# Elastostatics of Bernoulli–Euler Beams Resting on Displacement-Driven Nonlocal Foundation

^{*}

## Abstract

**:**

## 1. Introduction

#### Winkler and Wieghardt Foundations

## 2. Research Significance, Motivation, and Outline

## 3. Bernoulli–Euler Beams on Elastic Foundation

## 4. Elastic Equilibrium Problem of a Beam on Displacement-Driven Nonlocal Foundation

**Box 1.**Elastostatic structural problem of a beam on nonlocal displacement-driven DD elastic foundation.

**Proposition**

**1**

**Box 2.**Elastostatic structural differential problem of a beam on nonlocal DD elastic foundation.

#### Transverse Displacement of the Nonlocal Elastic Foundation Outside the Beam Interval

**Remark**

**1.**

**Remark**

**2.**

## 5. Numerical Applications

#### 5.1. Free Beam on a Nonlocal Foundation Subject to a Uniformly Distributed Load

#### 5.2. Simply Supported Beam on a Nonlocal Foundation Subject to a Uniformly Distributed Load

## 6. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## List of Acronyms

DD | Displacement-Driven |

FBCs | Foundation Boundary Conditions |

FF | Free-Free beam |

MFBCs | Modified Foundation Boundary Conditions |

MRD | Modified Reaction-Driven |

NEF | No Elastic Foundation |

SS | Simply Supported beam |

RD | Reaction-Driven |

RDFBCs | Reaction-Driven foundation boundary conditions |

## Appendix A

**Proposition**

**A1**

## Appendix B

**Proposition**

**A2**

**Proof.**

## References

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**Figure 2.**$FF-q$ beam. Plots of the non-dimensional transverse displacement ${v}^{*}$ of the surface of the elastic foundation for increasing values of the non-dimensional non-local parameter $\lambda $ in the set $\left(\right)$ with ${k}^{*}=10$: (

**a**) DD method, (

**b**) MRD method, (

**c**) zoom of the beam deflection using the DD method, (

**d**) zoom of the beam deflection using the MRD method.

**Figure 3.**$FF-q$ beam. DD method: plots of the non-dimensional reaction ${r}^{*}$ of the elastic foundation for increasing values of the non-dimensional non-local parameter $\lambda $ in the set $\left(\right)$ with ${k}^{*}=10$.

**Figure 4.**$FF-q$ beam. Plots for increasing values of the non-dimensional non-local parameter $\lambda $ in the set $\left(\right)$ with ${k}^{*}=10$ of (

**a**) non-dimensional bending moment ${M}^{*}$ using the DD method and (

**b**) non-dimensional shear force ${T}^{*}$ using the DD method.

**Figure 5.**$FF-q$ beam. Comparison of the DD and MRD methods for increasing values of the non-dimensional non-local parameter $\lambda $ in the set $\left(\right)$ with ${k}^{*}=\left(\right)open="\{"\; close="\}">0.4,2,10$: (

**a**) non-dimensional transverse midpoint displacement ${v}^{*}(1/2)$, (

**b**) dimensionless reaction ${r}^{*}(1/2)$, (

**c**) non-dimensional bending moment ${M}^{*}(1/2)$, and (

**d**) non-dimensional shear force ${T}^{*}(-1)$.

**Figure 6.**$SS-q$ beam. DD method: plots of the non-dimensional transverse displacement ${v}^{*}$ of the surface of the elastic foundation for increasing values of the non-dimensional non-local parameter $\lambda $ in the set $\left(\right)$ with ${k}^{*}=10$.

**Figure 7.**$SS-q$ beam. DD method: plots of the non-dimensional reaction ${r}^{*}$ of the elastic foundation for increasing values of the non-dimensional non-local parameter $\lambda $ in the set $\left(\right)$ with ${k}^{*}=10$.

**Figure 8.**$SS-q$ beam. Plots for increasing values of the non-dimensional non-local parameter $\lambda $ in the set $\left(\right)$ with ${k}^{*}=10$ of (

**a**) non-dimensional bending moment ${M}^{*}$ using the DD method and (

**b**) non-dimensional shear force ${T}^{*}$ using the DD method.

**Figure 9.**$SS-q$ beam. Comparison of the DD and MRD methods for increasing values of the non-dimensional non-local parameter $\lambda $ in the set $\left(\right)$ with ${k}^{*}=\left(\right)open="\{"\; close="\}">0.4,2,10$: (

**a**) non-dimensional transverse midpoint displacement ${v}^{*}(1/2)$, (

**b**) dimensionless reaction ${r}^{*}(1/2)$, (

**c**) non-dimensional bending moment ${M}^{*}(1/2)$, and (

**d**) non-dimensional shear force ${T}^{*}\left(1\right)$.

**Table 1.**Free beam subjected to a non-dimensional uniform load ${q}_{y}^{*}=-1$. Non-dimensional maximum displacement ${v}^{*}\left(\right)open="("\; close=")">1/2$ versus the non-dimensional length-scale parameter $\lambda $ evaluated by the non-dimensional Winkler parameter ${k}^{*}\in \left(\right)open="\{"\; close="\}">0.4,2,10,20$ in the DD and MRD models.

v${}^{*}$ (1/2) | |||||||||
---|---|---|---|---|---|---|---|---|---|

$\mathbf{\lambda}$ | DD | MRD | |||||||

k${}^{*}$ = 0.4 | k${}^{*}$ = 2 | k${}^{*}$ = 10 | k${}^{*}$ = 20 | k${}^{*}$ = 0.4 | k${}^{*}$ = 2 | k${}^{*}$ = 10 | k${}^{*}$ = 20 | ||

${0}^{+}$ | −2.5 | −0.5 | −0.1 | −0.05 | −2.5 | −0.5 | −0.1 | −0.05 | |

0.1 | −2.7775 | −0.555285 | −0.110845 | −0.0552931 | −2.08434 | −0.41767 | −0.0843029 | −0.0425966 | |

0.2 | −3.11944 | −0.623643 | −0.124485 | −0.0620923 | −1.78773 | −0.359129 | −0.0732901 | −0.0374405 | |

0.3 | −3.51736 | −0.703245 | −0.140422 | −0.0700698 | −1.56541 | −0.315348 | −0.0650625 | −0.0335266 | |

0.4 | −3.95023 | −0.789841 | −0.157764 | −0.0787552 | −1.39259 | −0.28135 | −0.0586113 | −0.030365 | |

0.5 | −4.40376 | −0.880569 | −0.175931 | −0.0878521 | −1.25437 | −0.254164 | −0.0533617 | −0.0277079 |

**Table 2.**Free beam subjected to a non-dimensional uniform load ${q}_{y}^{*}=-1$. Non-dimensional midpoint foundation reaction ${r}^{*}\left(\right)open="("\; close=")">1/2$ versus the non-dimensional length-scale parameter $\lambda $ evaluated by the non-dimensional Winkler parameter ${k}^{*}\in \left(\right)open="\{"\; close="\}">0.4,2,10,20$ in the DD and MRD models.

r${}^{*}$ (1/2) | |||||||||
---|---|---|---|---|---|---|---|---|---|

$\mathbf{\lambda}$ | DD | MRD | |||||||

k${}^{*}$ = 0.4 | k${}^{*}$ = 2 | k${}^{*}$ = 10 | k${}^{*}$ = 20 | k${}^{*}$ = 0.4 | k${}^{*}$ = 2 | k${}^{*}$ = 10 | k${}^{*}$ = 20 | ||

${0}^{+}$ | −1.0 | −1.0 | −1.0 | −1.0 | −1.0 | −1.0 | −1.0 | −1.0 | |

0.1 | −1.10354 | −1.10322 | −1.10166 | −1.09974 | −0.833821 | −0.835754 | −0.845019 | −0.85574 | |

0.2 | −1.14542 | −1.14522 | −1.14425 | −1.14305 | −0.71566 | −0.721065 | −0.746027 | −0.773076 | |

0.3 | −1.14127 | −1.14117 | −1.14065 | −1.14001 | −0.627843 | −0.638892 | −0.687364 | −0.735528 | |

0.4 | −1.12745 | −1.12739 | −1.1271 | −1.12673 | −0.560558 | −0.579707 | −0.658509 | −0.729175 | |

0.5 | −1.11353 | −1.1135 | −1.11332 | −1.1131 | −0.507919 | −0.537668 | −0.651312 | −0.742323 |

**Table 3.**Free beam subjected to a non-dimensional uniform load ${q}_{y}^{*}=-1$. Non-dimensional midpoint bending moment ${M}^{*}\left(\right)open="("\; close=")">1/2$ versus the non-dimensional length-scale parameter $\lambda $ evaluated by the non-dimensional Winkler parameter ${k}^{*}\in \left(\right)open="\{"\; close="\}">0.4,2,10,20$ in the Displacement-Driven (DD) and Modified Reaction-Driven (MRD) models.

M${}^{*}$ (1/2) | |||||||||
---|---|---|---|---|---|---|---|---|---|

$\mathbf{\lambda}$ | DD | MRD | |||||||

k${}^{*}$ = 0.4 | k${}^{*}$ = 2 | k${}^{*}$ = 10 | k${}^{*}$ = 20 | k${}^{*}$ = 0.4 | k${}^{*}$ = 2 | k${}^{*}$ = 10 | k${}^{*}$ = 20 | ||

${0}^{+}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

0.1 | −0.00840319 | −0.00838687 | −0.00830616 | −0.00820727 | 0.0207943 | 0.0206397 | 0.0198977 | 0.0190375 | |

0.2 | −0.00995599 | −0.00994611 | −0.00989699 | −0.00983623 | 0.0355889 | 0.0350958 | 0.0328139 | 0.0303315 | |

0.3 | −0.00922294 | −0.00921768 | −0.00919144 | −0.00915884 | 0.0465999 | 0.0455298 | 0.0408201 | 0.0361086 | |

0.4 | −0.00816841 | −0.00816543 | −0.00815058 | −0.00813209 | 0.0550571 | 0.0531471 | 0.0452469 | 0.038086 | |

0.5 | −0.00721239 | −0.00721058 | −0.00720154 | −0.00719027 | 0.0616985 | 0.0586827 | 0.0470779 | 0.037633 |

**Table 4.**Free beam subjected to a non-dimensional uniform load ${q}_{y}^{*}=-1$. Non-dimensional parameters ${A}_{1}^{*}={A}_{2}^{*}$ versus the non-dimensional length-scale parameter $\lambda $ evaluated by the non-dimensional Winkler parameter ${k}^{*}\in \left(\right)open="\{"\; close="\}">0.4,2,10,20$ in the MRD model.

${\mathit{A}}_{1}^{*}={\mathit{A}}_{2}^{*}$ | ||||
---|---|---|---|---|

$\mathbf{\lambda}$ | MRD | |||

k${}^{*}$ = 0.4 | k${}^{*}$ = 2 | k${}^{*}$ = 10 | k${}^{*}$ = 20 | |

$0.1$ | −0.0832594 | −0.0829661 | −0.0815572 | −0.0799202 |

$0.2$ | −0.142532 | −0.141251 | −0.135315 | −0.128836 |

$0.3$ | −0.186708 | −0.183624 | −0.170018 | −0.156337 |

$0.4$ | −0.220721 | −0.214963 | −0.191062 | −0.169231 |

$0.5$ | −0.247531 | −0.23823 | −0.202257 | −0.172651 |

**Table 5.**Free beam subjected to a non-dimensional uniform load ${q}_{y}^{*}=-1$ with the non-dimensional length-scale parameter $\lambda =0.5$. Non-dimensional displacement ${v}^{*}\left(\xi \right)$ evaluated by the non-dimensional Winkler parameter ${k}^{*}\in \left(\right)open="\{"\; close="\}">0.4,2,10,20$ in the DD and MRD models.

${\mathit{v}}^{*}\left(\mathit{\xi}\right)$ with $\phantom{\rule{4pt}{0ex}}\mathit{\lambda}=0.5$ | |||||||||
---|---|---|---|---|---|---|---|---|---|

$\mathbf{\xi}$ | DD | MRD | |||||||

k${}^{*}$ = 0.4 | k${}^{*}$ = 2 | k${}^{*}$ = 10 | k${}^{*}$ = 20 | k${}^{*}$ = 0.4 | k${}^{*}$ = 2 | k${}^{*}$ = 10 | k${}^{*}$ = 20 | ||

0.5 | −4.40376 | −0.880569 | −0.175931 | −0.0878521 | −1.25437 | −0.254164 | −0.0533617 | −0.0277079 | |

0.6 | −4.40379 | −0.880604 | −0.175967 | −0.0878876 | −1.25407 | −0.253872 | −0.0531278 | −0.0275208 | |

0.7 | −4.40389 | −0.880706 | −0.176068 | −0.0879886 | −1.25317 | −0.253021 | −0.0524435 | −0.0269726 | |

0.8 | −4.40405 | −0.880858 | −0.17622 | −0.0881401 | −1.25176 | −0.251679 | −0.0513622 | −0.0261034 | |

0.9 | −4.40423 | −0.88104 | −0.176402 | −0.0883215 | −1.24996 | −0.249964 | −0.049975 | −0.0249838 | |

1.0 | −4.40442 | −0.881232 | −0.176594 | −0.0885133 | −1.24794 | −0.24804 | −0.0484141 | −0.0237196 | |

1.2 | −2.95237 | −0.590707 | −0.118374 | −0.0593323 | −0.836521 | −0.166266 | −0.032453 | −0.0158997 | |

1.4 | −1.97903 | −0.395963 | −0.0793487 | −0.0397716 | −0.560737 | −0.111451 | −0.021753 | −0.0106579 | |

1.6 | −1.32659 | −0.265422 | −0.053189 | −0.0266597 | −0.375873 | −0.0747081 | −0.0145821 | −0.0071442 | |

1.8 | −0.889237 | −0.177918 | −0.0356536 | −0.0178705 | −0.251955 | −0.0500784 | −0.00977464 | −0.0047889 | |

2.0 | −0.596073 | −0.119262 | −0.0238994 | −0.011979 | −0.168891 | −0.0335685 | −0.00655214 | −0.00321009 | |

2.2 | −0.39956 | −0.0799436 | −0.0160202 | −0.00802975 | −0.113211 | −0.0225017 | −0.0043920 | −0.00215179 | |

2.4 | −0.267833 | −0.0535878 | −0.0107387 | −0.0053825 | −0.0758875 | −0.0150833 | −0.00294407 | −0.00144239 | |

2.6 | −0.179534 | −0.035921 | −0.00719835 | −0.003608 | −0.0508689 | −0.0101106 | −0.00197347 | −0.000966862 | |

2.8 | −0.120345 | −0.0240785 | −0.0048252 | −0.00241851 | −0.0340984 | −0.00677737 | −0.00132285 | −0.00064810 | |

3.0 | −0.0806698 | −0.0161403 | −0.00323443 | −0.00162118 | −0.0228569 | −0.00454301 | −0.000886735 | −0.000434439 |

**Table 6.**Simply supported beam subjected to a non-dimensional uniform load ${q}_{y}^{*}=-1$. Non-dimensional maximum displacement ${v}^{*}\left(\right)open="("\; close=")">1/2$ versus the non-dimensional length-scale parameter $\lambda $ evaluated by the non-dimensional Winkler parameter ${k}^{*}\in \left(\right)open="\{"\; close="\}">0.4,2,10,20$ in the DD and MRD models.

v${}^{*}$ (1/2) | |||||||||
---|---|---|---|---|---|---|---|---|---|

$\mathbf{\lambda}$ | DD | MRD | |||||||

k${}^{*}$ = 0.4 | k${}^{*}$ = 2 | k${}^{*}$ = 10 | k${}^{*}$ = 20 | k${}^{*}$ = 0.4 | k${}^{*}$ = 2 | k${}^{*}$ = 10 | k${}^{*}$ = 20 | ||

${0}^{+}$ | −0.0129674 | −0.0127579 | −0.011804 | −0.0107944 | −0.0129674 | −0.0127579 | −0.011804 | −0.0107944 | |

0.1 | −0.0129713 | −0.0127769 | −0.0118858 | −0.0109322 | −0.0129621 | −0.0127325 | −0.0116961 | −0.0106152 | |

0.2 | −0.0129781 | −0.0128101 | −0.012031 | −0.0111806 | −0.0129464 | −0.012657 | −0.0113839 | −0.0101115 | |

0.3 | −0.0129842 | −0.0128399 | −0.0121636 | −0.0114121 | −0.0129203 | −0.0125331 | −0.010899 | −0.00937019 | |

0.4 | −0.0129891 | −0.0128637 | −0.0122715 | −0.0116035 | −0.0128839 | −0.0123637 | −0.0102855 | −0.00849743 | |

0.5 | −0.0129929 | −0.0128826 | −0.0123577 | −0.0117587 | −0.0128374 | −0.0121524 | −0.00959096 | −0.00758798 |

**Table 7.**Simply supported beam subjected to a non-dimensional uniform load ${q}_{y}^{*}=-1$. Non-dimensional midpoint foundation reaction ${r}^{*}\left(\right)open="("\; close=")">1/2$ versus the non-dimensional length-scale parameter $\lambda $ evaluated by the non-dimensional Winkler parameter ${k}^{*}\in \left(\right)open="\{"\; close="\}">0.4,2,10,20$ in the DD and MRD models.

r${}^{*}$ (1/2) | |||||||||
---|---|---|---|---|---|---|---|---|---|

$\mathbf{\lambda}$ | DD | MRD | |||||||

k${}^{*}$ = 0.4 | k${}^{*}$ = 2 | k${}^{*}$ = 10 | k${}^{*}$ = 20 | k${}^{*}$ = 0.4 | k${}^{*}$ = 2 | k${}^{*}$ = 10 | k${}^{*}$ = 20 | ||

${0}^{+}$ | −0.00518695 | −0.0255157 | −0.11804 | −0.215888 | −0.00518695 | −0.0255157 | −0.11804 | −0.215888 | |

0.1 | −0.00473987 | −0.0233446 | −0.108595 | −0.199791 | −0.00568254 | −0.0279081 | −0.128154 | −0.232559 | |

0.2 | −0.00392519 | −0.0193724 | −0.0909833 | −0.169134 | −0.00716682 | −0.0350271 | −0.157386 | −0.279296 | |

0.3 | −0.00325168 | −0.016078 | −0.0761649 | −0.142938 | −0.00963245 | −0.0467014 | −0.202683 | −0.347708 | |

0.4 | −0.00275059 | −0.0136205 | −0.0649725 | −0.122886 | −0.0130672 | −0.0626567 | −0.259779 | −0.427556 | |

0.5 | −0.00237493 | −0.011774 | −0.0564754 | −0.107486 | −0.0174544 | −0.082531 | −0.324063 | −0.509754 |

**Table 8.**Simply supported beam subjected to a non-dimensional uniform load ${q}_{y}^{*}=-1$. Non-dimensional midpoint bending moment ${M}^{*}\left(\right)open="("\; close=")">1/2$ versus the non-dimensional length-scale parameter $\lambda $ evaluated by the non-dimensional Winkler parameter ${k}^{*}\in \left(\right)open="\{"\; close="\}">0.4,2,10,20$ in the DD and MRD models.

M${}^{*}$ (1/2) | |||||||||
---|---|---|---|---|---|---|---|---|---|

$\mathbf{\lambda}$ | DD | MRD | |||||||

k${}^{*}$ = 0.4 | k${}^{*}$ = 2 | k${}^{*}$ = 10 | k${}^{*}$ = 20 | k${}^{*}$ = 0.4 | k${}^{*}$ = 2 | k${}^{*}$ = 10 | k${}^{*}$ = 20 | ||

${0}^{+}$ | 0.124473 | 0.122406 | 0.112995 | 0.103036 | 0.124473 | 0.122406 | 0.112995 | 0.103036 | |

0.1 | 0.124512 | 0.122598 | 0.113826 | 0.104439 | 0.124421 | 0.122156 | 0.111935 | 0.101277 | |

0.2 | 0.124582 | 0.122936 | 0.115305 | 0.106976 | 0.124266 | 0.121414 | 0.108867 | 0.0963317 | |

0.3 | 0.124643 | 0.123235 | 0.116638 | 0.109306 | 0.124009 | 0.120195 | 0.104103 | 0.0890578 | |

0.4 | 0.124691 | 0.123472 | 0.11771 | 0.111212 | 0.123651 | 0.118529 | 0.0980778 | 0.0805024 | |

0.5 | 0.124729 | 0.123658 | 0.118562 | 0.112747 | 0.123194 | 0.116452 | 0.0912613 | 0.0715989 |

**Table 9.**Simply supported beam subjected to a non-dimensional uniform load ${q}_{y}^{*}=-1$. Non-dimensional shear force ${T}^{*}\left(1\right)$ versus the non-dimensional length-scale parameter $\lambda $ evaluated by the non-dimensional Winkler parameter ${k}^{*}\in \left(\right)open="\{"\; close="\}">0.4,2,10,20$ in the DD and MRD models.

T${}^{*}$ (1) | |||||||||
---|---|---|---|---|---|---|---|---|---|

$\mathbf{\lambda}$ | DD | MRD | |||||||

k${}^{*}$ = 0.4 | k${}^{*}$ = 2 | k${}^{*}$ = 10 | k${}^{*}$ = 20 | k${}^{*}$ = 0.4 | k${}^{*}$ = 2 | k${}^{*}$ = 10 | k${}^{*}$ = 20 | ||

${0}^{+}$ | 0.49834 | 0.491834 | 0.462207 | 0.430842 | 0.49834 | 0.491834 | 0.462207 | 0.430842 | |

0.1 | 0.498415 | 0.492192 | 0.463669 | 0.433139 | 0.498175 | 0.491035 | 0.4588 | 0.425162 | |

0.2 | 0.498576 | 0.492972 | 0.466991 | 0.43863 | 0.49768 | 0.488657 | 0.448937 | 0.409178 | |

0.3 | 0.498743 | 0.493787 | 0.470565 | 0.444756 | 0.496858 | 0.484753 | 0.433596 | 0.385589 | |

0.4 | 0.498889 | 0.494496 | 0.473746 | 0.450343 | 0.495712 | 0.479413 | 0.414148 | 0.357704 | |

0.5 | 0.499009 | 0.495087 | 0.476435 | 0.45515 | 0.494248 | 0.472753 | 0.392079 | 0.328478 |

**Table 10.**Simply supported beam subjected to a non-dimensional uniform load ${q}_{y}^{*}=-1$. Non-dimensional parameters ${A}_{1}^{*}={A}_{2}^{*}$ versus the non-dimensional length-scale parameter $\lambda $ evaluated by the non-dimensional Winkler parameter ${k}^{*}\in \left(\right)open="\{"\; close="\}">0.4,2,10,20$ in the MRD model.

${\mathit{A}}_{1}^{*}={\mathit{A}}_{2}^{*}$ | ||||
---|---|---|---|---|

$\mathbf{\lambda}$ | MRD | |||

k${}^{*}$ = 0.4 | k${}^{*}$ = 2 | k${}^{*}$ = 10 | k${}^{*}$ = 20 | |

$0.1$ | 0.000165929 | 0.00081521 | 0.00375032 | 0.00682117 |

$0.2$ | 0.000662923 | 0.00324183 | 0.0146081 | 0.0260151 |

$0.3$ | 0.00148862 | 0.00722391 | 0.0314937 | 0.0543281 |

$0.4$ | 0.0026391 | 0.0126718 | 0.0528961 | 0.087776 |

$0.5$ | 0.00410896 | 0.0194672 | 0.0771786 | 0.1228 |

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**MDPI and ACS Style**

Vaccaro, M.S.; Pinnola, F.P.; Marotti de Sciarra, F.; Barretta, R.
Elastostatics of Bernoulli–Euler Beams Resting on Displacement-Driven Nonlocal Foundation. *Nanomaterials* **2021**, *11*, 573.
https://doi.org/10.3390/nano11030573

**AMA Style**

Vaccaro MS, Pinnola FP, Marotti de Sciarra F, Barretta R.
Elastostatics of Bernoulli–Euler Beams Resting on Displacement-Driven Nonlocal Foundation. *Nanomaterials*. 2021; 11(3):573.
https://doi.org/10.3390/nano11030573

**Chicago/Turabian Style**

Vaccaro, Marzia Sara, Francesco Paolo Pinnola, Francesco Marotti de Sciarra, and Raffaele Barretta.
2021. "Elastostatics of Bernoulli–Euler Beams Resting on Displacement-Driven Nonlocal Foundation" *Nanomaterials* 11, no. 3: 573.
https://doi.org/10.3390/nano11030573