Green’s functions for soft materials containing a hard line inhomogeneity

2019 ◽  
Vol 24 (11) ◽  
pp. 3614-3631 ◽  
Author(s):  
Pengyu Pei ◽  
Guang Yang ◽  
Cun-Fa Gao
Keyword(s):  
Thermal Expansion ◽  
Heat Source ◽  
Closed Form ◽  
Hilbert Problem ◽  
Closed Form Solution ◽  
Soft Materials ◽  
Form Solution ◽  
Soft Material ◽  
Elastic Plane ◽  
Rigid Line

The linear elastic plane deformation of a soft material containing a rigid line inhomogeneity subjected to a concentrated force, a concentrated moment, and a point heat source was studied. Distinct from the existing rigid line inhomogeneity model which neglects the deformation of the inhomogeneity induced by both the mechanical stresses and thermal expansion, the current model allows for the thermal expansion-induced stretch and rotation of the inhomogeneity. In this context, we derive the closed-form solution for the full stress field in the soft material by solving the corresponding Riemann–Hilbert problem. In particular, our solution can serve as the Green’s function to establish other analytical solutions for more practical and complicated problems in this area. Several numerical examples are presented to illustrate our closed-form solution corresponding to the thermal loading. It is found that the presence of the heat source contributes significantly to the rigid rotation of inhomogeneity, and the thermal expansion-induced stretch of the inhomogeneity has a great impact on the stress intensity factors at the inhomogeneity tips.

scholarly journals Analysis of hyperbolic Pennes bioheat equation in perfused homogeneous biological tissue subject to the instantaneous moving heat source

2021 ◽  
Vol 3 (4) ◽  
Author(s):  
Ali Kabiri ◽  
Mohammad Reza Talaee
Keyword(s):  
Heat Source ◽  
Closed Form ◽  
Method Comparison ◽  
Closed Form Solution ◽  
Form Solution ◽  
Temperature Profiles ◽  
Moving Heat Source ◽  
Bioheat Equation ◽  
Perfusion Rate

AbstractThe one-dimensional hyperbolic Pennes bioheat equation under instantaneous moving heat source is solved analytically based on the Eigenvalue method. Comparison with results of in vivo experiments performed earlier by other authors shows the excellent prediction of the presented closed-form solution. We present three examples for calculating the Arrhenius equation to predict the tissue thermal damage analysis with our solution, i.e., characteristics of skin, liver, and kidney are modeled by using their thermophysical properties. Furthermore, the effects of moving velocity and perfusion rate on temperature profiles and thermal tissue damage are investigated. Results illustrate that the perfusion rate plays the cooling role in the heating source moving path. Also, increasing the moving velocity leads to a decrease in absorbed heat and temperature profiles. The closed-form analytical solution could be applied to verify the numerical heating model and optimize surgery planning parameters.

On the Changing Order of Singularity at a Crack Tip

1977 ◽  
Vol 44 (4) ◽  
pp. 625-630 ◽  
Author(s):  
R. J. Nuismer ◽  
G. P. Sendeckyj
Keyword(s):  
Closed Form ◽  
Stress Singularity ◽  
Closed Form Solution ◽  
Crack Tip ◽  
Form Solution ◽  
Square Root ◽  
Rigid Line Inclusion ◽  
Rigid Line ◽  
Line Inclusion ◽  
Crack Tip Stress

The nature of the transition in the crack tip stress singularity from an inverse square root to an inverse fractional power as a crack tip reaches a phase boundary or a geometrical discontinuity for interface cracks is examined. This is done by analyzing the simple closed-form solution to the problem of a rigid line inclusion with one side partially debonded for the case of antiplane deformation. For this example, the crack tip stress singularity changes from an inverse square root to an inverse three-quarters power as the crack tips approach the inclusion tips (i.e., when one face of the rigid line inclusion is completely debonded). A detailed analysis, based on series expansions of the closed-form solution, is used to show how the singularity transition occurs. Moreover, the expansions indicate difficulties that may be encountered when solving such problems by approximate methods.

A Closed-Form Solution for Temperature Profiles in Selective Laser Melting of Metal Additive Manufacturing

2020 ◽  
Vol 982 ◽  
pp. 98-105
Author(s):  
Steven Y. Liang ◽  
Jin Qiang Ning ◽  
Elham Mirkoohi
Keyword(s):  
Finite Element ◽  
Heat Source ◽  
Closed Form ◽  
Heat Loss ◽  
Heat Sink ◽  
Prediction Accuracy ◽  
Closed Form Solution ◽  
Form Solution ◽  
Laser Melting ◽  
Temperature Prediction

This paper presents a closed-form solution for the temperature prediction in selective laser melting (SLM). This solution is developed for the three-dimensional temperature prediction with consideration of heat input from a moving laser heat source, and heat loss from convection and radiation on the part top boundary. The consideration of heat transfer boundary condition and latent heat in the closed-form solution leads to an improvement on the understanding of thermal development and prediction accuracy in SLM, and thus the usefulness of the analytical model in the temperature prediction in real applications. A moving point heat source solution is used to calculate the temperature rise due to the heat input. A heat sink solution is used to calculate the temperature drop due to heat loss from convection and radiation on the part boundary. The heat sink solution is modified from a heat source solution with equivalent power due to heat loss from convection and radiation, and zero-moving velocity. The temperature solution is then constructed from the superposition of the linear heat source solution and linear heat sink solution. Latent heat is considered using a heat integration method. Ti-6Al-4V is chosen to test the presented model with the assumption of isotropic and homogeneous material. The predicted molten pool dimensions are compared to the documented values from the finite element method and experiments in the literature. The presented model has improved prediction accuracy and significantly higher computational efficiency compared to the finite element model.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations

scholarly journals A Closed-Form Solution to Planar Feature-Based Registration of LiDAR Point Clouds

2021 ◽  
Vol 10 (7) ◽  
pp. 435
Author(s):  
Yongbo Wang ◽  
Nanshan Zheng ◽  
Zhengfu Bian
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Simulated Data ◽  
Real Data ◽  
Point Clouds ◽  
Form Solution ◽  
Spatial Transformation ◽  
Dual Quaternions ◽  
Feature Based ◽  
Planar Feature

Since pairwise registration is a necessary step for the seamless fusion of point clouds from neighboring stations, a closed-form solution to planar feature-based registration of LiDAR (Light Detection and Ranging) point clouds is proposed in this paper. Based on the Plücker coordinate-based representation of linear features in three-dimensional space, a quad tuple-based representation of planar features is introduced, which makes it possible to directly determine the difference between any two planar features. Dual quaternions are employed to represent spatial transformation and operations between dual quaternions and the quad tuple-based representation of planar features are given, with which an error norm is constructed. Based on L2-norm-minimization, detailed derivations of the proposed solution are explained step by step. Two experiments were designed in which simulated data and real data were both used to verify the correctness and the feasibility of the proposed solution. With the simulated data, the calculated registration results were consistent with the pre-established parameters, which verifies the correctness of the presented solution. With the real data, the calculated registration results were consistent with the results calculated by iterative methods. Conclusions can be drawn from the two experiments: (1) The proposed solution does not require any initial estimates of the unknown parameters in advance, which assures the stability and robustness of the solution; (2) Using dual quaternions to represent spatial transformation greatly reduces the additional constraints in the estimation process.

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