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To extend the two atoms approach into solid, consider a simple model, say, a 1-D array of one element with interatomic distance of r, and the equilibrium distance is r 0. Its potential energy-interatomic distance relationship has similar form as the two atoms case, which reaches minimal at r 0, The Taylor expansion for this is:
The true interatomic interactions are quantum mechanical in nature, and there is no known way in which the true interactions described by the Schrödinger equation or Dirac equation for all electrons and nuclei could be cast into an analytical functional form. Hence all analytical interatomic potentials are by necessity approximations.
In physics, Hooke's law is an empirical law which states that the force (F) needed to extend or compress a spring by some distance (x) scales linearly with respect to that distance—that is, F s = kx, where k is a constant factor characteristic of the spring (i.e., its stiffness), and x is small compared to the total possible deformation of the spring.
Young's modulus is the slope of the linear part of the stress–strain curve for a material under tension or compression.. Young's modulus (or Young modulus) is a mechanical property of solid materials that measures the tensile or compressive stiffness when the force is applied lengthwise.
These are all examples of a class of problems called stiff (mathematical stiffness) systems of differential equations, due to their application in analyzing the motion of spring and mass systems having large spring constants (physical stiffness). [5] For example, the initial value problem
The shear modulus is one of several quantities for measuring the stiffness of materials. All of them arise in the generalized Hooke's law: . Young's modulus E describes the material's strain response to uniaxial stress in the direction of this stress (like pulling on the ends of a wire or putting a weight on top of a column, with the wire getting longer and the column losing height),
In theoretical chemistry, the Buckingham potential is a formula proposed by Richard Buckingham which describes the Pauli exclusion principle and van der Waals energy for the interaction of two atoms that are not directly bonded as a function of the interatomic distance.
Approximate specific stiffness for various materials. No attempt is made to correct for materials whose stiffness varies with their density. Material Young's modulus Density (g/cm 3) Young's modulus per density; specific stiffness (10 6 m 2 s −2) Young's modulus per density squared (10 3 m 5 kg −1 s −2) Young's modulus per density cubed ...