**5. Instructive examples**

Fig. 6. Local stress and strain patterns of RVEs having grain boundaries with with different properties.

polycrystal slip systems with *a*/2[1, 1, 0] and *a*/2[−1, 1, 0] Burgers vectors will be activated

Scale Bridging Modeling of Plastic Deformation and Damage Initiation in Polycrystals 19

Figure 6 shows the stress and strain distributions at about 2% tensile strain with a loading speed of 10 nm/s. These results are calculated with combinations of a normal interface strength of 1500 MPa and two tangential interface strengths; an strong one of 1500 MPa and an weak one of 300 MPa. One can easily see that the stronger grain boundary causes higher stress concentrations and strain heterogeneities inside the aggregate compared with the weaker grain boundary. For the weaker grain boundary, the strain localisation instead starts from the triple junction and tends to expand into the bulk material roughly along the maximum shear stress direction. Although this phenomenon is also observed for the stronger grain boundaries, the most obvious strain localisation is in some grains with a larger Schmid factor along the grain boundary direction and even extending to the grain center in some extreme cases. In both cases, cracks have initiated in the upper and lower boundaries and attempted to propagate along the vertical grain boundaries under tensile loading along the

Figure 7 shows the global stress-strain curves with respect to 5 different grain boundary strength conditions. From this plot, one can see that the combination of grain boundary opening and sliding can relax almost one third of the average stress level. Because the weak-normal-strong-tangential cohesive zone model and strong-normal-weak-tangential cohesive zone model produce almost the same global stress strain curve, it seems that the normal and tangential cohesive strengths have similar influences on the material load carrying

With respect to an small qusi-2D RVE with 5 grains in a 1*μm* × 1*μm* domain, the details of the global stress strain curve have been studied as shown in Figure 8. The RVE shows some unstable mechanical behaviors especially at about 0.5% tensile strain. Generally the material load carrying capacity loss infers the damage initialisation. From our calculations, one can see clearly that the kinks of the stress strain curve are mainly stemming from grain boundary opening and sliding near grain boundary triple junctions which can relax the locally

The current study with respective to nano-metere grain size polycrystals implies that grain boundary mediated deformation processes decisively change the global stress-strain response of the studied material. Since the grain boundary cohesive behavior is independent of the grain size, whereas the resistance for dislocation slip inside the bulk material points becomes smaller as grain size increase, we expect that the influence of grain boundary processes will

An RVE including 12 ferritic grains and one austenitic grain as shown in Figure 9 has been generated to investigate the TRIP behavior under different loading conditions. As shown in Figure 10, tensile and compression loadings induce different total martensitic volume fractions. This numerical result is consistent with experimental observations. During the martensite phase transformation, about 22% shear strain inside the habit plane and about 2% dilatation strain along the normal of the habit plane are needed to transfer from the FCC

**5.2 TRIP steel deformation modelling with the crystal plasticity method**

under tensile loads along the horizontal direction (Shaban et al., 2010).

horizontal direction.

accumulated stress efficiently.

gradually vanish for coarse-grained material.

capacity.

Fig. 7. Global stress-strain curves of RVEs having grain boundaries with with different properties.

Fig. 8. Global stress-strain curve and local stress and strain distribution.

#### **5.1 Polycrystal deformation modelling with bulk material slip and grain boundary material glide with the crystal plasticity method**

A qusi-2D RVE with 17 hexagonal shape grains with 80nm × 60nm × 2nm volume has been generated. Keeping *φ* = 0 and *ϕ*<sup>2</sup> = 0 we have assigned initial crystal orientations with an 5◦ increment for Euler angle *ϕ*<sup>1</sup> from grain 0◦ to 80◦ randomly. For the studied aluminum 16 Will-be-set-by-IN-TECH

Fig. 7. Global stress-strain curves of RVEs having grain boundaries with with different

Fig. 8. Global stress-strain curve and local stress and strain distribution.

**glide with the crystal plasticity method**

**5.1 Polycrystal deformation modelling with bulk material slip and grain boundary material**

A qusi-2D RVE with 17 hexagonal shape grains with 80nm × 60nm × 2nm volume has been generated. Keeping *φ* = 0 and *ϕ*<sup>2</sup> = 0 we have assigned initial crystal orientations with an 5◦ increment for Euler angle *ϕ*<sup>1</sup> from grain 0◦ to 80◦ randomly. For the studied aluminum

properties.

polycrystal slip systems with *a*/2[1, 1, 0] and *a*/2[−1, 1, 0] Burgers vectors will be activated under tensile loads along the horizontal direction (Shaban et al., 2010).

Figure 6 shows the stress and strain distributions at about 2% tensile strain with a loading speed of 10 nm/s. These results are calculated with combinations of a normal interface strength of 1500 MPa and two tangential interface strengths; an strong one of 1500 MPa and an weak one of 300 MPa. One can easily see that the stronger grain boundary causes higher stress concentrations and strain heterogeneities inside the aggregate compared with the weaker grain boundary. For the weaker grain boundary, the strain localisation instead starts from the triple junction and tends to expand into the bulk material roughly along the maximum shear stress direction. Although this phenomenon is also observed for the stronger grain boundaries, the most obvious strain localisation is in some grains with a larger Schmid factor along the grain boundary direction and even extending to the grain center in some extreme cases. In both cases, cracks have initiated in the upper and lower boundaries and attempted to propagate along the vertical grain boundaries under tensile loading along the horizontal direction.

Figure 7 shows the global stress-strain curves with respect to 5 different grain boundary strength conditions. From this plot, one can see that the combination of grain boundary opening and sliding can relax almost one third of the average stress level. Because the weak-normal-strong-tangential cohesive zone model and strong-normal-weak-tangential cohesive zone model produce almost the same global stress strain curve, it seems that the normal and tangential cohesive strengths have similar influences on the material load carrying capacity.

With respect to an small qusi-2D RVE with 5 grains in a 1*μm* × 1*μm* domain, the details of the global stress strain curve have been studied as shown in Figure 8. The RVE shows some unstable mechanical behaviors especially at about 0.5% tensile strain. Generally the material load carrying capacity loss infers the damage initialisation. From our calculations, one can see clearly that the kinks of the stress strain curve are mainly stemming from grain boundary opening and sliding near grain boundary triple junctions which can relax the locally accumulated stress efficiently.

The current study with respective to nano-metere grain size polycrystals implies that grain boundary mediated deformation processes decisively change the global stress-strain response of the studied material. Since the grain boundary cohesive behavior is independent of the grain size, whereas the resistance for dislocation slip inside the bulk material points becomes smaller as grain size increase, we expect that the influence of grain boundary processes will gradually vanish for coarse-grained material.

## **5.2 TRIP steel deformation modelling with the crystal plasticity method**

An RVE including 12 ferritic grains and one austenitic grain as shown in Figure 9 has been generated to investigate the TRIP behavior under different loading conditions. As shown in Figure 10, tensile and compression loadings induce different total martensitic volume fractions. This numerical result is consistent with experimental observations. During the martensite phase transformation, about 22% shear strain inside the habit plane and about 2% dilatation strain along the normal of the habit plane are needed to transfer from the FCC

Fig. 11. Martensitic volume fractions of specific transformation systems under tensile and

Scale Bridging Modeling of Plastic Deformation and Damage Initiation in Polycrystals 21

Fig. 12. Stress-strain curve comparison between deformations with and without martensite

With the help of the 13 grain RVE we have investigated into why martensite phase transformation can provide high ductility and high strength at the same time. As shown in Figure 12, the global stress-strain curves of the simulation with and without martensite transformation have a intersection point at about 14% tensile strain. Before the intersection point, the eigen strain of the phase transformation serves as a competing partner of dislocation

compression loads.

transformation.

Fig. 9. Initial orientations of ferritic and austenitic grains. The only austenitic grain has been highlighted and has a volume fraction of about 10%. For the compression calculation, the loading direction has been inversed.

Fig. 10. Total martensitic volume fractions under tensile and compression loading cases.

lattice to the BCT lattice. Modelling results support that the normal part of the resolved stress calculated by equation 37 strongly influences the transformation volume fraction evolution. As stated in literature (Kouznetsova & Geers, 2008; Stringfellow et al., 1992), this is the well known hydrostatic stress dependence of the martensite transformation. The current study shows that there are four dominating transformation systems under tensile and compression loading conditions as shown in Figure 11. Careful analysis of the magnitude of dilatation and shear resolved stresses under tensile and compression loads shows that the shear part is important to determine the activated transformation systems.

18 Will-be-set-by-IN-TECH

Fig. 9. Initial orientations of ferritic and austenitic grains. The only austenitic grain has been highlighted and has a volume fraction of about 10%. For the compression calculation, the

Fig. 10. Total martensitic volume fractions under tensile and compression loading cases.

important to determine the activated transformation systems.

lattice to the BCT lattice. Modelling results support that the normal part of the resolved stress calculated by equation 37 strongly influences the transformation volume fraction evolution. As stated in literature (Kouznetsova & Geers, 2008; Stringfellow et al., 1992), this is the well known hydrostatic stress dependence of the martensite transformation. The current study shows that there are four dominating transformation systems under tensile and compression loading conditions as shown in Figure 11. Careful analysis of the magnitude of dilatation and shear resolved stresses under tensile and compression loads shows that the shear part is

loading direction has been inversed.

Fig. 11. Martensitic volume fractions of specific transformation systems under tensile and compression loads.

Fig. 12. Stress-strain curve comparison between deformations with and without martensite transformation.

With the help of the 13 grain RVE we have investigated into why martensite phase transformation can provide high ductility and high strength at the same time. As shown in Figure 12, the global stress-strain curves of the simulation with and without martensite transformation have a intersection point at about 14% tensile strain. Before the intersection point, the eigen strain of the phase transformation serves as a competing partner of dislocation

Fig. 15. Stress-strain curve comparison among RVEs with different grain size and lamella

Scale Bridging Modeling of Plastic Deformation and Damage Initiation in Polycrystals 23

Fig. 16. Yield stress comparison among RVEs with different grain size and lamella thickness.

With respect to four grain numbers (189, 91, 35, 9) and three lamella thicknesses (16 nm, 31 nm, 47 nm), a total of 12 global stress-strain curves have been simulated to investigate the size effect on material mechanical behaviors. Figure 15 shows several stress-strain curves along the loading direction. From these results one can see easily *the smaller the stronger* rule often observed in experiments. Through determining the yield stress for different RVEs, the

thickness.

Fig. 13. Local stress distribution comparison between deformations with (right) and without (left) martensite transformation. The austenitic grain has been highlighted.

slip to reduce the external load potential, and as a direct result TRIP can increase the material ductility as shown in Figure 12. Because the phase transformation will exhaust the dislocation slip volume fraction and the martensite can only deform elastically, after the intersection point the hardening side of the TRIP mechanism will overcome the softening side and one can observe there is a enhanced tensile strength.

Figure 13 shows the local stress distribution comparison between simulations with and without phase transformation. As expected there are higher internal stresses inside the austenite grain when phase transformation exists. Through modelling the internal stress accumulation the current model system can be used to investigate the material damage and failure phenomena.

Fig. 14. Local strain distributions of an 189 grain RVE.

#### **5.3 Polycrystal with twin lamella deformation modelling with the crystal plasticity method**

As a mesh free method, the fast Fourier transformation approach can be used to model deformation of RVEs with very complicated microstructures. Several RVEs occupying a 1*μ*m×1*μ*m×1*μ*m space with nano-metere sized twin lamellas inside nano-metere sized grains have been generated and discretised to 64 × 64 × 64 regular grids. The initial crystal orientations have been assigned randomly. Based on equations 5 and 6, twin lamellas with different thickness have been generated. Figure 14 shows an RVE with 189 grains containing a 31 nm thickness lamella and the von-Mises equivlent strain distribution under tensile load and periodic boundary conditions. Here, the material parameters of pure aluminum have been used during the simulation.

20 Will-be-set-by-IN-TECH

Fig. 13. Local stress distribution comparison between deformations with (right) and without

slip to reduce the external load potential, and as a direct result TRIP can increase the material ductility as shown in Figure 12. Because the phase transformation will exhaust the dislocation slip volume fraction and the martensite can only deform elastically, after the intersection point the hardening side of the TRIP mechanism will overcome the softening side and one can

Figure 13 shows the local stress distribution comparison between simulations with and without phase transformation. As expected there are higher internal stresses inside the austenite grain when phase transformation exists. Through modelling the internal stress accumulation the current model system can be used to investigate the material damage and

**5.3 Polycrystal with twin lamella deformation modelling with the crystal plasticity method** As a mesh free method, the fast Fourier transformation approach can be used to model deformation of RVEs with very complicated microstructures. Several RVEs occupying a 1*μ*m×1*μ*m×1*μ*m space with nano-metere sized twin lamellas inside nano-metere sized grains have been generated and discretised to 64 × 64 × 64 regular grids. The initial crystal orientations have been assigned randomly. Based on equations 5 and 6, twin lamellas with different thickness have been generated. Figure 14 shows an RVE with 189 grains containing a 31 nm thickness lamella and the von-Mises equivlent strain distribution under tensile load and periodic boundary conditions. Here, the material parameters of pure aluminum have

(left) martensite transformation. The austenitic grain has been highlighted.

observe there is a enhanced tensile strength.

Fig. 14. Local strain distributions of an 189 grain RVE.

been used during the simulation.

failure phenomena.

Fig. 15. Stress-strain curve comparison among RVEs with different grain size and lamella thickness.

Fig. 16. Yield stress comparison among RVEs with different grain size and lamella thickness.

With respect to four grain numbers (189, 91, 35, 9) and three lamella thicknesses (16 nm, 31 nm, 47 nm), a total of 12 global stress-strain curves have been simulated to investigate the size effect on material mechanical behaviors. Figure 15 shows several stress-strain curves along the loading direction. From these results one can see easily *the smaller the stronger* rule often observed in experiments. Through determining the yield stress for different RVEs, the

Janisch, R., Ahmed, N. & Hartmaier, A. (2010). Ab initio tensile tests of Al bulk crystals

Scale Bridging Modeling of Plastic Deformation and Damage Initiation in Polycrystals 25

Kalidindi, S., Bronkhort, C. & Anand, L. (1992). Crystallographic texture evolution in bulk

Kocks, U., Argon, A. & Ashby, M. (1975). Thermodynamics and kinetics of slip, *Chalmers, B., Christian, J.W., Massalski, T.B. (Eds.), Progress in Materials Science* 19: 1–289. Kouznetsova, V. & Geers, M. (2008). A multi-scale model of martensitic transformation

Lebensohn, R. A. (2001). N-site modeling of a 3d viscoplastic polycrystal using fast fourier

Lu, K., Lu, L. & Suresh, S. (2009). Strengthening materials by engineering coherent internal

Lu, L., Shen, Y., Chen, X., Qian, L. & Lu, K. (2004). Ultrahigh strength and high electrical

Ma, A. (2006). *Modeling the constitutive behavior of polycrystalline metals based on dislocation*

Ma, A. & Roters, F. (2004). A constitutive model for fcc single crystals based on dislocation

Madec, R., Devincre, B., Kubin, L., Hoc, T. & Rodney, D. (2008). The role of collinear interaction

Michel, J., Moulinec, H. & Suquet, P. (2000). A computational method based on augmented

Michel, J., Moulinec, H. & Suquet, P. (2001). A computational scheme for linear and non-linear composites with arbitrary phase contrast, *Int. J. Numer. Methods Eng.* 52: 139. Nemat-Nasser, S., Luqun, N. & Okinaka, T. (1998). A constitutive model for fcc crystals with

Peirce, D., Asaro, R. & Needleman, A. (1982). An analysis of non-uniform and localized

Roters, F. (1999). *Realisierung eines Mehrebenenkonzeptes in der Plastizitätsmodellierung*, Phd

Schröder, J., Balzani, D. & Brands, D. (2010). Approximation of random microstructures

Shaban, A., Ma, A. & Hartmaier, A. (2010). Polycrystalline material deformation modeling

Stringfellow, R., Parks, D. & Olson, G. (1992). A constitutive model for transformation

by periodic statistically similar representative volume elements based on lineal-path

with grain boundary sliding and damage accumulation, *Proceedings of 18th European*

plasticity accompanying strain-induced martensitic transformation in metastable

application to polycrystalline ofhc copper, *Mech. Mater.* 30: 325–341. Nye, J. (1953). Some geometrical relations in dislocated crystals, *Acta Metall.* 1: 153–162. Olson, G. & Cohen, M. (1972). A mechanism for strain-induced nucleation of martensitic

deformation in ductile single crystals, *Acta Metall.* 30: 1087–1119.

densities and its application to uniaxial compression of aluminium single crystals,

lagrangians and fast fourier transforms for composites with high contrast, *Comput.*

deformation processing of fcc metals, *J. Mech. Phys. Solids* 40.

Lee, E. (1969). Elastic-plastic deformation at finite strains, *J Appl. Mech.* 36: 1–6.

plasticity, *Mechanics of Materials* 40: 641.

boundaries at the nanoscale, *Science* 324: 349.

in dislocation-induced hardening, *Science* 301: 1879.

conductivity in copper, *Science* 304: 422.

*mechanisms*, Phd thesis, RWTH Aachen.

transformations, *J. Less-Common Metals* 28.

functions, *Archive of Applied Mechanics* 81: 975.

austenitic steels, *Acta Metall. Mater.* 40: 1703.

*Acta Materialia* 52: 3603–3612.

*Model. Eng. Sci.* 1: 79.

thesis, RWTH Aachen.

*Conference on Fracture (ECF18)* .

transform, *Acta Materialia* 49: 2723.

81: 184108–1–6.

and grain boundaries: universality of mechanical behavior, *Physical Review B*

parameters *σ*<sup>0</sup> and *ky* of the Hall-Petch relation

$$
\sigma = \sigma\_0 + \frac{k\_y}{\sqrt{D}} \tag{60}
$$

for pure aluminum have been investigated carefully. From the numerical results shown in Figure 16 we found that equation 60 with parameters *<sup>σ</sup>*0=6MPa and *ky*=300MPa√*nm* fits the simulation data very well. Indeed, these two values almost fall nicely in the experimental measurement ranges *<sup>σ</sup>*0=6 <sup>±</sup> 2 MPa and *ky*=400 <sup>±</sup> 80MPa <sup>√</sup>*nm* in Bonetti et al. (1992).

#### **6. Summary**

In this work it has been demonstrated how information from several length scales can be integrated into representative volume element (RVE) models for the mechanical behaviour of heterogeneous materials, consisting of several grains and different phases. In particular, the relevance of phenomena on different scales, like atomic bonds that determine the mechanical properties of grain boundaries, or the interaction of dislocations with grain boundaries should be investigated carefully in future studies. The mechanisms occurring at such atomistic and microstructural scales need to be modelled in a suited way such that they can be taken into account in continuum simulations of RVE's. Once an RVE for a given microstructure is constructed and the critical deformation and damage mechanisms are included into the constitutive relations, this RVE can be applied to calculate stress-strain curves and other mechanical data. The advantage of this approach is that by conducting parametric studies the influence of several microstructural features, like for example grain size or strength of grain boundaries, on the macroscopic mechanical response of a material can be predicted.

#### **7. References**

ABAQUS (2009). *ABAQUS Version6.91*, Dassault Systemes.

Bhadeshia, K. (2002). Trip-assisted steels?, *ISIJ International* 42: 1059.


22 Will-be-set-by-IN-TECH

*ky*

<sup>√</sup>*<sup>D</sup>* (60)

*σ* = *σ*<sup>0</sup> +

for pure aluminum have been investigated carefully. From the numerical results shown in Figure 16 we found that equation 60 with parameters *<sup>σ</sup>*0=6MPa and *ky*=300MPa√*nm* fits the simulation data very well. Indeed, these two values almost fall nicely in the experimental measurement ranges *<sup>σ</sup>*0=6 <sup>±</sup> 2 MPa and *ky*=400 <sup>±</sup> 80MPa <sup>√</sup>*nm* in Bonetti et al. (1992).

In this work it has been demonstrated how information from several length scales can be integrated into representative volume element (RVE) models for the mechanical behaviour of heterogeneous materials, consisting of several grains and different phases. In particular, the relevance of phenomena on different scales, like atomic bonds that determine the mechanical properties of grain boundaries, or the interaction of dislocations with grain boundaries should be investigated carefully in future studies. The mechanisms occurring at such atomistic and microstructural scales need to be modelled in a suited way such that they can be taken into account in continuum simulations of RVE's. Once an RVE for a given microstructure is constructed and the critical deformation and damage mechanisms are included into the constitutive relations, this RVE can be applied to calculate stress-strain curves and other mechanical data. The advantage of this approach is that by conducting parametric studies the influence of several microstructural features, like for example grain size or strength of grain boundaries, on the macroscopic mechanical response of a material can be predicted.

Bhattacharya, K. (1993). Comparison of the geometrically nonlinear and linear theories of

Bonetti, E., Pasquini, L. & Sampaolesi, E. (1992). The influence of grain size on the mechanical properties of nanocrystalline aluminium, *Nanostructured Materials* 9: 611. Dai, H. & Parks, D. (1997). Geometrically-necessary dislocation density and scale-dependent

Devincre, B., Hoc, T. & Kubin, L. (2008). Dislocation mean free paths and strain hardening of

Gottstein, G. (2004). *Physical Foundations of Materials Science*, Springer Verlag,

Hane, K. & Shield, T. (1998). Symmetry and microstructure in martensites, *Philosophical*

Hane, K. & Shield, T. (1999). Microstructure in the cubic to monoclinic transition in

crystal plasticity, *Khan, A., (Ed.),Proceedings of Sixth International Symposium on*

parameters *σ*<sup>0</sup> and *ky* of the Hall-Petch relation

ABAQUS (2009). *ABAQUS Version6.91*, Dassault Systemes.

*Plasticity, Gordon and Breach* .

Berlin-Heidelberg, Germany.

crystals, *Science* 320: 1745.

*Magazine A* 78.

Bhadeshia, K. (2002). Trip-assisted steels?, *ISIJ International* 42: 1059.

titanium-nickel shape memory alloys, *Acta Mater.* 47. Hirth, J. & Lothe, J. (1992). *Theory of dislocations*, Krieger Pub Co.

martensitic-transformation, *Continuum Mech. Therm.* 5: 205.

**6. Summary**

**7. References**


**2** 

P.V. Galptshyan

*Republic of Armenia* 

*Institute of Mechanics, National Academy of Sciences of the Republic of Armenia, Erevan* 

**Strength of a Polycrystalline Material** 

There are numerous polycrystalline materials, including polycrystals whose crystals have a cubic symmetry. Polycrystals with cubic symmetry comprise minerals and metals such as cubic pyrites (FeS2), fluorite (CaF2), rock salt (NaCl), sylvite (KCl), iron (Fe), aluminum (Al),

It is assumed that many materials can be treated as a homogeneous and isotropic medium independently of the specific characteristics of their microstructure. It is clear that, in fact, this is impossible already because of the molecular structure of materials. For example, materials with polycrystalline structure, which consist of numerous chaotically located small crystals of different size and different orientation, cannot actually be homogeneous and isotropic. Each separate crystal of the metal is anisotropic. But if the volume contains very many chaotically located crystals, then the material as a whole can be treated as an isotropic material. Just in a similar way, if the geometric dimensions of a body are large compared with the dimensions of a single crystal, then, with a high degree of accuracy, one can assume that the material is

On the other hand, if the problem is considered in more detail, then the anisotropy both of the material and of separate crystals must be taken into account. For a body under the action of external forces, it is impossible to determine the stress-strain state theoretically with its

Assume that a body consists of crystals of the same material. Moreover, in general, the principal directions of elasticity of neighboring crystals do not coincide and are oriented arbitrarily. The following question arises: Can stress concentration exist near a corner point

To answer this question, it is convenient to replace the problem under study by several simplified problems each of which can reflect separate situations in which several

A similar problem for two orthotropic crystals having the shape of wedges rigidly connected along their jointing plane was considered in (Belubekyan, 2000). They have a common vertex, and their external faces are free. Both of the wedges consist of the same material. The wedges have common principal direction of elasticity of the same name, and the other elastic-equivalent principal directions form a nonzero angle. We consider

of the interface between neighboring crystals and near and edge of the interface?

longitudinal shear (out-of-plane strain) along the common principal direction.

copper (Cu), and tungsten (W) (Love, 1927; Vainstein et al., 1981).

homogeneous (Feodos'ev, 1979; Timoshenko & Goodyear, 1951).

polycrystalline structure taken into account.

neighboring crystals may occur.

**1. Introduction** 

