▁ ▁Since ▁, ▁we ▁have ▁ ▁Therefore , ▁ ▁The ▁engineering ▁stra in ▁is ▁. ▁ ▁The ▁engineering ▁stress ▁is ▁ ▁Plan ar ▁extension ▁ ▁Plan ar ▁extension ▁tests ▁are ▁carried ▁out ▁on ▁thin ▁spec im ens ▁which ▁are ▁const rained ▁from ▁de form ing ▁in ▁one ▁direction . ▁ ▁For ▁plan ar ▁extension ▁in ▁the ▁ ▁directions ▁with ▁the ▁ ▁direction ▁const rained , ▁the ▁principal ▁st ret ches ▁are ▁. ▁ ▁From ▁in com press ibility ▁. ▁ ▁Hence ▁. ▁▁ ▁Therefore , ▁ ▁The ▁left ▁C auch y - Green ▁de formation ▁tensor ▁can ▁then ▁be ▁expressed ▁as ▁ ▁If ▁the ▁directions ▁of ▁the ▁principal ▁st ret ches ▁are ▁orient ed ▁with ▁the ▁coordinate ▁basis ▁vectors , ▁we ▁have ▁ ▁Since ▁, ▁we ▁have ▁ ▁Therefore , ▁ ▁The ▁engineering ▁stra in ▁is ▁. ▁ ▁The ▁engineering ▁stress ▁is ▁ ▁Ye oh ▁model ▁for ▁compress ible ▁rub bers ▁ ▁A ▁version ▁of ▁the ▁Ye oh ▁model ▁that ▁includes ▁ ▁dependence ▁is ▁used ▁for ▁compress ible ▁rub bers . ▁ ▁The ▁stra in ▁energy ▁density ▁function ▁for ▁this ▁model ▁is ▁written ▁as ▁ ▁where ▁, ▁and ▁ ▁are ▁material ▁constants . ▁ ▁The ▁quantity ▁ ▁is ▁interpreted ▁as ▁half ▁the ▁initial ▁she ar ▁mod ulus , ▁while ▁ ▁is ▁interpreted ▁as ▁half ▁the ▁initial ▁bulk ▁mod ulus . ▁ ▁When ▁ ▁the ▁compress ible ▁Ye oh ▁model ▁reduces ▁to ▁the ▁ne o - H oo ke an ▁model ▁for ▁in com press ible ▁materials . ▁ ▁References ▁ ▁See ▁also ▁▁