smart technologies for safety engineering

VDM-Based Remodeling 223
6.2 Remodeling of Elastoplastic Structures
This section considers the coupled problem of simulation of material redistribution for structures
with elastoplastic material characteristics. Two cases are considered: modifications of the
element cross-sectional areas Ai, which couple the stiffness and mass matrices, and separate
independent modifications of element stiffnesses and masses.
6.2.1 VDM Formulation
The equations of motion for the modeled structures with virtual distortions simulating modifications
of material distribution and physical nonlinearities can be expressed similarly to
Equation (1) as
MNMüM(t) + G T Nili
Sii GiMu M(t) − ε 0 i (t) − β0 i (t) = fN (t) + f 0 N (t)
which, as in Equation (2), can also be expressed via strains rather than displacements:
MNMüM(t) + G T Nili
Sii εi(t) − ε 0 i (t) − β0 i (t) = fN (t) + f 0 N (t)
The virtual distortion method expresses the dynamic structural response as a superposition
of the linear response uL N and terms due to modifications, which are simulated by virtual
distortions:
u N (t) = u L
N (t) + B ε Nj (t − τ)ε0
j (τ) + B ε Nk (t − τ)β0
k (τ) + B f NM (t − τ) f 0 M (τ)
τ≤t
τ≤t
where the distortions ε0 i (t), β0 i (t) and f 0 N (t) simulate respectively the modifications of element
stiffness (strain distortions), physical nonlinearities (plastic distortions) and element mass
(force distortions). The index N denotes all degrees of freedom, j denotes the elements with
modified cross-sections, k denotes the plastified elements and M denotes the degrees of freedom
related to the elements with modified cross-sections. The corresponding formula for strain is
obtained by premultiplying Equation (18) by GiN:
εi(t) = ε L
i (t) + D ε ij (t − τ)ε0
j (τ) + D ε ik (t − τ)β0
k (τ) + D f iM (t − τ) f 0 M (τ)
τ≤t
which, by directly separating the increments of plastic distortions β 0 k
τ≤t
in successive time steps,
takes the form
εi(t) = ε L
i (t) + D ε ij (t − τ)ε0
j (τ) + D ε ik (t − κ)β 0
k (κ) + D f iM (t − τ) f 0 M (τ)
τ≤t
τ≤t κ≤τ
Let ε =t (t) denote the strains without the effect of strain distortions εi(t), distortion forces
f 0 N (t) and plastic distortion increments βi(t) in the current time step t:
τ≤t
τ≤t
τ≤t
(18)
(19)
εi(t) = ε =t
i (t) + D ε ij (0)ε0j (t) + Dε ik (0)β 0 k (t) + Df iM (0) f 0 M (t) (20)