11 Numerical Evaluation of Integrals in Laplace Domain Anisotropic …
169
The question of selecting required number of subintervals to attain desirable accuracy before performing the integration still remains open. For evaluations of dynamic
anisotropic elastic fundamental solutions in practical applications, it is possible to
employ an approach similar to the one presented by Fooladi and Kundu (2019a,
b). For orthotropic materials, they proposed a calibration strategy for choosing the
number of integration points for the dynamic part of the anisotropic Green’s function. Their technique involves determining an isotropic equivalent of the original
anisotropic elasticity tensor and then using it to find an optimum number of integration points for a user-defined set of distances between source and observation
points.
Acknowledgements The work is financially supported by the Russian Science Foundation under
grant No. 18-79-00082.
References
Alibert, J., Seppecher, P., & dell’Isola, F. (2003). Truss modular beams with deformation energy
depending on higher displacement gradients. Mathematics and Mechanics of Solids, 8, 51–73.
https://doi.org/10.1177/1081286503008001658.
Auffray, N., dell’Isola, F., Eremeyev, V., Madeo, A., & Rossi, G. (2013). Analytical continuum
mechanics à la Hamilton-Piola: Least action principle for second gradient continua and capillary
fluids. Mathematics and Mechanics of Solids, 20, 375–417. https://doi.org/10.1177/108128651
3497616.
Barchiesi, E., Spagnuolo, M., & Placidi, L. (2018). Mechanical metamaterials: A state of the art.
Mathematics and Mechanics of Solids, 24, 212–234. https://doi.org/10.1177/1081286517735695.
Barnett, D. (1972). The precise evaluation of derivatives of the anisotropic elastic Green’s functions.
Physica Status Solidi (B), 49, 741–748. https://doi.org/10.1002/pssb.2220490238.
Bogaert, I. (2014). Iteration-free computation of Gauss-Legendre quadrature nodes and weights.
SIAM Journal on Scientific Computing, 36, A1008–A1026. https://doi.org/10.1137/140954969.
Buroni, F., & Sáez, A. (2013). Unique and explicit formulas for green’s function in three-dimensional
anisotropic linear elasticity. Journal of Applied Mechanics, 80, 051018. https://doi.org/10.1115/
1.4023627.
Del Vescovo, D., & Giorgio, I. (2014). Dynamic problems for metamaterials: Review of existing
models and ideas for further research. International Journal of Engineering Science, 80, 153–172.
https://doi.org/10.1016/j.ijengsci.2014.02.022.
dell’Isola, F., Andreaus, U., & Placidi, L. (2015). At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical
contribution of Gabrio Piola. Mathematics and Mechanics of Solids, 20(8), 887–928. https://doi.
org/10.1177/1081286513509811.
dell’Isola, F., Cuomo, M., Greco, L., & Della Corte, A. (2017). Bias extension test for pantographic
sheets: Numerical simulations based on second gradient shear energies. Journal of Engineering
Mathematics, 103(1), 127–157. https://doi.org/10.1007/s10665-016-9865-7.
dell’Isola, F., Della Corte, A., & Giorgio, I. (2016a). Higher-gradient continua: The legacy of Piola,
Mindlin, Sedov and Toupin and some future research perspectives. Mathematics and Mechanics
of Solids, 22(4), 852–872. https://doi.org/10.1177/1081286515616034.
dell’Isola, F., Della Corte, A., Greco, L., & Luongo, A. (2016b). Plane bias extension test for
a continuum with two inextensible families of fibers: A variational treatment with Lagrange
169
The question of selecting required number of subintervals to attain desirable accuracy before performing the integration still remains open. For evaluations of dynamic
anisotropic elastic fundamental solutions in practical applications, it is possible to
employ an approach similar to the one presented by Fooladi and Kundu (2019a,
b). For orthotropic materials, they proposed a calibration strategy for choosing the
number of integration points for the dynamic part of the anisotropic Green’s function. Their technique involves determining an isotropic equivalent of the original
anisotropic elasticity tensor and then using it to find an optimum number of integration points for a user-defined set of distances between source and observation
points.
Acknowledgements The work is financially supported by the Russian Science Foundation under
grant No. 18-79-00082.
References
Alibert, J., Seppecher, P., & dell’Isola, F. (2003). Truss modular beams with deformation energy
depending on higher displacement gradients. Mathematics and Mechanics of Solids, 8, 51–73.
https://doi.org/10.1177/1081286503008001658.
Auffray, N., dell’Isola, F., Eremeyev, V., Madeo, A., & Rossi, G. (2013). Analytical continuum
mechanics à la Hamilton-Piola: Least action principle for second gradient continua and capillary
fluids. Mathematics and Mechanics of Solids, 20, 375–417. https://doi.org/10.1177/108128651
3497616.
Barchiesi, E., Spagnuolo, M., & Placidi, L. (2018). Mechanical metamaterials: A state of the art.
Mathematics and Mechanics of Solids, 24, 212–234. https://doi.org/10.1177/1081286517735695.
Barnett, D. (1972). The precise evaluation of derivatives of the anisotropic elastic Green’s functions.
Physica Status Solidi (B), 49, 741–748. https://doi.org/10.1002/pssb.2220490238.
Bogaert, I. (2014). Iteration-free computation of Gauss-Legendre quadrature nodes and weights.
SIAM Journal on Scientific Computing, 36, A1008–A1026. https://doi.org/10.1137/140954969.
Buroni, F., & Sáez, A. (2013). Unique and explicit formulas for green’s function in three-dimensional
anisotropic linear elasticity. Journal of Applied Mechanics, 80, 051018. https://doi.org/10.1115/
1.4023627.
Del Vescovo, D., & Giorgio, I. (2014). Dynamic problems for metamaterials: Review of existing
models and ideas for further research. International Journal of Engineering Science, 80, 153–172.
https://doi.org/10.1016/j.ijengsci.2014.02.022.
dell’Isola, F., Andreaus, U., & Placidi, L. (2015). At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical
contribution of Gabrio Piola. Mathematics and Mechanics of Solids, 20(8), 887–928. https://doi.
org/10.1177/1081286513509811.
dell’Isola, F., Cuomo, M., Greco, L., & Della Corte, A. (2017). Bias extension test for pantographic
sheets: Numerical simulations based on second gradient shear energies. Journal of Engineering
Mathematics, 103(1), 127–157. https://doi.org/10.1007/s10665-016-9865-7.
dell’Isola, F., Della Corte, A., & Giorgio, I. (2016a). Higher-gradient continua: The legacy of Piola,
Mindlin, Sedov and Toupin and some future research perspectives. Mathematics and Mechanics
of Solids, 22(4), 852–872. https://doi.org/10.1177/1081286515616034.
dell’Isola, F., Della Corte, A., Greco, L., & Luongo, A. (2016b). Plane bias extension test for
a continuum with two inextensible families of fibers: A variational treatment with Lagrange