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3.1 A suitable modeling approach for an efficient structural optimization
where i,j,k = 1,2,3 and the summation convention is also used. As the most
common material law, Hooke’s law can be used to express the components
of the stress tensor σ in terms of the components of the strain tensor ǫ as
σ ij = c ijkl ǫ kl , (3.5)
in which the summation convention is used again to shorten the occurring
expressions [12]. c ijkl are the components of a tensor C of fourth order containing
the material constants, where C = C(x) here. Inserting Hooke’s law
into the potential energy and considering the different tensors containing the
material constants in the domains D\H with the material constants C and H
with C+C results in
∫ ∫
( )
(C : ǫ) : ǫdV C : ǫ : ǫdV . (3.6)
U = 1 2
D
} {{ }
I
+ 1 2
H
} {{ }
II
Similarly, the kinetic energy (3.1) can be written as
T = 1 ∫
ρṗ(x,t)·ṗ(x,t)dV + 1 ∫
ρṗ(x,t)·ṗ(x,t)dV . (3.7)
2 D 2 H
} {{ } } {{ }
I
II
It is well known that for isotropic material behavior, the material constants
c ijkl can be expressed by two independent material constants [12], e.g. the
modulus of elasticity E and Poisson’s ratio ν leading to
σ ij =
νE
(1+ν)(1−2ν) ǫ kkδ ij + E
1+ν ǫ ij, (3.8)
making use ofδ ij as the well known Kronecker-Delta. In the case considered
here, the modulus of elasticity in the domain D\H is E and in the domain H
E +E. The expression for the potential energy (3.6) simplifies accordingly.
The discretized equations of motion for the system can be obtained using
the Ritz method, where the basic idea of the EMM is to use separate
discretization schemes for the domains I (denoted as basis structure) and II
(representing the modification) in the Ritz approximation of the system and
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