Jesús Aranda

Frank D Valencia

Cristian Versari

On the Expressive Power of Restriction and Priorities in CCS with replication

. We do this by showing that there is no encoding of RAMs into CCS -!ν ! which preserves and reflects convergence. We also prove that up to failures equivalence, there is no encoding from CCS ! into CCS -!ν ! nor from CCS -!ν ! α -→ P , then P | Q α -→ P | Q and if α does not involve x also (νx)P α -→ (νx)P .

into CCS -ν

! . As lemmata for the above results we prove that convergence is decidable for CCS -!ν ! and that language equivalence is decidable for CCS -ν ! . As corollary it follows that convergence is decidable for restriction-free CCS. Finally, we show the expressive power of priorities by providing an encoding of RAMs in CCS -!ν !+pr .

Introduction

As for other language-based formalisms (e.g., logic, formal grammars, λ-calculus, etc) a fundamental part of the research in process calculi involves the study of the expressiveness of syntactic fragments or variants of a given process calculus. Process calculi provide a language in which the structure of terms represents the structure of processes together with a transition relation to represent computational steps. Consider for example CCS [START_REF] Milner | Communication and Concurrency[END_REF]. The parallel composition term P |Q, which is built from the terms P and Q represents the process that results from the parallel execution of the processes P and Q. The restriction (νx)P represents a process P with a private/local/restricted/bound resource x-e.g., a location, a link, or a name. Processes with infinite behaviour are often specified with recursive expressions of the form µX.P which behaves as P [µX.P/X], i.e., P with the (free) occurrences of X replaced by µX.P . A transition semantics dictates that if P may have a transition into P by performing an action α, written P Classifying Criteria. One natural approach to comparing expressiveness of two given process calculus variants is by comparing them wrt some standard process equivalence . If there exists a computable function (encoding) [[•]] from the terms of a variant C into the terms of another variant C such that for every P in C we have P [[P ]], we say that C is at least as expressive as C up to .

Another way of classifying the variants of a given calculus is by considering the complexity or decidability of a fundamental property of processes. For example, the decidability of convergence (the existence of a terminating computation) or divergence (the existence of a non-terminating computation).

The CCS ! Calculus. The CCS ! calculus [START_REF] Busi | Replication vs. recursive definitions in channel based calculi[END_REF] is a variant of CCS which instead of using recursive expressions to specify infinite behaviour uses processes of form !P.

The replicated process !P can be thought of as abbreviating the parallel composition P | P | . . . of an unbounded number of P processes. In [START_REF] Busi | Comparing recursion, replication, and iteration in process calculi[END_REF] it is shown that CCS ! is less expressive than CCS wrt (weak) bisimilarity. It is also shown that convergence is undecidable for CCS ! while divergence, unlike for CCS, is decidable.

Turing Expressiveness and Convergence in CCS ! . A remarkable expressiveness result in [START_REF] Busi | Comparing recursion, replication, and iteration in process calculi[END_REF] states that, in spite of its being less expressive than CCS, CCS ! is in fact Turing powerful. This is done by encoding Random Access Machines (RAMs) [START_REF] Minsky | Computation: finite and infinite machines[END_REF]. The fundamental property of the encoding is that it preserves (and reflects) convergence; i.e., the RAM converges if and only if its encoding converges.

The CCS ! encoding of RAMs in [START_REF] Busi | Comparing recursion, replication, and iteration in process calculi[END_REF] uses an unbounded number of restrictions arising from having restriction operators under the scope of a replication operator as for example in !(νx)P . Similarly, the CCS encoding of RAMs in [START_REF] Busi | Replication vs. recursive definitions in channel based calculi[END_REF] involves also an unbounded number of restrictions arising from having restrictions under the scope of recursive expressions as for example in µX.(νx)(P | X). One then may wonder if the generation of unboundedly many names is necessary for Turing Expressiveness. This Work. In this paper we study the expressiveness of restriction and its interplay with replication. We do this by considering two syntactic fragments of CCS ! , namely CCS -!ν ! and CCS -ν ! which differ from CCS ! in the occurrences of restriction under the scope of replication. These fragments and a variant of CCS ! , CCS -!ν !+pr , as well as our classification criteria are described and motivated below.

Although different in nature, our work was inspired by the study of decidable classes (wrt satisfiability) of formulae involving the occurrence of existential quantifiers under the scope of universal quantification. E.g., Skolem showed that the class of formulae of the form ∀y 1 . . . y n ∃z 1 . . . z m F , where F is quantifier-free formula, is undecidable while from Gödel we know that its subclass ∀y 1 y 2 ∃z 1 . . . z m F is decidable [START_REF] Borger | The Classical Decision Problem[END_REF].

The CCS ! Variants. As explained above CCS ! allows processes with restriction under the scope of replication and hence they can generate an unbounded number of restricted names. In order to allow only processes with a number of restricted names bounded by their size, we consider CCS -!ν two restricted names. The reader familiar with CCS can verify that the set of (maximal) finite sequences of visible actions performed by P corresponds to the context-free language a n b n . A similar but slightly more complex example involves a CCS -!ν ! process with only five restricted names whose set of (maximal) finite sequences of visible actions corresponds to the context-sensitive language a n b n c n -see [START_REF] Aranda | Ccs with replication in the chomsky hierarchy: The expressive power of divergence[END_REF]. Now, one may wonder whether a process that uses only a number of restricted names bounded by its size, can be encoded, perhaps by introducing some additional non-restricted names, into one which uses none. For this purpose we shall also consider the restriction-free fragment of CCS ! , which shall denote as CCS -ν ! . Finally, we may also wonder whether some other natural process construct can replace the use in CCS ! of unboundedly many restrictions in achieving Turing expressiveness. For this purpose we shall consider CCS -!ν !+pr which is CCS -!ν ! extended with Phillips' priority guards construct [START_REF] Phillips | Ccs with priority guards[END_REF].

Classifying Criteria. Our main comparison criteria for the above variants are the decidability of convergence and their relative expressiveness wrt failures equivalence [START_REF] Brookes | A theory of communicating sequential processes[END_REF][START_REF] Milner | Communication and Concurrency[END_REF].

As mentioned before, convergence is a fundamental property of processes and its preservation and reflection are also fundamental properties of the encoding of RAMs in CCS ! . Furthermore, we choose it over divergence because the former is undecidable for CCS ! while the latter is already known to be decidable for CCS ! .

Failures equivalence is a well-established notion of process equivalence and we choose it over other equivalences because of its sensitivity to convergence. In fact unlike failures equivalence, other standard equivalences for observable behaviour such as weak-bisimilarity, must testing, trace equivalence and language equivalence may actually equate a convergent process with a non-convergent one. This claim about sensitivity to convergence will be shown later on in the paper (Section 3) once we fix our notation.

Contributions. Our main contributions are the following:

-We show that convergence is decidable for CCS -!ν ! and thus that there is no (computable) encoding, which preserves and reflects convergence, of RAMs using only a bounded number of restricted names. We do this by encoding CCS -!ν ! into Petri Nets. Thus convergence is also decidable for the fragment of CCS with no restrictions within recursive expressions, here refered to as CCS -µν , because of the convergence preserving and reflecting encoding into CCS -!ν ! given in [START_REF] Giambiagi | On the expressiveness of infinite behavior and name scoping in process calculi[END_REF].

-We show that, up to failures equivalence, CCS ! is strictly more expressive than CCS -!ν ! and, similarly, that CCS -!ν ! is strictly more expressive than CCS -ν ! . Thus up to failures equivalence, we cannot encode a process with an unbounded number of restrictions into one with a bounded number of restrictions, nor one with a bounded number of restrictions into a restriction-free process.

-We show that adding Phillips' priority guards to CCS -!ν ! renders the resulting calculus capable of encoding RAMs. Furthermore, unlike the encoding into CCS ! and just like the encoding into CCS, the encoding of RAMs into CCS -!ν !+pr preserves and reflects both convergence and divergence. This bears witness to the expressive power of Phillips' priority guards.

The classification of the various fragments mentioned above are summarized in Figure 1. The undecidability of convergence and decidability of divergence for CCS ! as well as the undecidability of both divergence and convergence for CCS were shown in [START_REF] Busi | Replication vs. recursive definitions in channel based calculi[END_REF][START_REF] Busi | Comparing recursion, replication, and iteration in process calculi[END_REF]. The other results are derived from the work here presented.

The Calculi

CCS processes can perform actions or synchronize on them. These actions can be either offering port names for communication, or the so-called silent action τ. We presuppose a countable set N of port names, ranged over by a, b, x, y . . . and their primed versions. We then introduce a set of co-names N = {a | a ∈ N } disjoint from N . The set of labels, ranged over by l and l , is L = N ∪ N . The set of actions Act, ranged over by α and β, extends L with a new symbol τ. Actions a and a are thought of as complementary, so we decree that a = a. We also decree that τ = τ .

The CCS ! processes [START_REF] Busi | Replication vs. recursive definitions in channel based calculi[END_REF] are defined as in CCS except that recursive expressions are replaced by replication expression of the form !P . Definition 1. Processes in CCS ! are built from names by the following syntax:

P, Q . . . := 0 | α.P | P + Q | P | Q | (νa)P | !P (1)
Convention 1 We use Σ i∈I P i where I = {i 1 . . . i n }, to denote P i1 + . . . + P in ., the order of the summands being insignificant. We use Π i∈I P i , where I = {i 1 . . . i n }, to denote P i1 | . . . | P in . Both Σ i∈I P i and Π i∈I P i are assumed to be 0 if I = ∅. The names a and ā in P are said to be bound in (νa)P . The bound names of P , bn(P ), are those with a bound occurrence in P , and the free names of P , fn(P ), are those with an not bound occurrence in P . The set of names of P , n(P ), is then given by fn(P ) ∪ bn(P ). We use (νa 1 . . . a n )P as an abbreviation of (νa 1 )(νa 2 ) . . . (νa n )P.

Process expressions are endowed with meaning by using the labeled transitions of the form P α -→ Q which intuitively says that P may perform α and evolve into Q. Similarly, P s =⇒ Q, where s ∈ L * , means that P can evolve into Q after zero or more transitions labeled with the elements of s without considering τ moves. Formally, Definition 2. The labeled transition relation

• -→ is given by the rules in Table 1. Define s =⇒, with s = α 1 . . . . α n ∈ L * , as ( τ -→) * α1 -→ ( τ -→) * . . . ( τ -→) * αn -→ ( τ -→) * . For the empty sequence s = , s =⇒ is defined as ( τ -→) * . PREF α.P α -→ P RES P α -→ P (ν a)P α -→ (ν a)P if α ∈ {a, a} SUM1 P α -→ P P + Q α -→ P SUM2 Q α -→ Q P + Q α -→ Q PAR1 P α -→ P P | Q α -→ P | Q PAR2 Q α -→ Q P | Q α -→ P | Q COM P l -→ P Q l -→ Q P | Q τ -→ P | Q REP P | !P α -→ P !P α -→ P Table 1. The labeled semantics of CCS ! .
Intuition and Basic Ideas. We shall now give some intuition and state some conventions on process expressions. We shall concentrate on the expressions !P and (νx)R and their interplay because they are central to our work.

The process P | Q represents the parallel execution of P and Q. Intuitively, P | Q may perform either an action performed by P , an action performed by Q, or if P and Q perform complementary actions (and thus synchronize), a τ action. Thus, a.

P | ā.Q a -→ P | ā.Q, a.P | ā.Q ā -→ a.P | Q, and a.P | ā.Q τ -→ P | Q.
The restriction process (νa)P behaves as P except that it can offer neither a nor ā to its environment. One may think of (νa)P as a local declaration of name a in P and thus can be used to restrict the possible synchronization (interactions) of a process. For example, (νa)(a.P | ā.Q) may not perfom a or ā though -crucially-it may perform a τ action resulting from the synchronization over the actions a and ā. Thus

(νa)(a.P | ā.Q) α -→ for α ∈ {a, ā} but (νa)(a.P | ā.Q) τ -→ (νa)(P | Q).
Replication is the only means to specify infinite behaviour in CCS ! . The replication !P behaves as P | P | . . . | !P ; unboundedly many P s in parallel.

We can use restriction under the scope of replication to specify an unbounded number of local names. It should be clear from the intuition above that (νa)P should behave exactly as (νb)(P [b/a]) where b is not free in P and P [b/a] is the process that results from replacing in P every free occurrence of a with b renaming bound names wherever needed to avoid capture (α-conversion). Thus, for example !(νa)P can be viewed as

(νa 1 )P [a 1 /a] | (νa 2 )P [a 2 /a] | . . . | !(νa)P
thus allowing the declaration of unboundedly many different restricted names a 1 , a 2 , . . .. As previously mentioned in the introduction, this sort of unbounded generation of restricted names appears to be crucial in the encodings of Turing-powerful formalisms.

The Sub-calculi. To understand the above-mentioned interplay between restriction and replication we shall consider two calculi which arise from syntactic constraints over CCS ! . Namely, CCS -!ν ! and CCS -ν ! .

Definition 3 (CCS -!ν ! and CCS -ν ! ). The processes of CCS -!ν ! are those CCS ! processes which do not have occurrences of a process of the form (νx)P within a process of the form !R. The processes of CCS -ν ! are those CCS ! processes with no occurrences of processes of the form (νx)P .

Convergence, Failures and Related Notions

In this section we shall introduce the notions we shall consider as classifying criteria, namely convergence and failures equivalence, as well as some other related notions.

Following [START_REF] Baeten | Decidability of bisimulation equivalence for processes generating context-free languages[END_REF], we say that a process generates a sequence of non-silent actions s if it can perform the actions of s in a finite maximal sequence of transitions. More precisely: Definition 4 (Sequence and language generation). The process P generates a sequence s ∈ L * if and only if there exists Q such that P s =⇒ Q and Q α -→ for any α ∈ Act. Define the language of (or generated by) a process P , L(P ), as the set of all sequences P generates. We say that P and Q are language equivalent, written

P ∼ L Q, iff L(P ) = L(Q).
We recall the notion of failure following [START_REF] Milner | Communication and Concurrency[END_REF]. We need the following notion: Definition 5. We say that P is stable iff P τ -→.

Intuitively we say that a pair e, L , with e ∈ L * and L ⊆ L, is failure of P if P can perform e and thereby reach a state in which no further action (including τ ) is possible if the environment will only allow actions in L.

Definition 6 (Failures).

A pair e, L , where e ∈ L * and L ⊆ L, is a failure of P iff there is P such that: (1) P e =⇒ P , (2) P l -→ for all l ∈ L, and (3) P is stable. Define Failures(P ) as the set of failures of a process P . We say that P and Q are failures equivalent, written

P ∼ F Q iff Failures(P ) = Failures(Q).
We recall the notions of convergence and divergence following [START_REF] Busi | Replication vs. recursive definitions in channel based calculi[END_REF][START_REF] Busi | Comparing recursion, replication, and iteration in process calculi[END_REF]. Intuitively, a process converges if it can reach a stable process after a sequence of τ moves. A process is deemed divergent iff it can perform an infinite sequence of τ moves.

Definition 7 (Convergence and Divergence). We say that P is convergent iff there is a stable process Q such that P ( τ -→) * Q. We say that P is divergent iff P ( τ -→) ω , i.e., there exists an infinite sequence P = P 0 τ -→ P 1 τ -→ . . . . We conclude this section by stating relations between the above notions which we shall use in the following sections.

Some Basic Properties of Failures

We claimed in the introduction that unlike other standard notions such as weak bisimilarity, must testing and trace equivalence, failures equivalence never equates a convergent process with a non-convergent one. In fact,

Proposition 1. Suppose that P ∼ F Q. Then P is convergent iff Q is convergent.
To justify the rest of the above claim, take P = τ.!a.0 and P =!τ.0. Clearly P converges but P does not, however they are both language equivalent. Now take Q = τ.!τ.0 + τ.0 and Q =!τ.0. Thus Q converges but Q does not. It can be verified that Q and Q are equated by these standard equivalences.

We shall use the fact that failures equivalence implies language equivalence.

Proposition 2. ∼ F ⊆ ∼ L .
4 Decidability of Convergence in CCS -!ν

!

In this section we show the decidability of convergence for CCS -!ν ! by a reduction to the same problem for a fragment of Petri Nets.

Convergence-invariant properties in fragments of CCS -!ν

!

Notice that decidability of convergence in CCS -!ν ! can be reduced to the decidability of convergence in CCS -ν ! . Proposition 3. For every P in CCS -!ν ! one can effectively construct a CCS -ν ! process P , such that P converges if only if P converges. Proof. (Outline) First α-convert P so that each bound name in P is replaced with a unique bound name. Then remove from the resulting process each occurrence of a "(νx)". Let P be the resulting restriction-free process. One can verify that P converges iff P converges.

Consequently, in what follows we reduce the convergence problem for CCS -ν ! to convergence problem in Petri nets.

In order to simplify the reduction to Petri Nets, we shall consider the fragment CCS -ν s! of those CCS -ν ! processes in which replication can only be applied to prefix or summation processses. Finitely Branching Transition System. In order to prove the decidability of convergence, we shall make use of an alternative but equivalent definition (up to failures equivalence) of the transition relation for CCS ! . The equivalent definition can be obtained by replacing Rule Rep in Table 1 with the rules in Table 2.

REPL1 P α -→ P !P α -→ P | !P REPL2 P α -→ P P α -→ P !P τ -→ P | P | !P Table 2.
One can verify that the resulting transition relation is finitely-branching. This is essential for being able to provide an effective Petri net construction for any given CCS -ν s! process.

The Reduction to Petri Nets

Here we shall provide a (Unlabelled Place/Transition) Petri Net semantics for CCS -ν s! which considers only the τ moves. For these Petri Nets convergence is decidable [START_REF] Esparza | Decidability issues for petri nets[END_REF].

Definition 8 (Petri Nets).

A Petri net is a 3-tuple (S, T, m 0 ), where S is a set of places, T is a set of transitions M fin (S) × M fin (S) with M fin (S) being a finite multiset of S called a marking. The (non-empty) multiset m 0 is the initial marking; for each place s ∈ S, there are m 0 (s) tokens. A transition (c, p) is written in the form c =⇒ p. A transition is enabled at a marking m if c ⊆ m. The execution of the transition produces the marking m = ( m \ c ) ⊕ p (where \ and ⊕ are the difference and the union operators on multisets). This is written as m £ m . If no transition is enable at m we say that m is a dead marking.

We say that the Petri net (S, T, m 0 ) converges iff there exists a dead marking m such that m 0 (£) * m .

Intuitively, we will associate to each CCS -ν s! a Petri net so that: -Places are identified as syntactic components reachable from P , -Markings are descriptions of processes reachable from P through τ -actions. The places and tokens in the marking represent different syntactic components and their number of occurrences in the process described. -Transitions represent the τ -actions enabled to be performed at certain process. Input places correspond to the components in the process involved in the τ -action and Output places are the components to be enabled once the τ -action has been executed.

Given a Petri net for P the elements of Sub(P ) below will be the syntactic components represented by places in the Petri net. Definition 9. Define Sub(P ), where P ∈ CCS -ν s! , as Sub(0) = {0}, Sub(Σ i∈I P i ) = {Σ i∈I P i } ∪ ( i∈I Sub(P i )), Sub(α.P ) = {α.P } ∪ Sub(P ), Sub(!P ) = {!P } ∪ Sub(P ) , Sub(P | Q) = Sub(P ) ∪ Sub(Q). Sub(P ) denotes the set all null, replicated, summation, prefix processes occurring in P. Since a process P may have several parallel occurrences of an element in Sub(P ) we use a multi-set Occur(P ) take into account its number of occurrences.

Definition 10 (Occurrence). Let P ∈ CCS -ν s! . The multiset of processes which occur in P, Occur(P ), is given by the following rule:

Occur(P ) = Occur(Q) ⊕ Occur(R) if P = Q | R else Occur(P ) = {P }.
Furthermore, we say that Q is an occurrence of a process P if and only if Q ∈ Occur(P ).

Occur(P ) associates to a CCS -ν

s! process P the multiset of its immediate parallel components (occurrences) and will be identified as the marking of P in the Petri net.

We are now ready to define our Petri net encoding of CCS -ν s! processes.

Definition 11 (Nets for CCS -ν s! ). Given a CCS -ν s! process P , we define its Petri net N P = (S, T, m 0 ) where S = {Q | Q ∈ Sub(P )}, m 0 = Occur(P ) and T = T 1 ∪ T 2 where:

T 1 = {{P } =⇒ Occur(P )| P τ -→ P } and T 2 = {{P, Q} =⇒ Occur(P ) ⊕ Occur(Q )|P α -→ P and Q α -→ Q }.
Clearly, given P , N P can be effectively constructed-here we use the finite-branching nature of the alternative transition semantics in Section 4.1.

Roughly speaking, the set of transitions T represents the possible τ moves to be performed and the initial marking Occur(P ) is the one which identifies the process P . In particular:

-T 1 : this type of transition reflects a τ move coming from one of the components, it is referred as P , going to the process P . Notice as a token representing P is consumed and the tokens representing P , there might be more than one component, are added, in this way the transition reflects the evolution from the component P into the process P . -T 2 : this type of transition reflects the τ -actions resulting from the synchronisation of two components P and Q, as a result of the synchronisation the processes P and Q are reached, in this case, a token associated to both P and another one associated to Q are consumed, the tokens representing P and Q are added.

Wee can now state the correctness of the encoding of CCS -ν ! into Petri nets.

Lemma 1 (Convergence-invariance property between CCS -ν ! and Petri nets ). For any CCS -ν s! process P , P converges if and only if the Petri net N P converges.

Since convergence is decidable for Petri nets [START_REF] Esparza | Decidability issues for petri nets[END_REF], we conclude from the above lemma and our effective construction of Petri Nets that convergence is also decidable for CCS -ν s! . Thus, from Propositions 3 and 4, we obtain the following corollary.

Theorem 2. Convergence is a decidable property for CCS -!ν ! processes.

Decidability of Language Equivalence in CCS -ν

!

We now prove that decidability of language equivalence for CCS -ν ! . The crucial observation is that up to language equivalence every occurrence of a replicated process !R in a CCS -ν ! process can be replaced with !τ.0 if R can perform at least an action, otherwise it can be replaced with 0. More precisely, let P [Q/R] the process that results from replacing in P every occurrence of R with Q.

Proposition 5. Let P be a CCS -ν ! process and suppose that !R occurs in P . Then

L(P ) = L(P [Q/!R]) where Q =!τ.0 if there exists α s.t., R α -→ else Q = 0.
Given any R in CCS ! one can effectively decide whether there exists α such that R α -→ (This can be proven using the alternative finitely-branching presentation of the transition relation in Section 4.1). We can then use the above proposition for proving the following statement.

Lemma 2. Let P be a CCS -ν ! process. One can effectively construct a process P such that L(P ) = L(P ) and P is either !τ.0 or a replication-free CCS -ν ! process.

Proof. (Sketch.) Notice that we can use systematically Proposition 5 to transform any CCS -ν ! process P into an language equivalent process Q whose replicated occurrences are all of the form !τ.0. Now a !τ.0 can occur either in a parallel composition, a summation or prefix process. Observe that (1) P | !τ.0 ∼ L !τ.0, (2) !τ.0 | P ∼ L !τ.0, (3) α.!τ.0 ∼ L !τ.0, (4) P +!τ.0 ∼ L P , (5) !τ.0 + P ∼ L P. One can apply (1-5) from left to right to systematically transform Q into the process P as required in the lemma.

From the above lemma, we conclude that every CCS -ν ! process can be effectively transformed into a language equivalent finite-state process. Hence, Theorem 3. Given P and Q in CCS -ν ! , the question of whether L(P ) = L(Q) is decidable.

6 Impossibility results for failure-preserving encodings in CCS ! , CCS -!ν ! and CCS -ν

!

In this section, we shall state the impossibility results about the existence of computable encodings from CCS ! into CCS -!ν ! and from CCS -!ν ! into CCS -ν ! which preserve and reflect failures equivalence. The separation results follow from our previous decidability results and the undecidability results in the literature.

The non-existence of failure-preserving encoding from CCS ! into CCS -!ν ! follows from Proposition 1, Theorem 2 and the undecidability of convergence for CCS ! [START_REF] Busi | Comparing recursion, replication, and iteration in process calculi[END_REF].

Theorem 4. There is no computable function [[•]] : CCS ! → CCS -!ν ! s.t [[P ]] ∼ F P.
To state the non-existence of failure-preserving encoding from CCS -!ν ! into CCS -ν ! we appeal to the undecidability of language equivalence for BPP processes [START_REF] Christensen | Decidability and Decomposition in Process Algebras[END_REF][START_REF] Hirshfeld | Petri nets and the equivalence problem[END_REF]. BPP processes form a subset of restriction-free CCS processes. Now we can use the encoding of [START_REF] Giambiagi | On the expressiveness of infinite behavior and name scoping in process calculi[END_REF] to transform a restriction-free CCS processes into CCS -!ν ! -the encoding is correct up to failures equivalence (see [START_REF] Aranda | On the expressive power of restriction and priorities in ccs with replication[END_REF]). We can therefore conclude, with the help of Proposition 2, that language-equivalence for CCS -!ν ! processes is undecidable. Proposition 6. Given P and Q in CCS -!ν ! , the problem of whether P ∼ L Q is undecidable.

From the above proposition, the decidability of language equivalence for CCS -ν ! (Theorem 3) and Propositions 1 and 2 we can conclude the following.

Theorem 5. There is no computable function [[•]] : CCS -!ν ! → CCS -ν ! s.t. [[P ]] ∼ F P.
Remark 1. We can use the encoding of [START_REF] Giambiagi | On the expressiveness of infinite behavior and name scoping in process calculi[END_REF] to transform any CCS process which uses no restriction within recursive expression into a failures equivalent CCS -!ν ! process [START_REF] Aranda | On the expressive power of restriction and priorities in ccs with replication[END_REF]. Thus, from Proposition 1 and Theorem 2 we can conclude that convergence is also decidable for CCS with no restriction within recursive expressions.

Expressiveness of Priorities

In this section we add Phillips' priority guards [START_REF] Phillips | Ccs with priority guards[END_REF] to CCS -!ν ! . We shall refer to the resulting calculus as CCS -!ν !+pr . This calculus corresponds to Phillips' Calculus of Priority Guards (CPG) with replication rather than recursion and no restrictions within the scope of replication-hence it cannot use an unbounded number of restrictions.

We show that CCS -!ν !+pr turns out to be Turing powerful in the sense of Busi et al [START_REF] Busi | Comparing recursion, replication, and iteration in process calculi[END_REF] (i.e., preserving and reflecting convergence), thus bearing witness to computational expressiveness of priority guards. Recall that from the previous sections CCS ! , and even CCS, cannot encode Turing machines, in the sense above, without using an unbounded number of restrictions (Theorem 2 and Remark 1).

CCS -!ν

!+pr

In CPG there are two sets of names: N which corresponds to the set of names used to represent the visible actions in CCS -!ν ! and a set of priority names U . Each set has a set of complementary actions : N and Ū , where Std = N ∪ N (the standard visible actions), Pri = U ∪ Ū (the priority actions), Vis = Std ∪ Pri (the visible actions), and Act = Vis ∪ τ (all actions). We let a, b, . . . range over N ∪ U ; u, v, . . . over Pri ; λ, . . . over Vis; and α, β, . . . over Act. Also S, T , . . . range over finite subsets of Vis, and U , V , . . . over finite subsets of Pri .

The syntax of processes in CCS -!ν !+pr is like that of CCS -ν ! , except for the summations which now take the form of priority-guarded summations: Σ i∈I S i : α i .P i where I and each S i are finite. The meaning of the priority guard S : α is that α can only be performed if the environment does not offer any action in S Pri (see [START_REF] Phillips | Ccs with priority guards[END_REF] for details).

Labelled Transition and Offers

We recall the set off (P ) of "higher priority" actions "offered" by P .

Definition 12 (Offers). Let P be a CCS -!ν !+pr process and u ∈ Pri . The relation P off u ( P offers u ) is given by the rules in Table 3 . We define off (P ) = {u ∈ Pri : P off u}. Finally, we say that P eschews U iff off (P ) ∩ Ū = ∅.

M + S : u.P + N off u if u / ∈ S P off u P | Q off u Q off u P | Q off u P off u (νa) P off u if a = name(u) P off u !P off u Table 3.
The transitions are conditional on offers from the environment. Intuitively, a transition of the form P α -→ U P means that P may perform α as long as the environment does not offer ū for any u ∈ U (i.e., the environment "eschews" U ). E.g. 4. A RAM can be seen as a program consisting of a finite sequence of instructions labeled with numbers (1 : I 1 ), (2 : I 2 ) . . . , (m : I m ) which modify the values of a finite set of non-negative registers r 1 , . . . , r n . The instructions are either Incr(r j ) which adds 1 to the contents of register r j and goes to the next instruction, or DecJump(r j , l) which tests the register r j value, if it is not zero then decreases it by 1 and goes to the next instruction, otherwise jumps to instruction l.

SUM M + S : α.P + N α -→S∩Pri P if α ∈ S ∩ Pri PAR1 P α -→U P Q eschews U P | Q α -→U P | Q PAR2 Q α -→U Q P eschews U P | Q α -→U P | Q REACT P λ -→U 1 P Q λ -→U 2 Q P eschews U2 Q eschews U1 P | Q τ -→U
A state of a RAM is given by (i, c 1 , . . . , c n ) where i is the program counter indicating the next instruction to be executed, and c 1 , . . . , c n are the current values of the registers r 1 , . . . , r n (resp.). Given a program its computation proceeds by executing the instructions as indicated by the program counter. The execution stops when the program counter reaches the value m + 1 where m is the label of the last instruction; in this case we say that the program terminates.

The Encoding. A register r j with value c j (written r j : c j ) is modeled by a corresponding number of processes of the form u j .

(r j : c

j ) = cj 1 u j
The program counter is modeled with the absence of p i (i.e., the action p i is eschwed by the encoding) indicating that the i-th instruction is the next to be executed. The initial value of the program counter is 1 so by using m+1 i=2 p i we indicate the absence of p 1 . The increasing instruction is modelled with a process (i : Incr(r j )) which is guarded by a τ -action which is only performed when there is an absence of p i .

(i : Incr(r j )) = !({p i } : τ.(p i | p i+1 | u j ))
Once activated, the instruction increases the register r j by offering u j , and goes to the next instruction by both disallowing the current one by offering p i and allowing the next one by performing p i+1 so that p i+1 can be consumed.

The decreasing instruction is defined similarly. In addition we consider the absence of u j to test for zero.

(i : DecJump(r j , l)) =!({p i } : u j .(p i | p i+1 ))|!({p i , u j } : τ.(p i | p l ))
The encoding of a RAM is given below. Without loss of generality we assume that initially the RAM has all its registers set to zero and its program counter is 1.

Definition 13. Let R be a RAM with program instructions (1 : I 1 ), . . . , (m : I m ) and registers r 1 , . . . , r n . We define its encoding into CCS -!ν !+pr as:

R = (νp 1 , . . . , p m+1 , u 1 , . . . , u n )( m i=1 (i : I i ) | n i=1 (r i : 0) | m i=2 p i )
The correctness of the encoding is stated as follows (see the extended version [START_REF] Aranda | On the expressive power of restriction and priorities in ccs with replication[END_REF]). As corollary we obtain that convergence and divergence are undecidable for CCS -!ν !+pr .

Concluding remarks and related Work

The most closely related works are [START_REF] Busi | Replication vs. recursive definitions in channel based calculi[END_REF][START_REF] Busi | Comparing recursion, replication, and iteration in process calculi[END_REF] and they were already discussed in the introduction. In [START_REF] Giambiagi | On the expressiveness of infinite behavior and name scoping in process calculi[END_REF] the authors study replication and recursion in CCS focusing on the role of restriction and name scoping. In particular they show that CCS ! is equivalent to CCS with recursion with static scoping. The standard CCS is shown to have dynamic scoping precisely because the use of restriction within recursive definitions. However, if no restriction appears within recursive expressions then there is no distinction between static and dynamic scoping. Hence, if no restriction is allowed within recursive expressions then we know from [START_REF] Giambiagi | On the expressiveness of infinite behavior and name scoping in process calculi[END_REF] that CCS can be encoded in CCS ! , without restriction under replication, while preserving and reflecting convergence. As for the other direction, clearly νX.(P |X) behaves as !P . Nevertheless, if recursion is required to be prefix guarded, it is not clear how to produce an encoding which preserves and reflects convergence-without appealing to the decidability results for CCS ! here presented. Consider e.g., E = νX.(P |α.X) and !P . If α = τ then E does not converge and !P may-take P = a.0. If α = τ then E may converge and !P may not-take P = τ.0. The authors in [START_REF] Christensen | Decidable subsets of ccs[END_REF] also pointed out the role of restriction in the expressiveness of CCS. They showed that strong bisimilarity is decidable for restriction-free CCS, in contrast with the undecidability result for CCS [START_REF] Taubner | Finite representation of CCS and TCSP programs by automata and Petri nets[END_REF]. It is not clear to us how to relate strong bisimilarity with convergence or failures equivalence.

The authors of [START_REF] Amadio | On decidability of the control reachability problem in the asynchronous π-calculus[END_REF] studied a fragment of the asynchronous π-calculus with restricted forms of bound name generation. A closely related result in of [START_REF] Amadio | On decidability of the control reachability problem in the asynchronous π-calculus[END_REF] is the decidability of the control reachability problem for restriction-free asynchronous π-calculus. This implies the decidability of the same problem for the restriction-free fragment of asynchronous CCS ! (i.e., only 0 can be prefixed with an output action). It is not obvious how to relate control reachability to failures equivalence or convergence. Also it is not clear how to encode our CCS ! fragment into restriction-free asynchronous CCS ! .

In [START_REF] Goltz | Ccs and petri nets[END_REF] a Petri net semantics is proposed for a subset of CCS without restriction and with guarded choice. Also in [START_REF] Taubner | Finite representation of CCS and TCSP programs by automata and Petri nets[END_REF] it was shown that the subset studied in [START_REF] Goltz | Ccs and petri nets[END_REF] can not be extended significantly. These works also presuppose guarded recursion in their fragments which seem crucial for their Petri net constructions. We do not restrict our Petri net construction to guarded sums. Furthermore, as explained above, it is not clear how to translate CCS ! into CCS with guarded recursion while preserving convergence.

In [START_REF] Zavattaro | Termination problems in chemical kinetics[END_REF] the authors show the decidability of convergence for a restriction-free calculus for the compositional description of chemical systems, called CFG which seems closely related to CCS. The calculus, however, presupposes guarded summation and guarded recursion and thus, as argued before, it is not clear how to encode CCS ! into such a calculus while preserving convergence.

In [START_REF] Phillips | Ccs with priority guards[END_REF] it was shown that priorities add expressive power to CCS by modelling electoral systems that cannot be modelled in CCS. Also [START_REF] Versari | On the expressive power of global and local priority in process calculi[END_REF] studies two process algebras enriched with different priority mechanisms. The work reveals the gap between the two prioritised calculi and the two non prioritised ones by modeling electoral systems. Both works state the impossibility of the existence of an encoding subject to certain structural requirements such as homomorphism wrt parallel composition and name invariance. Our derived impossibility result about the non-existence of convergent preserving encodings makes no structural assumptions on the encodings. Finally, we claim that our expressivity results involving priorities are also held by using other priority approaches as they provide the capability of processes to know if another process is ready to perform a synchronisation on some channel or not.

Fig. 1 .

 1 Fig. 1. A (crossed) arrow from C to C represents the (non) existence of an encoding from C into C preserving and reflecting failures equivalence. Convergence is/isn't decidable for each C in/outside the inner rectangle. Divergence is/isn't decidable for each C in/outside the outer rectangle.

Proposition 4 .

 4 For every CCS -ν ! process P , one can effectively construct a CCS -ν s! process Q such that P converges iff Q converges. Proof. (Outline) Systematically replace in P every occurrence of the form !!R with !R, !(Q|Q ) with !Q | !Q , and !0 with 0. The resulting process converges iff P converges.

a

  : b.P b -→ {a} P means that a : b.P may perform b as long as the environment does not offer a. Thus, a : b.P | b.Q could evolve into P | Q however the system a : b.P | b.Q | a could not evolve into P | Q | a as the presence of a prevents the execution of b and thus the τ -action resulting from (b, b) communication. This capability of processes to test the presence or the absence of a channel ready to be performed will be fundamental to represent the test for zero in the encoding of RAMs in CCS -!ν !+pr presented in the next subsection. Transitions are determined by the rules in Table

Theorem 7 .

 7 Let R be a RAM with program instructions (1 : I 1 ), . . . , (m : I m ) and registers r 1 , . . . , r n . Then, R terminates if and only if [[R]] converges. Furthermore, R does not terminate if and only if [[R]] diverges.

Table 4 .

 4 1 ∪U 2 P | Q An operational semantics for CCS -!ν !+pr .

	RES	REP P α -→U P if α / P | !P α -→U P !P α -→U P ∈ {a, a} (ν a)P α -→ U -{a,a} (ν a)P
	SUM	Σi∈I αi.Pi	a j -→ Pj	if j ∈ I
	Convention 6 We write P from the environment thus corresponding to a standard CCS ! transition). Thus, the no-α -→ ∅ P as P α -→ P (i.e., α is not constrained on offers tions of divergence and convergence for CCS -!ν !+pr are obtained as in Definition 7 by replacing τ -→ with τ -→ ∅ .
	7.2 Encoding RAMs in CCS -!ν !+pr

Acknowledgments. We would like to thank Ian Phillips and Uwe Nestmann for insightful discussions on the topics here studied. We are also grateful to Jorge A. Pérez and the anonymous reviewers for their remarks and suggestions.

The work of Jesús Aranda has been supported by COLCIENCIAS (Instituto Colombiano para el Desarrollo de la Ciencia y la Tecnología "Francisco José de Caldas") and INRIA Futurs. scope of replications. To illustrate the expressiveness of CCS -!ν ! take for example P = (νk)(νu)( k | !


Thomas Zillhardt
email: thomaszillhardt@cantab.net

Genoveva Burca

Wolfgang Ludwig

Dong Liu

T James

T James Marrow

In Situ Observation of Crystal Reorientation in Polygranular Graphite by Synchrotron X-Ray Diffraction and Neutron Imaging

Keywords: X-ray diffraction, crystal reorientation, nuclear graphite, Bragg-edge neutron imaging

come

In situ observation of crystal reorientation in polygranular graphite by synchrotron X-ray diffraction and neutron imaging

Introduction

Polygranular graphite is an important functional and structural material in energy applications. For instance, large blocks of synthetic graphite have been used in the steel refining industry for Electric Arc Furnaces (EAF) for many years [START_REF] Bystrov | Modelling of the thermal state and the melting loss of a graphite electrode in the conditions of the evaporative cooling in the arc furnace[END_REF] and as electrodes in Electro-Spark Deposition (ESD) to improve corrosion resistance of metals [START_REF] Kuptsov | Electro-spark deposition in vacuum using graphite electrode at different electrode polarities: Peculiarities of microstructure, electrochemical and tribological properties[END_REF]. Graphite is also widely used in the anodes of lithium-ion batteries, in which its in operando dimensional changes are important [START_REF] Sauerteig | Reversible and irreversible dilation of lithium-ion battery electrodes investigated by in-situ dilatometry[END_REF][START_REF] Michael | A Dilatometric Study of Graphite Electrodes during Cycling with X-ray Computed Tomography[END_REF]. For over half a century, graphite has been critical as a neutron moderator in gas-cooled nuclear fission reactors [START_REF] Marsden | Dimensional change, irradiation creep and thermal/mechanical property changes in nuclear graphite[END_REF] and novel graphite microstructures are now being investigated for use in next-generation advanced fission reactors [START_REF] Marsden | 14 -Graphite as a core material for Generation IV nuclear reactors[END_REF], particularly thermal neutron versions of Very High Temperature (VHTR) [START_REF] Zhou | Nuclear graphite for high temperature gas-cooled reactors[END_REF][START_REF] Zhou | Carbon materials in a high temperature gas-cooled reactor pebble-bed module[END_REF] and Molten Salt Reactors (MSR) [START_REF] Mignacca | Economics and finance of Molten Salt Reactors[END_REF]. Graphite's thermal properties led to its consideration in fusion reactors as a potential plasma facing component [START_REF] Miyahara | Graphite as plasma facing material[END_REF], and it is also of interest for its tritium recycling capabilities [START_REF] Vergari | Thermodynamics of hydrogen in graphite at high temperature and the effects of oxidation, irradiation and isotopics[END_REF].

The room temperature elastic properties of unirradiated polygranular graphite are nonlinear [START_REF] Yoda | Change in dynamic young's modulus of nuclear-grade isotropic graphite during tensile and compressive stressing[END_REF]. Fine scale microcracks in the microstructure ('Mrozowski' accommodation pores [START_REF] Mrozowski | Mechanical strength, thermal expansion and structure of cokes and carbons[END_REF]) have an important influence on its thermal expansion and elastic modulus, and it has been proposed that deformation mechanisms including microcracking decrease the elastic modulus with increasing tensile strain [START_REF] Mrozowski | Mechanical strength, thermal expansion and structure of cokes and carbons[END_REF][START_REF] Brocklehurst | Analysis of the dimensional changes and structural changes in polycrystalline graphite under fast neutron irradiation[END_REF]. In nuclear applications, significant dimensional and property changes occur with irradiation, which can lead to the development of internal stress and mechanical damage [START_REF] Marsden | Dimensional change, irradiation creep and thermal/mechanical property changes in nuclear graphite[END_REF].

Irradiation also removes the non-linearity of polygranular graphite's elastic deformation [START_REF] Brocklehurst | Analysis of the dimensional changes and structural changes in polycrystalline graphite under fast neutron irradiation[END_REF] and affects its strength, stiffness and fracture behaviour [START_REF] Jin | Assessment of the fracture toughness of neutron-irradiated nuclear graphite by 3D analysis of the crack displacement field[END_REF]. Polygranular synthetic graphite microstructures are complex and depend on the manufacturing processes and raw materials [START_REF] Lee | Bulk graphite: materials and manufacturing process[END_REF]. This work considers Gilsocarbon graphite (GCMB/IM1-24), which is used as moderator, reflector and fuel sleeves in the UK Advanced Gas-Cooled Reactors (AGR). The spherical Gilsonite filler particles, together with the vibration moulding manufacturing process, give it near-isotropic properties. Studies of graphite crystal properties are necessary to provide essential information for microstructure-based predictive models for the effects of irradiation and oxidation [START_REF] Vertyagina | A multi-scale three-dimensional Cellular Automata fracture model of radiolytically oxidised nuclear graphite[END_REF][START_REF] Hall | The microstructural modelling of nuclear grade graphite[END_REF]. Such models are required to understand the general effects of irradiation and temperature on graphite properties and will support the design and operation of advanced fission reactors for which new isotropic graphites are being developed [START_REF] Abram | Generation-IV nuclear power: A review of the state of the science[END_REF].

The crystals in synthetic graphite have Bernal ABAB stacking of hexagonal planes that can shear by basal dislocation glide [START_REF] Soule | Direct Basal-Plane Shear in Single-Crystal Graphite[END_REF] and kink band formation [START_REF] Barsoum | Kink bands, nonlinear elasticity and nanoindentations in graphite[END_REF][START_REF] Marrow | High temperature spherical nano-indentation of graphite crystals[END_REF]. Twinning can also occur in graphite [START_REF] Palache | Contributions to the mineralogy of sterling hill, new jersey: morphology of graphite, arsenopyrite, pyrite, and arsenic[END_REF], [START_REF] Freise | Twinning in Graphite[END_REF], and has been defined as a rotation of approximately 20° about < 11 ̅ 01 > [START_REF] Kennedy | Dislocations and Twinning in Graphite[END_REF]. Twinning under load in natural graphite flakes has been observed [START_REF] Skinner | The deformation and twinning of graphite crystals under a point load[END_REF] and it has also been reported in a micro-mechanical study of a neutron-irradiated synthetic graphite that twins recover once the load is removed [START_REF] Heard | Deformation and fracture of irradiated polygranular pile grade A reactor core graphite[END_REF]. Twinning is an important deformation mechanism in many materials, but there is limited knowledge of the role of twinning in polygranular graphite and no experiment to date has investigated its occurrence in situ under load in the bulk material.

The microstructure of synthetic polygranular graphite contains a high level of thermally induced residual stress due to the strong graphite crystal anisotropy and high processing temperatures [START_REF] Krishna | Residual stress measurements in polycrystalline graphite with micro-Raman spectroscopy[END_REF]. Synchrotron X-ray diffraction studies have shown that these internal stresses are relaxed upon mechanical loading [START_REF] Marrow | In situ measurement of the strains within a mechanically loaded polygranular graphite[END_REF]. Twinning, kink band formation and basal plane slip all are mechanisms by which crystals can deform in the microstructure, and local variations in residual stress are expected to affect their onset when the microstructure is subjected to additional strain.

To investigate these mechanisms, it is necessary to be able detect changes in the orientation of crystals in the polygranular microstructure, which requires advanced diffraction techniques. Electron backscatter diffraction (EBSD) is not readily applicable to graphite due to the difficulty of obtaining an adequate surface [START_REF] Haverty | The microstructure, texture and thermal expansion of nuclear graphite[END_REF]. Although grain mapping by Diffraction Contrast Tomography (DCT) has been used to provide threedimensional grain maps of metals and ceramics [START_REF] Gonzalez | Three-dimensional observation and image-based modelling of thermal strains in polycrystalline alumina[END_REF][START_REF] King | Observations of Intergranular Stress Corrosion Cracking in a Grain-Mapped Polycrystal[END_REF], tests by some of the authors that were completed at ESRF-ID11 [START_REF] Zillhardt | In-situ Mapping of Twinning in Graphite by Diffraction Contrast Tomography @ID11[END_REF] with Gilsocarbon graphite found that it did not provide the clear diffraction spots necessary for pair-matching and grain reconstruction.

Three-Dimensional Synchrotron X-ray Diffraction (3DXRD) was developed for lowresolution crystal orientation mapping of samples with large grains (of the order of a few hundred microns) [START_REF] Renversade | Analysis of in-grain orientation distributions in deformed aluminium using X-ray diffraction and finite elements[END_REF][START_REF] Hayashi | Polycrystal orientation mapping using scanning three-dimensional X-ray diffraction microscopy[END_REF]. Ideally, the materials investigated have well-ordered crystal domains from which segmentation of the diffracted spots and pair-matching can be used to obtain 3D maps of crystal shape and dimension, such as in Diffraction Contrast Tomography (DCT). In graphite, although there are large domains (>100's of µm) of crystallites with parallel basal (0001) planes [START_REF] Marsden | Dimensional change, irradiation creep and thermal/mechanical property changes in nuclear graphite[END_REF], these have a high degree of misorientation along [0001] [START_REF] März | Mesoscopic structure features in synthetic graphite[END_REF]. Consequently, diffraction patterns of sufficient quality for 3DXRD indexing and reconstruction are not attainable. However, a simuli-3DXRD experiment may sample a gauge volume in which a strong local texture from the graphite domain structure will give partial Debye-Scherrer cones, observed as partial rings on a 2D detector. The distribution of intensity around these rings should be sensitive to changes in grain orientation (e.g., kinks or twinning) and coherence (i.e., anisotropic plastic deformation).

Bragg-edge neutron imaging is a fast-developing acquisition tool that can provide energy-resolved information of internal structures in materials, and recent developments of in situ wavelength-dependent imaging allow accurate determination of crystal lattice strains [START_REF] Song | Characterization of Crystallographic Structures Using Bragg-Edge Neutron Imaging at the Spallation Neutron Source[END_REF][START_REF] Zillhardt | In Situ Mechanical Loading and Neutron Bragg-edge Imaging, Applied to Polygranular Graphite On IMAT@ISIS[END_REF]. A relatively unexplored use of Bragg-edge imaging is direct image processing and analysis of extinction spots for those crystals that are in the Bragg-condition. This application of energy-resolved neutron imaging is particularly suitable for materials with grains or domains over 100 µm in size as neutron detectors currently have quite low spatial resolution compared to X-ray imaging. Briefly, when the neutron wavelength is much larger than twice the interplanar spacing of the periodic crystal lattice planes, there is no coherent scattering. However, at wavelengths where the Bragg condition is satisfied by suitably oriented planes there is a marked change in the transmitted intensity (i.e., a Bragg edge) [START_REF] Fermi | The Transmission of Slow Neutrons through Microcrystalline Materials[END_REF]. On a radiograph of a polycrystalline material at a specific energy, extinction spots with reduced intensity arise from the grains in which the neutrons have been scattered at the corresponding Bragg edge. At energies close to the Bragg edge, the intensity of these spots is most sensitive to changes in crystal orientation or changes in scattering due to the disruption of coherency by plastic deformation [START_REF] Reid | Mapping of axial plastic zone for roller bearing overloads using neutron transmission imaging[END_REF]. A recent in situ study of the deformation of Gilsocarbon graphite at the ISIS spallation neutron and muon source on the IMAT (Imaging and MATerials science and engineering) instrument observed local changes in Bragg contrast images [START_REF] Zillhardt | In Situ Mechanical Loading and Neutron Bragg-edge Imaging, Applied to Polygranular Graphite On IMAT@ISIS[END_REF]. As Bragg contrast is affected by local changes in scattering, this may provide evidence for local grain reorientation or plastic deformation [START_REF] Gonzalez | Three-dimensional observation and image-based modelling of thermal strains in polycrystalline alumina[END_REF][START_REF] Renversade | Analysis of in-grain orientation distributions in deformed aluminium using X-ray diffraction and finite elements[END_REF][START_REF] Nakai | Observations of twinning and detwinning in magnesium alloy by synchrotron radiation DCT and EBSD[END_REF][START_REF] Proudhon | Incipient Bulk Polycrystal Plasticity Observed by Synchrotron In-Situ Topotomography[END_REF].

The objective of this study was therefore to apply the advanced and in situ characterisation techniques of Bragg-edge neutron imaging and 3DXRD to investigate the crystal deformation mechanisms by which the synthetic polygranular Gilsocarbon graphite accommodates strain. The study was conducted with un-irradiated graphite at room temperature during compressive loading, as the first step towards future investigations of the effects of neutron irradiation and elevated temperature that are relevant to its performance in nuclear engineering.

Experimental Details

Material

The material, supplied by EDF Energy Generation UK, is a moulded IM1-24 Gilsocarbon (GCMB grade) polygranular nuclear graphite that was manufactured by Graftech (formally UCAR). It has weakly anisotropic properties and the elastic Young's modulus is reported to be between approximately 11.6 and 11.9 GPa (Poisson ratio of 0.2), with tensile strength of ~20 MPa and compressive strength of ~75 MPa [START_REF] Preston | Changes in the coefficient of thermal expansion in stressed Gilsocarbon graphite[END_REF]. It is one of the graphite grades used in the core of the UK Advanced Gas-cooled Reactor fleet. The specimens, prepared using Electrical discharge machining (EDM), are presented schematically in Figure 1 with their loading methods.

Energy-resolved Neutron Imaging

The in-situ neutron Bragg-edge imaging experiment took place on the IMAT beamline at the STFC-ISIS facility under experiment number RB1910517-2 [START_REF] Marrow | RB1910517-2: High Resolution Neutron Bragg Imaging of Gilsocarbon Graphite for the Investigation of Crystal Reorientation @IMAT[END_REF]. Except where noted, it followed the setup and analysis described in the previous method paper [START_REF] Zillhardt | In Situ Mechanical Loading and Neutron Bragg-edge Imaging, Applied to Polygranular Graphite On IMAT@ISIS[END_REF].

The pinhole aperture and the neutron count (exposure time) were optimised to increase the counting statistics and the Signal-to-Noise Ratio (SNR). Two specimens of slightly different geometry were investigated (Figure 1 a and b). Specimen A was a round disc (20 mm diameter, 𝐷, × 10 mm thickness, 𝑇) that was manually loaded across its diameter using curved steel anvils (radius 42.5 mm) with a relative curvature to the specimen of  = 0.24, which is in line with recommendations for diametral compression testing of brittle materials [START_REF] Ch | The influence of jaw's curvature on the results of the Brazilian disc test[END_REF]. Specimen B adopted a slightly larger size disc design (25 mm diameter, 𝐷, × 12.5 mm thickness, 𝑇), with diametrically opposed flattened surfaces (width 4.26 mm) for a contact angle 2𝛼 of 20°. This was loaded in compression across its diameter using flat steel anvils. Both designs avoid premature failure from contact stresses [START_REF] Keles | Investigation of proper specimen geometry for mode I fracture toughness testing with flattened Brazilian disc method[END_REF][START_REF] Chen | Flattened Brazilian Disc Method for Determining the Dynamic Tensile Stress-Strain Curve of Low Strength Brittle Solids[END_REF], but have respective challenges of matching the anvil curvature to the deforming disc and ensuring parallelism of the flat surfaces.

The count current per scan that is the total neutron exposure (i.e., accounting for temporary loss of beam) for specimen A was 240 µA (approximately 6 hours per scan), which was twice that of the previous proof-of-concept experiment [START_REF] Zillhardt | In Situ Mechanical Loading and Neutron Bragg-edge Imaging, Applied to Polygranular Graphite On IMAT@ISIS[END_REF]. Due to time constraints, the count current for specimen B was 120 µA (3 hours, same as proof-ofconcept) but the dataset quality was improved at the expense of spatial resolution [START_REF] Zillhardt | In Situ Mechanical Loading and Neutron Bragg-edge Imaging, Applied to Polygranular Graphite On IMAT@ISIS[END_REF] by increasing the pinhole aperture from 40 mm to 60 mm. The 28×28 mm Micro-Channel Plate (MCP) neutron detector [START_REF] Tremsin | Unique capabilities and applications of Microchannel Plate (MCP) detectors with Medipix/Timepix readout[END_REF] with 2×2 Timepix ASICs for readout (256 × 256 pixels each), for a total array of 512 × 512 pixels, with a spacing of about 3 pixels between each chip was used. For specimen A, the Bragg-edge image resolution was 168 µm. The image resolution was approximately 276 µm in the configuration for specimen B due to an increase in the detector distance (to accommodate the driving screw of the loading rig) in addition to the effect of the larger pinhole aperture. Example radiographs at the same wavelength close to the (0002) Bragg edge are given in the Supplementary Information.

Both specimens A and B were observed at a reference preload (110 and 200 N respectively), then with in situ observations during loading and unloading. The changes in radiographs at energies near to the (0002) Bragg edge, relative to the preload radiograph, were extracted using the method developed in [START_REF] Zillhardt | In Situ Mechanical Loading and Neutron Bragg-edge Imaging, Applied to Polygranular Graphite On IMAT@ISIS[END_REF]. Briefly, the datasets were pre-processed in BEAn [START_REF] Liptak | Developments towards Bragg edge imaging on the IMAT beamline at the ISIS Pulsed Neutron and Muon Source: BEAn software[END_REF] and analysed in NEUTRONIX [START_REF] Zillhardt | NEUTRONIX: Program for the detection of grain reorientation with in-situ Bragg imaging[END_REF] [START_REF] März | Mesoscopic structure features in synthetic graphite[END_REF]. A digital image correlation (DIC) algorithm was applied to radiographs at energies away from the Bragg edge (i.e., with image contrast dominated by attenuation due to density changes from internal pores). This provided 2D displacement fields of the specimen's movements and deformation, which was used to precisely register and transform the radiographs that were recorded at energies near to the (0002) Bragg edge. It was then possible to map, via subtraction, the residual difference between Bragg edge radiographs of the deformed image and the unloaded reference at the same locations in the specimen in order to detect local changes in neutron scattering.

Three-dimensional X-ray Diffraction and Tomography

The experiment was completed on beamline ID11 at the European Synchrotron Radiation Facility (ESRF) under experiment number ME1504 [START_REF] Zillhardt | In-situ Mapping of Twinning in Graphite by Diffraction Contrast Tomography @ID11[END_REF] in Experimental Hutch 3 (EH3) with a monochromatic beam (40 keV) obtained using a Laue-Laue double bent silicon crystal monochromator. Specimen C (Figure 1c) was a cylinder (6 mm length, 𝐿, 3 mm diameter, 𝐷) that was loaded by compression along its longitudinal axis. For 3DXRD, a gauge window with a slit opening of 300 × 300 µm was centred on the longitudinal axis of the specimen, and the Far Field detector FReLoN-4M was used to collect diffraction patterns at 1° intervals (each with an exposure time of approximately 0.15 s) over a full 360° rotation. Following this 3DXRD acquisition, the scintillator of the Near Field FReLoN detector was placed in the beam and the slits were opened to approximately 3 × 3 mm for tomography (XCT) at 1.5 µm/voxel. The tomographs (2,000 3 voxels) were reconstructed using pyHST2's Paganin reconstruction routine [START_REF] Mirone | The PyHST2 hybrid distributed code for high speed tomographic reconstruction with iterative reconstruction and a priori knowledge capabilities[END_REF].

Specimen C was loaded in steps of ~40 N to a maximum of 300 N (~42 MPa compressive stress) and then unloaded in steps of ~50 N. Each loading step was followed by a wait time of a few seconds to stabilise the load before acquiring data by The variation of intensity with azimuth angle, 𝜒, depends on the texture of the diffracting crystallites in the gauge volume that defines the ROI, and changes in the texture will change the azimuth angle, 𝜒 𝑚 , at which there is maximum intensity.

Similarly, a change in the texture or orientation of the crystals will change the stage rotation angle, Φ, at which their Bragg diffraction condition is satisfied. The Bragg angle of the (0002) reflection at 40 keV is small (2𝜃 is 5.1 degrees), so for small angular changes in the crystal orientation the change in Φ that is required to bring the crystals back into diffraction is similar. Peak fitting and interpolation techniques were used to determine the rotation angle, Φ, of the maxima in azimuthal angle, 𝜒 𝑚 , of maximum intensity with sub-degree resolution. The change in Φ was tracked in each ROI as the compressive load was increased and then decreased in order to study the changes of orientation of the crystal domains in the ROI gauge volume. The analysis methods are described in more detail in the Supplementary Information.

Results

In Situ Bragg-edge neutron imaging

Total strains in the vertical x-y plane were mapped for both specimens using Digital Image Correlation (DIC) on neutron radiographs obtained at energies away from the Bragg edge, as described in [START_REF] Zillhardt | In Situ Mechanical Loading and Neutron Bragg-edge Imaging, Applied to Polygranular Graphite On IMAT@ISIS[END_REF]. Specimen A was a round disc loaded with curved anvils, for which the closest standard [START_REF]Suggested methods for determining tensile strength of rock materials[END_REF] advises equation ( 1) for calculating the tensile stress, 𝜎 𝑡 , acting at the centre of the disc due to the compressive applied load, 𝑃 𝑓𝑟𝑎𝑚𝑒 :

𝜎 𝑡 = 𝑃 𝑓𝑟𝑎𝑚𝑒 𝑇𝐷 × 𝑌 (1)
The ratio of disc radius to anvil radius, 𝜌, is an important factor that determines the constant, 𝑌, but its sensitivity to 𝜌 (here 0.24) is not significant in the range 0 to 0.67 [START_REF] Ch | The influence of jaw's curvature on the results of the Brazilian disc test[END_REF]. Equation [START_REF] Bystrov | Modelling of the thermal state and the melting loss of a graphite electrode in the conditions of the evaporative cooling in the arc furnace[END_REF], where 𝑌 = 0.636, leads to a result within 7% of the ASTM (American Society for Testing of Materials) standard D8289-19 [START_REF]ASTM D8289 -19 Standard Test Method for Tensile Strength Estimate by Disc Compression of Manufactured Graphite[END_REF] that is derived from Hondros [START_REF] Hondros | The Evaluation of Poisson's Ratio and Young's Modulus of Materials of a Low Tensile Resistance by the Brazilian Test[END_REF]. For point loading of the disc, the ratio, 𝑘, of the magnitudes of the compressive, 𝜎 𝑐 , to tensile, 𝜎 𝑡 , stresses at the disc centre is 3 [START_REF] Muskhelishvili | Some Basic Problems of the Mathematical Theory of Elasticity, Some Basic Problems of the Mathematical Theory of Elasticity[END_REF][START_REF] Markides | The stress field in a standardized Brazilian disc: The influence of the loading type acting on the actual contact length[END_REF], which is also correct for small angles of load contact [START_REF] Hondros | The Evaluation of Poisson's Ratio and Young's Modulus of Materials of a Low Tensile Resistance by the Brazilian Test[END_REF]. With the disc/anvil ratio 𝜌 used here and elastic contact with graphite's Young's modulus of ~11 GPa [START_REF] Marsden | Dimensional change, irradiation creep and thermal/mechanical property changes in nuclear graphite[END_REF] the contact semi-angle remains less than 6° and the biaxiality is within 2% of 𝑘 = 3 [START_REF] Markides | The influence of jaw's curvature on the results of the Brazilian disc test[END_REF]. Specimen B was a flattened Brazilian disc, loaded between flat, parallel platens. It is best described by the ASTM standard D3967-16 [START_REF] Ac | Standard test method for splitting tensile strength of intact rock core specimens[END_REF], where the tensile stress, 𝜎 𝑡 , at the specimen centre is calculated with equation ( 2) and finite element analysis has shown that for contact angle 2𝛼 = 20° the coefficient 𝑌(𝛼) = 0.964 [START_REF] Chen | Flattened Brazilian Disc Method for Determining the Dynamic Tensile Stress-Strain Curve of Low Strength Brittle Solids[END_REF][START_REF] Wang | Dynamic split tensile test of Flattened Brazilian Disc of rock with SHPB setup[END_REF][START_REF] Wang | The flattened Brazilian disc specimen used for testing elastic modulus, tensile strength and fracture toughness of brittle rocks: analytical and numerical results[END_REF].

𝜎 𝑡 = 2𝑃 𝑓𝑟𝑎𝑚𝑒 𝜋𝑇𝐷 × 𝑌(𝛼) (2)
Given the semi-angle for the flattened disc of 10° is constant and <15°, we make the same approximation that 𝑘 = 𝜎 𝑐 𝜎 𝑡 ⁄ = 3. Least Square Matching (LSM) at 69 pixel per subset and a step size of 3 pixels and relative to the preloaded reference. The load was increased from i) to iv), with a partial unload and then reload shown in v) and vi). Large compressive strains at the contact points are visible. Figure 3 presents the evolution of the average value of 𝜀 𝑦𝑦 in a central gauge area (approximately 4×4 mm) for both specimens A and B, where the biaxial strain state is quite uniform [START_REF] Liu | In situ measurement of elastic and total strains during ambient and high temperature deformation of a polygranular graphite[END_REF]. Both samples show essentially elastic loading.

They have a similar relation between the average strain and change in compressive stress, with some apparent hysteresis. Their behaviour is not quite identical, which may be due to the effects of nonlinear deformations near the loading surfaces in the two specimen geometries. A least-squares fit to the mean values in Figure 3b has a gradient of 12.10 ± 0.44 GPa. are persistent between the images, as was observed in the preliminary proof of concept study [START_REF] Zillhardt | In Situ Mechanical Loading and Neutron Bragg-edge Imaging, Applied to Polygranular Graphite On IMAT@ISIS[END_REF].

For both specimens, Figure 5 shows there is an apparent decrease in the correlation values as the compressive load magnitude is increased, and higher correlation values on unloadingError! Reference source not found.. This is quantified in Figure 6 using the median value of the correlation coefficient (values of 0 or NaN excluded) for the data within the central 4 × 4 mm ROI gauge area. The correlation coefficient values may be affected by differences in data noise for the imaging conditions of the two specimens. Nonetheless, both specimens show a slight decrease in the median correlation coefficient with increasing compressive load and a significant recovery (i.e. increase) when the load is reduced (Figure 6a). Figure 6b presents the same median correlation coefficient data against the change in mean compressive strain within the ROI (from the strains presented in Figure 3b) relative to the mean strain at the reference state. Both specimens show similar behaviour, where the correlation coefficient decreases slightly with increasing compressive strain change and increases significantly as the compressive strain is reduced.

In Situ Three-Dimensional Synchrotron X-ray Diffraction

Total and elastic strains

There were no microstructure changes nor damage visible in specimen C in response to loading. Example sections of high-resolution tomographs of ROI-A at the preload and maximum load are shown in Figure 7. The change in elastic strain within the graphite crystals (XRD) was obtained from the relative shifts of the (0002) peak, as in [START_REF] Liu | In situ measurement of elastic and total strains during ambient and high temperature deformation of a polygranular graphite[END_REF]. The 2D diffraction pattern (e.g. Figure 9a) was azimuthally caked in 1° bins, cross-correlated with the preload pattern to measure the peak shifts and the results fitted to obtain the 2D strain tensor (details are provided in the supplementary information). The results are presented in Figure 8a for both radial, 𝜀 𝑟 𝑋𝑅𝐷 , and longitudinal (compression, 𝜀 𝑦𝑦 𝑋𝑅𝐷 ) strains. The compressive behaviours of ROI-A and ROI-B are similar and appear elastic with no significant difference between loading and unloading. The transverse radial strains, 𝜀 𝑟 , are tensile. The ratio of the gradients of transverse to longitudinal strains with applied load is 0.19±0.7. Although the uncertainty is large, this is close to the Poisson's ratio of Gilsocarbon [START_REF] Marsden | Dimensional change, irradiation creep and thermal/mechanical property changes in nuclear graphite[END_REF][START_REF] Marrow | In situ measurement of the strains within a mechanically loaded polygranular graphite[END_REF].

The DVC analysis of the tomographs used the attenuation contrast due to microstructure porosity and density variations, achieving correlation coefficients above 0.7. The average longitudinal compressive strains (𝜀 𝑦𝑦 𝐷𝑉𝐶 , measured using the best fit to the local gradients of the displacement field across the analysed ROI) are presented in Figure 8b as a function of the change in magnitude of the compressive longitudinal stress. The behaviour appears elastic, and a fit to the combined data for stress vs.

𝜀 𝑦𝑦 𝐷𝑉𝐶 from ROI-A and ROI-B (for both loading and unloading) gives an apparent

Young's modulus in compression of 9.16 ± 0.67 GPa. The precision of the DVC analysis did not allow meaningful measurements of the transverse strains. Figure 8c presents the relationship between the longitudinal total strain (DVC, 𝜀 𝑦𝑦 𝐷𝑉𝐶 ) and elastic

Maximum load Preload

Crystal Orientation Tracking

An example diffraction image obtained at one rotation angle is shown in Figure 9a.

The contrast has been enhanced in Figure 9b to show two regions of the (0002)

Debye-Sherrer ring that have higher intensity at symmetric (high and low) azimuth angles (𝜒 -𝐻 and 𝜒 -𝐿). These regions, which are due to texture of the crystal domains, were analysed independently. The DVC analysis (see Supplementary Information) found the movements of the specimen were less than 50 µm at the highest load. This is small in comparison to the 300 µm slit size used for the diffraction acquisitions, so the sampled ROI for 3DXRD was essentially constant throughout the observations. The physical rotations were less than 0.3° up to the maximum load.

The azimuth angle, 𝜒 𝑚 , at which there was a maximum intensity around the (0002)

Debye-Sherrer ring varied with the rotation angle of the stage, Φ, to give a series of peaks (Figure 9c) due to different crystal domains within the gauge volume. Each peak is labelled by its azimuth position (high 𝐻 or low 𝐿) and by the approximate angle Φ (rounded to five) of the stage rotation for the peak (e.g. Φ -𝐿50°). Analysis of the 𝜒 𝑚 vs Φ relationship was used to identify and fit, with sub-degree precision, the stage angle, Φ, of the major peaks. This allowed measurement of the angular change in peak position, ∆Φ, with applied compressive stress. Further details of the analysis are in the Supplementary Information. Both disc compression specimens (A and B) showed similar elastic behaviour in compression, with some apparent hysteresis with unloading, and an effective Young's modulus of 12.10 ± 0.44 GPa (Figure 3). The axially compressed specimen in the 3DXRD study, loaded over the same range of compressive stress as the disc compression specimens, had similar Young's modulus (9.16 ± 0.67 GPa) (Figure 8b).

These moduli are lower than the value of ~15 GPa measured in tensile loading of the same graphite [START_REF] Marrow | In situ measurement of the strains within a mechanically loaded polygranular graphite[END_REF] but close to the expected value (10.8 GPa) of the Young' modulus for Gilsocarbon graphite [START_REF] Arregui-Mena | Spatial variability in the mechanical properties of Gilsocarbon[END_REF]. The differences between the moduli of the specimens may be due to material heterogeneity and small specimen size, and also the effect of the non-linear deformation of graphite on the determination of stress and strain in different specimen geometries.

The XRD analysis in this work has measured the change in the (0002) crystal lattice periodicity in order to determine the elastic strains acting perpendicular to the basal planes. The graphite crystal has very large elastic anisotropy and is most compliant perpendicular to the basal plane with an elastic modulus, 𝐶 33 , that is typically reported as 36 GPa [START_REF] Marsden | Dimensional change, irradiation creep and thermal/mechanical property changes in nuclear graphite[END_REF], although the value of 𝐶 33 of defect-free monocrystalline graphite may exceed 45 GPa [START_REF] Kusakabe | Interplanar stiffness in defect-free monocrystalline graphite[END_REF]. In the longitudinal compression specimen, the variation with applied stress of the elastic strain, 𝜀 𝑦𝑦 𝑋𝑅𝐷 , gave an effective diffraction modulus of 26.2 ± 0.6 GPa (Figure 8a). The elastic strain was a fraction (0.26 ± 0.02) of the applied compressive strain, 𝜀 𝑦𝑦 𝐷𝑉𝐶 , (Figure 8c). An XRD analysis of the compressive strains at the centre of a diametral compression specimen of the same Gilsocarbon graphite [START_REF] Markides | The influence of jaw's curvature on the results of the Brazilian disc test[END_REF] found a similar fraction of the elastic crystal strain to total strain (0.16±0.09). By inspection of the data, with calculation of stress via equation ( 2) and 𝑘 = 𝜎 𝑐 𝜎 𝑡 ⁄ = 3, its effective diffraction modulus for loading in compression was 44.1 ± 6.5 GPa.

The measurements discussed above were obtained during progressive loading.

Measurements during unloading of a tensile specimen of the same graphite found a significantly higher diffraction modulus of 61.2 ± 6.1 GPa [START_REF] Barhli | Synchrotron X-ray characterization of crack strain fields in polygranular graphite[END_REF], which is quite close to the average modulus expected for a random isotropic distribution of crystals (i.e., 𝜋 2 ⁄

x 36 GPa). Polygranular graphite is a porous network of connected crystalline domains of different orientations [START_REF] Lee | Bulk graphite: materials and manufacturing process[END_REF]. The significant difference between the crystal strains and bulk strains (i.e. the diffraction and Young's moduli) is too large to be accounted by the 20% porosity of the Gilsocarbon graphite [START_REF] Vertyagina | A multi-scale three-dimensional Cellular Automata fracture model of radiolytically oxidised nuclear graphite[END_REF]. This shows that non-linear deformation in the microstructure affects the development of the crystal strains during loading. This is not fully reversible, since the disparity between the diffraction moduli during loading and unloading shows there is a greater contribution to the microstructure's accommodation of applied strain during loading than is recovered during unloading.

This study is concerned with mechanisms for the accommodation of strain. The analysis of the in-situ energy-resolved neutron radiographs showed a systematic loss of correlation with increasing load between registered radiographs taken close to the (0002) Bragg edge (Figure 6). This represents a change in the pattern of image contrast due to changes in diffraction. Such local effects would be caused either by variations in grain orientation (i.e., texture change) or coherence (i.e., non-basal plane plastic deformation), and occurred in regions with a size of the order of several pixels (i.e. 100's of um) (Figure 4). The correlation increased as the load was removed, as the patterns of contrast become more similar to the reference. There was no clear relation between the spatial distributions of the correlation and the strain (Figure 5), although the median correlation coefficient clearly varied with applied strain (Figure 6). This may be due to relatively low gradients of strain in both specimens, except close to the anvil contacts, and the large correlation subset size (~3.8 mm) that was required due to data noise, which would be insensitive to the short-range strain gradients. These observations show that the mechanisms that caused the changes in local diffraction are i) affected by stress; ii) at least partially reversible; and iii) occur within regions that have a size of the order of the crystal domains in the graphite microstructure.

The neutron Bragg-Edge data had a large Field-of-View (FoV) that allowed many grains to be studied. The 3DXRD method used a smaller FoV (300 µm) that was of similar size to the extinction spots. This studied a gauge volume comparable to the scale of the Gilsocarbon microstructure (Figure 7), and so examined the combined behaviour of several domains of graphite crystals. The 3DXRD observations showed there were changes of the averaged direction, or orientation vector, of the diffracting crystal domains within the gauge volume with change in applied stress ( Figure 10). The change varied for different domains in the two regions of interest that were studied. Some domains deformed elastically, leading to small recoverable changes in orientation of the crystals that increased and then decreased with load.

The magnitude of this elastic reorientation would depend on the domain orientation, accommodation porosity and the local stress state in the heterogeneous microstructure. The reorientation was larger in some domains and was not fully recovered on unloading, which shows that inelastic behaviour occurred.

Both the neutron and X-ray observations show that non-linear and inelastic deformation mechanisms operate in unirradiated graphite at ambient temperature.

Possible inelastic deformation mechanisms in graphite are kink bands [START_REF] Barsoum | Kink bands, nonlinear elasticity and nanoindentations in graphite[END_REF], twinning [START_REF] Palache | Contributions to the mineralogy of sterling hill, new jersey: morphology of graphite, arsenopyrite, pyrite, and arsenic[END_REF][START_REF] Freise | Twinning in Graphite[END_REF], transformation to the rhombohedral phase [START_REF] Freise | The deformation of graphite crystals and the production of the rhombohedral form[END_REF] and basal dislocation glide [START_REF] Soule | Direct Basal-Plane Shear in Single-Crystal Graphite[END_REF] (glide of prismatic dislocations in graphite is difficult except at high temperatures [START_REF] Mchugh | Prismatic edge dislocations in graphite[END_REF] and their potential effect is neglected). Crystal reorientation due to kink bands, twins and phase transformation would introduce additional peaks to the 3DXRD 𝜒 𝑚 vs Φ relationship (e.g., Figure 10). No additional peaks were observed, which suggests that if these crystal reorientation mechanisms did occur, it was not in a significant volume fraction of the gauge volume.

Glide of basal dislocations occurs very readily due to the low shear resistance of the graphite crystal [START_REF] Soule | Direct Basal-Plane Shear in Single-Crystal Graphite[END_REF][START_REF] Telling | Stacking fault and dislocation glide on the basal plane of graphite[END_REF]. Basal glide would not affect the coherence of the basal planes.

Lattice rotation by glide is a common phenomenon in plastic deformation ('Taylor rotation') as grains deform to align the shear direction with the applied strain [START_REF] Abdolvand | On the state of deformation in a polycrystalline material in three-dimension: Elastic strains, lattice rotations, and deformation mechanisms[END_REF] and in metals it can lead to texture evolution. We propose that the observed lattice reorientations of the crystal domains are predominantly elastic, with a contribution from shear by basal dislocation glide during loading. On unloading, the shear component to the lattice rotation is not fully recovered. The accommodation of applied strain by such deformations in differently oriented and connected domains of the graphite microstructure would provide the non-linear behaviour that is observed. Incompatible deformations of adjacent domains may lead to high local stresses with strains then accommodated by other inelastic mechanisms including kink bands and twinning, but these are not detectable in the current study.

The shear modulus of the graphite crystal, 𝐶 44 , which varies with the degree of crystallite order [START_REF] Mason | The Young's modulus of carbon and graphite artefacts[END_REF], is strongly affected by pinning of basal dislocations from crystal defects [START_REF] Telling | Stacking fault and dislocation glide on the basal plane of graphite[END_REF], particularly irradiation-induced point defects [START_REF] Seldin | Elastic Constants and Electron-Microscope Observations of Neutron-Irradiated Compression-Annealed Pyrolytic and Single-Crystal Graphite[END_REF]. In the event of neutron irradiation, the contribution from basal dislocation glide to shear and rotation of the crystal domains would decrease significantly, with a consequent increase in the effective elastic modulus of the domain towards the intrinsic lattice stiffness. This can explain the observed increase with irradiation of both the linearity of deformation of polycrystalline graphite and its Young's modulus [START_REF] Brocklehurst | Analysis of the dimensional changes and structural changes in polycrystalline graphite under fast neutron irradiation[END_REF]. Future in situ studies to test this hypothesis will study the effects of irradiation and temperature on local changes in strain and crystal orientation, such as via high resolution Bragg-edge neutron tomography [START_REF] Busi | Bragg edge tomography characterization of additively manufactured 316L steel[END_REF].

Conclusion

In situ diffraction experiments at different length scales with neutron Bragg-edge imaging and 3DXDRD provide evidence for reorientation of the crystal domains in polycrystalline Gilsocarbon graphite under mechanical stress. The reorientations are predominantly elastic, with a contribution that is not fully recovered on unloading which is attributed to basal dislocation glide. This observation can explain the effect of fast neutron irradiation on both the linearity of deformation of polycrystalline graphite and its Young's modulus.

Figure 1 :

 1 Figure 1: Schematic of specimens A (a), B (b) and C (c), all loaded in compression along the vertical y-axis

  3DXRD and then XCT. Alternate loading steps placed the Field of View (FoV) in the centre of the specimen (ROI-A, reference pre-load 5.4 N) and then with the FoV centre translated horizontally along the cylinder radius by about 200 µm (ROI-B, reference pre-load 50 N). Local Digital Volume Correlation (L-DVC) of the tomographs in LaVision DaVis (Version 8.4.0 66797) was used to extract three-dimensional displacement fields to quantify the movement and deformation of specimen C. The correlation approach prioritised precision and accuracy of the measurements over spatial resolution. A Fast Fourier Transform (FFT) DVC was used at a final subset size of 256 3 voxels, with a search radius of 16 voxels, followed by a 4-step Direct Correlation (DC) at 64 3 voxel subset size at 75% overlap. Changes in the crystal orientations in the 3DXRD gauge volumes of ROI-A and ROI-B were quantified by examining the effect of the specimen rotation on the azimuthal variation of intensity in the 2D images of the (0002) Debye-Scherrer diffraction rings.

Figure 2

 2 Figure 2 maps the component of normal strain, 𝜀 𝑦𝑦 , parallel to the diametral loading

Figure 2 :

 2 Figure 2: Specimen A: maps of the strain 𝜺 𝒚𝒚 parallel to the vertical loading direction measured by DIC of radiographs with varying load from i) to vi). The colour scale is the strain. The arrows show the local displacements relative to the preload and are not corrected for rigid body translation or rotation. The arrow length is proportional to the displacement magnitude (1 pixel = 55 µm).

Figure 3 :

 3 Figure 3: Average strain 𝜺 𝒚𝒚 , measured by DIC at the centre of specimens A and B with loading and unloading as a function of a) load; b) change in compressive stress. The strain error bars represent the 95% confidence in the mean (sample size 200 measurements in 4x4 mm gauge area). Points have been added to indicate zero strain at the pre-load (i.e., zero stress change). Green arrows indicate the load/unload sequence

Figure 4 :

 4 Figure 4: Specimen B: (a) radiograph residuals at the (0002) Bragg edge, relative to the preload, with increasing load to 2.5 kN, then unloading to 1 kN. The marked area of interest is shown with higher magnification in (b), where the dash-dot/solid circles and dotted ellipse identify selected extinction spots that are described in the text

Figure 5 :Figure 6 :

 56 Figure 5: Maps of the correlation coefficient between the (0002) Bragg edge radiograph residuals (relative to the map at 1 kN): a) Specimen A with i) to iii) compressive loading (along the y-axis), iv) partial unload and v) reload, and b) Specimen B with i) to iii) compressive loading and iv) to vi) unloading

(

  XRD, 𝜀 𝑦𝑦 𝑋𝑅𝐷 ) strain in compression for both ROI. A linear fit to the combined data has a gradient (𝜀 𝑦𝑦 𝑋𝑅𝐷 𝜀 𝑦𝑦 𝐷𝑉𝐶 ⁄ ) of 0.26 ± 0.02.

Figure 7 :

 7 Figure 7: Example tomography slices (vertical sections) at preload (1 MPa) and maximum load (~40 MPa) for ROI-A in Specimen C. The compressive loading direction was vertical. Typical features of a coarse Gilsocarbon filler particle, matrix and porosity are identified.

Figure 8 :

 8 Figure 8: Strain measurement in both Regions of Interest (ROI-A and ROI-B) during axial compression of Specimen C: a) change in strain measured by XRD (radial 𝜺 𝒓 𝑿𝑹𝑫 and longitudinal 𝜺 𝒚𝒚 𝑿𝑹𝑫 ) with the change in compressive stress (relative to the preload state); b) change in strain measured by DVC (longitudinal 𝜺 𝒚𝒚 𝑫𝑽𝑪 ) with the change in compressive stress; c) comparison between the longitudinal strain changes measured by DVC (𝜺 𝒚𝒚 𝑫𝑽𝑪 ) and XRD (𝜺 𝒚𝒚 𝑿𝑹𝑫 ).

  Figure 10a and

  Figure

Figure 9 :

 9 Figure 9: Example analysis of the 3DXRD data for Specimen C: a) image of the Debye-Scherrer rings at one stage rotation angle, 𝚽; b) enhanced contrast to identify the 𝝌 -𝑯 and 𝝌 -𝑳 regions of the (0002) ring with higher intensity; c) variation of the azimuth angle of maximum intensity (𝝌 𝒎 ) of the low angle 𝝌 -𝑳 and high angle 𝝌 -𝑯 region of ROI-A as a function of stage rotation angle, 𝚽, to identify various peaks, 𝚽 -𝑳𝟓𝟎°, 𝚽 -𝑳𝟏𝟗𝟎° etc. The change of 𝚽 for each peak (∆𝚽) with applied stress was then determined by peak fitting.

Figure 10 :

 10 Figure 10: The relative change in stage rotation angle, ∆𝚽, for maximum intensity for selected (0002) peaks at high 𝑯 and low 𝑳 azimuth angles as a function of the change in compressive stress, for both ROI-A (a and b) and ROI-B (c and d). The cumulative fitting error is below 5% for all peaks tracked. Nominal points have been added to indicate the reference at the origin. The arrows represent the direction of loading.

Acknowledgments

The authors acknowledge the UK Science and Technology Facilities Council (STFC) for access to neutron beamtime on IMAT at ISIS (Experiment RB1910517), and the European Synchrotron Radiation Facility (ESRF), Grenoble, France for access to Xray beamline ID11 (Experiment ME1504). We thank Dr Abdalrhaman Mohamed Koko and Dr Yang Chen (University of Oxford), as well as Joachim Paul Forna-Kreutzer (University of Bristol) for their participation in experiment at the ESRF. We are also thankful to Evelina Ametova (Karlsruhe Institute of Technology) for helpful discussion and advice. Thomas Zillhardt is grateful for support from EDF Energy Generation UK.

Author Contributions

Conflicts of interest

The authors have no affiliation with any company nor have any financial interest in the material presented in this manuscript.