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The Timoshenko–Ehrenfest beam theory was developed by Stephen Timoshenko and Paul Ehrenfest [1] [2] [3] early in the 20th century. [ 4 ] [ 5 ] The model takes into account shear deformation and rotational bending effects, making it suitable for describing the behaviour of thick beams, sandwich composite beams , or beams subject to high ...
Timoshenko improved upon that theory in 1922 by adding the effect of shear into the beam equation. Shear deformations of the normal to the mid-surface of the beam are allowed in the Timoshenko–Rayleigh theory. The equation for the bending of a linear elastic, isotropic, homogeneous beam of constant cross-section under these assumptions is [7 ...
Castigliano's method for calculating displacements is an application of his second theorem, which states: If the strain energy of a linearly elastic structure can be expressed as a function of generalised force Q i then the partial derivative of the strain energy with respect to generalised force gives the generalised displacement q i in the direction of Q i.
The total elastic energy due to strain can be divided into two parts: one part causes change in volume, and the other part causes a change in shape. Distortion energy is the amount of energy that is needed to change the shape. Fracture mechanics was established by Alan Arnold Griffith and George Rankine Irwin. This important theory is also ...
The strain energy in the form of elastic deformation is mostly recoverable in the form of mechanical work. For example, the heat of combustion of cyclopropane (696 kJ/mol) is higher than that of propane (657 kJ/mol) for each additional CH 2 unit. Compounds with unusually large strain energy include tetrahedranes, propellanes, cubane-type ...
For example, at the coarsest level, a beam is just a 1D curve whose torque is a function of local curvature. At a more refined level, the beam is a 2D body whose stress-tensor is a function of local strain-tensor, and strain-tensor is a function of its deformation. The equations are then a PDE system.
The inertia effect (kinetic energy), which, unlike the stiffness effect (strain energy), of the damage has been neglected by researchers, is included in it. The framework is generic and is applicable to wide variety of engineering structures of different shapes with arbitrary boundary conditions which constitute self adjoint systems and also to ...
In such instances, if the new phase is constrained by the surrounding material, elastic or plastic deformation may occur, introducing a strain energy term. The interplay between these interfacial and strain energy terms significantly influences the kinetics of the transformation and the morphology of the resulting phase. Notably, in shuffle ...