The 36-year-old inside linebacker, speaking after Thursday's practice, would not say specifically why he was gone, only that it was ''just personal stuff.''
Abraham said he had faced tougher challenges, mentioning his mother's cancer and the death of his grandfather.
''A lot of stuff in my life hurt me more than this did, so this is another bump in the road but the bump is not as big as the hills I was going through,'' he said. ''... Give us a few weeks, a couple of months, the season starts nobody's going to be talking about this no more. That's how I'm going to go into it.''
Abraham's comments were far less contrite than the statement attributed to him and released by the team last week, when he apologized to his family, friends, the Cardinals and his fans ''for letting them down.''
He said in that statement that he understands the seriousness of his situation and was taking steps to deal with his ''personal business.''
Talking to reporters on Thursday, he indicated that his arrest in late June on a drunken driving charge in suburban Atlanta was not the reason, or at least not the only reason, he was away.
''Most people think it was because of the DUI situation, which I think that's going to be handled,'' he said. ''It really wasn't as big as people think but you know how the media gets ahold of stuff. I'm not trying to knock y'all but they made this thing like it was so much different than it really was.
''I'm just going to let that handle itself because I'm still going through that, but it was more just personal stuff, just handling personal stuff, just getting back so I have a clear mind when the season starts.''
The five-time Pro Bowl player is the NFL's active career sacks leader with 133 1/2 and ranks ninth on the league's all-time list. He had 11 1/2 sacks last year, his first season with the Cardinals. He needs 8 1/2 sacks to move past Michael Strahan to No. 5 on the career list.
Coach Bruce Arians said that, as long as he's healthy, the 14-year NFL veteran will start for the Cardinals and be an every-down player.
''There's no doubt about that,'' the coach said. ''He's never been out of those plans.''
Abraham said he was prepared to compete for the job but appreciates it already being given to him.
''I put the numbers up and they know when I get in football season I'm a football giant,'' he said.
Tyrann Mathieu, who missed an entire season of football because of marijuana-related problems before he was drafted by Arizona last year, said he thinks the Cardinals' reception of Abraham will be the same as it was for him.
''I think guys are going to rally around him, support him and try to encourage him,'' Mathieu said. ''Any other way to handle it won't benefit him and it definitely won't benefit us.''
The Cardinals certainly need a strong pass rusher off the edge.
''We know he had to go through what he did to get back to here,'' Arians said. ''We're all behind him. Right now it's just physical, getting in shape and ready to go.''
Asked if this experience had changed him, Abraham said ''as a person I'm still the same guy. It's just something that I had to handle for myself.''
Mathieu said he ''absolutely'' knows what Abraham is going to be dealing with.
''I think he's had enough time alone to himself to look in the mirror and right his wrongs,'' Mathieu said, ''because he's a great teammate. You don't worry about that, but hopefully when he's not around us he'll be able to handle himself in the right way.''
Police said Abraham was at the wheel of a car stopped in a traffic lane in Brookhaven, Georgia, on June 29. Police said Abraham remained asleep and didn't respond until officers beat on the car door for several minutes. Police said he refused to take a Breathalyzer test.
It was his second DUI arrest. The first was in 2003 in New York state.
''You know it's a situation that I've got to deal with,'' Abraham said. ''It don't hurt me as much because I know that I wasn't really wrong in the situation and how I handled it. So I'm just going to let it handle itself.''
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