Coupled Dislocations and Fracture dynamics at finite deformation:
model derivation, and physical questions

Define

$$F:=W^{-1},\qquad\qquad E:=\frac{1}{2}\left(F^{T}F-I\right),\qquad\qquad r:=F^{T}c,$$

(and we caution that $F$ is not, in general, the gradient of a deformation w.r.t.a fixed global reference configuration). With $\mathbb{I}$ the fourth-order identity tensor on the space of symmetric second order tensors, the intact elastic modulus given by $\mathbb{C}$, the damaged elastic modulus by $\widetilde{\mathbb{C}}$, $\lambda>0,\mu>0$ the intact Lamé parameters, and $\widetilde{\lambda}$ and $\widetilde{\mu}$ the Lamé parameters for the damaged material, define

$$\mathbb{C}:=\lambda I\otimes I+2\mu\mathbb{I},\qquad\qquad\widetilde{\mathbb{C}}:=\widetilde{\lambda}(|r|)I\otimes I+2\widetilde{\mu}(|r|)\mathbb{I},\qquad\qquad\Delta\mathbb{C}_{r}:=\mathbb{C}-\widetilde{\mathbb{C}},$$

where $\widetilde{\lambda},\widetilde{\mu}$ are positive, monotone decreasing functions of $|r|$ from the intact values of the parameters to some small (positive) residual values. Let

$$\widehat{r}:=\frac{r}{|r|},\qquad\qquad\mathbb{R}:=\widehat{r}\otimes\widehat{r}\otimes\widehat{r}\otimes\widehat{r},\qquad\qquad E_{r}:=\widehat{r}\cdot E\,\widehat{r},$$

$\rho_{0}>0$ be the mass density of the intact, unstretched elastic material, and $H(x)=0$ for $x\leq 0$ and $H(x)=1$ for $x>0$ be the Heaviside function (an appropriately smoothed representation will also suffice). We now define the strain energy density of the material accounting for damage due to cracking as (cf. [MA21])

	$\displaystyle 2\rho_{0}\psi_{e}(W,c)$	$\displaystyle=H(|r|)\,E:\left[\widetilde{\mathbb{C}}+(1-H(E_{r}))\left(\Delta\mathbb{C}_{r}\cdot_{4}\mathbb{R}\right)\mathbb{R}\right]:E+\left(1-H(|r|)\right)E:\mathbb{C}E$		(8)
		$\displaystyle=H(|r|)\left[E:\widetilde{\mathbb{C}}E+(1-H(E_{r}))(\Delta\mathbb{C}_{r}\cdot_{4}\mathbb{R})E_{r}^{2}\right]+\left(1-H(|r|)\right)E:\mathbb{C}E.$

$\psi_{e}$ has physical dimensions of energy per unit mass, and $A\cdot_{4}B:=A_{ijkl}B_{ijkl}$ for fourth-order tensors $A,B$.

It can be shown (using arguments given in [Ach11] and [MA21]) that for a crack-only model with $\psi=\psi_{e}(W,c)=\psi_{e}\left(F^{T}F,c\right)$ frame-indifferent, the ‘elastic-distortion driven’ part of the Cauchy stress, say $T_{(F)}$, is given by $T_{(F)}=-\rho W^{T}\partial_{W}\psi_{e}=\rho F\partial_{E}\psi_{e}F^{T}$, and the normal stress component to the crack, $|c|^{-2}c\cdot T_{(F)}c$, at a damaged point where $H(|r|)=1$ is given, up to a factor of $\frac{\rho|r|^{2}}{\rho_{0}|c|^{2}}$, by $\widetilde{\lambda}(tr(E)-E_{r})+\lambda E_{r}+2\mu E_{r}$ if the material point experiences compressive strain characterized by $H(E_{r})=0$, and by $\widetilde{\lambda}tr(E)+2\widetilde{\mu}E_{r}$ if the point experiences tensile normal strain perpendicular to the crack.

We also introduce a crack-resistance energy density function $\eta(|c|)$ with physical dimensions of energy per unit mass. A typical example reflecting no residual energy stored in damaged regions is

$$\rho\,\eta(|c|)=\begin{cases}a\left(1-\cos^{2}\left(\pi\frac{|c|}{c_{sat}}\right)\right),&0\leq|c|\leq c_{sat}\\ 0,&|c|\geq c_{sat},\end{cases}$$

(9)

where $a\geq 0$ (with physical dimensions of $stress$) and $c_{sat}>0$ (dimensionless) are material constants. Another example, modeling Griffith type ‘surface energy’ (but not dependent on crack-length) is

$$\rho\,\eta(|c|)=\begin{cases}a\left(1-\cos^{2}\left(\pi\frac{|c|}{c_{sat}}\right)\right),&0\leq|c|\leq\frac{c_{sat}}{2}\\ a,&|c|\geq\frac{c_{sat}}{2}.\end{cases}$$

(10)

With these constructs a physically reasonable constitutive assumption for the free energy density of our material is

$$\psi(W,\alpha,c,t,\rho)=\psi_{e}(W,c)+\eta(|c|)+\frac{\mu_{\alpha}\,l_{\alpha}^{2}}{2\rho}|\alpha|^{2}+\frac{\mu_{t}\,l_{t}^{2}}{2\rho}|t|^{2},$$

(11)

where $0<\mu_{\alpha},\mu_{t}=\mathcal{O}(\mu)$ are material constants with dimensions of $stress$, and $l_{\alpha},l_{t}>0$ are material constants with dimensions of $length$. The lengths involved are expected to be much smaller than typical macroscopic dimensions.

Turning to the constitutive equation for the dislocation velocity in the bulk, motivated by the ‘driving force’ for dislocation motion in (6), define

$$\mathcal{PK}:=X\left\{-\rho\,\partial_{W}\psi+\mbox{curl}\,(\rho\,\partial_{\alpha}\psi)\right\}^{T}\alpha,\qquad\qquad p:=\frac{X:(F\alpha)}{|X:(F\alpha)|},$$

and a dislocation mobility tensor of the form

$$M=\frac{1}{|\alpha|^{f}}\left(m_{gl}(I-p\otimes p)+m_{cl}\,p\otimes p\right),$$

(12)

where $f=0$ or $1$, and $m_{gl},m_{cl}\geq 0$ are material constants with physical dimensions of $\frac{length^{2}}{stress.time}$ for $f=0$ and $\frac{length}{stress.time}$ for $f=1$. Then we propose the constitutive assumption

$$V^{\alpha}=M\mathcal{PK},$$

(13)

and note that when $\psi$ is independent of $\alpha$,

$$\mathcal{PK}=X\left(T^{T}(F\alpha)\right),$$

which is the natural generalization of the form of the Peach-Koehler force of classical dislocation theory to finite deformation.

For the crack velocity we assume a simple isotropic mobility:

$$V^{t}=\frac{m_{cr}}{|t|^{f}}\left\{-\rho\partial_{c}\psi+\mbox{curl}\,(\rho\partial_{t}\psi)\right\}\times t,$$

(14)

where $m_{cr}>0$ (with same physical dimensions as $m_{cl}$ or $m_{gl}$) is a material constant reflecting crack mobility and $f=0$ or $1$. Ignoring the contribution of the second term in the crack driving force, We note that the crack velocity is not restricted to be in the direction of $t\times c$, allowing crack-tip motions off of the local crack-plane (defined by $c_{\perp}$).

Equations (7), (11), (13), and (14) form the constitutive assumptions of a specific model.

Consider the quasi-static approximation for balance of linear momentum without body force:

$$\mbox{div}\,T=0.$$

Since this holds for all times, it can be shown that this is equivalent to

$$\mbox{div}\,\left[(\mbox{div}\,v)T+\dot{T}-TL^{T}\right]=0,$$

(15)

with $\mbox{div}\,T=0$ initially. Assuming, for simplicity, that $\psi=\psi(F,c)$, we have that $T=T(F,c)$ so that $\dot{T}=\partial_{F}T:\dot{F}+\partial_{c}T\cdot\dot{c}$, and combining with (1c) written in the form

$$\dot{F}F^{-1}=L-(F\alpha)\times V^{\alpha}-FL^{p}$$

(16)

and (1d) we obtain

$$\dot{T}=\partial_{F}T:\left[L-(F\alpha)\times V^{\alpha}-FL^{p}\right]F+\partial_{c}T\cdot\left[-L^{T}c+t\times V^{t}\right].$$

(17)

Combining (17) and (15) and defining

	$\displaystyle\mathbb{L}$	$\displaystyle:=T\otimes I-\mathbb{A}+\partial_{F}TF^{T}-\mathbb{B}$
	$\displaystyle\mathbb{A}_{ikrj}$	$\displaystyle=T_{ij}\delta_{kr}$
	$\displaystyle\mathbb{B}_{ijmk}$	$\displaystyle=\partial_{c_{k}}T_{ij}\,c_{m},$

we have

$$\mbox{div}\,(\mathbb{L}:\nabla v)=\mbox{div}\,\left[\partial_{F}T:\left\{(F\alpha)\times V^{\alpha}+FL^{p}\right\}F-\partial_{c}T\cdot(t\times V^{t})\right].$$

(18)

Evidently, the evolution of cracks and dislocations play the role of a ‘body-force forcing’ in the evolution of the deformation of the body.

In the presence of dislocations and cracks in general, but in the absence of their motion relative to the material, we have

$$\mbox{div}\,(\mathbb{L}(F,c):\nabla v)=0$$

(19)

with standard combinations of Dirichlet b.c. on the velocity on the boundary and Neumann conditions related to the First Piola-Kirchhoff traction (w.r.t the current configuration as the reference) rate.

Evidently, (19) does not reduce to an isotropic $4^{th}$-order tensor acting on the stretching tensor $L_{sym}=D$, but this system is energetically and mechanically (in terms of applied loads) reversible, whereas ‘viscous Stokes flow’ is only mechanically reversible.

In an elastic solid, (dislocation) defects can be said to arise when the inverse elastic distortion is no longer curl-free, i.e.

$$-\mbox{curl}\,W=\alpha\neq 0.$$

In a fluid one might say that defects arise when the velocity gradient develops a ‘singular part,’ thinking, roughly, that the velocity field is discontinuous across 2-d surfaces.

What might be the connection between these two ideas? Can such a connection, in the context of a specific constitutive model, be used to study the transition of a solid to a fluid due to a proliferation of defects?

Noting (1c) rewritten in the suggestive form of (16) and assuming $L^{p}=0$ (a coarse-scale ‘homogenized’ effect), when the $\alpha$ field is a distribution of superposed core fields moving with velocity $V^{\alpha}$, $\alpha\times V^{\alpha}$ very much looks like a singular distribution (when viewed macroscopically), and then (16) suggests that $\dot{F}F^{-1}$ - the elastic part of the velocity gradient (of the solid (fluid?)) - is its ‘regular part’ (the absolutely continuous part), with $F(\alpha\times V^{\alpha})$ being its ‘singular’ part.

Here we consider a model with no cracks and $L^{p}=0$. Assume $\psi=\psi(F);T=T(F)$.

Here we consider a model with no dislocation or plasticity, $\alpha,L^{p}=0$, and $\psi=\psi(F,c);T=T(F,c)$. Here, $W$ is a gradient on the current configuration and $F$ is as well, on the reference defined by the inverse deformation which is a potential for $W$ since $\mbox{curl}\,W=0$.

This is a coupled crack-dislocation problem. The initial condition is that of an unloaded body with an edge crack as shown in Fig. 1.

• •

  Question: Under, say Mode I, loading, i.e., Dirichlet conditions on velocity $v_{2}\neq 0$ on top boundary with bottom fixed as shown, does the stress field of the crack with a concentration at the notch-tip nucleate a dislocation (or a dipole) in the body which then moves (expands) causing plasticity (ductile behavior), or does the crack propagate without any dislocation nucleation and propagation (brittle behavior)? Characterize based on material parameters of the model (for a large body).

Consider the dislocation-only model and define $\varepsilon:=\frac{l_{\alpha}}{H}$, recall (11), where $H$ is a representative dimension of the body and we will be interested in $\varepsilon\to 0$ with $l_{\alpha}$ fixed.

Consider the system

	$\displaystyle 0$	$\displaystyle=\mbox{div}\,T^{\varepsilon}$
	$\displaystyle\dot{W}^{\varepsilon}+W^{\varepsilon}L^{\varepsilon}$	$\displaystyle=-\mbox{curl}\,W^{\varepsilon}\times(V^{\alpha})^{\varepsilon}$
		$\displaystyle``\Leftrightarrow"$
	$\displaystyle\quad\dot{\alpha}^{\varepsilon}+tr(L^{\varepsilon})\alpha^{\varepsilon}-\alpha^{\varepsilon}(L^{\varepsilon})^{T}$	$\displaystyle=-\mbox{curl}\,\left[\alpha^{\varepsilon}\times(V^{\alpha})^{\varepsilon}\right]$

subjected to a constraint on the initial condition

$$\lim_{\varepsilon\to 0}\|\alpha_{0}^{\varepsilon}\|_{L^{2}(\Omega_{0})}=\mbox{constant}_{\alpha},$$

a boundary condition

$$v^{\varepsilon}(\cdot,t)=\bar{v}(\cdot,t)\mbox{ on }\partial\mathcal{C}$$

(22)

where $\bar{v}$ is a given function, and

$$\frac{1}{\mbox{vol}(\mathcal{C})}\int_{\mathcal{C}}\alpha^{\varepsilon}\,dv=0\quad\mbox{for all times}.$$

• •

  Question: (assuming existence of solutions for $\varepsilon>0$, or plausible demonstration of such in finite-dimensional computational settings) What is the limit model that arises as $\varepsilon\to 0$?

  Here, the ‘limit model’ is the question of what model the evolution of the weak limits (roughly, all smoothly weighted space-time averages) of the fundamental fields of the model, say $\alpha^{\varepsilon},v^{\varepsilon}$ here, satisfy. Since the microscopic equations are nonlinear and averages of nonlinear functions of field quantities ( say, e.g., $\alpha^{\varepsilon}\times V^{\varepsilon}$), do not equal the same functions evaluated on the averages of the said quantities, this requires the determination of the evolution of the weak limits of a further set of quantities, like say $|\alpha^{\varepsilon}|^{2}$, defined on a sequence of solutions of the microscopic model. Moreover, to be useful, such evolution of the limits must be ‘closed’ in the sense that it must need information only on the state of only the limits of these quantities at any given time.

  In particular, there is good physical intuition behind the expectation that $\lim_{\varepsilon\to 0}\alpha^{\varepsilon}\times(V^{\alpha})^{\varepsilon}$ produces an extra term in the limit, related to the plastic strain rate produced by the expansion of ‘sub-grid’ loops, the latter not sensed by $\lim_{\varepsilon\to 0}\alpha^{\varepsilon}$. In fact, this is the reason for the phenomenological introduction of the term $L^{p}$ (and only this term as representative of the plastic strain rate) in macroscopic models.

  Is the limit parametrized by constant_α?

Consider the crack-only model and define $\varepsilon:=\frac{l_{t}}{H}$, recall (11), where $H$ is a representative dimension of the body and we will be interested in $\varepsilon\to 0$ with $l_{t}$ fixed.

Consider the system

	$\displaystyle 0$	$\displaystyle=\mbox{div}\,T^{\varepsilon}$
	$\displaystyle\dot{c}^{\varepsilon}+(L^{\varepsilon})^{T}c^{\varepsilon}$	$\displaystyle=-\mbox{curl}\,c^{\varepsilon}\times(V^{t})^{\varepsilon}$
	$\displaystyle``$	$\displaystyle\Leftrightarrow"$
	$\displaystyle\dot{t}^{\varepsilon}+tr(L^{\varepsilon})t^{\varepsilon}-(L^{\varepsilon})t^{\varepsilon}$	$\displaystyle=-\mbox{curl}\,\left[t^{\varepsilon}\times(V^{t})^{\varepsilon}\right].$

In the above, the equivalence is not strict since the bottom equation implies the top one up to the gradient of a scalar field which is assumed to vanish based on the assumption that microscopically the crack-tip flux can occur only in the presence of a crack-tip at a point (much like microscopic plastic strain rate/slipping rate at a point can arise only if a dislocation is present at a point).

Let the system be subjected to a constraint on the initial condition

$$\lim_{\varepsilon\to 0}\|c_{0}^{\varepsilon}\|_{L^{2}(\Omega_{0})}=\mbox{constant}_{c},$$

a boundary condition

$$v^{\varepsilon}(\cdot,t)=\bar{v}(\cdot,t)\mbox{ on }\partial\mathcal{C}$$

where $\bar{v}$ is a given function, and

$$\frac{1}{\mbox{vol}(\mathcal{C})}\int_{\mathcal{C}}t^{\varepsilon}\,dv=0\quad\mbox{for all times}.$$

• •

  Question: As in Sec. 4.6, what is the limit model as $\varepsilon\to 0$? Does a natural connection arise with the type of coupled brittle-ductile model of fracture proposed in [Ach20]?

Consider the dislocation-only model from (1). The classical, phenomenological model of elasto-viscoplasticity is given by the system (1) with $V^{\alpha}=0$ and $L^{p}$ and $T(F)$ specified by constitutive assumptions. The strain-rate decomposition (16), that follows from the conservation of Burgers vector (3a), then takes the form

$$\dot{F}F^{-1}=L-FL^{p}.$$

Recall that the model does not involve a reference configuration of any sort and $F$ is not a deformation gradient in general (customarily it is written as $F^{e}$, but a ‘multiplicative decomposition’ of a deformation gradient from any reference plays no role in our development). Define

	$\displaystyle(\dot{F}F^{-1})_{sym}=:D^{e};$	$\displaystyle\qquad\qquad(\dot{F}F^{-1})_{skw}=:\omega^{e}$
	$\displaystyle L_{sym}=:D;$	$\displaystyle\qquad\qquad L_{skw}=:\omega$
	$\displaystyle(FL^{p})_{sym}=:D^{p};$	$\displaystyle\qquad\qquad(FL^{p})_{skw}=:\omega^{p}$

The constitutive equation for $L^{p}$ specifies $D^{p}$ and $\omega^{p}$; in describing the elastoviscoplasticity of polycrystals without texture, it is customary to assume

$$\omega^{p}=0$$

(but not for single crystals or strongly textured polycrystals).

Considering isotropic elasto-viscoplasticity for simplicity, a typical constitutive assumption for $D^{p}$ is

$$D^{p}=\frac{1}{\nu}\left(\frac{|T^{\prime}|}{g}\right)^{\frac{1}{m}}\frac{T^{\prime}}{|T^{\prime}|},$$

(23)

where $\nu$ has physical dimensions of $time$, $T^{\prime}$ is the stress deviator, $m>0$ is a dimensionless constant called the rate-sensitivity, and $g$, a scalar, is the strength which may itself evolve; a common expression for metals is

$$\dot{g}=\begin{cases}\theta_{0}\left(\frac{g_{s}-g}{g_{s}-g_{0}}\right)|D^{p}|&g<g_{s}\\ 0&g=g_{s},\end{cases}$$

(24)

where $g_{s}\geq g_{0}>0,\theta_{0}>0$ material constants, and $g_{0}$ is an initial value. For ice, the strength does not evolve, staying fixed at $g=g_{0}$. The rate-sensitivity, $m$, for ice is $\sim 4.0$, for metals usually small $\sim 0.01$.

One obtains the classical theory of (rigid) viscoplasticity under the

Assumption: the elastic strain rate is ‘small’, i.e.,

$$\frac{|D^{e}|}{|D^{p}|}\ll 1,$$

so that

$$D=D^{p}$$

is assumed. For the typical power-law constitutive behavior (23), one then has

$$\nu D=\left(\frac{|T^{\prime}|}{g}\right)^{\frac{1}{m}}\frac{T^{\prime}}{|T^{\prime}|}\Rightarrow|T^{\prime}|=g(\nu|D|)^{m};\qquad\qquad T^{\prime}=g(\nu|D|)^{m}\frac{D}{|D|}$$

which also implies incompressibility, and one obtains the constitutive behavior of a non-Newtonian viscous fluid

$$T=-pI+g(\nu|D|)^{m}\frac{D}{|D|},$$

where $p$ is the constitutively undetermined pressure.

• •

  Question: Can classical elasto-viscoplasticity and rigid-viscoplasticity, including the constitutive assumptions (23, 24), be recovered as particular limits of the model in Sec. 4.6 and, if so, under what conditions? Presumably one such condition is $m_{cl}=0$ in (12)?

• •

  Question: Rate-independent behavior is assumed to arise in these models as $m\to 0$ in (23). Can this be justified as a limit when the rate of loading in (22) $\dot{\bar{v}}\to 0$?