Brittle-Ductile Transitions in a Metallic Glass

TL;DR

This paper uses shear-transformation-zone theory to analyze how initial disorder affects brittle-ductile transitions in metallic glasses, revealing a sudden shift from brittle to ductile failure linked to plastic zone dynamics.

Contribution

It provides a theoretical framework explaining the brittle-ductile transition in metallic glasses based on initial disorder and plastic zone behavior.

Findings

Plastic zone shields the notch tip initially.

Sudden breakdown of the plastic zone causes brittle-ductile transition.

Theoretical predictions align with experimental observations.

Abstract

Recent computational and laboratory experiments have shown that the brittle-ductile transitions in metallic glasses such as Vitreloy1 are strongly sensitive to the initial effective disorder (or "fictive") temperature. Glasses with lower effective temperatures are weak and brittle; those with higher effective temperatures are strong and ductile. The analysis of this phenomenon presented here examines the onset of fracture at the tip of a slightly rounded notch as predicted by the shear-transformation-zone (STZ) theory of spatially varying plastic deformation. The central ingredient of this analysis is an approximation for the dynamics of the plastic zone formed by stress concentration at the notch tip. This zone first shields the tip but then breaks down suddenly producing a discontinuous transition between brittle and ductile failure, in agreement with the numerical and experimental observations.

Figures

a in [SCHetal-18 ](#bib.bib3). They should be directly comparable with the other two data sets shown in this figure.

b in [RB-12 ](#bib.bib1) shows a bulge with a radius of curvature roughly half that of the tip emerging from the front of the notch and substantially raising the effective temperature in its neighborhood. Rycroft has shown unpublished movies of simulated later stages of ductile yielding in which a bulge of that kind moves forward for a considerable distance before launching a fast crack. There are only hints of such behavior in the present theory. Note that the theoretical notch tip described by the graphs in Fig. [6](#S5.F6) does sharpen before reaching its failure limit.

b in [RB-12 ](#bib.bib1) shows a bulge with a radius of curvature roughly half that of the tip emerging from the front of the notch and substantially raising the effective temperature in its neighborhood. Rycroft has shown unpublished movies of simulated later stages of ductile yielding in which a bulge of that kind moves forward for a considerable distance before launching a fast crack. There are only hints of such behavior in the present theory. Note that the theoretical notch tip described by the graphs in Fig. [6](#S5.F6) does sharpen before reaching its failure limit.

Equations

$$x=W\left(\rho +\frac{m}{\rho }\right)\cos \theta ,y=W\left(\rho +\frac{m}{\rho }\right)\sin \theta .$$

$$m\approx 1-2ϵ;ϵ\equiv \sqrt{\frac{1}{2{\mathcal{K}}_{tip}W}}\ll 1,$$

$$s(\rho,\theta)\equiv s(\tilde{x},\theta)\cong{\sigma_{\infty}\epsilon^{2}\over(\epsilon+\tilde{x})^{3}}\,\Bigl{(}1-{2\,\theta^{2}\over\epsilon^{2}}\Bigr{)}.$$

$$\frac{s(0,0)}{s_y}=\frac{{\sigma }_{\infty }}{s_yϵ}\equiv \psi \sqrt{\kappa },$$

$$\psi \equiv \frac{{\sigma }_{\infty }}{s_y}\sqrt{\frac{2W}{a}};\kappa \equiv a{\mathcal{K}}_{tip}.$$

$$\tilde{x}_{max}=\epsilon\,[[\nu-1]],~{}~{}~{}\nu\equiv\Bigl{(}{\sigma_{\infty}\over\epsilon s_{y}}\Bigr{)}^{1/3}=(\psi\sqrt{\kappa})^{1/3}.$$

$$x_0=W(1+m)\cong 2W(1-ϵ),$$

$$\displaystyle x_{max}=W\,\Bigl{(}1+\tilde{x}_{max}+{1-2\epsilon\over 1+\tilde{x}_{max}}\Bigr{)}~{}~{}~{}$$

$$\cong x_0+\frac{1}{{\mathcal{K}}_{tip}}[[\nu -1]].$$

$$D_{\theta \theta }(\rho ,\theta )=-D_{\rho \rho }(\rho ,\theta )$$

$$\equiv D(\rho ,\theta )\cong \frac{1}{2\mu }\frac{ds(\rho ,\theta )}{dt}+D^{pl}(\rho ,\theta ),$$

$$D^{pl}(\rho ,\theta )\cong \frac{1}{{\tau }_{pl}}[[\frac{s(\rho ,\theta )}{s_y}-1]],$$

$$D^{pl}(\rho,\theta)\equiv D^{pl}(\tilde{x},\theta)\cong{1\over\tau_{pl}}\,[[{s(\tilde{x},0)\over s_{y}}-1]]\,\Bigl{(}1-{2\,\theta^{2}\over\epsilon^{2}}\Bigr{)}~{}~{}~{}~{}~{}$$

$${s(\tilde{x},0)\over s_{y}}-1\cong({s_{0}\over s_{y}}-1)\Bigl{(}1-{\tilde{x}\over\tilde{x}_{max}}\Bigr{)}.$$

$${\mathcal{D}}_{\rho \rho }$$

$${\mathcal{D}}_{\theta \theta }$$

$$D(\tilde{x},\theta)\cong D_{0}(\tilde{x})\Bigl{(}1-{2\,\theta^{2}\over\epsilon^{2}}\Bigr{)},$$

$$D_0(\tilde{x})=\frac{1}{2\mu }\dot{s}(\tilde{x},0)+D^{pl}(\tilde{x},0).$$

$$v_{tip}=v_{\rho }(0)=-{\int }_0^{\infty }d\tilde{x}\frac{d{v}_{\rho }}{d\tilde{x}}=2ϵW{\int }_0^{\infty }d\tilde{x}{D}_0(\tilde{x})$$

$$\displaystyle={a\over 2\kappa}(\bar{\nu}-1)\Bigl{(}{1\over 2\mu}\dot{s}_{0}+{1\over\tau_{pl}}[[{s_{0}\over s_{y}}-1]]\Bigr{)}$$

$$\displaystyle~{}~{}~{}~{}~{}+\Bigl{(}{a\over 2\,\kappa\,\bar{\nu}^{2}}\Bigr{)}\Bigl{(}{s_{y}\over 2\mu}\Bigr{)}\Bigl{(}{\dot{\sigma}_{\infty}\over\epsilon s_{y}}\Bigr{)}.$$

$$-{\dot{\cal K}_{tip}\over{\cal K}^{2}_{tip}}=v_{tip}+{1\over 2\,{\cal K}_{tip}\,W}\,{\partial^{2}v_{\rho}\over\partial\,\theta^{2}}\Bigr{|}_{\theta=0}.$$

$$v_{\rho }(\tilde{x},\theta )\equiv v_0(\tilde{x})+v_2(\tilde{x})\frac{{\theta }^2}{{ϵ}^2},$$

$$-{\dot{\kappa}\over\kappa}={\kappa\over a}\,\bigl{(}v_{0}(0)+2\,v_{2}(0)\bigr{)}.$$

$$\Bigl{(}{\partial v_{\theta}\over\partial\theta}\Bigr{)}_{\theta=0}=-{v_{0}(\tilde{x})\over\epsilon}+2\,\epsilon\,WD_{0}(\tilde{x}),$$

$$\frac{d{v}_2}{d\tilde{x}}=ϵW{D}_0(\tilde{x})+\frac{v_0(\tilde{x})}{ϵ}.$$

$$\frac{d{v}_0}{d\tilde{x}}=-2ϵW{D}_0(\tilde{x})$$

$$\frac{\dot{\kappa }}{\kappa }$$

$$\frac{\partial v}{\partial r}+\frac{v}{r}=0,$$

$$-{\partial v\over\partial r}+{v\over r}=2\,\Bigl{(}{\dot{s}\over 2\,\mu}+{1\over\tau_{pl}}[[{s\over s_{y}}-1]]\Bigr{)}.$$

$${s(r)\over s_{y}}\cong 1+\Bigl{(}{s_{0}\over s_{y}}-1\Bigr{)}\,{R_{1}-r\over R_{1}-R},~{}~{}~{}R<r<R_{1},$$

$${\dot{R}\over R}-{\dot{R}_{1}\over R_{1}}=\Bigl{(}{\dot{s}_{0}\over 2\,\mu}+{1\over\tau_{pl}}[[{s_{0}\over s_{y}}-1]]\Bigr{)}\,\lambda(\bar{\nu})$$

Full text

Recent computational and laboratory experiments have shown that the brittle-ductile transitions in metallic glasses such as Vitreloy1 are strongly sensitive to the initial effective disorder (or “fictive”) temperature. Glasses with lower effective temperatures are weak and brittle; those with higher effective temperatures are strong and ductile. The analysis of this phenomenon presented here examines the onset of fracture at the tip of a slightly rounded notch as predicted by the shear-transformation-zone (STZ) theory of spatially varying plastic deformation. The central ingredient of this analysis is an approximation for the dynamics of the plastic zone formed by stress concentration at the notch tip. This zone first shields the tip but then breaks down suddenly producing a discontinuous transition between brittle and ductile failure, in semiquantitative agreement with the numerical and experimental observations.

Brittle-Ductile Transitions in a Metallic Glass

J.S. Langer

Kavli Institute for Theoretical Physics, Kohn Hall, University of California, Santa Barbara, CA 93106

I Introduction

Two recent developments in fracture mechanics have interesting implications for materials theory. Specifically, the numerical simulations of amorphous crack-tip dynamics by Rycroft and Bouchbinder RB-12 ; VRB-16 and the related experimental results for metallic glasses by Ketkaew et al.SCHetal-18 both demonstrate that amorphous materials are embrittled by forming them with low densities of flow defects. References RB-12 ; VRB-16 show that notch-like indentations are weak and brittle at low effective disorder temperatures and correspondingly low initial densities of shear-transformation zones (STZ’s)FL ; FL-11 ; and that they become stronger and more ductile at higher effective temperatures. According to Ref. SCHetal-18 (see also SCHetal-13 ), crack formation in metallic glasses is enhanced by decreasing their fictive temperatures. That is, glasses are embrittled by quenching them slowly enough through their glass temperatures that they settle into states of relatively low disorder. Conversely, they remain tougher when quenched more quickly.

Fracture toughness is a central issue in materials science that has long been addressed primarily by phenomenology. However, we now have the STZ theory for amorphous plasticity FL ; FL-11 and the thermodynamic dislocation theory for crystalline materials LBL-10 ; JSL-17a ; JSL-19 , both of which are based on fundamental nonequilibrium statistical physics and have been tested by experiment in important but as-yet limited ways. With RB-12 and SCHetal-18 , we have simulational and experimental results directly relevant to the brittle-ductile problem. Thus the time seems ripe to look again at the basic theory of these phenomena and try to understand what is happening.

Here I describe an attempt to interpret the results of RB-12 and SCHetal-18 analytically, and thus to obtain some basic understanding of these phenomena. My strategy is to use an elliptical approximation to describe the tip of a notch in a sheet of material subject to an increasing, mode-I, opening stress. My main assumption is that a crack is launched near this tip when the tensile stress, i.e. the negative pressure in its neighborhood, reaches some material-dependent threshold. Rycroft and Bouchbinder RB-12 assume that cracks in metallic glasses are initiated by stress-induced cavitation events; but there are many other mechanisms that could be operative in other kinds of materials. The critical stress for crack initiation will be one of the important system-dependent parameters in this theory.

I start by considering simple Bingham plasticity (a special case of the STZ theory) with a linear increase in the rate of plastic deformation as a function of stress above a fixed yield stress; and I look at the onset of fracture near the tip of a notch where the rising stress is highly concentrated. I find that both the elastic and plastic dynamics drive the tip to move forward and to sharpen. Here I depart from the conventional wisdom that assumes plasticity always to be a blunting mechanism; but the sharpening effect is obvious from simple physics. The growing concentrated transverse stress in front of the tip moves it forward, and sharpening occurs because the stress concentration is larger at the tip than behind it. This behavior will be shown mathematically in what follows.

The Bingham analysis to be presented here tells us most – but not all – of what we need to know about glassy fracture toughness. As will be seen, the Bingham solid undergoes a smooth transition from brittle to ductile behaviors as the relative strength of plastic versus elastic deformation is increased. When the plastic response is much slower than the loading rate, the fracture threshold is reached by the elastic forces alone and thus the fracture toughness is a relatively small constant as a function of loading rate. This looks like brittle behavior.

With increased plasticity or slower driving, a plastic boundary layer forms at the notch tip and partially shields it from the external stress, thus suppressing the elasticity-induced fracture. Then, when the far-field stress exceeds the yield stress, the boundary layer expands rapidly and the tip stress grows suddenly, thus initiating ductile failure. This rapid expansion of the plastic zone is a well known feature of simple elasto-plastic theories (e.g. see BLLP-07 .) It plays a major role in the present theory. But it does not cause a sharp transition between brittle and ductile behaviors in the Bingham model, at least not in the analysis described here.

The inclusion of STZ dynamics markedly changes this picture. If the system has been quenched to a low effective temperature, then the work done by plastic deformation at the notch tip generates new STZ flow defects, increasing the local plastic deformation rate, and further increasing the STZ production rate. This nonlinear instability eventually produces a sharp transition between brittle and ductile behaviors. It is the central theme of this paper.

Some mathematical elements of this fracture-toughness theory are described in Sec. II. Sections III and IV present the Bingham analysis; Sections V and VI describe the effective-temperature analysis and its predictions. Section VII contains concluding remarks. Some mathematical details are provided in an Appendix.

II Mathematical Preliminaries: The Elliptical Model

Consider a plate of elasto-plastic material lying in the $x,y$ plane and containing an elliptical hole. The ellipse is highly elongated in the $x$ direction and thus has sharp tips at its ends on the $x$ axis. A mode-I stress $\sigma_{\infty}$ is imposed in the $y$ direction very far from the hole. If we assume symmetry about the $y$ axis, then this model is equivalent to a sharp notch with an opening stress $\sigma_{\infty}$.

My scheme is to use the elasto-plastic equations of motion to determine the behavior of this elliptical notch under steadily increasing values of $\sigma_{\infty}$. There is an obvious difficulty. We know that this shape does not remain elliptical under strong forcing; its irreversible motions must involve shape changes that cannot be described simply by time dependent values of the position and curvature of the tip. To minimize this difficulty, I focus only on the immediate neighborhood of the tip and look only at the early onset of plastic deformation there. By the end of this paper, we shall see important limitations of this strategy.

The first step is to transform from Cartesian coordinates $(x,y)$ to elliptical coordinates ($\rho$, $\theta$):

[TABLE]

Curves of constant $\rho$ are ellipses, and curves of constant $\theta$ are orthogonal hyperbolas. If we take the boundary of the elliptical hole to be at $\rho=1$, then the semi-major and semi-minor axes of the ellipse have lengths $W(1+m)$ and $W(1-m)$ respectively. Let $0<m<1$ so that the long axis of the ellipse lies in the $x$ direction, perpendicular to the applied stress, in analogy to a mode-I crack.

To produce the long, thin ellipse, let $W$ become much larger than any other length in the system, and set $m\leq 1$ so that the curvature at the tip, i.e. at $x=W(1+m)$, $y=0$, is large but finite. Denote this curvature by ${\cal K}_{tip}$. Then a calculation to leading order in $1/\sqrt{W}$ yields

[TABLE]

where $\epsilon$ will be the principal small parameter in this analysis.

The linearly elastic version of this problem has been solved by Muskhelishvili MUSK-63 . His general results are summarized in the Appendix, Eqs. (66 -72). For our purposes, his most important formula is the expression for the deviatoric stress $s(\rho,\theta)\equiv s_{\theta\theta}(\rho,\theta)=-s_{\rho\rho}(\rho,\theta)$ given in Eq.(A), which can be used to derive an approximation for $s(\rho,\theta)$ near the tip. Let $\rho=1+\tilde{x}$, use the definition of $\epsilon$ in Eq.(2), and assume that $\theta\ll\epsilon$. I find:

[TABLE]

For $\tilde{x}=0$ and $\theta=0$, Eq.(3) can be written

[TABLE]

where

[TABLE]

Here, $s_{y}$ is the plastic yield stress, which will play a prominent role in what follows, but which has been introduced here primarily for dimensional convenience. Similarly, $a$ is a length scale of the order of magnitude of the initial tip radius, also included for dimensional reasons. (Unlike $\kappa$, $a$ is not a dynamical variable.) Thus, $\psi$ is a dimensionless measure of the stress intensity factor, where the applied stress $\sigma_{\infty}$ is amplified by the large factor $\sqrt{2W/a}$.

At this point, we must begin to pay attention to the plastic zone that forms ahead of the tip when the stress given by Eq.(3) would exceed the plastic yield stress $s_{y}$. This happens at a nonzero value of $\tilde{x}$, say $\tilde{x}_{max}$. Within this zone, where $\tilde{x}<\tilde{x}_{max}$, Eq.(3) is not valid. For $\theta=0$,

[TABLE]

The onset of plastic deformation at the tip occurs when $\nu=1$; that is, when $\psi=\kappa^{-1/2}$.

I have introduced a notation here that will be convenient in much of what follows. For any quantity $f$, the double square brackets mean that $[[f]]=f$ if $f\geq 0$ and $[[f]]=0$ otherwise.

It is important to understand the significance of the quantity $[[\nu-1]]$. According to Eq.(1), the position of the tip on the $x$ axis is

[TABLE]

and the front edge of the plastic zone is at

[TABLE]

Thus, $[[\nu-1]]={\cal K}_{tip}(x_{max}-x_{0})$ is the thickness of the plastic zone in units of the radius of curvature at the tip.

III Bingham Elastoplasticity

The next stage of this investigation is to develop an analytic approximation for elasto-plastic dynamics near the notch tip using only the Bingham model.

For simplicity, assume that the material is incompressible. Also assume hypo-elasto-plasticity (additive decomposition of elastic and plastic rates of deformation). These assumptions imply that the diagonal elements of the rate-of-deformation tensor have the form

[TABLE]

where $s(\rho,\theta)=s_{\theta\theta}(\rho,\theta)=-s_{\rho\rho}(\rho,\theta)$ is the deviatoric stress. Bingham plasticity, with a yield stress $s_{y}$ and a constant plastic rate factor $1/\tau_{pl}$, means that

[TABLE]

where the double square brackets mean the same thing that they did when introduced in Eq.(6). For these purposes, we do not need to consider changes in direction of the stress field or even its values at large distances away from the $x$-axis. Assume that the important behavior is controlled by the elasto-plastic dynamics immediately ahead of the notch tip.

The next question is how to evaluate the stress $s(\rho,\theta)$ for values of $\rho$ and $\theta$ inside the plastic region. About a decade ago, my colleagues and I BLLP-07 considered STZ elasto-plasticity in the neighborhood of an expanding circular hole, where the problem could be solved analytically because variations in the size of the hole and in the neighboring elasto-plastic fields occur only in the radial direction. Our stated motivation was to gain some insight regarding the fracture problem. I shall use two ideas from BLLP-07 , the first being a boundary-layer approximation, and the second a circular approximation for the stress at the tip of the notch.

The boundary-layer approximation for Eq.(12), at and just ahead of the tip, is:

[TABLE]

where, for $0<\tilde{x}<\tilde{x}_{max}$ and $s(0,0)>s_{y}$,

[TABLE]

That is, $s(\tilde{x},0)$ is approximated by a linear function of $\tilde{x}$ across the boundary layer; and $s_{0}=s(0,0)$ is a time dependent boundary stress yet to be determined. This kind of approximation worked well in BLLP-07 ; I shall use it throughout this paper. Note also that, for mathematical consistency in Eq.(13), I have kept only the lowest order correction in the angle $\theta$, moving that dependence outside the double square brackets. The angular dependence of the boundary layer is a higher-order correction in the limit of small $\epsilon$ and $\theta$.

We now can use Eqs.(63) and (64) in the Appendix to express the rate of deformation tensor ${\cal D}$ in terms of the material velocities $v_{\rho}$ and $v_{\theta}$ near the crack tip, and thus use Eq.(III) to write equations of motion for those velocities. Using the same approximations for small $\tilde{x}$ and small $\theta$ used above, I find

[TABLE]

and

[TABLE]

where

[TABLE]

and

[TABLE]

In evaluating $D_{0}(\tilde{x})$, use Eq.(14) for the stress inside the plastic region, and assume that the time derivative of $s(\tilde{x})$ is adequately approximated simply by taking the time derivative of $s_{0}$ in that equation. Outside the plastic region, use Eq.(3) and take the time derivative of $\sigma_{\infty}$.

Set $\theta=0$ in Eq.(16), and use Eq.(20) to compute the tip velocity:

[TABLE]

The final result shown here is obtained by integrating separately over the plastic zone ($0<\tilde{x}<\tilde{x}_{max}$) and the elastic region ($\tilde{x}_{max}<\tilde{x}<\infty$) . The quantity $\nu$ is defined in Eq.(6) as a function of the stress-intensity factor and the tip curvature. The related quantity $\bar{\nu}$ is defined so as to distinguish contributions from inside and outside the plastic region in the integrals over $\tilde{x}$: $\bar{\nu}\equiv\nu$ if $\nu\geq 1$ and $\bar{\nu}\equiv 1$ if $\nu<1$.

Because the tip curvature $\kappa$ has become a time-dependent dynamical variable in these equations, we need an equation of motion for it. Start with the geometric formula JSL-87

[TABLE]

To evaluate this expression, it is useful to define

[TABLE]

so that Eq.(24) becomes

[TABLE]

Now use Eq.(18) at $\theta=0$ to write

[TABLE]

and insert this into Eq.(16). Collecting terms proportional to $\theta^{2}/\epsilon^{2}$, I find

[TABLE]

Then, using

[TABLE]

and combining terms, I find

[TABLE]

With equations of motion for the tip position and curvature, it remains to find an equation of motion for the tip stress $s_{0}$. It is here that I shall use a circular approximation, similar to but not the same as the ones used in VRB-16 and BLLP-07 . Consider a pair of concentric rings in a circular geometry with radial variable $r$ and a radial rate of deformation $v(r)$. The inner ring has a radius $R$ equal to the tip radius $a/\kappa$; and the outer ring is at the boundary of the plastic zone, thus at $R_{1}=\bar{\nu}R$. The analogs of the equations of motion, Eqs.(16) and (18), are

[TABLE]

and

[TABLE]

The first of these equations is the statement of incompressibility, which implies that $v(r)=R\,\dot{R}/r$. If we make the boundary-layer approximation analogous to Eq.(14),

[TABLE]

then we can integrate (34) to find

[TABLE]

where, using $R_{1}/R=\bar{\nu}$,

[TABLE]

Finally, use the expression for $v_{tip}$ in Eq.(III) to evaluate $\dot{R}$, integrate Eq.(29) to evaluate $\dot{R}_{1}$, insert these expressions into the left-hand side of Eq.(36), and solve for $\dot{s}_{0}$. The resulting equation of motion for the tip stress is

[TABLE]

where

[TABLE]

Despite appearances, $\Lambda(\bar{\nu})$ is continuous in both value and slope at the onset of plasticity at $\bar{\nu}=1$. Importantly, it diverges at $\bar{\nu}\cong 5$ describing – but only approximately – the sudden expansion of the plastic zone and rapid unshielding of the notch tip that occurs when the far-field stress exceeds the yield stress.

IV Solutions of the Bingham Equations

Equations (III), (32) and (38) provide a mathematically complete statement of the Bingham problem. It will be useful to restate them in dimensionless form using variables introduced in Sec. II.

Let the dimensionless stress intensity factor $\psi=(\sigma_{\infty}/s_{y})\sqrt{2W/a}$ be the principal independent variable, increasing linearly in time and thus serving as a time-like quantity. Therefore $\dot{\psi}=(\dot{\sigma}_{\infty}/s_{y})\sqrt{2W/a}\equiv 1/\tau_{ex}$ is a constant, and $1/\tau_{ex}$ is the external driving rate. Then, $\dot{\sigma}_{\infty}/s_{y}\,\epsilon=\sqrt{\kappa}/\tau_{ex}$. Measure stresses in units of $s_{y}$, so that $\tilde{s}_{0}\equiv s_{0}/s_{y}$; and define the dimensionless constant $c_{0}\equiv s_{y}/2\,\mu$. ($c_{0}\cong 0.01$ for Vitreloy1.) Define the ratio of time scales to be $\xi\equiv\tau_{ex}/\tau_{pl}$. (Note that this $\xi$ is not exactly the same as the $\xi$ defined in VRB-16 .). Importantly, define the critical failure stress in units of $s_{y}$ to be $\tilde{s}_{c}$. According to RB-12 ; VRB-16 , $\tilde{s}_{c}\cong 4.5$ for Vitreloy1.

Let $\tilde{u}_{tip}(\psi)$ be a dimensionless tip displacement for which $d\tilde{u}_{tip}/d\psi=(\tau_{ex}/a)\,v_{tip}$. Then Eq.(III) becomes an equation of motion for $\tilde{u}_{tip}$:

[TABLE]

Similarly, Eq.(32) becomes

[TABLE]

and the tip-stress equation is:

[TABLE]

where $\Lambda(\bar{\nu})$ is defined in Eq.(39). Also,

[TABLE]

Figures 1 and 2 show graphs of, from top to bottom, the dimensionless tip stress $\tilde{s}_{0}$ (red), the tip curvature $\kappa$ (blue), and the tip displacement $\tilde{u}_{tip}$ (orange dashed) as functions of the steadily increasing, dimensionless stress intensity factor $\psi$. In the first figure $\xi=0.003$; in the second $\xi=0.03$. The first situation looks brittle; the tip stress rises almost linearly with the applied stress and reaches its critical value for fracture, ($\tilde{s}_{c}\cong 4.5$) at $\psi\cong 5$. Both $\kappa$ and $\tilde{u}_{tip}$ diverge at a much larger value of $\psi$ where the system theoretically would undergo rapid plastic failure; but a crack has been launched elastically before the system reaches that state.

In Figure 2, where the plasticity strength $\xi$ is stronger by a factor of ten, the tip becomes shielded by a plastic boundary layer almost immediately as soon as the tip stress reaches the yield stress ($\tilde{s}_{0}\cong 1$), and the graph of $\tilde{s}_{0}(\psi)$ bends over smoothly but abruptly. As a result, the system undergoes ductile failure at a considerably larger value of $\psi$.

Figure 3 shows four $\tilde{s}_{0}(\psi)$ curves for $\xi=0.001,0.005,0.01,{\rm and}\,\,0.05$ along with a dashed line at $\tilde{s}_{0}=\tilde{s}_{c}=4.5$. That line intersects the $\tilde{s}_{0}(\psi)$ curves at the corresponding fracture-toughness values of $\psi$. The full range of those fracture-toughness values as a function of $\xi$ is shown by the semi-log plot in Fig.4. This is the advertised smooth brittle-ductile transition for the Bingham model.

V Effective Temperature Dynamics

To make contact with the Rycroft-Bouchbinder simulations RB-12 , we must introduce the space and time dependent effective temperature $\chi$ that determines the local density of flow defects, that is, the STZ’s. The basic assumption is that the plastic deformation rate is proportional to this density which, in turn, is proportional to an effective thermal activation factor. I write this modified rate factor in the form:

[TABLE]

where $e_{Z}$ is the STZ formation energy. The first factor, $1/\tau_{pl}$, is the same as the one introduced in Eq.(12) to describe Bingham plasticity. The second is the STZ activation factor, and the last term adjusts that factor so that, in the steady-state limit $\chi\to\chi_{\infty}$, we recover the Bingham result.

The effective temperature $\chi$ needs to be evaluated here only on the surface of the crack tip. Thus, I modify Eq.(13) to read

[TABLE]

Let

[TABLE]

and $\chi(t)\equiv\chi(\theta=0,t)$. Then

[TABLE]

and

[TABLE]

We next need equations of motion for $\chi(t)$ and $\gamma(t)$. The basic equation of motion for $\chi(\theta,t)$ has the form

[TABLE]

This is the effective heat-flow equation that has been conventional in STZ theory. $c_{e\!f\!f}$ is the effective specific heat; the product $\tilde{D}^{pl}\,\tilde{s}$ is the rate at which power is delivered to the tip region by the plastic deformation; and $\chi_{\infty}$ is the steady-state value of $\chi$. Eq.(V) tells us what to use for $\tilde{D}^{pl}(0,\theta)$ here; and $\tilde{s}(0,\theta)\approx s_{y}(1-2\theta^{2}/\epsilon^{2})$ is accurate enough for this purpose. Inserting these ingredients into Eq.(51) and setting $\theta=0$, we find

[TABLE]

Then, by equating coefficients of $\theta^{2}$ in Eq.(51), we obtain an equation of motion for the new angular variable $\gamma(t)$:

[TABLE]

As in earlier papers, it is convenient to introduce the notation $\tilde{\chi}=\chi/e_{Z}$ and $\tilde{\gamma}=\gamma/e_{Z}$. Making this substitution, and transforming to the dimensionless variables introduced in Sec.IV, I find for the tip displacement:

[TABLE]

where

[TABLE]

Equation (55) is the same as Eq.(40) except for the factor $\Gamma(\tilde{\chi})$ multiplying $\xi$. Similarly the tip-stress equation (42) is

[TABLE]

The curvature equation becomes

[TABLE]

. The equation of motion for $\tilde{\chi}$, Eq.(52), becomes

[TABLE]

where $c_{1}=(s_{y}/e_{Z}\,c_{e\!f\!f})$ is a dimensionless prefactor. Finally, the equation of motion for $\tilde{\gamma}$, Eq.(V), becomes

[TABLE]

VI Numerical Results for the Effective-Temperature Model

To solve Eqs.(55) - (V) and compare the results with the numerical simulations of RB-12 and experimental data from SCHetal-18 , we need only a small number of system parameters specific to Vitreloy 1. I already have noted in Sec.IV that $c_{0}=s_{y}/2\,\mu\cong 0.01$. Here we also need $c_{1}\cong 0.1$, a value that I deduce from VRB-16 . As will be seen, this fairly small value of $c_{1}$ means that the sharp increase in the STZ density does not occur until the final plasticity-dominated phase of ductile fracture initiation.

As in RB-12 , I denote values of $\tilde{\chi}$ by effective temperatures. Thus $\tilde{\chi}_{\infty}\equiv k_{B}T_{\infty}^{e\!f\!f}/e_{Z}$ and, according to RB-12 , $T_{\infty}^{e\!f\!f}\cong 900\,K$. Similarly, initial effective temperatures are denoted by $T_{0}^{e\!f\!f}$ with $\tilde{\chi}_{0}=k_{B}T_{0}^{e\!f\!f}/e_{Z}$.

The conversion from experimental units of fracture toughness to values of the dimensionless variable $\psi$ is easily accomplished just by fitting the single toughness value in the elastic limit at small $\xi$. Thus, I find that my values of fracture toughness are approximately the reported values $K_{Q}$ in units of ${\rm MPa}\,\sqrt{m}$ multiplied by a factor $0.06$. Similarly, the conversion from driving rate $\dot{K}_{I}$ to $\xi$ needs only a single fitting parameter, $\xi\cong 320/\dot{K}_{I}$.

Figures 5 and 6 are analogous to Figs. 1 and 2 in that they show $\tilde{s}_{0}$, $\tilde{u}_{tip}$, and $\kappa$ as functions of $\psi$. They also show graphs of the STZ density factor $\Gamma(\chi)$, defined in Eq.(56), multiplied in the figure by a factor of $100$ for visibility. Both of these figures are computed with an initial effective temperature $T_{0}^{e\!f\!f}=640\,K$. Figure 5 is plotted for a relatively small rate ratio, $\xi=2$. The peak in the tip stress looks much like the peak in Figure 2 of VRB-16 which was obtained via a circle approximation roughly similar to the one used here but without the boundary-layer dynamics or the relation to the tip parameters. Here, the top of the peak has no special significance; a brittle crack would have been launched earlier when the stress crossed the critical value of $\tilde{s}_{c}=4.5$.

Figure 6, for $\xi=20$, shows what happens on the ductile side of the transition. The peak in $\tilde{s}_{0}(\psi)$ at $\psi\cong 5$ has dropped below $\tilde{s}_{c}$ because of plastic shielding of the notch tip and, thus, the notch continues to elongate and sharpen until it reaches the ductile failure limit at $\psi\cong 8.5$.

Figure 7 shows a set of $\tilde{s}_{0}(\psi)$ curves, analogous to those shown for the Bingham model in Fig. 3, here for $\xi=2,\,5,\,20,\,{\rm and}\,\,100$ from left to right. Apparently, an abrupt brittle to ductile transition occurs for $\xi$ just slightly smaller than $5$, where the $\tilde{s}_{0}(\psi)$ curve is tangent to the horizontal line at $\tilde{s}_{c}=4.5$. Above that value of $\xi$, ductile failure occurs when $\tilde{s}_{0}(\psi)=\tilde{s}_{c}$ at larger values of $\psi$ where the plastic zone expands suddenly and the notch tip is no longer shielded.

Figures 8 and 9 show comparisons between predictions of the effective-temperature theory and the numerical simulation data shown in Fig. 4 of VRB-16 . These two figures are drawn from the data for fracture toughness as functions of driving rate $\dot{K}_{I}$ for $T_{0}^{e\!f\!f}=640\,K\,\,{\rm and}\,\,610\,K$ respectively. They translate into toughness as functions of the rate ratio $\xi$.

Finally, Fig. 10 shows toughness as a function of the initial effective temperature $T_{0}^{e\!f\!f}$ for a fixed driving rate, that is, for $\dot{K}_{I}=10\,{\rm MPa}\,\sqrt{m}/s$ in Fig. 4 of VRB-16 (square data points), and theoretically for $\xi=32$ (red curve). This is the one place where I can make a direct comparison with experimental data. The joined circles in Fig. 10 are taken from Fig. 1a in SCHetal-18 . They should be directly comparable with the other two data sets shown in this figure.

I find these admittedly rough comparisons to be both encouraging and thought provoking. The agreement between theory, simulation, and experiment is good in the sense that the magnitudes and positions of the brittle-ductile transitions are well predicted without the use of arbitrary fitting parameters. But there is obviously a substantial amount of uncertainty about all three data sets in Fig. 10; and whether the data rules out – for example – the discontinuity in the theory seems to me to be an open question.

In my opinion, the fact that the predicted transitions are mathematically discontinuous means that the theory probably is unrealistic in some way. Also, the clear non-monotonicity of the simulation data in Figs. 9 and 10 seems interesting, as has been pointed out by the authors of VRB-16 . But there is no hint of that non-monotonicity in the experiments.

I suspect that the main missing ingredient in this theory is a sufficiently detailed description of the changing shape of the notch tip. For example, Figure 2b in RB-12 shows a bulge with a radius of curvature roughly half that of the tip emerging from the front of the notch and substantially raising the effective temperature in its neighborhood. Rycroft has shown unpublished movies of simulated later stages of ductile yielding in which a bulge of that kind moves forward for a considerable distance before launching a fast crack. There are only hints of such behavior in the present theory. Note that the theoretical notch tip described by the graphs in Fig. 6 does sharpen before reaching its failure limit.

VII Remarks

I have long been skeptical about various aspects of conventional materials science, especially dislocation theory (e.g. see JSL-19 ), because results often are based on non-predictive phenomenology. The results presented here make me more optimistic about opportunities for improving the situation. There are many open issues.

Yielding Transitions. A key assumption throughout this analysis is that the plastic yielding transition is sharp and nonsingular, i.e. that it is Bingham-like near threshold. This is not true in many rheological models, for example, in Herschel-Bulkley models where the flow rate is proportional to the square root of the incremental stress above the yield stress $s_{y}$. There is also the possibility that yielding transitions may be critical phenomena accompanied by diverging fluctuations. That is what happens in athermal quasistatic models which ignore the fact that internal relaxation rates necessarily become faster than external driving rates when the latter vanish at a yield point.

My papers JSL-08 and JSL-15a were written in large part to explore the nature of yielding transitions in STZ theories of amorphous materials, especially metallic glasses. In JSL-15a , I argued from first principles that realistic yielding transitions of this kind are non-critical. I also showed in JSL-15a that the Bingham model can be derived as a limit of STZ theory.

Dislocations. One of my original motives for starting this project was the idea that the new thermodynamic dislocation theory LBL-10 ; JSL-17a ; JSL-19 must be relevant to fracture toughness. The problem of understanding brittleness and ductility in metals and alloys and other crystalline materials is far more complex than it is for amorphous materials. Just the existence of multiple slip systems and grain boundaries and the like makes this topic seem formidable. Nevertheless, important progress has been made in the last decade simply by realizing that dislocations in driven systems must obey the second law of thermodynamics and thus must be amenable to an effective-temperature analysis. That realization has led to successful first-principles theories of strain hardening and sharp yielding transitions, both of which are relevant to fracture.

The picture developed here of brittle fracture being initiated in a metallic glass at a low fictive temperature looks almost identical to the picture of a notch in an unhardened crystalline material with a low initial density of dislocations. The external stress generates dislocations at an effectively hot spot at the tip of the notch. It should be possible to use the new dislocation theory to predict how rapidly that happens and what happens next. There are many such opportunities for progress along these lines.

Fracture Dynamics. This theory of the onset of brittle or ductile fracture occupies just a tiny corner of the large field of fracture dynamics. It is not obvious how to bridge the gap between this corner and the rest of the field.

Note that my equations of motion in Secs. III and V do not look like those that appear in the conventional literature on fracture dynamics. In the conventional picture (e.g. see Freund FREUND ), we visualize a Griffiths-like crack moving on a well defined plane, driven by remote loading that causes elastic energy to flow to the crack tip where that energy is somehow dissipated. For many years, the most promising descriptions of the tip behavior seemed to be the cohesive-zone models of Dugdale and Barenblatt.DUGDALE ; BARENBLATT Sometimes those cohesive zones were called “plastic” zones; but the models never included realistic equations of motion for the plastic flow fields $v_{\rho}^{pl}$ and $v_{\theta}^{pl}$ that appear here. Moreover, it has been known for twenty years that most cohesive-zone models are intrinsically ill-posed; they produce strongly unstable cracks if they describe cracks at all.CLN ; monster

In my opinion, some of the most interesting recent developments in fracture dynamics are those described by Bouchbinder and colleagues in Refs.BFM-10 ; BK-17 ; BK-18 . These authors develop nonlinear field theories to describe the dynamics of fast cracks, and show that their theories predict high-speed behaviors, including instabilities and sidebranching, in agreement with experimental observations. Those theories are not – and cannot be – simple extensions of the quasistatic onset behavior studied here. Both of these related but qualitatively different classes of behavior – the onset behavior that determines brittleness and ductility, the late-stage behavior that determines large-scale failure, and the range of phenomena that lies between them – continue to be highly promising areas for research.

Appendix A Elliptical Formulas

For completeness, I list in this Appendix the formulas on which I have based my analyses. The elliptical coordinates are defined in Eq.(1).

First, there are expressions for the rate-of-deformation tensor ${\cal D}$ in terms of the elliptical material velocity components $v_{\rho}$ and $v_{\theta}$ as derived from more general formulas in Malvern MALVERN .

[TABLE]

[TABLE]

where the metric function is

[TABLE]

Then there are the formulas for incompressible, two-dimensional elasticity that I have derived from Mushkelishvili MUSK-63 . The following formulas assume vanishing normal stress on the surface of the elliptical hole, that is, at $\rho=1$. The stress tensor $\sigma$ is given by

[TABLE]

and

[TABLE]

where

[TABLE]

According to (A) the deviatoric stress has components

[TABLE]

Acknowledgements.

JSL was supported in part by the U.S. Department of Energy, Office of Basic Energy Sciences, Materials Science and Engineering Division, DE-AC05-00OR-22725, through a subcontract from Oak Ridge National Laboratory.