Microstructure Design of Multiphase Compositionally Complex Alloys I: Effects of Strength Contrast and Strain Hardening

Abstract

This work addresses the question, “Which microstructure yields the highest “toughness” for a two phase, compositionally complex alloy (CCA) composed of a ductile, face centered cubic (FCC) matrix phase and a stronger, intermetallic reinforcing phase?” A polycrystal plasticity model is used, which takes as input the phase fraction, grain sizes, and parameters describing the mechanical properties (stiffness, strength and strain hardening behavior) of each phase. In addition to assuming an elasto-viscoplastic matrix, the reinforcing phase is also assumed to be elasto-viscoplastic, albeit with a higher strength and lower strain hardening rate. The results reveal that, if the reinforcement remains elastic, the addition of such a reinforcement leads to an increase in strain hardening rate, in addition to the obvious increase in yield strength. If the reinforcement yields, the hardening rate sharply decreases, leading to earlier tensile instability. Therefore, when the strength contrast between the phases is high, addition of reinforcement is beneficial to toughness as well as strength. For lower contrast cases, the optimal toughness case may be the unreinforced matrix material. More specifically, only if the FCC matrix exhibits sustained linear hardening-type behavior typical of TWIP or TRIP effects (which correlate with very low stacking fault energies achievable in CCAs), will it exhibit an increase in toughness, as well as an increase in strength. Finally, the hard reinforcement is under a lower stress triaxiality due to pressure imposed by the ductile matrix, and this causes it to yield at a lower uniaxial stress than it would in isolation. The findings of this work provide a mechanistic understanding of the dependence of toughness on the microstructure of two-phase aggregates, and the approach is equally applicable to multiphase alloys.

Introduction

Recent compositionally complex alloy (CCA) developments have highlighted a demand for two-phase/multiphase alloys with higher specific strength (σ/ρ). If the reinforcing phases are low in density, they can lower the overall alloy density, while simultaneously achieving a balance of high strength and ductility. Several such alloys have been developed recently [1,2,3,4,5,6], in which high strength is achieved via load sharing among embedded phases (composite strengthening) [7, 8] as opposed to precipitate strengthening typical of age hardenable alloys. Whereas precipitation strengthening can have a profound effect on strength, it can also give rise to solute depleted regions which are detrimental to corrosion [6]. The present work seeks to establish alloy/microstructure design principles which would guide the creation of high-performance dual-phase alloys.

For improved ductility, a key ingredient is the resistance to localized deformation, i.e., necking, which is typically achieved via high strain hardening rate, and may be assessed by the material’s uniform elongation in tension. One can furthermore define a metric “toughness” as the product of ultimate tensile stress ${\sigma }_{u}$ and the uniform elongation ${\varepsilon }_{u}$, as obtained from the Considére criterion, $\frac{d\sigma }{d\varepsilon }=\sigma$. The present work seeks to establish the connection between the microstructure and the toughness, such that an optimum microstructure can be designed for two-phase CCAs.

The plasticity of two-phase alloys, where both phases are elasto-(visco)plastic, and their intrinsic constitutive responses are radically different, raises interesting challenges. When embedded within the aggregate, the mechanical response of the phases is not independent, rather the response of one phase depends on the response of the other. The aggregate level constitutive response is thus determined by the interactions between the phases [7, 9]. While it is well known that a higher strain hardening rate leads to a higher uniform elongation, the hardening rate of an aggregate containing multiple phases is unknown a priori. In this work, the constitutive response of the aggregate is modeled for a variety of individual phase responses. The present work aims to answer the following question: What microstructure yields the highest toughness? While there have been several studies focusing on structure–property relationships in two-phase materials, almost all have focused on a particular alloy e.g., α + β Ti alloys, α/β brasses [8], ferrite–martensitic dual-phase (DP) steels [10,11,12], ferrite-austenitic duplex steels [13], ferrite-bainitic steels [14], or metal matrix composites e.g., [15, 16]. The present modeling approach is most closely related to that of Delince et al. [12] where the authors employed a physics-based constitutive model to explain the experimentally observed optimum volume fraction of martensite and other microstructural features within DP steels. The present work aims to address general concepts, such that new alloys which benefit from compositional complexity can be designed based on the relationships developed.

For demonstration purposes, wherever quantitative values are used, they are chosen to represent FCC–CCA matrices resembling traditional austenitic stainless steels, Cantor-type alloys, or Fe–Mn and Fe–Mn–Al alloys. The reinforcing phase is modeled after a ferritic-type B2/L2₁-based low-density intermetallic. The purpose of this paper is to identify the microstructures and conditions under which a certain volume fraction of reinforcing phase is favorable for not only increasing strength, but also improving the toughness. Particular attention is devoted to the role of the matrix strain hardening behavior.

A connection between stacking fault energy (SFE) and the strain hardening behavior of FCC alloys is now established. For example, in the case of Fe–Mn and related alloys, it has been shown e.g., [17] that SFE $\ge$ 35 mJ/m² gives rise to “normal” dislocation-based parabolic hardening behavior; between 18 and 35 mJ/m² gives rise to twinning-induced plasticity (TWIP) and a linear (sustained) strain hardening behavior; and finally, even lower SFE values give rise to transformation-induced plasticity (TRIP) (e.g., FCC → HCP and sometimes subsequent HCP → BCC/BCT martensitic phase transformation), which also gives rise to a linear strain hardening response. A similar trend is also observed in FCC–CCAs [18,19,20,21,22]. In fact, recent explorations have highlighted the exciting potential of combined TWIP/TRIP CCAs which have very high toughness values due to an excellent combination of strength and strain hardening capacity [18, 19]. Although TWIP/TRIP effects are not explicitly modeled in this present work, the phenomenological effects of a sustained, linear strain hardening response, which these mechanisms induce, are explicitly examined and compared with the more normal parabolic hardening stemming from dislocation–dislocation interactions. It will be shown that in order for dual-phase reinforcement of the FCC–CCA to produce the desired effect (simultaneous improvement of strength and toughness) sustained, linear-type strain hardening behavior of the matrix is essential.

Modeling Framework

One could potentially account for the effect of microstructural variables within the J2 deformation theory framework [23, 24] to model the flow behavior and perform most of the analysis presented in the forthcoming sections. However, a polycrystal plasticity model is employed which has the following advantages. First, polycrystal plasticity models directly account for the elastic and plastic anisotropy, and the intergranular interactions between the individual grains and the surrounding polycrystalline aggregate are accounted for naturally. Second, besides the stress–strain response, the internal elastic (lattice) strains and texture evolution can be used for rigorous model validation and robust parametrization [25,26,27]. Third, the constitutive response under non-proportional loadings and strain path changes are naturally obtained via such models, additional parametrization is not required [e.g., 28]. The model presented in this work takes as input a set of microstructure variables e.g., dislocation density, grain sizes, phase morphology, phase fraction, initial crystallographic texture, and material parameters pertaining to the strength and strain hardening response of each phase. In what follows, the single-crystal constitutive model of the two phases is described, followed by a description of the polycrystal homogenization method.

Single-Crystal Model for the FCC–CCA Matrix

An FCC–CCA matrix is considered and is modeled according to the Mechanical Threshold Stress (MTS) model with a dislocation density-based strain hardening rule [29,30,31,32]. According to the MTS model [30, 32, 33], at 0 K, the resistance (i.e., the mechanical threshold) of slip system α, ${\widehat{\tau }}_{0}^{\alpha }$, is given by

$$\hat{\tau }_{0}^{\alpha } = \hat{\tau }_{a}^{\alpha } + \hat{\tau }_{t}^{\alpha ,i} + \hat{\tau }_{t}^{\alpha ,\varepsilon } = \hat{\tau }_{a}^{\alpha } + \mathop \sum \limits_{\kappa } \hat{\tau }_{t}^{\alpha ,\kappa }$$

(1)

where ${\widehat{\tau }}_{a}^{\alpha }$ is the athermal component, and ${\widehat{\tau }}_{t}^{\alpha ,i}$ and ${\widehat{\tau }}_{t}^{\alpha ,\varepsilon }$ are the thermally activated strain-independent and strain-dependent components, respectively. The athermal contribution arises from long-range obstacles, such as grain boundaries, and is usually given by the Hall–Petch relation:

$$\hat{\tau }_{a}^{\alpha } = \frac{{k_{{{\text{HP}}}}^{\alpha } }}{{\sqrt {d_{g}^{{{\text{FCC}}}} } }},$$

(2)

where ${k}_{\text{HP}}^{\alpha }$ is the Hall–Petch strength coefficient, and ${d}_{g}^{\text{FCC}}$ is the grain size of the FCC matrix, as demonstrated to be applicable to FCC–CCAs [34,35,36].

The strain-independent thermally activated component arises from the lattice resistance and solute strengthening. For FCC–CCAs, the model proposed by Varvenne et al. [37, 38] is employed which is able to predict the shear resistance of the disordered FCC matrix by approximating it as a compositionally weighted effective medium, in which dislocations are hindered by lattice distortions [39, 40] which are modeled as random fluctuations in local concentration and the associated misfit volume ${\Delta V}_{n}$ of the nth element. Dislocations adopt a wavy configuration, where the segments lie in regions of favorable solute fluctuations corresponding to minima in a potential energy landscape. The 0 K shear resistance is given by

$$\hat{\tau }_{t}^{\alpha ,i} = 0.01785\alpha^{{ - \frac{1}{3}}} \overline{\mu }\left( {\frac{{1 + \overline{\nu }}}{{1 - \overline{\nu }}}} \right)^{\frac{4}{3}} \left[ {\frac{{\mathop \sum \nolimits_{n} c_{n} {\Delta }V_{n}^{2} }}{{b^{\alpha 6} }}} \right]^{\frac{2}{3}} ,$$

(3)

where $\overline{\mu }$ and $\overline{\nu }$ are the compositionally weighted average shear modulus and Poisson’s ratio, respectively. The dislocation line tension is described as $\Gamma =\alpha \overline{\mu }{b}^{2},$ where α is a constant equal to 0.123, ${b}^{\alpha }$ is the average Burgers vector, ${\Delta V}_{n}$ is the misfit volume of the nth element approximated using Vegard’s law as ${\Delta V}_{n}={V}_{n}-\overline{V }=\frac{{a}_{n}^{3}}{4}-{\sum }_{n}{c}_{n}{\Delta V}_{n}$, where ${a}_{n}$ is the lattice constant of the pure FCC element and ${c}_{n}$ is the concentration.

The strain-dependent contribution stems from dislocation–dislocation interactions and is given by the Taylor relation:

$$\hat{\tau }_{t}^{\alpha ,\varepsilon } = \chi \mu b^{\alpha } \sqrt {\mathop \sum \limits_{\beta } h^{\alpha \beta } \rho^{\beta } } \cong \chi \mu b^{\alpha } \sqrt {\rho^{g} },$$

(4)

where $\chi$ is a constant, $\mu$ is the shear modulus, ${b}^{\alpha }$ is Burgers vector, h is the latent hardening matrix, and ${\rho }^{\beta }$ is dislocation density on slip system β. In the present work, all the latent hardening coefficients ${h}^{\alpha \beta }$ are set to 1, which implies that the Taylor hardening relation may be simplified as illustrated in Eq. 4, since the total dislocation density in the grain, ${\rho }^{g}=\sum_{\beta }{\rho }^{\beta }$.

At a given temperature T and strain rate $\dot{\varepsilon }$, the flow stress contribution ${\tau }_{t}^{\alpha ,\kappa }$ relates to the mechanical threshold through a temperature- and rate-dependent scaling factor ${S}_{t}^{\alpha ,\kappa }\left(\dot{\varepsilon },T\right)$ such that ${\tau }_{t}^{\alpha ,\kappa }={\tau }_{t}^{\alpha ,\kappa }{S}_{t}^{\alpha ,\kappa }\left(\dot{\varepsilon },T\right)$, where

$$S_{t}^{\alpha ,\kappa } \left( {\dot{\varepsilon },T} \right) = \left( {1 - \left( {\frac{kT}{{g_{0,\kappa } \mu b^{\alpha 3} }}ln\left( {\frac{{\dot{\varepsilon }_{0,\kappa } }}{{\dot{\varepsilon }}}} \right)} \right)^{{1/q_{\kappa } }} } \right)^{{1/p_{\kappa } }}$$

(5)

which is derived from the Orowan equation, where the velocity of the dislocations is described by an Arrhenius relationship with a phenomenological stress-dependent activation free energy [29]. Here, ${g}_{0,\kappa }$ is the normalized activation free energy required to overcome an obstacle of type $\kappa$ in the absence of any applied stress, and ${\dot{\varepsilon }}_{0,\kappa }$ is a reference strain rate, chosen as 10⁷ s⁻¹ [32]. ${p}_{\kappa }$ and ${q}_{\kappa }$ determine the shape of the energy barrier for dislocation motion.

For the strain-independent component of solute strengthening p = 1, q = 3/2 and the normalized activation free energy can be expressed in terms of the average shear modulus, Poisson’s ratio, and the misfit volume as [37, 41]

$$g_{0,i} = 1.5618\alpha^{\frac{1}{3}} \left( {\frac{{1 + \overline{\nu }}}{{1 - \overline{\nu }}}} \right)^{\frac{2}{3}} \left[ {\frac{{\mathop \sum \nolimits_{n} c_{n} {\Delta }V_{n}^{2} }}{{b^{6} }}} \right]^{\frac{1}{2}} .$$

(6)

Similarly, for the strain-dependent dislocation–dislocation interactions, the shape of the energy barrier is described by p = 2/3, q = 1 [30, 32, 41].

The dislocation density on slip system $\alpha$, ${\rho }^{\alpha }$, evolves according to the Kocks-Mecking relation:

$$d\rho^{\alpha } = k_{1}^{\alpha } \sqrt {\rho^{g} } d\gamma^{\alpha } - k_{2}^{\alpha } \left( {\dot{\varepsilon },T} \right)\rho^{\alpha } d\Gamma,$$

(7)

where the accumulated shear strain in a grain, $d\Gamma =\sum_{\beta }d{\gamma }^{\beta }$. ${k}_{1}^{\alpha }$ relates to the number of forest dislocations a mobile segment crosses before it is stored and ${k}_{2}^{\alpha }\left(\dot{\varepsilon },T\right)$ is the temperature- and rate-dependent dynamic recovery term, given by [42]:

$$k_{2}^{\alpha } (\dot{\varepsilon },T) = \frac{{\chi k_{1}^{\alpha } b^{\alpha } }}{{g^{\alpha } }}\left( {1 - \frac{kT}{{D^{\alpha } b^{\alpha 3} \ln }}\left( {\frac{{\dot{\varepsilon }}}{{\dot{\varepsilon }_{0} }}} \right)} \right),$$

(8)

where $g^{\alpha }$ is stress-independent activation energy, $D^{\alpha }$ is a drag stress, and $\dot{\varepsilon }_{0}$ is a reference strain rate. Here, the total shear increment, $d\Gamma$, is used in the recovery term instead of the shear increment of a particular system, $d\gamma^{\alpha }$, because thermally activated recovery in a given slip system can occur via mutual interactions due to shear activity on other systems [43, 44].

Single-Crystal Model for Reinforcement

The behavior of the reinforcement is also assumed to be elastic, viscoplastic though with a strain hardening rate considerably lower than the matrix (100 MPa). The empirical Voce strain hardening rule is employed for the evolution of the CRSS $\tau_{{{\text{ref}}}}^{\alpha }$:

$$\frac{{\partial \tau_{{{\text{ref}}}}^{\alpha } }}{\partial \Gamma } = \theta_{0}^{\alpha } \left( {1 - \left( {\frac{{\tau^{\alpha } - \tau_{0}^{\alpha } }}{{\tau_{1}^{\alpha } }}} \right) } \right),$$

(9)

where $\tau_{{{\text{ref}}}}^{\alpha }$ is the reference shear resistance of slip system $\alpha$, $\tau_{0}^{\alpha } + \tau_{1}^{\alpha }$ is the saturation stress, and θ₀ is the initial strain rate. The initial slip system shear resistance is given by the Hall–Petch relation:

$$\tau_{0}^{\alpha ,p} = \tau_{r}^{\alpha ,p} + \frac{{k_{{{\text{HP}}}}^{p} }}{{\sqrt {d_{g}^{p} } }},$$

(10)

where $\tau_{r}^{\alpha ,p}$ accounts for the combined contributions of lattice resistance and solute strengthening,$k_{{{\text{HP}}}}^{p}$ is the Hall–Petch strength coefficient, and $d_{g}^{p}$ is the grain size of the second phase. For this phase, the {110}〈111 〉 slip mode (like BCC) as well as the cube slip mode, {110}〈100  〉  (typical of many B2 compounds), are assumed to be active [45, 46]. In order to keep the number of parameters to a minimum, both these modes are assumed to have the same strength.

Polycrystal Homogenization

To obtain the mechanical response of the two-phase alloy, the above single-crystal constitutive models are incorporated in a Elasto-Viscoplastic Self-Consistent (EVPSC) polycrystal plasticity model [47,48,49]. Both the phases are represented as a collection of spherical grains with crystal orientations and volume fractions selected to reproduce a particular crystallographic texture. Notably, the self-consistent homogenization scheme leads to a realistic treatment of the strain partitioning and has been successfully applied to multiphase materials in the past, e.g., [7, 50]. To simulate uniaxial tension, straining increments of 2.5 × 10^–4 are imposed parallel to one direction, while the normal stresses along the other two perpendicular directions and all shear stresses are set to zero. From the stress–strain data, the previously introduced ‘toughness’ is obtained using the Considère criterion $\frac{d\sigma }{{d\varepsilon }} = \sigma$.

Assumptions and Limitations

The grain size-dependent strain hardening is not accounted for because this effect is prominent only at small grain sizes (~ few μm), and even then the hardening rate is affected only at small strain levels (~ few percent) [51]. Second, scale dependence is not considered and kinematic hardening due to geometrically necessary dislocations and strain gradient effects are not accounted for. Third, it is assumed that the strength of the reinforcing phase does not depend on the volume fraction of the reinforcement in the two-phase alloy. While this is a fair assumption in most instances, there are situations where this is not applicable, e.g., dual-phase steels where the chemistry (C content), volume fraction, and the strength of the martensite are interlinked [12, 52,53,54]. Fourth, the strain hardening behavior of the matrix is assumed to be dictated by dislocation–dislocation interactions. Although the explicit treatment of TWIP/TRIP effects is not considered in this present work, the effects of a sustained strain hardening response resembling twinning- and/or transformation-induced hardening effects are examined. Similarly, effects of short-range order are not considered as they have recently been shown to have negligible (second-order) effects on the constitutive response of FCC-based alloys [55]. Fifth, the intermetallic phases under consideration are assumed to be much stronger than the matrix and are assumed to have almost no strain hardening capacity, akin to martensite in dual-phase steels [12]. Assuming the converse, i.e., a more strongly hardening reinforcement phase, would lead to the trivial result in the present microstructure design context, where strength, and toughness would monotonically increase with increasing volume fraction of reinforcement. In reality, the hard phases are prone to strain localization and/or fracture which leads to a lowering of the aggregate hardening rate and degradation of ductility and toughness. Finally, the distribution of the reinforcing phase, reinforcement fracture, and ductile fracture of the matrix are not accounted for in part 1; however, effects of particle fracture are taken up in part 2 [56]. The model presented here enables a generic analysis of the microstructure–plastic property relationships of dual-phase type alloys.

Material Parameters

In this section, the procedure used to select the material parameters of the model outlined above is presented and listed in Table 1. The elastic constants of the matrix were selected to represent Al containing austenitic alloys and properties of a B2-structured FeAl-type intermetallic were chosen [57, 58]. These represent a broad class of alloys, where the elastic mismatch between the two phases is intermediate to DP steels (in which the elastic mismatch is negligible) and metal matrix composites e.g., Al–SiC (where the elastic mismatch is quite large). The single-crystal elastic constants were obtained from in situ neutron diffraction data analysis of lattice strains obtained from a Fe–21Mn–10Al–5Ni–1C alloy which contains an FCC matrix and an B2 FeAl intermetallic reinforcing phase [59]. The shear modulus is given by $\mu = \sqrt {C_{44} \left( {\frac{{C_{11} - C_{12} }}{2}} \right)}$.

Table 1 Description, symbol, and magnitude of the model parameters used in this study

The intrinsic lattice resistance for most FCC materials at room temperature is generally small, so it is assumed to be zero (i.e., absorbed within the solid solution strengthening term) for simplicity. The 0 K shear resistance for the solid solution strengthening term, $\hat{\tau }_{t}^{\alpha ,i}$, is 90 MPa and the corresponding normalized activation energy, $g_{0,i}$, is 0.6. These values are within the range of solid solution strengthening contribution of a series of Al–Cr–Fe–Mn–Mo–Ni–Ti FCC–CCAs [6, 59], calculated using the model developed by Varvenne et al. [37, 38] and described in Sect. 2.1. The normalized activation energy for strain-dependent dislocation interactions, $g_{0,\varepsilon }$, is set to 1.6 [30, 32]. The Hall–Petch coefficient given in Table 1 for the austenitic phase was chosen as the average of the values reported by Singh et al. [60], Follansbe [32], and Bronkhorst et al. [61]. Note, these values were obtained based on macroscopic $\sigma - \varepsilon$ data, so they were converted using a Taylor factor M = 3.06¹ appropriate for Eq. 2 (and Eq. 10). Note this average value of 461 MPa μm^1/2 is similar to the one used for Fe–29Mn–9.4Al–0.9C steel [62] and Fe–15Mn–10Al–0.8C–5Ni steels containing B2 reinforcement [58].

The initial dislocation density is taken as 1.0 × 10¹² m⁻² representative of a well-annealed material. The values pertaining to the dislocation density-based strain hardening model were chosen to represent 316L austenitic stainless steel [65], CCAs including B2- and L2₁-containing alloys [3, 58, 66, 67], and FeMn, FeMnAl alloys that exhibit traditional saturation-type strain hardening, as well as specific alloys which exhibit TWIP/TRIP effects which result in higher, nearly linear hardening behavior [18, 19, 68,69,70]. Notably, the initial strain hardening rates are within the range E/50–E/150, where E is the respective Young’s modulus, in all cases including the alloys that exhibit TWIP/TRIP effects. This is consistent with the athermal and rather material insensitive Stage II hardening rate of FCC polycrystals [71]. The hardening rates just prior to failure vary among the alloys with 316L and Cantor alloy having the lowest rates (E/250–E/300), while the TWIP/TRIP alloys exhibit a sustained strain hardening response (E/100), which is adequately modeled using a linear hardening rule. Based on this, six different strain hardening behaviors were explored with storage coefficients $\left( {k_{1} b^{\alpha } } \right)$ and dynamic recovery coefficients $\left( {k_{2} } \right)$ ranging from 0.026–0.078 and 0.8–4.24, respectively. Using the well-known relationships $\theta_{0} = \frac{1}{2}\chi \mu b^{\alpha } k_{1}$ and $\tau_{1} = \frac{{\chi \mu b^{\alpha } k_{1} }}{{k_{2} }}$, the corresponding initial hardening rates ranges between μ/256 and μ/85 and the saturation stress values fall between 159 and 811 MPa ($\frac{{\tau_{1} }}{\mu } =$ 0.003–0.015). The strain hardening parameters are shown in Table 2.

Table 2 Strain hardening parameters related to the accumulation and recovery of dislocations. $k_{1}$ is the athermal storage coefficient, $b^{\alpha }$ is the Burgers vector of slip system, $\alpha$, $\mu$ is the shear modulus, $D$ is a drag stress, and $k_{2}$ is the temperature- and rate-dependent dynamic recovery term.$\theta_{0}$ and $\tau_{1}$ are the initial hardening rate and the saturation stress, respectively

The reinforcing phase is assumed to have a high lattice resistance with a significant contribution from order strengthening besides the usual solute strengthening effect. The combined initial resistance is set to 1.0 GPa and a Hall–Petch coefficient of 502 MPa μm^1/2 was chosen based on the work of Crimp and Vedula for B2 FeAl alloys (adjusted by M = 2) [63].

A polycrystal comprising of 1000 FCC and 1000 reinforcement grains, where each phase is assumed to have an initial crystallographic texture similar to FeMn alloys, was deformed in uniaxial tension as described above. The grains are considered to be equiaxed and the grain sizes explored in this study range from 1 to 50 μm for both the phases. The results are discussed in terms of the contrast between the two phases, defined as the ratio of the CRSS of the reinforcement to that of the softer matrix. In the present study, for a given hardening behavior of the matrix, all other strengthening contributions as well as the thermal activation parameters are kept constant, and a change in the grain size is used to change the contrast between the phases. The lowest contrast explored in this work is 4.9 and it corresponds to a matrix grain size, $d_{m}$ = 1 μm and a reinforcement grain size, $d_{f}$ = 50 μm. The highest value of the contrast is 17 and it corresponds to $d_{m}$ = 50 μm and $d_{f}$ = 1 μm. Four reinforcement volume fractions ($v_{f} )$ were explored viz. 0 (pure matrix), 0.1, 0.2, 0.3, and 0.4.

Results

Results from selected cases viz. #1, #5 and #6 (Table 2) are presented in detail. The dependence of toughness, the $\sigma_{u}$ , and the $\varepsilon_{u}$ on reinforcement volume fraction for all the other cases are given in the Supplementary Data.

Case 1 considers a low strain hardening matrix. Figure 1a shows the stress–strain curve of the matrix and the hard phase in isolation (i.e., pure phases) for a $d_{m}$ = 1 μm and $d_{f}$ = 50 μm ,respectively. This combination of grain sizes represents the lowest contrast case. Figure 1b shows the stress–strain response as well as the strain hardening rate of the two-phase alloys containing different amounts of reinforcement content. The uniform elongation and the tensile strength as obtained via Considère criterion are also shown. It is evident that with addition of the hard phase, besides the obvious increase in the yield strength, the strain hardening rate also increases. As a result, the ultimate tensile strength increases, although this is accompanied by a ductility reduction (the Considère criterion is met at a lower strain). However, the toughness shows a monotonic increase with the reinforcement content (Fig. 1c). Figure 1d shows the strain in the hard phase. Notice that as the reinforcement content increases, the amount of strain partitioned to this phase also increases; however, it remains below 1% for the highest vol. fraction (0.4). This is because the low strain hardening rate of the matrix keeps the overall stresses low. The increased strain partitioning to the hard phase is responsible for the observed increase in the hardening rate and thus the toughness increases.

Fig. 1

The constitutive response when a low strain hardening matrix (case #1), exhibiting saturation behavior, and the lowest strength contrast is considered. a The stress–strain curves of the matrix and the reinforcing phase in isolation for a grain size of 1 and 50 μm, respectively. b The stress–strain response of the resulting aggregates, with different reinforcement volume fractions. Also shown are the corresponding strain hardening rates. The symbols mark the points where Considère criterion is satisfied and give the tensile strength and the uniform elongation for each case. c The variation of toughness as a function of the volume fraction of the reinforcing phase. d The amount of strain partitioned to the second phase for aggregates containing different amounts of the reinforcing phase. The results show that if the stresses remain low throughout such that the reinforcement remains elastic, the increased strength offsets the decrease in ductility, leading to an increase in toughness

Figure 2a shows another low contrast case, but in this case, the matrix hardening rate is higher (case 5). Like before, the yield strength, the strain hardening rate, and the tensile strength increase with increasing volume fraction of second phase, but notice that there is a much larger reduction in ductility (Fig. 2b). The toughness monotonically decreases with reinforcement addition (Fig. 2c). The strain partitioned to the second phase is significantly higher even for a reinforcement vol. fraction of only 0.1 (Fig. 2d). This is due to the high matrix hardening rate which increases the overall stress level of the aggregate. The larger strain partitioning results in yielding of the reinforcement which drastically reduces the strain hardening rate of the aggregate and, ultimately, results in a large reduction in uniform elongation, and consequently the toughness.

Fig. 2

The constitutive response when a high strain hardening matrix, exhibiting saturation behavior (case #5), and the lowest strength contrast is considered. a The stress–strain curves of the matrix and the reinforcing phase in isolation for a grain size of 1 and 50 μm, respectively. b The stress–strain response of the resulting aggregates, with different reinforcement volume fractions. Also shown are the corresponding strain hardening rates. The symbols mark the points where Considère criterion is satisfied and give the tensile strength and the uniform elongation for each case. c The variation of toughness as a function of the volume fraction of the reinforcing phase. d The amount of strain partitioned to the second phase for aggregates containing different amounts of the reinforcing phase. The results show that if the stresses are high, more strain is partitioned, leading to yielding of the reinforcement, and toughness degrades with reinforcement addition

Figure 3 shows the results for case 6, where the matrix exhibits an intermediate initial hardening rate (w.r.t cases 1 and 5), but has a sustained strain hardening response (Fig. 3a). The instance under consideration is the highest contrast case ($d_{m}$ = 50 μm and $d_{f}$ = 1 μm). Like case 1, as reinforcement content increases, the tensile strength increases and the ductility reduces (Fig. 3b). The toughness increases monotonically with increasing second phase fraction (Fig. 3c). Furthermore, the toughness values are much higher compared to cases 1 and 5, because the sustained hardening response of the matrix leads to high values of uniform elongation. The strain partitioned to the second phase also increases, but remains within a few percent implying that the reinforcement remains elastic up to the highest volume fraction explored (Fig. 3d). The increase in strength due to the increased strain hardening rate (since part of the material is elastic) compensates for the reduction in uniform elongation, and the toughness increases monotonically.

Fig. 3

The constitutive response when a matrix, exhibiting sustained strain hardening behavior, (case #6) and the highest strength contrast is considered. a The stress–strain curves of the matrix and the reinforcing phase in isolation for a grain size of 50 and 1 μm, respectively. b The stress–strain response of the resulting aggregates, with different reinforcement volume fractions. Also shown are the corresponding strain hardening rates. The symbols mark the points where Considère criterion is satisfied and give the tensile strength and the uniform elongation for each case. c The variation of toughness as a function of the volume fraction of the reinforcing phase. d The amount of strain partitioned to the second phase for aggregates containing different amounts of the reinforcing phase. The results show that the reinforcement remains elastic, and the strength increment overcompensates the ductility reduction leading to a monotonic increase in the toughness

Figure 4 shows an intermediate contrast case (Contrast = 10.7), for the same matrix as in case 6, where $d_{m}$ = 10 μm and $d_{f}$ = 5 μm. In this case, as long as the reinforcement content is low, it remains elastic and the toughness increases, because of the reasons mentioned above. At larger volume fractions (> 20% for this example), the reinforcement yields, the hardening rate sharply decreases, leading to a large decrease in uniform elongation, and consequently, the toughness degrades (Fig. 4c and d). From these four cases, it is evident that when the increase in strength compensates for decrease in ductility, the toughness increases and vice versa, and as long as the reinforcement remains elastic, a larger strain partitioning is beneficial for the toughness. As soon as the hard phase yields, the strain hardening rate drastically deteriorates, resulting in a degradation of the toughness.

Fig. 4

The constitutive response when a matrix, exhibiting sustained strain hardening behavior, (case #6) and intermediate strength contrast is considered. a The stress–strain curves of the matrix and the reinforcing phase in isolation for a grain size of 10 and 5 μm, respectively. b The stress–strain response of the resulting aggregates, with different reinforcement volume fractions. Also shown are the corresponding strain hardening rates. The symbols mark the points where Considère criterion is satisfied and give the tensile strength and the uniform elongation for each case. c The variation of toughness as a function of the volume fraction of the reinforcing phase. d The amount of strain partitioned to the second phase for aggregates containing different amounts of the reinforcing phase. The results show that with reinforcement addition, more strain is partitioned to the second phase and while it remains elastic, the toughness improves, but eventually it yields leading to toughness degradation

For a matrix showing sustained strain hardening (case 6), the toughness can monotonically increase, decrease, or exhibit a maximum, depending on the level of strength contrast between the matrix and reinforcement (Fig. 5). This implies that by altering the microstructure, a variety of responses can be obtained. Figure 5a shows the dependence of toughness on reinforcement content when the reinforcement grain size is 1 µm, for different matrix grain sizes. When the contrast is high, i.e., for microstructures with a coarse matrix grain size, the toughness monotonically increases with increasing volume fraction of the second phase. The strength increment with increasing reinforcement content is significant, since the reinforcement is much stronger than the matrix. The uniform elongation also does not decrease drastically due to the high strain hardening rate. These two factors lead to a monotonic increase in toughness with second phase content.

Fig. 5

The dependence of toughness on volume fraction of the second phase of aggregates comprising of a matrix, exhibiting sustained strain hardening behavior (case #6), for various strength contrast values which is controlled by the grain size of the two phases. Aggregates with different matrix grain sizes for a reinforcement grain size of a 1 μm and b 5 μm, c aggregates in which the grain sizes of the phases are equal and d aggregates with different reinforcement grain sizes for a matrix grain size of 1 μm. The toughness can either monotonically increase, decrease, or exhibit a maximum, depending on the microstructure. In general, a coarser matrix and fine second phase is preferable, because it enhances the contrast

As the reinforcement size is increased, the volume fraction at which the maximum toughness is obtained, i.e., the optimal volume fraction, decreases (Fig. 5b). For instance, if a matrix with grain size 1 µm is reinforced with a 1 µm second phase, then the optimal volume fraction is 0.3, whereas if the same matrix is reinforced with 5 µm second phase, the optimal volume fraction decreases to 0.1 (Fig. 5b). As the matrix grain size is reduced, the contrast decreases, and the optimal volume fraction also decreases (Fig. 5a and b). When both phases have equal grain sizes, a finer microstructure ($d_{m}$ =  $d_{f}$ = 1 µm vs. $d_{m}$ = $d_{f}$ = 5 µm) results in higher maximum toughness values, which are achieved at higher reinforcement contents (Fig. 5c).

For fine grained matrix, a stronger reinforcement (i.e., with a small grain size) leads to improved toughness (Fig. 5d). As the second phase grain size increases, the contrast decreases, and the optimal volume fraction also decreases. At one limit, it is found that the toughness monotonically decreases with the addition of the second phase and the pure matrix phase has the highest toughness, and any reinforcement addition has an adverse effect (Fig. 5d). As the reinforcement content in the aggregate increases, more strain is partitioned to the reinforcement. In such scenarios, the overall stresses build up such that the reinforcement yields, reducing the strain hardening rate and the ductility. The increase in strength is not large enough to compensate the loss in ductility since the reinforcement is not strong enough.

Discussion

Yield Strength of Two-Phase Alloys and The Stress State in the Constituent Phases

Note that the results presented in Sect. 3 show that the yield strength of the two-phase alloy, $\sigma_{c}^{Y}$, is not given according to the simple rule of mixtures of yield strength of the constituent phases: $\sigma_{c}^{Y} = \sigma_{m}^{Y} \left( {1 - v_{f} } \right) + \sigma_{f}^{Y} v_{f}$, as has been assumed in several studies [57, 58, 72]. This is because, when the applied uniaxial tensile stress reaches the yield strength of the softer phase, the softer phase starts to deform plastically, while the hard phase remains elastic. By the time the hard phase yields (if at all), the stress in the matrix has increased due to strain hardening. Macroscopic homogenization requires that

$$\sigma_{c} = \sigma_{m} \left( {1 - v_{f} } \right) + \sigma_{f} v_{f}$$

(11)

and

$$\varepsilon_{c} = \varepsilon_{m} \left( {1 - v_{f} } \right) + \varepsilon_{f} v_{f},$$

(12)

where $\sigma_{c}$, $\sigma_{m}$, $\sigma_{f}$ and $\varepsilon_{c}$, $\varepsilon_{m}$, $\varepsilon_{f}$ are the instantaneous stresses and strains in the overall alloy, matrix, and the reinforcement, respectively. Then, using Hooke’s law $\sigma_{f} = C_{f} \varepsilon_{f}$ and (9), we obtain:$\sigma_{c} = \sigma_{m} \left( {1 - v_{f} } \right) + C_{f} \varepsilon_{f} v_{f}$.

When combined with (10) the following is obtained:

$$\sigma_{c} = \sigma_{m} \left( {1 - v_{f} } \right) + \left( {\varepsilon_{c} - \varepsilon_{m} \left( {1 - v_{f} } \right)} \right)C_{f}$$

(13)

To proceed further, knowledge of the interaction stiffness, i.e., the level of the strain partitioning between the matrix and the aggregate, is required. If one assumes $\varepsilon_{c} = A\varepsilon_{m}$ and $A \le 1$, then

$$\sigma_{c} = \sigma_{m} \left( {1 - v_{f} } \right) + \left( {A - \left( {1 - v_{f} } \right)} \right)\varepsilon_{m} C_{f}$$

(14)

$A$ can be thought of as the scalar version of the localization tensor in micromechanics based on Eshelby inclusion formalism, whose deviation from identity (unity) quantifies the level of heterogeneity between the inclusion and homogeneous effective medium [73,74,75]. Equation 12 shows that the yield strength of the two-phase alloy is dictated by the strain partitioned to the matrix, and the stiffness of the hard phase. When A = 1 and $\varepsilon_{m} = \varepsilon_{f}$, i.e., stiffest possible interaction, the isostrain case is retrieved, where the yield strength of the aggregate is given by the rule of mixtures of the stresses (not yield strengths) in the two phases. In general, A < 1 and thus, the yield strength is lower than predicted by the isostrain case. Another point to note is that the yield strength of the aggregate strongly depends on the stiffness of the reinforcing phase, and as such, materials with larger elastic stiffness mismatch, e.g., Al–SiC and Al–Al₂O₃ composites, get a significant boost in their strength with the addition of reinforcement.

Finally, as the softer matrix undergoes plastic deformation, the difference in strain levels between the two phases increases. This leads to the buildup of lateral compressive stresses, as shown by the schematic in Fig. 6a, which aid the onset of plastic flow in the hard phase at an axial stress level which is lower than the yield strength of the hard phase in isolation. Another way of representing this concept is via the stress triaxialities (i.e., the ratio of hydrostatic stress to Von Mises equivalent stress, $\frac{{\sigma_{p} }}{{\sigma_{VM} }}$) of the two phases. The stress triaxiality within the hard phase is lowered as compared to uniaxial tension (Fig. 6b). The softer matrix phase develops a higher stress triaxiality in order to ensure that the volume averaged stress state of the aggregate is equivalent to the macroscopically enforced uniaxial tension. The difference in stress triaxiality is largest during the elastoplastic transition, while the hard reinforcement phase is still elastic. As straining progresses, the stress level in the matrix increases, and thus, the constraint decreases. This is evident from Fig. 6b, where the deviation of triaxiality from the globally enforced uniaxial condition decreases with straining. A key implication of this observation is that, under such circumstances, a brittle intermetallic reinforcing phase can survive larger deformation levels before fracture than it could have when tested in isolation. This has already been shown in the literature for NiAl, where the fracture strain increases under imposed pressure [76]. Conversely, the matrix is under higher stress triaxiality which can promote ductile fracture via void nucleation, growth, and coalescence.

Fig. 6

a A schematic showing the stress state of the individual phases in a dual-phase aggregate. The hard reinforcement shown in blue is under compressive stress $\sigma_{22}^{p}$ and $\sigma_{33}^{p}$, when loaded along the 1 direction, whereas the matrix is under hydrostatic tension. b The evolution of stress triaxiality within the matrix and the reinforcing phase with applied strain showing deviation from uniaxial tension as prescribed by the external boundary conditions

Toughness and Volume Fraction

Results from Sect. 3 have shown that the toughness can either monotonically increase, decrease, or exhibit a maximum, with increasing volume fraction of reinforcement, depending on microstructure. To understand why these various trends exist, one can begin from the definition: $T = \sigma_{u} \varepsilon_{u}$ where the toughness T is given by the product of the ultimate tensile stress $\sigma_{u}$ and the uniform elongation $\varepsilon_{u}$. To find the maximum, T is differentiated with respect to the microstructural variable of interest, e.g., the volume fraction of the second phase $v_{f}$, and set equal to zero:

$$\frac{dT}{{dv_{f} }} = \frac{{d\sigma_{u} }}{{dv_{f} }}\varepsilon_{u} + \frac{{d\varepsilon_{u} }}{{dv_{f} }}\sigma_{u} = 0 \;{\text{and}}\;\frac{{d^{2} T}}{{dv_{f}^{2} }} < 0.$$

(15)

Note that the practical maximum in toughness may also exist at one of the limits of volume fraction incorporation, rather than that given by Eq. 15. Here, the limits of exploration were set as $0 \le v_{f} \le 0.4$. In order to proceed further, the functional dependencies $\sigma_{u} \left( {v_{f} } \right)$ and $\varepsilon_{u} \left( {v_{f} } \right)$ are required. In principle, one could obtain these dependencies from a set of tensile experiments on samples containing different reinforcement contents or simulations. For example, a quadratic dependence is explored in the Appendix, and it is shown to well describe the present self-consistent polycrystal simulation results. However, the same general principles are obtained when one assumes a simple linear dependence: $\sigma_{u} = S_{1} v_{f} + S_{2}$ and $\varepsilon_{u} = E_{1} v_{f} + E_{2}$. Then,

$$\frac{dT}{{dv_{f} }} = S_{1} \left( {E_{1} v_{f} + E_{2} } \right) + E_{1} \left( {S_{1} v_{f} + S_{2} } \right) = 0$$

(16)

Solving for $v_{f}$ gives

$$v_{f} = v_{{{\text{fopt}}}} = - \frac{{S_{1} E_{2} + S_{2} E_{1} }}{{2S_{1} E_{1} }}.$$

(17)

Typically,$E_{1}$ is negative, while $E_{2}$, $S_{1}$, and $S_{2}$ are positive quantities, since strength increases and uniform elongation decreases with increasing volume fraction. The intercepts $S_{2}$ and $E_{2}$ give the strength and uniform elongation of the matrix. Equation 17 shows that in order for a maximum to exist at $v_{f} > 0, S_{1} E_{2} + S_{2} E_{1} > 0$.

That is, matrix materials that have a high uniform elongation ($E_{2}$) and low strength ($S_{2}$) can withstand larger second phase content without negatively affecting the toughness. A larger reduction in uniform elongation per volume fraction ($E_{1} )$ is detrimental, while a larger increase in strength per volume fraction ($S_{1}$) is beneficial. The first scenario arises when greater amount of strain is partitioned to the second phase, and if the strength contrast between the softer matrix and the hard phase is low, this leads to the onset of plasticity in the hard phase. The latter, positive scenario arises when the strength contrast is high and the reinforcement remains elastic to higher levels of aggregate strain.

Toughness and Grain Size

To understand the dependency of toughness on the grain sizes of the matrix and reinforcing phases, one can carry out a similar analysis as shown in Sect. 4.2. It will be shown that there is no optimal matrix or reinforcement grain size, assuming a general Hall–Petch-type grain size dependence, that leads to the maximum toughness. Rather, the optimal grain sizes will appear at the practical limits imposed by metallurgical processing and other factors. The toughness $T = \sigma_{u} \varepsilon_{u}$ is differentiated with respect to the grain size $d_{g}$, and set equal to zero:

$$\frac{dT}{{dd_{g} }} = \frac{{d\sigma_{u} }}{{dd_{g} }}\varepsilon_{u} + \frac{{d\varepsilon_{u} }}{{dd_{g} }}\sigma_{u} = 0\;{\text{and}}\;\frac{{d^{2} T}}{{dd_{g}^{2} }} < 0.$$

(18)

The functional dependencies $\sigma_{u} \left( {d_{g} } \right)$ and $\varepsilon_{u} \left( {d_{g} } \right)$ are assumed to be power-law, a generalization of the Hall–Petch relation, based on the results obtained in the present work (Fig. 7a and b). Therefore, $\sigma_{u} = S_{1} d_{g}^{m}$ and $\varepsilon_{u} = E_{1} d_{g}^{n}$ then,

$$\frac{dT}{{dd_{g} }} = mS_{1} d_{g}^{m - 1} E_{1} d_{g}^{n} + nE_{1} d_{g}^{n - 1} S_{1} d_{g}^{m} = 0$$

(19)

Fig. 7

Grain size effects. The variation of a the tensile strength and b the uniform elongation as a function of the matrix grain size, for aggregates with different reinforcement contents, but a constant grain size of 5 μm. The variation of toughness as a function of the matrix grain size reinforced with c 5 μm second phase d 1 μm second phase. The hardening behavior of the matrix under consideration corresponds to case #6

On simplifying and rearranging,

$$\frac{dT}{{dd_{g} }} = S_{1} E_{1} d_{g}^{m + n - 1} \left( {m + n} \right) = 0$$

(20)

The only possibility that arises is $\left( {m + n} \right) = 0$ since $S_{1}$, $E_{1}$, $d_{g} \ne 0$ which implies $m = - n$. In this case, there is no optimum grain size, since $\frac{dT}{{dd_{g} }} = 0$ for any $d_{g}$. This is indeed what is observed in the present study (Fig. 7c and d). A reinforcement with a finer grain size results in higher toughness values. Interestingly, for a given reinforcement grain size, the toughness increases as the matrix grain size increases. In short, the best toughness values are obtained for fine (strong) reinforcement grain size and a coarse (soft) matrix. This dependence is most prominent for small grain sizes (1–10 μm). At higher grain sizes, the toughness becomes relatively insensitive to the matrix grain size. Large matrix grain sizes lead to a lower strength via the Hall–Petch relation, and consequently increases the contrast and vice versa. The strong dependence at small grain sizes is due to the inverse square root dependence of the Hall–Petch relation.

Implications for Alloy Design—Strain Hardening Rate of the Matrix

Figure 8 summarizes the toughness dependency for the cases presented in Sect. 3. It is found that when the matrix hardening rate exhibits a traditional saturation-type behavior typical of dislocation–dislocation interactions, improvement of the toughness with addition of reinforcement is, at best, marginal. A higher hardening rate matrix can be beneficial if the contrast between the phases is sufficiently high. In other words, higher contrast allows sustaining high levels of stress as long as the reinforcement remains elastic. If the matrix exhibits sustained, linear-type strain hardening behavior, then addition of reinforcing phases can lead to markedly improvement in the toughness, depending on the microstructure. One strategy to obtain a such a sustained strain hardening behavior is via TRIP/TWIP effects e.g., [77, 78], which can be achieved by alloying strategies which lead to modification of the generalized stacking fault energy (GSFE) landscape or by inducing compositional undulations (short-range ordering) which may also suppress dynamic recovery [79].

Fig. 8

The variation of toughness as a function of the volume fraction of the reinforcing phase, for the cases shown in Figs. 1, 2, 3, and 4. Low saturation-type strain hardening behavior of the matrix leads to low toughness values. The toughness can be improved if the hardening rate of the matrix is increased. When the matrix exhibits a sustained strain hardening behavior, very high levels of toughness can be achieved, and this is further increased if the strength contrast between the matrix and the second phase is increased

If one plots the maximum toughness and the corresponding $\sigma_{u}$, normalized by the matrix CRSS, versus the contrast, a strong positive correlation is observed (Fig. 9a). This implies that increasing the contrast between the phases increases both the tensile strength and the toughness of the resulting aggregate. Figure 9b shows a plot of the uniform elongation $\varepsilon_{u}$ corresponding to the maximum toughness and the corresponding strain partitioned to the reinforcement. First, it is evident that for all cases, the maximum toughness is obtained only when a small amount of strain is partitioned to the reinforcing phase (1–4%). Second, a strong negative correlation is observed between $\varepsilon_{u}$ and the level of strain in the second phase. Thus, to optimize the tensile properties of two-phase alloys, the highest possible contrast must be employed and one strategy is to employ the smallest possible reinforcement crystallite size (because that leads to greatest Hall–Petch strengthening). In principle, one can also adjust the chemistry (composition) of the two phases to alter the contrast to achieve design goals.

Fig. 9

a Strong correlation between the strength contrast and the maximum toughness and the corresponding tensile strength (normalized by the matrix CRSS) for all the cases shown in Fig. 5. b Strong inverse correlation between the corresponding uniform elongation and the amount of strain partitioned to the second phase. These relationships strongly suggest that plasticity in a non-hardening reinforcement is detrimental (i.e., lowers) the strain hardening rate of aggregate

Conclusions

A polycrystal plasticity model is employed to evaluate the effect of strength contrast and matrix strain hardening behavior on the mechanical response of two-phase aggregates, which may serve as a guide to microstructure design. Both phases are considered to be elasto-viscoplastic, and the model takes as input the phase fraction, grain sizes, and parameters pertaining to the stiffness, strength, and strain hardening behavior of each phase. The following conclusions can be drawn from the results presented in this work.

• 1. Adding hard phase reinforcement obviously increases strength, but the present work highlights that, as long as the reinforcement remains elastic, the strain hardening rate also increases.

• 2. However, when an elastoplastic reinforcement with a low hardening rate relative to the matrix yields, the hardening rate sharply decreases leading to a decrease in uniform elongation.

• 3. When the increase in strength compensates for the decrease in uniform elongation, the toughness (here defined as the product between the two) increases & vice versa.

• 4. Second phase reinforcement can be added without adversely affecting toughness for very high contrast (high relative strength) reinforcements.

• 5. For lower contrast reinforcements, the maximum toughness occurs at a lower second phase content. At the limit, any second phase addition degrades toughness.

• 6. To simultaneously improve both strength and toughness by reinforcement, the matrix must possess a sustained (linear-type) hardening typical of TWIP/TRIP FCC–CCAs with very low stacking fault energies.

• 7. Assuming the strength of the second phase particles is governed by a Hall–Petch-type grain boundary mechanism, a fine second phase is preferable for obtaining higher toughness values, because it enhances the contrast, for a given matrix grain size.

• 8. Stress triaxiality within the phases deviates from uniaxial due to interactions between the grains and the surrounding medium. The hard reinforcement is under a lower stress triaxiality due to the imposed pressure from the surrounding matrix. As a result, the reinforcement yields at a lower axial stress than it would in isolation. This implies simple rule-of-mixtures approaches, which are frequently used for prediction of yield strength, are incorrect.

Appendix

Based on the simulations results obtained in this paper, it is found that a second-order polynomial fits the $\varepsilon_{u}$ vs $v_{f}$ data better than the linear approximation. Thus, if the following dependencies are assumed:

$\sigma_{u} = S_{1} v_{f} + S_{2}$ and $\varepsilon_{u} = E_{1} v_{f}^{2} + E_{2} v_{f} + E_{3}$.

Then,

$$\frac{dT}{{dv_{f} }} = S_{1} \varepsilon_{u} + \left( {2E_{1} v_{f} + E_{2} } \right)\sigma_{u} = 0$$

Upon substituting $\sigma_{u}$ and $\varepsilon_{u}$,

$$\frac{dT}{{dv_{f} }} = S_{1} \left( {E_{1} v_{f}^{2} + E_{2} v_{f} + E_{3} } \right) + \left( {2E_{1} v_{f} + E_{2} } \right)\left( { S_{1} v_{f} + S_{2} } \right) = 0$$

And simplifying

$$\frac{dT}{{dv_{f} }} = \left( {S_{1} E_{1} v_{f}^{2} + S_{1} E_{2} v_{f} + S_{1} E_{3} } \right) + \left( {2E_{1} v_{f} S_{1} v_{f} + 2E_{1} v_{f} S_{2} + E_{2} S_{1} v_{f} + E_{2} S_{2} } \right) = 0$$

One obtains the following quadric relation

$$\frac{dT}{{dv_{f} }} = 3S_{1} E_{1} v_{f}^{2} + 2\left( {S_{1} E_{2} + S_{2} E_{1} } \right)v_{f} + \left( {S_{1} E_{3} + E_{2} S_{2} } \right) = 0$$

Let $A = 3S_{1} E_{1} , B = 2\left( {S_{1} E_{2} + S_{2} E_{1} } \right), C = \left( {S_{1} E_{3} + E_{2} S_{2} } \right)$

Solving for $v_{f}$ that results in max. toughness:

$$v_{f} = v_{{{\text{fopt}}}} = \frac{{ - B \pm \sqrt {B^{2} - 4AC} }}{2A}$$

(A1)

The max toughness is given by

$$T_{{{\text{max}}}} = \left( {S_{1} v_{{{\text{fopt}}}} + S_{2} } \right)\left( {E_{1} v_{{{\text{fopt}}}}^{2} + E_{2} v_{{{\text{fopt}}}} + E_{3} } \right)$$

(A2)

Comparison between the predicted (using Eqs. A1 and A2) and the observed (based on Considère criterion of EVPSC simulations) volume fractions at which the maximum toughness occurs and the corresponding maximum toughness values are shown in Fig. 10. Note the simulations were performed at discrete specific volume fraction of reinforcements.

Fig. 10

Comparison between the predicted and the observed i.e., based on Considère criterion of EVPSC simulations a volume fractions at which the maximum toughness occurs, for different strength contrast values explored in this work and b the maximum toughness values. The matrix hardening rate corresponds to case #6