Abstract
An approximate solution capability is developed to handle three dimensional anisotropic elastostatic boundary value problems. The method depends crucially on the existence and explicit definition of a fundamental solution to the governing partial differential equations. The construction of this solution for the anisotropic elastostatic problem is presented as is the derivation of the expression for the surface tractions necessary to maintain the fundamental solution in a bounded region. After the fundamental solution and its associated surface tractions are determined, a real variable boundary integral formula is generated which can be solved numerically for the unknown surface tractions and displacements in a well-posed boundary value problem. Once all boundary quantities are known, the field solution is given by a Somigliana type integral formula. Techniques for numerically solving the integral equations are discussed.
Zusammenfassung
Es wird eine Näherungslösung entwickelt die es dreidimensionale anisotrope elastostatische Randwertprobleme zu lösen. Die Methode hängt entscheidend vom Vorhandensein und expliziten Bestimmung einer Grundlösung der zugeehörigen partiellen Differentialgleichungen. Die Êntwicklung dieser Lösung fur den Fall eines anisotropen elastostatischen Problems wird gegeben wie auch die Ableitung einer Formel fur Oberflächenspannung die erforderlich ist, um die Gultigkeit der Grundlösung im Randgebiet fortbestehen zu lassen. Nachdem die Grundlösung und die mit ihr verbundenen Oberflächenspannungen gefunden worden sind, wird eine Randwertformel in Integralform fur reelle Variablen entwickelt, die numerisch fur unbekannte Oberflächenspannungen und Verschiebungen jedoch mit gut definierten Randwerten gelöst werden kann. Nachdem alle Randwerte bekannt sind ist die Feldlösung durch eine Integralformel vom Typ Somigliana gegeben. Verfahren zur numerischen Lösung der Integralgleichungen werden erörtert.
Similar content being viewed by others
References
Rizzo F. J., “An Integral Equation Approach to Boundary Value Problems of Classical Elastostatics,” Quarterly of Applied Mathematics, 25 (1967) 83
Cruse T. A. and Rizzo F. J., “A Direct Formulation and Numerical Solution of the General Transient Elastodynamic Problem. I,” Journal of Mathematical Analysis and Applications, 22 (1968) 244
Cruse T. A., “A Direct Formulation and Numerical Solution of the General Transient Elastodynamic Problem. II,” Journal of Mathematical Analysis and Applications, 22 (1968) 341
Rizzo F. J. and Shippy D. J., “A Formulation and Solution Procedure for the General Non-Homogeneous Elastic Inclusion Problem,” International Journal of Solids and Structures, 4 (1968) 1161
Rizzo F. J. and Shippy D. J., “A Method for Stress Determination in Plane Anisotropic Elastic Bodies,” Journal of Composite Materials, 4 (1970) 36
Cruse T. A., “Numerical Solutions in Three Dimensional Elastostatics,” International Journal of Solids and Structures, 5 (1969) 1259
John F., Plane Waves and Spherical Means Applied to Partial Differential Equations, Interscience, New York 1955
Synge J. L., The Hypercircle in Mathematical Physics, Cambridge University Press, Cambridge 1957
Stakgold I., Boundary Value Problems of Mathematical Physics. Volume II, Macmillan, New York 1968
John F., Partial Differential Equations, Springer-Verlag, New York 1971
Willis J. R., “The Elastic Interaction Energy of Dislocation Loops in Anisotropic Media,” Quarterly Journal of Mechanics and Applied Mathematics, 18 (1965) 419
Nickerson H. K., Spencer D. C. and Steenrod N. E., Advanced Calculus, Van Nostrand, Princeton 1959
Gel'fand I. M., Graev M. I. and Vilenkin N. Ya., Generalized Functions: Volume 5 Integral Geometry and Representation Theory, Academic Press, New York 1966
Gel'fand I. M. and Shilov G. E., Generalized Functions: Volume 1 Properties and Operations, Academic Press, New York 1964.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Vogel, S.M., Rizzo, F.J. An integral equation formulation of three dimensional anisotropic elastostatic boundary value problems. J Elasticity 3, 203–216 (1973). https://doi.org/10.1007/BF00052894
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00052894