A generalized integro-differential theory of nonlocal elasticity of n-Helmholtz type—part II: boundary-value problems in the one-dimensional case

Abstract

This paper is the second in a series of two that deal with a generalized theory of nonlocal elasticity of n-Helmholtz type. This terminology is motivated by the fact that the attenuation function (kernel) of the integral type nonlocal constitutive equation is the Green function associated with a generalized Helmholtz differential operator of order n. In the first paper, the governing equations have been derived and supported by suitable thermodynamic arguments. In this second paper, the proposed nonlocal model is specialized for the one-dimensional case to solve boundary-value problems. First, the relevant higher-order nonstandard boundary conditions in the differential (or, more precisely, integro-differential) version of the theory are derived. These boundary conditions are consistent with the particular family of attenuation functions adopted in the integral formulation. Then, some simple applications to statics and dynamics problems are presented. In particular, the theory is used to capture the static response and to perform free vibration analysis of a discrete lattice model with periodic microstructure (mass-and-spring chain) featured by nearest neighbor and next nearest neighbor particle interactions. In the latter case, boundary effects arise at the two lattice ends that are well captured by the proposed nonlocal continuum formulation. The nonlocal material parameters are identified a priori by matching the dispersion curve of the discrete lattice model, and a comparison in terms of attenuation function is also presented.

Appendix A: Nonstandard boundary conditions of order n applied to nonlocal strain of order less than n

In this Appendix, we present some supplemental nonstandard boundary conditions that useful for the determination of the eigenfunctions $\phi (x)$ in the free vibration analysis. These supplemental relationships arise from applying the nonstandard boundary conditions of order n, i.e. involving n ${\mathscr{L}}$ differential operators, to nonlocal strain of order of order k less than n, i.e., $k = 1,2, \ldots ,n - 1$. Let us consider the expression of $\bar{\varepsilon }_{x}^{(1)} (x)$ given in (33). By exploiting the Leibniz integral rule (36), the nonstandard boundary conditions for $n = 2$ as per (41) and (42) applied to $\bar{\varepsilon }_{x}^{(1)} (x)$ leads to

$$\begin{aligned} & {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} ( - )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} ( - )}} \left. {\bar{\varepsilon }_{x}^{(1)} (x)} \right|_{x = 0} = \left. {\left[ {\left( {1 - (c_{1} + c_{2} )\frac{d}{dx} + c_{1} c_{2} \frac{{d^{2} }}{{dx^{2} }}} \right)\bar{\varepsilon }_{x}^{(1)} (x)} \right]} \right|_{x = 0} = - \frac{{c_{2} }}{{c_{1} }}\varepsilon_{x} (0), \\ & {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} ( - )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} (1)}} \left. {\bar{\varepsilon }_{x}^{(1)} (x)} \right|_{x = 0} = \left. {\left[ {\left( {1 - c_{2} \frac{d}{dx} - c_{1}^{2} \frac{{d^{2} }}{{dx^{2} }} + c_{1}^{2} c_{2} \frac{{d^{3} }}{{dx^{3} }}} \right)\bar{\varepsilon }_{x}^{(1)} (x)} \right]} \right|_{x = 0} = \varepsilon_{x} (0) - c_{2} \varepsilon_{x} '(0), \\ & {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} ( + )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} ( + )}} \left. {\bar{\varepsilon }_{x}^{(1)} (x)} \right|_{x = L} = \left. {\left[ {\left( {1 + (c_{1} + c_{2} )\frac{d}{dx} + c_{1} c_{2} \frac{{d^{2} }}{{dx^{2} }}} \right)\bar{\varepsilon }_{x}^{(1)} (x)} \right]} \right|_{x = L} = - \frac{{c_{2} }}{{c_{1} }}\varepsilon_{x} (L), \\ & {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} ( + )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} (1)}} \left. {\bar{\varepsilon }_{x}^{(1)} (x)} \right|_{x = L} = \left. {\left[ {\left( {1 + c_{2} \frac{d}{dx} - c_{1}^{2} \frac{{d^{2} }}{{dx^{2} }} - c_{1}^{2} c_{2} \frac{{d^{3} }}{{dx^{3} }}} \right)\bar{\varepsilon }_{x}^{(1)} (x)} \right]} \right|_{x = L} = \varepsilon_{x} (L) + c_{2} \varepsilon_{x} '(L). \\ \end{aligned}$$

(127)

By extending this result, the nonstandard boundary conditions for $n = 3$ as per (44) and (45) applied to $\bar{\varepsilon }_{x}^{(1)} (x)$ leads to

$$\begin{aligned} & {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;( - )}} \left. {\bar{\varepsilon }_{x}^{(1)} (x)} \right|_{x = 0} \\ & \quad \left. { = \left[ {\left( {1 - (c_{1} + c_{2} + c_{3} )\frac{d}{dx} + (c_{1} c_{2} + c_{1} c_{3} + c_{2} c_{3} )\frac{{d^{2} }}{{dx^{2} }} - c_{1} c_{2} c_{3} \frac{{d^{3} }}{{dx^{3} }}} \right)\bar{\varepsilon }_{x}^{(1)} (x)} \right]} \right|_{x = 0} \\ & \quad = - \frac{{\left( {c_{2} c_{3} + c_{1} c_{2} + c_{1} c_{3} } \right)}}{{c_{1}^{2} }}\varepsilon_{x} (0) + \frac{{c_{1} c_{2} c_{3} }}{{c_{1}^{2} }}\varepsilon_{x}^{'} (0), \\ & {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;(1)}} \left. {\bar{\varepsilon }_{x}^{(1)} (x)} \right|_{x = 0} \\ & \quad = \left. {\left[ {\left( {1 - (c_{2} + c_{3} )\frac{d}{dx} + (c_{2} c_{3} - c_{1}^{2} )\frac{{d^{2} }}{{dx^{2} }} + (c_{1}^{2} c_{2} + c_{1}^{2} c_{3} )\frac{{d^{3} }}{{dx^{3} }} - c_{1}^{2} c_{2} c_{3} \frac{{d^{4} }}{{dx^{4} }}} \right)\bar{\varepsilon }_{x}^{(1)} (x)} \right]} \right|_{x = 0} \\ & \quad = \varepsilon_{x} (0) - \left( {c_{2} + c_{3} } \right)\varepsilon_{x}^{'} (0) + c_{2} c_{3} \varepsilon_{x}^{''} (0), \\ & {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;(1)}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;(1)}} \left. {\bar{\varepsilon }_{x}^{(1)} (x)} \right|_{x = 0} = {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{1} ,c_{2} }}^{{{\kern 1pt} \;(2)}} \left. {\bar{\varepsilon }_{x}^{(1)} (x)} \right|_{x = 0} \\ & \quad = \left. {\left[ {\left( {1 - c_{3} \frac{d}{dx} - (c_{1}^{2} + c_{2}^{2} )\frac{{d^{2} }}{{dx^{2} }} + (c_{1}^{2} c_{3} + c_{2}^{2} c_{3} )\frac{{d^{3} }}{{dx^{3} }} + c_{1}^{2} c_{2}^{2} \frac{{d^{4} }}{{dx^{4} }} - c_{1}^{2} c_{2}^{2} c_{3} \frac{{d^{5} }}{{dx^{5} }}} \right)\bar{\varepsilon }_{x}^{(1)} (x)} \right]} \right|_{x = 0} \\ & \quad = \varepsilon_{x} (0) - c_{3} \varepsilon_{x}^{'} (0) - c_{2}^{2} \varepsilon_{x}^{''} (0) + c_{2}^{2} c_{3} \varepsilon_{x}^{'''} (0), \\ & {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;( + )}} \left. {\bar{\varepsilon }_{x}^{(1)} (x)} \right|_{x = L} \\ & \quad \left. { = \left[ {\left( {1 + (c_{1} + c_{2} + c_{3} )\frac{d}{dx} + (c_{1} c_{2} + c_{1} c_{3} + c_{2} c_{3} )\frac{{d^{2} }}{{dx^{2} }} + c_{1} c_{2} c_{3} \frac{{d^{3} }}{{dx^{3} }}} \right)\bar{\varepsilon }_{x}^{(1)} (x)} \right]} \right|_{x = L} \\ & \quad = - \frac{{\left( {c_{2} c_{3} + c_{1} c_{2} + c_{1} c_{3} } \right)}}{{c_{1}^{2} }}\varepsilon_{x} (L) + \frac{{c_{1} c_{2} c_{3} }}{{c_{1}^{2} }}\varepsilon_{x}^{'} (L), \\ & {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;(1)}} \left. {\bar{\varepsilon }_{x}^{(1)} (x)} \right|_{x = L} \\ & \quad \left. { = \left[ {\left( {1 + (c_{2} + c_{3} )\frac{d}{dx} + (c_{2} c_{3} - c_{1}^{2} )\frac{{d^{2} }}{{dx^{2} }} - (c_{1}^{2} c_{2} + c_{1}^{2} c_{3} )\frac{{d^{3} }}{{dx^{3} }} - c_{1}^{2} c_{2} c_{3} \frac{{d^{4} }}{{dx^{4} }}} \right)\bar{\varepsilon }_{x}^{(1)} (x)} \right]} \right|_{x = L} \\ & \quad = \varepsilon_{x} (L) + \left( {c_{2} + c_{3} } \right)\varepsilon_{x}^{'} (L) + c_{2} c_{3} \varepsilon_{x}^{''} (L), \\ & {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;(1)}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;(1)}} \left. {\bar{\varepsilon }_{x}^{(1)} (x)} \right|_{x = L} = {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{1} ,c_{2} }}^{{{\kern 1pt} \;(2)}} \left. {\bar{\varepsilon }_{x}^{(1)} (x)} \right|_{x = L} \\ & \quad \left. { = \left[ {\left( {1 + c_{3} \frac{d}{dx} - (c_{1}^{2} + c_{2}^{2} )\frac{{d^{2} }}{{dx^{2} }} - (c_{1}^{2} c_{3} + c_{2}^{2} c_{3} )\frac{{d^{3} }}{{dx^{3} }} + c_{1}^{2} c_{2}^{2} \frac{{d^{4} }}{{dx^{4} }} + c_{1}^{2} c_{2}^{2} c_{3} \frac{{d^{5} }}{{dx^{5} }}} \right)\bar{\varepsilon }_{x}^{(1)} (x)} \right]} \right|_{x = L} \\ & \quad = \varepsilon_{x} (L) + c_{3} \varepsilon_{x}^{'} (L) - c_{2}^{2} \varepsilon_{x}^{''} (L) - c_{2}^{2} c_{3} \varepsilon_{x}^{'''} (L). \\ \end{aligned}$$

(128)

And, finally, the nonstandard boundary conditions for $n = 4$ as per (46) applied to $\bar{\varepsilon }_{x}^{(1)} (x)$ leads to

$$\begin{aligned} & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;( - )}} \left. {\bar{\varepsilon }_{x}^{(1)} (x)} \right|_{x = 0} \\ & \quad = - \frac{{\left( {c_{2} c_{3} c_{4} + c_{1} c_{3} c_{4} + c_{1} c_{2} c_{3} + c_{1} c_{2} c_{4} + c_{1}^{2} \left( {c_{2} + c_{3} + c_{4} } \right)} \right)}}{{c_{1}^{3} }}\varepsilon_{x} (0) \\ & \quad - \frac{{\left( {c_{1}^{2} c_{2} c_{3} - c_{1}^{2} c_{2} c_{4} - c_{1}^{2} c_{3} c_{4} - c_{1} c_{2} c_{3} c_{4} } \right)}}{{c_{1}^{3} }}\varepsilon_{x}^{'} (0) - \frac{{\left( {c_{2} c_{3} c_{4} } \right)}}{{c_{1} }}\varepsilon_{x}^{''} (0), \\ & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;(1)}} \left. {\bar{\varepsilon }_{x}^{(1)} (x)} \right|_{x = 0} \\ & \quad = \varepsilon_{x} (0) - \left( {c_{2} + c_{3} + c_{4} } \right)\varepsilon_{x}^{'} (0) + \left( {c_{2} c_{3} + c_{2} c_{4} + c_{3} c_{4} } \right)\varepsilon_{x}^{''} (0) - \left( {c_{2} c_{3} c_{4} } \right)\varepsilon_{x}^{'''} (0), \\ & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;(1)}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;(1)}} \left. {\bar{\varepsilon }_{x}^{(1)} (x)} \right|_{x = 0} \\ & \quad = \varepsilon_{x} (0) - \left( {c_{3} + c_{4} } \right)\varepsilon_{x}^{'} (0) + \left( {c_{3} c_{4} - c_{2}^{2} } \right)\varepsilon_{x}^{''} (0) + \left( {c_{2}^{2} c_{3} + c_{2}^{2} c_{4} } \right)\varepsilon_{x}^{'''} (0) - \left( {c_{2}^{2} c_{3} c_{4} } \right)\varepsilon_{x}^{''''} (0), \\ & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;(1)}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;(1)}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;(1)}} \left. {\bar{\varepsilon }_{x}^{(1)} (x)} \right|_{x = 0} = {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{1} ,c_{2} ,c_{3} }}^{{{\kern 1pt} \;(3)}} \left. {\bar{\varepsilon }_{x}^{(1)} (x)} \right|_{x = 0} \\ & \quad = \varepsilon_{x} (0) - \left( {c_{4} } \right)\varepsilon_{x}^{'} (0) - \left( {c_{2}^{2} - c_{3}^{2} } \right)\varepsilon_{x}^{''} (0) + \left( {c_{2}^{2} c_{4} + c_{3}^{2} c_{4} } \right)\varepsilon_{x}^{'''} (0) + \left( {c_{2}^{2} c_{3}^{2} } \right)\varepsilon_{x}^{''''} (0) - \left( {c_{2}^{2} c_{3}^{2} c_{4} } \right)\varepsilon_{x}^{'''''} (0), \\ & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;( + )}} \left. {\bar{\varepsilon }_{x}^{(1)} (x)} \right|_{x = L} \\ & \quad = - \frac{{\left( {c_{2} c_{3} c_{4} + c_{1} c_{3} c_{4} + c_{1} c_{2} c_{3} + c_{1} c_{2} c_{4} + c_{1}^{2} \left( {c_{2} + c_{3} + c_{4} } \right)} \right)}}{{c_{1}^{3} }}\varepsilon_{x} (L) \\ & \quad + \frac{{\left( {c_{1}^{2} c_{2} c_{3} - c_{1}^{2} c_{2} c_{4} - c_{1}^{2} c_{3} c_{4} - c_{1} c_{2} c_{3} c_{4} } \right)}}{{c_{1}^{3} }}\varepsilon_{x}^{'} (0) - \frac{{\left( {c_{2} c_{3} c_{4} } \right)}}{{c_{1} }}\varepsilon_{x}^{''} (L), \\ & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;(1)}} \left. {\bar{\varepsilon }_{x}^{(4)} (x)} \right|_{x = L} \\ & \quad = \varepsilon_{x} (L) + \left( {c_{2} + c_{3} + c_{4} } \right)\varepsilon_{x}^{'} (L) + \left( {c_{2} c_{3} + c_{2} c_{4} + c_{3} c_{4} } \right)\varepsilon_{x}^{''} (L) + \left( {c_{2} c_{3} c_{4} } \right)\varepsilon_{x}^{'''} (L), \\ & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;(1)}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;(1)}} \left. {\bar{\varepsilon }_{x}^{(4)} (x)} \right|_{x = L} = {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{1} ,c_{2} }}^{{{\kern 1pt} \;(2)}} \left. {\bar{\varepsilon }_{x}^{(4)} (x)} \right|_{x = L} \\ & \quad = \varepsilon_{x} (L) + \left( {c_{3} + c_{4} } \right)\varepsilon_{x}^{'} (L) + \left( {c_{3} c_{4} - c_{2}^{2} } \right)\varepsilon_{x}^{''} (L) - \left( {c_{2}^{2} c_{3} + c_{2}^{2} c_{4} } \right)\varepsilon_{x}^{'''} (L) - \left( {c_{2}^{2} c_{3} c_{4} } \right)\varepsilon_{x}^{''''} (L), \\ & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;(1)}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;(1)}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;(1)}} \left. {\bar{\varepsilon }_{x}^{(4)} (x)} \right|_{x = L} = {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{1} ,c_{2} ,c_{3} }}^{{{\kern 1pt} \;(3)}} \left. {\bar{\varepsilon }_{x}^{(4)} (x)} \right|_{x = L} \\ & \quad = \varepsilon_{x} (0) + \left( {c_{4} } \right)\varepsilon_{x}^{'} (L) - \left( {c_{2}^{2} + c_{3}^{2} } \right)\varepsilon_{x}^{''} (L) - \left( {c_{2}^{2} c_{4} + c_{3}^{2} c_{4} } \right)\varepsilon_{x}^{'''} (L) + \left( {c_{2}^{2} c_{3}^{2} } \right)\varepsilon_{x}^{''''} (L) + \left( {c_{2}^{2} c_{3}^{2} c_{4} } \right)\varepsilon_{x}^{'''''} (L). \\ \end{aligned}$$

(129)

Now, let us consider the expression of $\bar{\varepsilon }_{x}^{(2)} (x)$ given in (37). The nonstandard boundary conditions for $n = 3$ as per (44) and (45) applied to $\bar{\varepsilon }_{x}^{(2)} (x)$ leads to

$$\begin{aligned} & {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;( - )}} \left. {\bar{\varepsilon }_{x}^{(2)} (x)} \right|_{x = 0} = \cdots = 0, \\ & {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;(1)}} \left. {\bar{\varepsilon }_{x}^{(2)} (x)} \right|_{x = 0} = \cdots = - \frac{{c_{3} }}{{c_{2} }}\varepsilon_{x} (0), \\ & {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;(1)}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;(1)}} \left. {\bar{\varepsilon }_{x}^{(2)} (x)} \right|_{x = 0} = {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{1} ,c_{2} }}^{{{\kern 1pt} \;(2)}} \left. {\bar{\varepsilon }_{x}^{(1)} (x)} \right|_{x = 0} = \cdots = \varepsilon_{x} (0) - c_{3} \varepsilon_{x}^{'} (0), \\ & {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;( + )}} \left. {\bar{\varepsilon }_{x}^{(2)} (x)} \right|_{x = L} = \cdots = 0, \\ & {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;(1)}} \left. {\bar{\varepsilon }_{x}^{(2)} (x)} \right|_{x = L} = \cdots = - \frac{{c_{3} }}{{c_{2} }}\varepsilon_{x} (L), \\ & {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;(1)}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;(1)}} \left. {\bar{\varepsilon }_{x}^{(2)} (x)} \right|_{x = L} = {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{1} ,c_{2} }}^{{{\kern 1pt} \;(2)}} \left. {\bar{\varepsilon }_{x}^{(1)} (x)} \right|_{x = L} = \cdots = \varepsilon_{x} (L) + c_{3} \varepsilon_{x}^{'} (L). \\ \end{aligned}$$

(130)

And, finally, the nonstandard boundary conditions for $n = 4$ as per (46) applied to $\bar{\varepsilon }_{x}^{(2)} (x)$ leads to

$$\begin{aligned} & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;( - )}} \left. {\bar{\varepsilon }_{x}^{(2)} (x)} \right|_{x = 0} = \cdots = \frac{{c_{3} c_{4} }}{{c_{1} c_{2} }}\varepsilon_{x} (0), \\ & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;(1)}} \left. {\bar{\varepsilon }_{x}^{(2)} (x)} \right|_{x = 0} = \cdots = - \frac{{\left( {c_{3} c_{4} + c_{2} c_{3} + c_{2} c_{4} } \right)}}{{c_{2}^{2} }}\varepsilon_{x} (0) + \frac{{\left( {c_{2} c_{3} c_{4} } \right)}}{{c_{2}^{2} }}\varepsilon_{x} '(0), \\ & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;(1)}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;(1)}} \left. {\bar{\varepsilon }_{x}^{(2)} (x)} \right|_{x = 0} = {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{1} ,c_{2} }}^{{{\kern 1pt} \;(2)}} \left. {\bar{\varepsilon }_{x}^{(2)} (x)} \right|_{x = 0} \\ & \quad = \varepsilon_{x} (0) - \left( {c_{3} + c_{4} } \right)\varepsilon_{x}^{'} (0) + \left( {c_{3} c_{4} } \right)\varepsilon_{x}^{''} (0), \\ & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;(1)}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;(1)}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;(1)}} \left. {\bar{\varepsilon }_{x}^{(2)} (x)} \right|_{x = 0} = {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} \;( - )}} {\mathscr{L}}_{{c_{1} ,c_{2} ,c_{3} }}^{{{\kern 1pt} \;(3)}} \left. {\bar{\varepsilon }_{x}^{(2)} (x)} \right|_{x = 0} \\ & \quad = \varepsilon_{x} (0) - c_{4} \varepsilon_{x}^{'} (0) - c_{3}^{2} \varepsilon_{x}^{''} (0) + \left( {c_{3}^{2} c_{4} } \right)\varepsilon_{x}^{'''} (0), \\ & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;( + )}} \left. {\bar{\varepsilon }_{x}^{(2)} (x)} \right|_{x = L} = \cdots = \frac{{c_{3} c_{4} }}{{c_{1} c_{2} }}\varepsilon_{x} (0), \\ & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;(1)}} \left. {\bar{\varepsilon }_{x}^{(2)} (x)} \right|_{x = L} = \cdots = - \frac{{\left( {c_{3} c_{4} + c_{2} c_{3} + c_{2} c_{4} } \right)}}{{c_{2}^{2} }}\varepsilon_{x} (L) - \frac{{\left( {c_{2} c_{3} c_{4} } \right)}}{{c_{2}^{2} }}\varepsilon_{x} '(L), \\ & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;(1)}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;(1)}} \left. {\bar{\varepsilon }_{x}^{(2)} (x)} \right|_{x = L} = {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{1} ,c_{2} }}^{{{\kern 1pt} \;(2)}} \left. {\bar{\varepsilon }_{x}^{(2)} (x)} \right|_{x = L} \\ & \quad = \varepsilon_{x} (L) + \left( {c_{3} + c_{4} } \right)\varepsilon_{x}^{'} (L) + \left( {c_{3} c_{4} } \right)\varepsilon_{x}^{''} (L), \\ & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} \;(1)}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} \;(1)}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} \;(1)}} \left. {\bar{\varepsilon }_{x}^{(2)} (x)} \right|_{x = L} = {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} \;( + )}} {\mathscr{L}}_{{c_{1} ,c_{2} ,c_{3} }}^{{{\kern 1pt} \;(3)}} \left. {\bar{\varepsilon }_{x}^{(2)} (x)} \right|_{x = L} \\ & \quad = \varepsilon_{x} (L) + c_{4} \varepsilon_{x}^{'} (L) - c_{3}^{2} \varepsilon_{x}^{''} (L) - \left( {c_{3}^{2} c_{4} } \right)\varepsilon_{x}^{'''} (L). \\ \end{aligned}$$

(131)

Now, let us consider the expression of $\bar{\varepsilon }_{x}^{(3)} (x)$ given in (43). The nonstandard boundary conditions for $n = 4$ as per (46) applied to $\bar{\varepsilon }_{x}^{(3)} (x)$ leads to

$$\begin{aligned} & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} ( - )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} ( - )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} ( - )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} ( - )}} \left. {\bar{\varepsilon }_{x}^{(3)} (x)} \right|_{x = 0} = 0, \\ & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} ( - )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} ( - )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} ( - )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} (1)}} \left. {\bar{\varepsilon }_{x}^{(3)} (x)} \right|_{x = 0} = 0, \\ & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} ( - )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} ( - )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} (1)}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} (1)}} \left. {\bar{\varepsilon }_{x}^{(3)} (x)} \right|_{x = 0} = {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} ( - )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} ( - )}} {\mathscr{L}}_{{c_{1} ,c_{2} }}^{{{\kern 1pt} (2)}} \left. {\bar{\varepsilon }_{x}^{(3)} (x)} \right|_{x = 0} = - \frac{{c_{4} }}{{c_{3} }}\varepsilon_{x} (0), \\ & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} ( - )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} (1)}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} (1)}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} (1)}} \left. {\bar{\varepsilon }_{x}^{(3)} (x)} \right|_{x = 0} = {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} ( - )}} {\mathscr{L}}_{{c_{1} ,c_{2} ,c_{3} }}^{{{\kern 1pt} (3)}} \left. {\bar{\varepsilon }_{x}^{(3)} (x)} \right|_{x = 0} = \varepsilon_{x} (0) - c_{4} \varepsilon_{x} '(0), \\ & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} ( + )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} ( + )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} ( + )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} ( + )}} \left. {\bar{\varepsilon }_{x}^{(3)} (x)} \right|_{x = L} = 0, \\ & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} ( + )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} ( + )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} ( + )}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} (1)}} \left. {\bar{\varepsilon }_{x}^{(3)} (x)} \right|_{x = L} = 0, \\ & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} ( + )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} ( + )}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} (1)}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} (1)}} \left. {\bar{\varepsilon }_{x}^{(3)} (x)} \right|_{x = L} = {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} ( + )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} ( + )}} {\mathscr{L}}_{{c_{1} ,c_{2} }}^{{{\kern 1pt} (2)}} \left. {\bar{\varepsilon }_{x}^{(3)} (x)} \right|_{x = L} = - \frac{{c_{4} }}{{c_{3} }}\varepsilon_{x} (L), \\ & {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} ( + )}} {\mathscr{L}}_{{c_{3} }}^{{{\kern 1pt} (1)}} {\mathscr{L}}_{{c_{2} }}^{{{\kern 1pt} (1)}} {\mathscr{L}}_{{c_{1} }}^{{{\kern 1pt} (1)}} \left. {\bar{\varepsilon }_{x}^{(3)} (x)} \right|_{x = L} = {\mathscr{L}}_{{c_{4} }}^{{{\kern 1pt} ( + )}} {\mathscr{L}}_{{c_{1} ,c_{2} ,c_{3} }}^{{{\kern 1pt} (3)}} \left. {\bar{\varepsilon }_{x}^{(3)} (x)} \right|_{x = L} = \varepsilon_{x} (L) + c_{4} \varepsilon_{x} '(L). \\ \end{aligned}$$

(132)

By extension, for a generic order n one should calculate the nonstandard boundary conditions applied to $\bar{\varepsilon }_{x}^{(k)} (x)\,(k = 1, \ldots ,n - 1)$ by repeatedly applying the following relations

$$\left\{ {\begin{array}{*{20}c} {{\mathscr{L}}_{{c_{n,} c_{n - 1} , \ldots ,c_{j} }}^{(p - )} {\mathscr{L}}_{{c_{1,} c_{2} , \ldots ,c_{j - 1} }}^{(j - 1)} \left. {\bar{\varepsilon }_{x}^{(k)} (x)} \right|_{x = 0} } \\ {{\mathscr{L}}_{{c_{n,} c_{n - 1} , \ldots ,c_{j} }}^{(p + )} {\mathscr{L}}_{{c_{1,} c_{2} , \ldots ,c_{j - 1} }}^{(j - 1)} \left. {\bar{\varepsilon }_{x}^{(k)} (x)} \right|_{x = L} } \\ \end{array} } \right.\quad \quad \left( {\begin{array}{*{20}c} {{\text{for }}j = 1,2, \ldots ,n - 1;} \\ \begin{aligned} p = n - j + 1;\quad n \ge j; \\ k = 1, \ldots ,n - 1 \\ \end{aligned} \\ \end{array} } \right)$$

(133)

where the differential operators ${\mathscr{L}}_{{c_{n,} c_{n - 1} , \ldots ,c_{j} }}^{(p - )}$ and ${\mathscr{L}}_{{c_{n,} c_{n - 1} , \ldots ,c_{j} }}^{(p + )}$ are defined in (48). These higher order boundary conditions are useful for the determination of the eigenfunctions $\phi (x)$ discussed in subsection 4.2. In this Appendix, we have explicitly reported the nonstandard boundary conditions up to order $n = 4$. It is worth noting that all the presented expressions are simplified in the case $c_{1} = c_{2} = \cdots = c_{n} = \ell$.