Extract Sub-Directory Path from Partially Known Directory

Let's start with your filename:

$ f=base/app/main/sub/first/tib1.ear

To extract the base name:

$ echo "${f##*/}"
tib1.ear

To extract the desired part of the directory name:

$ g=${f%/*}; echo "${g#base/app/}"
main/sub/first

${g#base/app/} and ${f##*/} are examples of prefix removal. ${f%/*} is an example of suffix removal.

Documentation

From man bash:

   ${parameter#word}
   ${parameter##word}
          Remove  matching prefix pattern.  The word is expanded to produce a pattern just as in pathname expansion.  If
          the pattern matches the beginning of the value of parameter, then the result of the expansion is the  expanded
          value  of  parameter  with the shortest matching pattern (the ``#'' case) or the longest matching pattern (the
          ``##'' case) deleted.  If parameter is @ or *, the pattern removal operation is  applied  to  each  positional
          parameter  in  turn,  and  the expansion is the resultant list.  If parameter is an array variable subscripted
          with @ or *, the pattern removal operation is applied to each member of the array in turn, and  the  expansion
          is the resultant list.

   ${parameter%word}
   ${parameter%%word}
          Remove  matching suffix pattern.  The word is expanded to produce a pattern just as in pathname expansion.  If
          the pattern matches a trailing portion of the expanded value of parameter, then the result of the expansion is
          the  expanded  value  of parameter with the shortest matching pattern (the ``%'' case) or the longest matching
          pattern (the ``%%'' case) deleted.  If parameter is @ or *, the pattern removal operation is applied  to  each
          positional parameter in turn, and the expansion is the resultant list.  If parameter is an array variable sub‐
          scripted with @ or *, the pattern removal operation is applied to each member of the array in  turn,  and  the
          expansion is the resultant list.

Alternatives

You may also want to consider the utilities basename and dirname:

$ basename "$f"
tib1.ear
$ dirname "$f"
base/app/main/sub/first

creating the test files

mkdir -p base/app/main/sub/{first,second}
touch base/app/main/sub/first/tib1.{ear,xml}
touch base/app/main/sub/second/tib2.{ear,xml}

finding the ear files with bash

shopt -s globstar nullglob
ear_files=( base/**/*.ear )
printf "%s\n" "${ear_files[@]}"

base/app/main/sub/first/tib1.ear
base/app/main/sub/second/tib2.ear

Iterate over the array and use John1024's answer to extract the necessary info from each path.

for f in "${ear_files[@]}"; do ...; done