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Spatiotemporal Klein–Gordon Nonlocal Elasticity with Strain Gradient Effects for Thermal Vibration Analysis of Rotating Nanobeams Subjected to Laser Irradiation

This study develops a novel spatiotemporal nonlocal elasticity model incorporating strain gradient effects and temperature-dependent thermal conductivity to accurately analyze the thermal vibrations of rotating nanobeams under laser irradiation, revealing that temporal memory and centrifugal stiffening significantly influence deflection and temperature distribution for improved design of nanoscale thermomechanical systems.

Original authors: Elhadi E.E. Elamir, Ahmed E Abouelregal, Mahmoud Abdelaziz

Published 2026-07-03

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Elhadi E.E. Elamir, Ahmed E Abouelregal, Mahmoud Abdelaziz

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a tiny, spinning beam, so small it's measured in nanometers (thousands of times thinner than a human hair). This isn't just any beam; it's part of a high-tech machine, like a microscopic turbine or gyroscope. Now, imagine someone blasts one end of this spinning beam with an incredibly fast, intense pulse of laser light.

The paper you're asking about is a sophisticated "instruction manual" for predicting exactly how this tiny beam will bend, vibrate, and heat up in that split second. The authors, Elhadi Elamir, Ahmed Abouelregal, and Mahmoud Abdelaziz, argue that the old rules of physics (which work great for bridges and cars) break down when things get this small and this fast.

Here is a breakdown of their new "rulebook" using simple analogies:

1. The Problem: The Old Rules Don't Work

Think of a standard steel beam. If you push it, it bends. If you heat it, it expands. Classical physics treats the beam like a smooth, continuous sheet of dough. But at the nanoscale, the beam is more like a crowd of people holding hands.

The "Long-Range" Effect: In a crowd, if someone at the back pushes, the person in the front feels it slightly because of the chain of people connecting them. Classical physics ignores this "chain reaction." The authors say we need to account for how distant parts of the beam influence each other.
The "Memory" Effect: When you stretch a rubber band, it snaps back instantly. But at the nanoscale, materials have a bit of "memory." They don't react immediately; they take a tiny fraction of a second to process the stress. The old rules assume instant reaction, which is wrong for ultra-fast laser pulses.

2. The New Solution: A Three-Part Recipe

To fix the old rules, the authors created a new model that mixes three advanced ingredients:

Ingredient A: The "Ghost Touch" (Spatial Nonlocality)
Imagine the beam is made of a special gel. When you poke one spot, the gel "feels" the poke from a few inches away, not just right where your finger is. This makes the beam act "softer" or more flexible than classical physics predicts. The authors use a mathematical tool called the Klein–Gordon operator to describe this ghostly connection between distant points.
Ingredient B: The "Stiffener" (Strain Gradient)
Now, imagine the beam is trying to bend very sharply, like a tight hairpin turn. At this scale, the material resists bending sharply because the "slope" of the bend changes so fast. This acts like a stiffener, making the beam harder to bend. The authors call this Strain Gradient theory. It's the opposite of the "Ghost Touch"; it makes the beam stiffer.
Ingredient C: The "Time Lag" (Temporal Memory)
This is the Klein–Gordon part again, but for time. If you hit the beam with a laser, the beam doesn't scream (vibrate) immediately. It hesitates for a tiny moment. The authors found that this "hesitation" (temporal memory) actually reduces the amount the beam bends by an extra 2–4% compared to just looking at space.

3. The Twist: The Beam is Spinning and Heating Up

The beam isn't just sitting there; it's spinning like a propeller.

Centrifugal Stiffening: Think of a spinning ice skater pulling their arms in. The spinning creates a tension that pulls the beam straight, making it harder to bend. The authors found this spinning effect can suppress the beam's bending by up to 15% at high speeds.
The Laser and the "Thickening" Heat: The laser heats the beam. Usually, we assume heat moves through metal at a constant speed. But the authors found that as the beam gets hotter, its ability to conduct heat changes.
- If the material gets worse at conducting heat as it gets hotter (like many real materials), the heat gets trapped, the beam gets hotter, and it bends more.
- If the material gets better at conducting heat as it gets hotter, the heat escapes faster, and the beam bends less.
- They found that ignoring this change in heat-conducting ability can lead to errors of up to 12% in predicting how much the beam will bend.

4. The Big Discovery: Heat and Motion are "Loosely Coupled"

One of the most interesting findings is about how the heat and the bending interact.

The authors discovered that the temperature of the beam is mostly determined by the laser and the material's heat properties. It doesn't care much about the fancy "ghost touch" or "stiffener" rules we use for the bending.
However, the bending of the beam does care about the temperature.
The Analogy: Think of the heat as the "engine" and the bending as the "car." The engine's performance (temperature) is mostly independent of the car's suspension (the fancy size-dependent rules). But the car's ride (bending) is heavily influenced by how hot the engine is. This means engineers can calculate the heat first using simple rules, then use that result to calculate the complex bending.

5. What They Actually Found (The Numbers)

The paper ran simulations using Silicon (the material in computer chips) and found:

Softening vs. Stiffening: If you only look at the "Ghost Touch" (spatial nonlocality), the beam bends more. If you only look at the "Stiffener" (strain gradient), it bends less. When you combine them, they fight each other, and the result depends on which one is stronger.
The "Hesitation" Matters: Adding the "Time Lag" (temporal memory) made the beam bend even less, proving that for ultra-fast lasers, time really does matter.
Spinning Helps: The faster the beam spins, the straighter it stays, reducing the bending caused by the laser.
Heat Conductivity is Key: If the material's ability to conduct heat drops as it gets hotter, the beam gets dangerously hot and bends significantly more.

Summary

The authors built a new, highly detailed mathematical model to predict how a microscopic, spinning beam reacts to a laser blast. They proved that to get accurate predictions, you can't just use old physics. You have to account for:

How distant parts of the beam talk to each other.
How the beam resists sharp bends.
The tiny delay in how the beam reacts to stress.
How the beam's ability to move heat changes as it gets hotter.

Their work provides a more accurate "instruction manual" for designing future nanomachines that spin and get hot, ensuring they don't bend too much or break under the stress.

Technical Summary: Spatiotemporal Klein–Gordon Nonlocal Elasticity with Strain Gradient Effects for Thermal Vibration Analysis of Rotating Nanobeams Subjected to Laser Irradiation

Problem Statement
Classical continuum theories fail to accurately predict the behavior of nanoscale systems under dynamic thermomechanical loading, particularly for rotating components subjected to ultrafast laser irradiation. Existing models often suffer from specific limitations: they may neglect strain gradient stiffening, ignore temporal memory effects (stress history dependence), or assume constant material properties. Furthermore, classical Fourier heat conduction predicts infinite thermal propagation speeds, which is physically unrealistic at nanoscales and ultrafast time scales. This study addresses the gap in analyzing rotating nanobeams where centrifugal stiffening, spatio-temporal nonlocality, strain gradients, and non-Fourier heat conduction interact simultaneously under temperature-dependent material properties.

Methodology
The authors develop a novel thermoelastic model for a rotating nanobeam based on the Euler–Bernoulli beam theory. The governing equations integrate three advanced mechanisms:

Spatiotemporal Nonlocal Elasticity: Based on the Klein–Gordon operator, this extends Eringen's integral theory to include both spatial nonlocality (long-range interatomic forces) and temporal memory (viscoelastic relaxation and phonon scattering).
Nonlocal Strain Gradient Theory (NSGT): This unifies Eringen's nonlocal stress (softening effect) with Mindlin's strain gradient stress (stiffening effect) within a single constitutive framework.
Dual-Phase-Lag (DPL) Thermoelasticity: This model introduces phase lags for both heat flux and temperature gradient to ensure finite thermal wave speeds.

The material properties include temperature-dependent thermal conductivity, modeled as a linear function of temperature change ( $K = K_0(1 + K_1\theta)$ ). The nonlinearity introduced by this dependency is linearized using the Kirchhoff transformation. The problem is formulated for a simply supported beam rotating with constant angular velocity $\Omega$ and subjected to a pulsed laser heat source at one end.

The governing coupled partial differential equations are solved analytically using the Laplace transform technique. The resulting system is reduced to an eighth-order ordinary differential equation in the spatial domain. The integration constants are determined by applying boundary conditions for transverse deflection, bending moment, and thermal fields. Time-domain solutions are recovered via numerical inversion using the Honig–Hirdes Fourier series expansion method.

Key Contributions

Unified Framework: This is the first study to integrate spatial nonlocality, strain gradient stiffening, and temporal memory (Klein–Gordon type) into a single model for rotating nanobeams under transient laser loading.
Decoupling Insight: The analysis demonstrates that the thermal governing equation remains effectively independent of mechanical nonlocal parameters, facilitating efficient solution strategies where the thermal field can be solved prior to the mechanical response.
Temperature-Dependent Conductivity: The model explicitly accounts for the variation of thermal conductivity with temperature, revealing significant impacts on the thermoelastic response that constant-property models miss.
Analytical Solution: The derivation of a closed-form analytical solution in the Laplace domain for this complex, coupled, size-dependent system.

Results
Numerical simulations were performed for a silicon nanobeam, comparing five continuum models: Classical Elastic Theory (CET), Eringen Nonlocal Theory (ENT), Klein–Gordon Nonlocal Theory (KGNT), Strain Gradient Theory (SGT), and the full Nonlocal Strain Gradient Theory (NSGT).

Size-Dependent Effects:
- Softening vs. Stiffening: Spatial nonlocality (ENT) reduces effective stiffness, increasing deflections. Conversely, strain gradient effects (SGT) increase stiffness, reducing deflections.
- Temporal Memory: The inclusion of temporal memory (KGNT) further reduces transverse deflections and bending moments by an additional 2–4% compared to spatial nonlocality alone, attributed to stress retardation under transient loading.
- Combined Effect: The NSGT model, balancing softening and stiffening, predicts the smallest deflections (approx. 16.8% reduction compared to CET) and bending moments.
- Centrifugal Stiffening: Rotation suppresses transverse deflection by up to 15% at high speeds due to axial tensile forces.
Temperature-Dependent Conductivity ( $K_1$ ):
- Negative $K_1$ (Conductivity decreases with temperature): Typical of many solids, this leads to higher peak temperatures (4–5% increase), larger thermal moments, and increased deflections and bending moments compared to constant-conductivity models.
- Positive $K_1$ (Conductivity increases with temperature): This results in lower peak temperatures and reduced mechanical responses.
- Sensitivity: The bending moment is the most sensitive field to $K_1$ , showing variations of up to 12.3% across the tested range, while temperature variations are smaller (approx. 4.4%).
Thermal-Mechanical Coupling: While the thermal field is primarily driven by boundary conditions, it is weakly coupled to mechanical curvature. Variations in mechanical size parameters (nonlocal/gradient) feed back into the temperature field, causing systematic differences of 1–5% in temperature predictions across models.

Significance and Claims
The paper claims that the proposed framework offers superior fidelity in capturing high-frequency vibrations and thermal wave propagation compared to classical or purely spatial nonlocal theories. The study provides direct design guidelines for nanothermomechanical systems (NTMS), such as nanoscale gyroscopes and turbines operating under intense laser-induced thermal loads.

The authors emphasize that accurate prediction of structural behavior in these systems requires the full nonlocal strain gradient framework with experimentally determined parameters. They note that neglecting temperature-dependent thermal conductivity can lead to errors of up to 12% in predicted bending moments and stresses, which is unacceptable for high-reliability applications. Conversely, for thermal analysis, the independence of the thermal equation from mechanical nonlocal parameters allows for simplified thermal modeling without calibrating mechanical size parameters.

The work concludes that size-dependent phenomena are critical for nanoscale structural behavior and that the inclusion of temporal memory is essential for short-pulse, high-frequency loading scenarios. Future work suggested by the authors includes extending the model to geometric nonlinearities, integrating surface energy effects, and experimental calibration of the nonlocal and memory parameters.