Modulus Of Rigidity Of Different Materials
Meri Thilmony
The shear modulus is concerned with the deformation of a solid when it experiences a force parallel to one of its surfaces while its opposite face experiences an opposing force (such as friction). In the case of an object shaped like a rectangular prism, it will deform into a parallelepiped. Anisotropic materials such as wood, paper and also essentially all single crystals exhibit differing material response to stress or strain when tested in different directions. In this case, one may need to use the full tensor-expression of the elastic constants, rather than a single scalar value.
The shear modulus of metals is usually observed to decrease with increasing temperature. At high pressures, the shear modulus also appears to increase with the applied pressure. Correlations between the melting temperature, vacancy formation energy, and the shear modulus have been observed in many metals.[13]
The Nadal-Le Poac (NP) shear modulus model is a modified version of the SCG model. The empirical temperature dependence of the shear modulus in the SCG model is replaced with an equation based on Lindemann melting theory. The NP shear modulus model has the form:
Fine Ceramics (also known as "advanced ceramics") possess high rigidity, which is measured by inspecting the elasticity of a specimen after applying a load. Materials that display less elastic deformation under load possess higher levels of rigidity. The coefficient of extension with respect to a load is called Young's modulus.
Using Young's modulus to measure rigidity, alumina and silicon carbide display nearly double the values of stainless steel. Why is high rigidity advantageous? It allows ceramic components to be manufactured to much higher levels of precision with regard to size and shape. In some cases, large mechanical stresses are generated on a material while it is being ground to final specifications. The less deformation that occurs during this process, the more precisely the parts can be processed.
Rigidity, also known as "stiffness," is generally measured using Young's modulus. It can be defined as the "force necessary to bend a material to a given degree."
As shown in the graph below, Fine Ceramics are highly rigid materials, according to Young's modulus. This makes their machining accuracy high enough to enable them to be used for high-precision parts.
Specific modulus is a materials property consisting of the elastic modulus per mass density of a material. It is also known as the stiffness to weight ratio or specific stiffness. High specific modulus materials find wide application in aerospace applications where minimum structural weight is required. The dimensional analysis yields units of distance squared per time squared. The equation can be written as:
To emphasize the point, consider the issue of choosing a material for building an airplane. Aluminum seems obvious because it is "lighter" than steel, but steel is stronger than aluminum, so one could imagine using thinner steel components to save weight without sacrificing (tensile) strength. The problem with this idea is that there would be a significant sacrifice of stiffness, allowing, e.g., wings to flex unacceptably. Because it is stiffness, not tensile strength, that drives this kind of decision for airplanes, we say that they are stiffness-driven.
The use of specific stiffness in tension applications is straightforward. Both stiffness in tension and total mass for a given length are directly proportional to cross-sectional area. Thus performance of a beam in tension will depend on Young's modulus divided by density.
Specific stiffness can be used in the design of beams subject to bending or Euler buckling, since bending and buckling are stiffness-driven. However, the role that density plays changes depending on the problem's constraints.
By contrast, if a beam's weight is fixed, its cross-sectional dimensions are unconstrained, and increased stiffness is the primary goal, the performance of the beam will depend on Young's modulus divided by either density squared or cubed. This is because a beam's overall stiffness, and thus its resistance to Euler buckling when subjected to an axial load and to deflection when subjected to a bending moment, is directly proportional to both the Young's modulus of the beam's material and the second moment of area (area moment of inertia) of the beam.
However, caution must be exercised in using this metric. Thin-walled beams are ultimately limited by local buckling and lateral-torsional buckling. These buckling modes depend on material properties other than stiffness and density, so the stiffness-over-density-cubed metric is at best a starting point for analysis. For example, most wood species score better than most metals on this metric, but many metals can be formed into useful beams with much thinner walls than could be achieved with wood, given wood's greater vulnerability to local buckling. The performance of thin-walled beams can also be greatly modified by relatively minor variations in geometry such as flanges and stiffeners.[1][2][3]
Note that the ultimate strength of a beam in bending depends on the ultimate strength of its material and its section modulus, not its stiffness and second moment of area. Its deflection, however, and thus its resistance to Euler buckling, will depend on these two latter values.
Aluminium is a material commonly used but carbon fiber provides a new solution for many construction engineers. This paper specifies differences between these materials and describes their strengths and weaknesses.
Carbon fiber is used in industries where high strength and rigidity are required in relation to weight. e.g. in aviation, automated machines, racing cars, professional bicycles, rehabilitation equipment.
Due to its unique design, carbon fiber is also used in the manufacture of luxury goods including watches, wallets etc. This material makes the product unique in the world of luxury and elegance and helps it to be one step ahead of the competition.
Strength and rigidity of a carbon fiber component is created by positioning fabrics in a specific way. This presents opportunities for the manufacturer but also requires great knowledge and expertise.
To explain rigidity in relation to weight, imagine a sheet 5 cm wide, 50 cm long and 2 mm thick. When you suspend a 5kg weight from the end of the sheet, the loading will result in bending and the degree of bending will correspond to rigidity. For different materials, a sheet of the same thickness will show different bending properties. The stiffer the material, the less bending will happen. After the load has been removed, the sheet will resume its original shape.
Material stiffness is measured with Young modulus. However this parameter alone is not sufficient to specify material stiffness, without taking into account the weight of the given element.
E.g. in the case of a bicycle frame (dimensions, geometry, wall thickness) made from two different metals: steel and aluminium, the steel one will exhibit 3 times more rigidity than the aluminium one. However if we also take into account the weight of elements, then the steel frame, although it has 3 times more rigidity than the aluminium one, it will also be 3 times heavier.
These numbers are approximate only as, in practice, a design engineer specifies geometry for the selected material, e.g. in case of an aluminium bicycle frame most often the frame diameter is increased, as well as wall thickness. In the case of a bicycle frame, stiffness and strength is directly linked to geometry and wall thickness (wall thickness increased x 2 times results in stiffness improved about 8 times).
The relation of stiffness to weight (namely a specific modulus) is in practice most effective to determine the rigidity of a material, as for most design engineers, both stiffness and weight are the most crucial parameters.
Weight for weight, carbon fiber offers 2 to 5 times more rigidity (depending on the fiber used) than aluminium and steel. In the case of specific components that will be stressed only along one plane, made from one-direction carbon fiber, its stiffness will be 5-10 times more than steel or aluminium (of the same weight).
The above statement shows many advantages provided by carbon fiber, as well as elements designed and made from carbon fiber. Fabrics of improved and highest modulus relate to special materials (unfortunately very costly) that offer rigidity characteristics x 2 or x 3 times more than standard carbon fiber, and they are used mostly in military applications and the aerospace industry.
To interpret the results shown in the table, imagine that a design engineer is going to construct a strong and lightweight carbon fiber sheet of 1m2 with maximum weight of 10 kg and he considers aluminium, steel and carbon fiber.
Note that, in practice, steel and aluminium have ultimate strength lower than that specified in the table. This is the due to the fact that before total destruction (calculation of ultimate strength was based on this moment) a metal element will undergo permanent deformation (will not restore its original dimensions).
With regard to interpretation of results in the table, it is evident that carbon fiber of highest modulus provides extraordinary rigidity. However, resistance to damage decreases as rigidity increases (higher modulus).
Weight is essential for many products. For example, reducing the weight of the arm / catcher of an automated heavy duty machine that works at a speed of 10 m/sec. will enable its speed to be increased and extend its working life. At an industrial scale, it may result in increased production plant capacity and significant savings.
Another example may be a wheelchair, where reducing its weight makes it easier to lift in and out of a car and also enables better control. It is very evident in the case of Formula 1 racing cars where aluminium replaced with carbon fiber resulted in reduction of weight, which is crucial in this sport.
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