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Deformation of a thin plate highlighting the displacement, the mid-surface (red) and the normal to the mid-surface (blue) The Kirchhoff–Love theory of plates is a two-dimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to forces and moments.
A single-displacement reaction, also known as single replacement reaction or exchange reaction, is an archaic concept in chemistry. It describes the stoichiometry of some chemical reactions in which one element or ligand is replaced by an atom or group. [1] [2] [3] It can be represented generically as:
A substitution reaction (also known as single displacement reaction or single substitution reaction) is a chemical reaction during which one functional group in a chemical compound is replaced by another functional group. [1] Substitution reactions are of prime importance in organic chemistry.
The member deformations can be expressed in terms of system nodal displacements r in order to ensure compatibility between members. This implies that r will be the primary unknowns. The member forces Q m {\displaystyle \mathbf {Q} ^{m}} help to the keep the nodes in equilibrium under the nodal forces R .
For infinitesimal deformations of a continuum body, in which the displacement gradient tensor (2nd order tensor) is small compared to unity, i.e. ‖ ‖, it is possible to perform a geometric linearization of any one of the finite strain tensors used in finite strain theory, e.g. the Lagrangian finite strain tensor, and the Eulerian finite strain tensor.
The equation is given by ¨ + ˙ + + = (), where the (unknown) function = is the displacement at time t, ˙ is the first derivative of with respect to time, i.e. velocity, and ¨ is the second time-derivative of , i.e. acceleration.
The threshold displacement energy is a materials property relevant during high-energy particle radiation of materials. The maximum energy that an irradiating particle can transfer in a binary collision to an atom in a material is given by (including relativistic effects)
Duhamel's principle is the result that the solution to an inhomogeneous, linear, partial differential equation can be solved by first finding the solution for a step input, and then superposing using Duhamel's integral. Suppose we have a constant coefficient, m-th order inhomogeneous ordinary differential equation.