Page 102 - Handbook of Materials Failure Analysis

2 Physical Nonlinearity of Materials 97

The state of elongation at a given point is represented by the equivalent strain, as
follows:
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2 2
ε ¼ ðÞ + ε 2 + ðÞ (5.1)
ðÞ + ε 3 +
ε 1 +
ðÞ
in which ε i + corresponds to the positive components of the main strain tensor.
These components are defined by: ε i + ½ ðÞ ¼ ε i if ε i > 0or
ðÞ ¼ ε i + ε i jjŠ=2 with ε i +
ðÞ ¼ 0if ε i 0.
ε i +
The criterion used to verify the material mechanical integrity is defined by:

f ¼ ε S DðÞ < 0 (5.2)

The variable S DðÞ represents the limit strain value as a function of the damage state.

At the beginning of the incremental-iterative process, S DðÞ corresponds to the

normal strain value of concrete tensile strength, ε d0 . For the following steps, S DðÞ

is updated considering the value of ε determined on the last converged load step, which
accounts for the updated damage state. Due to the nonsymmetric mechanical beha-
vior of concrete when subjected to tensile and compressive stresses, the damage
variable is composed by the sum of two independent parts: tensile portion, D T ,
and compressive portion, D C . Each of these portions indicates tensile and compres-
sive contributions to the local strain state. These portions are obtained as a function
of the equivalent strain and the internal parameters of the damage model, which are
defined by:
ð
ε d0 1 A T Þ A T
D T ¼ 1

ε e ½ B T ε ε d0ފ
ð
ð
ε d0 1 A C Þ A C
D C ¼ 1 (5.3)
ε e ½ B C ε ε d0ފ

ð
in which ε d0 ,A T ,B T ,A C ,B C are the internal parameters of Mazars’ damage criterion.
The subscripts T and C indicate tensile and compressive parts, respectively.
After the determination of each portion of damage, the final value of the damage
variable is composed as follows:
D ¼ α T D T + α C D C (5.4)
The coefficients α T and α C are calculated using the following expressions:
X X
ð ε T i + ð ε C i +
Þ
Þ
i i
α T ¼ , α C ¼ (5.5)
ε V + ε V +
are determined from the main stresses considering elastic mate-
in which ε T i and ε C i
+
rial and ε represents the total state of elongation, which is equal to:
V
X X
+
Þ +
Þ
ε ¼ ð ε T i + ð ε C i + (5.6)
V
i i
After the damage variable determination, the stress state for the structure (solid) is
calculated as follows:
σ ¼ 1 DÞEε
ð
τ ¼ 1 DÞGγ (5.7)
ð