Friday, April 09, 2010

Very interesing question I received through ALLEXPERTS.COM

Q: If stress is calculated as load / area. Why does the composite (for example a plate with single ply) has more stress in the direction of fiber? (Area calculated as length * thickness for a plate). What parameter of composite effect the change in stress?


My Answer: Load/Area is the definition of stress on a uniaxial test, or more generally in any situation when the stress can be calculated from the forces without worrying about the boundary conditions. These cases are called "isostatic" in the literature. A typical and practical example are pressure vessels (*1).


The stresses on a plate, supported on the boundary, and subjected to any kind of transverse load such as pressure is "hyperstatic", which is a fancy way of saying that the stresses cannot be calculated without first finding out what the deformations are. Deformations include displacements (i.e., deflections) and strains. Then, the stresses are computed from the strains through the constitutive equations, which is a fancy way of saying "using the material properties."

For sake of argument, let's think of a square plate made of a unidirectional (UD) lamina, simply supported on the boundary, and subjected to uniform pressure. The longitudinal direction is the direction of the fibers. The transverse direction is the weak/soft direction perpendicular to the fibers but still on the surface of the plate. The pressure acts on the 3rd direction, that is, perpendicular to the face of the plate.

Now, a unidirectional lamina, when deformed as a plate, does not deform too much more along the weak/soft direction (transverse to the fibers) than along the strong/stiff direction (fiber direction) because the fiber direction, which is stiff, will take up most of the load and hold the shape. The transverse direction, which has low stiffness, takes less load, but hey, the fiber direction is holding the shape so why stress yourself? It is like two teams competing tugging a rope; having a lazy guy on your team does weakens the team but if the rest of the guys, or gals, in your team are strong, it does not show. Well, the transverse direction is like the lazy guy, it does not take stress because the fibers area taking it.

Once the deformations, thus the strains, are known, you multiply the strain by the stiffness E to find the stress, roughly as stress=E*strain (this is a rough approximation but good enough for the argument at hand). Since the strains in the plate are roughly the same in the longitudinal and transverse directions, but the modulus is much lower in the transverse directions, ergo you have your answer!

(*1) The isostatic concept is used to great advantage in preliminary design, as shown in ISBN 1420079158, http://www.mae.wvu.edu/barbero/icmd/index.html

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