Elastostatic and elastodynamic Green's functions
For a brief review, please see the following paper in pdf and other references given there.
V. K. Tewary: "Elastic Green’s functions for anisotropic solids (review)," Proceedings of the NIST workshop on Green’s functions and boundary element analysis, NIST Special Publication SP 910 (1996). (Download, pdf)
Brief summary of research work
Movies - Results in wave propagation
The elastic Green's functions are used to solve the Christoffel's equation for elastic equilibrium with prescribed boundary conditions. At NIST, we have developed computationally efficient methods for calculating Green's functions for anisotropic solids.
We use the virtual-force method for satisfying the boundary conditions to account for discontinuities in the solids. The virtual-force method consists of applying a distribution of virtual forces just outside the domain of solution. The solution obtained by using the virtual-force distribution and the Green’s function gives a solution of the homogeneous equation. The virtual-force distribution is then determined by imposing the prescribed boundary conditions. This method is similar to the image-charge method in electrostatics and the boundary-element method for solving elastostatic problems.
The Fourier representation of the Green’s function is quite general and, subject to certain well-known conditions of integrability and convergence, can be used for most physical problems. In the case of elastodynamic Green’s functions, the Fourier representation is CPU intensive and is not computationally efficient for anisotropic solids.
Green’s function gives the response of a solid to a unit force. It gives the solution of the Christoffel equations of elastic equilibrium for prescribed boundary conditions and an arbitrary integrable force distribution. Since it gives the response of the whole solid, it provides a convenient mathematical technique to model the elastostatic and elastodynamic properties of anisotropic solids. The elastodynamic Green’s function is used to calculate the elastic waveforms and other characteristics of the elastic wave propagation induced by a pulsed load. The elastostatic Green’s function is used to calculate stress distribution due to a static load in solids containing cracks and/or other discontinuities such as interfaces and free surfaces. Combined with the boundary element analysis, the Green’s function method provides a powerful numerical tool for stress analysis of engineering materials of different geometrical shapes.
A major computational advantage of the Green’s function is that it is a characteristic of the material and its geometry, and is independent of the loading. The Green’s function can be calculated in steps of increasing geometrical complexities using the previous value as an input. Green’s function It is, therefore, possible to calculate and store the Green’s functions for basic geometrical shapes and different material parameters for use in further calculations. The strategy is that we calculate the Green’s function analytically for simple geometrical shapes and use the boundary element method for application to complicated geometrical shapes for engineering applications.
The Green’s function method is also applicable for perturbation solution of the nonlinear elastic problems. We are working on using the Green’s function method for obtaining approximate solutions of the nonlinear Christoffel equation for modeling the nonlinear elastic wave propagation in anisotropic solids. Study of nonlinear properties of elastic waves can be used for determination of microstructural characterization of advanced materials.
We calculate the Green’s functions by using a delta function representation that we had developed earlier. In the delta function representation, we use a linear combination of the space and time variables instead of using the two separately. This representation has been found to be computationally very efficient. We have already applied it to various problems on elastic wave propagation such as elastic waveforms in anisotropic half space solids and plates due to a pulsed load. We are trying to develop it further to solve the nonlinear Christoffel equation.
In elastostatics the delta function representation in two dimensions becomes similar to the Stroh’s representation. For solids containing elastic discontinuities, the requirement of satisfying the prescribed boundary conditions involves solution of a Hilbert problem. We have developed an orthogonal complex transform to solve the Hilbert problem and applied it to free edges, interfacial cracks, and cracks inclined at an arbitrary angle to the interface in bimaterial composites. Using the calculated Green’s functions, we have developed boundary element formulations for interface cracks and analysis of moire fields in anisotropic materials. In three dimensions, to which the Stroh’s representation is not applicable, the delta function representation provides a computationally efficient method for calculating the Green’s function and the stress distribution in anisotropic solids. These calculations will serve as the basic input to the three dimensional boundary element formulation and storage of Green’s functions for engineering applications.
Brief summary of research work
A computationally efficient delta function represntation has been developed for elastodynamic and elastostatic Green's functions of anisotropic solids.
Elastic waveforms were calculated for surface waves in tetragonal solids, and thick anisotropic plates with cubic and general orthorhombic symmetry. The theoretical results on plates were used to interpret the experimental results and to estimate the elastic constants. Waveforms in frequency space were calculated for harmonic loads in cubic and orthorhombic half-space solids. Mechanical impedance of anisotropic half-space solids was also calculated. It was shown that the singularity in the imaginary part of the mechanical admittance is an artifact of the continuum model and does not arise if a correction is introduced to account for the discrete structure of the lattice. corrections are
Elastic Green’s function was calculated for a damaged interface in anisotropic bimaterial composites. A boundary element formulation was developed for interfacial cracks and interpretation of moire field data in anisotropic bimaterial composites. The Green’s function and stress intensity factor was calculated for an crack inclined to the interface in an anisotropic bimaterial composite. It was found that the stress near the crack tip remains oscillatory for small angles but not for large angles between the cack and the interface.
Elastic waveforms were calculated for nonlinear bulk waves in cubic solids as function of frequency. At large frequencies in the range of experimental interest the results are very close to those reported in the literature using crude approximations. Similarly, a perturbation calculation of the nonlinear Rayleigh waves reproduces qualitatively the same result as reported in the literature.
A boundary-element formulation for the anisotropic crack problems based upon a Green’s function has been developed. The traction free boundary conditions on the crack faces are satisfied exactly with the Green’s function using virtual forces so that no discretization of the crack surfaces is necessary. The Green’s function contains both the inverse square root and oscillatory singularities associated with the elastic anisotropic interface crack problem. Results are obtained for interface cracking in a copper nickel bimaterial.
A procedure for analyzing moire fringe patterns using boundary elements is developed. The kernels of the boundary integrals are based on anisotropic elastic Green’s functions calculated earlier for bimaterial problems. The kernels are appropriate for homogeneous problems as well as degenerate isotropic problems. The moire fringe data provide full field displacement information and are analyzed in a least squares’ sense. The numerical procedure is shown to be a logical extension of the local collocation method developed for linear elastic fracture problems. It is found that moire fields associated with both displacement components are needed for an accurate analysis.
An exact transient thermoelastic Green’s function has been derived for an anisotropic bimaterial. The analysis is based upon an extension of the Stroh formalism for anisotropic elasticity to the anisotropic thermoelastic problem. The application of the exact Green’s function to boundary integral analysis of thermal barrier coatings is being studied.