Decomposition method in linear elastic problems with eigenstrain

Abstract

The general theory of linearized elasticity with eigenstrain is considered with applications to continuous, discrete and discretized structures. It is shown that any eigenstrain can be uniquely decomposed into impotent and nilpotent constituents. The proven theorem on decomposition is based on the concepts of functional analysis, in particular, with respect to Hilbert functional spaces. This unique decomposition allows for the individual and independent control of stress, strain and displacement (e.g. shape control). The associated algorithm avoids the cumbersome solution of boundary‐value problems with eigenstrain in connection with these control problems. Decomposition of eigenstrain opens the practically important opportunity to fully separate the control of strain and stress produced by force loading.

The basis of optimal active(static and dynamic) shape- and stress-control by means of smart materials

“…Equation (10) implies: there exists the orthogonal decomposition of the Hilbert (energy) space H into subspaces H u and H σ , see [12], H = H u ⊕ H σ . Further, the unique decomposition of the space of eigenstrains allows us to establish the significant properties of eigenstress and deformation induced by eigenstrain, see again Eq.…”

“…Accordingly, the space is thus called energy space. For discretized structures, two mutual orthogonal finite dimensional sub-spaces exist, i. e., any tensor of eigenstrain ε ∈ H existing in a body can be uniquely decomposed into its impotentε * and nilpotentε * * constituents, see [12] and [14],…”

“…The mass is lumped to the nodes, stiffness EA of the member rods is assumed to be constant. Since the load case is prescribed, we can directly calculate the quasi-static strains in the smart member rods, [21], and impose these strains as impotent eigenstrains, thus preserving the quasi-static member forces N For a rigorous formulation of the main theorem on eigenstress and eigenstrain, the Hilbert function space H is introduced [12], [14], as the space of second rank symmetric tensors where the components are real functions of spatial coordinates in the function space L 2 . Assume, that the eigenstrain tensors are the elements of the Hilbert space.…”

“…Such benchmark solutions with unlimited intensity of the actuators understood, serve for the best possible practical design. In [12] the crucial, unique decomposition of an eigenstrain tensor, e.g. of piezoelectric strain, is performed by means of the scalar product measure in Hilbert functional space, rendering its impotent part that does not produce stress, see also [13] and [14], and its complement, the nilpotent eigenstrain that renders stresses but does not produce deformation, see [10], [12] and [15].…”

Биомеханический Анализ Мостовидного Протеза Для Замещения Дефектов Зубного Ряда, Осложненных Вторичными Деформациями

“…В некоторых случаях полем напряжений можно управлять [9][10][11][12]. В любом случае необходимо как можно раньше исправлять патологии зубочелюстной системы.…”