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Relation between Poisson's ratio, Modulus of Elasticity and Modulus of Rigidity01:15

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Deformation occurs in axial and transverse directions when an axial load is applied to a slender bar. This deformation impacts the cubic element within the bar, transforming it into either a rectangular parallelepiped or a rhombus, contingent on its orientation. This transformation process induces shearing strain. Axial loading elicits both shearing and normal strains. Applying an axial load instigates equal normal and shearing stresses on elements oriented at a 45° angle to the load axis.
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Accurately determining beam deflection and slope under various loading conditions in structural engineering is crucial for ensuring safety and structural integrity. Singularity functions offer a streamlined approach to analyzing beams, especially when multiple loading functions complicate the bending moment equation.
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When analyzing the deformation of a symmetric prismatic member subjected to bending by equal and opposite couples, it becomes clear that as the member bends, the originally straight lines on its wider faces curve into circular arcs, with a constant radius centered at a point known as Point C. This phenomenon helps to understand the stress and strain distribution within the member more clearly.
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Castigliano's theorem analyzes displacements and rotations in elastic structures. It relates the derivative of elastic strain energy to the applied forces or moments, allowing for the calculation of deformations. The theorem states that the partial derivative of the total strain energy of a system with respect to a specific load results in the displacement at the point where the load is applied. This principle applies to both forces and moments.
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In a nonhomogeneous rod made up of steel and brass, restrained at both ends and subjected to a temperature change, several steps are involved in calculating the stress and compressive load. Due to the problem's static indeterminacy, one end support is disconnected, allowing the rod to experience the temperature change freely. Next, an unknown force is applied at the free end, triggering deformations in the rod's steel and brass portions. These deformations are then calculated and added...
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Sine-square deformation applied to classical Ising models.

Chisa Hotta1, Takashi Nakamaniwa1, Tota Nakamura2

  • 1Graduate School of Arts and Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 1538902, Japan.

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Sine-square deformation (SSD) applied to classical Ising models creates an extended canonical ensemble. This method allows a single calculation to yield physical quantities across various effective temperatures, mimicking infinite systems.

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Area of Science:

  • Statistical mechanics
  • Condensed matter physics
  • Computational physics

Background:

  • Sine-square deformation (SSD) is a technique used in quantum systems to modify Hamiltonians spatially.
  • SSD creates boundary conditions and reproduces physical quantities of infinite-size systems.
  • Classical Ising models are fundamental in understanding magnetism and phase transitions.

Purpose of the Study:

  • To investigate the application and effects of sine-square deformation (SSD) on classical Ising models.
  • To explore the concept of effective temperature in SSD-modified classical systems.
  • To determine if SSD can efficiently reproduce properties of infinite classical systems.

Main Methods:

  • Analytical calculations were performed on 1D and 2D classical Ising models.
  • Monte Carlo simulations were employed to study the behavior of the systems.
  • The concept of effective temperature was derived and analyzed.

Main Results:

  • The classical SSD system behaves as an extended canonical ensemble of local subsystems.
  • Each subsystem is characterized by a unique effective temperature, defined by the ratio of system temperature to the local energy scale.
  • A single simulation provides physical quantities across a range of effective temperatures.

Conclusions:

  • Sine-square deformation is a viable method for studying classical Ising models.
  • The effective temperature concept provides a novel way to access properties of systems at various temperatures from a single simulation.
  • SSD offers a computationally efficient approach to approximate the behavior of infinite classical systems.