# What is the correct form of the principle of virtual displacements applied to plane finite elastic element If Ve is the volume of element and se is its surface?

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What is the correct form of the principle of virtual displacements applied to plane finite elastic element If Ve is the volume of element and se is its surface?

(a) 0=$\int_{V_e}$(σijδεij+ρüiδui)dV-$\int_{V_e}$fiδuidV-∮se$\hat{t_i}$δuids

(b) 0=$\int_{V_e}$(σijδεij+ρu̇iδui)dV-$\int_{V_e}$fiδuidV-∮se $\hat{t_i}$δuids

(c) 0=$\int_{V_e}$(σijδεij+ρüiδui)dV+$\int_{V_e}$fiδuidV-∮se $\hat{t_i}$δuids

(d) 0=$\int_{V_e}$(σijδεij+ρu̇iδui)dV$\int_{V_e}$fiδuidV+∮se $\hat{t_i}$δuids

I had been asked this question by my college professor while I was bunking the class.

The origin of the question is Plane Elasticity in section Plane Elasticity of Finite Element Method

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The correct option is (a) 0=$\int_{V_e}$(σijδεij+ρüiδui)dV-$\int_{V_e}$fiδuidV-∮se$\hat{t_i}$δuids

The best explanation: The vector form of the principle of virtual displacements applied to plane finite elastic element with volume, Ve and surface, se  is 0=$\int_{V_e}$(σijδεij+ρüiδui)dV-$\int_{V_e}$fiδuidV-∮se$\hat{t_i}$ δuids, where “δ”  denotes the variational operator, (σij and εij  are the components of stress and strain tensors, respectively, and fi and ti are the components of the body force and boundary stress vectors, respectively.

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