Steel, aluminum, titanium and carbon fiber all attempt to achieve the above criteria, but differ from each other in strength, stiffness, weight, fatigue resistance, corrosion, etc. For example using aluminum or titanium in the same tube dimensions as a traditional steel frame would reduce weight but would produce excessive flexibility. So non-ferrous metal frames typically have larger tube diameters than steel ones to gain rigidity.
Metal frames usually do not fail due to a single catastrophic load but because of small, repeated stresses (called "fatigue"). Steel and titanium have defined minimum fatigue limits - if the stresses are smaller than these limits, these smaller forces generally don't shorten the fatigue life of the frame. Aluminum has no such specific endurance limit, so each stress cycle, however small, takes the material that much closer to fatigue failure. This sounds worse than it is, however - designers realize this limitation and attempt to "over build" their frames for a lifetime of use.
Titanium's high strength, light weight, resilience, and resistance to corrosion make it a well-suited frame material. However, being a metal, many of the same mechanical properties that limit steel and aluminum also limit titanium: metals are equally strong and stiff in all directions (a property called "isotropy"). Once the cross section geometry of a metal pipe is determined to meet strength or stiffness requirements in one plane, an engineer lacks the freedom to meet varying demands for strength or stiffness in any other plane. In metal tubes, by setting diameter and wall thicknesses to meet bending standards, this automatically determines torsional and lateral bending stiffness.
Metal frames are just variations on a single theme compared to composites. Composites consist of reinforcing fibers that are embedded in a matrix material. The most common composite is known as "glass", meaning polyester resin (the matrix) reinforced with fiberglass. Advanced composites are composed of engineered fibers such as carbon, polymer, metal, or ceramic. Usually these fibers are impregnated with a thermosetting resin like epoxy. Other matrix materials include thermoplastics, metals and even ceramics. These advanced composites make structures that are as strong and rigid as metal ones of equal size, but weigh much less. Furthermore, until the matrix material is hardened by a chemical reaction or heat, the resin-soaked fibers can be molded or formed into virtually any shape.
Unlike isotropic metals, composites are anisotropic - their strength and stiffness is only realized along the axis of the fibers which can be arranged in any desired pattern. Thus, to absorb the variable stresses of a given bicycle frame, composite frames can use multiple layers with different fiber angles for each. This puts strength only where it is needed while minimizing weight.
Along with traditional round tube and lug frame designs, composite frames can be molded with the use of internal bladders and foam in either one-piece ("monocoque construction") or multi-section frames. Also, they can be formed in a high pressure lamination process combining the frame tubes into one integral piece.
