In this paper, it is proposed to use a combination of the Coupled Criterion of Finite Fracture Mechanics (FFM) and the Equivalent Constraint Model (ECM) to predict the evolution of crack density with increasing applied load.
Section 2 provides an introduction to the Freedom and Constraint Topology (FACT) method as well as the equivalent constraint model of selected flexure primitive within the framework of the screw theory.
Since the equivalent constraint model subjects to its ratio of the length to the radius, for the equivalent wire constraint model as used in this paper, there is no doubt that the smaller the radius of the beam is, the more ideal the wire flexure approximation is.
By comparing the compliances of the wire flexure with circular cross section in different directions, the equivalent constraint model can be established for realizing the simplification from structure to topology.
Then the magnitudes of the actuation wrenches, the non-overconstraint wrenches, and the equivalent constraint wrenches can be solved from Eq. (17) as {varvec{f}}_{c} = left( {begin{array}{*{20}c} {{varvec{w}}_{text{a}}^{text{T}} } & {{varvec{f}}_{text{r,non}}^{text{T}} } & {{varvec{f}}_{text{e}}^{text{T}} } end{array} } right)^{text{T}} = {varvec{G}}_{c}^{ - 1} {not!{varvec{S}}}_{{varvec{F}}}.
With f ( P ) = P S Q - 1 P H + R T - 1, Lemma 4 can be applied to achieve the equivalent constraint P S Q - 1 P H + R T - 1 ≽ c λ max ( S R I R ) I T ⇔ P S Q - 1 P H ≽ c λ max ( S R I R ) I T - R T - 1 ⇔ P S Q - 1 P H ≽ [ c λ max ( S R I R ) I T - R T - 1 ] + (56).
Recasting the chance constraints into the equivalent deterministic constraints, the uncertain optimization problem of maximizing the flux of a desired product through gene up/down-regulation operations can be formulated for a system of arbitrary size consisting of N metabolites and M reactions.
All the probabilistic constraints have been transferred into the equivalent deterministic constraints by stochastic programming approach and a general transformation technique has used for the multi-choice cost coefficients of the objective functions using fuzzy programming technique which provides a compromise solution.
Since the solution algorithm of the FFP can transform multiple objective functions to equivalent constraints, it makes the FFP straightforward to deal with importance of multiple objectives.
For a periodically constrained beam the geometric constraints can be replaced by a set of equivalent constraints, each of which only involves the functions in one coupling group dictated by the coupling relationships when the stiffeners are also added at the constrained points.
