(3) A thin metallic wire deposited onto a flexible substrate such as PDMS, can be treated...

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(3) A thin metallic wire deposited onto a flexible substrate such as PDMS, can be treated as a beam on an elastic foundation. Standard beam theory can be used to address such problems by treating the distributed force q(x) to be a function of the amount of displacement into the substrate; specifically q(x) = Sv(x), where S is the stiffness of the foundation with units of force per area. (a) Use classical Euler-Bernoulli beam theory (i.e. assume axial displacements are neg- ligible) to show that the general solution of the displacement of a beam on an elastic foundation is v(x) = where = exx -x [C sin ax + C2 cos ax] + e [C3 sin ax + C4 cos ax] = S 4EI (b) Suppose there is a concentrated force, F, applied at the origin of the beam such that the beam is deflected into the foundation. Show that for x 0 that the displacment is v(x) = == -x -e [sin ax + cos ax]. 2S Hint: One way to solve this problem is to treat the applied force F as a shear force and note that one-half of F will be borne by the beam on each side of the origin. (3) A thin metallic wire deposited onto a flexible substrate such as PDMS, can be treated as a beam on an elastic foundation. Standard beam theory can be used to address such problems by treating the distributed force q(x) to be a function of the amount of displacement into the substrate; specifically q(x) = Sv(x), where S is the stiffness of the foundation with units of force per area. (a) Use classical Euler-Bernoulli beam theory (i.e. assume axial displacements are neg- ligible) to show that the general solution of the displacement of a beam on an elastic foundation is v(x) = where = exx -x [C sin ax + C2 cos ax] + e [C3 sin ax + C4 cos ax] = S 4EI (b) Suppose there is a concentrated force, F, applied at the origin of the beam such that the beam is deflected into the foundation. Show that for x 0 that the displacment is v(x) = == -x -e [sin ax + cos ax]. 2S Hint: One way to solve this problem is to treat the applied force F as a shear force and note that one-half of F will be borne by the beam on each side of the origin.