Abstract
A generalized theory of nonlocal elasticity is elaborated. The proposed integral type nonlocal formulation is based on attenuation functions being assumed as the convolution product of n first order (Eringen type) kernels. The theory stems from a generalized higher-order constitutive relation between the nonlocal stress and the local strain. Inspired by the Eringen two-phase local/nonlocal integral model, this theory can also be thought of as the constitutive relation for an (n + 1)-phase material, in which one phase has local elastic behavior, and the remaining n phases comply with nonlocal elasticity of higher order. The theory is supported by a suitable thermodynamic framework. In the spirit of Eringen’s 1983 paper, the particular family of attenuation functions adopted are the Green functions associated with generalized Helmholtz type differential operators of order n —which suggests denoting this model as a generalized nonlocal elasticity theory of n-Helmholtz type. Besides the integral type nonlocal formulation, elegant and compact expressions for the differential and integro-differential counterpart are derived. For n = 1 this formulation straightforwardly leads to the Aifantis 2003 implicit gradient elasticity theory with simultaneous stress gradients and strain gradients, which was postulated to eliminate stress and strain singularities from crack tips and dislocation lines. For n = 2 an implicit gradient elasticity formulation with bi-Helmholtz type stress and strain gradients is obtained. The paper is complemented by a companion Part II on the particularization of the generalized theory of nonlocal elasticity for the one-dimensional case, along with some applications in statics and dynamics.
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References
Askes H, Aifantis EC (2011) Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results. Int J Solids Struct 48:1962–1990
Challamel N, Rakotomanana L, Le Marrec L (2009) A dispersive wave equation using nonlocal elasticity. Compt Rendus Mec 337(8):591–595
Bacigalupo A, Gambarotta L (2017) Dispersive wave propagation in two-dimensional rigid periodic blocky materials with elastic interfaces. J Mech Phys Solids 102:165–186
Bacigalupo A, Gambarotta L (2014) Second-gradient homogenized model for wave propagation in heterogeneous periodic media. Int J Solids Struct 51(5):1052–1065
Andrianov IV, Awrejcewicz J, Weichert D. 2010 Improved continuous models for discrete media. Math Probl Eng 2010:986242. https://doi.org/10.1155/2010/986242
Andrianov I, Bolshakov V, Danishevskyy V, Weichert D (2008) Higher order asymptotic homogenization and wave propagation in periodic composite materials. Proc R Soc A 464:1181–1201
Yarnell J, Warren J, Wenzel R, Koenig S (1964) Phonon dispersion curves in bismuth. IBM J Res Dev 8:234–240
Yarnell J, Warren J, Wenzel R (1964) Lattice vibrations in diamond. Phys Rev Lett 13:13–15
Dederichs PH, Schober H, Sellmyer DJ (1981) Phonon states of elements. Electron states and fermi surfaces of alloys. Springer, Berlin, Heidelberg
Philippidis TP, Aggelis DG (2005) Experimental study of wave dispersion and attenuation in concrete. Ultrasonics 43:584–595
Owczarek S, Ghiba ID, d’Agostino MV, Neff P (2019) Nonstandard micro-inertia terms in the relaxed micromorphic model: well-posedness for dynamics. Math Mech Solids 24(10):3200–3215
Ansari R, Gholami R, Rouhi H (2012) Vibration analysis of single-walled carbon nanotubes using different gradient elasticity theories. Comp Part B Eng. 43:2985–2989
Ayad M, Karathanasopoulos N, Ganghoffer JF, Lakiss H (2020) Higher-gradient and micro-inertia contributions on the mechanical response of composite beam structures. Int J Eng Sci 154:103318
Challamel N, Aydogdu M, Elishakoff I (2018) Statics and dynamics of nanorods embedded in an elastic medium: nonlocal elasticity and lattice formulations. Eur J Mech A-Solids 67:254–271
Ghavanloo E, Fazelzadeh SA (2016) Evaluation of nonlocal parameter for single-walled carbon nanotubes with arbitrary chirality. Meccanica 51(1):41–54
Barretta R, Fazelzadeh SA, Feo L, Ghavanloo E, Luciano R (2018) Nonlocal inflected nano-beams: a stress-driven approach of bi-Helmholtz type. Comp Struct 200:239–245
Ghavanloo E, Fazelzadeh SA (2019) Wave propagation in one-dimensional infinite acoustic metamaterials with long-range interactions. Acta Mech 230(12):4453–4461
Madeo A, Neff P, Aifantis EC, Barbagallo G, d’Agostino MV (2017) On the role of micro-inertia in enriched continuum mechanics. Proc R Soc A 473(2198):20160722
Barretta R, Feo L, Luciano R, Marotti de Sciarra F (2016) Application of an enhanced version of the Eringen differential model to nanotechnology. Comp Part B Eng 96:274–280
Maugin GA, Metrikine AV (eds) (2010) Mechanics of generalized continua one hundred years after the cosserats. Springer, New York
Aifantis EC (2016) Internal length gradient (ILG) material mechanics across scales and disciplines. Adv Appli Mech 49:1–110
Kouris LAS, Bournas DA, Akintayo OT, Konstantinidis AA, Aifantis EC (2020) A gradient elastic homogenisation model for brick masonry. Eng Struct 208:110311
Triantafyllidis N, Aifantis EC (1986) A gradient approach to localization of deformation. I Hyperel Mater J Elast 16:225–237
Aifantis EC (1992) On the role of gradients in localization of deformation and fracture. Int J Eng Sci 30:1279–1299
Altan B, Aifantis EC (1997) On some aspects in the special theory of gradient elasticity. J Mech Behav Mat 8:231–282
Ru CQ, Aifantis EC (1993) A simple approach to solve boundary-value problems in gradient elasticity. Acta Mech 101:59–68
Mindlin R (1964) Micro-structure in linear elasticity. Arch Ration Mech Anal 16:51–78
Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54:4703–4710
Eringen AC (1987) Theory of nonlocal elasticity and some applications. Res Mech 21(4):313–342
Eringen AC (1992) Vistas of nonlocal continuum physics. Int J Eng Sci 30:1551–1565
Eringen AC (2002) Nonlocal continuum field theories. Springer-Verlag, New York
Kröner E (1967) Elasticity theory of materials with long range cohesive forces. Int J Solids Struct 3(5):731–742
Green AE, Rivlin RS (1965) Multipolar continuum mechanics: functional theory I. Proc R Soc A 284:303–324
Peddieson J, Buchanan GR, McNitt RP (2003) Application of nonlocal continuous models to nanotechnology. Int J Eng Sci 41:305–312
Zhang YQ, Liu GR, Xie XY (2005) Free transverse vibrations of double-walled carbon nanotubes using a theory of nonlocal elasticity. Phys Rev B 71(19):195404
Askes H, Gutiérrez MA (2006) Implicit gradient elasticity. Int J Numer Meth Eng 67:400–416
Reddy JN (2007) Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 45:288–307
Kumar D, Heinrich C, Waas AM (2008) Buckling analysis of carbon nanotubes modeled using nonlocal continuum theories. J Appl Phys 103(7):073521
Challamel N (2013) Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams. Compos Struct 105:351–368
Challamel N, Reddy JN, Wang CM (2016) Eringen’s stress gradient model for bending of nonlocal beams. J Eng Mech 142:04016095
Hozhabrossadati SM, Challamel N, Rezaiee-Pajand M, Sani AA (2018) Application of Green’s function method to bending of stress gradient nanobeams. Int J Solids Struct 143:209–217
Challamel N, Wang CM (2008) The small length scale effect for a non-local cantilever beam: a paradox solved. Nanotechnology 19(7):345703
Benvenuti E, Simone A (2013) One-dimensional nonlocal and gradient elasticity: closed-form solution and size effect. Mech Res Comm 48:46–51
Challamel N, Zhang Z, Wang CM, Reddy JN, Wang Q, Michelitsch T, Col-let B (2014) On non-conservativeness of Eringen’s nonlocal elasticity in beam mechanics: correction from a discrete-based approach. Arch Appl Mech 84:1275–1292
Li C, Yao L, Chen W, Li S (2015) Comments on nonlocal effects in nano-cantilever beams. Int J Eng Sci 87:47–57
Fernández-Sáez J, Zaera R, Loya JA, Reddy JN (2016) Bending of Euler-Bernoulli beams using Eringen’s integral formulation: a paradox resolved. Int J Eng Sci 99:107–116
Romano G, Barretta R, Diaco M, Marotti de Sciarra F (2017) Constitutive boundary conditions and paradoxes in nonlocal elastic nano-beams. Int J Mech Sci 121:151–156
Polizzotto C (2014) Stress gradient versus strain gradient constitutive models within elasticity. Int J Solids Struct 51(9):1809–1818
Polizzotto C (2015) A unifying variational framework for stress gradient and strain gradient elasticity theories. Eur J Mech-A/Solids 49:430–440
Lazar M, Maugin GM, Aifantis EC (2006) On a theory of nonlocal elasticity of bi-Helmoltz type and some applications. Int J Solids Struct 43:1404–1421
Lazar M, Maugin GA, Aifantis EC (2006) Dislocations in second strain gradient elasticity. Int J Solids Struct 43(6):1787–1817
Mousavi SM (2016) Dislocation-based fracture mechanics within nonlocal and gradient elasticity of bi-Helmholtz type—Part I: antiplane analysis. Int J Solids Struct 87:222–235
Mousavi SM (2016) Dislocation-based fracture mechanics within nonlocal and gradient elasticity of bi-Helmholtz type—Part II: inplane analysis. Int J Solids Struct 92:105–120
Fafalis DA, Filopoulos SP, Tsamasphyros GJ (2012) On the capability of generalized continuum theories to capture dispersion characteristics at the atomic scale. Eur J Mech-A/Solids 36:25–37
Polizzotto C (2013) A second strain gradient elasticity theory with second velocity gradient inertia—Part II: dynamic behavior. Int J Solids Struct 50(24):3766–3777
Koutsoumaris CC, Vogiatzis GG, Theodorou DN, Tsamasphyros GJ (2015) Application of bi-Helmholtz nonlocal elasticity and molecular simulations to the dynamical response of carbon nanotubes. AIP Conf Proc 1702(1):190011
Koutsoumaris CC, Eptaimeros KG (2018) A research into bi-Helmholtz type of nonlocal elasticity and a direct approach to Eringen’s nonlocal integral model in a finite body. Acta Mech 229(9):3629–3649
Aifantis EC (2003) Update on a class of gradient theories. Mech Mater 35:259–280
Aifantis EC (2011) On the gradient approach: relation to Eringen’s nonlocal theory. Int J Eng Science 49:1367–1377
Polizzotto C (2001) Nonlocal elasticity and related variational principles. Int J Solids Struct 38(42–43):7359–7380
Altan SB (1989) Uniqueness of the initial-value problem in nonlocal elastic solids. Int J Solids Struct 25:1271–1278
Pisano AA, Fuschi P (2003) Closed form solution for a nonlocal elastic bar in tension. Int J Solids Struct 40(1):13–23
Pisano AA, Sofi A, Fuschi P (2009) Nonlocal integral elasticity: 2D finite element based solutions. Int J Solids Struct 46(21):3836–3849
Abdollahi R, Boroomand B (2013) Benchmarks in nonlocal elasticity defined by Eringen’s integral model. Int J Solids Struct 50(18):2758–2771
Abdollahi R, Boroomand B (2014) Nonlocal elasticity defined by Eringen’s integral model: introduction of a boundary layer method. Int J Solids Struct 51(9):1758–1780
Fuschi P, Pisano AA, De Domenico D (2015) Plane stress problems in nonlocal elasticity: finite element solutions with a strain-difference-based formulation. J Math Analysis Appl 431:714–736
Failla I, Impollonia N, Ricciardi G. 2015 A local/nonlocal elasticity model for the Euler-Bernoulli beam. In: Proc 15th Int Conf Civ Struct Env Eng Comput 2015. J Kruis, Y. Tsompanakis and B.H.V. Topping, (Eds.). Civil-Comp Press, Stirlingshire, Scotland, vol. 108.
Khodabakhshi P, Reddy JN (2015) A unified integro-differential nonlocal model. Int J Eng Sci 95:60–75
Koutsoumaris CC, Eptaimeros KG, Tsamasphyros GJ (2017) A different approach to Eringen’s nonlocal integral stress model with applications for beams. Int J Solids Struct 112:222–238
Eptaimeros KG, Koutsoumaris CC, Tsamasphyros GJ (2016) Nonlocal integral approach to the dynamical response of nanobeams. Int J Mech Sci 115–116:68–80
Fernández-Sáez J, Zaera R (2017) Vibrations of Bernoulli-Euler beams using the two-phase nonlocal elasticity theory. Int J Eng Sci 119:232–248
Zhu X, Li L (2017) Longitudinal and torsional vibrations of size-dependent rods via nonlocal integral elasticity. Int J Mech Sci 133:639–650
Bažant ZP, Jirásek M (2002) Nonlocal integral formulations of plasticity and damage: survey of progress. J Eng Mech 128(11):1119–1149
Germain P, Nguyen QS, Suquet P (1983) Continuum thermodynamics. ASME J Appl Mech 50:1010–1021
Lemaitre J, Chaboche J-L (1990) Mechanics of Solid Materials. Cambridge University Press, Cambridge
Di Paola M, Pirrotta A, Zingales M (2010) Mechanically-based approach to nonlocal elasticity: variational principles. Int J Solids Struct 47(5):539–548
Di Paola M, Failla G, Sofi A, Zingales M (2011) A mechanically based approach to non-local beam theories. Int J Mech Sci 53:676–687
Edelen DGB, Laws N (1971) On the thermodynamics of systems with nonlocality. Arch Ration Mech Anal 43:24–35
Edelen DGB, Green AE, Laws N (1971) On the thermodynamics of systems with nonlocality. Arch Ration Mech Anal 43:36–44
Nguyen GD (2008) A thermodynamic approach to non-local damage modelling of concrete. Int J Solids Struct 45(7–8):1918–1934
Aifantis EC (2009) On scale invariance in anisotropic plasticity, gradient plasticity and gradient elasticity. Int J Eng Science 47(11–12):1089–1099
Metrikine AV (2006) On causality of the gradient elasticity models. J Sound Vibration 297:727–742
Askes H, Aifantis EC (2006) Gradient elasticity theories in statics and dynamics: a unification of approaches. Int J Fract 139:297–304
Papargyri-Beskou S, Polyzos D, Beskos DE (2009) Wave dispersion in gradient elastic solids and structures: a unified treatment. Int J Solids Struct 46:3751–3759
Challamel N, Rakotomanana L, Le Marrec L (2009) A dispersive wave equation using nonlocal elasticity. C R Mecanique 337:591–595
De Domenico D, Askes H (2016) A new multi-scale dispersive gradient elasticity model with micro-inertia: formulation and C0-finite element implementation. Int J Num Meth Engn 108(5):308–333
De Domenico D, Askes H (2017) Computational aspects of a new multi-scale dispersive gradient elasticity model with micro-inertia. Int J Num Meth Engn 109(1):52–72
De Domenico D, Askes H (2018) Stress gradient, strain gradient and inertia gradient beam theories for the simulation of flexural wave dispersion in carbon nanotubes. Comp Part B: Eng 153:285–294
De Domenico D, Askes H, Aifantis EC (2018) Capturing wave dispersion in heterogeneous and microstructured materials through a three-length-scale gradient elasticity formulation. J Mech Behav Mat 27:5–6
De Domenico D, Askes H (2018) Nano-scale wave dispersion beyond the first Brillouin Zone simulated with inertia gradient continua. J Appl Phys 124(20):205107
De Domenico D, Askes H, Aifantis EC (2019) Gradient elasticity and dispersive wave propagation: Model motivation and length scale identification procedures in concrete and composite laminates. Int J Solids Struct 158:176–190
Polizzotto C (2003) Gradient elasticity and nonstandard boundary conditions. Int J Solids Struct 40(26):7399–7423
Pisano A, Fuschi P, Polizzotto C (2020) A strain-difference based nonlocal elasticity theory for small-scale shear-deformable beams with parametric warping. Int J Multiscale Comput Eng 18(1):83–102
Fuschi P, Pisano AA, Polizzotto C (2019) Size effects of small-scale beams in bending addressed with a strain-difference based nonlocal elasticity theory. Int J Mech Sciences 151:661–671
Pisano AA, Fuschi P (2018) Structural symmetry and boundary conditions for nonlocal symmetrical problems. Meccanica 53(3):629–638
Pisano AA, Fuschi P (2017) Structural symmetry within nonlocal integral elasticity: theoretical issues and computational strategies. Curved Layer Struct 4(1):1–7
Hohn FE (1973) Elementary matrix algebra, 3rd edn. The Macmillan Company (ISBN: 978–0–02–355950–1)
Acknowledgements
This research work is the result of fruitful scientific discussions of the authors with Prof. Elias C. Aifantis. This article is dedicated to his academic career and achievements on the occasion of his 70th birthday.
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Appendices
Appendix 1 Expressions of the length scale parameters \(\ell_{k}^{2k} (k = 1, \ldots ,n)\) for the n-Helmholtz type differential operator with multiple length scales
In this Appendix, we consider the nth order Helmholtz differential operator with multiple length scales \(c_{1} \ne c_{2} \ne \ldots \ne c_{n}\). We will demonstrate Eq. (54) and we will derive the expressions of the length scale parameters \(\ell_{k}^{2k} (k = 1, \ldots ,n)\) (namely \(\ell_{1}^{2} ,\ell_{2}^{4} ,\ell_{3}^{6} ,\ell_{4}^{8} , \ldots\)) in terms of the length scale coefficients of the n individual (factorizing) Helmholtz operators \(c_{k} (k = 1,2, \ldots ,n)\) in compact form. Let us consider the definition of the nth order Helmholtz differential operator with the aid of the product operator as follows
in which \({\mathcal{L}}_{H} (c_{k} )[ \cdot ] = \left( {1 - c_{k}^{2} \nabla^{2} } \right)[ \cdot ]\) is the Helmholtz operator with length scale parameter \(c_{k}\). For n = 2, 3 and 4 the formula (75) leads to, respectively:
-
for n = 2
with \(\ell_{1}^{2} = c_{1}^{2} + c_{2}^{2}\) and \(\ell_{2}^{4} = c_{1}^{2} c_{2}^{2}\). This result was already presented in Eq. (49) for the bi-Helmholtz differential operator \({\mathcal{L}}_{bH}\);
-
for n = 3
$${\mathcal{L}}_{{c_{1} ,c_{2} ,c_{3} }}^{{{\kern 1pt} (3)}} = {\mathcal{L}}_{H} (c_{1} ){\mathcal{L}}_{H} (c_{2} ){\mathcal{L}}_{H} (c_{3} ) = \left( {1 - c_{1}^{2} \nabla^{2} } \right)\left( {1 - c_{2}^{2} \nabla^{2} } \right)\left( {1 - c_{3}^{2} \nabla^{2} } \right) = 1 - \ell_{1}^{2} \nabla^{2} + \ell_{2}^{4} \nabla^{4} - \ell_{3}^{6} \nabla^{6}$$(77)with \(\ell_{1}^{2} = c_{1}^{2} + c_{2}^{2} + c_{3}^{2}\), \(\ell_{2}^{4} = c_{1}^{2} c_{2}^{2} + c_{1}^{2} c_{3}^{2} + c_{2}^{2} c_{3}^{2}\) and \(\ell_{3}^{6} = c_{1}^{2} c_{2}^{2} c_{3}^{2}\);
-
for n = 4
with \(\ell_{1}^{2} = c_{1}^{2} + c_{2}^{2} + c_{3}^{2} + c_{4}^{2}\), \(\ell_{2}^{4} = c_{1}^{2} c_{2}^{2} + c_{1}^{2} c_{3}^{2} + c_{2}^{2} c_{3}^{2} + c_{1}^{2} c_{4}^{2} + c_{2}^{2} c_{4}^{2} + c_{3}^{2} c_{4}^{2}\),
\(\ell_{3}^{6} = c_{1}^{2} c_{2}^{2} c_{3}^{2} + c_{1}^{2} c_{2}^{2} c_{4}^{2} + c_{1}^{2} c_{3}^{2} c_{4}^{2} + c_{2}^{2} c_{3}^{2} c_{4}^{2}\), \(\ell_{4}^{8} = c_{1}^{2} c_{2}^{2} c_{3}^{2} c_{4}^{2}\).
To extend and generalize the above results up to order n, it is useful to introduce the \(n \times n\) diagonal matrix collecting the squared length scale parameters \(c_{j}^{2} \,(j = 1,2, \ldots ,n)\) of the individual n Helmholtz type differential operators \({\mathcal{L}}_{H} (c_{j} )\) as follows
The n length scale parameters \(\ell_{1} ,\ell_{2} , \ldots ,\ell_{n}\) of the resulting nth order Helmholtz differential operator can be expressed in terms of the \(c_{j}\) length scales in the following compact form
where the symbol \([{\mathbf{C}}]_{\,i}^{(k)}\) denotes the ith minor (determinant) of order k of the square diagonal matrix \({\mathbf{C}}\), while \({\text{tr(}}{\mathbf{C}})\) and \(\det \left( {\mathbf{C}} \right)\) denote the trace and determinant of the \({\mathbf{C}}\) matrix. It is well known that for a \(n \times n\) diagonal matrix there are a total of \(m_{n,k} = \left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)\) minors of order k [97], which represent the upper limit of the summations in (80). Thus, there are \(m_{n,1} = n\) minors of order 1, whose sum gives the trace of the \({\mathbf{C}}\) matrix, and only one minor of order n (\(m_{n,n} = 1\)), which coincides with the determinant of the \({\mathbf{C}}\) matrix according to (80). In conclusion, the generalization of the nth order Helmholtz differential operator (75) to the general case of multiple length scale parameters \(c_{k} (k = 1, \ldots ,n)\) in each individual Helmholtz operator leads to the following compact expression
where by \(\ell_{0} = 1\) and each length scale parameter \(\ell_{k}^{2k} (k = 1, \ldots ,n)\) is expressed in compact form in terms of the \(c_{k}\) length scales through the relationships (80).
While the length scale parameters \(\ell_{k}\) entering the nth order Helmholtz differential operator can be calibrated based on experimental data (for instance, matching the Born–von Kármán model of lattice dynamics), their physical meaning and admissibility must be checked afterwards in terms of the originating \(c_{k}\) length scale coefficients that must be all real for the single Helmholtz operators. This is done by inverting the (nonlinear) relations (80) between \(\ell_{k}\) and \(c_{k}\) and verifying that all \(c_{k}^{2}\) terms are positive. Admissibility conditions in analytical form were reported by Lazar et al. [50] for the simple case of n = 2. Numerical, rather than analytical, conditions can be derived for a generic order n.
To generalize the relationships (80) further, for the next derivations it is useful to introduce the following truncated \((n - j)^{{{\text{th}}}}\) order Helmholtz operator (with \(n > j\)) defined as
which represents the product of \(n - j\) Helmholtz operators. Indeed, expression (82) represents a truncated version of the nth order Helmholtz differential operator in (81) where the first j Helmholtz operators with the corresponding length scales \(c_{1} , \ldots ,c_{j}\) are eliminated, while the remaining \(n - j\) with length scales from \(c_{j + 1}\) up to \(c_{n}\) are retained. It can be demonstrated that such truncated \((n - j)^{{{\text{th}}}}\) order Helmholtz operator in (82) can be expressed in a similar format to Eq. (81) but with a slightly different definition of the length scale parameters, namely
where the new length scale parameters \(\ell_{j,k}^{2k} (k = 1, \ldots ,n - j)\) depend not only on the order \(k\) but also on the truncation order j being considered. As an example, for n = 4 Eq. (82) can be particularized in the following cases.
-
for j = 1
with \(\ell_{1,1}^{2} = c_{2}^{2} + c_{3}^{2} + c_{4}^{2}\), \(\ell_{1,2}^{4} = c_{2}^{2} c_{3}^{2} + c_{2}^{2} c_{4}^{2} + c_{3}^{2} c_{4}^{2}\), \(\ell_{1,3}^{6} = c_{2}^{2} c_{3}^{2} c_{4}^{2}\);
-
for j = 2
with \(\ell_{2,1}^{2} = c_{3}^{2} + c_{4}^{2}\), \(\ell_{2,2}^{4} = c_{3}^{2} c_{4}^{2}\);
for j = 3
with \(\ell_{3,1}^{2} = c_{4}^{2}\).
To extend and generalize the above results up to order n and truncation j, it is useful to introduce the following truncated \((n - j) \times (n - j)\) diagonal matrix collecting the squared length scale parameters from \(c_{j + 1}\) up to \(c_{n}\) as follows
For the previous example of n = 4 we have
It is worth noting that the diagonal \((n - j)\)-dimensional \({\mathbf{C}}^{(j)}\) matrix in (87) is a submatrix of the previously defined \({\mathbf{C}}\) matrix in (79) by eliminating the first j rows and columns. By considering the previous results (84), (85), (86) for \(j = 1,2,3\), respectively, it can easily be seen that the new length scale parameters \(\ell_{j,k}^{2k} (k = 1, \ldots ,n - j)\) entering the truncated \((n - j)^{{{\text{th}}}}\) order Helmholtz operator (83) are related to the minors (determinants) of the square diagonal matrix \({\mathbf{C}}^{(j)}\) in (87) through the following relationships
where the symbol \([{\mathbf{C}}^{(j)} ]_{\,i}^{(k)}\) denotes the ith minor (determinant) of order k of the square diagonal matrix \({\mathbf{C}}^{(j)}\) and \(m_{n - j,k} = \left( {\begin{array}{*{20}c} {n - j} \\ k \\ \end{array} } \right)\) is the number of minors of order k of the \((n - j)\)-dimensional \({\mathbf{C}}^{(j)}\) matrix [97].
The general nomenclature adopted in (89) includes the previous relationships (80) as special subcase for \(j = 0\): in this case, there is no truncation as the \({\mathbf{C}}^{(j)}\) matrix coincides with the entire \({\mathbf{C}}\) matrix in (79) (which can be viewed as a truncation of zero order), and the length scale coefficients in (89) \(\ell_{0,k}^{2k} \equiv \ell_{k}^{2k}\) whose expressions are given above in (80).
Appendix 2 Mathematical relationships between the Helmholtz type differential operator \({\mathcal{L}}^{\;(i)}\) and the nonlocal strain tensor \(\overline{\varepsilon }_{kl}^{(j)} ({\mathbf{x}})\)
In this Appendix, we report some mathematical properties of the differential operator \({\mathcal{L}}^{(i)}\) in relationship to the nonlocal strain tensor \(\overline{\varepsilon }_{kl}^{(j)} ({\mathbf{x}})\) for equal and different orders of differentiation and regularization i and j, respectively. As a first step, a more explicit expression for Eq. (57) is given below for a generic order n in both the differential operator and the nonlocal strain tensor (\(i = j = n\)), by recalling the definition of \(\overline{\varepsilon }_{kl}^{(n)} ({\mathbf{x}})\) in (16) and (17) for a multiple-length-scale kernel \(\alpha^{(n)} (|{\mathbf{x}} - {\mathbf{y}}|;c_{1} ,c_{2} , \ldots ,c_{n} )\), namely
where account has been taken of the Green-type equality (20) and of fact that \({\mathcal{L}}\) is, by hypothesis, a differential operator with constant coefficients. The derivations and the integrands in (90) proves the fact that the nth order kernel \(\alpha^{(n)}\) is the Green function of the nth order Helmholtz differential operator \({\mathcal{L}}^{\;(n)}\), as per Eq. (57).
As a second step, generic expressions for a different order in the differential operator \(i = n\) and in the regularization (integral) operator j (affecting the nonlocal strain tensor), with \(n > j\), are derived as follows
where account has been taken of the previous relationship (90) for equal order j for the term within round brackets. In Eq. (91) \({\mathcal{L}}_{{c_{j + 1} ,c_{j + 2} , \ldots ,c_{n} }}^{{\;{\kern 1pt} (n - j)}}\) represents the truncated \((n - j)^{{{\text{th}}}}\) order Helmholtz operator previously defined in (82), whose length scale parameters are given in (89) in terms of the \(c_{k}\) length scales. As a third step, another relationship can be derived as follows
where account has been taken of the previous relationship (90) for equal order \(n - j\) for the term within round brackets. These two relationships (91) and (92) are recalled in Eq. (59) of the paper and are useful for the derivation of the differential and the integro-differential form of the generalized theory of nonlocal elasticity of n-Helmholtz type.
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De Domenico, D., Ricciardi, G. & Askes, H. A generalized integro-differential theory of nonlocal elasticity of n-Helmholtz type: part I—analytical formulation and thermodynamic framework. Meccanica 56, 629–649 (2021). https://doi.org/10.1007/s11012-020-01297-w
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DOI: https://doi.org/10.1007/s11012-020-01297-w