Smith, who made two centuries in the first test at Edgbaston, was struck by a Jofra Archer bouncer in the first innings at Lords and was replaced by “concussion sub” Marnus Labuschagne for the second innings.
Australia coach Justin Langer said there was “no doubt” not having Smith was going to be a “blow” to his side adding that they were following concussion policy protocols.
It was the first time a player in international cricket was substituted under the new concussion rules.
Initially, Smith passed a concussion test and was allowed to return to the field and complete his innings, but his condition worsened the following day.
Smith scored 142, 144 and 92 in his three innings in the Ashes series and is ranked second in the ICC test batting rankings.
Labuschagne is the most likely option to replace Smith for the Headingley test starting on Thursday, having made 59 as Australia battled on Sunday to a draw which retained their 1-0 lead in the series.
Smith has two weeks to recover and prepare for the fourth test in Manchester, which begins on September 4.
Australia also play a three-day tour match in Derby from August 29-31.
(Production: Oliver Regan, Kurt Michael Hall)