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Bina's model

Problem description: Consider the rod shown for a transient creep analysis. The initial stress $\sigma_{xx}=\sigma_{xx}^1$ abruptly changes to $\sigma_{xx}=\sigma_{xx}^2$ at time t=t₁ after which the straining continues at the new load level until t=t₂. The temperature T is assumed to be constant throughout.

Mesh: Four hexahedra--see appendix B.1.

Material properties: $\alpha=10^{-5}$ 1/K, $E=1.81\times10^5$ MPa, $\nu=0.3$.
  Parameters for Bina's model:

E1	E2	E3
0.2152947E+6	-0.2502203E+6	0.1434375E+4
A1	A2	A3
-0.2079980E+3	-0.8215359E+2	-0.3412273E+0
A4	A5	A6
-0.1391403E+1	0.2730000E+3	0.1000000E-3
M1	M2	M3
0.1455834E+1	-0.1260596E+1	0.4886680E+2
M4	M5
-0.5002250E-1	0.1825170E+1
N	M	K
0.3260447E+0	-0.6814129E+0	0.1000000E+1

Support: Clamped at x=0. Statically determinate.

u=v=w=0 node:	 1 
u=w=0 nodes: 	 2 21       
u=v=0 nodes: 	 4 24       
u=0 nodes: 	 3 22 23      

Loading: $T=450^\circ$C, $t_1=1.001\times10^5$h, $t_2=1.998\times10^5$h.

$\sigma_{xx}=\left\{ \begin{array}{ll} \sigma_{xx}^1=230\,\mbox{MPa}&\mbox{for ... ...\ \sigma_{xx}^2=250\,\mbox{MPa}&\mbox{for }t_1 < t \le t_2 \end{array}\right.$

Solution: It is useful at this point to overview some important features of the Bina's material model omitting, however, unnecessary details. The most important issue concerns the compatibility between the model's data collected in a separate file with the extension DAT and the elastic parameters that have been entered independently in the I2 type file.

According to Bina's model the functional dependence of the Young modulus on temperature is suppossed to have the form

$$E(T)=E_1+E_2\exp(-E_3/T)$$

where the numerical values of the constants E₁, E₂, E₃ shown in the table on page correspond to physical units [MPa], [K] for the Young modulus and the thermodynamic temperature, respectively. It should be pointed out that the above expression applies to the evaluation of the uniaxial creep curves only, leaving the elastic constants used for the formation of the stiffness matrix unaltered. In order to achieve compatibility it is necessary to describe the Young modulus in the I2 file as a function of temperature that approximates the exponential relation of the creep model as closely as possible.

To this end, we may, for instance, estimate the upper and lower bounds for temperatures occurring in the body and define the piecewise linear function that will cover up the entire range. Examples of this technique can be found in the I2 input files shown in next sections. Of course, in this problems with the homogeneous temperature field we could have simply put

$$\begin{array}{rcl} E&=&0.2152947\times10^6-0.2502203\times10^... ...{273+450}\right) \\ &=& 1.81\times10^5\,\mbox{MPa} \end{array}$$

just as well.

Next, we turn our attention toward the analytical solution. In general it is difficult to obtain the closed form solution for a complex material model but in this particular case at least one variable--the one describing material damage--can be calculated rather easily. In the Bina model the damage parameter is defined as

$$\pi(t)=\frac{t}{t_r}~\in~(0,1)$$

where t_r is the time elapsed at rupture, estimated by the empiric formula

$$\begin{array}{rcl}\displaystyle \log t_r &=& A_1+A_2\log\left... ...T)\Bigr)+ A_4\log\Bigl(\sinh(A_6\sigma_e T)\Bigr) \end{array}$$

in which $\sigma_e$ is the effective stress in [MPa], T is the thermodynamic temperature in Kelvins, t_r the time of rupture in hours, and the coefficients A₁-A₆ are given in the table on page . Thus we can calculated the rupture times t_r¹, t_r² corresponding to two stress levels $\sigma_{xx}^1$, $\sigma_{xx}^2$ as

$$t_r^1=466518\,\mbox{h}~,~~~t_r^2=229746\,\mbox{h}$$

These rupture times pertain to situations with constant stress levels. In order to obtain the total damage for varying loading the increments of $\pi$ must be summed up according to a simple rule

$$\pi(t_2)=\frac{t_1}{t_r^1}+\frac{t_2-t_1}{t_r^2}= \frac{100100}{466518}+\frac{99700}{229746}=0.6485$$

This result should be compared with the DMG output variables shown in next sections (DMG=64.85%).

Finally, we must account for a sudden stress change at time t₁ when the stress point traverses from one creep curve to another one in the $\epsilon_c$-t plot. In principle, we have three approaches at hand to accomplish this transition: The strain hardening method, characterized by the constitutive equation for the equivalent creep strain having the form

$$\dot\epsilon_c=\dot\epsilon_c^{sh}(\epsilon_c,\sigma_e,T)$$

the time hardening method

$$\dot\epsilon_c=\dot\epsilon_c^{th}(t,\sigma_e,T)$$

and the damage hardening (softening) method

$$\dot\epsilon_c=\dot\epsilon_c^{dh}(\pi,\sigma_e,T)$$

One may expect that $\epsilon_c^{sh}<\epsilon_c^{dh}<\epsilon_c^{th}$ since the stress change had occured later in the softening stage when the creep curves were convex. This was indeed confirmed by the computational results as well as the fact the damage remained the same, independent of the choice of a transition method.