The studies of laminar unsteady boundary layer flows is crucial for understanding laminar-turbulence transition and origins of turbulence. However, the task of finding its solutions poses a significant challenge. In this paper, we propose a novel approach by introducing a similar transformation to convert the 2D unsteady laminar boundary layer equations into a single partial differential equation. By applying this transformation, we are able to obtain the "exact" similarity solutions for the 2D unsteady laminar boundary layer equations, specifically for the case of flat plate boundary flow. Notably, this is the first time that such an exact solution has been obtained. Application of this exact solution is being used to solve Stokes' first problem in 2D boundary layer. Perspectives on the transition fron Rayleigh solution to Blasius solution is provided.

The discovery of solitonlike coherent structure (SCS) in boundary layer flows is crucial for understanding turbulence origins and in particular for laminar-turbulence transition. However, the task of finding solutions for solitonlike coherent structure from the Navier-Stokes equations poses a significant challenge. In this paper, based on the author's previous work [B.H. Sun, Exact similarity solutions of unsteady laminar boundary layer flows, Preprints 2023, 2023092117], we are able to study convergent flow boundary layers, whose solution encompasses both shock wave and solitary wave solutions, and their superposition gives rise to solitary-like waves, namely solitonlike coherent structure. It is found that the solitonlike coherent structure can only be obtained by combining the Navier-Stokes equations and mass conservation, since the combined equations will have the third order derivatives respect to coordinate $x$, in other words, without the mass conservation condition, the Navier-Stokes equations does not contain solitonlike coherent structure by itself. Finally proved solitonlike coherent structure is a universal motion form of viscous fluid.

The formulation used by most of the studies on an elastic torus are either Reissner mixed formulation or Novozhilov's complex-form one, however, for vibration and some displacement boundary related problem of a torus, those formulations face a great challenge. It is highly demanded to have a displacement-type formulation for the torus. In this paper, I will carry on my previous work [ B.H. Sun, Closed-form solution of axisymmetric slender elastic toroidal shells. J. of Engineering Mechanics, 136 (2010) 1281-1288.], and with the help of my own maple code, I am able to simulate some typical problems and free vibration of the torus. The numerical results are verified by both finite element analysis and H. Reissner's formulation. My investigations show that both deformation and stress response of an elastic torus are sensitive to the radius ratio, and suggest that the analysis of a torus should be done by using the bending theory of a shell, and also reveal that the inner torus is stronger than outer torus due to the property of their Gaussian curvature. Regarding the free vibration of a torus, our analysis indicates that both initial in u and w direction must be included otherwise will cause big errors in eigenfrequency. One of the most intestine discovery is that the crowns of a torus are the turning point of the Gaussian curvature at the crown where the mechanics' response of inner and outer torus is almost separated.

A century-old scientific conundrum is solved in this paper. The Prandtl mixing length modelled plane boundary turbulent flow is described by: $\frac{du^+}{d{y^+} }+\kappa^2{y^+} ^2\left( \frac{du^+}{d{y^+} }\right)^2=1$, together with boundary condition $ {y^+} =0:\, u^+=0$. Only an approximate solution to this nonlinear ordinary differential equation (ODE) has been sought so far, however, the exact solution to this ODE has not been obtained. By introducing a transformation, $2\kappa y^+=\sinh \xi$, I successfully find the exact solution of the ODE as follows: $u^+=\frac{1}{\kappa}\ln(2\kappa {y^+} +\sqrt{1+4\kappa^2{y^+} ^2})-\frac{2{y^+} }{1+\sqrt{1+4\kappa^2{y^+} ^2}}$.

One open question remains regarding the theory of the generalized variational principle, that is, why the stress-strain relation still be derived from the generalized variational principle while the Lagrangian multiplier method is applied in vain? This study shows that the generalized variational principle can only be understood and implemented correctly within the framework of thermodynamics. As long as the functional has one of the combination $A(\epsilon_{ij})-\sigma_{ij}\epsilon_{ij}$ or $B(\sigma_{ij})-\sigma_{ij}\epsilon_{ij}$, its corresponding variational principle will produce the stress-strain relation without the need to introduce extra constraints by the Lagrangian multiplier method. It is proved herein that the Hu-Washizu functional $\Pi_{HW}[u_i,\epsilon_{ij},\sigma_{ij}]$ and Hu-Washizu variational principle comprise a real three-field functional.

The nonlinear deformation and stress analysis of a circular torus is a difficult undertaking due to its complicated topology and the variation of the Gauss curvature. A nonlinear deformation (only one term in strain is omitted) of Mindlin torus was formulated in terms of the generalized displacement, and a general Maple code was written for numerical simulations. Numerical investigations show that the results obtained by nonlinear Mindlin, linear Mindlin, nonlinear Kirchhoff-Love, and linear Kirchhoff-Love models are close to each other. The study further reveals that the linear Kirchhoff-Love modeling of the circular torus gives good accuracy and provides assurance that the nonlinear deformation and stress analysis (not dynamics) of a Mindlin torus can be replaced by a simpler formulation, such as a linear Kirchhoff-Love theory of the torus, which has not been reported in the literature.

For a given material, different shapes correspond to different rigidities. In this paper, the radii of the oblique elliptic torus are formulated, a nonlinear displacement formulation is presented and numerical simulations are carried out for circular, normal elliptic, and oblique tori, respectively. Our investigation shows that both the deformation and the stress response of an elastic torus are sensitive to the radius ratio, and indicate that the analysis of a torus should be done by using the bending theory of shells rather than membrance theory. A numerical study demonstrates that the inner region of the torus is stiffer than the outer region due to the Gauss curvature. The study also shows that an elastic torus deforms in a very specific manner, as the strain and stress concentration in two very narrow regions around the top and bottom crowns. The desired rigidity can be achieved by adjusting the ratio of minor and major radii and the oblique angle.

The catenary shells of revolution are widely used in constructions due to their unique mechanics' feature. However, no publications on this type of shells can be found in the literature. To have a better understanding of the deformation and stress of the catenary shells of revolution, we formulate the principal radii for two kinds of catenary shells of revolution and their displacement type governing equations. Numerical simulations are carried out based on both Reissner-Meissner mixed formulations and displacement formulations. Our investigations show that both deformation and stress response of elastic catenary shells of revolution are sensitive to its geometric parameter $c$, and reveal that the mechanics of the catenary shells of revolution does much better than the spherical shells. Two complete codes in Maple are provided.

By introducing a variable transformation $\xi=\frac{1}{2}(\sin \theta+1)$, a complex-form ordinary differential equation (ODE) for the small symmetrical deformation of an elastic torus is successfully transformed into the well-known Heun's ODE, whose exact solution is obtained in terms of Heun's functions. To overcome the computational difficulties of the complex-form ODE in dealing with boundary conditions, a real-form ODE system is proposed. A general code of numerical solution of the real-form ODE is written by using Maple. Some numerical studies are carried out and verified by both finite element analysis and H. Reissner's formulation. Our investigations show that both deformation and stress response of an elastic torus are sensitive to the radius ratio, and suggest that the analysis of a torus should be done by using the bending theory of a shell.

The Gol'denveizer problem of a torus can be described as follows: a toroidal shell is loaded under axial forces and the outer and inner equators are loaded with opposite balanced forces. Gol'denveizer pointed out that the membrane theory of shells is unable to predict deformation in this problem, as it yields diverging stress near the crowns. Although the problem has been studied by Audoly and Pomeau (2002) with the membrane theory of shells, the problem is still far from resolved within the framework of bending theory of shells. In this paper, the bending theory of shells is applied to formulate the Gol'denveizer problem of a torus. To overcome the computational difficulties of the governing complex-form ordinary differential equation (ODE), the complex-form ODE is converted into a real-form ODE system. Several numerical studies are carried out and verified by finite-element analysis. Investigations reveal that the deformation and stress of an elastic torus are sensitive to the radius ratio, and the Gol'denveizer problem of a torus can only be fully understood based on the bending theory of shells.

The formulation used by the most of studies on an elastic torus are either Reissner mixed formulation or Novozhilov's complex-form one, however, for vibration and some displacement boundary related problem of the torus, application of those formulations has encountered great difficulty. It is highly demanded to have a displacement-type formulation for the torus. In this paper, I will simulate some typical problems and free vibration of the torus. The numerical results are verified by both finite element analysis and H. Reissner's formulation. My investigations show that both deformation and stress response of an elastic torus are sensitive to the radius ratio, and suggest that the analysis of a torus should be done by using the bending theory of a shell, and also reveal that the inner torus is stronger than outer torus due to the property of their Gaussian curvature. Regarding the free vibration of the torus, our analysis indicates that both initial in u and w direction must be included otherwise will cause big errors in eigenfrequency.

This paper examines the influence of physical parameters on the collapse dynamics of a spherical bubble filled with diatomic gas ($\kappa=7/5$). The present numerical investigation shows that each physical parameter affects the bubble collapse dynamics differently. After comparing the contribution of each physical parameter, it appears that, of all the parameters, the surrounding liquid environment affects the bubble collapse dynamics the most. Meanwhile, surface tension has the weakest influence and can be ignored in the bubble collapse dynamics. However, surface tension must be retained in the initial analysis since this, as well as the pressure difference jointly control initial bubble formation. As an essential part of this study, a general Maple code is provided.

The hydraulic fracturing is studied by using dimensional analysis. A universal scaling law of the hydraulic fracturing is obtained. This simple relation has not been seen in the literature.

This paper revisits a classic problem in physics - Hertz elastic dynamics of two colliding elastic spheres. This study obtains impact period in terms of hypergeometric function and successfully combines Deresiewicz's three segmental solutions into one single solution. Our numerical investigation confirms that Deresiewicz's inversion is a good approximation. As an essential part of this study, a general Maple code is provided.

A solution for the Prandtl-Blasius equation is essential to all kinds of boundary layer problems. This paper revisits this classic problem and presents a general Maple code as its numerical solution. The solutions were obtained from the Maple code, using the Runge-Kutta method. The study also considers convergence radius expanding and an approximate analytic solution is proposed by curve fitting. Similarly, the study resolves some boundary layer related problems and provide relevant Maple codes for these.

Finding a solution for Euler's equations is a classic mechanics problem. This study revisits the problem with numerical approaches. For ease of teaching and research, a Maple code comprising 2 lines is written to find a numerical solution for the problem. The study's results are validated by comparing these with previous studies. Our results confirm the correctness of the principle of maximum moment of inertia of the rotating body, which is verified by thermodynamics. As an essential part of this study, the Maple code is provided.

For the solution existence condition of the Navier-Stokes equation, we propose a conjecture as follows: "\emph{The Navier-Stokes equation has a solution if and only if the determinant of flow velocity gradient is not zero, namely $\det (\bm \nabla \bm v)\neq 0$.}"

Among the best-known capillarity phenomena is a capillary rise, the understanding of which is essential in fluidics. Some capillary flows rise monotonically whereas others oscillate, but until now no criteria have been formulated for this scenario. In this paper, the Levine's capillary rise modelling is computed numerically, then the critical radius of the capillary tube is formulated by using the dimensional method and data fitting for identification of exponent index. The phase space diagram of capillary velocity versus height is obtained for the first time and shows that the phase transition from oscillating to monotonic rising happens when the phase trajectory decreases exponentially to somewhere other than the "attractor." Two general Maple codes of the problem are provided as an essential part of this paper.

The material derivative is important in continuum physics. This Letter shows that the expression $\frac{d }{dt}=\frac{\partial }{\partial t}+(\bm v\cdot \bm \nabla)$, used in most literature and textbooks, is incorrect. The correct expression $ \frac{d (:)}{dt}=\frac{\partial }{\partial t}(:)+\bm v\cdot [\bm \nabla (:)]$ is formulated. The solution existence condition of Navier-Stokes equation has been proposed from its form-solution, the conclusion is that "\emph{The Navier-Stokes equation has a solution if and only if the determinant of flow velocity gradient is not zero, namely $\det (\bm \nabla \bm v)\neq 0$.}"

By introducing a variable transformation $\xi=\frac{1}{2}(\sin \theta+1)$, the symmetrical deformation equation of elastic toroidal shells is successfully transferred into a well-known equation, namely Heun's equation of ordinary differential equation, whose exact solution is obtained in terms of Heun's functions. The computation of the problem can be carried out by symbolic software that is able to with the Heun's function, such as Maple. The Gauss curvature of the elastic toroidal shells shows that the internal portion of the toroidal shells has better bending capacity than the outer portion, which might be useful for the design of metamaterials with toroidal shells cells. Through numerical comparison study, the mechanics of elastic toroidal shells is sensitive to the radius ratio. By slightly adjustment of the ratio might get a desired high performance shell structure.

Computing the rotation tensor is vital in the analysis of deformable bodies. This paper describes an explicit expression for the SO(3) rotation tensor R of the deformation gradient F, and successfully establishes an intrinsic relation between the exponential mapping Q = exp A and the deformation F. As an application, Truesdell's simple shear deformation is revisited.

By using directed dimensional analysis and data fitting, an explicit universal scaling law for the velocity of dominoes toppling motion is formulated. The scaling law shows that domino propagational velocity is linearly proportional to the 1/2 power of domino separation and thickness, and -1/2 power of domino height and gravitation. The study also proved that dominoes width and mass have no influence on the domino wave traveling velocity. The scaling law obtained in this Letter is very useful to the dominoes game and will help the domino player to place the dominoes for fast speed and have a quick estimation on the speed without doing complicated multi-bodies dynamical simulation.

This Letter revisits the instability of jet produced by shaped charge. Scaling laws for shaped charge jet are derived by dimensional analysis. It shows that the scaling laws are universal when cross-sectional shrinkage rate is not included in the formulation.

This study revisits the Reynolds-averaged Navier--Stokes equations (RANS) and finds that the existing literature is erroneous regarding the primary unknowns and the number of independent unknowns in the RANS. The literature claims that the Reynolds stress tensor has six independent unknowns, but in fact the six unknowns can be reduced to three that are functions of the three velocity fluctuation components, because the Reynolds stress tensor is simply an integration of a second-order dyadic tensor of flow velocity fluctuations rather than a general symmetric tensor. This difficult situation is resolved by returning to the time of Reynolds in 1895 and revisiting Reynolds' averaging formulation of turbulence. The study of turbulence modeling could focus on the velocity fluctuations instead of on the Reynolds stress. An advantage of modeling the velocity fluctuations is, from both physical and experimental perspectives, that the velocity fluctuation components are observable whereas the Reynolds stress tensor is not.

This letter solves an open question of paper spring risen by Yoneda (2019). Universal scaling laws of a paper spring are proposed by using both dimensional analysis and data fitting. It is found that spring force obeys power square law of spring extension, however strong nonlinear to the total twist angle. Without doing any additional works, we have successfully generalize the scaling laws for Poisson ratio 0.3 to the materials with an arbitrary Poisson's ratio with the help of dimensional analysis.

In light of Liu \emph{at el.}'s original works, this paper revisits the solution of Burgers's nonlinear equation $u_t=a(u_x)^2+bu_{xx} $. The study found two exact and explicit solutions for groups $G_4$ and $G_6$, as well as a general solution. A numerical simulation is carried out. In the appendix a Maple code is provided

This letter attempts to clarify an issue regarding the proper definition of plastic dislocation density tensor. This study shows that the Ortiz’s and Berdichevsky’s plastic dislocation density tensor are equivalent with each other, but not with Kondo’s one. To fix the problem, we propose a modified version of Kondo’s plastic dislocation density tensor.

This paper proposed an explicit and simple representation of velocity fluctuation and the Reynolds stress tensor in terms of the mean velocity field. The proposed turbulence equations are closed. The proposed formulations reveal that the mean vorticity is the key source of producing turbulence. It is found that there are no velocity fluctuation and turbulence if there were no vorticity. As a natural consequence, the laminar- turbulence transition condition was obtained in a rational way.

This paper attempts to clarify an long-standing issue about the number of unknowns in the Reynolds-Averaged Navier-Stokes equations (RANS). This study shows that all perspectives regarding the numbers of unknowns in the RANS stem from the misinterpretation of the Reynolds stress tensor. The current literature consider that the Reynolds stress tensor has six unknown components; however, this study shows that the Reynolds stress tensor actually has only three unknown components, namely the three components of fluctuation velocity. This understanding might shed a light to understand the well-known closure problem of turbulence.

Inspired by amazing result obtained by Semay [1], this study revisits generalised Kepler's third law of an n-body system from the perspective of dimension analysis. To be compatible with Semay's quantum n-body result, this letter reports a conjecture which had not be included in author's early publication [2] but formulated in the author's research memo. The new conjecture for quantum N-body system is proposed as follows: Tq|Eq|^3/2 = πG/√2[(Σ^N_i=1 Σ^N_j=i+1 m_im_j)³/(Σ^N_k=1 m_k)]^1/2. This formulae is, of course, consistent with the Kepler's third law of 2-body system, and exact same as Semay's quantum result for identical bodies.

Wind tunnels are devices that enable researchers to study the flow over objects of interest, the forces acting on them and their interaction with the flow, which is nowadays playing an increasingly important role due to noise pollution. Since the first closed circuit wind tunnel with variable cross-section was built in G¨ottingen, its Prandtl configuration has little change. The wind tunnel with Prandtl configuration has four corners and vanes, more than 50% of the total pressure loss are caused by the corners and vanes. How to reduce the total pressure loss is a world class problem in the wind tunnel design. This study attempts to propose a novel configuration of wind tunnel, where the corners have been replaced by semi-circular tunnel. Sun wind tunnel 2 has only two corners and vanes, while Sun wind tunnel 1 has no corners and vanes at all. It is expected the new wind tunnel can reduce the total pressure loss from 50% to 10%.

To extend the calculation power of tensor analysis, we introduce four new definition of tensor calculations. Some useful tensor identities have been proved. We demonstrate the application of the tensor identities in continuum mechanics: momentum conservation law and deformation superposition.

This letter attempts to find splashing height of liquid-filled container drop impact to a solid surface by dimensional analysis (DA). Two solutions were obtained by both traditional DA and directed DA without solving any governing equations. It is found that the directed DA can provide much more useful information than the traditional one. This study shows that the central controlling parameter is called splash number $\mathrm{Sp}=\mathrm{Ga} \mathrm{La}^\beta=(\frac{gR^3}{\nu^2})(\frac{\sigma R}{\rho \nu^2})^\beta$, which is the collective performance of each quantity. The splash height is given by $ \frac{h}{H}=(\frac{\rho\nu^2}{\sigma R})^\alpha f[\frac{gR^3}{\nu^2}(\frac{R\sigma}{\rho\nu^2})^\beta]=\frac{1}{\mathrm{La}^\alpha}f(\mathrm{Ga}\cdot \mathrm{La}^\beta)$. From the physics of the splashing number, we can have a fair good picture on the physics of the liquid splashing as follows: the jets propagation will generate vortex streets from the container bottom due to sudden pressure increasing from drop impact (water-hammer effect), which will travel along the container sidewall to the centre of the container and subsequently excite a gravity wave on the liquid surface. The interaction between the gravitational force, surface force and viscous force is responsible for creating droplet splash at the liquid surface.

This paper attempts to propose a Riemann spring model via an analogy between the Riemann Zeta function of a complex number and the elastic spring in series.

This paper attempts to extend the Riemann Zeta function of a complex number to a function of a matrix and/or a tensor $A$, namely $$\zeta (A)=\sum _{n=1}^{\infty} \frac{1}{n^{A}}= \sum _{n=1}^{\infty} \sum_{k=1}^n\lambda_k A^k$$ and inverse $$A=\sum _{n=1}^{\infty} \sum_{k=1}^n \mu(n)\lambda_k \zeta(A^k)$$ where $\mu(n)$ is the M\"obius function, $A$ is a complex matrix or tensor with any order, and $\lambda_k$ is eigenvalue of the matri/tensor $A$. This kind of calculations on the Riemann Zeta function has never been seen in the literature. Some examples are provided.

This paper attempts to clarify an issue regarding the lasting unsolved problem of turbulence, namely the closure problem. This study shows that all perspectives regarding the numbers of un- known quantities in the Reynolds turbulence equations stem from the misunderstandings of physics of the Reynolds stress tensor. The current literatures have a consensus that the Reynolds stress tensor has six unknowns; however, this study shows that the Reynolds stress tensor actually has only three ones, namely the three components of fluctuation velocity. With this new understanding, the closed turbulence equations for incompressible flows are proposed.

Based on author's previous work [Sun, B. The Reynolds Navier-Stokes Turbulence Equations of Incompressible Flow Are Closed Rather Than Unclosed. Preprints 2018, 2018060461 (doi: 10.20944/preprints201806.0461.v1)], this paper proposed an explicit representation of velocity fluctuation and formulated the Reynolds stress tensor in terms of the mean velocity field. The proposed closed Reynolds Navier-Stokes turbulence formulations reveal that the mean vorticity is the key source of producing turbulence.

The dislocation density tensors of thin elastic shells have been formulated explicitly in terms of the Riemann curvature tensor. The formulation reveals that the dislocation density of the shells is proportional to KA3=2, where K is the Gauss curvature and A is the determinant of metric tensor of the middle surface.

This paper shown that turbulence closure problem is not an issue at all. All mistakes in the literature regarding the numbers of unknown quantities in the Reynolds turbulence equations stem from the misunderstandings of physics of the Reynolds stress tensor, i.e., all literatures have stated that the symmetric Reynolds stress tensor has six unknowns; however, it actually has only three unknowns, i.e., the three components of fluctuation velocity. We shown the integral-differential equations of the Reynolds mean and fluctuation equations have exactly eight equations, which equal to the numbers of quantities in total, namely, three components of mean velocity, three components of fluctuation velocity, one mean pressure and one fluctuation pressure. That is why we claim in this paper, that the Reynolds Navier-Stokes turbulence equations of incompressible flow are closed rather than unclosed. This study may help to solve the puzzle that has eluded scientists and mathematicians for centuries.

In this paper, for the Reynolds-averaged Navier-Stokes equations, a self-closed turbulence model without any adjustable parameter is formulated and a simple model is proposed. The validation rule for self-closed turbulence model is rigorously derived from the Reynolds-averaged Navier-Stokes equation. The rule is not effected by turbulence modelling on the Reynolds stresses. As an application, two dimensional turbulent boundary layers, and plate turbulent boundary layers have been formulated.

The turbulent Poiseuille flow between two parallel plates is one of the simplest possible physical situations, and it has been studied intensively. In this paper, we propose a modified Prandtl-van Driest mixing length that satisfies both boundary conditions and wall damping effects. With our new formulations, we numerically solve the problem and, moreover, propose an approximate analytical solution of mean velocity. As applications of our solution, an approximate analytical friction coefficient of turbulent Poiseuille flow is proposed.

The exact similarity solutions (also called as special exact solutions) of two dimensional laminar boundary layer were obtained by Blasius in 1908, however, no similarity solutions for two dimensional turbulent boundary layers have been reported in the literature. With the help of dimensional analysis and invariance principle, Prandtl mixing length $\ell=\kappa y$ for one dimensional turbulent boundary layer is extended to $\ell(x,y)=\kappa y (1-\frac{y}{\delta})\sqrt{\frac{\nu}{U\delta}} $ for the two dimensional turbulent boundary layers, furthermore, with a similarity transformation, we successfully transform the two dimensional turbulent boundary layers partial differential equations into a single ordinary differential equation $f'''+ ff''+\beta(1-f'^2)+\kappa^2[\eta^2(1-\eta)^2f''|f''|]'=0$. As an application, similarity solutions of the two dimensional turbulent boundary layer on a flat plate at zero incidence have been studied in detail. To solve the ordinary differential equation numerically, a complete Maple code is provided.

The exact similarity solutions of two dimensional laminar boundary layer were obtained by Blasius in 1908, however, for two dimensional turbulent boundary layers, no similarity solutions (special exact solutions) have ever been found. In the light of Blasius' pioneer works, we extend Blasius similarity transformation to the two dimensional turbulent boundary layers, and successfully transform the two dimensional turbulent boundary layers partial differential equations into a single ordinary differential equation. By author's Maple code, we numerically solve the ordinary differential equation and produce some useful quantities.

Friction, one of the oldest technologies harnessed by humanity, has thus far eluded a theoretical model capable of predicting kinetic friction forces across a broad range of sliding velocities. This study employs the first principle of thermodynamics to delve into sliding friction. For the first time, a general expression for the friction force has been derived, revealing that the damping of sliding friction is inversely proportional to the relative sliding velocity between the two interfaces in motion, contrary to the assumption in traditional models that it is directly proportional to the relative velocity. To simplify the general result, the concept of a friction boundary layer is introduced. An approximate expression is found to reveal the mechanism for reducing friction, which involves increasing the sliding velocity while decreasing the vibration velocity of the object perpendicular to the sliding interface.

Weaving technology can convert two-dimensional structures such as ribbons into three-dimensional structures by specific connections. However, most of the 3D structures fabricated by conventional weaving methods using straight ribbons have some topological defects. In order to obtain smoother continuous 3D surface structures, Baek et al. proposed a novel weaving method using naturally curved (in-plane) ribbons to fabricated three-dimensional curved structures and using this method to weave new spherical weave structures that are closer to perfect spheres. We believe that this new spherical weave structure with smooth geometric properties must correspond to new mechanical properties. To this end, we investigated the buckling characteristics of different types of spherical weave structures by the combination of test and finite element method. The results of calculations and experiments show that the failure mode of the spherical weave structure under vertical loading can be divided into two stages: a flat contact region forms between the spherical weave structure and the rigid plate and inward dimple of ribbons. The spherical weave structures using naturally curved (in-plane) ribbon weaving have better buckling stability than those woven with straight ribbons. The vertical buckling load of spherical weave structures using naturally curved ribbon increases with the width and thickness of the ribbon. In addition, this paper combines test, theoretical and finite element analysis to propose the buckling load equation and buckling correction factor equation for the new spherical weave structure under vertical compression load.

Weaving is an ancient and effective structural forming technique characterized by the ability to convert two-dimensional ribbons to three-dimensional structures. However, most 3D structures woven from straight ribbons have topological defects. Baek et al. proposed a method to weave smoother continuous 3D surface structures using naturally curved (in-plane) ribbons, obtained a new surface structure with relatively continuous variation of Gaussian curvature, and analyzed its geometric properties. We believe that this new 3D surface structure with smooth geometric properties must correspond to new mechanical properties. To this end, we investigated a 3D surface structure using naturally curved (in-plane) ribbon weaving, and the results of calculations and experiments show that such structures have better buckling stability than those woven with straight ribbons. It is observed that the number of ribbons influences the buckling behavior of different types of woven structures.

The cuttlebone-like structure is a complex porous bionic structure with asymmetric sinusoidal S-shaped wall structure connecting laminar septa, and studies have shown that the cuttlebone-like structure has light weight, high strength and excellent energy absorption capability. It has become an important research to investigate the mechanical mechanism of cuttlebone-like structure and to design a bionic structure beyond nature structure based on it. In order to investigate the mechanical properties of the static compression and shearing processes of the cuttlebone-like structure, this paper establishes theoretical formulas and analyzes the influence of dimensionless parameters such as height-to-thickness ratio $\lambda$, period-to-thickness ratio $\xi$ and amplitude to period ratio $\eta$ on the compression and shearing processes with the help of ABAQUS finite element software.The parameter sensitivity analysis method was used to compare the important influence degree of each dimensionless parameter on the mechanical properties of the structure and to determine the relative optimal parameters of the structural mechanical properties under compression and shearing. Based on the relatively optimal parameter of the cuttlebone-like structure, the cross-sectional geometry of the structure is improved to build a new cuttlebone-like structure.The experimental results showed that the new cuttlebone-like structure with revised cross-section improved the mechanical properties of the cuttlebone-like structure.The study results are of theoretical guidance to enhance the mechanical properties and geometric design of cuttlebone-like structures.

Gol'denveizer's problem of a torus has been analyzed by Audoly and Pomeau (2002) and Sun (2021). However, all of the investigations of Gol'denveizer's problem of an elastic torus have been linear. In this paper, the finite element method is used to more accurately address this problem. Furthermore, Sun (2021) cannot be solved by nonlinear analysis. We research the nonlinear mechanical properties of Gol'denveizer's problem of circular and elliptic tori, and relevant nephograms are given. We study the buckling of Gol'denveizer's problem of an elastic torus, and propose failure patterns and force-displacement curves of tori in the nonlinear range. Investigations reveal that circular tori have more rich buckling phenomena as the parameter a increases. Gol'denveizer's problem of the buckling of an elliptic torus is analyzed, and we find a new buckling phenomenon called a "skirt." As a/b increases, the collapse load of an elliptic torus of the Gol'denveizer problem is enhanced gradually.

The Gol'denveizer problem of a torus was studied analytically by Audoly and Pomeau (2002), and the accuracy of the Audoly and Pomeau linear law was confirmed numerically by Sun (2021). However, the law does not include the major radius R of the torus. To find the influence of the major radius, we used finite element numerical simulation to simulate different cases, and we propose a modified Audoly and Pomeau linear law for vertical deformation, which includes R. A linear law of horizontal deformation is presented as well. Our studies show that the Audoly and Pomeau linear law has high accuracy. With modified vertical and horizontal deformation, a displacement-compatible relation between them is formulated.

Snap-fit is a commonly used jointing/connention mechanism, and its easy installation and difficult removal are the result of the coordinated interaction between friction, geometric shape, and elasticity. This paper presents a detailed study on the assembly/disassembly forces of thin-walled torus snap-fit through finite element simulation, experimental testing, and data fitting of approximate analytical solutions. The research reveals that the non-zero Gaussian curvature of the torus has a significant impact on the mechanical performance of the torus snap fit. The findings in this study are of great significance for future design of high-performance structural connections.

In engineering, connections between components are often weak areas. Unreasonable connection methods can easily reduce the strength of components, resulting in unpredictable failure modes. In nature, numerous connection methods for biological structures with excellent mechanical properties have evolved. Studying the connection methods of organisms in nature can inspire new ideas for bionic connection methods. When the diabolical ironclad beetle is under pressure, the elytra are not easy to separate, which ensures the stability of the beetle's external structure, thus making the beetle extremely resistant to pressure. The reason for this is the interlocking and toughening effect of the unique jigsaw connection between the elytra. Therefore, in this paper a theoretical analysis model is established and used to analyze the mechanical behavior of the diabolical ironclad beetle's jigsaw connection during the drawing process and determine the influence of factors such as quantity, angle, and geometric characteristics on the mechanical properties of the jigsaw connection. The results of the theoretical analysis are then compared with the results of experiments and ABAQUS finite element simulation.

Holding an object by clamping force is a fundamental phenomena, layered or laminated architectures with internal sliding features are essential mechanism in natural and man-made structural system. In this paper, we combine the layered architecture and clamping mechanism to form a multilayered clamp and study the clamping force with internal friction. Our investigations show that the clamping force and energy dissipation are very mich depend on the number of layers, its geometry and elasticity, as well as internal friction. The central goal of studying the covered book is not only to predict the clamping force of the clamp, but also as a representative case to help finding some clue on the universal behaviours of multilayered architectures with internal friction.

Snap fit is a common mechanical mechanism. It uses the physical asymmetry that is easy to assemble but difficult to disassemble to provide a simple and fast link between objects. The ingenious combination of geometric shape, bending elasticity and friction of the snap fit is the mechanism behind the easy to assemble but difficult to disassemble disassemble. Yoshida and Wada (2020) has done a groundbreaking work in the analysis of the elastic snap fit. During our study of their paper, while we really enjoyed their research, unfortunately we detected several questioning formulations. After careful checking, we found that those formulations are not typographical, therefore it is necessary to make corrections. This paper reformulates the linear elasticity of a cylindrical snap fit, obtains an exact solution and proposes an accurate relation between the opening angle and bending tangent angle. Under the first order approximation, our formulations can reduced to the results of Yoshida and Wada and hence confirms the scientific correctness of Yoshida and Wada's work. Furthermore, this paper also derives a correct vertical displacement expression, and propose a new way of disassembly by bending for the first time and formulate a scaling law by data fitting. All formulations are validated by finite element simulation and experiment. The research here is helpful to the design of elastic snap fit or adjustable mechanical mechanism and metamaterial cell.

The common dandelion uses a bundle of drag-enhancing bristles (the pappus) that enables seed dispersal over formidable distances; however, the scaling laws of aerodynamic drag underpinning pappus-mediated flight remains unresolved. In this paper, we study the aerodynamic shape of dandelion, derive the scaling law of resistance, determine the Vogel exponent. In particular, we find that the total drag coefficient is proportional to the -2/3 power of the dandelion pappus Reynolds number, and obtain the terminal velocity of the dandelion seed under gravitation field.

The study of a flexible body immersed in a flowing medium is one of best way to find its aerodynamic shape. This Letter revisited the problem first studied by Alben, Shelley and Zhang (Nature 420, 479-481, 2002). The aerodynamic shape of the fibre is found by simpler approach and universal drag scaling laws of the flexible fibre in flowing medium are proposed by using dimensional analysis. The Alben scaling laws is being generalized and confirmed to be universal. Our study show that the Alben number is a measurement of maximum curvature of the fibre forced by dynamic pressure. A complete Maple code is provided for finding aerodynamic shape of the fibre in the flowing medium.

The biomimetic design of engineering structures is based on biological structures with excellent mechanical properties, which are the result of billions of years of evolution. However, current biomimetic structures, such as ordered lattice materials, are still inferior to many biological materials in terms of structural complexity and mechanical properties. For example, the structure of \textit{Euplectella aspergillum}, a type of deep-sea glass sponge, is an eye-catching source of inspiration for biomimetic design; however, guided by scientific theory, how to engineer structures surpassing the mechanical properties of \textit{E. aspergillum} remains an unsolved problem. The lattice structure of the skeleton of \textit{E. aspergillum} consists of vertically, horizontally, and diagonally oriented struts, which provide superior strength and flexural resistance compared with the conventional square lattice structure. Herein, the structure of \textit{E. aspergillum} was investigated in detail, and by using the theory of elasticity, a lattice structure inspired by the bionic structure was proposed. The mechanical properties of the sponge-inspired lattice structure surpassed the sponge structure under a variety of loading conditions, and the excellent performance of this configuration was verified experimentally. The proposed lattice structure can greatly improve the mechanical properties of engineering structures, and it improves strength without much redundancy of material. This study achieved the first surpassing of the mechanical properties of an existing sponge-mimicking design. This design can be applied to lattice structures, truss systems, and metamaterial cells.

Single-layer core corrugated sandwich panels generally consist of a corrugated core and two layers of panels, while multi-layer core corrugated sandwich panels are formed by stacking multiple layers of panels with multiple layers of core layers. In this study, integrated multilayer core corrugated sandwich panels with different shapes of corrugated cores (triangular, trapezoidal, and rectangular) and the different number of core layers were fabricated using 3D printing technology, and the mechanical behavior of such multilayer core corrugated sandwich panels under quasi-static three-point bending was investigated using experiments and numerical simulations. The effects of core shape and number of core layers on the bending deformation process, damage mode, load carrying capacity, and bending energy dissipation capacity of multilayer core sandwich panels are discussed. Parametric design of multilayer triangular core corrugated sandwich panels was also carried out by finite element software ABAQUS. It was found that a new multilayer corrugated sandwich panel with a multi-layer core is better than the single core shape multilayer corrugated sandwich panel in terms of bending load capacity, energy dissipation capacity and deformation capacity can be obtained through the combination design of different core shapes.

The morphological transformation of an arched ribbon driven by torsion is a scientific problem that is connected with daily life and requires thorough analysis. An arched ribbon can achieve an instantaneous high speed through energy transformation and then return to the original shape of the structure. In this paper, based on the characteristics of the ribbon structure, the dynamic mathematical model of the arched ribbon driven by torsion is established from the Kirchhoff rod equation. The variations of the Euler angle of each point on the center line of the ribbon with the arc coordinate s and the rotation angle of the supports $\phi$ was examined. The relationship between the internal force distribution of each point in the direction of $\hat{d}_{a}$ and the material, cross-sectional properties, and rotation angle of the supports was obtained. We used ABAQUS, a nonlinear finite element analysis tool, to simulate the morphological transformations of the ribbons, verified our theory with simulation results, and reproduced the experimental results of Sano. Furthermore, we redefined the concept of the ``critical flipping point'' of Sano. In this paper, the dimensional analysis method was used to fit the simulation data. The following relationship between the critical width $ w ^{*} $, thickness h, and the radius R of the ribbon with different cross sections was obtained: $w^{*}=A\cdot R\left(h/R\right)^{0.6}$, where A is 3.19 for rectangular cross sections and 3.06 for elliptical cross sections. By analyzing the simulation data, we determined the variation behavior of the out-of-plane deflection of the center point of the ribbon with the radius R, width w, and thickness h. Our research has guiding significance for understanding and designing arched ribbons driven by torsion, and the results can be applied to problems of different scales.

Why are pieces of spaghetti generally broken into three to ten segments instead of two as one thinks? How can one obtain the desired number of fracture segments? For these problems, in this paper a strand of spaghetti is considered an elastic rod, and the finite-element software ABAQUS is used to simulate the detailed fracture dynamics of the elastic rod. By changing the size (length and diameter) of the rod, the relevant data on the fracture limit curvature and the number of fractured segments of the elastic rod are obtained. The ABAQUS simulation results confirm the scientific judgment of B. Audoly and S. Neukirch (Fragmentation of rods by cascading cracks: Why spaghetti does not break in half. Phys Rev Lett 95: 095505). Using dimensional analysis to fit the finite-element data, two relations of the elastic rod fracture dynamics are obtained: (1) the relationship between the fracture limit curvature and the diameter, and (2) the relationship between the number of fracture segments and the diameter-length ratio. Results reveal that when the length is constant, the larger the diameter (the smaller the diameter-length ratio D/L), the smaller the limit curvature; and the larger the diameter-length ratio D/L, the fewer the number of fractured segments. The relevant formulations can be used to obtain the desired number of broken segments of spaghetti by changing the diameter-to-length ratio.

Why do stretched rubber bands not hit the hand after ejection? What is the mechanism behind the rubber band ejection dynamics? These questions represent a fascinating scientific problem. Because the size of a rubber band in the circumferential direction is much larger than that in the other two directions of its cross-section, we regard the rubber band as a slender beam and establish a mathematical model of the dynamics of the rubber band stretching and ejection. Furthermore, we obtain the dependence of the dynamic curvature of the rubber band on the arc length and time. We used the finite element software ABAQUS to simulate the dynamic process of a rubber band stretching and ejection. The simulation results and dimensional analysis were performed to examine the effect of the bending elastic rebound velocity. The mathematical model and simulation results revealed that the relationship between the curvature and time at the end of the rubber band ($s =0$) was as follows: $\kappa\sim t^{-{1}/{2}}$. This research has guiding significance for the design of rubber bands as elastic energy storage devices.

To have a better protection, strong toughness and good flexibility, all lives and plants must have skins, similarly, all books should have covers. In this paper, considering interlayer friction as a perturbation, we formulate a hardcover-book-like laminated plates with internal friction. For a quasi-static problem, detail analysis on the bending response and energy dissipation are carried out for a three-point-supported plates. Our numerical investigations show that the hardcover is more essential than the core layers in terms of bending response, and the energy dissipation within per loading-unloading cycle can be made to vary by a considerable amount. The central goal of studying is not only to predict the bending deformation of the book-like-plates, but also as a representative case to help finding some clue on the universal behaviours of multilayered architectures with internal friction.