strain gradient plasticity model for size effects in heterogeneous nano-microstructures

International Journal of Plasticity 97 (2017) 46e63

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Stress/strain gradient plasticity model for size effects in heterogeneous nano-microstructures Hao Lyu a, *, Mehdi Hamid a, Annie Ruimi b, Hussein M. Zbib a a b

School of Mechanical and Materials Engineering, Washington State University, Pullman, WA 99163, USA Department of Mechanical Engineering, Texas A&M University at Qatar, Doha, Qatar

a r t i c l e i n f o

a b s t r a c t

Article history: Received 2 February 2017 Received in revised form 15 May 2017 Accepted 23 May 2017 Available online 26 May 2017

Traditionally, modeling the effect of grain size on the mechanical behavior of crystalline materials is based on assuming an equivalent homogenous microstructure with strength being dependent on the average grain size, for example the well-known Hall-Petch relation. However, assuming an equivalent homogenized microstructure for a highly heterogeneous microstructure can lead to inaccurate prediction of strength and ductility, especially when the gradients in the spatial heterogeneity are severe. In this work, we employ a multiscale dislocation-based model combined with a strain/stress-gradient theory to investigate the effect of spatial heterogeneity of the microstructure on strength and ductility. We concentrate on understanding the effect of various grain size spatial distributions on the mechanical properties of interstitial free (IF)-steel. The results show that by controlling some parameters in the spatial distribution of the microstructure with regions composed of micro-grains and nano-grains one can achieve improved strength and ductility. Based on these results, it is suggested that the mechanical properties of gradient materials can be described by phenomenological relations that include two structural parameters, grain size and grain-size gradient, in contrast to Hall-Petch relation for homogenous materials where only grains size appears in the equation. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Dislocation theory Multiscale modeling Plasticity of metals Gradient microstructure Grain size gradient Stress/strain gradient

1. Introduction Numerous investigations have shown that the mechanical behavior of metallic materials can be significantly altered by refinement of the microstructure, e.g. (Calcagnotto et al., 2010; Estrin and Vinogradov, 2013). For instance, a polycrystalline material with nano-size grains will have a higher strength than its equivalent micron-size grains. However, the trade-off is that it will also become more brittle (Morris, 2010). For a material with a homogenous microstructure, it is difficult to attain both high strength and ductility. This has been reported by many researchers, e.g. Lu (2014), where it is shown that the strength versus ductility curve, for many classes of polycrystalline materials, follows a so-called ‘banana curve’, shown schematically in Fig. 1. However, recent experimental studies have shown that some heterogeneous microstructures with grain size in the range of a few nanometer up to micrometer can result in both high strength and increased ductility. As an example, a method called surface mechanical attrition treatment (SMAT) has been employed by many researches (e.g (Balusamy et al., 2013; Tao et al., 2003; Yin et al., 2016; Zhu et al., 2004)). To produce materials with heterogeneous

* Corresponding author. School of Mechanical and Materials Engineering, Washington State University, PO BOX 642920, Pullman, WA, USA. E-mail address: [email protected] (H. Lyu). http://dx.doi.org/10.1016/j.ijplas.2017.05.009 0749-6419/© 2017 Elsevier Ltd. All rights reserved.

H. Lyu et al. / International Journal of Plasticity 97 (2017) 46e63

47

Fig. 1. Strength vs. ductility plot.

microstructures. Other experimental studies showed that this process leads to producing a texture with gradient of grain size ranging from nano grain size close to the surface layer up to coarse grain size about a few micrometers around the middle of the layer. This grain size distribution results in a material having both high strength and high ductility along the tensile direction (Lu and Lu, 2004; Lu et al., 2000; Wu et al., 2002; Yang et al., 2016; Zhao et al., 2006). Furthermore, results obtained from tensile tests showed that when material with a microstructure composed largely of micro-grains is processed so that it also contains nano-regions composed of nano-grains strength and ductility can be improved due to the gradient microstructure (Grange, 1971; Hufnagel et al., 2002). These results suggest that it may be possible to design material with heterogeneous microstructure that may possess both optimum high strength and ductility, as illustrated in the hypothetical curve shown in Fig. 1 (dash line). Conventional continuum plasticity models cannot capture the effect of the grain size in heterogeneous microstructure. This is because these conventional models do not account for internal length scales and spatial gradient effects. In order to capture size effect, two gradient models, strain-gradient and stress-gradient models, which are based on different dislocation mechanisms and phenomena, have been proposed in the literature. Strain-gradient models account for hardening resulting from the formation of so-called geometrically necessary dislocations (GNDs) to accommodate lattice curvature during nonuniform deformation (Huang et al., 2004; Rhee et al., 1994; Shizawa and Zbib, 1999; Taylor et al., 2002; Wulfinghoff and € hlke, 2015; Zhu et al., 1997). The effect of the GNDs on flow stress and hardening can be incorporated directly in hardBo ening laws, or in the expression for the dislocation mean-free path, e.g. (Lyu et al., 2015; Ohashi, 2005). The counterpart to the strain-gradient plasticity theory is the stress-gradient plasticity, which is based on the mechanism of dislocation pile-ups against grain boundaries or obstacles under inhomogeneous state of stress (Chakravarthy and Curtin, 2011; Hirth, 2006; Taheri-Nassaj and Zbib, 2015). In a previous study it was shown that these two theories are complementary to each other (Liu et al., 2014) and can be employed in a combined strain/stress gradient model to capture the grain size effect over a wide range of length scale. Modeling the effect of the grain size in a heterogeneous microstructure requires the introduction of representative volume elements (RVEs) that account for the statistical nature of the microstructure. Currently, there are a few models that account for the statistical nature of the grain size, e.g. (Berbenni et al., 2007; Lehto et al., 2014; Quested and Greer, 2004). Some models discretize the domain into a continuous heterogeneous domain and combine strain-gradient or stress-gradient theory with a finite element method coupled with a Voronoi cell (VCFEM), e.g. (Kabiri and Vernerey, 2013; Vernerey and Kabiri, 2012). However, computational efficiency limits the number of grains that can be considered in the simulations, and this is seen as a problem for modeling heterogeneous microstructure with nano and micron size grain for which a large number of grains needs to be considered. Therefore, in the present work we develop and employ an efficient multi-scale model that combines a visco-plastic self-consistent model (VPSC) with a continuum dislocation dynamic model (CDD) and discrete dislocation dynamic (DDD) model (see flow chart in appendix A). By implementing both strain-gradient and stress-gradient theories in a 2D Voronoi tessellation diagram, with each Voroni cell representing an individual grain, we are able to capture the grain size effect in a heterogeneous microstructure. Although the framework and the theories developed in this work as described in the next section are applicable for any type of polycrystalline material, the focus in this paper is on a typical BCC metal-interstitial free (IF)-steel. There are numerous experimental and numerical studies that have been performed on this material and

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provide needed material parameters (Askari et al., 2014; Wei et al., 2004). Next, we investigate the strength and ductility of IF steel for various grain size and orientation spatial distributions. Heterogeneous microstructures that result in a combination of both high strength and high ductility are studied and compared to their equivalent homogeneous microstructures. 2. Methodology We employ a multi-scale framework that couples the VPSC model developed by Lebenson and Tome (Lebensohn and , 1993, 1994) and the continuum dislocation density (CDD) model developed in (Askari et al., 2014; Li et al., 2014). Tome Generally, self-consistent polycrystalline plasticity models are based on evaluation of the interaction between a single crystal grain, undergoing anisotropic plastic deformation, and a “ homogenized matrix” representing the rest of the material. The , 1993, 1994) and adopted in this work has the advantage of including the VPSC model developed in (Lebensohn and Tome effect of grain shape in the calculations by assuming each grain is an ellipsoidal inclusion surrounded by a homogeneous matrix. The model uses the anisotropic visoplastic response of the single crystal to solve for local deformation quantities and then homogenizes these local results to calculate the response of the overall polycrystalline material. This model can also predict of the texture evolution during deformation calculation of the contribution of different deformation mechanisms during the process. The response of each grain is derived from a rate dependent crystal plasticity formulation including slip and twinning mechanisms, and the activation of these systems is determined according to the Schmid's law (Lebensohn and , 1993). The complete formulation of the VPSC can be found in many papers, e.g.(Lebensohn and Tome , 1993; Zhang et al., Tome 2012). Here we provide a brief description of the basic equations. For a single grain, the plastic part of the velocity gradient tensor Lp is given by the following equation.

Lp ¼

N X g_ a sa 5na

(1)

a¼1

where N is the total number of slip systems, sa is a unit vector in the slip direction of slip system a, na is a unit vector normal to the slip plane, and g_ a is the plastic shearing on the slip system. Furthermore, the velocity gradient can be decomposed into plastic stretching tensor Dp and plastic spin tensor W p. i.e., Lp ¼ Dp þ W p , where

Dp ¼

N X g_ a ma

(2)

a¼1

Wp ¼

N X g_ a qa

(3)

a¼1

with ma and qa being the symmetric and anti-symmetric parts of the orientation tensor sa 5na, respectively. The next step in the VPSC method is a self-consistent homogenization scheme, which is used to calculate the macroscopic mechanical behavior and texture evolution of a polycrystalline material composed of many grains. The basic equations involved in the development of the self-consistent homogenization method are summarized in appendix B, and main derivations are given in , 1993, 1994). (Lebensohn and Tome Within a phenomenological approach, a constitutive equation is normally assumed for the plastic shear strain rate g_ a , appearing in Equations (2) and (3), which relates shearing to the resolved shear stress (e.g. a power law relation). However, here the plastic shearing of each grain is determined through a set of non-linear rate sensitive equations and by coupling a physically-based continuum dislocation dynamics model (CDD) with the VPSC model (Askari et al., 2014; Li et al., 2014). The CDD relates the plastic deformation of each grain to internal variables such as mobile and immobile dislocations densities and dislocation velocity. This is done by using the Orowan relation (Orowan, 1940) g_ a ¼ ram bvag , where b is the magnitude of the Burgers vector, ram is the mobile dislocation density on slip system a, and vag is the average dislocation glide velocity. Here, the average dislocation glide velocity is calculated using the following formula.

    1=m vag ¼ v0 ta ta  signðta Þ; cr

ta ¼ s : ma

(4)

Here, it is assumed that vag is directly related to the ratio of the resolved shear stress ta to the critical resolved shear stress along the slip system a, tacr . In the above equation, s is the Cauchy stress tensor, v0 is the reference velocity, and m is the strain rate sensitivity exponent. The two parameters, v0 and, are numerical constants that are obtained from experimental data. Here v0 is set in the order of 105 (Askari et al., 2014) and m is set as 0.05 (Wei et al., 2004). As discussed in (Askari et al., 2014; Li et al., 2014), the critical shear stress is a combination of three contributions, i.e.

H. Lyu et al. / International Journal of Plasticity 97 (2017) 46e63

tacr ¼ ta0 þ taH þ taS

49

(5)

where ta0 is the reference shear stress which is the lattice friction stress, taH is the forest dislocation hardening term which accounts for interactions between intersecting dislocations, and taS is a size-dependent term which is obtained from the stress-gradient theory (Taheri-Nassaj and Zbib, 2015). The dislocation hardening term taH in Equation (5) is determined by the Bailey-Hirsch relation (Bailey and Hirsch, 1960),

taH ¼ a* bm

N qffiffiffiffiffiffiffi X Uas rs

(6)

TS

s¼1

where m is the elastic shear modulus, a* is a numerical factor in the order of 0.1 (Lyu et al., 2015) and Uas is a dislocation interaction matrix that describes the interaction between slip system s and slip system a. Since we assume isotropic hardening, all the elements of this matrix are set to unity (Ohashi et al., 2007). The term rsTS in Equation (6) is the total statistically stored dislocation density (SDD) on slip system s. This term is further decomposed into two terms: rsTS ¼ rsM þ rsI where the subscripts M and I stand for mobile dislocation density and immobile dislocation density, respectively. The evolution laws for the mobile and immobile dislocations are given by the following equations (Li et al., 2014):

.

.

.

   a a a r_ aM ¼ a1 raM vag ~lg  a2 2Rc raM raM vag  a3 raM vag ~lg þ a4 jta j tacr r raI vag ~lg þ a5

N X

b¼1

.

.

   a a r_ aI ¼ a3 raM vag ~lg  a4 jta j tacr r raI vag ~lg  a6 Rc raM raI vag

.a P ba rbM vag ~lg  a6 Rc raM raI vag

(7)

(8)

In the evolution equations above, there are six terms each having a parameter that reflects the dislocation interactions during plastic deformations (Askari et al., 2014; Li et al., 2014; Lyu et al., 2015). For instance, the parameter a1 can be found from the mechanisms of multiplication and growth of resident dislocations as well as the production of new dislocations from Frank-Reed sources in the slip system a. The parameter a2 is related to the mutual annihilation of two mobile edge or screw dislocations with opposite signs in the slip system a, while a3 contains transition from the mobile type to the immobile type dislocations. The parameter a4 is a parameter that captures the mobilization of immobile dislocations due to the breakup of junctions, dipoles, pinning parts, etc., at critical stress conditions, and a5 is relevant to cross-slip that is when screw dislocation segments on one slip plane move to another glide plane during plastic deformation. Finally, the a6 parameter represents annihilation between mobile and immobile dislocations. The values for these parameters for the material system a considered in this study have been evaluated in a previous study (Li et al., 2014). Furthermore, in Equations (7) and (8), ~lg is the mean free path of mobile dislocations in a plane, Rc is the critical radius for interaction between to dislocations which is set as fifteen times the Burgers vector; and P ba is a matrix obtained from Monte-Carlo analysis (Li et al., 2014) that describes the probability for a screw dislocation to cross-slip from b-slip plane to a-slip plane and r is a numerical constant set to 0.5. 2.1. Combined strain-gradient and stress-gradient model Models based on stress-gradient and strain-gradient theories can predict size effects unlike the conventional models of crystal plasticity. As discussed above, these two theories attempt to predict size effects that arise from two different dislocation mechanisms. In most strain-gradient theories (Fleck and Hutchinson, 1993; Gao and Huang, 2003; Huang et al., 2004), the so-called geometrically necessary dislocations (GND) are treated as the main origin of size effect. Generally, the GNDs are related to the spatial gradients of the plastic deformation and their effect can be directly incorporated into hardening laws. Here, within the dislocation model approach, the GNDs are considered as obstacles to dislocation motion and thus are a incorporated in the dislocation evolution equations (7) and (8) through the mean-free path ~lg [20, 21]:

c* ~la ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g ffi P ja  a w rT þ raGND

(9)

where c* is a constant, here we set it equal to unity, wja is a general weight matrix which can be simplified as the identity matrix (Ohashi, 2005), and raGND denotes the density norm of the GNDs on slip system a and expressed as

raGND ¼

1 pffiffiffiffiffiffiffiffiffiffiffi aij aij b

(10)

where aij is the Nye's tensor (Nye, 1953). For large deformations, the rate of the Nye's tensor is given by Shizawa and Zbib (1999) as

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a_ ¼ curl ðLp Þ

(11)

Size effect in the stress-gradient plasticity theory originates from the problem of dislocation pile-ups against grain boundaries and or obstacles under inhomogeneous stress state (Akarapu and Hirth, 2013; Chakravarthy and Curtin, 2011). Thus, the average obstacle spacing as well as grain size both emerge as material length scales that affect hardening. TaheriNassaj and Zbib (2015) (Taheri-Nassaj and Zbib, 2015) developed a higher order stress-gradient plasticity theory for general inhomogeneous state of stress. Here we use the linear version of the theory for the size-dependent stress-gradient hardening term taS .



K

taS ¼ pffiffiffi 1 þ L

0

L jVtj 4t

 (12)

where K is the Hall-Petch constant, t is the effective stress and Vt is the spatial gradient of the effective shear stress, L denotes 0 the grain size and L is defined as the average length of dislocation obstacles spacing. Here, we assume that the only obstacles 0 to the dislocation motion are the grain boundaries and thus L is also equal to the grain size, more details can be found in (Lyu et al., 2016). 2.2. Implementation of strain/stress gradient model The original development of the VPSC model is dimensionless with no need to specify spatial positions of grains and domain size. However, in order to capture the strain and stress spatial distributions, a spatial representative domain representing the polycrystalline material is defined and appropriately discretized to represent grain size and distribution. In order to investigate the stochastic nature of the microstructure, a controlled Poisson Voronoi tessellation (CPVT) model is applied to generate a two-dimensional virtual grain structure. As discussed in (Zhang et al., 2011, 2012), in this model a parameter d is defined as the minimum distance among the neighbor grains as

d ¼ adreg

(13)

where a 2 [0,1] represents a “regularity” parameter used to represent proper grain size distribution and dreg is the distance between any two adjacent seeds, namely

 2A0 1=2 pffiffiffi N 3

 dreg ¼

(14)

with A0 being the domain area and N the number of hexagons in the domain. If the grain size values are normalized by the mean grain size, then a one-parameter gamma distribution function can also provide proper fitting for grain size distribution in terms of area as below:

Px;xþdx ¼

cc c1 cx x e dx GðcÞ

Z∞

GðcÞ ¼

x>0

xc1 ecx dx

(15)

(16)

0

where P is grain size distribution function, GðcÞ is the gamma function and the parameter c (c > 1) is the only distribution variable in the above functions (CAO et al., 2009). The relation between the parameter c and regularity a is given by

aðcÞ ¼ AðzðcÞ  z0 ÞkþnzðcÞ ; c0  c

(17)

where zðcÞ ¼ c=cm , z0 ¼ c0 =cm , c0 ¼ 3:555 cm ¼ 47:524, A ¼ 0:738895, k ¼ 0:323911 and n ¼ 0:4114367. In this work we generate the Voronoi tessellation by controlling the regularity parameter a which changes between 0 and 1. The details for determining the distribution parameter c can be found in (Zhang et al., 2011). In the future, we will consider how the parameter c affects the generation of the Voronoi space. In a two-dimensional domain, the space can be discretized using the CVPT model described above, resulting into a triangular mesh with each element in the mesh representing a grain. Then each grain is assigned a crystallographic orientation randomly, and its neighbors can be determined according to its spatial locations in the mesh. The stress state, strain and dislocation densities in each element can be computed using the coupled CDD-VPSC model. Then, the spatial gradient of the stress and strain fields can be evaluated numerically. Here we use a moving least square method to find approximate values of stress and strain gradients (Armentano and Dur an, 2001; Liu et al., 1997). We assume a strain or stress (local) function (

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51

4ðiÞ ðcÞ) at each node of the two dimensional domain and approximate it by another set of functions pðiÞ (c), where the superscript refers to grain i. The state of strain or stress at each individual grain i is assigned to an integration node located at the center of the grain ci and is defined as pðiÞ ¼ pðiÞ ðci Þ. Then the stress or strain function ð4ðiÞ ðcÞÞ may be approximated by

4ðiÞ ðcÞ ¼

m X

ðiÞ

ðiÞ

pj ðcÞaj ðcÞ ¼ pðiÞT ðcÞaðiÞ ðcÞ

(18)

j¼1

where c denotes the coordinates of the node, 4ðiÞ ðcÞ represents the stress or strain function at grain i, pðiÞT ðcÞ is a monomial basis function determined by the surrounding neighbors of grain i, m is the size of the basis functions, aðiÞ ðcÞ is a matrix that contains unknown parameters for grain i, and the symbol T denotes the transpose operator. Here, for grains with less than five neighbors such as those located at the boundary of the Voronoi diagram, a linear basis function is used, i.e.

pðiÞT ðcÞ ¼ ð1; x; yÞ

(19)

For computational efficiency, a quadratic basis function for grains located in the center of domain is defined as

pðiÞT ðcÞ ¼ 1; x; y; x2 ; xy; y2

(20)

Next, and in order to find the parameters a’s appearing in Equation (18), we define a function J at grain i as

J

ðiÞ

¼

n X

" wðdl Þ

l¼1

m X

#2 ðiÞ ðiÞ ðiÞ pk ðcl Þ$ak ðcÞ  4k ðcl Þ

(21)

k¼1

The derivative of this function with respect to the coefficients a’s leads to finding these coefficients with minimized error, leading to n X l¼1

ðiÞ al ðcÞ

¼

n X

!1

ðiÞ ðiÞT wðdl Þpl ðcl Þpl ðcl Þ

l¼1

n X

ðiÞT

wðdl Þpl

ðiÞ

ðcl Þ4k ðcl Þ

(22)

l¼1

Using the method described above, both the stress field and strain field at grain i can be constructed as functions of the grain coordinates. Then, the stress and strain gradient for each grain can be computed. In this work, we didn't apply any boundary condition for grains located at the domain borders. As those grains own less neighbors to be used for approximation, this may lead to less accurate results. This will be addressed in the future by applying other boundary conditions. The CPVT method can also be extended to three spatial dimensions, with increased computational effort. It is necessary to mention that, In this work CPVT doesn't have anything to do with strain and stress calculation. We have applied standard VPSC to calculate the amount of stress and strain. CPVT is applied to dimensionalize grains in microstructure since VPSC is dimensionless. So this is the main reason in the first place we have CPVT in our work. Having the spatial location of each grain provides the ability to calculate the gradient of stress and strain in our work which is explained in the main body of manuscript by applying stress gradient and strain gradient theory. 3. Results and discussion We analyze the mechanical behavior and ductility of an IF-steel with banded heterogeneous nano-microstructure. IF-steel is an interstitial free body center cubic (BCC) ferrite steel. Here, we consider all possible 48-slip systems existing in the BCC lattice. Each slip system can be activated depending on the state of stress. A series of simulations is conducted under the following assumptions: i) the initial dislocation density of 1012 m2 for mobile and immobile dislocations are assumed to be the same for all slip modes, ii) there are no initial GNDs at the beginning of the deformation, and iii) the initial texture is a random texture as shown in Fig. 2. In this study, the ductility is defined as the strain at the onset of necking instability. In uniaxial loading conditions, the onset of necking instability is when the true stress s satisfies s ¼ ds=dε, which also corresponds to the peak stress in the engineering stress vs. strain curve. All parameters used for the simulations are summarized in Table 1. Two series of nano microstructures are assumed and investigated. Each series consists of five different gradient microstructures with grain size ranging from 100 nm to 10 mm. In order to design the layered microstructures like cases I to X each layer in these structures are produced separately and eventually all the layers were combined to have a complete microstructure. For example, case I has several layers with different grain size. The range of grain size in layers of this microstructure is from 100 nm up to 10 mm, and these range is the same for all the produced structures in this study. So each of these layers are produced with CPVT separately on its proper geometry and position and eventually by combining all these layers a microstructure like case I is produced. It is worth nothing that the position and geometry of these layers varies in different

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Fig. 2. Pole figure of initial texture.

Table 1 Simulation parameters. Symbol

Ferrite (Unit)

Reference

a* (Bailey-Hirsh hardening coefficient) t0 (internal friction) on [112]

0.1 11 MPa 2.5 2 242 GPa 150 GPa 112 GPa 80 GPa 2.76 MPa/mm1/2 4  105 m/s 0.05 2.54 Å 15 b 0.02 1.0 0.002 0.002 0.018 1.0 0.1

(Luo et al., 2004; Lyu et al., 2015) (Li et al., 2014; Lyu et al., 2015)

f1 (Anisotropic factor for slip resistance between [112] and [110] f2 (Anisotropic factor for slip resistance between [110] and [123] C11 (Anisotropic elasticity constant) C12 (Anisotropic elasticity constant) C44 (Anisotropic elasticity constant) m (Shear modulus) K (Hall-Petch constant) v0 (Reference strain rate) m (Strain rate sensitivity) b (Magnitude of burger vector) Rc (Critical radius for annihilation coefficient)

a1 a2 a3 a4 a5 a6 Uij (i ¼ 1,48; j ¼ 1,48) (Interaction matrix)

(Kim and Johnson, 2007)

(Calcagnotto et al., 2011) (Li et al., 2014)

(Queyreau et al., 2009; Terentyev et al., 2008)

microstructures. So cases in group I which are cases I to V, the gradient layer starts from the corners in the microstructure so it produces inclusion like microstructures. To be more clear it should be emphasis that those inclusions like shapes in the corners of cases I to V are layers with grain sizes in the nanometer range. As mentioned above, these gradient materials can be produced by a method called surface mechanical attrition treatment (SMAT (Tao et al., 2003)). The first series (Group I) are microstructures composed of band-like both regions and localized inclusions located at two corners as shown in Fig. 3 Cases IV. These regions contain grains that are much smaller than the grains in the other parts of the cell. Mechanisms leading to higher strength can be attributed, among other things, to precipitate or inclusion hardening (Ardell, 1985). The second series (Group II) are microstructures without inclusions as shown in Fig. 3 Cases VI-X. We focus on generating microstructures with grain-size gradient along the diagonal direction. Since the movement of dislocations in polycrystalline materials cause plastic deformation (Taylor, 1934), obstacles to dislocation motion results in increased strength. This process is also known to increase friction stress in the Hall-Petch relation (Rashid, 1980). Because we assumed free boundary condition in the simulations, the corner nano layers can be considered as inclusions in large scale samples. 3.1. Effect of gradient microstructure on the stress-strain response The mechanical response and ductility of the 10 different cases were studied by conducting tensile test under a constant strain rate of 1  103 s1. The stress-strain responses obtained from the simulations of IF-steel with different gradient microstructures for the Cases I, III, V, VI and VIII are shown in Fig. 4 (solid lines); similar curves were also obtained for the other cases but not shown here for brevity. Also shown in Fig. 4 simulation results for cases corresponding to equivalent homogenous microstructure, i.e. micro~ in a 2D domain can be calculated using the relation structures with homogenous grain size. The average grain size d ~ ¼ P d =N, where d is the effective grain size, and N is the number of grain in this domain. Here, we calculated d ~ for d N N N different microstructures and built equivalent homogeneous microstructures with the same average grain size. This is summarized in Table 2 while the corresponding stress-strain curves are represented with dashed lines in Fig. 4. For the sake of brevity, we only plotted three engineering stress-strain curves representing these homogenous structures (Fig. 4). The curves show that for a homogeneous microstructure, the strength increases with decreasing grain size and

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53

follows the Hall-Petch relation. However, for gradient microstructure, the strength does not correlate with the average grain size, as will be shown below. The strength value obtained for Case I with larger equivalent grain size is higher than that of Case

Fig. 3. Different microstructures gradients: Group I: Case I  CaseV; Group II: Case VI  Case X.

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H. Lyu et al. / International Journal of Plasticity 97 (2017) 46e63

Fig. 3. (Continued)

Fig. 4. Stress-strain response of heterogeneous microstructures (solid lines) and equivalent homogeneous microstructures (dash dot lines).

III and Case V with much smaller grain size. We also find that Case VI shows a weak strength compared to the others cases, which means the position of the nano structure can also impact the mechanical response. Thus, additional microstructural parameters such as the grain size distribution should also be considered when studying the material strength. Fig. 5 shows

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55

Table 2 Homogeneous and heterogeneous microstructure cases. Homogeneous case A B C D E F G H I

Heterogeneous case

Average grain size d (nm)

Case Case Case Case Case Case Case Case Case Case

200 300 400 545 591 594 718 821 1200 769 714 626 619

V III IV II I VI VII VIII IX X

Fig. 5. Grain size distribution for Case V and Case VI (solid lines) and equivalent homogeneous microstructure for Case V and Case VI (dash dot lines).

that the gradient microstructure has a different grain size distribution than the one obtained for the homogeneous microstructure and has a bimodal distribution instead of unimodal distribution (as in the case of a homogeneous microstructure). This indicates the possibility that microstructures with the same mean grain size but different grain size distribution would result in a different mechanical behavior. As can be seen from Fig. 5, Case V has a higher peak around hundreds of nano meters than that of Case VI, which may explain the higher strength.

3.2. Effect of gradient microstructure on the strength-ductility map In order to compare strength and ductility for different microstructures, we normalized the strength (strength/shear modulus) and the ductility (ductility/strain rate sensitivity) curves for homogenous and gradient microstructures. The results are shown in Fig. 6. For the homogenous microstructure cases, the strength-ductility curve has the expected “banana” shape as shown in the figure. For the gradient microstructure the values for strength and ductility are both higher when compared to the homogenous cases as can be deduced from the figure. A closer look at the results in the figure show that the microstructure with inclusions (Group I) have higher strength than the materials with homogeneous microstructures or microstructures as those in Group II, the density of nano regions and the location at which they are inserted have a strong effect on the ductility. This can be seen from Fig. 6: ductility increases when the number of nano regions increases and when the nano regions are inserted in the center (Case VI), the average grain size of the structure decreases and the ductility increases. This is in contrast with the accepted thought that increasing the grain size will increase ductility. When more nano regions are inserted into the center, grain size is in the micron scale and a higher value of ductility is achieved. This is noteworthy, inserting more nano layers into the structure reduces the average grain size and ductility increases. This can be attributed to a transfer of strain from the center to the nano regions, which have been inserted at different locations. The nano regions that have been inserted behave like barriers and resist the movement of the dislocations. This reduces the dislocation density in

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Fig. 6. Strength vs. ductility curve for materials with homogeneous microstructure and gradient microstructure.

the entire system and postpones the instability that causes necking. Thus, by inserting a certain number of gradient layers into the microstructure, a better combination of ductility and strength in the material can be achieved. Material with nano layers in the corner region (such as in Group I) result in a better combination of strength and ductility. A similar behavior has been reported in the recent literature (Young, 2015; Young et al., 2015) for structures produced by high strain rate rolling (HSRR). During HSRR, refined grains originate from coarse grains through the formation of a high density of deformation twins and dynamic recrystallization (DRX) taking place at these sites. Thus, those preferential twin boundaries with refined gradient grains resulting from DRX produce gradient layers inside the microstructure and according to Young (2015). These structures have a higher tolerance for stress and strain. The predicted strength vs. ductility shown in Fig. 6 is comparable to that obtained experimentally for microstructures (Wu et al., 2002) possessing nano structures in the boundary (such as sandwich-like structure). Thus, it is possible to achieve an optimum strength and ductility by inserting a sufficient number of gradient layers into the microstructure. But to fully understand how the microstructure parameters affect the strength/ductility and optimize the trend, more studies on the effect of distribution/density and number of the nano layers are needed. The effect of layers with nano-grains on ductility can be further explained by examining the stress and strain fields shown for two cases in Fig. 7. Fig. 7a shows contour plots of the effective stress and effective strain distributions corresponding to Case I with coarse grains, while Fig. 7b corresponds to Case V which has more inserted nano regions. In the former case, the localization of strain is confined in the center layer, while for the latter the strain is distributed into several regions between the nano layers (see figures displayed on the right). This strain localization can be attributed to dislocation pile-up at the grain boundary which is captured by the stress-gradient theory. In addition, GNDs form within the grain in order to accommodate the lattice curvature due to the inhomogeneous deformation of grains with different grain size and orientations. These dislocations act as obstacles and prevent the motion of mobile dislocation. Therefore, it can be argued that the amount of deformation gradient increases with increasing the number (density) of the nano layer regions, and thereby more GNDs appears and the density of mobile dislocations increases. Shown in Fig. 8 plots of dislocation density vs. strain for Case I and Case V. It can be readily seen from the figure that Case V has a much higher mobile dislocation density than for Case I, which also explains why Case I has a much higher strength than the other cases. In the same spirit, inserting nano regions into the material (Case II and Case III) increase the ductility with decreasing the strength. 3.3. Gradients and size effect In this section we elucidate on the origin of size effect that influences strength and ductility. Plotted in Fig. 9a and b the simulation results for strength versus average grain size and ductility versus average grain size, respectively, for the cases listed in Table 2. As can be seen from the figures, for the homogenous cases, the results for both strength ðsÞ and ductility (D) scale with the average grain size d, and follow a power law relationship, while the results for the heterogeneous cases are scattered. The prediction of size effect arises from the strain/stress gradient theory that combines the effect of both strain gradients (Equations (9) and (11)) and stress gradients (Equation (12)), which introduce lengths scales that are based on specific dislocation mechanisms ((Lyu et al., 2015, 2016). Namely, one length scale arises from the formation of geometrically

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57

necessary dislocations (GNDs) accommodated by strain gradients, and enters directly in the expression for the dislocation mean free path which in turn affect the evolution of mobile and immobile dislocation densities. The second length scale (grain

Fig. 7. Contour plots of (a) effective strain and (b) effective stress (c) GND density (d) stress gradient for Case I and e) effective strain and (f) effective stress (g) GND density (h) stress gradient for Case V at strain 0.1.

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Fig. 7. (Continued)

Fig. 8. Dislocation density vs. strain for case I and case V.

size) arises from dislocation pile-ups at grain boundaries that accommodate local stress and stress gradients. For the homogenous cases, since the microstructure is homogenous and for a homogenous remote stress field, local perturbations in the strain and stress fields may not be significant and thus the stress and strain gradients are not as severe as for the heterogeneous cases as can be deduced from Fig. 7. These gradients, in turn, contribute to strength as predicted by the stress/strain gradient model (Equations (6)e(9)), and consequently to ductility. The results in Fig. 9a and b shows that the predictions of strength s and ductility D for the heterogeneous cases differ significantly from their equivalent homogenous cases, and the differences between them may depend on the degree of spatial gradients of the microstructure. Therefore, we decompose each of these two properties, into a homogenous part and a gradient part, i.e.

s ¼ sh þ Dsg ; D ¼ Dh þ DDg

(23)

where sh and Dh are the homogenous parts and Dsg and DDg are the gradient parts. The homogenous parts can be fitted to n m power laws, i.e. sh ; f d ; Dh f d , where it turns out that n ¼ 0.54 for strength, m ¼ 0.88 for ductility. It is noted that although this power law relationship for the homogenous microstructures is a phenomenological equation that relates strength and ductility to a structural parameter (average grain size), it is based on data predicted by physically-based models (stress-gradient and strain-gradient plasticity) for basic dislocation mechanisms: dislocation pileups and GNDs.

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Fig. 9. (a) Strength vs. grain size (b) Ductility vs. grain size.

Fig. 10. Grain size spatial distribution along diagonal direction.

For the cases of heterogeneous microstructures, one can argue that a corresponding phenomenological equation should depend not only on the average grain size but also on the spatial gradient of grain size distribution: ‘grain size gradient’. Typical grain size spatial distributions along the diagonal direction for Cases I, II and III are shown in Fig. 10. As discussed in the section above, materials with a gradient microstructure (inclusion-like) present at the corners will have higher strength. Adding, nano-layers into the microstructure results in a change of strength and ductility in the material depending where the nano-structures are inserted. Thus, it would be useful to define a parameter that measures the grain-size gradient and correlate it to the strength and ductility of the material. We define a parameter g that can be used to measure the “average” grain-size gradient in the microstructure, defined as

g¼

1 2A0

  Z    N  1 X  e  ~ Vd  dAy Vdi dA   2A0 i  

(24)

~ is the first gradient of the spatial distribution of the grain size. For a given where A0 is the total area of the domain, and Vd i ~ represents the first gradient of grain “i”, the parameter “g” can be approximated numerically using equation (24), where Vd i grain size calculated at grain i. For each of the heterogeneous cases, we evaluate the average grain size d, grain-size gradient g,

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and its corresponding gradient strength and gradient ductility parts Dsg and DDg , respectively (see Fig. 9). The results are mapped in Fig. 11a and b. The maps in Fig. 11a and b, show that both Dsg and DDg are general functions of both average grain size and grain-size gradient, i.e.







Dsg ¼ F d;g ; DDg ¼ H d; g



(25)

By definition for homogenous microstructures g ¼ 0, Dsg ¼ 0 and DDg ¼ 0, and therefore, Fðd; 0Þ ¼ 0 and Hðd; 0Þ ¼ 0. As a first approximation, we assume that both F and H are linear in g such that





Dsg ¼ g f d ; DDg ¼ g h d

(26)

Using this assumption and the results for the heterogonous cases, we plot in Fig. 12a and b f and h versus grain size d. Although the data in the figures are scattered, and within the limits of the given data, there is a general trend of increasing f and h with increasing g. The data can be fitted to power law relations as follow:

Fig. 11. Effect of gradient of grain size g on (a) strength and (b) ductility.

Fig. 12. Effect of grain size and gradient of grain size on (a) strength and (b) ductility, assuming linear dependence on g.

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a

b

Dsg ¼ a g d ; DDg ¼ b g d

61

(27)

Combining equation (27) with (23) and the results for the homogenous parts, we obtain the following power law relations for the strength and ductility for their dependence on grain size and grain-size gradient. n

a

m

b

s ¼ Kd þ agd ; D ¼ kd þ bgd

(28)

where n¼-0.54, m¼0.88, a ¼ 0:24 and b ¼ 2:03. It should be emphasized that although these equations are phenomenological they are based on results obtained from a physically-based model. These simple power law equations relate mechanical properties of gradient materials to two structural parameters, grain-size and gradient of grain size, in contrast to Hall-Petch power law for homogenous materials where only grain size appears in the equation. 4. Conclusions In this work, we developed a physically-based multi-scale modeling framework to investigate materials with gradient microstructures in the form of nano-microstructures. Stress/strain gradient models based on dislocation mechanisms were employed to capture size and gradient effects. A series of simulations were conducted to study the effect of various microstructural gradients. The results obtained for various microstructures were analyzed and compared to experimental data found in the literature. Conclusions that can be drawn from this study include the following:  Compared to materials with homogeneous microstructure, a combination of high strength and high ductility can be attained simultaneously in materials with gradient microstructure.  Materials with banded nano-microstructure can achieve higher ductility by delaying strain localization.  A microstructural parameter g is proposed to measure the grain size gradient. It is suggested that the mechanical properties of gradient materials can be described by phenomenological relations that include two structural parameters, grain size and grain-size gradient, in contrast to Hall-Petch relation for homogenous materials where only grains size appears in the equation.

Acknowledgement This work was supported by Qatar National Research Fund (a member of Qatar Foundation) under Grant No. 05-1294-2559. The statements made herein are solely the responsibility of the authors. Appendix A The flow chart of this multi-scale model is shown as below.

Fig. A1. Flow Chart of Multi-scale model.

Appendix B The combination of equations (2)e(4) in the manuscript yields

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DðxÞ ¼

 a a 1 N1 N X X m s ðxÞm  signðta Þ g_ a ma ¼ bv0 ram ma   a a

a

(A.1)

tcr

, 1993), for a polycrystal For a material point x within a given grain. As discussed in (Eshelby, 1957; Lebensohn and Tome aggregates with the mentioned viscoplastic local constitutive equation, to linearize the equation of deformation rate A.1 in a , 1993) the given grain (s), an affine linearization method is applied. As shown in (Eshelby, 1957; Lebensohn and Tome deviatoric strain-rate can be written as:

DðxÞ ¼ M ðsÞ : sa ðxÞ þ DoðsÞ

(A.2)

where M ðsÞ is a fourth-order compliance tensor and DoðsÞ is a back-extrapolated term. The same logic is assumed for the effective medium in polycrystals:

DðxÞ ¼ M

ðsÞ

: sa ðxÞ þ D

oðsÞ

(A.3) ðsÞ

oðsÞ

In which D and sa are the macroscopic magnitudes and the M and D are macroscopic compliance and back extrapolated term which are unknown and need to be adjusted self-consistently by applying the idea of the equivalent in, 1994). This process begins by using an initial Taylor guess for calculating the initial values of the clusion (Lebensohn and Tome local deviatoric stress, compliance tensor and back-extrapolated for each grain. Hence, the general macroscopic constitutive law can be calculated, while the Eshelby tensor, which includes the deviation effect from the grain shape, can also be obtained. By applying these tensors, a new estimates of macroscopic compliance tensor and back-extrapolated term can be reached until the convergence on the macroscopic moduli are achieved, then the next deformation step starts. More details , 1993, 1994). can be found in the literature (Lebensohn and Tome References Akarapu, S., Hirth, J., 2013. Dislocation pile-ups in stress gradients revisited. 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