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Finite Element Analysis in Structural Mechanics

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Bina model

The total creep strain in percent in time $t$ for the specified stress $\sigma$ and temperature $T$ is expressed as $$\varepsilon_\text{tot}(t|\sigma,T) = \varepsilon_0\left(\frac{\varepsilon_m}{\varepsilon_0}\right)^{g(\pi)},$$ where $\varepsilon_0$ is the initial strain, $\varepsilon_m$ is the limit strain, and $g(\pi)$ is the hardening function.

The value of the initial strain depends on the material and is given as

  1. type 2a) \begin{align} \varepsilon_0 &= 100\frac{\sigma}{E(T)}\\ E(T) &= E_1+E_2\exp\left(-\frac{E_3}{T}\right) \end{align}
  2. type 2b) \begin{align} \varepsilon_0 &= 100\frac{\sigma}{E(T)}\left[A\tanh(B\sigma)\exp\left(\frac{Q}{T^n}\right)\right]\\ E(T) &= E_1+E_2\exp\left(-\frac{E_3}{T}\right) \end{align}
  3. type 2c) \begin{align} \varepsilon_0 &= 100\frac{\sigma}{E(T)}\left[A\left(\frac{\sigma}{\sigma_m(T)}\right)^{m(T)}\exp\left(\frac{Q}{T^n}\right)\right]\\ E(T) &= E_1+E_2\exp\left(-\frac{E_3}{T}\right)\\ \sigma_m(T) &= B_1+B_2\exp\left(\frac{B_3}{T}\right)\\ m(T) &= N_1+N_2T+N_3T^2+N_4T^3+N_5T^4 \end{align}

The limit strain is defined as $$\varepsilon_m = \exp\left\{M_1+M_2\tanh\left[\frac{\ln(t_r)-M_3-M_4T}{M_5}\right]\right\}+100\frac{\sigma}{E(T)}.$$

The time to fracture $t_r$ is determined from $$\log(t_r) = A_1+A_2\log\left|\frac{1}{T}-\frac{1}{A_5}\right|+A_3\log\left|\frac{1}{T}-\frac{1}{A_5}\right|\log\left[\sinh(A_6\sigma T)\right]+A_4\log\left[\sinh(A_6\sigma T)\right].$$

The hardening function $g(\pi)$ is then defined as $$g(\pi) = \pi^N\left[\frac{1+\exp\left(-2\pi^{K(T)}\right)}{1+\exp(-2)}\right]^M,$$ where $\pi$ is the damage $\pi=t/t_r,$ $N$ a $M$ are material constants, and parameter $K$ is defined using constants $K_1$ and $K_2$ as $$K(T)=\exp\left(K_1+\frac{K_2}{T}\right).$$

The material constants $E_1$ to $E_3,$ $A,$ $B,$ $Q,$ $n,$ $B_1$ to $B_3,$ $N_1$ to $N_5,$ $A_1$ to $A_6,$ $M_1$ to $M_5,$ $N,$ $M,$ $K_1,$ and $K_2$ are specified in separate input files, see below.

Input quantities

The Bina model is activated by the parameter $\mathtt{KCRP}=pqr$ in the file name.iP, where:

  • the first digit $p=2$
  • the second digit $q\in\{1,2,3\}$ specifies the calculation of initial strain
    1. type 2a)
    2. type 2b)
    3. type 2c)
  • the third digit $r\in\{1,2,3,4\}$ specifies the transition between creep curves
    1. “Strain Hardening” theory
    2. “Time Hardening” theory
    3. “Life Fraction Rule” theory
    4. “Strain Fraction Rule” theory

All materials used must be assigned to the elements using the file name.DAT. The material parameter files have the following structure:

* * Comments *

POCATECNI DEFORMACE $E_1$ $E_2$ $E_3$

PEVNOST PRI TECENI $A_1$ $A_2$ $A_3$ $A_4$ $A_5$ $A_6$

2B) $A$ $Q$ $B$ $n$

2C) $A$ $Q$ $n$ $B_1$ $B_2$ $B_3$ $N_1$ $N_2$ $N_3$ $N_4$ $N_5$

MEZNA DEFORMACE $M_1$ $M_2$ $M_3$ $M_4$ $M_5$

FUNKCE ZPEVNENI $N$ $M$ $K_1$ $K_2$

  • The number of initial comment lines is unlimited.
  • There must be an empty line before each block.
  • The block POCATECNI DEFORMACE contains the constants for the calculation of the initial strain.
  • The block 2B) contains further constants for the calculation of the initial strain; if $k_B\in\{1,3\},$ this block must be omitted.
  • The block 2C) contains further constants for the calculation of the initial strain; if $k_B\in\{1,2\},$ this block must be omitted.
  • The block PEVNOST PRI TECENI contains the constants for the calculation of the time to fracture.
  • The block MEZNA DEFORMACE contains the constants for the calculation of the limit strain.
  • The block FUNKCE ZPEVNENI contains the constants for the calculation of the damage function.

The program checks, according to the specified values of $k_B,$ if each material parameter file contains — or does not contain — the blocks 2B) and 2C). This prevents a possible error in the parameter $\mathtt{KCRP}$ in the file name.iP or on the material line in the file name.DAT.

Example

There is only one material $k_B=1$ used in the creep problem.

name.iP
; KREST NLC NCYC KMOD KCRP KLARG KCNT
IP  1    11   1    0   213   0     0  3*0  11*2
RP 10*0 0 1 100 1000 10000 30000 80000 120000 160000 200000 250000
EN
EN
name.DAT
number 1
mat.dat

Example

There are three materials used in the creep problem. Let for the material number 1 be $k_B=3,$ for the material number 2 be $k_B=2,$ and the material number 3 be $k_B=1.$

name.iP
; KREST NLC NCYC KMOD KCRP KLARG KCNT
IP  1    11   1    0   233   0     0  3*0  11*2
RP 10*0 0 1 100 1000 10000 30000 80000 120000 160000 200000 250000
EN
EN
name.DAT
number 3
mat1.dat
mat2.dat
mat3.dat
material 2 2 8
10 13 14:15 17:20
material 3 1 37
12 21:30 35:60

Example

The sample material parameter file for a single material $k_B=1.$

15128_5.dat
************************************************************************
*
*          15128.5   Z-89-6013 CSN 1.3.1979, (470-900/2.5E5)
*
*
*            QUANTITY                                UNIT
*            ----------------------------------------------
*            Temperature                             [K]
*            Stress                                  [MPa]
*            Initial strain, Limit s., Creep s.      [%]
*            Time to fracture                        [h]
*            Creep strain rate                       [%/h]
*
*     POCATECNI DEFORMACE  parameters  E(1) - E(3)
*     DOBA DO LOMU         parameters  A(1) - A(6)
*     MEZNA DEFORMACE      parameters  M(1) - M(5)
*     FUNKCE ZPEVNENI      parameters  N, M, K(1), K(2)
*
************************************************************************

POCATECNI DEFORMACE
  0.21425035E+6
 -0.45038419E+6
  0.19371094E+4

PEVNOST PRI TECENI
 -0.1840487E+2
 -0.5906108E+1
  0.7682633E+1
  0.2298323E+2
  0.6730000E+3
  0.4000000E-5

MEZNA DEFORMACE
  0.144927e+1
  0.0E+0
  0.0E+0
  0.0E+0
  1.0E+0

FUNKCE ZPEVNENI
  0.26069593E+0
 -0.80546546E+0
 -0.51082559E+0
  0.0
en/ref/d/2.txt · Last modified: 2024-11-12 11:03 by Petr Pařík