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We consider polynomial approximations of z ¯ to better understand the torsional rigidity of polygons. Our main focus is on low degree approximations and associated extremal problems that are analogous to Pólya's Conjecture for torsional rigidity of polygons. We also present some numerics in support of Pólya's Conjecture on the torsional rigidity of pentagons. [ABSTRACT FROM AUTHOR]

We consider families of polynomial lemniscates in the complex plane and determine if they bound a Jordan domain. This allows us to find examples of regions for which we can calculate the projection of z ¯ to the Bergman space of the bounded region. Such knowledge has applications to the calculation of torsional rigidity. [ABSTRACT FROM AUTHOR]

We prove a lower bound on the sharp Poincaré–Sobolev embedding constants for general open sets, in terms of their inradius. We consider the following two situations: planar sets with given topology; open sets in any dimension, under the restriction that points are not removable sets. In the first case, we get an estimate which optimally depends on the topology of the sets, thus generalizing a result by Croke, Osserman and Taylor, originally devised for the first eigenvalue of the Dirichlet–Laplacian. We also consider some limit situations, like the sharp Moser–Trudinger constant and the Cheeger constant. As a byproduct of our discussion, we also obtain a Buser-type inequality for open subsets of the plane, with given topology. An interesting problem on the sharp constant for this inequality is presented. [ABSTRACT FROM AUTHOR]

Motivated by the connection between the first eigenvalue of the Dirichlet–Laplacian and the torsional rigidity, the aim of this paper is to find a physically coherent and mathematically interesting new concept for boundary torsional rigidity, closely related to the Steklov eigenvalue. From a variational point of view, such a new object corresponds to the sharp constant for the trace embedding of W 1 , 2 (Ω) into L 1 (∂ Ω). We obtain various equivalent variational formulations, present some properties of the state function and obtain some sharp geometric estimates, both for planar simply connected sets and for convex sets in any dimension. [ABSTRACT FROM AUTHOR]

We formulate a plausible conjecture for the optimal Ehrhard-type inequality for convex symmetric sets with respect to the Gaussian measure. Namely, letting J_{k-1}(s)=\int ^s_0 t^{k-1} e^{-\frac {t^2}{2}}dt and c_{k-1}=J_{k-1}(+\infty), we conjecture that the function F:[0,1]\rightarrow \mathbb {R}, given by \begin{equation*} F(a)= \sum _{k=1}^n 1_{a\in E_k}\cdot (\beta _k J_{k-1}^{-1}(c_{k-1} a)+\alpha _k) \end{equation*} (with an appropriate choice of a decomposition [0,1]=\cup _{i} E_i and coefficients \alpha _i, \beta _i) satisfies, for all symmetric convex sets K and L, and any \lambda \in [0,1], \begin{equation*} F\left (\gamma (\lambda K+(1-\lambda)L)\right)\geq \lambda F\left (\gamma (K)\right)+(1-\lambda) F\left (\gamma (L)\right). \end{equation*} We explain that this conjecture is "the most optimistic possible", and is equivalent to the fact that for any symmetric convex set K, its Gaussian concavity power p_s(K,\gamma) is greater than or equal to p_s(RB^k_2\times \mathbb {R}^{n-k},\gamma), for some k\in \{1,\dots,n\}. We call the sets RB^k_2\times \mathbb {R}^{n-k} round k-cylinders ; they also appear as the conjectured Gaussian isoperimetric minimizers for symmetric sets, see Heilman [Amer. J. Math. 143 (2021), pp. 53–94]. In this manuscript, we make progress towards this question, and show that for any symmetric convex set K in \mathbb {R}^n, \begin{equation*} p_s(K,\gamma)\geq \sup _{F\in L^2(K,\gamma)\cap Lip(K):\,\int F=1} \left (2T_{\gamma }^F(K)-Var(F)\right)+\frac {1}{n-\mathbb {E}X^2}, \end{equation*} where T_{\gamma }^F(K) is the F-torsional rigidity of K with respect to the Gaussian measure. Moreover, the equality holds if and only if K=RB^k_2\times \mathbb {R}^{n-k} for some R>0 and k=1,\dots,n. As a consequence, we get \begin{equation*} p_s(K,\gamma)\geq Q(\mathbb {E}|X|^2, \mathbb {E}\|X\|_K^4, \mathbb {E}\|X\|^2_K, r(K)), \end{equation*} where Q is a certain rational function of degree 2, the expectation is taken with respect to the restriction of the Gaussian measure onto K, \|\cdot \|_K is the Minkowski functional of K, and r(K) is the in-radius of K. The result follows via a combination of some novel estimates, the L2 method (previously studied by several authors, notably Kolesnikov and Milman [J. Geom. Anal. 27 (2017), pp. 1680–1702; Amer. J. Math. 140 (2018), pp. 1147–1185; Geometric aspects of functional analysis , Springer, Cham, 2017; Mem. Amer. Math. Soc. 277 (2022), v+78 pp.], Kolesnikov and the author [Adv. Math. 384 (2021), 23 pp.], Hosle, Kolesnikov, and the author [J. Geom. Anal. 31 (2021), pp. 5799–5836], Colesanti [Commun. Contemp. Math. 10 (2008), pp. 765–772], Colesanti, the author, and Marsiglietti [J. Funct. Anal. 273 (2017), pp. 1120–1139], Eskenazis and Moschidis [J. Funct. Anal. 280 (2021), 19 pp.]), and the analysis of the Gaussian torsional rigidity. As an auxiliary result on the way to the equality case characterization, we characterize the equality cases in the "convex set version" of the Brascamp-Lieb inequality, and moreover, obtain a quantitative stability version in the case of the standard Gaussian measure; this may be of independent interest. All the equality case characterizations rely on the careful analysis of the smooth case, the stability versions via trace theory, and local approximation arguments. In addition, we provide a non-sharp estimate for a function F whose composition with \gamma (K) is concave in the Minkowski sense for all symmetric convex sets. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the torsion log-Minkowski problem without symmetry assumptions via an approximation argument. [ABSTRACT FROM AUTHOR]

Upper bounds are obtained for the Newtonian capacity of compact sets in ℝd,d ≥ 3 in terms of the perimeter of the r-parallel neighborhood of K. For compact, convex sets in ℝd,d ≥ 3 with a C2 boundary the Newtonian capacity is bounded from above by (d − 2)M(K), where M(K) > 0 is the integral of the mean curvature over the boundary of K with equality if K is a ball. For compact, convex sets in ℝd,d ≥ 3 with non-empty interior the Newtonian capacity is bounded from above by (d−2)P(K)2 d|K| with equality if K is a ball. Here, P(K) is the perimeter of K and |K| is its measure. A quantitative refinement of the latter inequality in terms of the Fraenkel asymmetry is also obtained. An upper bound is obtained for expected Newtonian capacity of the Wiener sausage in ℝd,d ≥ 5 with radius 휀 and time length t. [ABSTRACT FROM AUTHOR]

In this paper, we consider parametric generalizations of a St. Venant type functional, defined on domains of the Euclidean space of dimension and connected with the torsional rigidity of a domain as well as with integrals of powers of the St. Venant stress function over simply connected plane domains. We give several estimates for the generalized functionals. In particular, for bounded convex domains we obtain an essential improvement of two known results proved by R. Bañuelos, M. van den Berg, and T. Carrol (see J. London Math. Soc. 66 (2), 499–512 (2002)) and by R.G. Salahudinov (see Russian Math. (Iz. VUZ) 50 (3), 39–46 (2006)). In addition, we examine these functionals over non convex domains in two cases when a domain has uniformly perfect boundary or it is close to convex domains in a certain sense. For such a domain we prove several new estimates using power boundary moments of domains. [ABSTRACT FROM AUTHOR]

In this paper, we consider the shape optimization problems for the quantities λ (Ω) /Tq where Ω varies among open sets of Rd with a prescribed Lebesgue measure. While the characterization of the infimum is completely clear, the same does not happen for the maximization in the case q > 1. We prove that for q large enough a maximizing domain exists among quasi-open sets and that the ball is optimal among nearly spherical domains. [ABSTRACT FROM AUTHOR]

In this paper, we establish a class of generalized Poincaré-type inequalities for torsional rigidity on the boundary of a convex body of class C + 2 in R n by using the concavity of related Brunn-Minkowski inequality. [ABSTRACT FROM AUTHOR]

This paper is devoted to reconsidering the Minkowski problem for torsional rigidity introduced by Colesanti-Fimiani [9]. Employing a Gauss curvature flow, we establish the new existence of smooth solutions of the torsional Minkowski problem under the premise of giving an initial convex body. [ABSTRACT FROM AUTHOR]

We deepen the study of Dirichlet eigenvalues in bounded domains where a thin tube is attached to the boundary. As its section shrinks to a point, the problem is spectrally stable and we quantitatively investigate the rate of convergence of the perturbed eigenvalues. We detect the proper quantity which sharply measures the perturbation's magnitude. It is a sort of torsional rigidity of the tube's section relative to the domain. This allows us to sharply describe the asymptotic behavior of the perturbed spectrum, even when eigenvalues converge to a multiple one. The final asymptotics of eigenbranches depend on the local behavior near the junction of eigenfunctions chosen in a proper way. The present techniques also apply when the perturbation of the Dirichlet eigenvalue problem consists in prescribing homogeneous Neumann boundary conditions on a small portion of the boundary of the domain. [ABSTRACT FROM AUTHOR]

We prove explicit upper and lower bounds for the Poisson hierarchy, the averaged L1-moment spectra A k B R M vol S R M k = 1 ∞ , and the torsional rigidity A 1 (B R M) of a geodesic ball B R M in a Riemannian manifold Mn which satisfies that the mean curvatures of the geodesic spheres S r M included in it, (up to the boundary S R M ), are controlled by the radial mean curvature of the geodesic spheres S r ω (o ω) with same radius centered at the center oω of a rotationally symmetric model space M ω n . As a consecuence, we prove a first Dirichlet eigenvalue λ 1 (B R M) comparison theorem and show that equality with the bound λ 1 (B R ω (o ω)) , (where B r ω (o ω) is the geodesic r-ball in M ω n ), characterizes the L1-moment spectrum A k (B R M) k = 1 ∞ as the sequence A k (B R ω) k = 1 ∞ and vice-versa. [ABSTRACT FROM AUTHOR]

A method for refining the transmission gear ratios of vehicles by coordinating the load driving conditions with the design parameters of the transmission ensuring optimal speed and the ability to overcome road resistance is proposed. Overcoming the forces of resistance to motion is considered from the conditions of energy consumption of the vehicle. Correlations of traction and dynamic properties are obtained based on refined transmission parameters to determine the basic patterns characterizing the movement of a vehicle. The parameters are close to the values of transmission parameters, thereby ensuring maximum speed and minimum fuel consumption of the vehicle. [ABSTRACT FROM AUTHOR]

Most small SUVs in the automotive market are equipped with torsion beam suspension for the rear wheels. Torsion beam suspension consists of a cross-member and a trailing arm. The cross-member plays a crucial role in preventing the vehicle from twisting; therefore, a shape that can withstand loads is essential. In this study, various shapes of cross-member reinforcements were added to the existing torsion beam suspension to analyze its structural strength when subjected to arbitrary forces. Analysis results were obtained for stiffness and driving stability factors such as smooth road shake, impact hardness, and memory shake. Based on these results, we identified the optimal cross-member shape with low torsional stiffness and a small side view swing arm angle by examining the changes in driving stability. [ABSTRACT FROM AUTHOR]

In this paper, we obtain two-sided estimates for the Euclidean moment of inertia I2(G) of a convex domain G on the plane in terms of geometric characteristics of this domain similar to the Pólya–Szegő and Makai inequalities for the torsional rigidity. [ABSTRACT FROM AUTHOR]

We investigate the existence of a maximiser among open, bounded, convex sets in R d , d ≥ 3 , for the product of torsional rigidity and Newtonian capacity (or logarithmic capacity if d = 2 ), with constraints involving Lebesgue measure or a combination of Lebesgue measure and perimeter. [ABSTRACT FROM AUTHOR]

In this paper we provide a comparison result between the solutions to the torsion problem for the Hermite operator with Robin boundary conditions and the one of a suitable symmetrized problem. [ABSTRACT FROM AUTHOR]

We show that for any even log-concave probability measure \mu on \mathbb {R}^n, any pair of symmetric convex sets K and L, and any \lambda \in [0,1], \begin{equation*} \mu ((1-\lambda) K+\lambda L)^{c_n}\geq (1-\lambda) \mu (K)^{c_n}+\lambda \mu (L)^{c_n}, \end{equation*} where c_n\geq n^{-4-o(1)}. This constitutes progress towards the dimensional Brunn-Minkowski conjecture (see Richard J. Gardner and Artem Zvavitch [Tran. Amer. Math. Soc. 362 (2010), pp. 5333–5353]; Andrea Colesanti, Galyna V. Livshyts, Arnaud Marsiglietti [J. Funct. Anal. 273 (2017), pp. 1120–1139]). Moreover, our bound improves for various special classes of log-concave measures. [ABSTRACT FROM AUTHOR]

The advancement of automotive technology is rapid in this era, as evidenced by the existence of autopilot vehicles that have been developed by a scientist. This progress is balanced with the knowledge that continues to develop in the world of education. Many prestigious automotive competitions are held to be a venue for student creativity and research in developing automotive technology, one of which is the Formula Student SAE. This is the background of a study to develop an engineered electric vehicle chassis, especially in Formula Student. This study aims to produce a chassis design that has torsional rigidity based on the selection of materials that have stiffness, strength, lightweight, and optimization of material cost. The structure of the vehicle was designed following Formula Student SAE regulations. To select material, initial screening was used by the Ashby method which produce 4 material types. Optimum of selecting the material used the Simple Additive Weighting (SAW) method. Meanwhile, chassis with material selected was analyzed by using Solidworks Simulation Education software. The results of this study produced Aluminum Alloys 7075-T6 material and torsional rigidity value of 552.65 x 103 Nmm/degree of chassis, which could achieve the minimum torsional rigidity value set at 500 x 103 Nmm/degree.

The Saint-Venant torsion of cylindrical orthotropic homogeneous linearly elastic bar is considered. The cross section of the bar is bounded by two straight lines and a curved arc. The geometry of the boundary curve of cross section depends on the shear moduli of the cylindrical orthotropic bar. An analytical method is presented to obtain the Prandtl's stress functions, warping function, shearing stresses and the torsional rigidity. [ABSTRACT FROM AUTHOR]

In 1934, Timoshenko derived and published the torsional rigidity of a rectangular bar of isotropic material by using the membrane analogy. The rigidity depends on the bar's shear modulus, width and thickness. In 1950, Lekhnitskii followed Timoshenko's process to derive the torsional rigidity for an orthotropic bar of unidirectional composite laminate, where the rigidity depends on the laminate's width, the thickness and the shear moduli. In 1990, Tsai, Daniel and Yaniv solved the same case, deriving a quasi-exact solution of torsional rigidity. In the same year, Tsai and Daniel verified their result through multiple experiments. All these rigidities become different when the definitions of thickness and width are swapped. However, they remain identical numerically, lacking mathematic proof for nearly a century. In 2022, Tsai et al. resolved Timoshenko's case by considering all the conditions and energy minimization criterion. By a completely different approach, the torsional rigidity was derived in a completely different form from that of Timoshenko but numerically identical. Moreover, the solution satisfies the rule of swapping, which makes perfect sense physically. This work applies Tsai's process to Lekhnitskii's case. A general solution is derived that satisfies the rule of swapping involving the swapping of shear moduli. [ABSTRACT FROM AUTHOR]

In [J Diff Equ, 269: 8549–8572, 2020], Li and Zhu studied the Orlicz–Minkowski problem for torsional rigidity, and among other things, they proved the existence of solutions to the problem regarding a continuous function φ satisfying lim x → 0 + φ (x) = ∞ . In this paper, with the motivation of complementing their results, we prove a new existence of solutions to the problem regarding a strictly increasing, continuously differentiable function φ satisfying lim x → 0 + φ (x) = 0 . [ABSTRACT FROM AUTHOR]

The paper deals with the torsional rigidity of homogenous and orthotropic beam with rectangular cross section. The torsional rigidity of the considered beam is defined in the framework of the Saint-Venant theory of uniform torsion. Exact and approximate solutions are given to the determination of the torsional rigidity. The shape of cross section is determined which gives maximum value of the torsional rigidity for a given cross-sectional area. The dependence of torsional rigidity as a function of the ratio shear moduli of beam is also studied. [ABSTRACT FROM AUTHOR]

Timoshenko presented the torsional rigidity of an isotropic rectangular bar, and Lekhnitskii presented that of an orthotropic rectangular bar. The solutions of Timoshenko and Lekhnitskii (T&L) are functions of the bar's length, width, thickness and shear modulus or moduli. However, the functions of T&L solutions become different from their original ones when the width and thickness are swapped. Swapping the width and thickness definitions does not alter the bar's physical properties, named the “rule of swapping” by the authors. In the last century, no research has shown the T&L solutions to satisfy the rule of swapping, an observation hereinafter referred to as the “Timoshenko & Lekhnitskii Puzzle”. Roughly 90 years later, Tsai et al. re-solved T&L cases using the TSAI technique. The derived solutions are nearly if not completely identical to T&L's numerically and satisfy the rule of swapping automatically. The rule of swapping is a novel issue and has never been mentioned before. Based on the Weierstrass factorization theorem, this study mathematically proves that they are identical for isotropic and orthotropic bars and satisfy the rule of swapping. The result of a torsional pendulum test is analyzed to support the rule.

The paper deals with the torsional rigidity of homogenous and orthotropic beam with rectangular cross section. The torsional rigidity of the considered beam is defined in the framework of the Saint-Venant theory of uniform torsion. Exact and approximate solutions are given to the determination of the torsional rigidity. The shape of cross section is determined which gives maximum value of the torsional rigidity for a given cross-sectional area. The dependence of torsional rigidity as a function of the ratio shear moduli of beam is also studied.

We consider the sharp Sobolev-Poincaré constant for the embedding of $ W^{1, 2}_0(\Omega) $ into $ L^q(\Omega) $. We show that such a constant exhibits an unexpected dual variational formulation, in the range $ 1 < q < 2 $. Namely, this can be written as a convex minimization problem, under a divergence–type constraint. This is particularly useful in order to prove lower bounds. The result generalizes what happens for the torsional rigidity (corresponding to $ q = 1 $) and extends up to the case of the first eigenvalue of the Dirichlet-Laplacian (i.e., to $ q = 2 $).

Most small SUVs in the automotive market are equipped with torsion beam suspension for the rear wheels. Torsion beam suspension consists of a cross-member and a trailing arm. The cross-member plays a crucial role in preventing the vehicle from twisting; therefore, a shape that can withstand loads is essential. In this study, various shapes of cross-member reinforcements were added to the existing torsion beam suspension to analyze its structural strength when subjected to arbitrary forces. Analysis results were obtained for stiffness and driving stability factors such as smooth road shake, impact hardness, and memory shake. Based on these results, we identified the optimal cross-member shape with low torsional stiffness and a small side view swing arm angle by examining the changes in driving stability.

We establish generalizations of the classical Polya–Szegö and Makai inequalities which estimate the torsional rigidity of a convex domain. The main idea of the proof is to apply a new exact isoperimetric inequality for Euclidean moments of a domain with respect to its boundary. This inequality has a wide class of extremal domains and is of independent interest itself. [ABSTRACT FROM AUTHOR]

We consider the Saint Venant functional for the torsional rigidity in simply connected plane domains. We prove new lower and upper estimates of using integrals of the conformal radius defined at any point of the domain and considered as a function. Applications of Davenport's formula for the torsional rigidity are discussed. In addition, we present a short https://doi.org/ of the Saint Venant–Pólya isoperimetric inequality. [ABSTRACT FROM AUTHOR]

Using the membrane analogy, in 1934 Timoshenko derived the torsional rigidity of a rectangular prism of isotropic material as a function of its material shear modulus, width and thickness. However, he did not consider the energy conservation criterion, as it could be either unnecessary or replaced by other criteria in Timoshenko's process. To confirm the correctness of Timoshenko's solution, this work re-derives the torsional rigidity by considering all the equilibrium conditions, boundary conditions, symmetric and anti-symmetric conditions of displacement and stress, the energy conservation criterion, and even the energy minimization criterion. Using the TSAI technique, exact solutions for the displacements, strains, stresses and the torsional rigidity are derived perfectly. The derived torsional rigidity is in a completely different form from that derived by Timoshenko and is numerically identical. Interestingly, the solutions derived in this work verify that, when the values of the width and thickness of the rectangular prism are swapped, the value of the torsional rigidity remains the same, which makes perfect sense physically but is not discussed in Timoshenko's process or any other research. This work presents a procedure considering all the mathematical details and the results remain correct when the width and thickness of the prism swap. This fact makes perfect sense physically, though has never been expounded before in Timoshenko's or other researchers' solutions, either for torsional rigidity or for the induced shear stresses and displacements. [ABSTRACT FROM AUTHOR]

The control accuracy of industrial robot depends on the comprehensive performance of RV reducer， which is the core component of industrial robot. At present， there is a lack of testing products for the performance of RV reducer in the market， the existing testing system has single function，which can't satisfy the requirements of industrial robot production and research. By analyzing the structure and working principle of RV reducer， the measurement method of performance parameters such as transmission error， transmission efficiency and torsional stiffness of RV reducer is intensively studied， an automatic measurement platform of RV reducer dynamic transmission performance is designed， it provides reference and basis for mechanical design and performance test of RV reducer.

The article presents the study which examines the mechanical behaviour of four brands of engine-driven nickel-titanium endodontic instruments, HyFlex CM, Twisted File, ProFile Vortex, and RaCe. It explores the materials and methods used in the study, which involves testing of instruments for bending resistance, cyclic fatigue, torsional resistance, Vickers hardness and toughness.

This article proposes a simplified method for determining the elastic radius ratio of the multi-storey reinforced concrete building. The elastic radius ratio is the benchmark parameter of the buildings in determining torsional stability during an earthquake. When buildings are torsionally flexible, the torsional component of seismic response amplifies the overall response of the building. Because of the numbers of simplified assumptions such as the adoption of the single-storey model, much of the published articles have a very limited range of application. Quantifying the interaction of different forces in multi-story non-proportional buildings has been the main challenge of these studies. The proposed “shear and bending combination method” solves this by introducing parameters that can determine the relative influence of individual actions. Moreover, the proposed method applies to buildings with all type of structural systems, having asymmetry, and accidental eccentricity. The method is validated through a parametric study consisting of eighty-one building models and using computer analysis. The proposed method and the research findings of this study are useful in determining the torsional stability of the building, in verifying the results of the computer-based analysis, and in optimizing the structural system in the buildings.

Motivated by some properties of the geometric measures for compact convex sets in the Brunn–Minkowski theory, such as the properties of the volume, the p -capacity (1 < p < n) and the torsional rigidity for compact convex sets, we introduce a more general geometric invariant, called the compatible functional F. Inspired also by the L p Minkowski problem associated with the volume, the p -capacity and the torsional rigidity for compact convex sets, we pose the L p Minkowski problem associated with the compatible functional F and prove the existence of the solutions to this problem for p > 0. We will show that the volume, the p -capacity (1 < p < 2) and the torsional rigidity for compact convex sets are the compatible functionals. Thus, as an application, we provide the solution to the L p Minkowski problem (0 < p < 1) for arbitrary measure associated with p -capacity (1 < p < 2). [ABSTRACT FROM AUTHOR]

We consider shape optimization problems involving functionals depending on perimeter, torsional rigidity and Lebesgue measure. The scaling free cost functionals are of the form P (Ω) T q (Ω) | Ω | - 2 q - 1 / 2 , and the class of admissible domains consists of two-dimensional open sets Ω satisfying the topological constraints of having a prescribed number k of bounded connected components of the complementary set. A relaxed procedure is needed to have a well-posed problem, and we show that when q < 1 / 2 an optimal relaxed domain exists. When q > 1 / 2 , the problem is ill-posed, and for q = 1 / 2 , the explicit value of the infimum is provided in the cases k = 0 and k = 1 . [ABSTRACT FROM AUTHOR]

In this paper, we consider the optimization problems associated with the nonhomogeneous and homogeneous Orlicz mixed torsional rigidities by investigating the properties of the corresponding mixed torsional rigidity. As the main results, the existence and the continuity of the solutions to these problems are proved.

Objective and accurate assessment of bone union after a fracture, arthrodesis, or osteotomy is relevant for scientific and clinical purposes. Bone union is most accurately imaged with computed tomography (CT), but no consensus exists about objective assessment of bone union from CT images. It is unclear which CT-generated parameters are most suitable for bone union assessment. The aim of this review of animal studies is to find which CT-generated parameters are associated most strongly with actual bone union. Scientific databases were systematically searched. Eligible studies were studies that (1) were animal studies, (2) created a fracture, (3) assessed bone union with CT, (4) performed mechanical or histological testing as measure of actual bone union, and (5) associated CT-generated outcomes to mechanical or histological testing results. Two authors selected eligible studies and performed risk of bias assessment with QUADAS-2 tool. From 2567 studies that were screened, thirteen studies were included. Most common CT parameters that were investigated were bone mineral density, bone volume, and total callus volume. Studies showed conflicting results concerning the associations of these parameters with actual bone union. CT-assessed torsional rigidity (assessed by three studies) and callus density (assessed by two studies) showed best results. The studies investigating these two parameters reported moderate to strong associations with actual bone union. CT-assessed torsional rigidity and callus density seem the most promising parameters to represent actual bone union after a fracture, arthrodesis, or osteotomy. Prospero trial registration number: CRD42020164733 [ABSTRACT FROM AUTHOR]

In this paper, we prove a reverse Hölder inequality for the eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with the integral Ricci curvature condition. We also prove an isoperimetric inequality for the torsional rigidity of such domains. These results extend some recent work of Gamara et al. (Open Math. 13(1), 557–570, 2015) and Colladay et al. (J. Geom. Anal. 28(4), 3906–3927, 2018) from the pointwise lower Ricci curvature bound to the integral Ricci curvature condition. We also extend the results from Laplacian to p-Laplacian. [ABSTRACT FROM AUTHOR]

In this paper, we concern the Orlicz–Minkowski problem for torsional rigidity. Our purpose is to obtain the new existence results for this problem in the smooth case via a Gauss curvature flow. Furthermore, using a parabolic approximation method, we give the existence results of the general Orlicz–Minkowski problem for torsional rigidity. [ABSTRACT FROM AUTHOR]

We study Blaschke–Santaló diagrams associated with the torsional rigidity and the first eigenvalue of the Laplacian with Dirichlet boundary conditions. We work under convexity and volume constraints, in both strong (volume exactly one) and weak (volume at most one) form. We discuss some topological (closedness, simply connectedness) and geometric (shape of the boundaries, slopes near the point corresponding to the ball) properties of these diagrams, also providing a list of conjectures. [ABSTRACT FROM AUTHOR]

In this paper two inequalities are presented for the torsional rigidity of homogeneous monoclinic piezoelectric beams. All results of the paper are based on the Saint-Venant theory of uniform torsion. The cross section of the considered elastic and piezoelectric beams may be simply connected or multiply connected two-dimensional bounded plane domain. Examples illustrate the proven inequality relations. [ABSTRACT FROM AUTHOR]

This paper presents the torsional behavior of inverted T-beams strengthened with carbon fiber-reinforced polymer (CFRP) sheets when simultaneously subjected to shear- and flexure-combined loading. Two types of CFRP bonding are tested: complete wrapping and side-bonding with single and double layers at variable coverage areas. The capacity of the beams with completely wrapped sheets is higher than that of the beams with side-bonded sheets, whereas such a tendency diminishes as the intensity of the combined loading increases. Moreover, the torsional stress of the test beams is in part relieved by the interaction with the flexural component. For the side-bonded beams, the cross-sectional area of the CFRP controls the load-carrying capacity. While the completely wrapped sheets enhance the ductility of the beams, the wide side-coverage of the CFRP benefits their polar moment of inertia that is concerned with the torsional rigidity and rotations. The degree of stress redistribution under the combined loading is not as significant as the case under pure torsion. Crack localization leads to the failure of the beams subjected to pure torsion, accompanied by partial delamination of the side-bonded sheets; however, torsional cracks spread over the loading span with the presence of shear and bending. The loading scheme affects the magnitude of the effective strain in the CFRP, and the applicability of existing equations is assessed. Based on the proposed effective confinement index, the influence of CFRP layers on the shear-flow path of the concrete section is examined. For design recommendations, analytical modeling with uncertainty simulations generates interaction envelopes, which demarcate the safe and unsafe regions of the strengthened beams under the combined loading. [ABSTRACT FROM AUTHOR]

Based on the conventional solution of Prandtl's stress function for an orthotropic rectangular bar under Saint-Venant's torsion, one can derive displacements, shear strains, shear stresses and torsional rigidity. The conventional solution of Prandtl's stress function has a hyperbolic function for the coordinate in the bar's thickness direction, and a trigonometric function for the coordinate in the width direction. This paper raises questions about the solution. Why is the solution not arranged in the opposite way? Why is the hyperbolic function not for the coordinate in the width direction and the trigonometric function for the coordinate in the thickness direction? How is it that these obvious questions have never been addressed? This study rearranges the solution of the conventional Prandtl's stress function using the TSAI technique and finds that the solution is multi-phased, indicating that the coordinates in the width and thickness directions and their corresponding parameters are swappable, a phenomenon proposed as the general rule of swapping., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2024 The Author.)

The strength and behavior of fiber reinforced concrete (FRC) members subjected to torsion has received little attention in the literature. The primary objective of including fibers in concrete is to bridge cracks once they form, and in doing so, provide some post‐cracking resistance to the otherwise brittle concrete. This and the accompanying paper that follows present the results of a comprehensive experimental and analytical study aimed at describing the behavior and strength of FRC members subjected to torsion. In this paper, results are presented on large scale pure torsion tests which have been conducted on eighteen 2.7 m long by 0.3 m wide by 0.3 m high beams with varying transverse and longitudinal reinforcement ratios along with varying steel fiber types and dosages. The results of this study demonstrates that the addition of steel fibers significantly increases the stiffness, rigidity and the maximum resisting torque and maximum twist when compared to the same specimen without fibers. The addition of fibers substantially reduced crack widths and crack spacings induced by torsion. The complementary behavior of specimens containing fibers and stirrups is explored along with a critical discussion on members containing low amounts of conventional longitudinal and/or transverse reinforcement. [ABSTRACT FROM AUTHOR]

We consider the problem of minimizing or maximizing the quantity λ (Ω) T q (Ω) on the class of open sets of prescribed Lebesgue measure. Here q > 0 is fixed, λ (Ω) denotes the first eigenvalue of the Dirichlet Laplacian on H 0 1 (Ω) , while T (Ω) is the torsional rigidity of Ω. The optimization problem above is considered in the class of all domains Ω , in the class of convex domains Ω , and in the class of thin domains. The full Blaschke–Santaló diagram for λ (Ω) and T (Ω) is obtained in dimension one, while for higher dimensions we provide some bounds. [ABSTRACT FROM AUTHOR]