The present paper is concerned with the analysis of the deformation systems in single crystal magnesium at the micro-scale and with the resulting texture evolution in a polycrystal representing the macroscopic mechanical response. For that purpose, a variationally consistent approach based on energy minimization is proposed. It is suitable for the modeling of crystal plasticity at finite strains including the phase transition associated with deformation-induced twinning. The method relies strongly on the variational structure of crystal plasticity theory, i.e., an incremental minimization principle can be derived which allows to determine the unknown slip rates by computing the stationarity conditions of a (pseudo) potential. Phase transition associated with twinning is modeled in a similar fashion. More precisely, a solid-solid phase transition corresponding to twinning is assumed, if this is energetically favorable. Mathematically speaking, the aforementioned transition can be interpreted as a certain rank-one convexification. Since such a scheme is computationally very expensive and thus, it cannot be applied to the analysis of a polycrystal, a computationally more efficient approximation is elaborated. Within this approximation, the deformation induced by twinning is decomposed into the reorientation of the crystal lattice and simple shear. The latter is assumed to be governed by means of a standard Schmid-type plasticity law (pseudo-dislocation), while the reorientation of the crystal lattice is considered, when the respective plastic shear strain reaches a certain threshold value. The underlying idea is in line with experimental observations, where dislocation slip within the twinned domain is most frequently seen, if the twin laminate reaches a critical volume. The resulting model predicts a stress-strain response in good agreement

In this study, fatigue behavior of polypropylene fiber reinforced concrete has been studied under constant and variable amplitude loading. Crack length was measured during flexural fatigue test under three loads. Accordingly, damage curves were determined as function of stress levels. The results ascertained that the presence of 1 wt% polypropylene fibers considerably increases resistance against fatigue crack growth under constant amplitude loading. Under variable amplitude loading, it was found that the damage curve approach predicts the fatigue life with 13–15.8% estimation error, whereas linear model predicts the fatigue life with 36.8–56.5% estimation error.

The description of the complex interplay between deformation-induced twinning and dislocation slip, typical for metals showing an hcp structure such as magnesium, is of utmost importance for understanding their deformation behavior. In the present paper, an incremental energy principle is presented for that purpose. Within this principle, dislocation slip is modeled by crystal plasticity theory, while the phase decomposition associated with twinning is considered by a mixture theory. This mixture theory naturally avoids the decomposition of the twinning process into so-called pseudo-dislocations followed by a reorientation of the total crystal. By way of contrast, the proposed model captures the transformation of the crystal lattice due to twinning in a continuous fashion by simultaneously taking dislocation slip within both, possibly co-existent, phases into account. The shear strain induced by twinning as well as the deformation history are consistently included within the twinned domain by an enhanced multiplicative decomposition of the deformation gradient. Kinematic compatibility between the different phases is enforced by a Hadamard-type compatibility condition, while compatibility with respect to the boundary conditions requires the introduction of a boundary layer. The evolution of all state variables such as the twinning volume and the plastic strains associated with dislocation slip follow jointly and conveniently from minimizing the stress power of the total crystal. This canonical variational principle is closely related to the postulate of maximum dissipation and guarantees thermodynamical consistency of the resulting model. Particularly, the second law of thermodynamics is fulfilled. In sharp contrast to previous models suitable for the analysis of the deformation systems in magnesium, the Helmholtz energy of the twinning interfaces and that of the aforementioned boundary layer are considered. Analogously, the energy due to twinning nucleation and that related to twinning growth are accounted for by suitable dissipation functionals. By doing so, the number of twinning laminates becomes an additional unknown within the minimization principle and thus, the thickness of the lamellas can be computed. Interestingly, by interpreting this thickness as the mean free path of dislocations, a size effect of Hall-Petch-type can naturally be included within the novel model. The predictive capabilities of the resulting approach are finally demonstrated by analyzing the channel die test. For that purpose, a certain rank-two laminate structure is considered. However, it bears emphasis that the proposed framework is very general and consequently, it can also be applied to other materials.

In the present contribution, a viscoplastic rate-dependent constitutive model based on Schmid's law has been used to designate the activity level of pure magnesium deformation systems. A crystal plasticity model in the framework of finite elements was used to investigate the deformation systems on the scale of single-and polycrystals. Twinning has been modelled by coupling between finite shear and subsequent lattice rotation. It is shown for pure magnesium, that twinning can affect the slip systems activities and therefore changes the hardening mode within the crystal as a result of crystal reorientation. Furthermore, the twinning evolution during in plane and through-thickness deformation of pure Mg polycrystal rolled plate reveals that tensile {10-12} and compression {10-11} twinning have different effects on the mechanical response.

Magnesium and its alloys are promising materials for lightweight applications. Unfortunately, the macroscopic formability of such materials is relatively poor at room temperature and these metals are characterized by a complex mechanical response. This response is a result of the interplay between different deformation modes at the microscale. Since magnesium is a material showing a hexagonal close‐packed (hcp) structure of the underlying atomic lattice, plasticity caused by dislocations and deformation‐induced twinning are the most relevant deformation modes. Within the present paper, two different recently advocated modeling approaches suitable for capturing such modes at the microscale are analyzed. It is shown that both models can be rewritten into a variationally consistent format where every aspect is naturally driven by energy minimization. In addition to this already known feature, it turns out that both models are based on the same minimization problem. The difference between the models results from different constraints enforced within the variational principle. For getting further insight into the interaction between dislocations and twinning interfaces, accompanying atomistic simulations based on molecular dynamics are also performed. The results of such simulations enter the micromechanical model through the initial plastic deformation within the twinned phase (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)