PS43.22 Mechanical Behavior of materials PS4 Due: March, 9, 2004 (Tuesday) before class (10:00am) 1. For the case of an epoxy + 70 v/o (volume percent) S-glass long fiber composite, what is the elastic modulus of the composite both parallel and perpendicular to the axis of the fibers? Assume = 3 GPa and = 85 GPa. epoxyEglassE 2. This problem is to estimate the modulus of a particulate composite using a “cube in a cube” model as shown below. Cube ① and ② have Young’s moduli E1 and E2, respectively. (a) Show that the modulus of the composite shown above is: ()221212211212212)(11+−+−=AAAAAAAAEEEEEEc. Cross section near the center ℓ2ℓ1 ②①ℓ2 ℓ1 ① ② (b) Show that this is close to lower bound : 32312323132121)(AAAAAEEEEEc+−= by plotting both equations on a plot of vs. V (= A ). In plotting, assume that and . cE13231/ AGPaE 2001= GPaE 102= 13. A bilayer is made up of a 0.5 mm thick layer of aluminum on a 0.8mm thick layer of silicon. At 320℃, the bilayer has zero curvature. The temperature is then decreased to ows: , 20℃. Assume that the bilayer can be treated as a beam. The properties of aluminum and silicon are as foll1×Aluminum: E=70GPaν=0.33 α=2360−/℃ Silicon: E=130GPa, ν=0.28 α=3610−× /℃ (a) What is the curvature of the bilayer at 20℃? Sketch the shape of the curvature. (b) What is the maximum stress in the bilayer at 20℃? At what location through the ickness of the bilayer does this stress occur? Is the stress tensile or compressive? stress-free at the deposition temperature. For simplicity, assume the film is isot(a)and stress of the film with respect to the substrate at describe the variation of curvature with temperature fluctuations over this range. ssume the film is isotropic and the mate film. ss to substrate thickness, a1/a2 constant. Does this change your th 4. Consider a thin film of aluminum, 1㎛ in thickness, which is deposited uniformly on a Si substrate, 500㎛ in thickness and 200mm in diameter, at a temperature of 50℃. The properties of the film and the substrate are the same as in Problem3. The film-substrate system is ropic. Find the mismatch strain room temperature (20℃). (b) Find the curvature and the shape of the curvature. (c) If the yield strength of the Al film is 180 MPa in both tension and compression over the temperature range 20-50℃, 5. An aluminum film with a thickness a1 of 1㎛ is deposited on a Si substrate with a thickness a2 of 500㎛ at an elevated temperature. The silicon substrate is a circular disk that is large compared to the thickness. After cooling to room temperature, an X-ray diffraction measurement shows that the d-spacing in the direction perpendicular to the film-substrate interface has decreased by 1%. Aerial properties are the same as in problem 3. (a) Calculate the strain and the stress in th(b) Calculate the curvature of the wafer. (c) It is determined that the curvature is too large for the wafer to be accepted for the intended application. As a first suggestion to decrease the curvature, you recommend that the film thickness be decreased if possible. The wafer manufacturer says that they can decrease the film thickness, but for this application they must keep the ratio of the film thicknerecommendation? 26. Consider the problem of themal stresses generated in a bimaterial strip subjected to temperature variations. Let the strip be made by diffusion bonding between two isotropic materials whose Young’s modulus are E1 and E2, Poisson’s ratio are 1ν and 2ν, and linear thermal expansion coefficient are 1α and 2α. The yield strength of layer 1, which may be assumed to be elastic-perfectly plastic, is σy1. Layer 2 is a polycrystalline ceramic. The interface between the two layer is a perfect mechanical bond. The thickness of layer 1 and 2 are a1 and a2, respectively, and the other dimensions of the two layers are muc(a)nge, h larger than a1 and a2. Calculate the critical temperature cha*T∆, at which layer 1 will begin to (b)t w ch yielding commences. Describe the spread of plastic yield plastically. (Ignore edge effects) Indicate the location a hizone with increasing T∆ . (c) What is the curvature of the strip for a temperature change of *T∆? If the strip were to be subjected to a uniform bending moment M instead of a temperature change, what is the location at whic(d)h yielding commences and how (e)asi and does the plastic zone spread with increasing M? Schematically sketch the variation of temperature versus curvature for both incre ng decreasingT∆ for the following two cases: (i) T∆<*T∆ , and (ii) T∆>*T∆ , where all ∆ quantities are absolute values.