Surprisingly, composite and/or laminated materials used in manufactured goods predate the founding of ancient Rome. For example, 3,500 years ago, the Egyptians improved the durability of mud bricks by adding chopped straw to a matrix of soft clay prior to molding and drying. Similarly, plywood (adhered laminated veneers of cross-grained wood) has had a millennial history. However, it was not until the 1930s, with the development and use of synthetic phenolic adhesives in place of animal and plant glues, that laminated wood materials emerged as load-bearing, waterproof structures. The British de Havilland Mosquito combat airplane of World War II featured such bonded plywood wings, among other components. At the same time, the US Navy navigated the Pacific Ocean in patrol torpedo boats made of plywood. Today, composites such as carbon-fiber golf club shafts and filament-wound rocket motor cases capture the imagination of the public, while the everyday pneumatic tire, composed of cord and rubber, is perceived as a commodity.
The tire is in fact a highly engineered composite structure. Reinforcing cords of fabric and steel are its principal load-carrying components; skim stock surrounding the cords acts as a non-structural binder whose principal purposes are to contain the pressurized air and transmit applied loads to the bonded cords via interply and intraply shearing stresses. Tread rubber is non-structural, but provides wear resistance and needed traction.
Fundamentally, a composite material consists of two or more phases with unrelated physical properties that, when combined, produce a substance with the desired characteristics. Like all composite products, a tire is designed to deliver more than the sum of its constituent parts.
Specifically, tires contain critical subassemblies, such as belts, beads and sidewalls, that serve predetermined structural purposes. Of particular interest to engineers are the tire component stiffness characteristics (elastic constants known as extensional and shear moduli) and ultimate strength properties. For example, the belt plies and bead filler (apex) of PCR tires are designed primarily for stiffness, while strength is the dominant property required for the beads and carcass plies.
To calculate its elastic properties, a tire’s internal components can be viewed and modeled at three distinct levels: as carbon black and/or silica particles dispersed in a rubber matrix; as one ply of continuous parallel cords embedded in rubber; and as laminated plies of cord and rubber stacked in various orientations throughout the structure. Particle reinforced rubber and unidirectional plies of cord bonded to rubber are analyzed most simply for stiffnesses using a ‘rule of mixtures’ approach – a formula based on the moduli of the two constituents contributing to load sharing based on their weighted-average volume fractions. While compounders specify weight fractions of chemical additives in formulating rubber recipes, volume fractions are required for these calculations.
At the next level (stacked plies of cord and rubber) one can derive algebraic expressions for the requisite elastic constants needed to predict selected tire performance parameters. That is, the laminated tire structure is analyzed as a pantographing network of inextensible cords embedded in an incompressible rubber matrix using netting analysis. This approach is aided by the large modulus mismatch between cord and rubber compared with rigid composite materials.
While netting analysis is best-suited for flexible fabric structures such as parachutes and sails, it works reasonably well for compliant composites subjected to simple loading conditions. For example, calculations and measurements show that the two-ply belt of a radial tire is stretched circumferentially due to inflation pressure, causing the belt to narrow and thicken while producing extremely large inter-laminar shearing stresses at the belt edges.
Similar baseline analyses can be used advantageously: to predict ply steer forces during the design stage; to design mold contours based on a tire’s natural or equilibrium profile; to estimate cord tensions due to inflation and centrifugal forces; and to calculate out-of-plane flexural rigidities of belt packages with and without cap-plies, among other tire properties of interest.
While seemingly old-school, I’ve often found that it’s useful to have such ‘back-of-the-envelope’ tire results at hand when analyzing the voluminous digital output of more complex computer simulations.