Investigation on a full coupling between damage and other thermomechanical behaviours in the standard thermodynamic framework including environmental effects

Abstract

In the present paper, constitutive equations accounting for coupled damage-thermo-elasto-(visco)plastic and diffusion at small deformation are proposed in the standard thermodynamics of irreversible processes framework. One main objective is to include the diffusion phenomenon in the models with ductile damage. The model is developed in the framework of thermodynamics of irreversible processes with a set of internal state variables. For illustration, we consider hydrogen diffusion in both normal interstitial lattice sites and trapping sites. The damage modelling of microvoids or microcracks is introduced by the use of Continuum Damage Mechanics framework leading to the definition of effective state variables on fictive undamaged configuration based on the total energy equivalence assumption. Consequently, the full coupling concerns not only the elastic and inelastic behaviour with hardening (isotopic and kinematic), but also the thermal and diffusion phenomena. The concept of the total energy equivalence is thus extended to define effective temperature, entropy, and effective state variables associated with diffusion. It enables to obtain different couplings between thermal phenomena, diffusion phenomena, and the mechanical behaviour, especially isotropic ductile damage. Such a full coupled model is then applied to a representative volume element subject to some typical simple loading paths for illustration and test of such couplings.

Appendix

The partial derivatives $\displaystyle {\frac{\partial c_T}{\partial \varvec{\varepsilon }^e}}$, $\displaystyle {\frac{\partial c_T}{\partial p}}$, $\displaystyle {\frac{\partial c_T}{\partial D}}$, $\displaystyle {\frac{\partial c_T}{\partial T}}$ of Eq. (61) are calculated here.

$c_T$ depends on $\varvec{\varepsilon }^e$ only by $K^T$. So, using (59) and (60) we have:

$$\begin{aligned} \frac{\partial c_T}{\partial \varvec{\varepsilon }^e} = \frac{\partial c_T}{\partial K^T} \frac{\partial K^T}{\partial \varvec{\varepsilon }^e} \end{aligned}$$

(67)

with

$$\begin{aligned} \frac{\partial c_T}{\partial K^T}= & {} \frac{\sqrt{1-D} N^T c_L N^L}{(N^L + \sqrt{1-D} c_L K^T)^2} \frac{\partial K^T}{\partial T}, \end{aligned}$$

(68)

$$\begin{aligned} \frac{\partial K^T}{\partial \varvec{\varepsilon }^e}= & {} - \frac{9K \beta \sqrt{1-D}(\sqrt{1-D}-1)}{\sqrt{1-hD} \mathcal {R}_g T} K^T. \end{aligned}$$

(69)

$c_T$ depends on p only by $N^T$,

$$\begin{aligned} \frac{\partial c_T}{\partial p} = \frac{\partial c_T}{\partial N^T} \frac{\partial N^T}{\partial p}, \end{aligned}$$

(70)

with

$$\begin{aligned} \frac{\partial c_T}{\partial N^T} = \theta _T . \end{aligned}$$

(71)

$c_T$ depends on T only by $K^T$, and using Eqs. (59) and (60) we have:

$$\begin{aligned} \frac{\partial c_T}{\partial T} = \frac{\partial c_T}{\partial K^T} \frac{\partial K^T}{\partial T} \end{aligned}$$

(72)

with

$$\begin{aligned} \frac{\partial K^T}{\partial T} = \frac{K^T}{\sqrt{1-hD} \mathcal {R}_g T^2} (-\varDelta \mu _b + 3K\beta \sqrt{1-D} (\sqrt{1-D}-1)(\varvec{\varepsilon }^e : \varvec{1})) \end{aligned}$$

(73)

and $\displaystyle {\frac{\partial {c_T}}{\partial K^T}}$ given by Eq. (68).

$\displaystyle {\frac{\partial c_T}{\partial D}}$ is calculated using Eqs. (59) and (60):

$$\begin{aligned} \frac{\partial c_T}{\partial D} = \frac{N^T (1+\theta _T)\left( -\theta _T + 2(1-D)^{3/2} (1-\theta _T) \theta _L \frac{\partial K^T}{\partial D}\right) }{2(1-D)} \end{aligned}$$

(74)

with

$$\begin{aligned} \frac{\partial K^T}{\partial D}= & {} \frac{\left( -(1-h)\mu _{L0} - h \sqrt{1-D} \mu _{T0}\right) K^T}{2\sqrt{1-D} \sqrt{1-hD} (1-hD) \mathcal {R}_g T} \nonumber \\&- \frac{3K \beta \sqrt{1-D} \left( \sqrt{1-D} (h-2) + (1-h)\right) (\varvec{\varepsilon }^e : \varvec{1}) K^T}{2\sqrt{1-hD} (1-hD) \mathcal {R}_g T}. \end{aligned}$$

(75)