Albert Sakhapov
email: a.sahapov@innopolis.ru

Joseph Alexander Brown
email: j.brown@innopolis.ru

Players Perception of Loot Boxes

Keywords: Loot Boxes, Gambling, Gaming, User Survey

Introduction

The phenomenon that is called loot box has existed for more than a decade. Originally, loot boxes were created to make the players game process easier or interesting for a certain amount of money. At the next stage, loot boxes have become a tool to keep the players interest within the game by offering new virtual items such as new characters, guns, or skins. The item varies from one game to another game, and it might be either customization for a players character that does not affect the balance of the game or booster that makes the game process easier and more comfortable for a target player [START_REF] Lawrence | The troubling psychology of pay-to-loot systems[END_REF].

The study presented in this paper discusses the perception of fairness on loot boxes by the essential people in game development -players. Currently, there is little data on how they perceive loot boxes from the perspective of fairness. Moreover, players opinions might differ according to certain factors of loot boxes, such as the cost of the box, the odds of particular drops, and the outcome. This study aims at finding out and analyzing how players perceive loot boxes and what they expect and want from developers to be adjusted. Our goal is to find the data and learn how players opinions change regarding the factors.

Behind the term loot box, there is a concept which Pawel Grabarczyk and Rune Nielsen named as random reward mechanisms (RRM) [START_REF] Grabarczyk | Are loot boxes gambling? random reward mechanisms in video games[END_REF]. Even though loot boxes appeared about 15 years ago, RRM has existed for decades and first appeared in baseball collecting cards in 1875 in the United States. One card pack contains several randomly selected pieces, ranging from standard to rare. Rare cards can reach prices in the millions of dollars. At that time, children and adolescents were spending hundreds and even thousands of dollars to receive cards of their favourite player, and now they are trying to open their favourite weapon or character skin in video games. In video games RRM came with the first RPG games like Diablo, where killing an enemy gives the player some random reward. That system was used in many games because it is an easy way to introduce the feeling of novelty and difference [START_REF] Wright | The evolution of loot boxes[END_REF]. Moreover, Blizzard Entertainment had an idea to sell individual game assets pack, which would contain several items of different rarity. It was inspired by a collectible card game of Magic: The Gathering that offers players to build their own physical desk from booster packs. However, due to the unavailability of fast Internet technologies and the high cost of distribution of these booster packs on physical discs, the idea was not adopted in the final version of the game [START_REF] Brown | Pitching diablo[END_REF].

After a decade, with the progress of technologies, the concept of RRM began to be used in mobile games. Since most of them are free to download and to play, companies added "microtransactions" that have been an alternative monetization for developers by providing extra items and functions that make the game process more fun and exciting for some amount of money in turn. Currently, 30 games out of 31 contain in-app purchases [START_REF] Shibuya | Systematic analysis of in-game purchases and social features of mobile social games in japan[END_REF]. Microtransactions have become an integral part of game marketing and monetization [START_REF] Stenros | Commoditization of helping players play: Rise of the service paradigm[END_REF].

To increase profits developers started to add loot boxes even in full-priced video games. This has made it easier for companies to recoup multimillion budgets for games, while keeping the initial purchase prices of games relatively the same. The reasons people buy loot boxes were explained by Juho Hamari et al. [START_REF] Hamari | Why do players buy in-game content? an empirical study on concrete purchase motivations[END_REF]. They highlighted six main reasons for opening loot boxes and buying in-game content: Performance, Personalization, Obtainment Achievements, "Showing-off", Low Perceived Cost, and Unlocking Content. The behaviour of players is explained in a Hooked Model and examines how companies keep peoples interest [START_REF] Eyal | Hooked: How to Build Habit-Forming Products[END_REF].

In the Autumn of 2017, EA Games went further than other companies, by partially locking the content in their upcoming game Battlefront II, so that the players spent their money or time to unlock it. Players researched that to open all the content, the player should play at least 4578 hours or spend 2100 dollars [START_REF] Frank | Star wars battlefront 2 content might take years to unlock, but ea wont say[END_REF]. This situation led to a huge video game scandal, and since then, governments of many countries began to regulate loot boxes [START_REF] Cross | How the legal battle around loot boxes will change video games forever[END_REF]. Since loot boxes use a random chance of winning and players pay real-world funds, games may be considered as a form of gambling [START_REF] Olivetti | The perfect ten: The truth about lockboxes[END_REF]. As a result of the scandal, Gaming Commissions of Belgium called game companies to have a "clear indication of the chances of winning for the various item values" [9]. The same requirement made by Chinas Ministry of Culture in December 2016 [START_REF]China's Ministry of Culture: Online game regulations[END_REF]. Singapore's Parliament passed The Remote Gambling Act, which obliges game companies to obtain a license to operate if their games have random awards and real-money payments [START_REF] Times | Remote gambling bill could have 'negative effects' for digital games[END_REF]. Australian Gambling Commission of Victoria State suggested restricting the games that contain loot boxes for children under eighteen [START_REF] Walker | Victoria's gambling regulator: Loot boxes 'constitute gambling[END_REF]. On the other hand of the dispute is UK Gambling Commission that concluded that loot boxes do not need regulations because virtual items are prizes, and they cannot be used outside the game [START_REF]Gambling Commission of UK: Virtual currencies[END_REF]. Entertainment Software Ratings Board (ESRB) organization that provides video game age and content ratings for North America agreed with UKs Gambling Commission and added that the players are guaranteed to receive the in-game item. In May 2019, a bill to ban loot boxes and pay-to-win microtransactions in video games for kids under 18 years was introduced in the US. Senator Josh Hawley explained that game developers should not monetize addiction when a game is designed for kids. After a day, The Entertainment Software Association stated that numerous countries do not consider loot boxes as gambling, and the ESRB organization "does not consider loot boxes to be gambling They desired to share with the senator the tools "that keeps the control of in-game spending in parents hands" [START_REF] Schreier | U.S. senator introduces bill to ban loot boxes and payto-win microtransactions[END_REF]. Though researchers have recommended game developers to avoid giving players real money predicting this could lead to problems with gambling laws [START_REF] Fields | Mobile and Social Game Design: Monetization Methods and Design[END_REF] and associations with real gambling [START_REF] Brooks | Associations between loot box use, problematic gaming and gambling, and gambling-related cognitions[END_REF]. The study presented brings into a discussion this controversial topicloot boxes, namely, how players perceive them.

Methodology

An experiment was started with the development of software that simulates opening a loot box. This system represents a loot box that provides a random item. As the idea was to examine players of Innopolis University, all items hypothetically can be used only within the University, so that students can easily get involved in the system and estimate its fairness without long introductions.

To cover all aspects and not miss important factors, we studied the principles of loot box system in details. The following are loot box factors that were applied in the software: Payment Type. There are three types of payment of loot boxes in video games: fully free, ones that can be purchased with in-game currency, and that can be bought with real-world currencies. Sometimes developers combine all types to cover a wider audience. Outcome. Game development companies use many different outcomes in their loot boxes. In our software, there are three types of outcome: Innopolis University apparel that represents itself an analogy to skins in video games, items that make the process more interesting, but are optional and do not introduce imbalance, and grades for the hypothetical course at University are like boosters that directly affect the result in video games. Rarity. Mostly in all video games, items in loot boxes are divided by rarity. It usually varies from very common to very rare, where the higher rarity, the more valuable and desired item becomes. All rarity types are highlighted in some colour so that players could easily see and identify the rarity. Odds Distribution. Items of different rarity have different odds distribution. Therefore, common items have the highest probability of drop, whereas rare items -the lowest. Odds Display. As was mentioned in the previous section, some governments require game companies to show loot box odds in video games. Developers added odds display only in those countries in which it was obliged, whereas players from other sides of our planet still cannot see the probabilities with which items fall. This factor is important because it might be the new standard in loot box systems. Therefore, it is present in software that was developed and can be enabled or disabled in its settings.

The second part of the study implies gathering of the information of participants. Subjects were asked questions about their experience and preferences in loot boxes regarding fairness in probabilities, rarity division, and outcome type. We collected this information to make overall analysis and to see in the end how participants views have changed during the experiment. Most of the questions in surveys use a scale with seven choices, where one completely disagrees/no, 4 is the neutral answer and 7 -completely agree/yes. This scale clearly shows the opinions of participants and allows us to assess and compare the results of the experiment easily.

All subjects that participated in initial data collection were divided into several groups. Each group covers some of the factors of loot boxes. We chose two the most important and widely discussed factors -items drop probabilities and displaying them. These factors were the subject of discussion of the governments because they form the opinion of people whether to open loot boxes. Thus, we developed two probabilities distributions:

Fair. Probabilities: Common -0.600, Uncommon -0.350 Rare -0.050 Unfair. Probabilities: ,Common -0.849, Uncommon -0.150, Rare -0.001 as a result we examined a full factorial design on the factors of probabilities and showing the values. To gather real information and avoid manipulating results distribution of participants between groups was random. All the participants received their version of the game according to their experiment group.

The final round of collection started with asking participants their opinion of loot boxes and whether they see loot boxes as gambling. These questions were already asked in the initial data collection and would be compared to their previous responses. Further, the participants were asked to describe their ideal loot boxes. The type that they would not mind if removing loot boxes from video games is not an option. Answers on this question might allow players, game companies, and governments to come to a single solution of loot box problem if the requirements and interests of all three sides will be satisfied.

Analysis

The main idea of the study was to understand the players perception of loot boxed, considering many factors of the loot box system. During the experiment, we were trying to influence participants opinions by changing two loot box factors mentioned in previous section. As a result of the experiment, we expected to see the change of participants views.

We created a list of questions on which we will look for changes of opinions: 1) What is your opinion about loot boxes?, 2) Do you agree with the statement: Loot boxes are gambling?, 3) Would you like to see the probabilities of item drops?, 4) Do you agree with the statement: Probability distributions in loot boxes are fair?, 5) Would you like to see the history of your previous openings?, [START_REF] Fields | Mobile and Social Game Design: Monetization Methods and Design[END_REF]) Do you like that items are divided into different groups of rarity?, 7) Which type of items do you like more?, and 8) Which type of loot box payment do you prefer?

Group 1 In the first group with fair probabilities and without display, we did not expect changes in opinion, because odds distribution was hidden from the participants, and they did not know the probabilities. But results were unexpected. Participants raised their opinion regarding fairness in loot box probabilities.

Group 2

The second group had fair probabilities and they were displayed. We observed changes in opinion regarding fairness of probabilities in loot boxes. Three participants in this group changed their opinion from one that equals to "strongly disagree that probabilities in loot boxes are fair" to seven that stands for "strongly agree".

Group 3

The third group had unfair probabilities without showing them. For the participants being in this group was challenging, we received feedback from 10 participants from this group that they did not open even one rare item. Some of them thought that the actual probability of getting a rare item in the game is zero. As a result, 71.4% of the group raised their desire to see probabilities in loot boxes.

Group 4

The fourth group had unfair probabilities with showing them, resulted with a probability distribution for the respondents which was found to be not fair, and 72.2% of the participants in the fourth group decreased their trust in the fairness of loot box odds.

The results showed significance in question four that is stated: Do you agree with the statement: Probability distributions in loot boxes are fair?. The significance level were if p < .05, where p value is an evidence against our hypothesis. For this question, we hypothesized that with different probabilities distributions in groups 1,2 and 3,4, there would be changes in opinion regarding the fairness of odds. For the first and second groups, we expected to see an increasing level of trust and for third and fourth groups -decreasing level. So, players perception of fairness of probability distributions is formed when they face and try loot boxes. The amount of dropped valuable items and odds distribution display helps them to decide if the probability distribution is fair in loot boxes or not.

We observed an increasing level of trust in loot box probabilities in groups one and two, and a decreasing level in groups three and four. The biggest changes were in the fourth group, where the participants saw probabilities, their average changes are -1.44, proving the hypothesis.

After analyzing the third question, we received p value equals to .000974 that makes results significant. The hypothesis for this question was that participants in groups with shown probabilities would increase their opinion regarding the display of probabilities. Therefore, our hypothesis was rejected because we observed changes conversely in groups that do not show probabilities. There was not a significant result that manipulations with probabilities and their display can influence players opinions whether they consider loot boxes as gambling. It was considered that players dislike loot boxes because loot boxes deteriorate video games. However, the distribution is evenly split across all opinions. They explained their belief that loot boxes are an additional way for developers to monetize video games and to earn money. Moreover, if game companies do not overuse loot boxes, players mostly are not against the presence of loot boxes in video games. Moreover, some of the participants noted that sometimes the process of opening loot boxes could be fun and enjoyable if the odds distribution of dropped items is balanced, and loot boxes do not affect game balance. Negative parts of loot boxes that were mentioned by participants are that loot boxes are very similar to gambling and are present in almost every game. However, even though players are neutral to loot boxes, overwhelming majority of the participants -73.6% -agreed that loot boxes are the form of gambling and must be regulated. They mentioned that it could lead to addiction. Only 15.1% of interviewed students did not agree or slightly disagree.

Overall satisfaction of results of loot box openings was also highly rated. So that, only 22.6% of the participants were not happy with results of their loot box trials, 26.4% were neither happy nor sad about the outcome, and 51% -happy and would open loot boxes in future. Even though 73.6% of the participants agreed with the statement that loot boxes are the form of gambling, most of the participants still open them some time, because they can give new experience, and the opening process might be fun and joyful. Only 25.6% spend money on loot boxes; others earn them in video games without investing money. As for preferences in loot boxes, all the questions with amount of people regarding each answer are provided in Table 1.

Compromise Loot Boxes

Loot boxes provide additional monetization for game companies so they could pay back the money spent on development and supporting the game even if it was not very successful. This is why loot boxes appear in almost every modern video game, and game companies start to overuse them. Sometimes it might negatively affect overall video game quality because companies focus on earnings from loot boxes. Such cases lead to an adverse reaction among the gaming community.

Payment Type-88.6% of the participants said the loot boxes should be not only paid with real-world money but should be received during the game.

Outcome-As for this factor, the participants chose different outcome types for two types of online games: PvP and PvE. These types are contrasting, so in PvP games, the participants offered to use cosmetics that do not affect a players statistics and game balance as the only outcome in loot boxes. In PvE, game companies can use any of the outcomes, including booster packs and unique items that can help to improve game character because in PvE, all the players act together, and such things will not spoil the game experience for the others.

Rarity-Participants responses yield that 90.5% claim items distribution in different groups of a rarity to be important for loot boxes. If all the items were the same type of rarity, players would be much less likely to buy them, because there would not be rare, valuable items that players wanted to get.

Odds Distribution-As for odds distribution, participants wanted to be odds distribution fair, so that they do not need to open thousands of loot boxes to get one rare item. However, fairness is a subjective concept, so participants were asked to say the number of loot boxes they agree to open to get a rare item. Participants would like to see one rare item at least every 30 loot box.

Odds Display-88.6% of the participants said that compromise loot boxes should show odds to the players.

Conclusions

In the study conducted, we discussed one of the controversial phenomenons in the video game industry -loot boxes. The majority of the participants face loot boxes almost every day in video games. Opinions are devised on if loot boxes negatively affect the game industry. However, there is the thing that unites those in the survey -the indifference to the game industry. The overall number of people who participated in the study is fifty-three, forty-six male and six females. All of the respondents had experience in video games and faced with loot boxes at least once. The perception of fairness in the study has highlighted many factors that broaden the understanding of peoples preferences in loot boxes. The results so far from the available sample size and covered loot box factors suggest that respondents' loot box experience influenced their preferences. Finally, the participants proposed their vision of ideal loot boxes that can satisfy all the sides of the conflict -giving a picture of ideal loot box that would generate income for game companies, meet governments' regulations and be accepted by video games community.

Table 1 .

 1 Distribution of answers regarding loot box preferences questions.

	Question		Strong No No Leaning No Neutral Leaning Yes Yes Strong Yes
	Would you like to see	0	5	0	1	7		9	31
	probabilities of item drops?
	Do you think probability distributions	4	6	13	12	11		3	4
	in loot boxes are fair?
	Would you like to see the	0	1	4	5	5		9	29
	history of your previous openings?
	Do you like that items are divided	3	2	1	3	7	18	19
	in different groups of rarity?
	Question	Only items	Prefer		Slightly for	Neutral	Slightly	Prefer	Only items
		that affect	Items that	items that		for	items	Items	that do not
		balance	affect		affect bal-		that	do	that	do	affect bal-
			balance		ance		not	affect	not	affect	ance
							balance	balance
	Which type of items do you	3	1		2	3	5		8	31
	like more?
	Question	Only real-	Prefer		Slightly for	Neutral	Slightly	Prefer In-	Only	in-
		World Cur-	Real-World	real-World		for	in-	Game Cur-	Game Cur-
		rency	Currency		Currency		Game Cur-	rency/Free	rency/Free
							rency/Free
	Which type of loot box pay-	2	0		0	2	4		18	27
	ment do you prefer?


Tarun Kansal

Fundamental Solutions in Generalized Theory of Thermoelastic Diffusion with Triple Porosity

Keywords: Thermoelastic diffusion, Triple porosity, Pores, Steady oscillations

The main aim of the paper is to derive the basic governing equations for anisotropic thermoelastic medium with mass diffusion and triple porosity. Additionally, the fundamental solutions of system of equations in cases of steady, pseudo-, quasi-static oscillations and equilibrium are also constructed.

Introduction

Diffusion is the transfer of mass of a substance from the high concentration regions to low concentration regions. Nowacki [START_REF] Nowacki | Dynamical problems of thermodiffusion in solids-I[END_REF][START_REF] Nowacki | Dynamical problems of thermodiffusion in solids-II[END_REF][START_REF] Nowacki | Dynamical problems of thermodiffusion in solids-III[END_REF][START_REF] Nowacki | Dynamical problems of thermodiffusion in solids[END_REF] developed the theory of thermoelasticity with mass diffusion based upon classical Fourier's and Fick's laws. Sherief et al. [START_REF] Sherief | The theory of generalized thermoelastic diffusion[END_REF] established generalized theory of thermoelasticity with mass diffusion by modifying Fourier's and Fick's laws.

The first model for single porosity deformable solid was given by Biot [START_REF] Biot | General theory of three-dimensional consolidation[END_REF]. Aifantis and co-workers [START_REF] Wilson | On the theory of consolidation with double porosity-I[END_REF][START_REF] Khaled | On the theory of consolidation with double porosity-III[END_REF][START_REF] Beskos | On the theory of consolidation with double porosity-II[END_REF] developed the theory for deformable materials with double porosity. A double porosity elastic material has macro pores in the body but in addition there is a micro porosity which arises because of fissures or cracks in the solid skeleton. Khalili and Selvadurai [START_REF] Khalili | A fully coupled constitutive model for thermo-hydro-mechanical analysis in elastic media with double porosity[END_REF][START_REF] Khalili | On the constitutive modelling of thermohydro-mechanical coupling in elastic media with double porosity[END_REF] and Gelet et al. [START_REF] Gelet | A thermo-hydro-mechanical coupled model in local thermal non-equilibrium for fractured HDR reservoir with double porosity[END_REF] established the basic governing homogeneous equations in the linear theory of thermoelasticity for solids with double porosity. A lots of work has been done in this field. All the theories developed by the above mentioned authors were based upon Darcy's law. Without using this law, Iesan and Quintanilla [START_REF] Iesan | On a theory of thermoelastic materials with a double porosity structure[END_REF] derived a non-linear theory of thermoelastic solids with double porosity structure. In the similar manner, Kansal [START_REF] Kansal | Generalized theory of thermoelastic diffusion with double porosity[END_REF] developed a linear generalized theory of thermoelastic diffusion with double porosity. Svanadze [START_REF] Svanadze | Boundary value problems of steady vibrations in the theory of thermoelasticity for materials with a double porosity structure[END_REF] developed the classical potential method in the linear theory of thermoelasticity for materials with a double porosity structure based ---E-mail address: tarun1 kansal@yahoo.co.in on the mechanics of materials with voids. Marin et al. [START_REF] Marin | A generalization of the Saint-Venant's principle for an elastic body with dipolar structure[END_REF] approached transient elastic processes and the steady state in a cylinder consisting of linear elastic body with dipolar structure which is only subjected to some boundary restrictions at a plane end. Emin et al. [START_REF] Amin | Some uniqueness results for thermoelastic materials with double porosity structure[END_REF] obtained new uniqueness results for the anisotropic thermoelastic bodies with double porosity structure based on the Betti reciprocity relation that involve some thermoelastic processes. In triple porosity elastic material, the body possesses three levels of pore structures. The first is the largest visible porosity known as macro porosity, the second represents an intermediate case which is known as meso porosity, and the final scenario is referred to as a micro porosity. Svanadze [START_REF] Svanadze | Fundamental solutions in the theory of elasticity for triple porosity materials[END_REF] and Straughan [START_REF] Straughan | Uniqueness and stability in triple porosity thermoelasticity[END_REF] presented the governing equations in the theories of elasticity and thermoelasticity with triple porosity respectively.

Svanadze [START_REF] Svanadze | External boundary value problems in the quasi static theory of elasticity for triple porosity materials[END_REF][START_REF] Svanadze | Boundary Value problems in the theory of thermoelasticity for triple porosity materials[END_REF][START_REF] Svanadze | External boundary value problems in the quasi static theory of triple porosity thermoelasticity[END_REF][START_REF] Svanadze | Potential method in the theory of elasticity for triple porosity materials[END_REF][START_REF] Svanadze | Potential method in the linear theory of triple porosity thermoelasticity[END_REF][START_REF] Svanadze | On the linear equilibrium theory of elasticity for materials with triple voids[END_REF] studied various boundary value problems on elastic solids and thermoelastic solids with triple porosity. The need for theories for multiple porosity elasticity and the associated mathematical, physical and numerical analysis which accompanies such theories is undoubtedly driven by the myriad of applications which exist and which are coming to light continuously. The first application area is in mathematical biology and the associated field of health. Replacement of damaged long bones in human beings is a major problem for a surgeon since the porosity of the bone can vary from 14% in the outer layer bone to 42% in the inner layer. Indeed to adequately model a long bone one may require a multi-porosity theory which is applicable to a graded porosity material. Another very important area of application for multiple porosity elasticity is in geophysics. For example, a careful description of landslides may well require employment of double porosity theory. Straughan [START_REF] Straughan | Mathematical Aspects of Multi-Porosity Continua[END_REF] discussed various applications of multiple porosity in his book.

The fundamental solutions play an important role in solving various boundary value problems. The reason is that an integral representation of solution of a boundary value problem by fundamental solution is often more easily solved by numerical methods rather than a differential equation with specified boundary and initial conditions. For investigating boundary value problems of the theories of elasticity and thermoelasticity by potential method, it is necessary to construct fundamental solutions of corresponding systems of partial differential equations and to establish their basic properties.Various authors [START_REF] Svanadze | Fundamental solution in the theory of consolidation with double porosity[END_REF][START_REF] Svanadze | Fundamental solutions in the full coupled linear theory of elasticity for solid with double porosity[END_REF][START_REF] Svanadze | Fundamental Solution in the linear theory of consolidation for elastic solids with double porosity[END_REF][START_REF] Scarpetta | Fundamental solutions in the theory of thermoelasticity for solids with double porosity[END_REF][START_REF] Kumar | Some considerations of fundamental solution in micropolar thermoelastic materials with double porosity[END_REF][START_REF] Kansal | Fundamental solution in the theory of thermoelastic diffusion materials with double porosity[END_REF][START_REF] Svanadze | Fundamental solutions in the linear theory of thermoelasticity for solids with triple porosity[END_REF] constructed the fundamental solutions by means of elementary functions in the different theories of elasticity and thermoelasticity with double porosity.

In this paper, the constitutive relations and field equations for anisotropic generalized thermoelastic diffusion with triple porosity are derived in section 2. After reducing the anisotropic system of equations into isotropic system of equations, the fundamental solution in case of steady oscillations is constructed in terms of elementary functions in sections 3 and 4. In section 5, the fundamental solutions in cases of pseudo-, quasi-static oscillations and equilibrium are constructed. Finally, some basic properties of fundamental matrix are also established in section 6.

Basic Equations

Based upon Iesan and Quintanilla [START_REF] Iesan | On a theory of thermoelastic materials with a double porosity structure[END_REF], the law of conservation of energy for an arbitrary material volume V bounded by a surface B at time t can be written as

∫ V ρ[ ui üi + κ 1 ν1 ν1 + κ 2 ν2 ν2 + κ 3 ν3 ν3 + U ]dV = ∫ V ρ[F i ui + Λ α να ]dV + ∫ B [f i ui + Ω (α) β ϖ β να -q i ϖ i ]dB, ( 1
)
where U is the internal energy per unit mass, ρ is the density, q i are the components of heat flux vector q, F i are the components of the external force per unit mass, u i are the components of displacement vector u, f i are the components of surface traction vector f occurring on the surface B, ν α are the volume fraction fields corresponding to macro-, meso-, micro-pores respectively, κ α are the coefficients of equilibrated inertia, Λ α are extrinsic equilibrated body forces per unit mass associated to macro-, meso-, micro-pores respectively, Ω The components f i are connected to stress vectors by the relation

f i = σ ji ϖ j , (2)
where σ ji (= σ ij ) are the components of stress tensor.

Using equation (2) in equation ( 1) and applying divergence theorem, we acquire

∫ V ρ[ ui üi + κ 1 ν1 ν1 + κ 2 ν2 ν2 + κ 3 ν3 ν3 + U ]dV = ∫ V ρ[F i ui + Λ α να ]dV + ∫ V [σ ji,j ui + σ ji ui,j + Ω (α) β,β να + Ω (α) β να,β -q i,i ]dV. (3)
Since equation ( 3) is valid for every part of the body, therefore local form of conservation of energy is obtained as

ρ[ ui üi + κ 1 ν1 ν1 + κ 2 ν2 ν2 + κ 3 ν3 ν3 + U ] = ρ[F i ui + Λ α να ]+ σ ji,j ui + σ ji ui,j + Ω (α) β,β να + Ω (α) β να,β -q i,i . (4)
Let us consider a second motion which differs from the given motion only by a constant superposed rigid body translational velocity. We assume that κ α , U, Λ α , ρ,

Ω (α)
β , q i , F i , σ ji do not vary by such superposed rigid body velocity. Equation ( 4) is also true when ui is replaced by ui + ℘ i , where ℘ i are arbitrary constants, all other terms being unchanged. Therefore, from equation( 4), we have

ρ[( ui + ℘ i )ü i + κ 1 ν1 ν1 + κ 2 ν2 ν2 + κ 3 ν3 ν3 + U ] = ρ[F i ( ui + ℘ i ) + Λ α να ]+ σ ji,j ( ui + ℘ i ) + σ ji ui,j + Ω (α) β,β να + Ω (α) β να,β -q i,i . ( 5
)
Subtracting equation (4) from equation (5), we get

℘ i [σ ji,j + ρF i -ρü i ] = 0. ( 6
)
Since the quantities in the square brackets are independent of ℘ i , therefore from equation ( 6), we obtain

σ ji,j + ρF i = ρ üi . (7)
Equation ( 4) with the assistance of equation ( 7) yields a simplified form of conservation of energy

ρ U = σ ji ui,j + Ω (α) β να,β -q i,i -Υ α να , (8)
where Υ α , α = 1, 2, 3 satisfy the relations Ω (1)

β,β + Υ 1 + ρΛ 1 = ρκ 1 ν1 , Ω (2)
β,β + Υ 2 + ρΛ 2 = ρκ 2 ν2 , Ω (3)
β,β + Υ 3 + ρΛ 3 = ρκ 3 ν3 . ( 9
)
Following Nowacki [START_REF] Nowacki | The theory of asymmetric elasticity[END_REF], the balance of entropy can be composed as

∫ V ρ ṠdV + ∫ B ( q i T ) ϖ i dB - ∫ B ( P η i T ) ϖ i dB = ∫ V [ - q i T 2 T ,i - P ,i T η i + P T 2 η i T ,i ] dV.
(10) where S, P , are entropy and chemical potential per unit mass respectively, η i are the components of mass diffusion flux vector η, T is absolute temperature.

The equation [START_REF] Khalili | A fully coupled constitutive model for thermo-hydro-mechanical analysis in elastic media with double porosity[END_REF] can be written in the local form

ρ Ṡ + ( q i T ) ,i - ( P η i T ) ,i = - q i T 2 T ,i - P ,i T η i + P T 2 η i T ,i . ( 11
)
The right hand side of equation ( 11) is the entropy source

ℜ = - q i T 2 T ,i - P ,i T η i + P T 2 η i T ,i ≥ 0.
On the basis of above relation, equation [START_REF] Khalili | On the constitutive modelling of thermohydro-mechanical coupling in elastic media with double porosity[END_REF] can be represented in the form of an inequality called Clausius-Duhem inequality

ρ Ṡ + q i,i T - q i T 2 T ,i - P T η i,i - P ,i T η i + P T 2 η i T ,i ≥ 0. ( 12
)
The equation of conservation of mass is

η j,j = -Ċ, ( 13
)
where C is the concentration of the diffusion material in the elastic body.

Equation ( 12) with the help of equations ( 8) and ( 13) becomes

ρT Ṡ -ρ U + σ ij ėij + Ω (α) β να,β -Υ α να - q i T T ,i + P Ċ -P ,i η i + P T η i T ,i ≥ 0. ( 14
)
where e ij = 1 2 (u i,j + u j,i ) are the components of strain tensor.

Helmholtz free energy function Γ is stated as

Γ = U -T S. ( 15
)
Applying equation [START_REF] Svanadze | Boundary value problems of steady vibrations in the theory of thermoelasticity for materials with a double porosity structure[END_REF] in the equation ( 14), we get

-ρ[ Γ + Ṫ S] + σ ij ėij + Ω (α) β να,β -Υ α να - q i T T ,i + P Ċ -P ,i η i + P T η i T ,i ≥ 0. ( 16
)
The function Γ can be expressed in terms of independent variables e ij , ν α , ν α,β ,T, T ,i , C

and C ,i . Therefore, we have

Γ = ∂Γ ∂e ij ėij + ∂Γ ∂ν α να + ∂Γ ∂ν α,β να,β + ∂Γ ∂T Ṫ + ∂Γ ∂T ,i Ṫ,i + ∂Γ ∂C Ċ + ∂Γ ∂C ,i Ċ,i . ( 17
)
Equation ( 16) with the help of equation ( 17) becomes

[ σ ij -ρ ∂Γ ∂e ij ] ėij + [ Ω (α) β -ρ ∂Γ ∂ν α,β ] να,β - [ Υ α + ρ ∂Γ ∂ν α ] να -ρ [ S + ∂Γ ∂T ] Ṫ + [ P -ρ ∂Γ ∂C ] Ċ -ρ ∂Γ ∂T ,i Ṫ,i -ρ ∂Γ ∂C ,i Ċ,i - q i T T ,i -P ,i η i + P T η i T ,i ≥ 0. ( 18
)
The inequality should be convinced for all rates ėij , να , να,β , Ṫ , Ṫ,i , Ċ and Ċ,i .

Hence the coefficients of above variables must vanish, that is,

σ ij = ρ ∂Γ ∂e ij , (19)
Ω (α) β = ρ ∂Γ ∂ν α,β , (20)
Υ α = -ρ ∂Γ ∂ν α , ( 21
) S = - ∂Γ ∂T , ( 22
)
P = ρ ∂Γ ∂C , ( 23
)
∂Γ ∂T ,i = ∂Γ ∂C ,i = 0, (24)
-

q i T T ,i -P ,i η i + P T η i T ,i ≥ 0. ( 25
)
Let us introduce the notations

ϕ α = ν α -(ν α ) 0 , θ = T -T 0 , α = 1, 2, 3 (26)
where T 0 is the reference temperature of the body chosen such that

| θ T 0 | ≪ 1, (ν α ) 0
are the volume fraction fields in reference configuration.

In the linear theory, the independent variables are e ij , ϕ α , ϕ α,β , θ and C. It is assumed that the undeformed body is free from stresses and has zero intrinsic equilibrated body forces and entropy. If the body has a centre of symmetry, then we have

2ρΓ = c ijkl e ij e kl + 2ϑ (α) ij e ij ϕ α -2a ij e ij θ -2b ij e ij C + Υ (αα) ij ϕ α,i ϕ α,j + 2Υ (12) ij ϕ 1,i ϕ 2,j +2Υ (13) ij ϕ 1,i ϕ 3,j +2Υ (23) ij ϕ 2,i ϕ 3,j +λ αβ ϕ α ϕ β -2ℓ α ϕ α θ-2ε α ϕ α C- ρC e θ 2 T 0 -2aθC+bC 2 . ( 27
) where c ijkl = c klij = c jikl , ϑ (α) ij = ϑ (α) ji , a ij = a ji , b ij = b ji , Υ (αα) ij = Υ (αα) ji , λ αβ = λ βα .
Using above equation in the equations ( 19)-( 23), the following constitutive equations are obtained:

σ ij = c ijkl e kl + ϑ (α) ij ϕ α -a ij θ -b ij C, (28)
Ω (α) β = Υ (αγ) βp ϕ γ,p , (29)
Υ α = -ϑ (α) jk e jk -λ αβ ϕ β + ℓ α θ + ε α C, ( 30
)
ρS = a ij e ij + ℓ α ϕ α + ρC e θ T 0 + aC, ( 31
)
P = -b ij e ij -ε α ϕ α -aθ + bC. ( 32
)
Equations ( 7) and ( 9) with the aid of equations ( 28)-( 30) become

c ijkl e kl,j + ϑ (α) ij ϕ α,j -a ij θ ,j -b ij C ,j + ρF i = ρü i , (33) -ϑ (1
)

ij e ij + Υ (1α) ij ϕ α,ij -λ 1α ϕ α + ℓ 1 θ + ε 1 C + ρΛ 1 = ρκ 1 φ1 , -ϑ (2) ij e ij + Υ (2α) ij ϕ α,ij -λ 2α ϕ α + ℓ 2 θ + ε 2 C + ρΛ 2 = ρκ 2 φ2 , -ϑ (3) ij e ij + Υ (3α) ij ϕ α,ij -λ 3α ϕ α + ℓ 3 θ + ε 3 C + ρΛ 3 = ρκ 3 φ3 . ( 34
)
The linearized form of equation ( 11) is

ρT 0 Ṡ = -q i,i . ( 35
)
Using equation [START_REF] Kumar | Some considerations of fundamental solution in micropolar thermoelastic materials with double porosity[END_REF] in equation ( 35), we get

T 0 [ a ij ėij + ℓ α φα + a Ċ] + ρC e θ = -q i,i . ( 36
)
Generalized Fourier's law of heat conduction equation is

q i + τ 0 qi = -K ij θ ,j , (37)
where K ij are coefficients of thermal conductivity tensor, τ 0 is the thermal relaxation time which will ensure that the heat conduction equation will predict finite speeds of heat propagation.

Equation (37) with the help of equation ( 36) becomes

( ∂ ∂t + τ 0 ∂ 2 ∂t 2 )[ T 0 (a ij e ij + ℓ α ϕ α + aC) + ρC e θ ] = K ij θ ,ij . ( 38
)
Similar to equation (37), generalized Fick's law of mass diffusion is

η i + τ 0 ηi = -D ij P ,j , (39)
where D ij are coefficients of diffusion tensor, τ 0 is the diffusion relaxation time which ensures that the equation satisfied by the concentration will also predict finite speeds of propagation of matter from one medium to the other.

Using equations ( 13) and [START_REF] Kansal | Fundamental solution in the theory of thermoelastic diffusion materials with double porosity[END_REF] in equation ( 39), we get

-D ij [b kl e kl,ij + ε α ϕ α,ij + aθ ,ij -bC ,ij ] = Ċ + τ 0 C. ( 40
)
If we take

c ijkl = λδ ij δ kl + µδ ik δ jl + µδ il δ jk , a ij = ϑ 1 δ ij , b ij = ϑ 2 δ ij , ϑ (α) ij = ℜ α δ ij , Υ (αγ) ij = A αγ δ ij , K ij = Kδ ij , D ij = Dδ ij ,
where δ ij is Kronecker's delta, in the equations ( 33), ( 34), ( 38) and (40), the governing equations for homogeneous isotropic generalized thermoelastic diffusion with triple porosity in absence of body forces are obtained as

µ∆u + (λ + µ)∇ div u + ℜ α ∇ϕ α -ϑ 1 ∇θ -ϑ 2 ∇C = ρü, ( 41
)
-ℜ 1 div u + A 1α ∆ϕ α -λ 1α ϕ α + ℓ 1 θ + ε 1 C = ρκ 1 φ1 , -ℜ 2 div u + A 2α ∆ϕ α -λ 2α ϕ α + ℓ 2 θ + ε 2 C = ρκ 2 φ2 , -ℜ 3 div u + A 3α ∆ϕ α -λ 3α ϕ α + ℓ 3 θ + ε 3 C = ρκ 3 φ3 , ( 42
)
( ∂ ∂t + τ 0 ∂ 2 ∂t 2 )[ T 0 (ϑ 1 div u + ℓ α ϕ α + aC) + ρC e θ ] = K∆θ, ( 43
)
D∆[ϑ 2 div u + ε α ϕ α + aθ -bC] + ( ∂ ∂t + τ 0 ∂ 2 ∂t 2 ) C = 0, (44)
where ∆, ∇ are respectively, Laplacian and Del operators.

In the upcoming sections, the chemical potential has been used as a state variable rather than concentration. In isotropic medium, equation [START_REF] Kansal | Fundamental solution in the theory of thermoelastic diffusion materials with double porosity[END_REF] becomes

P = -ϑ 2 div u -ε β ϕ β -aθ + bC. ( 45
)
The equations ( 41)-(44) with the aid of equation ( 45) can be rewritten as

µ∆u + (λ ′ + µ)∇ div u + σ α ∇ϕ α -ζ 1 ∇θ -ζ 2 ∇P = ρü, ( 46
)
-σ 1 div u + A 1α ∆ϕ α -L 1α ϕ α + ξ 1 θ + v 1 P = ρκ 1 φ1 , -σ 2 div u + A 2α ∆ϕ α -L 2α ϕ α + ξ 2 θ + v 2 P = ρκ 2 φ2 , -σ 3 div u + A 3α ∆ϕ α -L 3α ϕ α + ξ 3 θ + v 3 P = ρκ 3 φ3 , ( 47
) - ( ∂ ∂t + τ 0 ∂ 2 ∂t 2 ) T 0 [ ζ 1 div u + ξ α ϕ α + ηθ + ςP ] + K∆θ = 0, (48)
-

( ∂ ∂t + τ 0 ∂ 2 ∂t 2 )[ ζ 2 div u + v α ϕ α + ςθ + ϖP ] + D∆P = 0, (49)
where

ϖ = b -1 , ζ 2 = ϖϑ 2 , ζ 1 = ϑ 1 + a ζ 2 , σ α = ℜ α -ε α ζ 2 , λ ′ = λ -ζ 2 ϑ 2 , ς = aϖ, v α = ε α ϖ, L βα = λ βα -ε β v α , ξ α = ℓ α + ςε α , η = ρC e T 0 + aς.

Steady Oscillations

Let x = (x 1 , x 2 , x 3 ) be the point of the Euclidean three-dimensional space E 3 ,

|x| = (x 2 1 + x 2 2 + x 2 3 ) 1 2 , D x = ( ∂ ∂x 1 , ∂ ∂x 2 , ∂ ∂x 3 ).
The displacement vector, volume fraction fields, temperature change and chemical potential functions are assumed as:

[ u(x, t), ϕ α (x, t), θ(x, t), P (x, t)

] = Re [ (u * , ϕ * α , θ * , P * )e -ιωt ] , ( 50
)
where ω is oscillation frequency.

Using equation (50) in equations ( 46)-(49) and omitting asterisk (*) for simplicity, the system of equations of steady oscillations are obtained as

µ∆u + [(λ ′ + µ)∇ div + ρω 2 ]u + σ α ∇ϕ α -ζ 1 ∇θ -ζ 2 ∇P = 0, -σ 1 div u + A 1α ∆ϕ α -M 1α ϕ α + ξ 1 θ + v 1 P = 0, -σ 2 div u + A 2α ∆ϕ α -M 2α ϕ α + ξ 2 θ + v 2 P = 0, -σ 3 div u + A 3α ∆ϕ α -M 3α ϕ α + ξ 3 θ + v 3 P = 0, τ 1 T 0 [ζ 1 div u + ξ α ϕ α ] + [K∆ + τ 1 ηT 0 ]θ + τ 1 ςT 0 P = 0, τ 1 [ζ 2 div u + v α ϕ α + ςθ] + [D∆ + τ 1 ϖ]P = 0, (51)
where

M βα = L βα -ρω 2 κ β ϕ β δ βα , τ 1 = ιω(1 -ιωτ 0 ), τ 1 = ιω(1 -ιωτ 0 ).
If we replace ω by -ιτ , where τ is a complex number and Re(τ ) > 0 in the equations (51), we obtain the system of equations of pseudo-oscillations as:

µ∆u + [(λ ′ + µ)∇ div -ρτ 2 ]u + σ α ∇ϕ α -ζ 1 ∇θ -ζ 2 ∇P = 0, -σ 1 div u + A 1α ∆ϕ α -M1α ϕ α + ξ 1 θ + v 1 P = 0, -σ 2 div u + A 2α ∆ϕ α -M2α ϕ α + ξ 2 θ + v 2 P = 0, -σ 3 div u + A 3α ∆ϕ α -M3α ϕ α + ξ 3 θ + v 3 P = 0, τ1 T 0 [ζ 1 div u + ξ α ϕ α ] + [K∆ + τ1 ηT 0 ]θ + τ1 ςT 0 P = 0, τ 1 [ζ 2 div u + v α ϕ α + ςθ] + [D∆ + τ 1 ϖ]P = 0, (52)
where

Mβα = L βα + ρτ 2 κ β ϕ β δ βα , τ1 = τ (1 -τ τ 0 ), τ 1 = τ (1 -τ τ 0 ).
If we put ρ = 0 i.e. neglecting inertial effect in the equations (51), we obtain the system of equations of quasi-static oscillations as:

[µ∆ + (λ ′ + µ)∇ div]u + σ α ∇ϕ α -ζ 1 ∇θ -ζ 2 ∇P = 0, -σ 1 div u + A 1α ∆ϕ α -L 1α ϕ α + ξ 1 θ + v 1 P = 0, -σ 2 div u + A 2α ∆ϕ α -L 2α ϕ α + ξ 2 θ + v 2 P = 0, -σ 3 div u + A 3α ∆ϕ α -L 3α ϕ α + ξ 3 θ + v 3 P = 0, τ 1 T 0 [ζ 1 div u + ξ α ϕ α ] + [K∆ + τ 1 ηT 0 ]θ + τ 1 ςT 0 P = 0, τ 1 [ζ 2 div u + v α ϕ α + ςθ] + [D∆ + τ 1 ϖ]P = 0, (53)
If we put ω = 0 in the equations (51), we obtain the system of equations in equilibrium theory of thermoelastic diffusion with triple porosity as:

[µ∆ + (λ ′ + µ)∇ div]u + σ α ∇ϕ α -ζ 1 ∇θ -ζ 2 ∇P = 0, -σ 1 div u + A 1α ∆ϕ α -L 1α ϕ α + ξ 1 θ + v 1 P = 0, -σ 2 div u + A 2α ∆ϕ α -L 2α ϕ α + ξ 2 θ + v 2 P = 0, -σ 3 div u + A 3α ∆ϕ α -L 3α ϕ α + ξ 3 θ + v 3 P = 0, K∆θ = 0, D∆P = 0. ( 54
)
We introduce the second order matrix differential operators with constant coefficients

F (i) (D x ) = ( F (i) gh (D x ) ) 8×8 ,
where

F (1) αβ (D x ) = [µ∆+ρω 2 ]δ αβ +(λ ′ +µ) ∂ 2 ∂x α ∂x β , F (1)
α;β+3 (D x ) = -F (1) α+3;β (D x ) = σ α ∂ ∂x β , F (1) α7 (D x ) = -ζ 1 ∂ ∂x α , F (1) α8 (D x ) = -ζ 2 ∂ ∂x α , F (1) β+3;α+3 (D x ) = A βα ∆ -M βα , F (1) α+3;7 (D x ) = ξ α , F (1) α+3;8 (D x ) = v α , F (1)
7α

(D x ) = τ 1 ζ 1 T 0 ∂ ∂x α , F (1) 7
;α+3 (D x ) = τ 1 ξ α T 0 , F (1)
77

(D x ) = K∆ + τ 1 ηT 0 , F (1) 78 (D x ) = τ 1 ςT 0 , F (1) 8α (D x ) = τ 1 ζ 2 ∂ ∂x α , F (1) 8
;α+3 (D x ) = τ 1 v α , F (1)
87

(D x ) = τ 1 ς, F (1)
88 (D x ) = D∆ + τ 1 ϖ α, β = 1, 2, 3. Here i = 1, 2, 3, 4 corresponds to static, pseudo-, quasi-static oscillations and equilibrium theory of thermoelastic diffusion with triple porosity respectively. The matrices F (i) (D x ), i = 2, 3, 4 can be obtained from matrix F (1) , where

Fαβ (D x ) = µ∆δ αβ + (λ ′ + µ) ∂ 2 ∂x α ∂x β , Fβ+3;α+3 (D x ) = A βα ∆, F77 (D x ) = K∆, F88 (D x ) = D∆, Fα;β+3 (D x ) = Fβ+3;α (D x ) = 0 Fkl (D x ) = Flk (D x ) = F78 (D x ) = F87 (D x ) = 0 α, β = 1, 2, 3 l = 1, ..., 6 k = 7, 8
The system of equations ( 51)-( 54) can be represented as

F (i) (D x )U(x) = 0, i = 1, 2, 3, 4
where U = (u, ϕ α , θ, P ) is a eight-component vector function on E 3 . The matrix

F(D x ) is called the principal part of operator F (i) (D x ).
Definition 1: The operator

F (i) (D x ), i = 1, 2, 3, 4 is said to be elliptic if | F(m)| ̸ = 0, where m = (m 1 , m 2 , m 3 ). Since | F(m)| = µ 2 (λ ′ + 2µ)KDϱ|m| 16 , ϱ = |A βα | 3×3 , therefore operator F (i) (D x )
is an elliptic differential operator iff

µ(λ ′ + 2µ)KDϱ ̸ = 0. ( 55
)
Definition 2: The fundamental solutions of the system of equations ( 51)-( 54)

(fundamental matrices of operators F (i) ) are the matrices

G (i) (x) = ( G (i) gh (x) ) 8×8
satisfying conditions

F (i) (D x )G (i) (x) = δ(x) I(x), i = 1, 2, 3, 4 (56)
where δ(x) is the Dirac delta, I = (δ gh ) 8×8 is the unit matrix and x ∈ E 3 .

Construction of G (1) (x) in terms of Elementary Functions

Let us consider the system of non-homogeneous equations

µ∆u + [(λ ′ + µ)∇ div + ρω 2 ]u -σ α ∇ϕ α + τ 1 ζ 1 T 0 ∇θ + τ 1 ζ 2 ∇P = H, σ 1 div u + A 1α ∆ϕ α -M 1α ϕ α + τ 1 T 0 ξ 1 θ + τ 1 v 1 P = X 1 , σ 2 div u + A 2α ∆ϕ α -M 2α ϕ α + τ 1 T 0 ξ 2 θ + τ 1 v 2 P = X 2 , σ 3 div u + A 3α ∆ϕ α -M 3α ϕ α + τ 1 T 0 ξ 3 θ + τ 1 v 3 P = X 3 , -ζ 1 div u + ξ α ϕ α + [K∆ + τ 1 ηT 0 ]θ + τ 1 ςP = Y, -ζ 2 div u + v α ϕ α + τ 1 ςT 0 θ + [D∆ + τ 1 ϖ]P = Z, ( 57
)
where H is a three-component vector function on E 3 ; X α , Y and Z are scalar functions on E 3 .

The system of equations (57) may also be written in the form

F (1) tr (D x )U(x) = Q(x), (58)
where F (1) tr is the transpose of matrix F (1) , Q = (H, X α , Y, Z) and x ∈ E 3 .

Applying operator div to the equation (57) 1 , we obtain

[(λ ′ + 2µ)∆ + ρω 2 ] div u -σ α ∆ϕ α + τ 1 ζ 1 T 0 ∆θ + τ 1 ζ 2 ∆P = div H, (59)
The equations (57) 2 -(57) 4 and (59) may be expressed in the form

N (1) (∆)S = Q, ( 60
)
where S = (div u, ϕ α , θ, P ), Q = (φ 1 , ....., φ 6 ) = (div H, X α , Y, Z) and

N (1) (∆) = ( N (1) gh (∆) ) 6×6
, N

(1)

11 = (λ ′ + 2µ)∆ + ρω 2 , N (1)
1;α+1 = -σ α ∆,

N (1) 15 = τ 1 ζ 1 T 0 ∆, N (1) 16 = τ 1 ζ 2 ∆, N (1)
α+1;1 = σ α , N

(1)

β+1;α+1 = A βα ∆ -M βα , N (1) α+1;5 = τ 1 ξ α T 0 , N (1) α+1;6 = τ 1 v α , N (1) 51 = -ζ 1 , N (1)
5;α+1 = ξ α , N

(1)

55 = K∆ + τ 1 ηT 0 , N (1) 56 = τ 1 ς, N (1) 61 = -ζ 2 , N (1) 6
;α+1 = v α , N (1) 65 = τ 1 ςT 0 , N (1) 66 = D∆ + τ 1 ϖ α, β = 1, 2, 3 (61)
The equations (57) 2 -(57) 4 and (59) may also be written as

Γ (1) (∆)S = Ψ, (62)
where

Ψ = (Ψ 1 , ......, Ψ 6 ), Ψ p = 1 Ã 6 ∑ i=1 Ñ (1) ip φ i , Γ (1) (∆) = 1 Ã |N (1) (∆)|, Ã = (λ ′ + 2µ)KDϱ p = 1, ....., 6 (63)
and Ñ (1) ip is the cofactor of the element N

(1)

ip of the matrix N (1) .

From equations ( 61) and (63), we see that

Γ (1) (∆) = 6 ∏ i=1 (∆ + λ 2 i )
where λ 2 i , i = 1, ...., 6 are the roots of the equation Γ (1) (-m) = 0(with respect to m).

Applying operator Γ (1) (∆) to the equation (57) 1 and using equation ( 63), we get

Γ (1) (∆)(∆ + λ 2 7 )u = Ψ ′ , λ 2 7 = ρω 2 µ ( 64
)
where

Ψ ′ = 1 µ [ Γ (1) (∆)H -∇ [ (λ ′ + µ)Ψ 1 -σ α Ψ α+1 + τ 1 ζ 1 T 0 Ψ 5 + τ 1 ζ 2 Ψ 6 ]] . ( 65
)
From equations ( 62) and (64), we obtain

Θ (1) (∆)U(x) = Ψ(x), ( 66
)
where Ψ = (Ψ ′ , Ψ 2 , ......, Ψ 6 ) and

Θ (1) (∆) = ( Θ (1)
gh (∆)

) 8×8 , Θ (1) pp (∆) = Γ (1) (∆)(∆ + λ 2 7 ) = 7 ∏ i=1 (∆ + λ 2 i ), Θ (1)
ll (∆) = Γ (1) (∆) = 6 ∏ i=1 (∆ + λ 2 i ), Θ (1)
gh (∆) = 0, p = 1, 2, 3 g, h = 1, ....., 8 l = 4, ...., 8 g ̸ = h
The equations ( 63) and ( 65) can be rewritten in the form

Ψ ′ = [ 1 µ Γ (1) (∆)J + w (1) 11 (∆) ∇ div ] H + 6 ∑ i=2 w (1) i1 (∆)∇φ i , Ψ l = w (1) 1l (∆) div H + 6 ∑ i=2 w (1) i2 (∆)φ i , l = 2, ..., 6 (67)
where J = (δ ij ) 3×3 is the unit matrix. In equation (67), the following notations have been used:

w (1) p1 (∆) = - 1 õ [ (λ ′ + µ) Ñ (1) p1 (∆) -σ α Ñ (1) p;α+1 (∆) + τ 1 ζ 1 T 0 Ñ (1) p5 (∆) + τ 1 ζ 2 Ñ (1) p6 (∆) ] , w (1) pl (∆) = Ñ (1) pl (∆) à p = 1, ...., 6 l = 2, ..., 6
From equations (67), we have

Ψ(x) = R (1) tr (D x )Q(x), (68)
where R (1)

(D x ) = ( R (1) gh (D x ) ) 8×8 , R (1) ij (D x ) = 1 µ Γ (1) (∆)δ ij + w (1)
11 (∆)

∂ 2 ∂x i ∂x j , R (1) i;p+2 (D x ) = w (1) 1p (∆) ∂ ∂x i , R (1)
p+2;i (D x ) = w (1) p1 (∆) ∂ ∂x i , R (1) p+2;l+2 (D x ) = w (1) pl (∆) i, j = 1, 2, 3 p, l = 2, ..., 6 (69)
From equations ( 58), ( 66) and (68), we obtain

Θ (1) U = R (1) tr F (1) tr U.
The above relation implies R (1) tr F (1) tr = Θ (1) .

Therefore, we obtain

F (1) (D x )R (1) (D x ) = Θ (1) (∆). (70)
We assume that

λ 2 p ̸ = λ 2 l ̸ = 0 p, l = 1, ....., 7 p ̸ = l, õ ̸ = 0. Let Y (1) (x) = ( Y (1) ij (x) ) 8×8 , Y (1) pp (x) = 7 ∑ g=1 r (1) 1g ς g (x), Y (1)
ll (x) = 6 ∑ g=1 r (1) 2g ς g (x), Y (1) ij (x) = 0 p = 1, 2, 3 l = 4, ..., 8 i, j = 1, ....., 8 i ̸ = j where ς g (x) = - e ιλg|x| 4π|x| , r (1) 1g
= 7 ∏ i=1,i̸ =g (λ 2 i -λ 2 g ) -1 , r (1) 2h
= 6 ∏ i=1,i̸ =h (λ 2 i -λ 2 h ) -1 g = 1, ....., 7 h = 1, ...., 6 (71)
Lemma 1: The matrix Y (1) defined above is the fundamental matrix of operator Θ (1) (∆), i.e.

Θ (1) (∆)Y (1)

(x) = δ(x) I(x) (72)
Proof: To prove the lemma, it is sufficient to prove that

Γ (1) (∆)(∆ + λ 2 7 )Y (1) 11
(x) = δ(x), Γ (1) (∆)Y (1) 44 (x) = δ(x). ( 73
) Consider 7 ∑ i=1 r (1) 1i = ∑ 7 j=1 (-1) j+1 z j z 8 ,
where

z 1 = 7 ∏ i=3 (λ 2 2 -λ 2 i ) 7 ∏ j=4 (λ 2 3 -λ 2 j ) 7 ∏ l=5 (λ 2 4 -λ 2 l ) 7 ∏ p=6 (λ 2 5 -λ 2 p )(λ 2 6 -λ 2 7 )
,

z 2 = 7 ∏ i=3 (λ 2 1 -λ 2 i ) 7 ∏ j=4 (λ 2 3 -λ 2 j ) 7 ∏ l=5 (λ 2 4 -λ 2 l ) 7 ∏ p=6 (λ 2 5 -λ 2 p )(λ 2 6 -λ 2 7 ), z 3 = 7 ∏ i=2,i̸ =3 (λ 2 1 -λ 2 i ) 7 ∏ j=4 (λ 2 2 -λ 2 j ) 7 ∏ l=5 (λ 2 4 -λ 2 l ) 7 ∏ p=6 (λ 2 5 -λ 2 p )(λ 2 6 -λ 2 7 ), z 4 = 7 ∏ i=2,i̸ =4 (λ 2 1 -λ 2 i ) 7 ∏ j=3,j̸ =4 (λ 2 2 -λ 2 j ) 7 ∏ l=5 (λ 2 3 -λ 2 l ) 7 ∏ p=6 (λ 2 5 -λ 2 p )(λ 2 6 -λ 2 7 ), z 5 = 7 ∏ i=2,i̸ =5 (λ 2 1 -λ 2 i ) 7 ∏ j=3,j̸ =5 (λ 2 2 -λ 2 j ) 7 ∏ l=4,l̸ =5 (λ 2 3 -λ 2 l ) 7 ∏ p=6 (λ 2 4 -λ 2 p )(λ 2 6 -λ 2 7
),

z 6 = 7 ∏ i=2,i̸ =6 (λ 2 1 -λ 2 i ) 7 ∏ j=3,j̸ =6 (λ 2 2 -λ 2 j ) 7 ∏ l=4,l̸ =6 (λ 2 3 -λ 2 l ) 7 ∏ p=5,p̸ =6 (λ 2 4 -λ 2 p )(λ 2 5 -λ 2 7
),

z 7 = 6 ∏ i=2 (λ 2 1 -λ 2 i ) 6 ∏ j=3 (λ 2 2 -λ 2 j ) 6 ∏ l=4 (λ 2 3 -λ 2 l ) 6 ∏ p=5 (λ 2 4 -λ 2 p )(λ 2 5 -λ 2 6 ), z 8 = 7 ∏ i=2 (λ 2 1 -λ 2 i ) 7 ∏ j=3 (λ 2 2 -λ 2 j ) 7 ∏ l=4 (λ 2 3 -λ 2 l ) 7 ∏ p=5 (λ 2 4 -λ 2 p ) 7 ∏ k=6 (λ 2 5 -λ 2 k )(λ 2 6 -λ 2 7 ).
On simplifying the right hand side of above relation, we obtain

7 ∑ i=1 r (1) 1i = 0. (74)
Similarly, we find that

7 ∑ i=2 r (1) 1i (λ 2 1 -λ 2 i ) = 0, 7 ∑ i=3 r (1) 1i [ 2 ∏ j=1 (λ 2 j -λ 2 i ) ] = 0, 7 ∑ i=4 r (1) 1i [ 3 ∏ j=1 (λ 2 j -λ 2 i ) ] = 0, 7 ∑ i=5 r (1) 1i [ 4 ∏ j=1 (λ 2 j -λ 2 i ) ] = 0, 7 ∑ i=6 r (1) 1i [ 5 ∏ j=1 (λ 2 j -λ 2 i ) ] = 0, 6 ∏ j=1 r (1) 17 (λ 2 j -λ 2 7 ) = 1. (75)
Also,

(∆ + λ 2 p )ς g (x) = δ(x) + (λ 2 p -λ 2 g )ς g (x) p, g = 1, ..., 7. (76)
Now consider

Γ (1) (∆)(∆ + λ 2 7 )Y (1) 11 (x) = 7 ∏ i=1 (∆ + λ 2 i ) 7 ∑ g=1 r (1) 1g ς g (x) = 7 ∏ i=2 (∆ + λ 2 i ) 7 ∑ g=1 r (1) 1g [ δ(x) + (λ 2 1 -λ 2 g )ς g (x) ] = 7 ∏ i=2 (∆ + λ 2 i ) [ δ(x) 7 ∑ g=1 r (1) 1g + 7 ∑ g=2 r (1) 1g (λ 2 1 -λ 2 g )ς g (x)
]

Using equations ( 74) and (75) in the above relation, we obtain

Γ (1) (∆)(∆ + λ 2 7 )Y (1) 11 (x) = 7 ∏ i=2 (∆ + λ 2 i ) [ 7 ∑ g=2 r (1) 1g (λ 2 1 -λ 2 g )ς g (x) ] = 7 ∏ i=3 (∆ + λ 2 i ) [ 7 ∑ g=2 r (1) 1g (λ 2 1 -λ 2 g ) [ δ(x) + (λ 2 2 -λ 2 g )ς g (x) ]] = 7 ∏ i=3 (∆ + λ 2 i ) [ 7 ∑ g=3 r (1) 1g [ 2 ∏ j=1 (λ 2 j -λ 2 g ) ] ς g (x) ] = 7 ∏ i=4 (∆ + λ 2 i ) [ 7 ∑ g=3 r (1) 1g [ 2 ∏ j=1 (λ 2 j -λ 2 g ) ][ δ(x) + (λ 2 3 -λ 2 g )ς g (x)
]]

= 7 ∏ i=4 (∆ + λ 2 i ) [ 7 ∑ g=4 r (1) 1g [ 3 ∏ j=1 (λ 2 j -λ 2 g ) ] ς g (x) ] = 7 ∏ i=5 (∆ + λ 2 i ) [ 7 ∑ g=4 r (1) 1g [ 3 ∏ j=1 (λ 2 j -λ 2 g ) ][ δ(x) + (λ 2 4 -λ 2 g )ς g (x)
]]

= 7 ∏ i=5 (∆ + λ 2 i ) [ 7 ∑ g=5 r (1) 1g [ 4 ∏ j=1 (λ 2 j -λ 2 g ) ] ς g (x) ] = 7 ∏ i=6 (∆ + λ 2 i ) [ 7 ∑ g=5 r (1) 1g [ 4 ∏ j=1 (λ 2 j -λ 2 g ) ][ δ(x) + (λ 2 5 -λ 2 g )ς g (x)
]]

= 7 ∏ i=6 (∆ + λ 2 i ) [ 7 ∑ g=6 r (1) 1g [ 5 ∏ j=1 (λ 2 j -λ 2 g ) ] ς g (x) ] = (∆ + λ 2 7 ) [ 7 ∑ g=6 r (1) 1g [ 5 ∏ j=1 (λ 2 j -λ 2 g ) ][ δ(x) + (λ 2 6 -λ 2 g )ς g (x)
]]

= (∆ + λ 2 7 )ς 7 (x) = δ(x).
The equation (73) 2 can be proved in the similar way.

We introduce the matrix G (1) (x) = R (1)

(D x )Y (1) (x). ( 77
)
From equations (70), ( 72) and (77), we obtain

F (1) (D x )G (1) (x) = F (1) (D x )R (1) (D x )Y (1) (x) = Θ (1) (∆)Y (1) (x) = δ(x) I(x)
Hence, G (1) (x) is a solution to equation (56) 1 .

Theorem 1: If the condition (55) is satisfied, then the matrix G (1) (x) defined by the equation ( 77) is the fundamental solution of the system of equations ( 51) and the matrix G (1) (x) is represented in the following form:

G (1) (x) = ( G (1) pk (x) ) 8×8 , G (1)
gh (x) = R (1) gh (D x )Y (1) 11 (x), G (1)
gl (x) = R (1) gl (D x )Y (1) 44 (x), Θ (4) pp (∆) = Γ (4) (∆)∆ = ∆ 4 3 ∏ i=1 (∆ + ω 2 i ), Θ (4)
ll (∆) = Γ (4) (∆) = ∆ 3 3 ∏ i=1 (∆ + ω 2 i ), Θ (4)
gh (∆) = 0,

p = 1, 2, 3 g, h = 1, ....., 8 l = 4, ..., 8 g ̸ = h (iv) w (4) p1 (∆) = - 1 õ [ (λ ′ + µ) Ñ (4) p1 (∆) -σ α Ñ (4) p;α+1 (∆) ] , w (4) pl (∆) = Ñ (4) pl (∆) à , w (4) p6 (∆) = Ñ (4) p6 (∆) à , w (4) 55 (∆) = Ñ (4) 55 (∆) à , w (4) 66 (∆) = Ñ (4) 66 (∆) à p = 1, 2, 3, 4 l = 2, 3, 4, 5
where Ñ (4) ij is the cofactor of the element N (4) ij of the matrix N (4) .

(v) R (4)

(D x ) = ( R (4) gh (D x ) ) 8×8 , R (4) ij (D x ) = 1 µ Γ (4) (∆)δ ij + w (4) 11 (∆) ∂ 2 ∂x i ∂x j , R (4)
i;p+2 (D x ) = w (4) 1p (∆) ∂ ∂x i , R (4) l+2;i (D x ) = w (4) l1 (∆) ∂ ∂x i , R (4) l+2;p+2 (D x ) = w (4) lp (∆), R (4) ki (D x ) = R (4) k;i+3 (D x ) = 0, R (4) 77 (D x ) = w (4) 55 (∆), R (4) 78 (D x ) = R (4) 87 (D x ) = 0, R (4) 88 (D x ) = w (4) 66 (∆) i, j = 1, 2, 3 p = 2, ..., 6 k = 7, 8 l = 2, 3, 4 (vi) Y (4) (x) = ( Y (4) ij (x) ) 8×8 , Y (4) pp (x) = r (4) 11 ς * 1 (x)+r (4) 12 ς * 2 (x)+ 3 ∑ g=1 r (4) 1;g+2 ςg (x), Y (4) kk (x) = r (4) 21 ς * 1 (x) + r (4) 22 ς * 2 (x) + 3 ∑ g=1 r (4) 2;g+2 ςg (x), Y (4) ij (x) = 0 p = 1, 2, 3 k = 4, ..., 8 i, j = 1, ....., 8 i ̸ = j G (1) g;l+2 (x) = ∂ ∂x g 6 ∑ j=1 x 1lj ς j (x), G (1) l+2;g (x) = ∂ ∂x g 6 ∑ j=1 x l1j ς j (x), G (1) l+2;k+2 (x) = 6 ∑ j=1 x lkj ς j (x) g, h = 1, 2, 3 l, k = 2, ...., 6
where

x l1j = - r (1) 2j Ãλ 2 j Ñ (1) l1 (-λ 2 j ), x lpj = r (1) 2j Ã Ñ (1) lp (-λ 2 j ), x 117 = 1 ρω 2 = 1 µλ 2 7 j, l = 1, .., 6 p = 2, ...., 6 (83)
Proof: From equation (76),

∆ς j (x) = -λ 2 j ς j (x) j = 1, ...., 7 (84)
Thus, we have

- 1 λ 2 j ( ∂ 2 ∂x g ∂x h -Rgh ) ς j (x) = δ gh ς j (x), x ̸ = 0 (85) Consider G (1) gh (x) = R (1) gh (D x )Y (1) 11 (x) = [ 1 µ Γ (1) (∆)δ gh + w (1) 11 (∆) ∂ 2 ∂x g ∂x h ] 7 ∑ j=1 r (1) 1j ς j (x) = 7 ∑ j=1 r (1) 1j {[ - 1 µλ 2 j Γ (1) (-λ 2 j ) + w (1) 11 (-λ 2 j ) ] ∂ 2 ∂x g ∂x h + 1 µλ 2 j Γ (1) (-λ 2 j ) Rgh } ς j (x) (86) From equation (82), we have w (1) 11 (-λ 2 j ) = - 1 Ãλ 2 j (-λ 2 j + λ 2 7 ) Ñ (1) 11 (-λ 2 j ) + 1 µλ 2 j Γ (1) (-λ 2 j ). ( 87
)
Using equation (87) in equation ( 86), we get

G (1) gh (x) = 7 ∑ j=1 r (1) 1j {[ - 1 Ãλ 2 j (-λ 2 j +λ 2 7 ) Ñ (1) 11 (-λ 2 j ) ] ∂ 2 ∂x g ∂x h + 1 µλ 2 j Γ (1) (-λ 2 j ) Rgh } ς j (x) (88) Now,
Γ (1) (-λ 2 j )r

(1)

1j = 0 j = 1, ..., 6 Γ (1) (-λ 2 j )r (1)
1j = 1 j = 7 and

(-λ 2 j + λ 2 7 )r (1) 1j = r (1) 2j j = 1, ..., 6 (-λ 2 j + λ 2 7 )r (1)
1j = 0 j = 7 (89)
By virtue of equation ( 89), equation (88) becomes

G (1) gh (x) = ∂ 2 ∂x g ∂x h 6 ∑ j=1 [ - 1 Ãλ 2 j r (1) 2j Ñ (1) 11 (-λ 2 j ) ] ς j (x) + Rgh 1 µλ 2 7 ς 7 (x) = ∂ 2 ∂x g ∂x h 6 ∑ j=1 x 11j ς j (x) + Rgh x 117 ς 7 (x)
The remaining formulae of above theorem can be proved in the similar way.

where A p , p = 1, ..., 4 are coefficients, independent of λ j and skipped due to lengthy calculations.

It is easier to prove the relations (90) using equation (71).

From equations (90) and ( 92), we get

Consider

∂ ∂x i ( 1 |x| ) = - x i |x| 3 , ∂ 2 ∂x 2 i ( 1 |x| ) = [ 3x 2 i |x| 5 - 1 |x| 3 ] Hence, ∆ 1 |x| = 3 ∑ i=1 ∂ 2 ∂x 2 i ( 1 |x|
) = 0.
Therefore,

( ∂ 2 ∂x p ∂x l -Rpl ) 1 |x| = δ pl ∆ 1 |x| = 0. (100)

  β are the components of equilibrated stress vectors corresponding to ν α measured per unit area of surface B respectively, ϖ β are the components of outward unit normal vector ϖ to the surface B.

F

  (D x ) by taking ω = -ιτ, ρ = 0 and ω = 0 respectively. and

Lemma 3 :x 6 ∑x-KDϱλ 10 j + A 1 λ 8 j + A 2 λ 6 j + A 3 λ 4 j + A 4 λ 2

 362 If the condition (55) is satisfied, then 11j = -(ρω 2 ) -1 , j=1 11j λ 2 j = -(λ ′ + 2µ) -1 .

1 )

 1 j +A 1 λ 8 j +A 2 λ 6 j +A 3 λ 4 j +A 4 λ 2 j pl (x) -W pl (x) = constant + O(|x|) p, l = 1, ...., 8(93)hold in the neighborhood of the origin.Proof: For p, l = 1, 2, 3, considerG (pl (x) -W pl (x) = ∂ 2 ∂x p ∂x l Ȳ11 (x) + Rpl Ȳ22 (x), ) = O(|x| 2 ), ∂ ∂x k Ȳhh (x) = O(|x|), ∂ 2 ∂x k ∂x i Ȳhh (x) = constant + O(|x|) k, i = 1, 2, 3 h = 3, 4

g = 1, ...., 8 h = 1, 2, 3 l = 4, ..., 8 5. Construction of Matrices G (i) (x) i = 2, 3, 4

Pseudo-Oscillations

We introduce the matrix G (2) (x) = R (2)

where, the matrices R (2) (D x ) and Y (2) (x) can be obtained from matrices R (1)

and Y (1) (x) respectively by taking ω = -ιτ and repeating the above procedure after equation (56).

Theorem 2: If the condition (55) is satisfied, then the matrix G (2) (x) defined by the equation ( 78) is the fundamental solution of the system of equations (52).

Quasi-Static Oscillations

In this case, the matrix N (3) (∆), operator Γ (3) (∆) and matrix operators Θ (3) (∆), R (3) (D x ), Y (3) (x) and G (3) (x) are obtained as

α+1;1 (∆),

where (iii)

where Ñ (3) ij is the cofactor of the element N

(3) ij of the matrix N (3) .

(v) R (3)

On introducing the matrix

we obtain

Hence, G (3) (x) is a fundamental solution to equation (56) 3 .

Theorem 3: If the condition (55) is satisfied, then the matrix G (3) (x) defined by the equation ( 79) is the fundamental solution of the system of equations (53).

Equilibrium Theory

In this case, the matrix N (4) (∆), operator Γ (4) (∆) and matrix operators Θ (4) (∆), R (4) (D x ), Y (4) (x) and G (4) (x) are obtained as

where

If we introduce the matrix

then, we obtain

Hence, G (4) (x) is a solution to equation (56) 4 .

Theorem 4: If the condition (55) is satisfied, then the matrix G (4) (x) defined by the equation ( 80) is the fundamental solution of the system of equations (54).

Basic Properties of G (1) (x)

Theorem 5: Each column of the matrix G (1) (x) is a solution of the system of equations (51) at every point x ∈ E 3 except the origin.

Theorem 6: If the condition (55) is satisfied, then the fundamental solution of the system F(D x )U(x) = 0 is the matrix

where Ãβα is the cofactor of the element A βα of the symmetric matrix A = (A βα ) 3×3 .

Lemma 2: If condition (55) is satisfied, then ∆w

Proof: Consider

Therefore, ∆w

Equation (94) with the aid of equations ( 91),( 96)-(100) becomes

Similarly other formulae of equation ( 93) can be proved.

Therefore, matrix W(x) is the singular part of the fundamental matrix G (1) (x) in the neighborhood of the origin.

Conclusions:

The linear theory of thermoelastic diffusion with triple porosity is derived without utilizing Darcy's law in the current paper. After reducing the governing equations in isotropic medium, the fundamental solution G (1) (x) of system of equations (51) in case of steady oscillations is obtained. Additionally, the fundamental solutions G (i) (x), i = 2, 3, 4 of system of equations ( 52)-(54) in the cases of pseudo-, quasi-static oscillations and equilibrium are also obtained. The fundamental solution G (1) (x) of system of equations (51) makes it possible to investigate three-dimensional boundary value problems of the theory of triple porosity thermoelastic diffusion elastic solids with potential method. Also by this method, it is possible to construct the fundamental solutions of the system of equations in the linear theory of isotropic thermoelastic materials with triple porosity.


Marie-Eve Lefebvre

Critical Autism Studies (CAS) within Disabilities Studies

Critical Autism Studies (CAS) within Disabilities Studies

Marie-Eve Lefebvre, marie-eve.lefebvre.9@umontreal.ca , University of Montreal, Canada Background: Internationally, research in disability sciences on autism revolves mainly around autistic children. When autistic adults are the group of interest, the research methods are chosen often involve those around them, including the family and the intervening staff, but very rarely the autistic people themselves [START_REF] Elias | Autism goes to college: Understanding the needs of a student population on the rise[END_REF].

Several studies in disability sciences observe the lack of representativeness of the heterogeneity of autism [START_REF] Runswick-Cole | Transition From school to adulthood for youth with autism spectrum disorder[END_REF][START_REF] Woods | Redefining Critical Autism Studies: a more inclusive interpretation[END_REF]. Several autistic researchers, since 2010, have carried out critical autism studies (CAS) [START_REF] Woods | Redefining Critical Autism Studies: a more inclusive interpretation[END_REF].

Objectives: As part of this presentation, we wish to elaborate on CAS as a research movement still unknown in research through a literature review.

Methods: This literature review was carried on articles released since 2010. In fact, CAS emerged from the neurodiversity promoting movement in the UK, created in opposition to the biomedical model that attempts to normalize, reduce and eliminate as many as possible behaviors associated with autism [START_REF] Kapp | Deficit, difference, or both? Autism and neurodiversity[END_REF].

For our descriptive analysis, we considered a total of 13 documents internationally.

Results: CAS is distinguished by three characteristics:

1. the exploration of power dynamics, 2. the development of discourses challenging the dominant medical model in autism research, and; 3. the theoretical and methodological creation of emancipatory approaches.

Some authors argue that CAS necessarily calls into question the scientific validity of the diagnosis of autism spectrum disorder (ASD) (APA, 2013) and the need to use labels associated with this diagnosis in order to emphasize the experiences associated with recognizing oneself as an autistic person.

Conclusions: CAS are emancipatory spaces for the voice of people with autism through their experiential narratives that they reappropriate, which can positively, through the co-construction of knowledge which positively influence: the results of research, recommendations in terms of services; educational, social, financial, residential, identity, family and professional needs; and policies and laws governing the services offered to adults with autism. In addition, by sharing their experiences, the discourse of adults with autism can shed light on the different oppressions and power dynamics of which they are victims on a daily basis, making it possible to become aware of them and to propose compensatory mechanisms (e.g. political, social, financial) in society.

Consistent with the extraction of the richness, depth, nuances, and complexity of the experience shared by the autistic participants through ethnographic methodologies (Mason, 2017), CAS are interdisciplinary in order to consider the intersectional nature of the experiences of people with autism, who live with other types of oppression related to gender and sexual orientation, ethnicity, their education, and socioeconomic status. Known through the perspectives of participants, adults with autism challenge stereotypes and stigma associated with being diagnosed with autism [START_REF] Runswick-Cole | Transition From school to adulthood for youth with autism spectrum disorder[END_REF][START_REF] Woods | Autism: a new introduction to psychological theory and current debate[END_REF]

Current autism research

Potential autism knowledge