154
I. P. Markov and M. V. Markina
2016a, b, c; Auffray et al. 2013; dell’Isola et al. 2015; Rahali et al. 2015). Pantographic structures are an example of the mechanical metamaterials (Placidi et al.
2016; dell’Isola et al. 2019a, b, c). Identification of constitutive parameters and
validation of theoretical predictions of built models are only a few subjects where
numerical simulations are used on everyday basis (dell’Isola et al. 2016a, b, c, 2017;
Placidi et al. 2017).
Green’s functions or fundamental solutions are extensively used in the solution of
boundary value problems, and they have many applications in different fields, such
as acoustics, earthquake engineering, nondestructive evaluation of the materials,
studying propagation and scattering of elastic waves in elastic media and so on.
They are essential for the development of various formulations of Boundary Element
Method (BEM), which is a very powerful numerical tool for engineering analysis
(dell’Isola et al. 2019a). However, when the elastic media with arbitrary degree of
anisotropy is considered, implementation of any method based on integral equations
is somewhat hindered because static and dynamic anisotropic elastic fundamental
solutions are not available in the explicit and closed-form analytical expressions.
Efficiency of any BEM formulation utilizing the anisotropic elastic fundamental
solutions heavily depends on computation efficiency of the method employed to
evaluate said fundamental solutions.
Frequency and Laplace transformed dynamic anisotropic elastic fundamental
solutions can be represented as a sum of static (singular) and dynamic (regular)
parts. Static part does not depend on frequency and was intensely studied starting
with Fredholm (1900). Over the years, several different approaches to calculate the
static anisotropic elastic fundamental solutions and their derivatives were proposed:
using integral expressions (Lifshitz and Rozenzweig 1947; Barnett 1972), employing
the residue calculus (Sales and Gray 1998; Phan et al. 2004, 2005), expressions in
terms of the Stroh eigenvalues (Ting 1997; Lee 2009; Shiah et al. 2010; Buroni and
Sáez 2013; Xie et al. 2016) or eigenvectors (Malén 1971; Nakamura and Tanuma
1997), solution in form of uniformly convergent series (Mura and Kinoshita 1971), a
double Fourier-series expansion (Shiah et al. 2012; Tan et al. 2013), an interpolation
scheme using stored values (Wilson and Cruse 1978). A detailed review of works on
static anisotropic elastic Greens’ functions is provided by Pan and Chen (2015).
Regarding the dynamic parts of anisotropic elastic fundamental solutions, practically important results belong to Wang and Achenbach (1994, 1995) who used
Radon transform and obtained expressions in terms of integrals over the surface of
a half of a unit sphere. However, due to the oscillatory nature of the integrands of
these integrals, they cannot be efficiently calculated using traditional quadrature rules
(e.g., Newton–Cotes, Gauss–Legendre, Clenshaw–Curtis, etc.) for high frequencies
or large distances between source and observation points. Asymptotic high-frequency
anisotropic fundamental solutions in the far field were obtained by Vavryˇ cuk (2007).
For the particular case of transversely isotropic materials, a number of works are
dedicated to somewhat alleviate this problem: Sáez and Dom´ ınguez (1999) derived
expressions more suitable for numerical evaluation and later presented far field fundamental solutions (Sáez and Dom´ ınguez 2000), and Fooladi and Kundu (2019a, b)
were able to reduce integration domain from half of a unit sphere to a quarter.
I. P. Markov and M. V. Markina
2016a, b, c; Auffray et al. 2013; dell’Isola et al. 2015; Rahali et al. 2015). Pantographic structures are an example of the mechanical metamaterials (Placidi et al.
2016; dell’Isola et al. 2019a, b, c). Identification of constitutive parameters and
validation of theoretical predictions of built models are only a few subjects where
numerical simulations are used on everyday basis (dell’Isola et al. 2016a, b, c, 2017;
Placidi et al. 2017).
Green’s functions or fundamental solutions are extensively used in the solution of
boundary value problems, and they have many applications in different fields, such
as acoustics, earthquake engineering, nondestructive evaluation of the materials,
studying propagation and scattering of elastic waves in elastic media and so on.
They are essential for the development of various formulations of Boundary Element
Method (BEM), which is a very powerful numerical tool for engineering analysis
(dell’Isola et al. 2019a). However, when the elastic media with arbitrary degree of
anisotropy is considered, implementation of any method based on integral equations
is somewhat hindered because static and dynamic anisotropic elastic fundamental
solutions are not available in the explicit and closed-form analytical expressions.
Efficiency of any BEM formulation utilizing the anisotropic elastic fundamental
solutions heavily depends on computation efficiency of the method employed to
evaluate said fundamental solutions.
Frequency and Laplace transformed dynamic anisotropic elastic fundamental
solutions can be represented as a sum of static (singular) and dynamic (regular)
parts. Static part does not depend on frequency and was intensely studied starting
with Fredholm (1900). Over the years, several different approaches to calculate the
static anisotropic elastic fundamental solutions and their derivatives were proposed:
using integral expressions (Lifshitz and Rozenzweig 1947; Barnett 1972), employing
the residue calculus (Sales and Gray 1998; Phan et al. 2004, 2005), expressions in
terms of the Stroh eigenvalues (Ting 1997; Lee 2009; Shiah et al. 2010; Buroni and
Sáez 2013; Xie et al. 2016) or eigenvectors (Malén 1971; Nakamura and Tanuma
1997), solution in form of uniformly convergent series (Mura and Kinoshita 1971), a
double Fourier-series expansion (Shiah et al. 2012; Tan et al. 2013), an interpolation
scheme using stored values (Wilson and Cruse 1978). A detailed review of works on
static anisotropic elastic Greens’ functions is provided by Pan and Chen (2015).
Regarding the dynamic parts of anisotropic elastic fundamental solutions, practically important results belong to Wang and Achenbach (1994, 1995) who used
Radon transform and obtained expressions in terms of integrals over the surface of
a half of a unit sphere. However, due to the oscillatory nature of the integrands of
these integrals, they cannot be efficiently calculated using traditional quadrature rules
(e.g., Newton–Cotes, Gauss–Legendre, Clenshaw–Curtis, etc.) for high frequencies
or large distances between source and observation points. Asymptotic high-frequency
anisotropic fundamental solutions in the far field were obtained by Vavryˇ cuk (2007).
For the particular case of transversely isotropic materials, a number of works are
dedicated to somewhat alleviate this problem: Sáez and Dom´ ınguez (1999) derived
expressions more suitable for numerical evaluation and later presented far field fundamental solutions (Sáez and Dom´ ınguez 2000), and Fooladi and Kundu (2019a, b)
were able to reduce integration domain from half of a unit sphere to a quarter.