"Bernard" is first mentioned in Book XI. Ivan, Chapter IV, in an exchange between Dmitri and Alyosha:
“Ethics?” asked Alyosha, wondering.
“Yes; is it a science?”
“Yes, there is such a science ... but ... I confess I can't explain to you what sort of science it is.”
“Rakitin knows. Rakitin knows a lot, damn him! He's not going to be a monk. He means to go to Petersburg. There he'll go in for criticism of an elevating tendency. Who knows, he may be of use and make his own career, too. Ough! they are first-rate, these people, at making a career! Damn ethics, I am done for, Alexey, I am, you man of God! I love you more than any one. It makes my heart yearn to look at you. Who was Karl Bernard?”
“Karl Bernard?” Alyosha was surprised again.
“No, not Karl. Stay, I made a mistake. Claude Bernard. What was he? Chemist or what?”
“He must be a savant,” answered Alyosha; “but I confess I can't tell you much about him, either. I've heard of him as a savant, but what sort I don't know.”
“Well, damn him, then! I don't know either,” swore Mitya. “A scoundrel of some sort, most likely. They are all scoundrels. And Rakitin will make his way. Rakitin will get on anywhere; he is another Bernard. Ugh, these Bernards! They are all over the place.”
"Bernard" is then used throughout the rest of book as a insult by Dmitri. For example:
What's the use of the counsel? I told him all about it. He's a soft, city-bred rogue—a Bernard! But he doesn't believe me—not a bit of it.
Since I've been arrested, he has borrowed money from me! He is a contemptible Bernard and opportunist, and he doesn't believe in God; he took the bishop in!
What's the significance of using "Bernard" as an insult?