Angel Asensio Cepn

Macroeconomic trouble and policy challenges in the wake of the financial bust

Contrasting with the 1929 great crisis, authorities intervened forcefully in 2008 to stop the disintegration of the financial system. Governments and central banks then sought to revise the prudential regulation in depth. It would be optimistic, however, to believe that prudential measures, alone, could deliver full economic recovery, for the collapse of the 'state of confidence' has fed depressive forces and policy challenges which could hold for a while, even once the financial sector is made safe. On the one hand, the economic slowdown and the direct and indirect assistance provided by the governments to the private sectors are having a heavy impact on public finances, meanwhile, on the other hand, the massive amounts of money which artificially inflated the prices of housing and financial products could produce inflationary pressures in the post-crisis period, unless a new assets bubble is allowed for.

Authorities could therefore be facing high unemployment in a damaged context of public deficits and inflationary pressures. The chapter aims at discussing these new challenges. The inadequacy of inflation targets and fiscal orthodoxy in a depressed economy is emphasized, and the outlines of a Post Keynesian alternative policy are examined.

Introduction

Contrasting with the 1929 great crisis, authorities intervened forcefully in 2008 to stop the disintegration of the financial system. This was obtained by means of massive liquidity pumping and state guarantees over private assets which eventually reintroduced some confidence within the financial system, and stopped the process of depreciation of assets. A salient feature at the end of 2009 was the rapid recovery in the financial sector as compared with the lasting difficulties in the goods and labor markets. It was as if the pre-crisis financial euphoria had come back, but in a depressed context, at least in the countries that had been involved in the financial turmoil. Such a discrepancy raises the nagging question of whether and how a depressed economic system could feed such high financial returns. In other words: did the authorities intervention fix the bubble and allow assets markets to start inflating again? It is not clear whether the markets practices are going to be severely constrained, but if the change in regulation principles is not constraining enough, old practices will hardly disappear spontaneously, so that a new bubble could hide difficulties one more time, until the next crisis. Of course, this should not be considered a safe solution for the authorities capacity to save the financial system and support the whole economy has been severely weakened, and authorities could hardly provide such a massive support again before having recovered, the knowledge of which would, in case of a new financial collapse, produce much more panics and catastrophic outcomes than in the previous case. Hence, although such a new bubble scenario looks plausible, this chapter concentrates on the case where, possibly after a future replica, safer financial regulation principles are eventually implemented so that the excess liquidity starts fuelling inflationary pressures in real estate, durables and speculative goods, instead of financial products 1 . This scenario involves new challenges in terms of monetary and fiscal policy, for inflationary pressures in this case are concomitant of serious public finance deterioration, so that authorities are tempted to implement restrictive policies in a context of fragile economic recovery.

The chapter first argues that, although depressive forces look to have been successfully stopped, there is little hope that the economies involved in the financial turmoil recover spontaneously the pre-crisis path. Then it is argued that monetary and fiscal policies have inherited difficulties in terms of public debt and excess liquidity that tend to induce restrictive responses. Finally, the inadequacy of orthodox monetary and fiscal policies in a context of depressed economies is emphasized and the outlines of an appropriate policy are examined.

1 Post Keynesians are not acquainted with the idea that inflationary pressures may arise below full employment, but as Victoria Chick pointed out, "there is nothing in [The General Theory] which actually impedes understanding of the conjunction of unemployment and inflation" (Chick 1983, p 280). In fact, it can hardly be argued that financial assets inflation may develop below full employment and goods and services inflation cannot. The point is discussed further below.

I. Sluggish recovery

The cause of the financial crisis triggered by the subprime episode has been located in the failure of prudential regulation [START_REF] Davidson | Is The Current Financial Distress Caused By The Sub Prime Mortgage Crisis A Minsky moment? Or Is It The Result of Attempting To Securitize Illiquid Non Commercial Mortgage Loans?[END_REF][START_REF] Davidson | Interview with Paul Davidson, Regarding the Crisis[END_REF] 2 , but the necessary palliative which permitted getting out of the financial turmoil (though not to get out of lasting severe difficulties) was an unusual and massive intervention of public institutions aimed at protecting the financial system against complete chaos and the whole economy against much deeper depression. In fact, these defensive reactions (including support to aggregate demand) were not strictly aimed at stimulating the economy, but rather at stopping both the financial system disintegration and the deflationary process. That is, the expected result was obviously not to improve the performances of a safe economic system, but to save a dying patient at all costs, until the cause of the disease starts being dealt with. This is an important point for, although the financial markets and institutions looked much better after such an emergency medication, and although depressive forces looked to have been successfully stopped, there is little hope that the economy recovers spontaneously the pre-crisis path.

The post-crisis growth path will be determined by the aggregate demand dynamism, which itself depends on financing conditions. As regards financial institutions, the memory of the collapse will certainly make banks and non-banks more prudent and selective as far as financing activities are concerned. This will be reinforced, more or less, by the new prudential regulations that are progressively implemented, so that one can reasonably expect some part of the riskier projects, which would have been financed in the pre-crisis euphoric climate, to be rejected in the future. Of course, it is not necessarily a bad thing, unless some projects are unduly rejected as a result of a kind of precautionary principle. Now, even if such a perverse effect can be avoided, it remains that financing conditions will hardly be easier than before the financial turmoil, and should actually be less generous. A memory effect is likely to influence portfolio decisions as well, for the post-crisis balance sheets of private and public financial institutions (including treasuries and central banks) should remain stuffed with lots of toxic assets for a while, so that further financial turmoil cannot be discarded. And such a 2 Palley (2009) "traces the roots of the current financial crisis to a faulty U.S. macroeconomic paradigm. One flaw in this paradigm was the neo-liberal growth model adopted after 1980 that relied on debt and asset price inflation to drive demand.[...] Deregulation and financial excess are important parts of the story, but they are not the ultimate cause of the crisis." threat is likely to boost the liquidity preference, people wanting to limit the share of risky assets in their portfolio. As a matter of consequence, while the financial turmoil has ended, the long-term interest rates and the required return on equities could remain relatively high as compared with their pre-crisis level. Moreover, in spite of the stocks of reserves banks have accumulated during the rescue episode, bank-credit rates as well could take advantage of the situation and remain relatively high, for banks will be able to sell more credit without reducing their interest rates substantially as far as bonds and non-bank loan rates remain relatively high.

Those worsened financing conditions in turn should make it more difficult for firms and households to maintain the pre-crisis level of expenditures as regards productive investments, housing and durable credit-financed consumption good. In addition, a degradation of the expected return on productive investments can also be feared. The inducement to invest should therefore be weakened both because of the sluggishness of the households' demand for goods and because of the weakened "entrepreneurial spirit."

Hence, the illness has not been dealt with yet, and the cure requires both new financial regulations and public support to aggregate demand. As for the new financial regulations, the message seems to have been well understood by authorities, which committed to revise profoundly the rules of the game, at least in 2008-2009 until the markets started showing some resistance against constraining prudential regulation. We will see whether markets practices are going to be deeply constrained or not, but in the meantime, as the emergency measures got to save the patient by providing the private sector with the required liquidity and the required guaranty, both the central banks and governments have accumulated or guaranteed huge amounts of doubtful private assets. And this is likely to harm their capacity to support further the economy subsequently.

II. Bogged down monetary and fiscal policies

Dealing with public debts in depressed economies

Public finances use to be severely damaged after a depression, both because of the collapse of fiscal revenues and because of the huge increase in the public expenditures that have been directly and indirectly induced by the support to the financial sector, to private companies and to households (Bank for International Settlements, 2009, pp 91-115). Fiscal authorities accordingly seems to have a limited room for manoeuvre to counter the sluggish conditions of the private sector recovery (see however the discussion below), for markets tend to raise the risk premium on public debt interest rate for those countries which exhibit strongly deteriorated public accounts and repayment difficulties. As an increased risk premium would deteriorate further the situation and could trigger a dangerous vicious circle, governments in such circumstances are very tempted not to increase expenditures or reduce taxes, thereby renouncing to support the private sector.

Of course, post-crisis economic recoveries use to alleviate public accounts spontaneously to some extent, but even in the optimistic case of a rapid return to the pre-crisis growth trajectory combined with a rapid reduction of the budget deficit, the level of the public debt is likely to remain higher than the pre-crisis one, for the past deficits do weight the debt burden as long as surpluses do not compensate for them. In the less favorable (but very plausible) circumstances where the recovery process spreads over a long period of time, owed to the weak aggregate demand dynamism and the weak public support, the post crisis growth regime should make it still more difficult for governments to balance their budget, which in turn allows for a growing public debt (fig. 1). sluggish recovery increasing public (weak public support) debt lasting budget deficits

Figure 1

The current context in addition involves a worsening factor, which results from the different kind of government guarantees and purchases of private assets that have been made necessary in order to save the financial sectors, whatever it has operated directly (nationalization, recapitalization, loans...), or indirectly (state-guaranteed bad banks and other ad hoc specialized institutions). Part of these bad assets have been purchased or guaranteed when the markets lost their bearing in the storm. They are likely to be sealable at a price which will not involve any loss. But there are amounts of either purchased or state guaranteed private debts which will appear to be irrecoverable, or which market price could hardly reach again the level at which they were purchased when some 'financial innovations' unduly inflated their value. Hence, unless a new bubble boosts again these assets value, public finances are likely to be affected by the depreciation of the stock of bad debts they purchased 3 . The problem is all the more detrimental since the corresponding amount of public money was not spent in any welfare-improving public goods or services. It was spent in depreciated assets, part of which will probably not bring back the money they have cost. And as no supplementary taxes or borrowed resources have been collected in order to pay for these assets, they simply contribute to make the public debt heavier 4 . Which finally makes the pernicious process illustrated in figure 1 more difficult to get round.

Bubbles and excess money

Central banks also accepted bad collaterals in exchange of the liquidity which has been pumped into the banking sector. According to the Laeven and Valencia (2008, p 24) database on systemic banking crises, "Fiscal costs, net of recoveries, associated with crisis management can be substantial, averaging about 13.3 percent of GDP on average, and can be as high as 55.1 percent of GDP. Recoveries of fiscal outlays vary widely as well, with the average recovery rate reaching 18.2 percent of gross fiscal costs. While countries that used asset management companies seem to achieve slightly higher recovery rates, the correlation is very small, at about 10 percent. Finally, output losses (measured as deviations from trend GDP) of systemic banking crises can be large, averaging about 20 percent of GDP on average during the first four years of the crisis, and ranging from a low of 0 percent to a high of 98 percent of GDP."

these exotic financial assets so we will make a profit on it, which says that these things have a value, but we don't know what that value is." Horn (2008, p 6) however reported that " [...] it is not guaranteed that the ECB can sell them with profits. The ECB takes this into account to some extent by applying a uniform haircut add-on between 5 and 10 % and other down payments."

Anyway, those possible losses are the tree which hides the wood. Even if the central banks were able to withdraw all the liquidities they pumped into the banking sector during the rescue episode 5 , it remains that banking systems, especially in the U.S., created huge amounts of money to finance private purchases (notably house purchases and real estate investments) in exchange of repay commitments that eventually were not honoured, or only partially.

Hence, important amounts of bad money had been (endogenously) pumped in exchange of bad debts for a long time when the crisis started. The bubble in the housing sector was obviously fuelled by such an excess credit-money 6 . The massive "securitization" subsequently helped to disseminate asset-prices inflation internationally, since the bad quality of the toxic assets was dissimulated, thereby allowing for overvaluation in international markets.

When the bubble burst, and the real value of real and financial assets collapsed, the excess money which still was circulating (since it had not been withdrawn, as part of the related private debts could not be honoured), was first absorbed by a dramatic increase in the demand 5 It might be not that easy in the fragile post-crisis context, because it could be dangerous to put back lots of toxic assets into circulation in order to withdraw liquidity. Horn (2008, pp 8) statement that "The ECB just has to sell the assets she has bought during the crisis" sounds therefore somewhat optimistic too.

6 Excess money does not refer here to an excess of the supply over the demand for money, which would be inconsistent with the Post Keynesian approach to endogenous money, where the money supply sticks to the demand; it refers to the notion of 'bad money' as defined above. , that is, money that has been provided endogenously but that subsequently has turned out to be in excess because of some counterpart failure.

for liquidity and other secure holdings (quasi-money, low-risk liquid assets, gold ...). But, as the new financial regulations were not implemented rapidly and therefore financial product became very attractive again (all the more since authorities had provided state guaranty in order to rescue the financial sector), the asset-prices inflation process started again 7 .

The financial assets overvaluation might continue, or even develop, until another financial collapse stops that unsafe process once again. But let us examine the implications if the financial rules are strongly enough reinforced so that the financial markets become much less attractive. Liquidities in this case would try to make profit from purchases of durable and capital goods, oil, gold... instead of financial assets. It is true that, according to the Post Keynesian literature, inflation cannot arise below full employment. This is obviously true in normal times where credit-money endogenously finances safe productive investments, so that the return on investment which is required to repay the money initially borrowed eventually proves to be sufficient. In this case, the money inflows and outflows ensure that no excess money circulates in the economy. Now, in troubled times where amounts of credit-money which has been previously created cannot be destroyed by a normal outflow, an excess purchasing power unduly remains in the economy, thereby allowing for an increase in the money value of some goods and real assets (unless a shift in the liquidity preference increases the money demand proportionally). Hence, with respect to the current situation, when the recovery process reaches the pre-crisis level, producers will hardly be able to provide within normal delays the additional amount of goods that the excess amount of money could purchase at previous prices, all the more so as the existing capacity is likely to have suffered during the depression (International Monetary Fund, World Economic Outlook 2009, p 31-7 The financial markets recovery of 2009 is documented for example in International Monetary Fund 2010, p 1. 33) 8 . A Demand-pull inflation process therefore may very well develop in a context of unemployment, insofar as the pre-crisis level of unemployment is below full employment 9 ,

The magnitude of such inflationary pressures (or the magnitude of the policy measures that could be adopted in order to fight them) depends on the authorities capacity to withdraw the bad money rapidly enough as to offset the decrease of the liquidity preference 10 . There is technically no major problem to withdraw liquidity from the economy, but as stated above, it

is not clear at all that such a policy could be implemented without damaging the financial system and the fragile economic recovery. It happens finally that central banks should have to navigate in turbulent water, between repressing inflationary pressures and preserving the financial system stability.

Collectivizing private losses: who is paying the bill?

As discussed above, the 2008 massive public interventions involved a transfer to the public sector (central banks and governments) of big amounts of private bad debts part of which will probably not be recovered. Also, public deficits were magnified because of the fall in fiscal revenues and induced expenditures. This raises the question of the distribution of the burden of the socialized losses. The answer actually depends on the way authorities decide to deal with the problems mentioned in the previous section.

8 In spite of a 12% unemployment rate in the UK, Keynes (1937) expressed his concern with possible inflationary pressures owing to the government proposal to finance rearmament partly by means of borrowed money. Today's possible inflationary pressures are related to the fact that the expansion of productive capacity is limited by the necessary delay for capital accumulation of course, not by the allocation of productive resources to the production of arms. 9 See [START_REF] Asensio | Bad money and distributive conflict: is stagflation coming back after the great depression?[END_REF] for a detailed discussion. Note that an excess of money is also a demand-pull inflationary process in the neoclassical view, but it may happen before full employment in the Post Keynesian approach (while it cannot in the neoclassical one). Also, as Post Keynesians use to emphasize, inflation always has a distributive conflict side. In the present case the conflict holds in the incompatible claims for goods that results, at previous prices, from the excess amount of money. As regards excess money, insofar as investments in extended capacities require some delay, authorities may decide to let inflation develop temporarily, until the real value of money has depreciated enough, in relation to the amount of unrecoverable debts. This solution preserves the economic activity and employment to the detriment of creditors, whose assets real value depreciate in proportion to the increase in prices. If on the other hand authorities decide not to allow for such a temporary increase in prices, unemployment is likely to be hit, as well as debtors, for interest rates have to go up in order to repress the inflationary pressures.

As regard public deficits, governments for their part may decide to increase taxes so that fiscal revenues increase enough as to balance the government purchase of unrecoverable private debts and offset the fiscal losses the economic depression has generated. As government purchases of unrecoverable debts do not support economic activity at all, while taxes certainly reduce the private demand for goods and services, this solution eventually puts the burden on redundant people and taxpayers. If on the other hand current taxes do not compensate totally for the unrecoverable debt purchases and fiscal losses (which means an increase in public debt or a decrease in public capital holdings), unemployment rises to a lesser extent, and some inflation develops.

Hence, central banks and governments should be faced with the well known inflation/unemployment dilemma, though with differentiated distributional effects. Notice that the fiscal and monetary responses which might be induced by the process of collectivization of losses per se would hardly support economic activity. It should on the contrary have depressive effects, for even if inflation and budget deficit were preferred, it would only spare restrictive measures such as tax and/or interest rates increases. The collectivization of losses therefore induces either depressive or stagflationary forces.

III. Comparing fiscal orthodoxy and interventionism à la Keynes in the post-crisis context

Restrictive fiscal policies: the next big threat

The financial crisis thereby has seriously weakened the central banks and governments capacity to really support a recovery; worse, it started compelling several countries to implement restrictive measures and several others to seriously think about it. The mainstream policy principles indeed recommend a reduction of the public deficit and even a budget surplus aimed at reducing the stock of the public debt, and thereby the sovereign debt risk premium. Such principles could make sense in a safe, full employment economy, as a way to stop the well known vicious circle: budget deficit -increased debt -increased risk premiumbudget deficit... But in depressed economies, it is counterproductive to strive to reduce the budget deficit before the economy has recovered his strengths: as authorities become impatient to remove the budget deficit, the recovery becomes more difficult, fiscal revenues go down, inducing more spending cuts, weaker recovery and again less fiscal revenues... (fig.

2) 11 . Many countries experienced such pernicious dynamics when they were compelled to implement the so called 'stabilization programs'.

sluggish recovery (spending cuts) depressed fiscal revenues

Figure 2

Hence, spending cuts may worry financial markets much more, instead of reassuring them, for markets are aware that increasing idle capacities owed to spending cuts mean lower returns on investments, lower economic growth and possibly higher public deficits. Mainstream recipes could therefore increase the risk premia instead of reducing them. Only when the private aggregate demand engine is firmly launched could prudent cuts have a chance of reducing the public debts. Public debts have been magnified because of the exceptional depression caused by the financial turmoil. They will go down when normal conditions are recovered. If restrictive measures then remained necessary, they would be much weaker, much more bearable and therefore much easier to implement. Politicians must consider that public spending cuts in a period where the social needs for public assistance explode could produce serious troubles and damage further the state of confidence and the whole financial and economic system.

Looking for solutions

11 At a national level, external demand may compensate for the depressed domestic demand if the international context is favorable, but such opportunity is much more unlikely to happen in the current international context where restrictive policies tend to become standard.

In the first times of a recovery, idle capacities use to allow firms to absorb increases in expected aggregate demand until the economy rejoins approximately the pre-crisis level of activity (or maybe an inferior level since some capacities may have been destroyed during the crisis). In this respect, the public support could be decisive, for it has been argued above that the private sector per se is likely to experiment a weakened 'entrepreneurial spirit' in the postcrisis period. What is needed here is a public stimulus capable of sustaining the expected aggregate demand until it stands on its own two feet and the confidence in the production viability and profitability is completely restored. Several factors are likely to make expansionary policies much more effective in a depressed system. First, since the public stimulus helps restoring the expected profitability of the private productive investments, it should make the Keynes multiplier higher. Second, the share of the irrecoverable debts is likely to be finally much lower in a context of economic recovery, which reduces the source of the troubles. Third, strong economic recoveries induce fiscal revenues which may help governments to socialize private losses without having to increase taxes. Hence, insofar as a fast recovery generates important fiscal revenues, public deficits would not necessarily increase and could even decrease if the initial impact of the public support does not overcome the induced fiscal revenues (figure 3) 12 .

rapid recovery stabilizing (public support) public debt balancing public accounts

Once the economy is back on the rails, the challenge for authorities, as suggested above, is to escape the depressive / stagflationary forces the collectivization of losses involves. The need then is to favour the extension of aggregate demand and the related productive capacities in relation with the part of the excess money which has not been withdrawn for whatever reason.

12 Since the purpose is only to boost temporarily the state of affairs private initiatives, the initial deficit is only temporary while the future revenues are likely to persist once the growth trajectory is consolidated.

As the capital accumulation and the raising productive capacity might not be as fast as to cancel inflationary pressures immediately, it might be of interest to consider the possibility for governments to borrow from money holders, and then start progressively financing public investments and fiscal incentives for private investments, spreading expenditures over time so that the productive capacity of capital goods is never overwhelmed 13 .

Central Banks could have a major part in this respect if they got to keep long-term interest rates low enough 14 so that investment financing is made as cheaper as possible (provided the credit-money finances safe/non inflationary investments of course). The danger would rather be that a restricted credit policy put the burden of past mistakes on current safe economic projects 15 . It is possible nevertheless that low interest rates per se do not stimulate private investments very much, as investments also depend on the expected return on productive capital. Therefore, it is important that authorities show a determined commitment to engage in measures aimed at supporting private and public investments, especially in the domains where productive capacities are going to be insufficient, so that the expected demand and related profits increase substantially.

Low long-term interest rates also prevent the dynamics of the debt to be unduly boosted 16 13 This amounts to transfer private purchasing power to the future in the spirit of the 'deferred pay' Keynes (1940) advocated in order to avoid the inflationary pressures owed to the shortage in productive capacities (since capacities were used in the armament industry instead of the production of goods and services).

. It would be unjustifiable to let those markets which have caused great troubles and deteriorated public finances to decide punitive risk premia on sovereign debts, all the more since it would 14 This is not assured according to Keynes's theory, for it requires the central bank policy to preserve the markets confidence (see Asensio 2009a for a detailed discussion). It might also require some capital controls in an open system.

15 Interestingly, when there is an excess money that cannot be withdrawn, it is possible to fight the related inflationary pressures by means of lower interest rates (and public borrowing for public investments), since inflationary pressures can be considered the result of an insufficient productive capacity.

16 On the complementary roles of the central bank and the Treasury in managing the public debt and the long term interest rates see Tily (2006, 2007 pp. 201-206).

suffice that central banks commit themselves to refinance public debts at normal rates for the markets calm down. It is the central banks responsability to prevent a situation where governments have no choice but to adopt depressive measures under the markets pressure in a fragile post-crisis context. Remember that public finances become safer as the recovery goes along, provided restrictive policies do not postpone the recovery (which would make the problem worse in the future). At the end of the day, who would approve that the central bank denied support to the government effort in favour of recovery and fragilized people, whereas the private financial sector was bailed out massively?

Conclusion

The first response authorities brought to the financial crisis got to stop the depressive effects on output and employment, but the public support to aggregate demand did not compensate for the private demand slump, as attested by the severe depression in those countries that have been involved in the financial turmoil. These countries accordingly struggle to reach the precrisis utilization of productive resources, for, in the one hand, the dynamism of aggregate demand and the expected profits have been hit lastingly and, in the other hand, macro policies have been facing serious difficulties that reduced their room for manoeuvre. There is accordingly a temptation to implement restrictive policies in spite of the fragile recovery context. This is confining economies in a sluggish growth regime and could soon generate instability in the international monetary system, for it will be tempting to export difficulties by means of a currency competitive devaluations in the country where the internal demand is apathetic.

What is needed, besides a safer financial regulation, differs in the immediate post-crisis recovery period and in the subsequent growth regime. The post-crisis recovery calls for a public stimulus capable of sustaining the expected aggregate demand until it stands on its own two feet and the confidence in the production profitability is completely restored. In the current context of depressed economies, this would be much more capable to reinforce banks and firms balance sheets, generate fiscal revenues and reduce social expenditures, and, eventually, to restore public finances and strengthen the economic growth. Afterwards, when the capacity utilization rejoins the pre-crisis level, the need will be to favour the extension of aggregate demand and the related productive capacities in relation with that part of the excess money which has not been withdrawn. Combining a low interest rate policy, provided it is possible, with fiscal support would benefit to both the private inducement to invest and the public debt burden, thereby improving the policy effectiveness. It would be much better that countries could co-operate so that interest rates differentials and exchanges rates remain under control. Capital controls however are likely to be a necessary condition for national authorities having, at least temporarily, the capacity to resist to the market forces. The sooner aggregate demand will recover, the less capacities losses will damage the economies.


G Legrain

- N Moës

- E Verron

Stress analysis around crack tips in finite strain problems using the eXtended Finite Element Method

Fracture of rubber-like materials is still an open problem. Indeed, it deals with modeling issues (crack growth law, bulk behaviour) and computational issues (robust crack growth in 2D and 3D, incompressibility). The present study focuses on the application of the eXtended Finite Element Method (X-FEM) to large strain fracture mechanics for plane stress problems. Two important issues are investigated: the choice of the formulation used to solve the problem and the determination of suitable enrichment functions. It is demonstrated that the results obtained with the method are in good agreement with previously published works.

INTRODUCTION

As pointed out in [START_REF] Charrier | Theorical and numerical limitations for the simulation of crack propagation in natural rubber components[END_REF], fracture of rubber-like materials is still an open problem. Indeed, very few tools exist to solve this particular kind of problems. They cannot avoid the remeshing issue coming from crack growth, which for some of them must be handled by the user at each step of the propagation of the crack. These problems contrast with linear fracture mechanics, where simulation tools have improved over the past years. Meshless methods [START_REF] Fleming | Enriched element-free galerkin methods for crack tip fields[END_REF] has been developed to avoid remeshing by enriching the approximation basis with discontinuous modes. Finite element approaches have also been modified within the partition of unity framework [START_REF] Babuska | The partition of unity finite element method[END_REF]. One of these methods is the eXtended Finite Element Method (referred as X-FEM throughout the rest of the paper). This method was first proposed as a response to the remeshing issue in crack propagation in the context of linear fracture mechanics [START_REF] Belytschko | Elastic crack growth in finite elements with minimal remeshing[END_REF][START_REF] Moës | A finite element method for crack growth without remeshing[END_REF]. The X-FEM uses the Partition of Unity in two ways: first to take into account the displacement jump across the crack faces far away from the crack tip, and second to enrich the approximation close to the tip considering the asymptotic fields. This method exhibits advantages which are in common with meshless methods (enrichment of the interpolation basis) and with the finite element method (mesh based approximations). Several fracture mechanics applications have been solved with the X-FEM approach: crack growth with friction [START_REF] Dolbow | An extended finite element method for modeling crack growth with frictional contact[END_REF], arbitrary branched and intersecting cracks [START_REF] Daux | Arbitrary branched and intersecting cracks with the eXtended Finite Element Method[END_REF], three dimensional crack propagation [START_REF] Moës | Non-planar 3D crack growth by the extended finite element and level sets. part I: Mechanical model[END_REF][START_REF] Gravouil | Non-planar 3d crack growth by the extended finite element and level sets. part II: level set update[END_REF] and cohesive crack simulation [START_REF] Wells | A new method for modelling cohesive cracks using finite elements[END_REF][START_REF] Moës | Extended finite element method for cohesive crack growth[END_REF]. The method has also been coupled with the level set approach [START_REF] Sethian | Fast marching methods and level sets methods for propagating interfaces[END_REF][START_REF] Stolarska | Modelling crack growth with level-set and the extended finite element method[END_REF] for greater robustness and versatility. The main purpose of the present paper is to show how to solve nonlinear fracture mechanics problems with the X-FEM, particularly for rubberlike materials. It seems that only one paper has been published on this problem: this paper, presented by Dolbow and Devan [START_REF] Dolbow | Enrichment of enhanced assumed strain approximations for representing strong discontinuities : Addressing volumetric incompressibility and the discontinuous patch test[END_REF] concerns geometrically nonlinear fracture with the X-FEM. More precisely, it investigates the locking issue occurring in plane strain analysis. Here, emphasis is placed on the enrichment around the crack tip in nonlinear fracture mechanics for plane stress case. The robustness and versatility of the X-FEM in nonlinear fracture mechanics is demonstrated. This paper is organized as follows: in the next section, the governing equations of the problem are discussed. Then, technical aspects of the implementation of the eXtended Finite Element Method to nonlinear elasticity are presented, followed by numerical examples of this implementation which show the versatility of the method. Finally, we will conclude and present possible extensions to this work. Consider a thin cracked sheet B defined by its mid-plane surface Ω and its thickness distribution H in the reference undeformed configuration (C 0 ). Under plane mechanical loading, B deforms and occupies the configuration (C). The corresponding mid-plane surface, boundary and thickness distribution are respectively denoted ω, ∂ω and h. Figure 1 presents the notations. The motion between (C 0 ) and (C) can be described by the mapping φ which relates the current position of a particle, x, at time t, to its initial position X :

GOVERNING EQUATIONS 2.1 Deformation of a cracked body in large strain

x = φ(X, t) (1)
In the deformed configuration, the boundary of B can be split into two disjoint sets: ∂ω u on which the displacement field is enforced (Dirichlet boundary condition), and ∂ω T on which the surface traction is enforced (Neumann boundary condition

Div X P = 0 in Ω u(x) = u d on ∂Ω u P • N = T d on ∂Ω T \ ((Γ C1 ) ∪ (Γ C2 )) P • N = 0 on Γ C1 and Γ C2 (2)
In these equations P is the first Piola-Kirchhoff stress tensor, u is the displacement field, u d is the prescribed displacement field, N stands for the unit outward normal vector to the boundary ∂Ω T and T d is the prescribed Piola-Kirchhoff traction vector, i.e. the force measured per unit of reference boundary length. According to these equations, the principle of virtual work in the material description is expressed as :

F(u, δu) = Ω H(X) S : δE dΩ - ∂Γ T T d • δu dΓ = 0 ∀δu (3)
in which S is the second Piola-Kirchhoff stress tensor, E the Green-Lagrange strain tensor and δu represents the virtual displacement field. In the case of rubber-like materials, the above equation is highly nonlinear (due to both large strain and material nonlinearities) and its linearization is an essential prerequisite for the use of a Newton-like algorithm :

DF(u, δu)[u] = Ω H(X) DE[δu] : C : DE[u] dΩ + Ω H(X) S : (Grad X u) T • (Grad X δu) dΩ (4)
where D• is the directional derivative operator and C is the material elasticity tensor.

Here, the prescribed tractions are assumed not to depend on the displacement field. Finally, the constitutive equation of the material is examined. The material is supposed homogeneous, isotropic and incompressible. Moreover the general theory of hyperelasticity is considered. Here the material is assumed to obey the Neo-Hookean constitutive equation. The corresponding strain energy function is :

W = µ 2 (Tr C -3) ( 5
)
where µ is the shear modulus and C stands for the right Cauchy-Green dilatation tensor (C = I + 2E). The second Piola-Kirchhoff stress tensor is given by :

S = 2 ∂W ∂C -pC -1 = µI -pC -1 (6)
In this equation p is the hydrostatic pressure due to the incompressibility of the material, it is determined using the equilibrium equations. Because of the small thickness of the sheet and plane loading, the plane stress assumption is adopted. In this case, the thickness direction is irrelevant and the thickness variation is simply given by :

h 2 = H 2 det( C) ( 7
)
where h is the thickness distribution in the deformed configuration and C is the in-plane dilatation tensor. As a consequence, the hydrostatic pressure can explicitly be evaluated and the in-plane second Piola-Kirchhoff stress tensor is :

S = µ Ī -det( C) -1 C-1 ( 8
)
where Ī is the 2 × 2 identity tensor. Finally the 4 th order elasticity tensor C of Eq. ( 4) reduces to :

C = 2µ det( C) -1 C-1 ⊗ C-1 + I (9)
where

I = ∂ C-1 /∂ C-1 .

Fracture of rubber-like materials

Rivlin and Thomas proposed an extension of the energy release rate initially proposed by Griffith [START_REF] Griffith | The phenomena of rupture and flow in solids[END_REF] to rubber-like materials [START_REF] Rivlin | Rupture of rubber. i. characteristic energy for tearing[END_REF]. In their work, they showed that the fracture of rubbers is controlled by the tearing energy T defined by :

T = - ∂W ∂A l ( 10
)
where W stands for the strain energy, A the area of one surface of the crack, and the subscript l indicates that the differentiation is carried out for constant displacement on the parts of the boundary which are not traction free. Considering that A = c H (where H stands for the thickness of the plate and c the crack length), Eq. ( 10) becomes :

T = - 1 H ∂W ∂c (11)
Numerically, the computation of this tearing energy will be performed using the Rice integral J which is commonly used in linear fracture mechanics to quantify the singularity of the stress field near the crack tip :

J = γ w n 1 -n • σ • ∂u ∂x 1 dγ (12)
in which, γ is the integration contour, n is the unit normal outward vector, and w the strain energy function per unit of deformed volume as shown in Figure 2. For large strain problems, it is possible to express the J-integral in the material description. It can be established by the use of a pull-back operation on variables involved in Eq. [START_REF] Sethian | Fast marching methods and level sets methods for propagating interfaces[END_REF] and described in Fig. 2 :

φ (C ) N 0 (C) n E 1 E 2 Γ d Γ γ γ d e 1 e 2
J = Γ W N 1 -N • P • ∂u ∂X 1 dS (13)
In order to demonstrate this result, consider the material expression of the Eshelby stress tensor for hyperelastic materials [START_REF] Steinmann | Application of material forces to hyperelastostatic fracture mechanics. i. continuum mechanical setting[END_REF] :

M = W I -F T • P (14)
Then, in the case of unloaded crack surfaces and crack tip, the following expression is equivalent to Eq. ( 13) :

J = Γ E 1 • M • NdΓ (15)
with :

M = M -P ( 16
)
where N is the outward normal vector to Γ, and E 1 the tangential vector to the crack. This expression of the J-integral is not well-suited to the finite element context because integration over a contour is difficult in FEA: the interpolation of material variables on the contour leads to numerical errors whereas those material variables are well-known at the quadrature points inside the elements. In fact, expressing J in domain form is better suited to finite element computation. Following the procedure proposed in [START_REF] Ph | Quelques remarques sur la mecanique de la rupture elastique[END_REF][START_REF] Moran | Crack tip and associated domain integrals from momentum and energy balance[END_REF], it is possible to express J in domain form :

J = - S M • Grad X (α) dΩ • E 1 ( 17
)
where S is the area bounded by Γ, and α represents a weight function that is sufficiently smooth on S, vanishes on Γ, and is equal to 1 at the crack-tip. This domain expression of the J-integral is equivalent to the contour integral Eq. ( 15), and may be rewritten as :

J = S ∂u ∂X T • P -W I • Grad X (α) dΩ • E 1 (18)

APPLICATION OF THE X-FEM TO LARGE STRAIN FRACTURE MECHANICS

In this section, we describe how the X-FEM can be adapted to nonlinear fracture mechanics. The X-FEM concept has already been applied in the finite strain context in [START_REF] Dolbow | Enrichment of enhanced assumed strain approximations for representing strong discontinuities : Addressing volumetric incompressibility and the discontinuous patch test[END_REF] where the locking issue in plane strain was discussed. However, the authors did not provide detailed implementation aspects of the X-FEM, focusing on the implementation of the enhanced assumed strain method. The present section focuses on the implementation issue in a more detailed manner. First, generalities on the X-FEM are reviewed.

Then, an appropriate enrichment strategy for nonlinear fracture mechanics is proposed, and finally the choice of the total lagrangian formulation is justified in regard to the numerical procedure.

X-FEM discretization

The formulation used in this study is the total lagrangian formulation i.e. the displacement field is approximated on the initial configuration. With classical finite elements, the approximation of a vector field u(X) on an element Ω e is written as:

u(X)| Ωe = n α=1 u α N α (X) ( 19
)
where n is the number of coefficients describing the approximation over the element, u α is the α st nodal value of this approximation and N α is the classical vectorial shape function associated with dof α. Within the partition of unity, the approximation is enriched as:

u(X)| Ωe = n α=1 N α   u α + ne β=1 a α β φ β (X)   ( 20
)
where n e is the number of enrichment modes, a α β is the additional dof associated to dof α and φ β stands for the β th scalar enrichment function. Two different types of enrichment are considered in fracture mechanics (see [START_REF] Moës | A finite element method for crack growth without remeshing[END_REF]), they are presented in Figure 3: A discontinuous enrichment for nodes whose support is bisected by the crack. In this case, the interpolation of the displacement field is discontinuous across the crack. A Heaviside jump function is considered to model the discontinuity: this function is equal to +1 on one side of the crack and -1 on the other side. The associated dof manages the magnitude of this displacement discontinuity, as seen in Fig. 3(a). The Heaviside function is computed using the level set representation of the crack [START_REF] Stolarska | Modelling crack growth by level sets and the extended finite element method[END_REF][START_REF] Moës | Non-planar 3D crack growth by the extended finite element and level sets. part I: Mechanical model[END_REF].

The second type of enrichment is the near tip enrichment for nodes whose support contains the crack tip. In linear elastic fracture mechanics, the enrichment functions are determined using the asymptotic displacement field near the tip of the crack (see Fig. 3(b)). The problem in nonlinear elasticity is the determination of this asymptotic field. This will be discussed in the next section.

Finally, we note that the modified Gauss quadrature scheme described in [START_REF] Moës | A finite element method for crack growth without remeshing[END_REF] is used to integrate both discontinuous and non-polynomial functions over the elements.

Near tip enrichment

As pointed out in various works [START_REF] Dolbow | Modeling fracture in Mindlin-Reissner plates with the eXtended finite element method[END_REF][START_REF] Goangseup | New crack-tip elements for x-fem and applications to cohesive cracks[END_REF], the choice of the right enrichment function is fundamental in the X-FEM in order to achieve precision even with coarse meshes. Usually, enrichment functions are related to the asymptotic displacement field around the crack tip. The determination of this field is a complex topic since the problem is highly nonlinear. Theoretical results were established in the incompressible plane stress case by Geubelle and Knauss [START_REF] Geubelle | Finite strains at the tip oh a crack in a sheet of hyperelastic material : I. homogeneous case[END_REF][START_REF] Geubelle | Finite strains at the tip oh a crack in a sheet of hyperelastic material : Ii. special bimaterial case[END_REF][START_REF] Geubelle | Finite strains at the tip oh a crack in a sheet of hyperelastic material : Iii. general bimaterial case[END_REF], in the general plane strain case by Knowles and Sternberg [START_REF] Knowles | An asymptotic finite-deformation analysis of the elastostatic field near the tip of a crack[END_REF][START_REF] Knowles | Finite-deformation analysis of the elastostatic field near the tip of a crack : Reconsideration of higher order results[END_REF] and in the incompressible plane strain case by Stephenson [START_REF] Stephenson | The equilibrium field near the tip of a crack for finite plane strain of incompressible elastic matreials[END_REF]. These studies were conducted using the generalized Neo-Hookean strain energy: recall that in the present case, the incompressible plane stress assumption is adopted and the classical Neo-Hookean material is considered. Thus, the corresponding asymptotic field is given by [START_REF] Geubelle | Finite strains at the tip oh a crack in a sheet of hyperelastic material : I. homogeneous case[END_REF]:

v 1 (r, θ) ≈ r cos(θ) v 2 (r, θ) = r 1/2 sin(θ/2) ( 21
)
where the v 1 and v 2 are the displacement vector projected respectively on E 1 and E 2 , and (r, θ) describes the crack front coodinate system (see Fig. 2). Moreover, if second order terms are needed, it can be shown that only one additional term appears along the second axis. However, this term is defined by an implicit equation, thus it is not straightforward to use it numerically as enrichment. Finally the enrichment function that will be used in the following is:

φ = r 1/2 sin(θ/2) ( 22
)
As a comparison, in linear fracture mechanics, the enrichment functions are:

φ = r 1/2 sin (θ/2) , r 1/2 cos (θ/2) , r 1/2 sin (θ/2) sin(θ), r 1/2 cos (θ/2) sin(θ) (23)
Indeed, it should be noted that the term on E 1 of Eq. ( 21) reduces to ′ x ′ when expressed in the crack front coordinate system. Since this function is linear, it is already contained in the polynomial approximation space. If this term was used in the enrichment basis, it would lead to an ill-conditioned system. Consquently, it is not considered in the following and won't be used as an enrichment function. Note also that ( 22) is similar to the discontinuous term of Eq. [START_REF] Geubelle | Finite strains at the tip oh a crack in a sheet of hyperelastic material : I. homogeneous case[END_REF].

Remark on the stress singularity around the crack tip: Using the asymptotic displacement field near the crack tip [START_REF] Goangseup | New crack-tip elements for x-fem and applications to cohesive cracks[END_REF], the asymptotic stress and strain fields are:

σ -→ 1 r as r → 0 (24) P -→ 1 √ r (25) det( C) -→ 1 √ r (26)
As shown in Eqs. ( 24) and ( 25), the order of the stress singularity differs between the reference and the current configurations: it is higher in the current configuration. This stems from the fact that the energy release rate, when evaluated on the actual state, must remain finite although the current thickness tends to zero at crack tip (h = H/ det( C)). Thus, the change of the singularity order of the stress field balances the thickness evolution.

Solution procedure

As presented above, the total lagrangian formulation of governing equations is considered here. Before detailing the solution procedure itself, the choice of this formulation should be justified. Classically, the integration of the weak form is performed on parent elements. The determination of integrals on a given configuration necessitates the use of the mapping which relates the real element to its parent counterpart. For an ordinary finite element, this mapping is polynomial and well-known for every configuration, particularly the reference and the current configurations. In the context of the X-FEM, additional difficulties exist as shown in Figure 4: due to the enrichment functions, the mapping between the deformed element and its parent counterpart is non polynomial. Nevertheless the mapping between the undeformed element and its parent counterpart remains polynomial. Thus the integration of the weak form is easier over the reference configuration, i.e. using a total lagrangian formulation. The integration configuration now being chosen, the use of the interpolated displacement field in Eq. ( 3) leads to the discretized system to be solved; and its use in Eq. ( 4) gives the tangent stiffness matrix.

NUMERICAL EXAMPLES

In this section, some basic examples are first presented to validate the use of the X-FEM for large strain elastic problems. Then nonlinear fracture mechanics examples are considered to exhibit the importance of enrichment. Finally the efficiency of the present method is highlighted.

Preliminary validations

The aim of the following four examples is the validation of the method. The two first examples are devoted to nonlinear structural problems, whereas the last two deal with nonlinear fracture mechanics.

Mesh independence

Consider the uniaxial extension of a single edge notched specimen under tensile loading (SET) as presented in Figure 5(a). Its height and width are both equal to 2mm. The material is assumed to be Neo-Hookean with µ = 0.4225 MPa, which corresponds to an unfilled natural rubber. The bottom edge is fixed vertically and an additional node is blocked horizontally to prevent rigid body motions. Two different meshes are considered: the first one is a structured 32 × 16 triangular elements mesh shown in Figure 5(b), and the second one is a structured 32 × 15 triangular elements mesh (see Figure 5(c)). For the first mesh (32 × 16) the crack is aligned with element edges and for the second mesh (32 × 15) the crack crosses finite elements. Vertical displacement of the nodes situated on the top edge are compared for the two meshes and the corresponding discrepancy versus the initial horizontal position of these top edge nodes is plotted in Figure 6(a). The difference between the two displacements is smaller than 0.18% even for large strain as illustrated by the deformed mesh in Figure 6(b). This example demonstrates one of the greatest advantages of the X-FEM: the part can be discretized without taking into account the crack position during mesh generation.

Comparison with a standard finite element code

In this example, X-FEM and Abaqus [START_REF] Hks | Abaqus/Standard user manual[END_REF] results are compared. A SET specimen (dimensions = 2 mm×6 mm and crack length = 1 mm) is subjected to a prescribed displacement on its top edge equal to 4 mm and its bottom edge is fixed. The material is identical to the one considered in section 4.1.1. The mesh used in Abaqus conforms to the crack sur- face and has the same element density as the X-FEM mesh which is unstructured (the crack runs through elements). Deformed configurations are compared. Qualitatively, they are similar as shown in Figure 7(a) and 7(b). To more precisely compare them, the vertical displacement of the crack surface nodes versus their initial horizontal position are plotted in Figure 7(c). The deformed shape of the crack is very similar for the two simulations, and results obtained with X-FEM are reliable.

Domain independence of the J-integral

The aim of this example is to ensure that the computation of the tearing energy does not depend on the domain chosen to calculate the J-integral. The problem considered here is identical to the previous one. The J-integral is computed for different stretch levels and considering different circular integration domains. More precisely, an element is selected in the integration domain if one of its nodes is contained in the circular region. Figure 8 presents the evolution of the J-integral (normalized with respect to its converged value) as a function of the integration domain radius (normalized with respect to the characteristic length of the element that contains the crack tip) for various stretch levels. In all cases, the J-integral computation converges rapidly (at about 2 layers of elements) and the convergence rate is not significantly influenced by the stretch level.

Robustness of the enrichment process

As a last example, we examine the accuracy of the tearing energy computation with respect to the set of enriched nodes. As mentioned in Section 3.1, the number of enriched nodes in fracture problems is small. A number of studies have demonstrated that an accurate computation of the tearing energy is ensured by a sufficiently refined mesh. Nevertheless, for nonlinear fracture problems, computing time highly increases with the number of dofs because of the iterative solving method, and then the tearing energy computation has to be accurate even with coarse meshes in order to be able to investigate complex parts. Consider a SET specimen (7 mm × 6 mm) discretized by 84 triangular 1 mm × 1 mm elements as shown in Figure 9(a). The crack tip position varies from (-1; 0) to (+1; 0) and the crack emerges from the left edge of the membrane. The set of enriched nodes is represented by gray squares in Figure 9. The bottom edge is fixed, and the top face is subjected to a vertical displacement leading to a stretch of 157%. The radius of the domain used to compute the J-integral is 0.9 mm. The evolution of the tearing energy is plotted with respect to the tip position in Figure 10. It corresponds to the curve denoted "J classical" in Fig. 10. In the neighborhood of X = ±0.5, J varies significantly; it corresponds to the change in the set of enriched nodes set when the crack tip moves from an element to another. In order to smooth the evolution of J with the set of enriched nodes, additionally enriched nodes are incorporated to the set by considering a virtual circle around the crack tip (see Figures 9(b) and (c) where gray square represent enriched nodes). Two different circle sizes are studied, their radii are respectively two and three times the characteristic length of the element containing the crack tip. The corresponding result curves are presented in Figure 10. In both cases, the curves are sufficiently smooth to ensure the accuracy of the J-integral computation. Moreover, the set of 2 layers is enough for the present problem, as the corresponding results agree well with the fourth curve in Fig. 10 where a twice refined mesh was considered. Consequently, for a smooth variation of the evaluated J-integral as the crack grows, several layers of enrichment should be considered for coarse meshes. We remark that in the following Sections, only one layer of elements will be enriched, because meshes will be considered to be sufficiently refined.

Simple loading problems

In this section, classical problems are considered to study the influence of both enrichment and mesh density. For each example in this section, the evolution of the tearing energy with respect to the stretch level, the mesh size and the enrichment type is examined.

Nonlinear fracture mechanics study for rubber-like material really started in the 1950's with the work of Rivlin and Thomas [START_REF] Rivlin | Rupture of rubber. i. characteristic energy for tearing[END_REF], who showed that the fracture of rubber is controlled by one parameter: the tearing energy. This parameter is equivalent to the energy release rate, and thus to the J-Integral proposed later by Rice [START_REF] Rice | A path independent integral and the approximation analysis of strain concentration by notches and cracks[END_REF]. They also introduced a parameter (k) which relates the energy loss due to the crack and the strain

k = - ∆U W Hc 2 (27)
where W stands for the uniform strain energy density in the domain before the introduction of the crack, c is the half length of the crack, H is the thickness of the plate in its reference configuration, and ∆U represents the loss of energy which stems from the crack. Introducing the J-integral, it reduces to :

k = J 2 W c (28)
Finally, Rivlin & Thomas also argued that this parameter should only depend on the stretch level.

Comparison with Greensmith experiments

Greensmith performed experiments on SET samples [START_REF] Greensmith | Rupture of rubber part 10 : The changes in stored energy on making a small cut in a test piece held in simple extension[END_REF] held in simple extension. His results were used by Lindley in addition to finite element simulations to propose an empirical relation between k and the stretch level in the tensile direction λ [START_REF] Lindley | Energy for crack growth in model rubber components[END_REF] :

k = 2.95 -0.008(1 -λ) √ λ (29)
The reference mesh (12958 dofs) presented in Figure 11(a) is used to establish a reference ( solution. Evolution of the J-integral J NL and of its geometrically linear counterpart J LIN (assuming small displacements and strains i.e. replacing P by the Cauchy stress tensor, and performing derivations with respect to the current configuration) are computed with respect to the stretching level. The corresponding results are presented in Figure 12.

The difference between J NL and J LIN curves shows the influence of the "blunting" of the crack tip with large displacements; this leads to a lower value for the nonlinear expression of the tearing energy. However, linear and nonlinear curves match at small strains. We also remark that J NL is close to J Lindley , which shows the reliability of the method. We study also the influence of the enrichment by considering a fine (2664 dofs) and a coarse (912 dofs) mesh presented respectively in Figure 11(b) and(c). The errors on the Jintegrals between the results obtained with those meshes and the reference solution are shown in Figure 13 as a function of the stretch level, with or without near tip enrichment. The two curves ("Fine mesh (1664 dofs) -H(X) & r 1/2 sin(θ/2)" and "Fine mesh (2664 dofs) -H(X)) describe the influence of the enrichment on the fine mesh, and the two other ( "Coarse mesh (912 dofs) H(X) and r 1/2 sin(θ/2)" and "Coarse mesh (912 dofs) H(X)) exhibits its influence on the coarse mesh. As seen, this influence is much more significant when dealing with coarse meshes (decrease of 2.7% on the error for the coarse mesh and 0.6% for the fine one).

1 1,02 1,04 1, 06 1,08 1,1 1,12 1,14 1,16 1,18 Elongation (%

Griffith problem

Consider a Griffith cracked membrane in nonlinear elasticity as shown in Figure 14(a).

The "infinite" membrane has a center crack of length equal to 0.5 mm. It is subjected to three fundamental loadings as depicted in Figure 14: (a) uniaxial extension, (b) equibiaxial extension and (c) pure shear [START_REF] Treloar | Stress-strain data for vulcanised rubber under various types of deformation[END_REF]. Dimensions of the domain are chosen so that the crack can be considered in detail, i.e. h = w = 6 mm for uniaxial and equibiaxial extension and h = 6 mm and w = 60 mm for pure shear. For the two first problems, a 1 MPa tensile loading is prescribed and rigid body motions are blocked. For pure shear, a 6 mm vertical displacement is prescribed on the bottom edge of the domain. Problem symmetries are not taken into account in the model in order to check if the J-integral results are similar for both tips of the crack. Three different meshes are considered: a very fine mesh (19758 dofs) to compute the reference solution (see Figure 15), a fine mesh (7234 dofs),and a coarse mesh (2176 dofs) depicted in Figures 16 and17 respectively. The domain used to compute the tearing energy is a 0.2 mm radius circle in all cases. First, it is to mention that the difference of J-integral values obtained at the two tips is below 0.02% in all cases. The evolution of the tearing energy versus the stretch level is plotted for uniaxial tension in Figure 18. The results are compared with the approximation of k proposed by Lake for central cracks held in simple tension [START_REF] Lake | Application of fracture mechanics to failure in rubber articles, with particulary reference to groove cracking in tyres[END_REF]. The factor is approximated by equating the energy loss due to crack opening to the energy required to close it again :

k = π √ λ (30)
Note that this expression was obtained considering a geometrically nonlinear approach, but a linear constitutive equation. As a consequence, Eq. ( 30) is not accurate for large strains (greater than 100%). Lindley's approximation of J has also been plotted (as in ref. [START_REF] Yeoh | Relation between crack surface displacement and strain energy release rate in thin rubber sheets[END_REF]) even if both curves cannot be compared (the SET boundary conditions are different from those of a symmetric Griffith problem). As in the previous Section, the influence of the crack tip blunting is noticeable and all the curves overlap at for small strains.

The influence of the enrichment is studied by considering the evolution of the J-integral as a function of stretch level for the two meshes and for various enrichment functions. More precisely, in the enrichment functions in Eq. ( 22), the singularity degree of u 2 is changed from 1/2 to 1/4, 1/3, 1 and 2. The discrepancy between the corresponding results and the reference solution is shown in Figure 19. The singularity degree strongly influences the accuracy of the solution: as the singularity degree tends to its theoretical value, i.e. 1/2 the results approach the reference solution. The best enrichment functions are clearly those obtained by considering analytical results. Moreover, the case without enrichment is also considered in Figure 19: the results show that for large strains (due to blunting), it is better to not enrich the crack tip than make a poor choice of enrichment.

In order to examine the validity of our developments under different loading conditions, equibiaxial extension and simple shear loading cases presented above in Fig. 14(b) and (c) are considered. For equibiaxial extension, the coarse mesh (n˚2) is used and a 4372 dofs mesh is considered for the pure shear case. The evolution of k as a function of stretch level is presented in Figure 20(a) and (b) for equibiaxial and pure shear respectively. These results are compared with those obtained by Yeoh [START_REF] Yeoh | Relation between crack surface displacement and strain energy release rate in thin rubber sheets[END_REF], who approximates k using a method based on the crack surface displacement [START_REF] Key | A relation between crack surface displacement and the strain energy release rate[END_REF], assuming that the deformed crack admits an elliptical shape. The numerical results are in good agreement, even if the X-FEM values are slightly different because of the enrichment. Moreover, the calculated small strain limit of k agrees well with theoretical values as shown in Table 1.

Loading conditions Theoretical values X-FEM values Uniaxial extension

k = π k = 3.17 Equibiaxial extension k = π k = 3.16 Pure shear k = 4 3 π ≃ 4.18 k = 4.08
Table 1: Small strain value of k.

Membrane with an inclined center crack

As a last example, we consider an "infinite membrane" with an inclined center crack subjected to uniaxial tension. The membrane, presented in Figure 21, is loaded on its top and bottom faces. The corresponding Piola-Kirchhoff (P ) traction evolves from 0 to 0.7 MPa. A crack of length 2a parametrized by its inclination angle β (β ∈ [0˚; 90˚]) is located at the center of the membrane. The membrane is considered as a square (h = w) whose dimensions are far greater than those of the crack (2a/w < 0.1). A 12734 dofs fine mesh is considered. The evolution of the tearing energy non-dimensionalized with respect to its value for β = 0 is plotted as a function of β for various stretch levels in Figure 22(a). In the present case, this evolution does not depend on the stretch level. Moreover, this result is close to the linear elastic fracture mechanics solution for which the tearing energy is related to β by : where σ is the applied traction on the top edge, E is the Young Modulus, and β is the crack inclination angle. J can be written as :

J = σ 2 π E a cos 4 (β) + sin(β) 2 cos(β) 2 (31)
J(σ, a, β) = J σ (σ) • J C (a, β) (32)
The difference between the large strain solutions of Fig. 22(a), and the linear solution Eq. ( 31) is presented in Figure 22(b). It remains below 6%. According to these results, we investigate the possibility of establishing a nonlinear counterpart to Eq. [START_REF] Greensmith | Rupture of rubber part 10 : The changes in stored energy on making a small cut in a test piece held in simple extension[END_REF]. In regards to Fig. 22(b) the dependence of J NL on β is identical to the dependence of J on β in Eq. [START_REF] Greensmith | Rupture of rubber part 10 : The changes in stored energy on making a small cut in a test piece held in simple extension[END_REF]. Thus only the dependence of J NL on P and a should be studied. The corresponding results are given in Figure 23(a) and (b) for the dependence on P and a respectively. Examining these figures, it is possible to separate the nonlinear tearing energy into the product of a function of the stress state only and a function of the crack geometry only:

J NL (P, a, β) = J N L-P (P ) × J N L-C (a, β)

= b P 2.1 J N L-C (a, β)

where b is a factor that depends on the material behaviour. As a final remark, it should be noted that the previous results may not be valid for other constitutive equations and for plane strain or 3D problems, and that other geometries have not been checked.

CONCLUSION

This work has presented an improvement of the eXtended Finite Element Method for large strain hyperelastic fracture mechanics. The asymptotic displacement field near the tip of a crack in a Neo-Hookean material has been discussed to obtain suitable enrichment functions for plane stress problems. Numerical examples showed the reliability of the approach and the influence of enrichment functions on results obtained with coarse meshes. Validations also demonstrated that interesting results could be easily obtained with this method. The present study is therefore a first step to overcome the lack of simulation tools for rubber fracture mechanics pointed out in [START_REF] Charrier | Theorical and numerical limitations for the simulation of crack propagation in natural rubber components[END_REF]. However, some particular points still have to be investigated. First, the enrichment function considered here is only valid for the Neo-Hookean material constitutive equation and for plane stress problems. Proper enrichment functions should be investigated in order to consider more realistic material models, such as Mooney-Rivlin or Ogden constitutive equations. Another point to investigate is the enforcement of the incompressibility constraint for 3D problems. A first work has already been proposed by Dolbow and Devan [START_REF] Dolbow | Enrichment of enhanced assumed strain approximations for representing strong discontinuities : Addressing volumetric incompressibility and the discontinuous patch test[END_REF] for plane strain problems, adapting the enhanced assumed strain method to the eXtended Finite Element Method .

Figure 1 :

 1 Figure 1: Notations of the model problem.

Figure 2 :

 2 Figure 2: Contour used for the J-integral.

Figure 3 :

 3 Figure 3: (a) Discontinuous approximation function obtained by the product of the regular function (N 1 ) and the enrichment function (φ), (b) Near tip approximation function obtained by the product of the regular function (N 1 ) and the enrichment function (φ).

Figure 4 :

 4 Figure 4: Mappings associated to the eXtended Finite Element Method for an enriched element with large deformations.

Figure 5 :

 5 Figure 5: (a) Single edge notched specimen under uniaxial tensile loading, (b) 32 × 16 mesh, (c) 32 × 15 mesh.

Figure 6 :

 6 Figure 6: (a) Error on the displacement of the top edge nodes between the two meshes results, (b) Deformed shape of the SET.

Figure 7 :

 7 Figure 7: (a) Abaqus deformed configuration; (b) X-FEM deformed configuration; (c) Comparison of the two crack vertical displacements.

Figure 8 :

 8 Figure 8: Evolution of the J-integral with respect to both domain size and stretch level (λ).

Figure 9 :

 9 Figure 9: The mesh used and the near-tip enriched nodes (gray squares).

JFigure 10 :

 10 Figure 10: Evolution of the tearing energy with respect to the crack tip position.

Figure 11 :

 11 Figure 11: (a) Reference mesh (12958 dofs); (b) Fine mesh (2664 dofs) ; (c) Coarse mesh (912 dofs).

Figure 12 :

 12 Figure 12: Evolution of the tearing energy with respect to the stretch level.

Figure 13 :

 13 Figure 13: Influence of the mesh and the enrichment on the tearing energy with respect to the stretch level.

Figure 14 :Figure 15 :

 1415 Figure 14: Nonlinear Griffith problem: (a) uniaxial extension, (b) equibiaxial extension, (c) pure shear.

Figure 16 :Figure 17 :

 1617 Figure 16: (a) The "fine" FE mesh (N˚1) (7234 dofs), (b) Zoom at the crack area.

Figure 18 :

 18 Figure 18: Evolution of J LIN and J NL (uniaxial extension).

Figure 19 :

 19 Figure 19: Influence of the enrichment for the two meshes: error on the value of the energy release rate (%).

Figure 20 :

 20 Figure 20: Evolution of k with respect of the elongation.

Figure 21 :

 21 Figure 21: Plate with a center crack at angle β.

Figure 22 :

 22 Figure 22: (a) Evolution of normalized J, (b) Evolution of the error on (31).

Figure 23 :

 23 Figure 23: (a) log(J N L-P ) vs. log(P), (b) J N L-P vs. crack length.

  ). More precisely, ∂ω T includes the crack faces γ C1 and γ C2 . The corresponding parts of the boundaries in the reference configuration are denoted ∂Ω u (Dirichlet boundary condition), and ∂Ω T (Neumann boundary condition), and Γ C1 and Γ C2 for both crack faces. Mathematically, these crack boundaries are different because of the bijectivity of the mapping φ, even if they are superimposed in (C 0 ), as shown in Fig.1. Considering that there is no body forces, the strong form of the boundary value problem in the material description is :

ACKNOWLEDGMENTS :

The authors greatfully acknowledge Prof. J. Dolbow for useful discussions.