MOSCOW — Russia, facing the toughest round of Western sanctions imposed since the Ukraine crisis erupted, has adopted a nonchalant public stance, with President Vladimir V. Putin emphasizing the importance of self-reliance and a new poll released Tuesday indicating a “What, me worry?” attitude among the bulk of the population.

But beneath that calm facade, there is growing alarm in Russia that the festering turmoil in Ukraine and the new round of far more punitive sanctions — announced Tuesday by both European nations and the United States — will have an impact on Russia’s relations with the West for years to come and damage the economy to the extent that ordinary Russians feel it.

Until now, Mr. Putin’s tactics seemed to be working. Russia was feeding the separatist insurgency in Ukraine without leaving distinct fingerprints — able to press Kiev to come to terms while avoiding a rupture with Europe that would alienate Russia’s business elite. But that strategy is beginning to crumble, battered under successive shock waves generated by the crisis.

More frequent and prominent critics are saying that Mr. Putin and the hard-line leaders in the Kremlin overreached by suggesting that Russia, far more dependent than the old Soviet Union on international trade and financial markets, could thrive without the West.