Problem 2. Consider an elastic membrane D fixed at its boundary bnd (D). The potential energy...

 

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Problem 2. Consider an elastic membrane D fixed at its boundary bnd (D). The potential energy of the stretched membrane is defined through the relation dUxdS-dA, where dU is the differential of the potential energy, dS is the surface differential in the stretched state, and dA is the area differential in the resting state. Recall that a quantity A is proportional to another quantity B, denoted as Ax B, if A=kB. Assume that the stretch function is given by z= u(x, y). a) Show that for sufficiently small u, the following relation holds when is small. dUx|vu|dA. Hint: you may wish to use the linear approximation 1+y=1+ + 1/27/ b) The total potential energy of the membrane can be calculated as =ff at Let D be a membrane in the shape of the unit disk stretched according to the function u(x, y)=sin(7(1-x - y)). U= dU. Find the total potential energy of the membrane if dU = - = |Vu|dA for a positive constant k. Problem 2. Consider an elastic membrane D fixed at its boundary bnd (D). The potential energy of the stretched membrane is defined through the relation dUxdS-dA, where dU is the differential of the potential energy, dS is the surface differential in the stretched state, and dA is the area differential in the resting state. Recall that a quantity A is proportional to another quantity B, denoted as Ax B, if A=kB. Assume that the stretch function is given by z= u(x, y). a) Show that for sufficiently small u, the following relation holds when is small. dUx|vu|dA. Hint: you may wish to use the linear approximation 1+y=1+ + 1/27/ b) The total potential energy of the membrane can be calculated as =ff at Let D be a membrane in the shape of the unit disk stretched according to the function u(x, y)=sin(7(1-x - y)). U= dU. Find the total potential energy of the membrane if dU = - = |Vu|dA for a positive constant k.