High Temperature Creep Property of Nanoparticle Reinforced Composite by Discrete Dislocation Dynamic Method

Abstract

Metal-matrix composites (MMCs) reinforced by nanoparticles exert much fine mechanical performance such as high strength, high modulus and good conductivities. The creep property of nanoparticle reinforced MMCs is investigated by discrete dislocation dynamic (DDD) method in this study. Both motions of dislocation glide and climb are involved in the present scheme and a dual time step strategy is adopted to deal with the velocity gap between dislocation glide and climb. The results reveal a transition of creep mechanism as the creep stress increases, and a negative correlation between the threshold stress and the environmental temperature.

1 Introduction

Metal-matrix reinforced with elastic particles composite materials have drawn much interests due to their high strengthss, high modulus and good conductivities, which can be served effectively for the extreme environments, especially at elevated temperature [1,2,3,4]. It has been found that the mechanical performance of metal-matrix composites (MMCs) tends to be improved with the size of particles decreasing, especially when the particle size reduces to nanometer scale [5]. It was found that the tensile strength of aluminum-matrix composites with $1{\text{ vol}}\%$ nanoparticles is equivalent to that with $15{\text{ vol}}\%$ microparticles [6]. The creep property of particulate-reinforced MMCs at high temperature has also been investigated by experimental tests [1,2,3,4,5, 7], and the results reveal high values of stress exponent and activation energy held in these composite materials. By introducing the concept of threshold stress $\sigma_{0}$, the creep data can be well rationalized to the values for the common alloys, and thus the power law relation for creep of MMCs can be regulated as the form [8]:

$$ \dot{\varepsilon } \propto \left( {\sigma - \sigma_{0} } \right)^{n} $$

(1)

where $\dot{\varepsilon }$ is the creep strain rate, $\sigma$ the applied stress and $n$ the stress exponent. It has been pointed out that different values of the stress exponent $n$ corresponds to different creep mechanism [5]: $n = 3, \, 5, \, 8$ corresponds to the mechanism of viscous dislocation glide controlled creep, dislocation climb controlled creep and lattice diffusion controlled creep, respectively. And the threshold stress $\sigma_{0}$ in Eq. (1) is proved to be strongly dependent on temperature by in-situ creep experiments [5].

To figure out the intrinsic mechanical creep mechanisms of nanoparticle reinforced MMCs at elevated temperature, numerical researches are urgently required. Discrete dislocation dynamics (DDD), which tracks the motions of dislocations, can be taken as an effective numerical method to study the mechanical mechanism of micro/nano materials. Since the size of dislocations in the matrix and the size of particles are both fallen into the range of mesoscale, DDD method is suitable to model the creep deformation. In practical, the plastic deformation of MMCs almost originates from the plastic deformation in matrix while the particles are usually assumed to be rigid, so that the interactions between dislocations and particles are critical in the simulation. In recent years, DDD method has been widely applied to investigate the mechanical behaviors of MMCs, such as particle reinforced single crystals [9] and nickel-based single crystal superalloy [10]. Traditional DDD methods only take dislocation glide motion into account, as done by [11,12,13,14,15]. Yet at high temperature the motion of dislocation climb coupled with vacancy diffusion also exerts dominated effects on the plastic deformation of metal materials [16,17,18,19]. The time separation between dislocation glide and climb is the major challenge held in DDD simulation [16, 20,21,22], which dues to that the glide velocity is about 4~5 magnitudes larger than the climb velocity. For creep simulations, an adaptive, staggered time stepping approach is adopted by several researcheres [5, 10, 20, 21] to simulate the creep behavior of single crystal materials and nickel-based single crystal superalloy. This approach assumes the motions of dislocation climb and glide are carried out separately and a frozen state for dislocations is attained when dislocation climb is under modeling, which is appropriate because the applied loading is fixed at a desired value.

However, litter work has been conducted to investigate the creep behaviors and the mechanical mechanism of nanoparticle reinforced MMCs. Thus, this paper focus on the creep behavior of nanoparticle reinforced MMCs at elevated temperature by DDD method, and the results of the creep simulations are carefully analyzed.

2 Theoretical Method

A two-dimensional (2D) single crystal model containing numerous randomly distributed nanoparticles is built, with an aspect ratio $H/D = 2$, as showed in Fig. 1. Pure edge dislocations are assumed to glide on two slip systems oriented at $\pm 60^\circ$, and the motion of dislocation climb is also involved to assist the dislocations cross the nanoparticles. The climb velocity for dislocations can be given by [20, 21]:

$$ V_{c}^{{}} = - \frac{{2\pi D_{v} }}{{b\ln \left( {\left( {\sqrt {D_{v} t^{i} } + b} \right)/b} \right)}}\left[ {{\mathbf{C}}_{{\mathbf{0}}} exp\left( { - \frac{{f_{c}^{{}} \Omega }}{bkT}} \right) - {\mathbf{C}}({\mathbf{R}}_{i} ,t) \, } \right] $$

(2)

Fig. 1

2D single crystal model containing nanoparticles

where $D_{v} = D_{0} \exp \left( { - \frac{{E_{m} }}{kT}} \right)$ is the diffusion constant with $E_{m}$ the migration energy of vacancy, and $b$ is the magnitude of burgers vector, $\Omega$ the atomic volume, and $f_{c} = {\mathbf{s}} \cdot \sigma \cdot {\mathbf{b}}$ the Peach–Koehler force on the climb direction, with ${\mathbf{s}}$ the unit vector along the slip direction and ${\mathbf{b}}$ the burgers vector. The parameter ${\mathbf{C}}_{{\mathbf{0}}} = \exp \left( { - \frac{{E_{f} }}{kT}} \right)$ is the equilibrium vacancy concentration at temperature $T$, with $E_{f}$ the formation energy of vacancy, $k$ the Boltzmann constant. As showed in Eq. (2), $C\left( {R_{i} ,t} \right)$ in the last term of the bracket represents the remote vacancy concentration for $i$ th dislocation, which can be obtained by interpolating the vacancy concentration fields at time $t$ to the position of $i$ th dislocation. The diffusion equation of vacancy concentration can be given by [15, 20]:

$$ \left\{ \begin{gathered} \dot{C} = D_{v} \nabla^{2} C - \frac{{D\Omega_{v} }}{kT}\nabla \cdot \left( {C\nabla p} \right) \, in \, V \hfill \\ C = C_{0} \exp \left( {\frac{{\sigma_{0} \Omega }}{kT}} \right) \, on \, \partial V \hfill \\ \end{gathered} \right. $$

(3)

where $\Omega_{v}$ is the vacancy relaxation volume, $p$ the hydrostatic stress, and $\sigma_{0}$ is the applied stress, which can be taken as the creep stress in the present creep problem. To account for the contribution of dislocation climb on the vacancy concentration fields, an additional term $\Delta C_{s} = - b^{2} \sum\limits_{{i \in V_{e} }} {s_{c}^{i} } { /}V_{e}$ should be introduced into the numerical module of vacancy diffusion equation. This term $\Delta C_{s}$ represents the contributed quantity of vacancy concentration for every dislocation $i$ in element $V_{e}$ during the climb process, which should be distributed into the mesh nodes of element $V_{e}$ at the end of staggered climb step as means of initial conditions for next climb process.

Besides climb, dislocations can also cross the particles by the thermally activated mechanism: dislocations are allowed to jump over the particles and continue to glide if the thermally activated probability $P$ satisfies [20]:

$$ P = \exp \left[ { - \frac{\Delta F}{{kT}}\left( {1 - \frac{{\left| {f_{g} } \right|}}{{\tau_{p} b}}} \right)} \right] \le RN $$

(4)

where $RN \in [0,1]$ $[0,1]$ is a random number generated at every glide increment, $\Delta F = 2\mu b^{3}$ the activation energy for dislocations to jump over the nanoparticles, and $\tau_{p} = \frac{\mu b}{l} = \mu b\sqrt \rho$ represents the athermal strength for particles, with $l$ the particle spacing and $\rho$ the particle density, respectively. As described above, the time separation exists between the motions of dislocation glide and climb. In the present scheme, a dual time step procedure is adopted to overcome this challenge. Two time steps are introduced into the model: dislocation glide is simulated with time step $\Delta t_{g}$ while dislocation climb as well as the evolutions of vacancy concentration fields is simulated with a lager time step $\Delta t_{c}$, and the detailed strategy is stated below: the nanoparticle reinforced composite structure containing numerous initial dislocations and sources is constructed and performed under a stress free condition to relax the microstructure of dislocations until the system reaches an equilibrium state. In this process, only the motion of dislocation glide is considered with the glide velocity $V_{g} = \frac{{f_{g} }}{B}$, where $f_{g} = {\mathbf{n}} \cdot \sigma \cdot {\mathbf{b}}$ is the Peach–Koehler force along the slip direction and $B = 3.2 \, \mu {\text{ Pa}} \cdot {\text{s}}$ is the drag coefficient, and the time step is set to be $\Delta t_{g}$; a uniform strain-controlled load is applied on the vertical direction until reaching the desired creep stress. The motion of dislocation glide as well as the short range effect between dislocations like dislocation nucleation, annihilate is simulated. Dislocations can cross the obstacles only by the thermally activated mechanism in this glide-restricted motion. The data transition is holds in this process: the plastic strain induced by dislocation glide is obtained and distributed into the Guassian points of the FEM module, and the stress fields are solved and transferred back to the DDD module to drive the evolutions of dislocations; when the equilibrium state of the system is attained (i.e. the frozen state for dislocations is reached), the dislocations involving the model are assuming to be immobile and the time step is set to be $\Delta t_{c}$. This state means the dislocations are obstructed by the nanoparticles, and the motion of climb starts to exert its effects by assisting dislocations cross the nanoparticles. The vacancy diffusion Eq. (3) is solved to get the vacancy concentration fields which would be used to determine the climb velocity of dislocations, and the disturbed term due to dislocation climb is introduced into the diffusion equation after the climb process is ended. When the climb distance of any dislocations reaches the spacing of slip plane, the frozen state for dislocations is relieved. The time step is switched back to $\Delta t_{g}$ and the iterative step for dislocation glide is repeated.

3 Results and Discussion

In the present simulation, the equilibrium state is considered to be attained if the average strain rate is less than a threshold value over $100$ glide increments. The composite is assumed to be elastically isotropic for both matrix and particles, with the elastic constant $\mu = \left( {3 \times 10^{4} - 16 \times T} \right){\text{ MPa}}$ [23] and $\nu = 0.35$. As noted, the shear modulus for the composite is related to the environmental temperature. The density of Frank-Read sources is set to be $120\,\mu m^{{2}}$. A Gaussian distribution is assigned to the nucleation strength of the sources with the mean strength $\overline{\tau }_{nuc} = 50{\text{ MPa}}$ and the standard deviation $\delta \tau = 1{\text{ MPa}}$; the density of initial dislocations is $80\,\mu m^{{2}}$. The composite contains a distribution of nanoparticles with a density $113\,\mu m^{{2}}$ and diameter $7.5\,{\text{nm}}$ to match the $2\%$ volume fraction for nanoparticles. The time step for glide is $\Delta t_{g} = 0.5{\text{ ns}}$. Compared with the velocity of dislocation glide, the dislocation climb velocity is much slower so that the time step for dislocation climb $\Delta t_{c}$ is at least $10^{2}$ larger than the glide time step. The material parameters for vacancy diffusion are given by: $D_{0} = 1.51 \times 10^{ - 5} {\text{ m}}^{{2}} {\text{/s}}$, $E_{f} = 0.67{\text{ ev}}$, $E_{m} = 0.61{\text{ ev}}$.

The tensile responses for the nanoparticle reinforced MMCs are first studied, and the results are obtained by executing the simulations without dislocation climb being considered under the applied strain rate $\varepsilon = 100{\text{ s}}^{{ - 1}}$. The stress–strain curves of the model with $D = 2\,\mu m$ are plotted in Fig. 2a, with and without nanoparticles containing in the model both being considered for comparison. As seen, the values of yields stress for the model with nanoparticles and without them are $68{\text{ MPa}}$ and $50{\text{ MPa}}$, respectively. The determination of yields stress provides the range for choosing creep stress for nanoparticle reinforced MMCs, that is, the desired creep stress should be lower than the yields stress. The strain hardening is captured for both cases, which is common for small-size materials. In addition, it can be clearly found in Fig. 2a that the both the initial yields stress and the flow stress are enhanced when the nanoparticles are involved in the model, and this is because the existence of nanoparticles improves the resistance to dislocation glide, leading to the strengthening of materials. The evolutions of dislocation density plotted in Fig. 2b indicated that the evolution rates of dislocation density are larger in the model with nanoparticles contained than that without nanoparticles, and this is due to the block effects of nanoparticles on dislocations keeping more dislocations inside the model.

Fig. 2

a Stress–strain curves and b Evolutions of dislocation density for model with and without nanoparticles contained

Figure 3a-c shows the creep response of MMCs with same size for different values of the creep stress in the range 20~30 Mpa at temperature $T = 673{\text{ K}}$, $T = 623{\text{ K}}$ and $T = 573{\text{ K}}$. The values of creep stress chosen for nanoparticle reinforced MMCs are all lower than the yields stress, and the computation strategy described above is carried out to obtain these results. Primary creep (stage 1) and steady state creep (stage 2) are both captured in Fig. 3. With the temperature being fixed, the creep strain is higher for higher creep stress at a given time. As the temperature rises, the creep strain at a given creep stress is higher at higher temperature. In addition, at any one of these three chosen temperatures the steady creep rate (creep rate in stage 2) is small when the creep stress is small while in the high stress region the steady creep rate turns to be extremely large.

Fig. 3

Creep curves for nanoparticle reinforced MMCs at a $T = 673{\text{K}}$, b $T = 623{\text{K}}$ and c $T = 573{\text{K}}$, and d logarithmic plots of $\dot{\varepsilon }\sim \sigma$

To obtain the stress exponent $n$, the logarithmic variations of steady creep rate versus stress for nanoparticle reinforced MMCs at different values of temperature are plotted in Fig. 3d and the scatter for the obtained creep rates is under calculations at least three realizations of the initial configurations of dislocations, sources and obstacles. A two stage creep is captured for all temperatures chose: in the low stress region, the steady stress exponent $n$ is approaching to $2$; in the high stress region, the steady stress exponent $n$ exceeds $10$. For those of pure aluminum and solid solution aluminum alloys, the stress exponent $n$ for creep is fallen in the range $n = 3\sim 5$ [22], which is higher than the value in the low stress region and smaller than the value in the high stress region. This suggests the existence of threshold stress, which is resulted from the effective barriers of nanoparticles for dislocation motion. In the low stress region, dislocations cannot break away the obstruction of nanoparticles and the climb rate of dislocations is also slow, while in the high stress region, the thermally activated probability as well as the climb rate of dislocations increases significantly. The threshold stress at every temperature can be obtained by linearly fitting the $\dot{\varepsilon }^{1/n} \sim \sigma$ in high stress region, where $n = 3,5,8$ corresponding to viscous dislocation glide controlled creep, dislocation climb controlled creep, and lattice diffusion controlled creep, respectively. In the present simulation, $n$ is chosen to be $5$ because neither viscous dislocation glide deformation mechanism nor lattice diffusion deformation mechanism is considered. With $n$ being fixed, the linearly fitting lines $\dot{\varepsilon }^{1/n} \sim \sigma$ can be get and the threshold stress yields the value of stress at zero strain rate (i.e. the stress of the intersection point between the fitting line and the line $\dot{\varepsilon } = 0$). According to this method, the values of threshold stress at temperature $T = 673{\text{ K}}$, $T = 623{\text{ K}}$ and $T = 573{\text{ K}}$ are $38.25{\text{ MPa}}$, $41.86{\text{ MPa}}$ and ${42}{\text{.48 MPa}}$, respectively. It is clear that that the threshold stress tends to decrease with increasing temperature.

4 Conclusion

In summary, the creep property of nanoparticle reinforced MMCs at high temperature is investigated by DDD method in this paper. Vacancy diffusion coupled dislocation climb is involved in the scheme and a dual time step strategy is adopted to overcome the velocity gap between dislocation climb and glide. The results show a transition of creep mechanism as the creep stress increases and it is revealed that the threshold stress tends to decrease with increasing temperature.