Green’s Functions for Holes/Cracks in Laminates With Stretching-Bending Coupling

2005 ◽  
Vol 72 (2) ◽  
pp. 282-289 ◽  
Author(s):  
Chyanbin Hwu
Keyword(s):  
Closed Form ◽  
Composite Laminates ◽  
Green's Functions ◽  
Green’S Functions ◽  
Continuation Method ◽  
Elliptical Hole ◽  
Arbitrary Point ◽  
Two Dimensional ◽  
Concentrated Force ◽  
Concentrated Forces

Consider an infinite composite laminate containing a traction-free elliptical hole subjected to concentrated forces and moments at an arbitrary point outside the hole. This problem for two-dimensional deformation has been solved analytically in the literature, while for the general unsymmetric composite laminates stretching and bending coupling may occur and due to the mathematical complexity the associated Green’s functions have never been found for complete loading cases. Recently, by employing Stroh-like formalism for coupled stretching-bending analysis, the Green’s functions for the infinite laminates (without holes) were obtained in closed-form. Based upon the nonhole Green’s functions, through the use of analytical continuation method the Green’s functions for holes are now obtained in explicit closed-form for complete loading cases and are valid for the full fields. The Green’s functions for cracks are then obtained by letting the minor axis of ellipse be zero. By proper differentiation, the stress resultants and moments along the hole boundary and the stress intensity factors of cracks are also solved explicitly. Like the Green’s functions for the infinite laminates, only the solutions associated with the in-plane concentrated forces f^1,f^2 and out-of-plane concentrated moments m^1,m^2 have exactly the same form as those of the corresponding two-dimensional problems. For the cases under the concentrated force f^3 and torsion m^3, new types of solutions are obtained.

Cracks in Laminates Subjected to Concentrated Forces and Moments

2006 ◽  
Vol 306-308 ◽  
pp. 1-6
Author(s):  
Chyan Bin Hwu
Keyword(s):  
Closed Form ◽  
Green's Functions ◽  
Green’S Functions ◽  
Continuation Method ◽  
Experimental Studies ◽  
Arbitrary Point ◽  
Kernel Functions ◽  
Bending Analysis ◽  
Crack Problems ◽  
Concentrated Forces

The crack problems are important not only in macromechanics but also in micromechanics. Because of its importance a lot of analytical, numerical and experimental studies have been published in journals and books. Among them, the study of Green’s function attracts many researchers’ attention because analytically it may provide solutions for arbitrary loading through superposition and numerically it can be employed as the fundamental solutions for boundary element method and as the kernel functions of integral equations to consider crack interaction problems. Although a lot of Green’s functions have been presented in the literature, due to mathematical infeasibility most of them are restricted to two-dimensional problems and very few of them consider possible coupled stretching-bending analysis which may occur for general unsymmetric composite laminates subjected inplane and/or out-of-plane forces and moments. In this paper we consider an infinite composite laminate containing a traction-free crack subjected to concentrated forces and moments at an arbitrary point of the laminate. By employing Stroh-like formalism for the coupled stretching-bending analysis, recently the Green’s functions for the infinite laminates (without holes) were obtained in closed-form. Based upon the non-hole Green’s functions, through the use of analytical continuation method the Green’s functions for cracks are now obtained in explicit closed-form and are valid for the full fields. By proper differentiation, the associated stress intensity factors are also solved explicitly.

Temperature field evaluation of a two-dimensional partial heat flow problem through Green's Functions

2019 ◽  
Author(s):  
Guilherme Ramalho Costa ◽  
José Aguiar santos junior ◽  
José Ricardo Ferreira Oliveira ◽  
Jefferson Gomes do Nascimento ◽  
Gilmar Guimaraes
Keyword(s):  
Temperature Field ◽  
Heat Flow ◽  
Green's Functions ◽  
Flow Problem ◽  
Green’S Functions ◽  
Field Evaluation ◽  
Two Dimensional

THREE‐DIMENSIONAL INDUCED POLARIZATION AND ELECTROMAGNETIC MODELING

Geophysics ◽  
1975 ◽  
Vol 40 (2) ◽  
pp. 309-324 ◽  
Author(s):  
Gerald W. Hohmann
Keyword(s):  
Integral Equation ◽  
Green's Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
Induced Polarization ◽  
The Body ◽  
Scale Model ◽  
Two Dimensional ◽  
Electromagnetic Modeling ◽  
The Earth

The induced polarization (IP) and electromagnetic (EM) responses of a three‐dimensional body in the earth can be calculated using an integral equation solution. The problem is formulated by replacing the body by a volume of polarization or scattering current. The integral equation is reduced to a matrix equation, which is solved numerically for the electric field in the body. Then the electric and magnetic fields outside the inhomogeneity can be found by integrating the appropriate dyadic Green’s functions over the scattering current. Because half‐space Green’s functions are used, it is only necessary to solve for scattering currents in the body—not throughout the earth. Numerical results for a number of practical cases show, for example, that for moderate conductivity contrasts the dipole‐dipole IP response of a body five units in strike length approximates that of a two‐dimensional body. Moving an IP line off the center of a body produces an effect similar to that of increasing the depth. IP response varies significantly with conductivity contrast; the peak response occurs at higher contrasts for two‐dimensional bodies than for bodies of limited length. Very conductive bodies can produce negative IP response due to EM induction. An electrically polarizable body produces a small magnetic field, so that it is possible to measure IP with a sensitive magnetometer. Calculations show that horizontal loop EM response is enhanced when the background resistivity in the earth is reduced, thus confirming scale model results.

scholarly journals Green's functions for unsymmetric composite laminates with inclusions

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190437
Author(s):  
Chia-Wen Hsu ◽  
Chyanbin Hwu
Keyword(s):  
Composite Laminates ◽  
Green's Functions ◽  
Green’S Functions ◽  
Closed Form Solution ◽  
Bending Moment ◽  
Form Solution ◽  
Displacement Discontinuity ◽  
Computational Time ◽  
Out Of Plane ◽  
Logarithmic Functions

It is known that the stretching and bending deformations will be coupled together for the unsymmetric composite laminates under in-plane force and/or out-of-plane bending moment. Although Green's functions for unsymmetric composite laminates with elliptical elastic inclusions have been obtained by using Stroh-like formalism around 10 years ago, due to the ignoring of inconsistent rigid body movements of matrix and inclusion, the existing solution may lead to displacement discontinuity across the interface between matrix and inclusion. Due to the multi-valued characteristics of complex logarithmic functions appeared in Green's functions, special attention should be made on the proper selection of branch cuts of mapped variables. To solve these problems, in this study, the existing Green's functions are corrected and a simple way to correctly evaluate the mapped complex variable logarithmic functions is suggested. Moreover, to apply the obtained solutions to boundary element method, we also derive the explicit closed-form solution for Green's function of deflection. Since the continuity conditions along the interface have been satisfied in Green's functions, no meshes are required along the interface, which will save a lot of computational time and the results are much more accurate than any other numerical methods.

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