▁in com press ible ▁materials . ▁ ▁For ▁consist ency ▁with ▁linear ▁el astic ity ▁the ▁Ye oh ▁model ▁has ▁to ▁satisfy ▁the ▁condition ▁ ▁where ▁ ▁is ▁the ▁she ar ▁mod ulus ▁of ▁the ▁material . ▁Now , ▁at ▁, ▁▁ ▁Therefore , ▁the ▁consist ency ▁condition ▁for ▁the ▁Ye oh ▁model ▁is ▁ ▁Str ess - de formation ▁relations ▁ ▁The ▁C auch y ▁stress ▁for ▁the ▁in com press ible ▁Ye oh ▁model ▁is ▁given ▁by ▁ ▁Un ia x ial ▁extension ▁ ▁For ▁un ia x ial ▁extension ▁in ▁the ▁- direction , ▁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 ▁ ▁Equ ib ia x ial ▁extension ▁ ▁For ▁equ ib ia x ial ▁extension ▁in ▁the ▁ ▁and ▁ ▁directions , ▁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