Effect of loading rate

In this section, the effect of the loading rate on plastic deformation is studied, both with and without thermomechanical coupling. Loading rates of 0.06M P a·ns⁻¹, 0.12M P a·ns⁻¹, 0.24M P a· ns⁻¹ , and 0.32M P a· ns⁻¹ are studied. The strain rate at strain of 2· 10⁻³ is given for the corresponding loading rate in Table 4.1.

The former case is shown in Fig.4.10, and demonstrates how an increase in the loading

Loading rate, ˙σ (MPa·ns⁻¹) Strain rate, ˙ε (s⁻¹)

0.06 9.1·10²

0.12 5.6·10³

0.24 1.1·10⁴

0.32 4.8·10⁴

Table 4.1: The strain rates at strain  = 2· 10⁻³ for the corresponding loading rates.

rate leads to an increase in the flow stress, as observed in experiments [152]. This trend is also consistent with effect of the Zener-Hollomon parameter on copper experimentally studied in [153]. The latter case is shown in Fig.4.11 where the flow stress increases with increasing loading rate, but the overall response is softer. Furthermore, we observe that the loading rate significantly influences the magnitude of the localized temperature rise.

The maximum temperature rise is reported in Fig.4.11 and varies from 489K for a loading rate of 0.06M P a· ns⁻¹ to 1131K for a loading rate of 0.32M P a· ns⁻¹ . The temperature rises observed in these simulations are of the same order as those reported in experimental studies of high strain rate plasticity [110, 154]. The observed response is attributed to the increase deformation rate, thus causing an increase in the rate of heat generation that exceeds the rate of increase in heat conduction.

The developed DD model targets high strain rate plastic deformations; however, we have neglected inertial forces in our governing equations, which is a common assumption in DD. Such limitation was recently discussed in a dislocation dynamics study, where it was pointed out to have only limited impact when the dynamics are governed by dislocation

generation and annihilation [109]. However, at high loading rates this assumption may be invalid and requires further investigation before it can be considered conclusive.

The presented results are dependent on location of randomly distributed Frank–Read sources and their randomly assigned strength which means that a different distribution of the sources and their strength may lead to slightly different results. Nonetheless, the overall trends are expected to be unchanged.

Simulations were carried out with strain rates which are commonly used in DD, whereas the typical experimental strain rates are of the order of 10⁻³ to 10⁻⁴s⁻¹; which makes the direct comparison with simulations challenging. However, the observed softening behavior and temperatures rises are consistent with the expected trends and observations from other DD studies [10, 109, 155].

4.4 Concluding remarks

The first fully coupled two–dimensional thermomechanical dislocation dynamics model based on eXtended Finite Element Method was developed. The distinct feature of this model is that the equilibrium equations are coupled with the heat equation, where the heat source is a function of the work done by the motion of dislocations. The work done by each moving dislocation is calculated using the Peach–Koehler force and dislocation velocity at every step of the simulation. The results of our simulations showed that during high rate deformation, common in DD, the significant local heating is present.

Further-0 1 2 3 4 5 6 x 10⁻³ 0

50 100 150 200 250

εeng

σ MPa

˙σ=0.32 MPa/ns

˙σ=0.24 MPa/ns

˙σ=0.12 MPa/ns

˙σ=0.06 MPa/ns

Figure 4.10: The stress-strain curve at the different loading rates without thermocoupling.

more, the heat equation should be incorporated into DD models to avoid overestimation of the stresses. The hardening of the material was observed due to temperature dependent dislocation drag. The simulations showed that the effect of temperature dependent drag increases with deformation rate. The softening effect was found to be stronger in the case of the adiabatic heating versus isothermal conditions on boundary of the domain. It was also observed that higher deformation rates resulted in both higher flow stresses and higher temperature rises. The proposed two–dimensional coupled thermomechanical discrete dis-location dynamics model provides a framework for further investigation of multiphysical phenomena of plastic deformation. It takes a first step in addressing concerns related to

0 1 2 3 4 5 x 10⁻³ 0

50 100 150 200 250

εeng

σ MPa

˙σ=0.32 MPa/ns; maxΘ=1131 K

˙σ=0.24 MPa/ns; maxΘ=1087 K

˙σ=0.12 MPa/ns; maxΘ=585 K

˙σ=0.06 MPa/ns; maxΘ=489 K

Figure 4.11: The stress-strain curve at the different loading rates.

the underestimation of thermal softening effects at high strain rates as well as the induced inter-play of temperature, dislocation drag, and hardening.

Chapter 5 Conclusions

Two new Discrete Dislocation Dynamics (DD) models that incorporate the effect of mul-tiphysics were developed based on the eXtended Finite Element Method (XFEM). The discontinuities in the displacement field across the glide planes created by the dislocations were modeled by introducing additional basis functions into the approximation. The driv-ing force on a dislocation, or Peach–Koehler force, in both models was calculated usdriv-ing the J-integral. The main contributions of this thesis are:

 The first fully coupled two-dimensional electromechanical (EM) Discrete Dislocation Dynamics model for the plastic deformation of anisotropic piezoelectric crystalline solids was developed.

• The piezoelectric effect on the Peach–Koehler force is found to be significant and can affect both the direction and magnitude of the force. As a result, it was discovered that the piezoelectric effect is important when modeling the plastic

behavior of piezoelectric materials in the presence of electrical and mechanical fields.

• The simulations of the motion of many edge dislocations in a small domain demonstrated that, for piezoelectric materials, the plastic response of the mate-rial differs considerably when various electric potential differences are applied, illustrating that the physics of plasticity under electromechanical load is more complex than in purely mechanical systems.

• The developed EM–XFEM–DD model has great potential as a tool to study the behavior of the piezoelectric materials.

 The first two–dimensional fully coupled thermomechanical (TM) Discrete Dislocation Dynamics model for high strain rate plastic deformation of crystalline solids was developed.

• The TM–XFEM-DD model is able to simulate the heat generated by the motion of the dislocations under the high strain rate plastic deformation.

• The TM–XFEM–DD model takes the temperature dependency of dislocation drag into account. As a result, the simulations showed that the effect of tem-perature dependent drag increases with deformation rate.

• The softening effect predicted by TM–XFEM–DD is stronger in the case of adi-abatic heating versus fixed temperature change on the boundary of the domain.

• The simulations showed that higher deformation rates resulted in both higher flow stresses and higher temperature rises.

• The developed TM–XFEM–DD is the first model to address concerns related to the underestimation of thermal softening effects at high strain rates at the mesoscale.

The thesis has qualitatively shown that when modeling plastic deformation of piezoelec-tric materials at the dislocation dynamics level, the coupling effects between elecpiezoelec-tric and mechanical fields should be taken into account. The developed EM–XFEM–DD model is the first DD model that is able to account for such electromechanical coupling effects. I have also shown that when modeling the high strain rate plastic deformation of crystalline solids on dislocation dynamics level, the thermal effects can not be neglected. The perfor-mance of the crystalline solids under a high rate of strain is different than the perforperfor-mance under quasi-static conditions. The developed TM-XFEM-DD model is the first DD model which incorporates thermal effects, including heating induced by dislocation motion, and can be used to more effectively study the plastic behavior of metals under high strain rate deformation than the previous DD models.

To summarize, the goal of this thesis was achieved with the development of the EM–

XFEM–DD and TM–XFEM–DD models. These new DD models bridge the gap between two scales, atomic- and micro-scale, by incorporating the effect of multiphysics, enabling the analysis of plastic phenomena in crystalline solids at the mesoscale, which was previously

not possible.

Chapter 6