Abstract
Damage accumulation and fracture of structures represented an actual mechanical problem that is needed in development of theoretical and computational methods. One of the most important problems in this direction is the problem of optimal structural design, when in optimization process it is necessary to take into account initial structural defects, arising cracks and damage accumulation. This problem is characterized by incomplete information concerning initial cracks size, cracks position and its orientation. In this context it is necessary to develop the statements of the optimization problems based on guaranteed (mini–max), probabilistic and mixed probabilistic-guaranteed approaches for considered problems with incomplete information. For many realistic it is reasonable to use variants of the mini–max optimization, named as optimization for “the worst case scenario” (see Banichuk et al. Mech Struct Mach 26(1):149–188, 1997; Mech Based Des Struct Mach 31(4):459–474, 2003; Meccanica 40:135–145, 2005a; Mech Based Des Struct Mach 33(2): 253–269, 2005b). Considered problem consist in finding the shape and thickness distribution of axisymmetric quasi-brittle shells with arising cracks in such a way, that the cost functional (volume or weight of the shell material) reaches the minimum, while satisfying some constraints on the stress intensity factor and geometrical constraints. In the case of cycling loadings we consider the number of loading cycles before fracture as the main constraint. Some examples of problems formulations, analytical and numerical solutions based on genetic algorithm are presented.
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Abbreviations
- α, β, γ, hmin, σ1, σ2, V0:
-
Given dimensionless parameters of optimization problems
- C,m:
-
Material constants
- \({\varepsilon}\) :
-
Positive value
- hmin, hmax, L, r m :
-
Geometrical parameters of the shell
- h (x), r(x):
-
Thickness distribution and shape function—design variables
- J, Ja, J i :
-
Functionals in the optimization problem
- \({K_{1}, K_{1C}, K_{1\varepsilon}}\) :
-
Stress intensity factor and quasi-brittle strength constants
- l, l i , l cr :
-
Geometrical parameters of cracks
- Λ, Λ q , Λξ, Λ ω :
-
Given sets, characterizing the incompleteness of information
- μ i :
-
Arbitrary positive constants
- n, n cr , n*:
-
Parameters of cyclic loading
- n pop :
-
The number of populations (iterations)
- r g (x):
-
Given function
- ψ i :
-
Dimensionless functions introduced in Eqs. 10–15
- Ψ i :
-
Dimensionless functions introduced in Eqs. 17–18
- ω :
-
Vector of unknown parameters
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Banichuk, N.V., Ivanova, S.Y. & Ragnedda, F. Design of fracture resistant structures. Int J Fract 150, 213–220 (2008). https://doi.org/10.1007/s10704-008-9222-6
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DOI: https://doi.org/10.1007/s10704-008-9222-6