80
T. Krykowski
˙
σ = C : ˙
ε
e
= C :
˙
ε − ˙
ε
p
− ˙
ε
f
− ˙
ε
V
,
(7)
where C is elastic tensor of material.
3 Determining Parameters for the Model Using the Monte
Carlo Method
Material parameters for the model were randomly selected (using the Monte Carlo
simulation) for the purpose of this paper. Calculations were made to evaluate the
impact of corrosion products on an increase in volume strains. Material parameters
were considered as dependent parameters and were to be determined in accordance
with the approach described in the paper [18] (correlated random variables could be
determined on the basis of uncorrelated variables) using the following relationship
X = T · Y , X
T
= {X 1 , X 2 , . . . , X n }, Y
T
= {Y 1 , Y 2 , . . . , Y n },
(8)
where X is the matrix, whose elements are correlated random variables, T is the
transformation matrix, Y is the matrix, whose elements are uncorrelated random
variables. The transformation matrix T can be determined by analysing the eigen
problem of the covariance matrix C x (eigenvectors of the matrix C x are its elements
T ). The covariance matrix C y for uncorrelated random variables Y and the vector of
expected values for uncorrelated random variables Y μ can be determined from the
following relationship:
C y = T
T C x T,
(9)
Y μ = T
T X μ ,
(10)
where X μ is the vector of expected values for correlated random variables. Elements
of the covariance matrix for uncorrelated random variables were defined by the
following relationship
C X ik = [cov(X i , X k )] = ρ X ik σ X i σ X k .
(11)
In the Eq. (11), ρ X ik are coefficients of correlation of correlated random variables
X i .
The distribution of correlated random variables X can be determined by the relationship (8). In this case, the vector Y is the matrix with rows containing elements
that are randomly selected uncorrelated random variables with expected values μ Y i
and standard deviation σ Y i (those distributions were determined using generators of
random numbers integrated in Matlab software). Value X is the matrix with rows,
T. Krykowski
˙
σ = C : ˙
ε
e
= C :
˙
ε − ˙
ε
p
− ˙
ε
f
− ˙
ε
V
,
(7)
where C is elastic tensor of material.
3 Determining Parameters for the Model Using the Monte
Carlo Method
Material parameters for the model were randomly selected (using the Monte Carlo
simulation) for the purpose of this paper. Calculations were made to evaluate the
impact of corrosion products on an increase in volume strains. Material parameters
were considered as dependent parameters and were to be determined in accordance
with the approach described in the paper [18] (correlated random variables could be
determined on the basis of uncorrelated variables) using the following relationship
X = T · Y , X
T
= {X 1 , X 2 , . . . , X n }, Y
T
= {Y 1 , Y 2 , . . . , Y n },
(8)
where X is the matrix, whose elements are correlated random variables, T is the
transformation matrix, Y is the matrix, whose elements are uncorrelated random
variables. The transformation matrix T can be determined by analysing the eigen
problem of the covariance matrix C x (eigenvectors of the matrix C x are its elements
T ). The covariance matrix C y for uncorrelated random variables Y and the vector of
expected values for uncorrelated random variables Y μ can be determined from the
following relationship:
C y = T
T C x T,
(9)
Y μ = T
T X μ ,
(10)
where X μ is the vector of expected values for correlated random variables. Elements
of the covariance matrix for uncorrelated random variables were defined by the
following relationship
C X ik = [cov(X i , X k )] = ρ X ik σ X i σ X k .
(11)
In the Eq. (11), ρ X ik are coefficients of correlation of correlated random variables
X i .
The distribution of correlated random variables X can be determined by the relationship (8). In this case, the vector Y is the matrix with rows containing elements
that are randomly selected uncorrelated random variables with expected values μ Y i
and standard deviation σ Y i (those distributions were determined using generators of
random numbers integrated in Matlab software). Value X is the matrix with rows,