Various architectures are instances of the framework, including AVATAR, labeled splitting, and SMT with quantifiers.AVATAR is an elegant and effective way to split clauses in a saturation prover using a SAT solver. But is it refutationally complete? And how does it relate to other splitting architectures? To answer these questions, we present a unifying framework that extends a saturation calculus  with splitting and that embeds the result in a prover guided by a SAT solver. The framework also allows us to study  One of the great strengths of saturation calculi such as resolution  and supetracking , 32, spltracking , labeledtracking , and AVAtracking .The AVATAR architecture, which is based on a satisfiability (SAT) solver, is of particular interest because it is so successful. Voronkov reported that an AVATAR-enabled Vampire could solve 421 TPTP problems that had never been solved before by any system , Sect. 9C is enabled only in models in which the associated propositional variable [C] is true. A SAT solver is then run to choose a model To illustrate the approach, we follow the key steps of an AVATAR-enabled resolution prover on the initial clause set containing saturation framework by Waldmann, Tourret, Robillard, and Blanchette [splitting framework, and use it to show the completeness of an AVATAR-like architecture. The framework is currently a pen-and-paper creature; a formalization using Isabelle/HOL [What about refutational completeness? Far from being a purely theoretical concern, establishing completeness—or finding counterexamples—could yield insights into splitting and perhaps lead to an even stronger AVATAR. Before we can answer this open question, we must mathematize splitting. Our starting point is the anchette , based oanchette . It coveanchette , p. 332.elle/HOL  is underbase calculus, such as resolution or superposition. It must be presentable as an inference system and a redundancy criterion, as required by the saturation framework, and it must be refutationally complete.Our framework has five layers, linked by refinement. The first layer consists of a splitting calculus  solvers might be tempted to claim that Voronkov merely reinvented SMT. To investigate such questions, we apply our framework to splitting without backtracking, labeled splitting, and SMT with quantifiers Sect. . This giA shorter version of this article was presented at CADE-28 . This arOur framework is parameterized by abstract notions of formulas, consequence relations, inferences, and redundancy. We largely follow the conventions of Waldmann et al. . A-formuformulas, ranged over by consequence relation if if if The intended interpretation of N will be useful to define splittability abstractly in Sect. A set For their saturation framework, Waldmann et al. instead consider a fully conjunctive version of the consequence relation, with different properties. The incompatibility can easily be repaired: Given a consequence relation The Let This holds by (D2) and (D3) due to the definition of Following the saturation framework , p. 318,C is a finite multiset of literals L is either an atom or its negation  complete , it suffices to show N is saturated w.r.t. N is saturated w.r.t. Given a redundancy criterion entclass1pt{minimasequenceX is a function from X that maps each limit inferior is limit superior is persistent. A sequence weakly fair w.r.t. strongly fair if i and ever after eventually be performed or become redundant for another reason. Strong fairness requires the same from all inferences that are possible infinitely often, even if not continuously so. Both can be used to ensure that some limit is saturated.A derivation is a sequence of X elements such that i. Finite runs can be extended to derivations by repeating the final state infinitely. We must then ensure that terminates if i.Given a relation Let dynamically  complete  and A different encoding is used to exploit the theories of an SMT solver . With a (R5)(R6)(R7)These requirements can easily be met by a well-designed redundancy criterion. Requirement (R5) is called reducedness by Waldmann et al. [Let entclass1pt{minimasplitting calculusFrom a base calculus, we will define an induced We start with the mandatory inference rules.splitting inference systemThe Base, the side condition is Unsat, the side condition is that For Split, are simplifications; they replace their premises with their conclusions in the current A-formula set. The premises’ removal is justified by In addition, the following optional inference rules can be used if desired; the completeness proof does not depend on their application. Rules identified by double bars, such as Split rule, we require that i. A-formula C is splittable into formulas i.In the Split performs an n-way case analysis on C. Each case n other conclusions assume x, whereas In a clausal prover, typically Collect, we require Trim, we require For Collect removes A-formulas whose assertions cannot be satisfied by any model of the propositional clauses—a form of garbage collection. Similarly, Trim removes assertions that are entailed by existing propositional clauses.StrongUnsat, we require Approx, we require Tauto, we require For StrongUnsat is a variant of Unsat that uses StrongUnsat if desired, but only Unsat is necessary for completeness. In practice, T solver   case, we can assume that C(x) does not hold when considering the D(y) case. This can be achieved by adding the A-clause Tauto. If we use an SMT solver that is strong enough to determine that D(y) are inconsistent, we can then apply StrongUnsat immediately, skipping the D(y) branch altogether. This would be the case if we took C(x) in the context of labeled splitting [Consider a splitting calculus obeying the AVATAR conventions of Example plitting , Sect. 2Approx make these clauses visible to the SMT solver, as the propositional clause set StrongUnsat, detects the unsatisfiability.Consider a splitting calculus whose propositional solver is an SMT solver supporting linear arithmetic. Suppose that we are given the inconsistent clause set The splitting inference system commutes nicely with the enabled projection:(Unsat inferences. It remains to show that the enabled projection of a Base inference is an The condition (Soundness) The rules Unsat, Split, Collect, Trim, StrongUnsat, Approx, and Tauto are sound w.r.t. Base is sound w.r.t. CasesUnsat, StrongUnsat, Tauto:Trivial.CaseSplit:For the left conclusion, by definition of i. Notice that For the right conclusions, we must show CaseCollect:We must show CaseTrim:Only the right conclusion is nontrivial. Let CaseApprox:The proof is as for the left conclusion of Split.CaseBase:To show Next, we lift the base redundancy criterion.splitting redundancy criterionthere exists an A-formula An inference (3)Base inference and (4)Unsat inference and The Split and Collect rules, as we will see below. Condition (2) is used to justify Trim. We will use Condition (1) lifts Since By Lemma Base inferences is a redundancy criterion. Then we will consider Unsat inferences.We will first show that the restriction intersection of lifted redundancy criteria with tiebreaker orders. Then we can simply invoke Theorem 37 and Lemma 19 from their technical report [We start by showing that n et al. , Sect. 3l report .tiebreaker ordergrounding functionsTo strengthen the redundancy criterion, we define a We must show that We also need to check that the consequence relation Condition (G1) obviously holds, and (G3) holds by property (R4) of Unsat inferences.To extend the above result to Unsat inference (R2)Given an Unsat inference (R3)Given an Unsat inference (R4)Given an A. If the base criterion D is strictly subsumed by C and n et al. ), this a(Simplification)  For every Split, Collect, or Trim inference, the conclusions collectively make the premises redundant according to CaseSplit:We must show i,  this follows from Split’s side condition CaseCollect:We must show i,  this follows from Collect’s side condition that CaseTrim:We must show x, y are variables, but this is not covered by condition (2) of Annoyingly, the redundancy criterion approach , Sect. 4We will now prove that the splitting calculus is statically complete and therefore dynamically complete. Unfortunately, derivations produced by most practical splitting architectures violate the fairness condition associated with dynamic completeness. Nevertheless, the standard completeness notions are useful stepping stones, so we start with them.Let Base, in Base inference Assuming n et al. . First n(Static completeness)  Assume Suppose First, we show Unsat, we obtain Hence Thanks to the requirements on the redundancy criterion, we obtain dynamic completeness as a corollary:(Dynamic completeness)  Assume This immediately follows from Theorem The above completeness result, about n et al. , Sect. 4Unsat rule to derive local and standard saturations global.To fully capture splitting, we need to start with a weaker notion of saturation. If an A-formula set is consistent, it should suffice to saturate w.r.t. a single propositional model. In other words, if no A-formula locally saturated w.r.t. A set strong static completeness:Local saturation works in tandem with (Strong static completeness)  Assume We show Consider the following A-clause set expressed using AVATAR conventions:We also need a notion of local fairness that works in tandem with local saturation.locally fair w.r.t. i or there exists a propositional model A sequence Let The proof is by case analysis on the condition by which n et al. , and thestrong dynamic completeness.Local fairness works in tandem with (Strong dynamic completeness)  Assume i.i. We connect the dynamic and static points of view along the lines of the proof of Lemma 6 in the technical report of Waldmann et al. . First, An alternative proof based on dynamic completeness follows:i by case analysis on the condition by which i. By Lemma We show entclass1pt{minimaIn Sects. The transition system model-guided proverDerive rule can add new A-formulas  i or (2) i and there exists a limit point An Switch forever ) or to starve off Derive by performing only Switch transitions ). Also note that we make no assumptions about the order in which propositional models are enumerated; the propositional solver is given carte blanche.Until all limit points We might at first expect that a realistic prover would ensure the inclusion Fairness of i. Then for every i such that Let k. For every Switch transition, which has Let i,  and let Let i such that Let In the spirit of refinement, we have that fairness of an (Fairness) Let i is trivial. Otherwise, we have that i and there is a limit point The case where (Dynamic completeness)  Assume i.By Theorem A well-behaved propositional solver, as in labeled splitting, enumerates potential models in a systematic way and always gives rise to a single limit point By contrast, an unconstrained solver, as supported by AVATAR, can produce multiple limit points; in particular, the restart feature of SAT solvers  could prSuppose that we leave out Consider the initial clause set consisting of entclass1pt{minimaBy contrast, in a splitting prover, we might split the first clause, yielding the A-clauses Switch and StrongUnsat transitions periodically. Suppose that the inconsistent initial clause set of Example It is crucial to invoke the SAT solver often enough—in other words, to take Consider the consistent set consisting of mentclass2pt{minimC from each copy We build a clause set from two copies of Example entclass1pt{minimaWith both AVATAR and labeled splitting, an enabled A-clause can be redundant locally, w.r.t. the current interpretation, and yet nonredundant globally. Both architectures provide mechanisms to temporarily lock away such A-clauses and unlock them when coming back to an interpretation where they are no longer locally redundant. In AVATAR, conditionally deleted A-clauses are stored in the locked set; in labeled splitting, they are stored in the split stack. We will refine the model-guided prover into a locking prover that captures these mechanisms.locking prover is defined by two transition rules:Lift rule performs an Lock rule can be used to lock A-formulas that are locally redundant.The states of a locking derivation are triples Let Lift transition clearly corresponds to an Lock transition corresponds to a Derive transition with Every Let There are three things to note. First, if we had simply thrown away the clause k,  then Locking can cause incompleteness, because an A-formula can be locally redundant at every point in the derivation and yet not be so at any limit point, thereby breaking local saturation. For example, if we have derived i. Following an age-based selection heuristic and maximal splitting, the second clause the prover derives is Lock applies, and Consider the inconsistent initial clause setNext, In the derivation in Example The calculus is ordered resolution with selection using the precedence i is even;i is odd.The derivation thus alternates between two families of interpretations, with even and odd indices, giving rise to two limit points.The initial clauses are as follows. First, we have a splittable clause After the clause In the first family, mutatis mutandis for The clause Given the right sequence of propositional interpretations returned by the SAT solver, the derivation in Example Our solution to the issues encountered above is as follows. Let fair if (A) i and there exists a subsequence An Fairness of an (Fairness) Let C, is not in that set; that is, We already showed that By (a) we have that B. If j,  then necessarily j, we also have i by the side condition of the Lock transition. Therefore, by reducedness of Hence (Dynamic completeness)  Assume i.By Theorems AVATAR was unveiled in 2014 by Voronkov , althouggiven A-formula to move is guided by timestamps indicating when the A-formulas were derived, to ensure fairness.We will define an abstract AVATAR-based prover that extends the locking prover timestamped A-formulas. (We will often omit the adjective “timestamped.”) Given a subset Let Using the saturation framework , Sect. 3Let This follows directly from the definition in Waldmann et al. . \documeactive A-formulas passive A-formulas Whenever we write a tuple A state is a tuple Split produce disabled and propositional clauses, which are put into The input formulas are first put in the passive set The division of nonactive A-formulas into the sets AVATAR-based proverLockP rule that is identical to LockA except that it starts in the state well timestamped if every A-formula introduced by a rule is assigned a unique timestamp. In practice, a prover would ensure well-timestampedness by assigning timestamps monotonically, but this is not necessary for the fairness and completeness proofs.The Let Infer and Process map to a Lift of Derive. The transitions map directly to the corresponding transitions in i.Let The invariant is preserved by each transition. Let us redo the Let us redo the In contrast with nonsplitting provers, for Infer makes Switch disables it; Infer makes Switch disables it. Even though A problematic scenario involves two premises More concretely, make two copies of the clause setj is maximal among such clauses, and that thus Let the models alternate between the With this strategy, the prover will never select In Example We believe that this counterexample cannot arise with Vampire, because Vampire alternates between age-based and weight-based selection using a fixed ratio (the “age–weight ratio” or “pick–given ratio”). In contrast, our example requires a highly unrestricted heuristic selection, where we choose young, large A-clauses such as Unrelated to completeness, we might expect that under a reasonable strategy an Take the following consistent A-clause set:After switching to the next model This process is then repeated infinitely often. In the model The subsequence fair if (A) i and there exists a subsequence An Condition (3) ensures that all inferences involving passive A-formulas are redundant at the limit point. It would not suffice to simply require (Fairness) Let We trivially have A, t) w.r.t. the lexicographic order. If Process and thus First we show Now let (Dynamic completeness)  Assume i.By Theorems Assuming the restriction on locking we already required for X be a set. A selection order X is an irreflexive and transitive relation such that Let X only contains finitely many A-formulas with the same timestamp. Define X so that Let Every selection order is a well-founded relation, but not every well-founded relation is a selection order. A well order is a selection order if and only if its order type is less than Selection orders on entclass1pt{minimafootnote , Sect. 4C| denotes the sum of the number of nonvariable positions in C. Then n nonvariable positions for any n. This selection order corresponds to a simple weight-based selection scheme.Let The intersection of two orders X. Then Let Irreflexivity and transitivity are preserved by intersections, and note that X. Then for all elements x and y,  there exists an element z such that Let The set To ensure completeness of the given clause procedure, we must restrict the inferences that the prover may perform; otherwise, it could derive infinitely many A-formulas with different assertions, causing it to switch between two branches of the split tree without making progress as in Example strongly finitary if (1) A function strongly finitary if the function strongly finitary if the set of inferences it defines is strongly finitary. We can extend a strongly finitary function F to sets of base formulas by taking F and F and G are strongly finitary, then so is A set of B in Definition Base, Unsat, Collect, Trim, and StrongUnsat, we can set Split and Approx require a nonempty B(N),  containing the assertions chosen for the split A-formulas. Deterministic splitting rules (where the chosen assertions are fully determined by the base formula), such as AVATAR’s, are thus also strongly finitary because then B(N) is finite. The optional inference rule Tauto is not strongly finitary.The function I restricting the inferences: The prover may perform an inference only if its conclusion is in Infer transition. This restriction will allow us to rule out the case where For the completeness of the given clause procedure, Lemma u. If simplifications were to produce infinitely many new splittable clauses, the prover might split clauses and switch propositional interpretations forever without making progress.Simplification rules used by the prover must be restricted even more to ensure completeness, because they can lead to new splits and assertions, and hence switching to new models. For example, simplifying I is a set of binary inferences such that I can still produce infinitely many base formulas. This is because in an Even if I in a different way each time, the first time to 1, then to 2, and so on. We hence need to ensure that, in the entire derivation, each formula is simplified in at most a finite number of ways.As a slightly abstract example, consider strongly finitary simplification bound for Let The prover may simplify an A-formula S be a strongly finitary simplification bound. For every ith iterate of S. Then Let Clearly, S is a strongly finitary simplification bound, i and that To show that B for S. Then set C), and we have Thus C and S is a strongly finitary simplification bound, because (1) C and (2) each clause has only finitely many subclauses. This S covers many simplification techniques, including elimination of duplicate literals (simplifying C), and subsumption resolution , always rewriting using the oldest available equation. Also assume that if t, there exist only finitely many terms t in one step. The term i be the index of l| be the number of nonvariable positions in l, and let We will show that for every term C, thus decreasing the i. Otherwise, it applies at the same position as r and l.) If u decreases. If l| stays the same, u might decrease, and If the prover derives a new ordered equation Based on the above definitions, we introduce a fairness criterion that is more concrete and easier to apply than the definition of fairness of I be a strongly finitary function, and let S be a strongly finitary simplification bound. Then a well-timestamped Infer transition, either for every Infer transition, for every Process transition, for every Switch or StrongUnsat occurs;if Infer, Switch, or StrongUnsat occurs;if i such that either Infer chooses a i;there are infinitely many indices Let StrongUnsat transition never occurs, and therefore Switch is eventually applied if the propositional clauses are not satisfied by the interpretation. Hence i,  thus satisfying condition (C)(2) of Definition If Case 1:The set of Infer transition for some state j is unbounded in Infer transition from state j such that the selected A-formula Infer transitions clearly form an infinite subsequence. By Lemma j such that Infer transition, a contradiction.Case 2:The set of Infer transition for some state j is unbounded in Case 3:Neither case 1 nor case 2 apply. Then the set of Infer transitions is bounded and hence finite since Infer transitions is finite as well. Let Infer transition. So I are strongly finitary, only a finite number of new assertions are introduced, and Switch transitions can occur. Thus there exists an index N such that no Switch transitions occur at states j with a Infer transition such that a N steps, every A-formula is selected only once  In either case, the A-formula is removed from the passive set for the rest of the derivation. Newly derived A-formulas have a different timestamp due to the well-timestampedness requirement. Therefore, once an A-formula is in the active set, it will not come back into the passive set, and we have Now take the whole derivation as subsequence. We have Recall the abstract counterexample from Sect. We could refine Even the concrete criterion offered by Lemma AVATAR may be the most natural application of our framework, but it is not the only one. We will complete the picture below by studying splitting without backtracking, labeled splitting, and SMT with quantifiers.Before the invention of AVATAR, Riazanov and Voronkov  had alreentclass1pt{minimaC and D share no variables and The basic idea of splitting without backtracking is to extend the signature blocking function allows for the selection of parallel function selects only maximal Riazanov and Voronkov also extend the selection function of the base calculus to support lightweight AVATAR. Lightweight AVATAR amounts to the splitting architecture Cruanes implemented in Zipperposition and confusingly called “AVATAR” [The calculus entclass1pt{minimaC, D. By Theorem with the side conditions that Unsat rule) and to eliminate assertions that are implied at the SAT solver’s top level (corresponding to our Trim rule).Like splitting without backtracking but unlike the real AVATAR, Cruanes’s architecture is not guided by a propositional model. It is essentially an instance of dulation ), which The calculi Another dissimilarity is that C is a As positive results, we will show that M, N, we have Given the M and N are ground.Since both entailments are defined via grounding, it suffices to consider the case where M. Let C contains the complementary N is true in N are false in N must contain a true For the forward direction, we must show that M, where N is true in N that is true in N is enabled in N is also true in For the backward direction, we must show that Base inference, then Given an inference Base inference. By the definition of ordered resolution, none of the Let C, if (1) Given a D, since For (2), let For (1), essentially the same line of argumentation applies, with N be a set of N is saturated w.r.t. (Saturation)  Let Let CaseBase:We need to show that N is saturated, the inference is redundant—i.e., By Lemma D and N is saturated,Let D true and then N. In both cases, CaseUnsat:The inference derives labeled clauses—clauses annotated with finite sets of natural numbers.Labeled splitting, as originally described by Fietzke and Weidenbach  and implWe model labeled splitting as an instance of the locking prover mentclass2pt{minimsplit levels. Given a split on k, the propositional variables Splits are identified by unique C or D, which Fietzke and Weidenbach avoid. We solve this as follows. Let StrongUnsat rule then knows that C and D. A-clauses are then split using the following binary variant of Split:When splitting, if we simply added the propositional clause C and D share no variables and k is the next split level. Unlike AVATAR, labeled splitting keeps the premise and might split it again with another level. We rely on locking to ensure that the premise is not split within either branch.where Base (via Lift and Derive) to perform an inference from the enabled A-clauses. If an enabled Switch or StrongUnsat.Apply Derive (via Lift) to delete a redundant enabled A-clause.Use Lock to temporarily remove a locally redundant enabled A-clause.Use Derive (via Lift) to simplify an enabled A-clause. If the original A-clause is made redundant, delete it; otherwise, use Lock to temporarily remove it. If an enabled Switch or StrongUnsat.Use SoftSplit (via Lift and Derive) with split level k on an A-clause Switch to enable the left branch and apply Lock on Apply Disabled A-clauses are the ones that occur in branches other than the current one and in disabled splits. As such, they should not be used when exploring the current branch.To emulate the original, the locking prover based on the Switch is powerful enough to support all of Fietzke and Weidenbach’s backtracking rules, but to explore the tree in the same order as they do, we must choose the new model carefully. If a left branch is closed, the model must be updated so as to disable the splits that were not used to close this branch and to enable the right branch. If a right branch is closed, the split must be disabled, and the model must switch to the right branch of the closest enabled split above it with an enabled left branch. If a right branch is closed but there is no split above with an enabled left branch, the entire tree has been visited. Then, a propositional clause StrongUnsat can finish the refutation by exploiting We illustrate the strategy on an example.SoftSplit. Then a Switch transition replaces Lock removes Let Base inference produces Switch disables The first backtracking step happens after a Switch rule is invoked, producing the model The second backtracking step happens after The Split. Clearly, it entails StrongUnsat derives Now, the resulting A-clause set contains the propositional clauses Switch rule requires that all propositional clauses are satisfied.By following the strategy presented above, What about fairness? The above strategy helps achieve fairness by ensuring that there exists at most one limit point. It also uses locks in a well-behaved way. This means we can considerably simplify the notion of fairness for For terminating derivations, their fairness criterion coincides with ours: Both require that the final A-clause set is locally saturated and all propositional clauses are satisfied by the interpretation. For diverging derivations, Fietzke and Weidenbach construct a limit subsequence Alternating age-based and unrestricted heuristic selection is incomplete for labeled splitting just as it is for AVATAR Example . To see This process can be repeated to give rise to infinitely many splittable clauses of the form T) [SMT solvers based on DPLL(T)  combine T) , veriT [T) , and Z3 T) , also suComplete instantiation strategies have been developed for various fragments of first-order logic , 23, 24.Let entclass1pt{minimaM is finite, then M is inconsistent modulo linear integer arithmetic. Both completeness and soundness of a concrete prover are statements about the finite set of input formulas, so using a compactified version of the consequence relation is purely an implementation detail and poses no restriction.Some theories such as linear integer arithmetic are not compact and thus cannot directly be used for the consequence relation. Instead, we define C and D from a premise N if t is some ground term; the instantiation strategy determines which ground terms must be tried and in which order. A lot of complexity hidden in The clausification rules work on logical symbols outside quantifiers; they derive subunit if C is not a disjunction. Whenever a (ground) disjunction Split. This delegates clausal reasoning to the SAT solver. It then suffices to assume that As with AVATAR, the initial problem is expressed using  Example . Abusing(Dynamic completeness)  Assume j.The proof is analogous to that of Theorem Care must be taken to design a practical fair strategy. Like AVATAR-based provers, SMT solvers will typically not perform all Consider the initial clause seti. However, at the limit point— where i— none of these conflicts is enabled. The only conflict which exists at the limit point is between Iterating this process, we see that all conflicts are of the form For decidable theories, a practical fair strategy is to first clausify and detect theory conflicts and to instantiate quantifiers only if no other rules are applicable. A similar case analysis as in the proof of Lemma j, contradicting the strategy. Otherwise, First consider the case where quantifier instantiation is invoked infinitely often. Then there exists an infinite subsequence Otherwise, quantifier instantiation is only invoked finitely often—either because every encountered model had a theory conflict, or because there was nothing to instantiate. Here, it suffices to assume that clausification is a strongly finitary simplification bound (which means that a formula can only be clausified in a finite number of ways). Under this assumption only finitely many base formulas will be derived; this implies that only a finite number of models will be considered. The last model will then be saturated due to the strategy.Approx systematically. Then There is also the question of model soundness. If the SMT solver starts with the Unsat.Consider an SMT solver equipped with the theory of uninterpreted functions and linear arithmetic. Letn-ary and quantifier instantiation is unary, whereas superposition is binary. Nevertheless, their completeness follows from the same principles.Our mathematization of AVATAR and SMT with quantifiers exposes their dissimilarities. With SMT, splitting is mandatory, and there is no subsumption or simplification, locking, or active and passive sets. And of course, theory inferences are Our framework captures splitting calculi and provers in a general way, independently of the base calculus. Users can conveniently derive a dynamic refutational completeness result for a splitting prover based on a given statically refutationally complete calculus. As we developed the framework, we faced some tension between constraining the SAT solver’s behavior and the saturation prover’s. It seemed preferable to constrain the prover, because the prover is typically easier to modify than an off-the-shelf SAT solver. To our surprise, we discovered counterexamples related to locking, formula selection, and simplification, which may affect Vampire’s AVATAR implementation, depending on the SAT solver and prover heuristics used. We proposed some restrictions, but alternatives could be investigated.We found that labeled splitting can be seen as a variant of AVATAR where the SAT solver follows a strict strategy and propositional variables are not reused across branches. A benefit of the strict strategy is that locking preserves completeness. As for the relationship between AVATAR and SMT, there are some glaring differences, including that splitting is necessary to support disjunctions in SMT but fully optional in AVATAR. For future work, we could try to complete the picture by considering other related architectures , 11, 12.


This study aimed to evaluate the effects of the participant’s attention target during repetitive passive movement (RPM) intervention on reciprocal inhibition (RI) and joint movement function. Twenty healthy adults participated in two experiments involving four attention conditions  during 10-min RPM interventions on the ankle joint. Counting tasks were included to ensure the participant’s attention remained on the target during the intervention. In Experiment 1, RI was measured before, immediately after, and 5, 10, 15, 20, and 30 min after the RPM intervention. In Experiment 2, we evaluated ankle joint movement function at the same time points before and after RPM intervention. The maximum ankle dorsiflexion movement (from 30° plantar flexion to 10° dorsiflexion) was measured, reflecting RI. In Experiment 1, the RI function reciprocal Ia inhibition was enhanced for 10 min after RPM under all attention conditions (excluding the control condition. D1 inhibition was enhanced for 20 min after RPM in the forward and monitor attention conditions and 30 min after RPM in the ankle joint attention condition. In Experiment 2, the joint movement function decreased under the forward and monitor attention conditions but improved under the ankle joint attention condition. This study is the first to demonstrate that the participant’s attention target affected the intervention effect of the RI enhancement method, which has implications for improving the intervention effect of rehabilitation. Decreases in spinal reciprocal inhibition (RI) function are observed in patients with upper motor neuron disorders –3 and olIn recent years, several studies have used techniques to enhance RI to inhibit excessive simultaneous muscle activity –23. ApprOur research group focuses on simpler attention techniques than motor imagery training. The active attention process has been shown to play an important role in motor learning , 27 and Like brain stimulation and motor imagery, attention to the intervention target during RI enhancement may increase the intervention effect of peripheral stimuli. From previous studies , 13, we Therefore, the purpose of this study was to examine the effect of RI enhancement by focusing attention on the target during the RPM intervention and evaluate the intervention effect via joint movement function. The hypothesis of this study is that attention to the intervention target during RPM intervention will inhibit SICI, increase corticospinal tract excitability, activate RI-inhibitory interneurons, and enhance RI potentiation. Moreover, we hypothesize that the joint movement function improvements will be modulated by RI enhancement and corticospinal tract excitability.We conducted two experiments to investigate how focusing attention on the target during RPM intervention affects RI and joint movement function. In Experiment 1, we set attention targets under four conditions during the RPM RI enhancement intervention and assessed RI before and after the intervention. In Experiment 2, the same four conditions were set, and the joint movement function was examined before and after the RPM intervention to examine the effect of the intervention.Twenty healthy adults  provided written informed consent to participate in this study. The study was approved by the Ethics Committee at the Niigata University of Health and Welfare . All experiments were conducted in accordance with the ethical standards of the Niigata University of Health and Welfare and the 1964 Helsinki Declaration and its later amendments.The experimental protocol is illustrated in Fig. The distance between the Ag/AgCl electrodes  for the surface electromyogram was 20 mm. The electrodes were placed on the tibialis anterior (TA) and soleus (Sol) muscles according to the surface EMG for non-invasive assessment of muscles protocols . A grounWe adopted the RPM parameters that most effectively enhanced RI in our previous studies , 13; speFour attention conditions  were randomly applied during a 10 min RPM intervention, with an interval of at least 3 days between each condition. For the control condition, the participant was sat in a chair for 10 min with attention to the front wall (without RPM intervention). For the forward attention condition, the participant’s attention was directed to the anterior wall during the 10-min RPM intervention. For the monitor attention condition, the participant’s attention was directed to the monitor in front of them during the 10-min RPM intervention, and they were asked to count the circles displayed on the monitor at 1 s intervals. The circle was randomly displayed 25–35 times, hidden for 10 s, and then displayed again  under each stimulus condition were averaged, and the Sol H reflection amplitude value and the M wave amplitude value were calculated. The RI was calculated by dividing the Sol H reflection amplitude value by the maximum M wave amplitude value in percent notation (Sol H-reflex amplitude as the percentage of Mmax). When comparing the changes over time in each attention condition, the H-reflex amplitude value of the test stimulus with the conditioned stimulus was divided by the H-reflex amplitude value of the test stimulus and presented as a percentage ([Amplitude of conditioned H-reflex amplitude/test H-reflex amplitude] × 100).Iant is the area of the waveform on which the antagonist muscles acted, calculated using the following equation:t1 to t2 indicate the period in which Sol EMG is less active than TA EMG, and t2 to t3 indicate the period in which TA EMG is less active than Sol EMG. Itotal is the total integral value of Sol and TA EMG calculated using the following formula:The analysis items for the ankle dorsiflexion task were TA and Sol EMG, co-contraction index (CI), ankle dorsiflexion peak torque (PT), and rate of joint movement development (RJD). The analysis sections are (1), from the start of TA EMG to the start of the joint movement; (2), from the start of the joint movement to the end of the joint movement; and (3), from the start of TA EMG to the end of the joint movement. The starting point of TA EMG was defined as the time point when the resting mean EMG ± 3 standard deviations (SD) was exceeded. The EMG of TA and Sol was calculated as the average EMG of each analysis range divided by the maximum voluntary contraction (MVC). The ankle dorsiflexion PT analyzed the maximum value of the ankle dorsiflexion torque of the task movement. The RJD was calculated by dividing the joint angle (40°) from the start to the end of the movement task by the time (s). The CI calculation method is as follows :\document test. To compare the measurement time of each attention condition, the Bonferroni correction was applied to the paired t test. For Experiment 2, paired t tests were used to compare the pre and post results in each analysis interval. The significance level was set to 5% for all analyses.For Experiment 1, repeated measures three-way ANOVA was performed to compare the attention condition  × the stimulus condition  × the measurement time . As a post-test, the comparison of the stimulus conditions of each attention condition was performed by applying the Bonferroni correction to the paired F = 2.575, p = 0.063, partial η2 = 0.119], but detected significant effects of the stimulus condition , the measurement time , and the interaction of three factors .Sol background EMG, Sol Mmax amplitude values, and TA M wave amplitude values are shown in Tables There was no significant difference in the H-reflex amplitude value under the single condition at each measurement time under each attention condition Table . Therefop < 0.001, Table The single condition obtained at each measurement time of each attention condition was compared with the CT-interval conditions. The H-reflex amplitude value was significantly lower under the CT-interval 2 ms and CT-interval 20 ms conditions than under the single condition at all measurement times under all attention conditions (p < 0.001). Similarly, the H-reflex amplitude values under the control condition did not significantly differ from Pre under CT-interval 20 ms. In the forward attention condition and the monitor attention condition, the H-reflex amplitude value was significantly lower at Post 5, Post 10, and Post 20 compared with Pre (p < 0.001). As for the ankle joint attention condition, the H-reflex amplitude value was significantly reduced at Post 5, Post 10, Post 20, and Post 30 compared with Pre (p < 0.001).The H-reflex amplitude value at the measurement times under CT-interval 2 ms and CT-interval 20 ms were compared with the Pre measurement Fig. A, B. Undp < 0.01) and a significant decrease in EMG for Sol (p < 0.05) for all analysis sessions in the ankle joint attention condition . Under the ankle joint attention condition, the ankle dorsiflexion PT was significantly increased (p < 0.001).Comparing the Pre and Post ankle dorsiflexion PT within each attention condition, the ankle dorsiflexion PT was significantly reduced under the forward attention condition and the monitor attention condition (p < 0.001). The RJD significantly increased under the ankle joint attention condition (p < 0.001).Comparing the Pre and Post RJD within each attention condition, the RJD significantly decreased under the forward attention condition and the monitor attention condition (p < 0.01) and (3) (p < 0.001).Comparing the pre- and post-CI within each attention condition, the CI decreased significantly in the analysis sessions of ankle joint attention condition (2)  were similar to those of the forward attention condition and the monitor attention condition, which involved counting the appearances of a circle on the monitor, indicating the effect of counting was negligible. This finding is consistent with the results of a previous report , who shoIn this study, the RI enhancement effect of RPM intervention supported the results of our previous study , 13, i.eIn addition, the activity of motor-associated cortical areas activates RI-inhibitory interneurons and contributes to RI enhancement, and the intervention effect of peripheral stimulation is enhanced by brain stimulation , 21, 23.The joint movement function evaluation in Experiment 2 revealed that the joint movement function decreased under the forward attention condition and the monitor attention condition and was improved under the ankle joint attention condition. In addition, this is the first study to evaluate joint movement function as a post-intervention effect of the RPM RI enhancement method. In the forward attention and monitor attention conditions, the RI was increased by the RPM intervention, but the joint movement function decreased, contrary to our hypothesis. However, under the ankle joint attention condition, the RI was increased by the RPM intervention, and the joint movement function was improved, supporting our hypothesis. The reason for the decrease in joint movement function under the forward attention and monitor attention conditions could be the decrease in the corticospinal tract excitability of the agonist muscle, which may be caused by post-exercise depression (PED). It has been reported that PED is induced by non-fatigue active movement and passive movement –51. PED Under the ankle joint attention condition, TA EMG increased during dorsiflexion after RPM intervention, the CI decreased due to the decrease in Sol EMG, and the joint movement function improved due to the increase in ankle dorsiflexion PT and RJD. The results of this study suggest that in addition to the RI enhancement (antagonist muscle EMG decrease during joint movement) by RPM, attention to the RPM intervention target may have increased M1 activity and thus the muscle output of the main movement muscle, improving the joint movement function. In a previous study , focusinOne limitation of this study was that the excitability of the corticospinal tract corresponding to the muscle targeted for RPM intervention could not be evaluated. However, in previous studies , 52, 57,RPM is an effective and efficient RI enhancement intervention method. In addition, the findings of this study suggest that simply focusing attention on the RPM intervention target can enhance RI potentiation and improve joint movement function, which may be an effective adjunct therapy for functional recovery after central nervous system injury. RI-enhancing methods other than RPM have only shown effects for 10 min after interventions of 15 min or longer –23. In tIn this study, we examined RI and joint movement function after RPM intervention, during which the participant’s attention was focused on different targets. As a result, it was clarified that the RPM intervention effect could be improved by focusing the participant’s attention on the intervention target  during the 10-min RPM. Furthermore, the RI enhancement lasted for 30 min after the intervention, and the joint movement function improved.


We prove the existence of stationary solutions for the density of an infinitely extended plasma interacting with an arbitrary configuration of background charges. Furthermore, we show that the solution cannot be unique if the total charge of the background is attractive. In this case, infinitely many different stationary solutions exist. The non-uniqueness can be explained by the presence of trapped particles orbiting the attractive background charge. This phenomenon can be modeled by the following nonlinear stationary Vlasov–Poisson equation for the plasma electron density mentclass2pt{minimA number of results have been shown for the nonlinear Vlasov–Poisson equation in the case of finite mass and finite energy. In this case, the conserved quantities can be used to study existence and stability of stationary solutions (cf. ). We alsA closely related problem is the existence and stability of stationary solutions to the Vlasov–Poisson Boltzmann equation. In the presence of collisions, the natural boundary condition entclass1pt{minimarefer to  for detaf exist as soon as the total charge of the background measure is attractive.The key point of this paper is to show that stationary solutions to  –1.3) ex ex1.3) eThe main result is contained in the following theorem.Q is understood in the weak sense, and the boundary condition (Let  problem –1.3). H. H\docum problem –(1.3).F of We split the proof of Theorem entclass1pt{minimaf and the compact embedding In , the eximentclass2pt{minimThe remainder of the paper is devoted to the proof of Theorem f of the formF in normalization (ii)decay condition (iii)stability condition: (Velocity distribution at infinity) The boundary condition For future reference, we defineg defined in ((i)normalization: (ii)differentiability and monotonicity: (iii)sub-differential at zero: (iv)growth condition: for some Before we establish the existence of solutions under the conditions  , , \documefined by  and (2.4Step 1. By construction we have ′>0 (cf. ).Step 2. The normalization condition  satisfin g (cf. –2.4) fo fo\documduced in  and for B follows from the subdifferential condition  Let n g (cf. ) satisfin g (cf. –2.4) fo fo\documn system –1.3) in in\documStep 1. Recall B[S] satisfiesS is a weak solution toH byStep 2. We pick the coefficient K is a continuous compact operator. Continuity follows from the continuity of B shown in Lemma K is precompact. To this end, consider a sequence fined in , and letion (cf. ), we knolight of , B[S] saaring in  and3.6\dStep 3. We infer from Step 2 the existence of a non-negative solution K maps into a bounded set in K, i.e. R satisfying   Step 5. ave used . \documeIn the preceeding sections we have constructed solutions of the Vlasov–Poisson equation of the form(Non-Uniqueness) Let Let further e system –1.3) pr pr\documStep 1. We first prove QStep 2. The functions Step 3. Finally A has positive measure. Then we estimate


The second sentence previously stated, “The alpha-band powers in the audiovisual  and that in the auditory condition , indicated with asterisk (*).The alpha-band power in the audiovisual condition was significantly lower than that in the resting condition  and that in the auditory condition , indicated with asterisk (*).Figure 2. Band powers at the central electrode site (Cz) in the three experimental conditions . The alpha-band power in the audiovisual condition was significantly lower than that in the resting condition (Cohen's The authors apologize for these errors and state that this does not change the scientific conclusions of the article in any way.The original article has been updated.


This article presents a review on the recent advances in the field of ternary diglycidyl ether of bisphenol A epoxy nanocomposites containing nanoparticles and other modifiers. Particular attention is paid to their mechanical and thermal properties. The properties of epoxy resins were improved by incorporating various single toughening agents, in solid or liquid states. This latter process often resulted in the improvement in some properties at the expense of others. The use of two appropriate modifiers for the preparation of hybrid composites, possibly will show a synergistic effect on the performance properties of the composites. Due to the huge amount of modifiers that were used, the present paper will focus mainly on largely employed nanoclays with modifiers in a liquid and solid state. The former modifier contributes to an increase in the flexibility of the matrix, while the latter modifier is intended to improve other properties of the polymer depending on its structure. Various studies which were carried out on hybrid epoxy nanocomposites confirmed the occurrence of a synergistic effect within the tested performance properties of the epoxy matrix. Nevertheless, there are still ongoing research works using other nanoparticles and other modifiers aiming at enhancing the mechanical and thermal properties of epoxy resins. Despite numerous studies carried out so far to assess the fracture toughness of epoxy hybrid nanocomposites, some problems still remain unresolved. Many research groups are dealing with many aspects of the subject, namely the choice of modifiers and preparation methods, while taking into account the protection of the environment and the use of components from natural resources. Epoxy resins are known for their excellent chemical resistance, high specific strength, good dimensional stability and adhesion properties ,2,3,4,5.The modification of epoxy resins to improve the above-mentioned properties has been the subject of intense research interest in the last four decades ,10,11,12Due to the very large number of potential modifiers to be used in the preparation of hybrid epoxy composites, this review focuses on nanofillers  and other modifiers such as thermoplastics, liquid rubbers, thermosets, diluents. Very interesting reviews have already been published in recent years on hybrid epoxy nanocomposites ,68,69,70Nanoparticles can be classified into different categories depending on their shape, size, morphology and properties. Based on their shapes, nanofillers are classified into three categories ,72: nanoIt is well known that the dispersion of nanoparticles plays a key role in improving the mechanical and thermal properties of polymer nanocomposites. This can be achieved by the intensive mixing of nanoparticles in the polymer matrix, aiming at the exfoliation and/or intercalation of the former, resulting in a higher contact surface between the constituents.Epoxy resins were modified with different liquid modifiers such as liquid natural rubbers ,76,77,78One of the most promising methods to toughen epoxy resin was the use of liquid rubbers in combination with nanoparticles. Rubber toughening of epoxy resin has been considered for over five decades to be one of the most effective methods for reducing brittleness and increasing other matrix properties. In general, epoxy resins modified with liquid rubbers exhibit improved flexibility and impact resistance. The inclusion of nanofillers in the epoxy matrix along with rubber is expected to improve its strength, stiffness and glass transition temperature, without, however, reducing the decrease in its stiffness, strength or glass transition temperature.In fact, most of the conducted investigations using nanoparticles with liquid rubbers showed an improvement in epoxy performance properties. The epoxy/rubber nanocomposites have been used in various fields such as adhesives, coatings, transducers and in electromagnetic systems.Hydroxyl-terminated polybutadiene (HTPB) is a telechelic, translucent liquid rubber which finds a wide range of applications. It is characterized by a low glass transition temperature of about −75 °C, an average molecular weight and a high level of reactive functionality, making it an excellent potential curing modifier for epoxy resin. Various researchers used both nanofillers and HTPB to improve the properties of diglycidyl ether of bisphenol A (DGEBA) epoxy resin ,81,82,83C in epoxy/nanosilica/rubber microparticles, epoxy/nanosilica/rubber nanoparticle composites and epoxy/CNT/rubber nanoparticle composites. The simultaneous incorporation of microscale elastomers/thermoplastics and nanoscale fillers for the modification of epoxy systems was also investigated and confirmed the complicated interaction between the modifier at the micro- and nanoscales and the inherent problems of synergy [Marouf et al.  confirme synergy .+) to improve the mechanical and thermal properties of epoxy resin [C) of the hybrid epoxy composite containing nanosilica and liquid rubber. The toughness of the ternary composite was higher than those of the binary epoxy systems, due to the obvious positive interactions between the epoxy hybrid constituents.Due to its good thermal oxidation resistance, oil resistance and damping properties, acrylic rubber (ACM) particles were combined with montmorillonite  have become one of the most attractive nanofillers for polymer matrices. They exhibit a very high mechanical strength with 1.0 TPa in Young’s modulus, 130 GPa in ultimate strength, high thermal conductivity of about 5000 Wmstrength . The tenstrength .Still, with the aim of achieving an improvement in the mechanical properties of the matrix, another rubber, namely acrylonitrile butadiene terminated by carboxyl (CTBN), was used with graphene nanoplatelets (GnPs) with different diameters . The addA very successful approach that has been widely used by various researchers to toughen diglycidyl ether of Bisphenol A (DGEBA) modified with nanoparticles involved the addition of a liquid copolymer of acrylonitrile butadiene terminated by carboxyl (CTBN), amine (ATBN) and epoxy (ETBN) groups ,96,97,98C and GC) maximally increased by 2.2 and 7.6 times, respectively, at 6 phr organoclay (Nanomer I.30E) and 20 phr CTBN loading compared to the pristine resin. A positive superposition effect was obtained on the fracture toughness of the epoxy matrix modified with both rubber and organoclay. The ultimate strength, yield strength and glass transition temperature of the epoxy resin modified with rubber and organoclay have also been improved. Similar findings were obtained by Chonkaew et al. [Using high pressure and direct mixing methods, Liu et al.  showed tw et al.  regardinw et al. . Cavitatw et al. ,93.C and GC) of ternary epoxy/CTBN/SN composites were higher than those of epoxy/SN and epoxy/CTBN binary systems [C value of the hybrid composite containing 15 wt% CTBN and 5 wt% SN more than doubled compared to the virgin matrix with a distinct positive synergistic effect between the CTBN and SN nanofillers on the fracture energy, GC. Liang and Pearson [C, was also obtained with more than 5 wt% SN in epoxy, most probably due to particle agglomeration.Moreover, it was shown in a separate study that the fracture toughness parameters  were also used in combination with CTBN liquid rubber to increase the fracture toughness of an epoxy resin. Hosseini and his research group [2O3 nanoparticles (0–10 phr). Hybrid nanocomposites containing 15 phr of CTBN and 7 phr of Al2O3 exhibited the maximum values of the critical stress intensity factor (K C) and critical strain energy release rate (GC).Metal oxides such as Al systems . Aluminach group  preparedC) of an epoxy matrix due to the synergistic effect of silica nanoparticles (SNs) and rubber particles. The critical strain energy release rate parameter, GC, of the epoxy resin increased from 77 to 191 J/m2 and 671 J/m2 by adding 15 wt% SN and 9 wt% of rubber particles, respectively. However, it increased up to 965 J/m2 for the ternary epoxy nanocomposites containing both SNs and rubber microparticles. The improvement in the toughness of the epoxy matrix containing soft and solid particles is explained by the same mechanism reported previously [In a comprehensive review, Kausar  comparedeviously ,88,93.C) values of EP containing 1 wt% nanoclay increased approximately by 200% and 75%, respectively, in relation to the neat EP. In another study, the matrix was modified with Nanobent ZW1 and ATBN with different amine equivalent weights (ATBN-16 with 18% acrylonitrile and ATBN-21 containing 10 wt% acrylonitrile) [Similarly to CTBN, liquid butadiene acrylonitrile terminated with amine groups (ATBN) was also used to toughen epoxy nanocomposites ,102,103.C) of an epoxy composite. It is seen that the hybrid composite can be prepared by fixing the content of one modifier with the highest value and by varying the amount of the second modifier.Kong et al.  preparedDue to its reactive groups, polysulfide is considered, along with liquid rubbers such as CTBN, ATBN and hydroxyl-terminated polybutadiene, as the most effective toughening agent for epoxy resins. The reaction between the thiol and the epoxy groups is expected to enhance the flexibility and impact strength (IS) of the epoxy matrix but decrease certain properties such as the glass transition temperature (Tg) and modulus. Pristine and carboxyl-functionalized multiwalled carbon nanotubes (p-MWCNTs and f-MWCNTs) in addition to polysulfide were used to prepare an amine-cured epoxy resin with improved mechanical properties . The incOther liquid rubbers with specific physical properties have also been used in combination with nanoparticles to improve the properties of epoxy nanocomposites ,110,111.C) with a decrease in the glass transition temperature  and Young’s modulus. However, the addition of SNs led to a modest improvement in the toughness and Tg but a significant increase in the modulus. The ternary hybrid epoxy composites exhibited improvements in the KC, stiffness and Tg. The maximum improvement in the fracture toughness, including a synergistic effect, was obtained with an epoxy matrix containing a defined amount of SNs and ENR. Morphology analysis revealed the existence of the cavitation of rubber particles together with matrix shear yielding, particle debonding and an increase in the damage zone size, which was associated with the dissipation of more energy during the crack propagation. Recently, Kam et al. [C), as compared with LNR and rNR, due to better interfacial adhesion between the epoxy/NR/GNP system. Similar results were obtained on the properties of epoxy resin modified with natural rubber and graphene nanoplatelets [Epoxidized natural rubber (ENR) has been used by various researchers to modify epoxy resins, for its polarity and good compatibility with several polar polymers ,111,112.m et al.  studied The effect of preformed powdered rubber (PR) nanoparticles and chemically reduced graphene oxide (CRGO) was also evaluated on the mechanical and thermal properties of epoxy resin . The incC, of the modified epoxy was measured at ambient and low temperatures (−40 °C and −80 °C). The results showed that hybrid composites with 10 wt% SN and 10 wt% CSR particles exhibited the maximum improvement in the GC, but with no dual strengthening of the matrix, most probably due to the large size of the modifier particles. A similar study was performed with epoxy resin modified with CSR combined with SNs but different results improved the properties and mechanisms [A very interesting approach to toughen epoxy resin was the use of core–shell rubber (CSR) particles, which constitute the specific group of rubber of the effective toughening agent for epoxy resins. The particles are generally based on an elastomeric core such as a homopolymer of butadiene or a styrene–butadiene copolymer and a thin epoxy-compatible shell. The particular advantage of CSR particles compared to liquid rubbers is their compatibility with EPS. Tsang and Taylor  solved tchanisms . The sofchanisms .2 resulted in a simultaneous improvement in the tensile and fracture properties. The fracture toughness parameter, KC, and the fracture energy parameter, GC, of hybrid systems containing 5 wt% CSR and 10 wt% SiO2 increased to a maximum of about 220% and 900%, respectively, compared to the virgin polymer matrix. However, an increase of 165% was reported for the KC of the epoxy hybrid based on 0.075 wt% MWCNTs and 10 wt% SiO2. The so-called core–shell impact modifier particles (CSIMPs) were added with multiwalled carbon nanotubes (MWCNTs) to improve the toughness and tensile properties of the epoxy resin [C, of the hybrid epoxy nanocomposite was higher than those of the matrix binary systems, due to the synergistic effect associated with the shear band formation along with the crack deflection. Similar impressive toughening was achieved by Mehrabi-Kooshki and Jalali-Arani [The MWCNT modifiers have attracted great interest as they generally lead to excellent mechanical, thermal and electrical properties in polymer composites. Furthermore, CSR particles have the ability to promote compatibility between incorporated modifiers with the matrix. Epoxy hybridization with CSR and SiOxy resin . The sofxy resin . Howeverxy resin  used corxy resin . MWCNT mli-Arani  with hybli-Arani . The incMost of the hybrid epoxy resin composites based on solid nanoparticles and liquid modifiers included nanoparticle debonding and void formation yielding of the polymer matrix. The created voids by the microparticles slow down the crack propagation and facilitate the plastic deformation of the epoxy matrix. Due to their specific mechanical, physical, biological and chemical properties, polyurethanes (PUs) are a class of versatile materials which can be used in different applications. Various studies concerned the toughening of epoxy resins by using polyurethanes and nanoparticles ,128,129.C) of the epoxy matrix containing 1 wt% and 2 wt% of nanoclay (Nanobent ZR1) are shown in The impact strength (IS) and critical stress intensity factor  and the epoxy/polyurethane interpenetrating polymer network (EP/PU IPN), leading to a further improvement in the properties of the polymer blend and that of the epoxy matrix .−1.Moreover, it should be remembered that modified montmorillonite (Cloisite 30B) contains quaternary ammonium ions with two hydroxyethyl groups which are able to react with the isocyanate groups of polyurethane, as shown in Mesua ferrea L. seed oil (MFLSO) was used with organically modified bentonite to improve the matrix properties [Recently, much attention has been paid to the toughening of epoxy resin using nanoparticles in combination with vegetable oil-based polyurethanes ,128,129.The last decades have seen a very significant growth in interest in the use of bioderived products, which has been driven by the need to replace petroleum-based materials and produce eco-friendly materials. Recent research has focused on the use of environmentally friendly and non-toxic routes for the synthesis of polyurethanes, which avoid the use of diisocyanates. The procedure offers a number of advantages, including the use of environmentally friendly, inexpensive and biobased components. However, there remains a great challenge to produce epoxy hybrid nanocomposites based on renewable resources with improved performance properties and taking into account environmental protection. Despite the various studies conducted so far to assess the fracture resistance of epoxy nanocomposites, some issues still remain unclear.Białkowska et al.  preparedA green polyurethane (NIPU) was synthesized from sunflower oil via an environment-friendly and non-toxic route and used as a modifier for epoxy resin containing amine-functionalized graphene oxide (af-GO) . The proThe occurrence of chemical reactions between components in hybrid epoxy nanocomposites is possible due to the presence of urethane linkage in the polyurethane structure. Therefore, the order of the addition of components is expected to have a crucial effect on the properties of epoxy composites ,133. CheThe addition of flexible polyurethane chains to an epoxy matrix has been shown to result in the formation of an interpenetrating polymer network (IPN) structure with improved ductility but also toughness with hydrogen bonding and grafting reactions. Moreover, the incorporation of solid nanoparticles to the IPN system contributes to the further reinforcement of the polymer matrix.The concept of incorporating both thermoplastic polymers and nanoclay in the epoxy matrix has been advanced as an efficient method to produce composites with improved mechanical and thermal properties. Thermoplastics have been intensively studied for about four decades as toughening agents because they generally do not cause a significant reduction in the modulus, yield strength and glass transition temperature, as in the case of liquid rubbers. Various ductile thermoplastics were investigated as an alternative to reactive rubber for improving the mechanical properties of the epoxy resins ,155,156.C, of the hybrid nanocomposite increased by approximately 95%, most probably due to the positive effects generated from the combination of two dissimilar toughening mechanisms, but also the prepared suspension of nanoparticles which is supposed to ensure the uniform dispersion and exfoliation of the latter.Although they have a lower strength and toughness than engineering plastics, commodity plastics have been employed in combination with nanoparticles to reinforce the brittle thermosetting epoxy material. Park and Jana  were thePoly vinyl chloride (PVC), which belongs to the important group of commodity thermoplastics with interesting properties, was also used to toughen epoxy nanocomposites. Kaushal et al.  studied C, and the elongation at break of epoxy ternary composites containing 5 phr PSU and 0.2 phr GO were maximally increased by 90% and 55%, respectively, compared to the pure matrix, while the tensile strength, Young’s modulus and thermal stability were not affected. The current strategy of using both PSU and GO as dual toughening agents is promising for effectively toughening epoxy resins without sacrificing mechanical and thermal properties. A similar improvement in mechanical properties was obtained by Rajasekaran et al. [C, and good thermal performance compared to the solvent method. The dual addition of PES and organoclay resulted in a synergistic effect on the fracture toughness of the epoxy resin, regardless of the method used [C of the EP/PES blend modified with 1 wt.% organoclay increased by about 98% compared to the neat epoxy matrix, while it was higher by 49% and 58% compared to the EP/PES and EP/nanoclay systems, respectively. A different toughening mechanism has been proposed compared to the above studies, and which would involve the debonding of the exfoliated nanoparticles but also a bifurcation of the cracks as well as an induced plastic deformation of the matrix [C of epoxy resin composites containing PES, graphene nanoplatelets (GNPs) and carbon nanotubes (CNTs). The addition of 5 wt% PES improved the mechanical properties and thermal stability including 35% and 40% increases in the tensile strength and fracture toughness, KC, respectively. Hybrid epoxy nanocomposites based on 5 wt% PES, GNPs and CNTs showed a synergistic toughening on the KC toward the binary matrix systems, with a remarkably higher Young’s modulus in all epoxy/PES/CNT hybrid epoxy composites. The toughening of the GNP/PESU hybrid epoxy composites was attributed mainly to crack deflection by the PESU and GNP particles, as well as crack bifurcation by the GNP clusters.Another alternative to toughen epoxy resins was the use of tough engineering thermoplastics, such as polysulfone (PSU), polyethersulphone (PES), poly (ether ether ketone) (PEEK), polyetherimide (PEI), acrylonitrile–butadiene–styrene copolymer (ABS), polycarbonate (PC), poly(styrene-co-acrylonitrile) (SAN) and high impact polystyrene (HIPS) ,146,147.n et al.  with tern et al. ,144 conchod used . The KC e matrix  for the If PSU and PES with similar molecular structures were used for their relatively high rigidity and glass transition temperatures, polyetherimide, poly(ether ether ketone) and acrylonitrile–butadiene–styrene terpolymer have been widely used due to the excellent combination of physical and mechanical properties including toughness, stiffness, chemical and solvent resistance as well as high glass transition temperature.2–MWCNTs) or carboxyl-functionalised (COOH–MWCNTs) have been carried out by Chen al. [2–MWCNTs. Ternary composites with 5 phr PEI and 0.4 phr NH2–MWCNTs showed the maximum improvement with a synergistic effect in the fracture toughness, KC [C, referred to as the fracture toughness of composites based on (COOH–CNTs), significantly increased, while the glass transition temperature and the storage modulus remained unaffected [C of the matrix. The debonding of the nanotubes and the resulting increase in the crack path can explain the toughening of the epoxy matrix. Hydroxyl-terminated poly (ether ether ketone) oligomer with pendant methyl groups (PEEKMOH) was used by Asif et al. [C, and strain at break of epoxy composites based on 1 phr oMMT and 5 phr PEEKMOH increased by 65% and 85%, respectively, compared to the neat matrix, without synergistic reinforcement. This could be related to gel formation during the curing of the matrix.Studies on epoxy resin toughening using polyetherimide (PEI) combined with amino-grafted multiwalled carbon nanotubes  is a high-performance commercial thermoplastic with a specific structure, a rubbery polybutadiene dispersed in a rigid styrene–acrylonitrile copolymer system. Due to its excellent mechanical and thermal properties, ABS was used with nanoparticles to improve the properties of epoxy resin by several authors ,150. MirTo further improve the properties, the epoxy resin was modified with methyl phenyl silicone (MPS), before the incorporation of ABS and graphene oxide (GO) . The tenC) and flexural strength was obtained with an epoxy composite based on 1 wt% montmorillonite (Cloisite 30B) and 5 wt% by weight of PC [C, while the flexural strength was slightly improved compared to the unmodified epoxy resin. Hybrid composites containing nanofillers in combination with another thermoplastic not only showed a higher impact strength and resistance to crack propagation, but also showed a synergistic effect with respect to the fracture energy [Polycarbonate (PC) is an amorphous engineering thermoplastic which has good thermal stability, outstanding impact resistance and good transparency. The maximum improvement in the impact strength (IS), fracture toughness  and graphene oxide (GO) nanosheets grafted with PA6 [2O3 nanocomposites showed an improved tensile and impact strength combined with an excellent dielectric and heat resistance [Although polyamide (PA) is of great interest as a compatible thermoplastic reinforcing agent for brittle epoxy resin and because of the possible formation of chemical bonds, there have not been enough research studies of its use in hybrid epoxy nanocomposites. Gul et al.  conductewith PA6 . The frawith PA6 . High imwith PA6 . A signiwith PA6  obtainedsistance . The impsistance  used polC, and impact strength of an epoxy resin containing 10 phr poly(methyl methacrylate)-b-poly(butyl acrylate)-b-poly(methyl methacrylate) triblock copolymer increased by 91.5% and 83.5%, respectively. They explained the increase in the mentioned properties by the nanophase structure obtained by self-assembly during the curing process. It was further confirmed that the toughness was improved to a large extent without significantly decreasing the glass transition temperature of the blends. In another work, the addition of 5 wt% poly-b-poly(ethylene oxide) (PEP-PEO) (BCP) block copolymer to a DGEBA epoxy resin led to a 100% improvement in the fracture toughness (KC), without affecting the modulus and Tg [C increase. As in most of the works previously described [C, by 20% and the critical strain energy release rate, GC, by 25%, while the use of only 5 wt% BCP resulted in a significant improvement in the KC (115%) and GC (∼400%), compared to the virgin epoxy matrix. Moreover, the KC and GC parameters were further improved when simultaneously incorporating 10 wt% nanosilica and 5 wt% BCP into the epoxy resin, beyond those obtained with the EP/nanosilica systems and EP/BCP, individually. However, various investigations have confirmed that a synergistic improvement is obtained in the properties of hybrid epoxy nanocomposites when specific conditions are met. Jojibabu et al. [Block copolymers offer several advantages over other copolymers due to their specific structures which consist of blocks of chemically dissimilar homopolymers linked together by covalent bonds. The effectiveness of a small amount of block copolymers as toughening agents for epoxy resins has been confirmed by various researchers ,162,163.escribed ,144,145,escribed . The addu et al.  investigC, of 115% and critical energy release rate, GC, of 410% compared to the unmodified matrix [C of 34% compared to the neat epoxy. The fracture toughness of the hybrid composite was not significantly improved compared to the binary epoxy systems, due to the difference in the mechanisms for the improvement in the properties and the inadequate proportion of the hybrid composite constituents. It seemed that the proportion of nanoparticles in the composite was not optimal to achieve a synergistic reinforcement of the matrix.The addition of 5 phr polystyrene–polybutadiene–poly(methyl methacrylate) triblock copolymers with unfunctionalized carbon nanotubes (CNTs) resulted in an increase in the fracture toughness, Kd matrix . Interesd matrix  were alsCK and CG, of the epoxy composite containing 6 wt% copolymer and 0.075 wt% MWCNTs increased from 0.57 to 1.41 MPa.m1/2 and 0.096 to 0.59 kJ/m2, respectively, but with no synergistic toughening effect. On the other hand, the tensile strength, strain at yield and modulus decreased with the increasing amount of copolymer particles, due to the softness of the particles. On the other hand, Zhang et al. [C). The fracture parameter maximally increased with the introduction of 5 wt% copolymer or 0.25 wt% MWCNTs, compared to the neat matrix. With the simultaneous incorporation of 5 wt% copolymer and 0.25 wt% MWCNTs, the KC value was significantly increased and was much higher than the binary system EP/copolymer and EP/MWCNTs. This was generated by a very good dispersion of the MWCNTs in the matrix, favored by the presence of the block copolymer, but also by the pull-out of MWCNT particles, crack bridging and matrix shear yielding. This is consistent with the positive results of Li et al. [C, over the neat epoxy, while the incorporation of 5 wt% BCP enhanced the parameter, GC, of the epoxy resin by a factor of 12. The combination of the rigid fGO and soft BCP particles at 0.04 wt% and 5 wt%, respectively, led to an 18 times increase in the GC over the unmodified matrix and a 31% improvement over the binary systems.A similar trend was observed when a functionalized PMMA-bloc-PbuA-bloc-PMMA copolymer was used in combination with MWCNT nanofillers to develop hybrid epoxy nanocomposites . The frag et al.  have reci et al. , who preCK) were significantly enhanced compared to the neat epoxy, BCP/epoxy and CSP/epoxy systems, while the thermal stability was maintained almost unchanged [A synergistic effect was also confirmed in the study on epoxy toughening by the simultaneous inclusion of block copolymers (BCPs) and core–shell particles (CSPs). It was shown that the epoxy/BCP/CSP flexural strength and fracture toughness  and 10 phr polyether polyol. Furthermore, the hybrid nanocomposite showed a synergistic effect on the KC and the flexural fracture energy of the epoxy matrix. The crack deflection, void formation with particle cavitation, nanoparticle debonding and induced shear yielding result in a drastic increase in the properties of the matrix [It was reported that the inclusion of soft modifiers such as plasticizers or reactive diluents to epoxy resin generally deteriorate the mechanical properties, including the yield strength, stiffness and glass transition temperature. Plasticization of the epoxy matrix has also been considered as a possible method to increase some properties of epoxy resin by the flexibilization resulting from the increased free volume. Recently, reactive diluents have been considered as a potential modifier to improve the elasticity and impact strength of brittle epoxy resins. Rahman et al.  showed te matrix . A studye matrix  confirmee matrix . Howevere matrix .C, of 255% and 334%, respectively, compared to the neat matrix when the epoxy matrix was modified with 0.1 wt% graphene oxide (GO) or 0.1% wt% of (PEG-grafted GO (GO-g-PEG). Moreover, the thermal degradation temperatures of epoxy nanocomposites based on GO and PEG-grafted GO have been increased by 50 °C [Due to the presence of flexible ether linkages, polyethylene glycol (PEG) is expected to increase the toughness of epoxy resin. Chozhan et al.  reportedby 50 °C .Furthermore, Chang et al.  studied C) improvement was much more significant with 25 wt% of a reactive diluent and 16 wt% CSR nanoparticles with a synergistic effect on the GC. The hybrid epoxy nanocomposites with both SNs and CSR showed only a modest increase in the GC. The increase in the toughness of the hybrid materials was related to the good dispersion of the nanoparticles in the epoxy polymer matrix. Another study concerned the contribution of 1,6-hexanediol diglycidyl ether (reactive diluent) in the toughening of DGEBA epoxy resin modified with graphene nanoplatelets (GNPs) and carboxyl-terminated acrylonitrile butadiene rubber (CTBN) or core–shell rubber (CSR) [C, of the matrix increased by approximately 430% with the incorporation of 9 wt% CTBN and 12.5% of the reactive diluent. In addition, the incorporation of a reactive diluent has been shown to improve the toughness of the epoxy polymer modified with CSR particles, due to the increased ductility of the epoxy matrix. The corresponding toughening mechanisms are those described previously [C) and fracture energy (GC) of the DGEBA/ECO/Cloisite (at a ratio of 8:2:0.1) system which were found to be higher than the other epoxy systems. The improved hybrid properties have been explained mainly by the good dispersion of the nanoparticles as well as their interactions with the matrix, with the respective reactive groups. However, it was found that the use of epoxidized vegetable oil from Mesua ferrea L. seeds in an epoxy resin containing nanoparticles decreased the viscosity as expected but improved the mechanical and adhesive properties of the cured resin [The effect of hexanediol diglycidylether as a reactive diluent for an epoxy resin cured with an anhydride hardener and modified with silica nanoparticles (SNs) and core–shell rubber (CSR) nanoparticles was studied by Carolan et al. . The adder (CSR) . The fraeviously ,173. In eviously , thermoped resin . A signiLiquid modifiers and solid nanoparticles are known to affect the properties of brittle epoxy resins differently, by flexibilization and strengthening, respectively. However, their simultaneous incorporation in the epoxy matrix can lead to positive toughening ,73,81,83Unsaturated polyester (UP) resins, one of the most widely used thermosetting resins, have very good chemical and corrosion resistance, high thermal stability, good processability and low cost. Although cured UP resins are brittle with a low impact strength and poor resistance to crack propagation, they are expected to act as a toughening agent for epoxy resin. It has already been confirmed that oMMT improves the mechanical properties of the epoxy matrix through the exfoliation/intercalation processes ,59. MoreVarious researchers used organomodified montmorillonite (oMMT) in combination with UP to toughen epoxy resins ,187,188.Studies have demonstrated that the increase in properties can be achieved by an optimum mixing of composite constituents ,191. In Vinyl ester and phenolic resins have also been tested as potential toughening agents for epoxy/nanoclay composites ,194. VinAs in the case of unsaturated polyester, cyanate ester resin possesses attractive performance properties but exhibits poor flexibility and high brittleness, which limits its use in many areas. It is expected that a mixture of epoxy resin (EP) and cyanate ester (CE) reinforced with selected nanoparticles would result in improved mechanical properties. EP/CE blends reinforced with functionalized multiwalled carbon nanotubes (MWCNTs) have been prepared by Li et al.  and theiThe studies indicated that the introduction of unsaturated polyester resin into the epoxy resin and further reinforcement with nanoparticles improved the mechanical and damping properties to an appreciable extent but with no reported positive toughening and synergistic effect.Depending on the improved properties, the prepared hybrid epoxy nanocomposites can be used in various fields. Epoxy/rubber nanocomposites can used as adhesives, while those based on engineering thermoplastics can be intended for the fabrication of advanced composite materials in the building, automotive and aerospace industries. Epoxy/unsaturated polyester systems are employed as a matrix for high-performance composites for engineering applications. The mentioned hybrid epoxy nanocomposites can also be used in existing fields but under severe conditions.This review presents advances on the improvement in the properties of epoxy nanocomposites modified with different types of modifiers. It focused on the toughening of diglycidyl ether of bisphenol A (DGEBA) epoxy resin by using solid nanoparticles with other modifiers such as rubbers, thermoplastics, thermosetting resins and diluents. It has been demonstrated that the performance properties of hybrid nanocomposites depend not only on the type and amount of components, and their eventual interactions, but also on the processing conditions ,198,199.The major challenges to be met in the context of obtaining hybrid polymer composites with high-performance properties remain in the choice of compatible polymers and nanofillers, the appropriate processing method of the constituents as well as the appropriate functionalization and dispersion of the nanofillers. The search for eco-friendly or green polymer nanocomposites is of great importance for researchers because of the necessary protection of the environment. The use of biobased and/or biodegradable modifiers seems to be a logical path toward reducing the problems of recycling ever-increasing quantities of plastic products.Nevertheless, due to the very convincing results recorded during the last decade with remarkable synergies on the mechanical properties of epoxy resin, it would be wise to explore the method of reinforcing epoxy resin with only nanofillers such as montmorillonite, graphene platelets and multiwalled carbon nanotubes ,202,203.


To examine the effects of varying needle heights  and exposure times  on GelMA MNs, the height, tip radius, and angle of the needles were measured; their morphological and mechanical properties were also characterized. It was observed that as the exposure time increased, the height of the MNs increased; moreover, sharper tips were obtained and tip angles decreased. In addition, GelMA MNs exhibited good mechanical performance with no breakage up to 0.3 mm displacement. These results indicate that 3D printed GelMA MNs have great potential for transdermal delivery of various therapeutics.Microneedles (MNs) are micrometer-sized arrays that can penetrate the skin in a minimally invasive manner; these devices offer tremendous potential for the transdermal delivery of therapeutic molecules. Although there are many conventional techniques for manufacturing MNs, most of them are complicated and can only fabricate MNs with specific geometries, which restricts the ability to adjust the performance of the MNs. Herein, we present the fabrication of gelatin methacryloyl (GelMA) MN arrays using the vat photopolymerization 3D printing technique. This technique allows for the fabrication of high-resolution and smooth surface MNs with desired geometries. The existence of methacryloyl groups bonded to the GelMA was verified by  Microneedles (MNs) are skin-penetrating painless transdermal drug delivery systems, which consist of sub-millimeter-sized needles . These dConventional MN fabrication techniques include molding-based techniques , photoliThe first-generation materials used in MN fabrication include silicon, metals, ceramics, and glasses . These mWhen using photopolymers in 3D printing systems, the use of a photoinitiator is necessary to facilitate the crosslinking process . During Herein, GelMA MN arrays with the desired design at different exposure times were developed using the DLP-based 3D printing technique. As far as is known, no previous studies have been reported in the literature regarding DLP-based GelMA MN arrays fabricated at different heights and exposure times. In the current study, after the GelMA was synthesized, the design, printing conditions, and post-printing processes of MNs were optimized. Mechanically and morphologically optimized GelMA MNs may represent an attractive component in new types of transdermal drug delivery systems. The findings of this study can offer insight into design applications aimed at optimizing 3D-printed MNs for adjustable and customizable drug delivery.Gelatin Type A obtained from porcine skin, methacrylic anhydride (MAA), lithium phenyl-2,4,6-trimethyl-benzoyl phosphinate (LAP), and dialysis membrane  were purchased from Sigma-Aldrich . Sodium carbonate, sodium hydroxide, and hydrochloric acid fuming 37% were obtained from Merck KGaA . Sodium hydrogen carbonate (>99.7%) was obtained from ISOLAB . Phosphate-buffered saline  was purchased from ChemBio (Turkey).10% (w/v) solution of type A gelatin was prepared in 0.1 M carbonate bicarbonate buffer  at 60°C. Then, 0.1 ml of methacrylic anhydride (MAA) per Gram of gelatin was added to the gelatin solution and allowed to react for 3 h at 50°C under constant stirring. The reaction was then terminated by adjusting the pH to 7.4 . The obt1H NMR spectroscopy . Both gelatin and GelMA were dissolved at a 10 mg/ml concentration in D2O; 1H NMR spectra were obtained at a frequency of 600 MHz and at room temperature. The DS of GelMA was calculated according to the following equation:The degree of substitution (DS) of GelMA was investigated using −1 range at a resolution of 4 cm−1.The chemical structure of gelatin and GelMA were investigated using Fourier transform infrared spectroscopy  at room temperature. Spectra were obtained between 450 and 4000 cmThe rheological behavior of GelMA hydrogel was analyzed using a digital rheometer . The oscillation mode was selected to determine the temperature dependence of the shear modulus of the GelMA hydrogel. The temperature ramp test was performed over a range of 37°C–15°C (heating rate 1°C/min) with a frequency of 1 Hz and strain of 1%. The frequency sweep test was carried out at an angular frequency of 0.01–100 rad/s with a constant frequency (1 Hz) and strain (1%). The viscosity of GelMA hydrogel was measured by varying the shear rate from 1 to 100 1/s.2 light intensity and 405 nm light wavelength were used for printing the MNs. Subsequently, MNs with varying heights were subjected to exposure times of 30, 50, and 70 s, respectively.Computer-aided design (CAD) files of conical MN arrays with three different needle heights of 1000, 750, and 500 µm were prepared using SolidWorks 2020 . MNs were designed to be 600 µm wide at the base and were attached to a solid 10 × 10 × 1 mm substrate consisting of a 6 × 6 array. The MN designs were converted to the stl file format and sliced using Chitubox , the software for the 3D printer . A commeFor the printing of MNs, a 10% (w/v) GelMA solution was prepared with PBS solution under constant stirring at 40°C for 30 min. The photoinitiator (LAP) at a 0.5% concentration was added to the GelMA solution and mixed for 10 min. Next, the mixture was brought to room temperature (∼25°C) and transferred to the tank of the DLP printer . MNs werThe height, tip radius, angle, and surface characterization of each MN were evaluated using a scanning electron microscope (SEM) . Prior to analysis, the surfaces of the MNs were coated with gold using a spray coating machine  for 120 s.10% GelMA hydrogels prepared with different exposure times  were characterized in terms of their compression stiffness using a compression testing machine . Cylindrical specimens of GelMA (8 mm in diameter and 6 mm in height) were tested; a rate of 1 mm/min and a maximum strain of 60% were used in these studies. Compressive modulus values were calculated from the initial linear region (0%–20% of strain) of the obtained stress-strain curves. Each measurement was performed in triplicate and results are reported as mean ± standard deviation values.The mechanical strength of GelMA MNs was analyzed with a compression testing machine . MNs were placed on a stainless steel plate at a distance of 2 mm; an axial force was applied at a constant rate of 0.1 mm/min perpendicular to the axis of the MNs. The mechanical characteristics of MNs with different needle heights  and different exposure times  were profiled. All tests were performed in triplicate.p ≤ 0.05 is considered statistically significant, and additional significance is indicated by ** for p < 0.01 and *** for p < 0.001.The experiments were carried out at least in triplicate, and data are expressed as mean ± standard deviation (SD). Post-hoc one-way ANOVA with a Tukey-Kramer pair-wise comparison were employed for statistical analysis. A value of 1H NMR analysis was performed to verify the successful substitution of gelatin with methacryloyl groups , green (c) and purple (d). The signals at around chemical shifts of 5.3 and 5.6 ppm (a+b) were attributed to the acrylic protons (2H) of methacryloyl group grafted to lysine and hydroxylysine residues of the gelatin backbone. This result indicates the existence of C=C bonding in the anhydride structure, which is related to presence of the vinyl groups of methacrylate anhydride  (−1 is attributed to the N–H bending groups of the amide II bond (labelled as green); the peak at 1244 cm−1 is related to the C–N stretching and N-H bending of the amide III bond (labelled as pink) (−1 (labelled as blue) detected in GelMA is associated with the existence of peptide bonds (N–H stretching) and -OH functional groups  . The peaas pink) . Furtherl groups . The shil groups .To determine the viscoelastic properties of 10% (w/v) GelMA hydrogel, its shear moduli and viscosity were evaluated by rheological testing . The hydGelMA MNs designed with three different needle heights  were fabricated with a DLP printer by utilizing three different exposure times ; SEM images of the obtained hydrogel MNs are shown in In order to better observe the influences of changing needle sizes and exposure times on the printability of MNs, three parameters  of the printed MNs were measured and presented in Another important parameter, that is, necessary to ensure penetration of the MNs into the skin is the tip radius. The tip radius determines the sharpness of the MN . FiguresThe θ angle was the last design parameter measured. Regardless of the tip radius, this value measures the angle between needle sides. According to The mechanical properties of MNs are critical for successful skin penetration . The mecMoreover, 1H NMR analysis proved the existence of methacryloyl groups attached to GelMA; these results were also supported by FTIR analysis. According to the morphological analysis, it was observed that MNs had higher needle heights and sharper tips as the exposure time increased; the tip angles decreased accordingly. Furthermore, the compression test results showed that increasing the exposure time decreased the amount of displacement. Therefore, it can be said that GelMA MNs at lower heights showed less displacement when equal force was applied. In addition, GelMA MNs demonstrated good mechanical performance without any breakage up to 0.3 mm displacement. These findings indicate that the 3D-printed GelMA MNs have the potential for use in a variety of transdermal drug delivery systems.In this study, GelMA MNs were successfully fabricated with desired geometries at high resolution using the DLP-based 3D printing technique. Smooth surface MNs with different heights  and different UV exposure times  were obtained.