi-material interface crack, a double-lap shear test, and a prenotched four-point bending masonry beam are then considered. Validated by analytical solutions, the lattice model uses two distinct postprocessing approaches to derive the energy release rates and configurational forces directly at bimaterial interface crack tips. While the first method takes advantage of the change of the lattice mesh's global stiffness matrix before and after crack growth without any stress–strain calculations to obtain crack tip driving forces, the second approach analyzes the configurational forces opposing the crack tip motion using the Eshelby stress tensor and local force balance law in cracked and heterogeneous domains. It is demonstrated that the discrete material forces at crack tips are closely equal to the tip driving forces for the three fracture problems, confirming that the lattice is an appropriate numerical tool to analyze fracture properties of evolving interface cracks. Satisfying C1 continuity by including rotational displacements for frame struts, there is also no need for the lattice to update interior computational points in the mesh to eliminate spurious material forces away from the tip.

The 3rd Generation of Dental Acrylic Intraoral Splint for Immobilization of Mandibular Fracture

This study was divided into 3 phases. The 1st phase was the experiment in seven cadaver skulls to compare force from 2 techniques of weight load; interdental wiring and interdental wiring together with dental acrylic, with 1, 2 and 3 kilogram bar for 10 minutes. The evaluation was done by measuring the gap of the fracture line in a lateral radiograph. The force was calculated from the gap according to momentum and force balance law. Force after the interdental wiring technique and force after intraoral acrylic splint together with interdental wiring were compared by paired t-test. The results found significant difference (P<0.05) (-22.76 and -3.58). The 2nd phase’s objective was to test tissue inflammation when use dental acrylic in the oral cavity. Upper canine teeth of 3 experiment dogs were banded with dental acrylic for 14 days. Then the gum was checked for inflammation. There was very mild to no inflammation. The last phase was done in mandibular body fractures of 2 dogs using the intraoral splint together with dental acrylic and which was followed up until mandibular bone was healed. Both of them had progressive bone healing at 2 months and clinical bone union at 4 months after surgery.

The stored energy of cold work, thermal annealing, and other thermodynamic issues in single crystal plasticity at small length scales

This paper develops a thermodynamically consistent gradient theory of single-crystal plasticity using the principle of virtual power as a paradigm to develop appropriate balance laws for forces and energy. The resulting theory leads to a system of microscopic force balances, one balance for each slip system, and to an energy balance that accounts for power expended during plastic flow via microscopic forces acting in concert with slip-rates and slip-rate gradients. Central to the theory are an internal energy and entropy, plastic in nature, dependent on densities that account for the accumulation of glide dislocations as well as geometrically necessary dislocations – and that, consequently, represent quantities associated with cold work. Our theory allows us to discuss – within the framework of a gradient theory – the fraction of plastic stress-power that goes into heating, as well as the reduction of the dislocation density in a cold-worked material upon subsequent (or concurrent) thermal annealing.

Statistical Foundations of Liquid-Crystal Theory. I: Discrete Systems of Rod-Like Molecules

We develop a mechanical theory for systems of rod-like particles. Central to our approach is the assumption that the external power expenditure for any subsystem of rods is independent of the underlying frame of reference. This assumption is used to derive the basic balance laws for forces and torques. By considering inertial forces on par with other forces, these laws hold relative to any frame of reference, inertial or noninertial. Finally, we introduce a simple set of constitutive relations to govern the interactions between rods and find restrictions necessary and sufficient for these laws to be consistent with thermodynamics. Our framework provides a foundation for a statistical mechanical derivation of the macroscopic balance laws governing liquid crystals.

Implications of the equipotential field line approximation for equatorial spread F analysis

Three different approaches to the evaluation of the electrostatic potential in the ionosphere under equatorial spread F(ESF) conditions are considered. First, we calculate the potential using an analytical approach, applying force balance laws to a simplified ionosphere. Second, we compute the potential around a cylinder‐like plasma depletion in an idealized ionosphere using both the equipotential field line (EFL) approach and the full 3‐D solution to the electrostatic potential problem. Our third approach involves an initial boundary value simulation in a realistic ionosphere using both EFL and 3‐D potential solutions. The results show that the equipotential field line assumption does not fully capture the 3‐D structure of the ionospheric current system and leads to an underestimation of the growth rate of ESF irregularities in numerical simulations.

A coupled theory of fluid permeation and large deformations for elastomeric materials

An elastomeric gel is a cross-linked polymer network swollen with a solvent (fluid). A continuum-mechanical theory to describe the various coupled aspects of fluid permeation and large deformations (e.g., swelling and squeezing) of elastomeric gels is formulated. The basic mechanical force balance laws and the balance law for the fluid content are reviewed, and the constitutive theory that we develop is consistent with modern treatments of continuum thermodynamics, and material frame-indifference. In discussing special constitutive equations we limit our attention to isotropic materials, and consider a model for the free energy based on a Flory–Huggins model for the free energy change due to mixing of the fluid with the polymer network, coupled with a non-Gaussian statistical–mechanical model for the change in configurational entropy—a model which accounts for the limited extensibility of polymer chains. As representative examples of application of the theory, we study (a) three-dimensional swelling-equilibrium of an elastomeric gel in an unconstrained, stress-free state; and (b) the following one-dimensional transient problems: (i) free-swelling of a gel; (ii) consolidation of an already swollen gel; and (iii) pressure-difference-driven diffusion of organic solvents across elastomeric membranes.

Mechanics and hemodynamics of esophageal varices during peristaltic contraction.

Our hypothesis states that variceal pressure and wall tension increase dramatically during esophageal peristaltic contractions. This increase in pressure and wall tension is a natural consequence of the anatomy and physiology of the esophagus and of the esophageal venous plexus. The purpose of this study was to evaluate variceal hemodynamics during peristaltic contraction. A simultaneous ultrasound probe and manometry catheter was placed in the distal esophagus in nine patients with esophageal varices. Simultaneous esophageal luminal pressure and ultrasound images of varices were recorded during peristaltic contraction. Maximum variceal cross-sectional area and esophageal luminal pressures at which the varix flattened, closed, and opened were measured. The esophageal lumen pressure equals the intravariceal pressure at variceal flattening due to force balance laws. The mean flattening pressures (40.11 +/- 16.77 mmHg) were significantly higher than the mean opening pressures (11.56 +/- 25.56 mmHg) (P < or = 0.0001). Flattening pressures >80 mmHg were generated during peristaltic contractions in 15.5% of the swallows. Variceal cross-sectional area increased a mean of 41% above baseline (range 7-89%, P < 0.0001) during swallowing. The peak closing pressures in patients that experience future variceal bleeding were significantly higher than the peak closing pressures in patients that did not experience variceal bleeding (P < 0.04). Patients with a mean peak closing pressure >61 mmHg were more likely to bleed. In this study, accuracy of predicting future variceal bleeding, based on these criteria, was 100%. Variceal models were developed, and it was demonstrated that during peristaltic contraction there was a significant increase in intravariceal pressure over baseline intravariceal pressure and that the peak intravariceal pressures were directly proportional to the resistance at the gastroesophageal junction. In conclusion, esophageal peristalsis in combination with high resistance to blood flow through the gastroesophageal junction leads to distension of the esophageal varices and an increase in intravariceal pressure and wall tension.

The role of the configurational force balance in the nonequilibrium epitaxy of films

We discuss the epitaxial growth of an elastic film, allowing for stress and diffusion within the film surface as well as nonequilibrium interactions between the film and the vapor. Our approach, which relies on recent ideas concerning configurational forces, is based on: (i) standard (Newtonian) balance laws for forces and moments together with an independent balance law for configurational forces; (ii) atomic balances, one for each species of mobile atoms; (iii) a mechanical version of the second law that accounts for temporal changes in free energy, energy flows due to atomic transport, and power expended by both standard and configurational forces; (iv) thermodynamically consistent constitutive relations for the film surface and for the interaction between the surface and the vapor environment. The normal component of the configurational force balance at the surface represents a generalization, to a dynamical context involving dissipation, of a condition that would arise in equilibrium by considering variations of the total free energy with respect to the configuration of the film surface. Our final results consist of partial differential equations that govern the evolution of the film surface.

Pointwise Balances and the Construction of Stress Fields in Dielectrics

We study the problem of deriving a pointwise version of the force-balance law for a dielectric solid. Our analysis hinges on an explicit treatment of self forces, those which a body part may exert on itself.