(version 1.1 [v2])

§.§.§ Disclaimer

The views expressed in this work are the personal views of the presenters and do not necessarily reflect the views or policies of current or previous employers.

§ ABSTRACT

We propose a lean and functional transaction scheme to establish a secure delivery-versus-payment across two blockchains, where a) no intermediary is required and b) the operator of the payment chain/payment system has a small overhead and does not need to store state.
The main idea comes with two requirements: First, the payment chain operator hosts a stateless decryption service that allows decrypting messages with his secret key. Second, a "Payment Contract" is deployed on the payment chain that implements a function

transferAndDecrypt(uint id, address from, address to,
string keyEncryptedSuccess, string keyEncryptedFail)

that processes the (trigger-based) payment and emits the decrypted key depending on the success or failure of the transaction. The respective key can then trigger an associated transaction, e.g. claiming delivery by the buyer or re-claiming the locked asset by the seller.
The core idea is depicted in the following slightly simplified sequence diagram:

inststyle+=[bottom color=white, top color=white, rounded corners=3mm]
[blue30]sSeller of Asset
[red30]bBuyer of Asset
inststyle+=[bottom color=white, top color=white, rounded corners=0mm]
[2]aAsset Contract
[1]pPayment Contract
[2]oPayment Operator

sencrypt BE(B)

bencrypt SE(S)

binceptTransfer(id, from, E(S))aemit (id, AssetIncepted)
astore E(S)

sinceptTransfer(id, amount, from, E(B), E(S))pemit (id, PaymentIncepted)

sconfirmTransfer(id, to, E(B))aemit (id, AssetConfirmed)
astore E(B)

alock asset tokens

Alt: cancel (option for seller)
[t]0.7Seller may cancel transaction if buyer fails to call PaymentChain::transferAndDecrypt. We leave this optional termination of the delivery-versus-payment out here.

For details see below or <Ref>.
[opacity=0.0] (s)+(-0.1,0) – (o)+(0.5,0);

btransferAndDecrypt(id, amount, to, E(B), E(S))pemit (id, KeyReleased, K)
pprocess payment orderp[l]E(K) = E(B) if successful,
E(K) = E(S) if failed

pemit (id, KeyRequested, E(K))oreleaseKey(id,K)
o[l]decrypt E(K)
with private keyK

Alt: successK = B (buyer key)
btransferWithKey(id, B)aemit (id, AssetClaimed)
aencrypt K with public key and check E(K) equals stored E(B)transfer asset to buyer

Alt: failedK = S (sellers key)
stransferWithKey(id, S)aemit (id, AssetReclaimed)
aencrypt K with public key and check E(K) equals stored E(S)transfer asset to seller

Sequence diagram of a delivery-versus-payment transaction between a buyer and a seller (of the asset), interacting on the asset and the payment chain. Note the low involvement of the payment chain and payment operator. For a complete version see <Ref>.

§ INTRODUCTION

Within the domain of financial transactions and distributed ledger technology (DLT), the Hash-Linked Contract (HLC) concept has been recognized as valuable and has been thoroughly investigated, see <cit.>. The concept may help to solve the challenge of delivery-versus-payment (DvP), especially in cases where the asset chain and payment system (which may be a chain, too) are separated. The proposed solutions are based on an API-based interaction mechanism which bridges the communication between a so-called Asset Chain and a corresponding Payment System or require complex and problematic time-locks (<cit.>). We believe that an even more lightweight interaction across both systems is possible, especially when the payment system is also based on a DLT infrastructure.

§.§ Acknowledgement

We like to thank Giuseppe Galano and Stephan Mögelin for their valuable feedback.

§ SETUP

Consider the setup of having two chains. The first one is managing tokenized assets. We call this chain the Asset Chain. The second chain can run smart contracts that trigger and verify payments. We call this chain the Payment Chain.

The following setup is assumed:

* Trading parties interact on the asset chain over an Asset Contract and agree on trade terms in a functional way.

* Comparable to HLC, the asset contract can lock asset balances and will transfer them upon certain conditions are met.

* Trading parties interact with the payment chain either by their private keys or via an API interface.

* The payment chain is linked to the payment system and is operated by a payment operator - e.g. a central bank.

* Payment orders are initiated over a payment contract and may be processed, e.g. via the trigger concept <cit.> (not depicted in the diagram above).

* The payment chain operator (or, alternatively, a dedicated service provider) provides a public key that can be used to encrypt “documents”. This key is known to the trading parties interacting on the asset chain.

* The payment operator (or, alternatively, a dedicated service provider) offers a single additional functionality, namely to decrypt given documents (in the following called keys), when the smart contract on the payment chain emits a certain message.

§.§ Notation

The method proposed here relies on a service that will decrypt documents observed in messages emitted by the smart contract on the payment chain. In the following, we call these documents key, because they are used to unlock transactions (on the asset chain). There is a key for the buyer and the seller. In Figure <ref>, these keys are denoted by $B$ and $S$, respectively. The encrypted keys are denoted by $E(B)$ and $E(S)$.

§ REALIZING DELIVERY-VS-PAYMENT WITHOUT EXTERNAL STATE

Based on the assumptions above, we propose a lean function pattern on both chains to implement Delivery-vs-Payment.

§.§ Functionality on Payment Chain:  and

On the payment chain the following function is needed:

* The payment chain operator offers functionality to decrypt any encrypted string (a key).

* The payment is associated with two encrypted keys keyEncryptedSuccess and keyEncryptedFail. Both encrypted key strings have been encrypted using the payment chain operator's public key.[It is assumed that the keys are large and random, such that a collision with $E(B)=E(S)$ is excluded. Otherwise, the inceptTransfer / confirmTransfer with equal encrypted keys is rejected.]

* If and only if the associated payment is successful, the keyEncryptedSuccess will be decrypted to the original key keySuccess and will be published by the payment contract as part of a message that the buyer can use to unlock the assets. In case of a failed payment transfer keyEncryptedFail is decrypted to keyFail, and the keyFail will be emitted by the payment contract, which the seller can use to retrieve back the locked assets.

This functionality is encapsulated in two functions: The first one is initiating the payment details. The second one is performing the payment transfer:

* PaymentChain::inceptTransfer(uint id, int amount, address from, string keyEncryptedSuccess, string encryptedKeyFail) - the receiver of the payment (address: to) is implicitly the function caller: to = msg.sender.

* PaymentChain::transferAndDecrypt(uint id) - requirements: sender == from.

The first one is sent from the seller of the asset; the second one is sent from the buyer of the asset. The ordering of the transactions as given in <Ref> is relevant.

Important: The smart contract on the payment chain does not allow the inception of a payment transfer containing a key that has been used in a previous or current transaction. This is required to avoid a decrypted key being obtained from a different payment transaction.
There are different ways of ensuring this.

§.§ Functionality on the Asset Chain:  and

On the asset chain, the following three functions are needed:

* AssetChain::inceptTransfer(uint id, int amount, address from, string keyEncryptedSeller) - called by the buyer to initiate the trade.

* AssetChain::confirmTransfer(uint id, int amount, address to, string keyEncryptedBuyer) - called by the seller to confirm the trading terms. This call will result in a locking of the asset to be transferred.

* AssetChain::transferWithKey(uint id, string key) - transfers the asset to the buyer if encrypt(key) = keyEncryptedBuyer or to the seller if encrypt(key) = keyEncryptedSeller.

§.§ Workflow

§.§.§ Buyer to Asset Chain

* Buyer generates a random key (string): keySeller.

* Buyer uses the payment chain operator's public key to encrypt the key keySeller to keyEncyptedSeller (E(S)).

* Buyer executes on asset chain AssetChain::inceptTransfer(uint id, int amount, address from, string keyEncyptedSeller)

§.§.§ Seller to Asset Chain

* Seller generates a random key (string):keyBuyer.

* Seller uses the payment chain operator's public key to encrypt the key keyBuyer to keyEncyptedBuyer (E(B))

* Seller executes on asset chain AssetChain::confirmTransfer(uint id, int amount, address to, string keyEncyptedBuyer).

The asset will now be locked by the asset contract to be transferred to the buyer (upon successful payment) or to be transferred back to the seller (upon failed payment).

Seller and buyer can now observe the tuple (keyEncyptedSuccess, keyEncyptedFail) on the asset chain (we implicitly assume that it can be verified that both are well-formed encrypted keys, i.e., for each encrypted key $E(K)$ the owner of the secret key can generate $K$ such that encrypting $K$ with the public key results in $E(K)$.

§.§.§ Seller to Payment Chain

* Seller executes on payment chain PaymentChain::inceptTransfer(uint id, int amount, address from, string keyEncryptedBuyer, string encryptedKeySeller).

Calling the payment contract, the buyer can now verify that the payment transfer has been incepted with the proper parameters.

§.§.§ Buyer to Payment Chain

* Buyer executes PaymentChain::transferAndDecrypt(uint id, address from, address to, string keyEncryptedBuyer, string encryptedKeySeller).

Upon Success:

* Payment contract emits message with keyBuyer (decrypted keyEncryptedBuyer).

* On the asset contract, the buyer is now able to execute AssetChain::transferWithKey(uint id, string key) with key = keyBuyer.

Upon Failure:

* Payment contract emits message with keySeller  (decrypted encryptedKeySeller).

* On the asset contract the seller is now able to execute AssetChain::transferWithKey(uint id, string key) with key = keySeller.

The asset contract will now apply the encryption algorithm with the payment operator's public key to convert the given key to keyEncrypted.

* The asset will be transferred to the buyer if keyEncrypted == keyEncryptedBuyer.

* The asset will be transferred (back) to the seller if keyEncrypted == keyEncryptedSeller.

[5]

§.§ Function Calls

To summarise, there are two interactions with the asset chain establishing the trade, two interactions with the payment chain preparing and executing the payment and one additional interaction with the asset chain to claim or re-claim the asset.

* Buyer to Asset Chain:AssetChain::inceptTransfer(uint id, int amount, address from, string keyEncryptedSeller).

* Seller to Asset Chain:AssetChain::confirmTransfer(uint id, int amount, address to, string keyEncryptedBuyer).

* Seller to Payment Chain:PaymentChain::inceptTransfer(uint id, int amount, address from, string keyEncryptedBuyer, string keyEncryptedSeller)

* Seller to Payment Chain:PaymentChain::cancelTransfer(uint id, int amount, address from, string keyEncryptedBuyer, string keyEncryptedSeller) (optional)

* Buyer to Payment Chain:PaymentChain::transferAndDecrypt(uint id, address from, address to, keyEncryptedBuyer, string keyEncryptedSeller)

* Buyer of Seller to Asset Chain:AssetChain::transferWithKey(uint id, string key), where key is the observed keyBuyer  or keySeller.

§.§ Cancellation

After PaymentChain::inceptTransfer and before PaymentChain::transferAndDecrypt, the seller can cancel the transaction by calling PaymentChain::cancelAndDecrypt.This will decrypt the keySellerEncrypted  to keySeller, and the seller can re-claim his asset.

§ INTERFACES

§.§ Smart Contract on the Asset Chain

inceptTransfer(uint id, int amount, address from, string keyEncryptedSeller)

confirmTransfer(uint id, int amount, address to, string keyEncryptedBuyer)

transferWithKey(uint id, string key)

§.§ Smart Contract on the Payment Chain

inceptTransfer(uint id, int amount, address from, string keyEncryptedBuyer, string keyEncryptedSeller)

transferAndDecrypt(uint id, address from, address to, keyEncryptedBuyer, string keyEncryptedSeller)

cancelAndDecrypt(uint id, address from, address to, keyEncryptedBuyer, string keyEncryptedSeller)

§ ADVANTAGES

Regarding unlocking assets by providing an appropriate secrete, the approach is similar to the Concept of a "Hash-linked Contract". Compared to this and other techniques, the main advantage of the present approach is:

* No Intermediary Service Holding State: Hashes or keys are not required to be stored by a third-party service. Hence, there is no additional point of failure. The payment operator's public key serves as the encryption key.

Besides this, other advantage, where some, but not all, are shared with variants of hash-linked contracts, are:

* No Centralized Key Generation: Keys are generated mutually by the trading parties at the trade inception phase and will not be needed afterwards.

* No Timeout Scheme: The transaction is not required to complete in a given time window, hence no timeout. The timing is up to the two counterparties. Either the buyer initiates the payment (via PaymentChain::transferAndDecrypt) or, in case of absence of payment initiation, the seller cancels the offer and re-claims the asset (via PaymentChain::cancelAndDecrypt).

* No Coupling: The payment chain and the payment chain operator do not need any knowledge of the associated asset. They only offer the possibility to perform a decryption with the payment operator's private key. The smart contract on the payment chain will trigger the decryption.

* Lean Interaction: The function workflow is structured and only consists of three main interactions:

* generate encrypted keys and lock assets,

* send payment order with encrypted keys,

* retrieve decrypted keys and unlock assets.

§ EXTENSIONS AND VARIANTS

§.§ Avoiding a Complex Encryption on the Asset Chain

The above scheme requires that the asset chain smart contract can encrypt the given key with the payment operator's public key and compare the result to the given keyEncrypted.
This is somewhat involved since it requires the implementation of an encryption algorithm on the chain. If this is not an option, the scheme can be modified to rely only on the given key's simple hashing (e.g., sha256).

In the proposed setup above, the encrypted keys $E(S)$ and $E(B)$ serve as a hash. The asset will be released if $S$ or $B$ are presented, and the corresponding encrypted value matches the stored value. This functionality can be implemented with a simple hashing algorithm. In that case, the asset chain will store hashed values $H(S)$ and $H(B)$ and release the asset to the corresponding owner if the hash of the presented key matches the stored hash value.

Thus, the function calls to the asset chain are performed with hashes of the keys, i.e., (keyEncrypted is replaced with keyHashed).

The encrypted keys, $E(S)$ and $E(B)$, are still needed to retrieve the key upon successful payment.

However, in this modified scheme, it has to be ensured that the hashes correspond to the encrypted key. So $H(K)$ corresponds to $E(K)$, i.e., both are derived from the same $K$. To allow verification that a hashed key $H(K)$ (keyHashed) corresponds to the encrypted key $E(K)$ (keyEncrypted), the payment system operator (the owner of the secret key) offers the simple function that returns the hashed key for a given encrypted key without exposing the decrypted key. So it provides the function $f$ with $f(E(K)) = H(K)$ without exposing $K$.

By that additional function, the buyer and seller can verify that the hashes stored on the asset chain are in accordance with the encrypted keys that initialized the payment transfer.

§ PRE-TRADE VERSUS POST-TRADE

If the buyer does not initiate PaymentChain::tranferAndDecrypt, this might be considered a failure to pay. However, this depends on interpreting which actions are deemed part of the trade inception phase and which are regarded post-trade.

Assume that we interpret the first three transactions, i.e.,

* AssetChain:inceptTransfer,

* PaymentChain:inceptTransfer,

* AssetChain:confirmTransfer,

as being pre-trade, manifesting a quote and the seller's intention to offer the asset for the agreed price.

We may interpret the fourth transaction, i.e.,

* PaymentChain::transferAnDecrypt,

as manifesting the trade inception and initiating the post-trade phase. So we could interpret this transaction as a PaymentChain:confirmTransfer that also immediately triggers the completion of the transaction.

If we take the two interpretations above, then PaymentChain::transferAnDecrypt marks the boundary of trade event and post-trade transactions. This interpretation implies that a lack of PaymentChain::transferAnDecrypt does not represent a failure to pay and a PaymentChain::cancelAndDecrypt, initiated by the seller, has the straightforward interpretation of invalidating a pre-trade quote (or offer).

It may be disputed if a failed payment resulting from a PaymentChain::transferAnDecrypt represents a failed inception of the trade (pre-trade) or a failure to pay (post-trade). In any case, the seller is not facing the risk of losing the asset without a payment, and the buyer is not facing the risk of losing the payment without a delivery of the asset.

Note that introducing a locking scheme for the payment is either unnecessary (because the transfer can be completed immediately) or raises a corresponding possibility of a failure-to-deliver (if the asset locking occurs after the payment locking).

§ CONCLUSION

We proposed a decentralized transaction scheme that allows to realize a secure delivery-versus-payment across two blockchain infrastructures without the requirement to hold the state outside the chains. The requirements for the payment system operator are comparably small and are detached from the corresponding asset transaction.

We invite the community to verify that the scheme does not permit attack scenarios or to jointly elaborate the conditions to prevent them. We mentioned one implementation requirement in this direction: the smart contract on the payment chain has to reject opening a payment transaction (PaymentChain::inceptTransfer) that contains an encrypted key that was part of a previous transaction or is currently part of an open payment transaction. This is required because otherwise, it would allow the decryption of a key in a payment transaction (e.g., one with a smaller amount) that can be used to unlock an asset associated with the other payment transaction.

inststyle+=[bottom color=white, top color=white, rounded corners=3mm]
[blue30]sSeller of Asset
[red30]bBuyer of Asset
inststyle+=[bottom color=white, top color=white, rounded corners=0mm]
[2]aAsset Contract
[1]pPayment Contract
[2]oPayment Operator

sencrypt BE(B)

bencrypt SE(S)

binceptTransfer(id, from, E(S))aemit (id, AssetIncepted)
astore E(S)

sinceptTransfer(id, amount, from, E(B), E(S))pemit (id, PaymentIncepted)

sconfirmTransfer(id, to, E(B))aemit (id, AssetConfirmed)
astore E(B)

alock asset tokens

Alt: cancel (if payment is not initiated by buyer)K = S (sellers key)
scancelAndDecrypt(id, amount, to, E(B), E(S))pemit (id, KeyReleased, S)
pemit (id, KeyRequested, E(K))oreleaseKey(id,K)
o[l]decrypt E(K)
with private keyK

Alt: payment
btransferAndDecrypt(id, amount, to, E(B), E(S))pemit (id, KeyReleased, K)
pprocess payment orderp[l]E(K) = E(B) if successful,
E(K) = E(S) if failed

pemit (id, KeyRequested, E(K))oreleaseKey(id,K)
o[l]decrypt E(K)
with private keyK

Alt: successK = B (buyer key)
btransferWithKey(id, B)aemit (id, AssetClaimed)
aencrypt K with public key and check E(K) equals stored E(B)transfer asset to buyer

Alt: failedK = S (sellers key)
stransferWithKey(id, S)aemit (id, AssetReclaimed)
aencrypt K with public key and check E(K) equals stored E(S)transfer asset to seller

Complete Sequence diagram of a delivery-versus-payment transaction between a buyer and a seller (of the asset), interacting on the asset and the payment chain. Note the low involvement of the payment chain and payment operator.


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

Amit Acharya Dept. of Civil & Environmental Engineering, and Center for
Nonlinear Analysis, Carnegie Mellon University, Pittsburgh,USA, email:
<EMAIL_ADDRESS>
###### Abstract

A continuum mechanical model of coupled dislocation based plasticity and
fracture at finite deformation is proposed. Motivating questions and target
applications of the model are sketched.

This paper is dedicated to Professor Nasr Ghoniem on the occasion of his
retirement.

##  1 Introduction

Based on experience and insights gathered from the partial differential
equation (PDE) based modeling of dislocation dynamics in [ZAWB15, AA20, AZA20,
GAM15] and fracture [Ach18, Ach20, MA21], a coupled model of fracture and
dislocation based plasticity at finite deformation is explored. Even though
plasticity, whether fundamentally rooted in the mechanics of dislocations or
in the phenomenology of slip, and fracture are much studied subjects, e.g.
[Fre98, Hut79, HL82, BC06, Asa83, Hav92, and the literature reviews in the
papers mentioned above], to our knowledge, a full-blown continuum PDE model
for their coupled mechanics does not exist and can be useful in the
understanding of the deformation, flow, and fracture of solids (e.g., metals
or glaciers), and the mutual interactions of these phenomena as, e.g.,
addressed in the seminal work [RT74]. While a full thermodynamically
consistent model is presented, it is recognized that this is merely a
beginning that sets the stage for future computation and analysis of a simply
stated, but intricate, nonlinear model which is expected to have some bearing
on its target applications.

An outline of the paper is as follows: in Sec. 2 the mechanical equations of
the model are presented. In Sec. 3 a possible set of thermodynamically
consistent constitutive equations are proposed. In Sec. 4 some target problems
motivating the development of the theory are sketched. It is understood that
most of the questions posed are beyond the reach of rigorous methods of PDE
analysis, but it is felt that demonstrating a dynamical theoretical setup
where such questions can at least begin to be clearly posed and, consequently,
at least be approached with finite-dimensional methods of approximation and
rigorous mathematical guidance, even if short of the ‘proven-theorem’ variety,
can be helpful in advancing the science of deformation, flow, and fracture of
solids.

A few words on notation: tensor components (when invoked) are written with
respect to the basis of a fixed Rectangular Cartesian coordinate system. All
spatial differential operators are w.r.t. position on the current
configuration. A superposed dot represents a material time derivative. $X$
will be the alternating tensor and the $curl$ operator acting on tensor fields
may simply be thought of as row-wise curls of the corresponding matrix field
of components.

##  2 Governing Equations: Mechanics

Based on the detailed kinematic motivations presented in [Ach11, Ach18,
Ach20], the governing equations of the model are given by

$\displaystyle\dot{\rho}+\rho\,\mbox{div}\,v$ $\displaystyle=0\qquad\qquad$
(Balance of mass) (1a) $\displaystyle\rho\dot{v}$
$\displaystyle=\mbox{div}\,T+\rho f\qquad\qquad$ (Balance of linear momentum)
(1b) $\displaystyle\dot{W}+WL$ $\displaystyle=-\mbox{curl}\,W\times
V^{\alpha}+L^{p}\qquad\qquad$ (Evolution of inverse elastic distortion) (1c)
$\displaystyle\dot{c}+L^{T}c$ $\displaystyle=-\mbox{curl}\,c\times
V^{t},\qquad\qquad$ (Evolution of the crack field) (1d)

where $T=T^{T}$ ensures balance of angular momentum, and

$L:=\nabla v$

is the velocity gradient. In the above, $T$ is the Cauchy stress, $\rho$ is
the mass density, $v$ is the material velocity, $f$ is the prescribed body
force density, $W$ is the inverse elastic distortion (a 2-point tensor field),
$V^{\alpha}$ is the dislocation velocity (vector field), $L^{p}$ is a meso-
macroscale construct not used in the fundamental microscale theory, the
plastic distortion rate of dislocations (tensor field) that are ‘averaged out’
in terms of their charge (the meaning of this can be made precise in terms of
microscopic quantities), $c$ is the crack (vector field), and $V^{t}$ is the
crack-tip velocity (vector field). The magnitude of the crack vector field
encapsulates the degree of damage at a material point, while its orientation
reflects that of the crack face normal at that point. An independent vector-
valued field representing the crack-face normal as a fundamental kinematic
ingredient in a PDE model of fracture was introduced in [Ach18, SK19], and is
beginning to be used [MA21, HADMC22, SSK22]. The dislocation and crack-tip
velocity fields are relative velocities of the motion of the dislocation
density field $\alpha$ and the crack-tip field $t$, respectively, w.r.t. the
material velocity. Defining the _dislocation_ and _crack-tip_ line density
fields

$-\mbox{curl}\,W=:\alpha;\qquad-\mbox{curl}\,c=:t$ (2)

(1c) and (1d) imply

$\displaystyle\mathring{\alpha}$
$\displaystyle:=\dot{\alpha}+\mbox{tr}(L)\alpha-\alpha
L^{T}=-\mbox{curl}\,\left(\alpha\times V^{\alpha}+L^{p}\right)$ (3a)
$\displaystyle\mathring{t}$
$\displaystyle:=\dot{t}+\mbox{tr}(L)t-Lt=-\mbox{curl}\,\left(t\times
V^{t}\right)$ (3b)

Physically, (1c, 1d) are motivated from the conservation of topological charge
statements (3a, 3b) on assuming that a ‘free’ gradient that arises in the
process in each case vanishes.

In the above, $W$ and $c$ are considered to be dimensionless physical
quantities.

##  3 Guidance for constitutive assumptions from the Second Law of
Thermodynamics

We consider the free-energy density per unit mass

$\psi=\psi(W,\alpha,c,t,\rho)$ (4)

and require that the power supplied by external agents be greater than or
equal to the rate of change of the sum of the free energy and kinetic energy
of the body:

$\begin{split}&\int_{\partial\mathcal{C}}(Tn)\cdot
v\,da+\int_{\mathcal{C}}\rho f\cdot
v\,dv\geq\frac{d}{dt}\left(\int_{\mathcal{C}}\rho\psi\,dv+\int_{\mathcal{C}}\frac{1}{2}\rho|v|^{2}\,dv\right)\\\
&\Rightarrow\int_{\mathcal{C}}T:L\,dv-\int_{\mathcal{C}}\rho\dot{\psi}\,dv\geq
0\end{split}$ (5)

for any process in which the mechanical equations hold, where $\mathcal{C}$ is
the (time-varying) current configuration of the body; and this is ensured by
choosing constitutive assumptions for $T,V^{\alpha},V^{c},L^{p}$ that
guarantee (5).

Now,

$\begin{split}\rho\dot{\psi}=&\,\rho\left(\partial_{W}\psi:\dot{W}+\partial_{\alpha}\psi:\dot{\alpha}+\partial_{c}\psi\cdot\dot{c}+\partial_{t}\psi\cdot\dot{t}+\partial_{\rho}\psi\dot{\rho}\right)\\\
=&\,\rho\,\partial_{W}\psi:\left(-WL+\alpha\times V^{\alpha}+L^{p}\right)\\\
&+\rho\,\partial_{\alpha}\psi:\left(-\alpha(L:I)+\alpha
L^{T}-\mbox{curl}\,(\alpha\times V^{\alpha}+L^{p})\right)\\\
&+\rho\,\partial_{c}\psi\cdot(-L^{T}c+t\times V^{t})\\\
&+\rho\,\partial_{t}\psi\cdot(-t(L:I)+Lt-\mbox{curl}\,(t\times V^{t}))\\\
&+\rho\partial_{\rho}\psi(-\rho(L:I)).\end{split}$

so that (5) can be expressed as

$\begin{split}&\int_{\mathcal{C}}\left[T+{\color[rgb]{0,0,1}\rho\left\\{W^{T}\partial_{W}\psi+(\partial_{\alpha}\psi:\alpha)I+c\otimes\partial_{c}\psi-\partial_{\alpha}\psi^{T}\alpha+(\partial_{t}\psi\cdot
t)I-\partial_{t}\psi\otimes t+\rho\partial_{\rho}\psi
I\right\\}}\right]:L\,dv\\\
&+\int_{\mathcal{C}}\left[-\rho\,\partial_{W}\psi+\mbox{curl}\,(\rho\,\partial_{\alpha}\psi)\right]:L^{p}\,dv\\\
&+\int_{\mathcal{C}}\left[X\left\\{-\rho\,\partial_{W}\psi+\mbox{curl}\,(\rho\,\partial_{\alpha}\psi)\right\\}^{T}\alpha\right]\cdot{V^{\alpha}}\,dv\\\
&+\int_{\mathcal{C}}\left[\left\\{-\rho\partial_{c}\psi+\mbox{curl}\,(\rho\partial_{t}\psi)\right\\}\times
t\,\right]\cdot V^{t}\,dv\\\
&+\int_{\partial\mathcal{C}}(\rho\partial_{\alpha}\psi\times
n):L^{p}\,da+\int_{\partial\mathcal{C}}\left[\left(\rho\partial_{\alpha}\psi\times
n\right)^{T}\alpha\right]\cdot
V^{\alpha}\,da+\int_{\partial\mathcal{C}}[(\rho\partial_{t}\psi\times n)\times
t]\cdot V^{t}\,da\\\ &\quad\geq 0.\end{split}$ (6)

Since the response function for $\psi$ is invariant under superposed rigid
body motions, it can be shown (see Appendix) that the term highlighted in blue
in (6) is symmetric. Since the stress is symmetric due to balance of angular
momentum, this also implies that the spin ($L_{skw}$) does not appear in the
dissipation for the model making the latter invariant under superposed rigid
motions. This is an important consistency check on the kinematic structure of
the model.

Viewing $L^{p},V^{\alpha}$, and $V^{t}$ (in the bulk and at the boundary) as
the sole dissipative mechanisms of the model, one recovers the stress-relation
of the model from the consideration of energetically reversible, purely
elastic processes:

$T=-{\rho\left\\{W^{T}\partial_{W}\psi+(\partial_{\alpha}\psi:\alpha)I+c\otimes\partial_{c}\psi-\partial_{\alpha}\psi^{T}\alpha+(\partial_{t}\psi\cdot
t)I-\partial_{t}\psi\otimes t+\rho\partial_{\rho}\psi I\right\\}}_{sym}.$ (7)

Finally, a sufficient condition for non-negative dissipation is obtained by
choosing the constitutive assumptions for the dissipative mechanisms to be in
the ‘direction’ of their respective ‘driving forces,’ as exposed in (6), in
the bulk and at the boundary.

###  3.1 A specific set of thermodynamically consistent constitutive
assumptions

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.

##  4 Motivating questions for the development of the model

Here, we outline some fundamental physical problems that served as the
motivation for the development of the theory, and which can be used to
evaluate its predictive capability through analysis and computation in the
future.

###  4.1 ‘Stokes flow’ from nonlinear elasticity with defects

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.

###  4.2 Relation between defects in elastic solids and viscous fluids, and
the mechanical load induced solid-fluid transition

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.

###  4.3 Dislocation nucleation

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

####  4.3.1 Quasi-static balance of forces

The governing equations for $v,W,\alpha$ are:

$\displaystyle\mbox{div}\,\left[\left(T\otimes
I-\mathbb{A}+\partial_{F}TF^{T}\right):\nabla v\right]$
$\displaystyle=\mbox{div}\,\left[\partial_{F}T:\left\\{(F\alpha)\times
V^{\alpha}\right\\}\right]$ (20a) $\displaystyle\dot{W}+WL$
$\displaystyle=\alpha\times V^{\alpha};\qquad\mbox{curl}\,W=-\alpha$ (20b)
$\displaystyle\dot{\alpha}+(\mbox{div}\,v)\alpha-\alpha L^{T}$
$\displaystyle=-\mbox{curl}\,\left(\alpha\times V^{\alpha}\right).$ (20c)

(although the fields $v,W$ suffice, intuition for nucleation related questions
based on prior work suggests working with the $\alpha$ equation for this
question).

  * •

Question: Do perturbations in $\alpha$ from a dislocation-free state
$\alpha=0$ grow? Characterize the instability in terms of the class of elastic
distortion fields $F$ and energy densities $\psi(F)$. The constitutive choices
for $V^{\alpha}$ can be as in (13) and further simplified as necessary, e.g.
assume isotropic mobility.

The initial state satisfies $\mbox{div}\,T=0$ and loading is required.

####  4.3.2 Dynamic balance of forces

In (20) replace (20a) with balance of linear momentum and balance of mass

$\mbox{div}\,T=\rho\dot{v};\qquad\dot{\rho}+\rho\,\mbox{div}\,v=0$

and ask the same question as in Sec. 4.3.1.

###  4.4 Crack nucleation

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 $\mbox{curl}\,W=0$.

####  4.4.1 Quasi-static balance of forces

The governing equations for $v,W,c,t$ are:

$\displaystyle\mbox{div}\,\left[\left(T\otimes
I-\mathbb{A}+\partial_{F}TF^{T}-\mathbb{B}\right):\nabla v\right]$
$\displaystyle=-\mbox{div}\,\left[\partial_{c}T\cdot\left(t\times
V^{t}\right)\right]$ (21a) $\displaystyle\dot{c}+L^{T}c$
$\displaystyle=t\times V^{t}$ (21b) $\displaystyle\dot{t}+(\mbox{div}\,v)t-Lt$
$\displaystyle=-\mbox{curl}\,\left(t\times V^{t}\right).$ (21c)

In the above $\dot{W}+WL=0\Rightarrow\dot{F}F^{-1}=L=\nabla v$. As in the
dislocation case, one of (21b) and (21c) suffices, but can be used as
necessary.

  * •

Question: Do perturbations in $t$ from a crack-free state $c=0\Longrightarrow
t=0$ grow? Characterize the instability in terms of the class of elastic
distortion fields $F$ and energy densities $\psi(F)$. The constitutive choices
for $V^{t}$ can be as in (14).

The initial state satisfies $\mbox{div}\,T=0$ and and loads are required.

####  4.4.2 Dynamic balance of forces

In (21) replace (21a) with balance of linear momentum and balance of mass

$\mbox{div}\,T=\rho\dot{v};\qquad\dot{\rho}+\rho\,\mbox{div}\,v=0$

and ask the same question as in Sec. 4.4.1.

###### Remark 1

For dislocation or crack nucleation from an undislocated or uncracked state,
respectively, the corresponding evolution equations for the perturbation in
dislocation density ($\widetilde{\alpha}$) and crack-tip density
($\widetilde{t}$) are given by

$\dot{\widetilde{\alpha}}_{i3}=-v_{r,r}\,\widetilde{\alpha}_{i3},\qquad\dot{\widetilde{t}}_{3}=-v_{r,r}\,\widetilde{t}_{3}$

where, for simplicity, we assume that $i=1,2$ and only straight
dislocation/crack-tips in the 3-direction are allowed.

Based on the above, it seems that the distinction between crack and
dislocation nucleation in this ansatz is a matter of nonlinear stability. We
note that for the purposes of linear stability, $\mathbb{B}=0$ at the crack-
free state.

###  4.5 Brittle-ductile transition

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.

Figure 1: Schematic of a body with an edge crack under load. In the brittle-
ductile transition, the question is whether under load the crack propagates or
a dislocation (dipole) nucleates and moves [RT74].

  * •

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).

###  4.6 Macroscopic model of elasto-viscoplasticity

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 constantα?

###  4.7 Macroscopic model of damage

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]?

###  4.8 Classical elasto-viscoplasticity, viscoplasticity (a non-Newtonian
viscous fluid), as limit models

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) $\dot{\bar{v}}\to 0$?

## Acknowledgment

This work was supported by the Center for Extreme Events in Structurally
Evolving Materials, Army Research Laboratory Contract No. W911NF2320073, and
NSF OIA-DMR grant # 2021019. It is a pleasure to acknowledge discussions with
Vladimir Sverak.

## Appendix

The argument used here may be called the ‘Ericksen equality’ for the theory,
extending an argument to dynamics of Ericksen [Eri61, sec. VII] in the context
of continuum mechanics of nematic liquid crystals.

Consider a superposed rigid motion of the body on a given motion. For any pair
of such motions, the value of the free energy density function at any material
point remains unchanged at any arbitrarily chosen instant of time, $s$, i.e.,

$\psi(x(X,s))=\psi(R(s)x(X,s)+d(s))\qquad\forall\mbox{ motions
}x(\cdot\cdot,\cdot),\forall R(\cdot),d(\cdot).$ (25)

Under such a superposed rigid motion, the fields $W,\alpha,c,t,\rho$ transform
as follows:

$W(s)\to W(s)R^{T}(s),\qquad\alpha(s)=\alpha(s)R^{T}(s),\qquad c(s)\to
R(s)c(s),\qquad t(s)=R(s)t(s),\qquad\rho(s)\to\rho(s).$

These transformation rules are consistent with the evolution statements
(1c),(1d),(2),(3), if the field $V^{\alpha},V^{t}$ transform as objective
vectors (and $L^{p}$ transforms as an objective 2-point tensor from the
current configuration to a local elastic reference that is unaffected by rigid
motions of the body).

Now consider the free energy density (11), and arbitrarily fixed state
$(W,\alpha,c,t,\rho)$ at an arbitrary instant of time $s$ and compute
$\dot{\psi}(s)$ on a pair of rigidly associated motions as described above for
which $R(s)=I$ and $\dot{R}(s)=S$, where $S$ is an arbitrarily fixed skew
tensor, so that $\dot{R}(s)R^{T}(s)=S$. By (25) the value of $\dot{\psi}(s)$
on both motions have to be equal which implies

$\displaystyle\left[\partial_{W}\psi:\dot{W}+\partial_{\alpha}\psi:\dot{\alpha}+\partial_{c}\psi:\dot{c}+\partial_{t}\psi:\dot{t}+\partial_{\rho}\psi:\dot{\rho}\right]=\dot{\psi}=$
$\displaystyle\left[\partial_{W}\psi:\dot{W}+\partial_{\alpha}\psi:\dot{\alpha}+\partial_{c}\psi:\dot{c}+\partial_{t}\psi:\dot{t}+\partial_{\rho}\psi:\dot{\rho}\right]-\left[\partial_{W}\psi:WS+\partial_{\alpha}\psi:\alpha
S-\partial_{c}\psi\cdot Sc-\partial_{t}\psi\cdot St\right]$
$\displaystyle\Rightarrow\left[W^{T}\partial_{W}\psi+\alpha^{T}\partial_{\alpha}\psi-\partial_{c}\psi\otimes
c-\partial_{t}\psi\otimes t\right]:S=0$
$\displaystyle\Rightarrow\frac{1}{2}\left[\left(W^{T}\partial_{W}\psi-\partial_{W}\psi^{T}W\right)+\left(\alpha^{T}\partial_{\alpha}\psi-\partial_{\alpha}\psi^{T}\alpha\right)-\left(\partial_{c}\psi\otimes
c-c\otimes\partial_{c}\psi\right)-\left(\partial_{t}\psi\otimes
t-t\otimes\partial_{t}\psi\right)\right]=0.$

But this is exactly the skew part of the term highlighted in blue in (6).

## Author Contribution

All work was performed by the sole author.

## Conflict of Interest

I declare that I, the author, have no competing interests as defined by
Springer, or other interests that might be perceived to influence the results
and/or discussion reported in this paper.

## Data Availability

Not applicable. There is no relevant data generated for the paper.

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