SQL server Like pattern match with CONCAT() always fails
You are only matching on Author, not B_Name.  Change code to:

CREATE PROCEDURE [dbo].[Search_By_Author_And_Name] 
(
    @Author_name AS [varchar](50)
)
AS
BEGIN
    Select B.*, R.R_number From Book AS B , Rack As R
    where (B.Author LIKE CONCAT('%', @Author_name, '%') OR
    B.B_Name LIKE CONCAT('%', @Author_name, '%'))
    AND B.B_Id=R.B_Id
END 
GO