Thin Solid Plate Research Articles

Optimal design of elastic plates

This paper is concerned with optimal design problems in the setting of the Kirchhoff–Love model for pure bending of a thin solid symmetric plate under a transverse load. For two isotropic elastic materials with a prescribed amount, the goal is to find their rearrangement within the domain that forms a least compliant structure. The homogenization method is used as a relaxation tool to overcome the lack of a classical solution of optimal design problem. Neccessary conditions of optimality were derived and an optimality criteria method for the single state compliance minimization problems is developed and tested on several examples.

Higher-Order Finite Element Vibration Analysis of Circular Raft on Winkler Foundation

On a Winkler foundation, solid circular plate vibration is examined using a higher-order finite element in polar co-ordinate system. The present formulation has developed a Mat-lab code to handle any boundary conditions. Validation of the code is carried out after the convergence studies. The results are compared to other researchers and show excellent conformity. Furthermore, a parametric analysis gave the first 10 natural frequency characteristics in tabular and graphical form. The authors conclude that the present formulation is straightforward, behaves exceptionally well for thin solid circular plates on elastic foundations with reasonable convergence rate and accuracy, and requires less computational effort, resources, and time.

Wrinkling Instability in Unsupported Epithelial Sheets.

We investigate the elasticity of an unsupported epithelial monolayer and we discover that unlike a thin solid plate, which wrinkles if geometrically incompatible with the underlying substrate, the epithelium may do so even in the absence of the substrate. From a cell-based model, we derive an exact elasticity theory and discover wrinkling driven by the differential apico-basal surface tension. Our theory is mapped onto that for supported plates by introducing a phantom substrate whose stiffness is finite beyond a critical differential tension. This suggests a new mechanism for an autonomous control of tissues over the length scale of their surface patterns.

Numerical Simulation of Nonlinear Pulse Compression of High-energy Ytterbium-doped Femtosecond Laser Based on Multiple Thin Solid Plates

Numerical Simulation of Nonlinear Pulse Compression of High-energy Ytterbium-doped Femtosecond Laser Based on Multiple Thin Solid Plates

Small‐amplitude homogenization of elastic plates via H‐measures

AbstractSmall‐amplitude homogenization is a procedure for computing the first correction in the homogenization limit. By using H‐measures, we calculate the first correction in the small‐amplitude homogenization limit of a non‐periodic sequence of tensors describing material properties in the Kirchhoff‐Love model for pure bending of a thin solid symmetric plate under a transverse load.

Demonstration of contrast improvement and spectral broadening in thin solid plates

Here we demonstrate that the pulses in thin solid plates (TSPs) can simultaneously realize contrast improvement and spectral broadening. In a proof-of-principle experiment, we used nine thin fused silica plates to make the beam form a series of foci in sequence, by which the divergence of the beam of the main pulse was seriously changed. After the last plate, the intense center spot of the output beam was picked out, and the energy of about 129 µJ was obtained for the 400 µJ input pulses, corresponding to a total transmission efficiency of more than 30%. The contrast measurement showed that the contrast was improved by 2 orders of magnitude. Meanwhile, the cleaned pulses were broadened spectrally, spanning from 680 to 930 nm at the -23dB intensity level and giving a compressed pulse of 11.3 fs. These characteristics make the TSP method suitable to generate broadband seed pulses for high-contrast, few-cycle intense lasers.

Geometric theory of non-regular separation of variables and the bi-Helmholtz equation

The geometric theory of additive separation of variables is applied to the search for multiplicative separated solutions of the bi-Helmholtz equation. It is shown that the equation does not admit regular separation in any coordinate system in any pseudo-Riemannian space. The equation is studied in the four coordinate systems in the Euclidean plane where the Helmholtz equation and hence the bi-Helmholtz equation is separable. It is shown that the bi-Helmoltz equation admits non-trivial non-regular separation in both Cartesian and polar coordinates, while it possesses only trivial separability in parabolic and elliptic–hyperbolic coordinates. The results are applied to the study of small vibrations of a thin solid circular plate of uniform density which is governed by the bi-Helmholtz equation.

Different scenarios of shrinking surface soap bubbles

We discuss a simple experiment investigating the shrinkage of surface soap bubbles sitting on a thin solid plate with a circular orifice located under the apex of the bubble. We identify three different shrinking regimes, the occurrence of which depends on a combination of key parameters that include the ratio between initial bubble and orifice sizes and physicochemical properties of the fluid system. For low-viscosity liquids and/or large ratios, a bubble remains quasi-hemispherical as shrinking proceeds. In contrast, for liquids with sufficiently large viscosities and/or small geometric ratios, a bubble seeks the shape of a spherical cap while the air inside it escapes through the orifice. In this case, shrinking proceeds with a bubble foot that either recedes over time or does not move for the largest viscosities and/or smallest ratios. We use basic physical arguments to rationalize the three identified regimes and to explain the shrinking dynamics. Specifically, this model which captures observations and measurements is based on Bernoulli's principle for the air flow, volume conservation, and a friction law that accounts for viscous dissipation at the moving bubble foot.

Compact and robust supercontinuum generation and post-compression using multiple thin plates

Abstract We report on compact and robust supercontinuum generation and post-compression using transmission of light through multiple thin solid plates at the SwissFEL X-ray free-electron laser facility. A single stage consisting of three thin plates followed by a chirped mirror compressor achieves compression of initially 30-fs pulses with 800-nm center wavelength to sub-10-fs duration. We also demonstrate a two-stage implementation to compress the pulses further to sub-5-fs duration. With the two-stage setup, the generated supercontinuum includes wavelengths ranging from 500 to 1100 nm. The multi-plate setup is compact, robust, and stable, which makes it ideal for applications at free-electron laser facilities such as pump-probe experiments and laser-arrival timing tools.

Determination of the efforts in two-way slabs of concrete through the finite difference method

From the implementation of analytical solutions for thin rectangular slabs using the Finite Difference Method, the present paper was developed with the purpose of comparing the determined efforts in a concrete plate with those that would be found using the tables originated from the plates theory of authors established in the literature. For this, the programming language Python was used, taking as a case study, a slab with dimensions of 5 meters long by 3 meters wide and thickness equal to 0.10 meters. In the analysis of the efforts, several support conditions were considered, in which the internal efforts and displacements obtained by the Finite Difference Method were compared with the results obtained through the tables proposed in literature. It was verified that the Finite Difference Method constitutes a good alternative for the resolution of thin solid plates, since the results were similar to the classic solution proposed in literature. The implemented program allows the visualization of the efforts through of spectrums of zone which facilitates the understanding of the distribution of the efforts along the slab, differing slightly from the uniform distribution adopted in the tables consulted.

Microscale Observation via High-Speed X-ray Diffraction of Alloy 718 During In Situ Laser Melting

The laser melting process is accompanied by rapid evolution in temperature, phase, structure, and strain because of its high heating and cooling rates. In this study, the evolution of grains within a thin solid plate of Ni alloy 718 during laser processing was probed with in situ high-energy x-ray diffraction experiments. The high temporal and spatial resolution available in the measurement allowed us to study the rapid evolution of the melted region beneath the surface of the sample. The characterization of the evolution of secondary phases, i.e., Laves and carbide, was captured despite the weak diffracted peaks caused by small volume fractions. Thermal history was estimated based on changes in the lattice spacing from the thermal contraction upon cooling. The temporal variation in 2θ with azimuthal direction revealed the evolution in anisotropy of lattice spacing and thus of the mechanical state during laser processing.

On the effective properties of composite elastic plate

We consider the fourth-order elliptic equation describing the Kirchhoff-Love model for pure bending of a thin solid symmetric plate under a transverse load. Following the results of Antonić and Balenović (1999, 2000) [4,5], and Burazin, Jankov and Vrdoljak (2018) [12], we show the local character of the set of all possible composites, also called the G-closure, and prove that the set of composites obtained by periodic homogenization is dense in that set. Moreover, we derive expressions for elastic coefficients of composite plate obtained by mixing two materials in thin layers, also known as laminated materials, and for mixing two materials in the low-contrast regime. Additionally, we derive optimal bounds on the effective energy of a composite material, known as Hashin-Shtrikman bounds. In the setting of two-phase isotropic materials in dimension d=2, explicit optimal Hashin-Shtrikman bounds are calculated.

High-contrast, intense single-cycle pulses from an all thin-solid-plate setup

High-contrast, intense single-cycle pulses are highly desirable tools in ultrafast science, enabling highest temporal resolution, pushing matter to extreme conditions, and serving as drivers in petahertz electronics. In this Letter, we use thin solid plates in a double multi-plate supercontinuum configuration, delivering a broadband spectrum spanning from ∼ 400 to ∼ 1000 n m at the − 20 d B intensity level to produce a single-cycle pulse. We show that the spectral broadening by self-phase modulation with few-cycle pulses is more suitable for compression than the single-cycle limit than with multi-cycle pulses. The pulses are compressed to 2.6 fs pulses, close to the transform limit of 2.55 fs, with an energy of 0.235 mJ. They exhibit an excellent power stability of 0.5% rms over 3 h and a beam profile. The obtained single-cycle pulses can be utilized in many applications, such as generation of isolated attosecond pulses via high-order harmonic generation, investigation of ultrafast phenomena with extreme temporal resolution, or high-intensity laser-solid experiments.

Multi-mode ultrasonic waves focusing in a variable focus technique for simultaneous sound-velocity and thickness measurement

A double-focus technique, using a spherically focused immersion transducer, has been developed for simultaneous sound-velocity and thickness measurement. This is an effective method that does not require pre-experiments or predictive parameters. In previous studies, the single-mode wave model, usually used, cannot explain the additional reflection peaks on the V(z,t) curve, which denote two-dimensional amplitudes as a function of the movement along distance z and the time t. In this research, a multi-mode wave model, interpreting the results of depth scanning of thin solid plates, using a spherically focused ultrasound transducer in pulse-echo mode, is presented. The existence of additional reflection peaks in the solid plate due to mode conversion, which are usually neglected, are described in the new model. A comprehensive interpretation of the measurements is proposed, instead of ignoring the transverse waves and mode conversion. The distribution of energy among multiple reflected focuses and the vertical displacements of the transducer are numerically analysed. The experiments on a 0.5 mm thick stainless steel and a 0.5 mm thick aluminium alloy have been carried out to verify the model. This study contributes to the identification of longitudinal-wave focuses, especially in cases of simultaneous measurement on multi-layered materials.

Small-amplitude homogenization of elastic plate equation

ABSTRACT In this paper we explicitly calculate the first correction in the small-amplitude homogenization limit of a sequence of periodic tensors describing material properties in the Kirchhoff model for pure bending of a thin solid symmetric plate under a transverse load. We also discuss smooth dependence of H-limit on a parameter and calculate H-limit of a periodic sequence of tensors.

HOMOGENIZATION OF ELASTIC PLATE EQUATION∗

We are interested in general homogenization theory for fourth-order elliptic equation describing the Kirchhoff model for pure bending of a thin solid symmetric plate under a transverse load. Such theory is well-developed for second-order elliptic problems, while some results for general elliptic equations were established by Zhikov, Kozlov, Oleinik and Ngoan (1979). We push forward an approach of Antoni´c and Balenovi´c (1999, 2000) by proving a number of properties of H-convergence for stationary plate equation.

Research progress on octave supercontinuum generation in solid medium

When a short laser pulse passes through transparent medium, the spectrum may be broadened due to nonlinear optical effects, and a coherent octave supercontinuum may be generated under certain conditions. Such a supercontinuum may be compressed into a femtosecond few-cycle pulse, which has many applications in ultrafast optics and beyond. Spectral broadening has been achieved experimentally in gases, liquids, and solids. Current mainstream technique of supercontinuum generation is to send multi-cycle femtosecond pulses through inert-gas-filled hollow-core fibers. However, due to the limitation of the core diameter, the hollow-core fiber cannot work with high-energy laser pulses. With a much higher nonlinear index of refraction, solid-state material is naturally a more promising candidate for supercontinuum generation, but it is difficult to obtain a near-octave spectrum in one piece of solid without filamentation. The optical Kerr effect in solids triggers self-phase modulation (SPM) which induces desired spectral broadening as well as self-focusing, thus causing the laser intensity to rise drastically with substaintial multiphoton excitation and ionization leading to plasma formation. This behavior results in filamentation and optical breakdown, and eventually permanent damage to the material occurs if the laser pulse energy is high enough. Using a thin plate of dielectrics may minimize the effect of self-focusing-the beam exits from the nonlinear medium before it starts to shrink and causes damage. However, one thin plate does not provide enough nonlinear effect to generate a broad spectrum. To prevent disastrous self-focusing while achieving spectral broadening, using multiple Kerr elements has been proposed theoretically and demonstrated experimentally at microjoule to millijoule level. In such a configuration, a femtosecond laser pulse is being spectrally broadened via SPM in the thin plates, while self-focusing converges the beam in each plate but the focal spot is located outside the plate. Once the converging beam passes through its focal spot in air, the beam diverges and enters the next plate to repeat this process until the spectral broadening stops after several elements. Using this method, octave supercontinuum with energies at microjoule to millijoule level has been experimentally obtained in a spectral range covering near-ultraviolet to mid-infrared. In this paper, we review the development of supercontinuum generation in multiple thin solid plates, outline the principle of supercontinuum generation in this new type of thin solid medium, brief the experiments using this new method in recent years, and look into the prospects for its development.

Measurement of the environmental temperature using the sol-air thermometer

Heat flow measurement with a heat flow meter is a standardized method (ISO 9869-1) to estimate thermal transmittance (U-value) of a building element. The heat flow meter is a thin plate mounted on top of the surface of the element, and measures the heat flux q through the plate. The measured q is the product of the difference in temperatures between exterior and interior environment, and the U-value. The heat transferred from the element is based on the radiant and the convective heat transfer.ISO 9869-1 specifies that the environment temperature Te “is a notional temperature" and it "cannot be measured directly” (section A.3.1). The air temperature Ta is proposed as a reasonable approximation for the indoor environment, while overcast conditions and absence of significant solar radiation are specified conditions for replacing Te with Ta for the exterior environment.The sol-air thermometer (SAT) measures the sol-air temperature Tsa, i.e. the equivalent temperature of the convective and the radiative environment. In the absence of solar radiation, Te = Tsa. SAT is a sensor consisting of a thin flat solid plate, of high thermal conductivity. The front side of the sensor is exposed to the environment, whose Tsa is to be measured, and the backside is thermally insulated. The temperature of the SAT-plate equals Tsa.In this work we propose introduction of the measured Te in the existing standard (ISO 9869-1). The method for measurement of Tsa, using the SAT, has been demonstrated experimentally for different periods, without solar radiation present and under stable climatic conditions.

Numerical simulation of the turbulence effect on heat transfer between fluid and thin plates of a solid material

We present the results of numerical simulation of conjugate heat transfer between thin (100-1000 nm) solid plates and periodically heated air flow. Several air inflow conditions were studied with different turbulence intensity and integral scale. The thermal response in the plate was calculated showing that inflow turbulence interacting with the plate generates significant temperature fluctuations in the plate volume. These fluctuations amplitudes grow along the plate and with time for several periods, indicating the existence of some feedback process, with turbulent air flow acting as a medium, transferring thermal signals between different sections of the plate. The described spatial and temporal fluctuations may be used for pyroelectric energy harvesting applications.

Supercontinuum generation in a multi-plate medium

We analyze femtosecond supercontinuum generation in a distribution of thin solid plates to show that the distributed scheme inhibits processes leading to pulse breakup while allowing spectral expansion to proceed as desired. We introduce basic criteria for setting the plate thickness or initial laser intensity and the location of each plate in the laser beam path and confirm that under these conditions a fully-coherent and intense supercontinuum can be generated for input peak power of as much as two thousand times the critical power for self-focusing of the solid medium.