4.1d) creates a torque T given by
(4.6)
The angle of twist θ is the angle of rotation in the circumferential plane of one face relative to the other and the displacement δ of one face relative to the other is given by
(4.7)
where, θ is the angle in radians. The maximum shear stress at the outer surface of the cylinder τ is given by
where, J is called the polar moment of inertia. For a cylinder of radius r, it is given by
The shear strain is given by
(4.10)
where, θ is the angle of twist (in radians).
Flexural Deformation (Bending)
Because of their high hardness, most ceramics are difficult and expensive to machine into the shape required for tensile testing. Ceramic specimens are also more difficult to grip stably at their ends when compared to metals and polymers. The ease of testing in a flexural (bending) mode, coupled with the simple geometry and low cost of the specimens makes this test very popular for ceramics. There are two ways of performing flexural testing, referred to as three‐point bending and four‐point bending, but four‐point bending is often preferred and commonly used in testing ceramics when design data are required.
In four‐point bending, the specimen, commonly in the shape of a beam, is supported on two rollers, a distance l apart (called the outer span), the load (force) F is applied by means of two upper rollers that are symmetrically arranged, and the deflection at the center of the beam is measured (Figure 4.1e). In a common testing configuration, the distance d between the upper rollers (called the inner span) is equal to l/2. As the specimen is bent, the lower surface of the beam is in tension whereas the upper surface is in compression. Halfway across the thickness, called the neutral axis, the length of the beam is unchanged and, thus, the resultant stress is zero. The highest tensile stress σ is at the bottom surface of the beam and is constant over the inner span. It is given by
where, h is the thickness and w the width of the beam. The deflection of the center of the beam δ, relative to the outer support points, is given by
(4.12)
where, E is the Young’s modulus.
In three‐point bending, the load is applied by means of a single roller midway between the two lower rollers. The corresponding relations for the highest tensile stress σ (at the bottom surface below the upper roller) and deflection at the center of the beam are
(4.14)
4.2.2 Elastic Modulus
For small deformations, less than ~0.1% strain, the measured stress of most solids is proportional to the measured strain, in accordance with Hooke’s law. In this region, the stress versus strain follows a straight line, and the slope of this straight line is called the elastic modulus. This type of behavior is sometimes described as perfectly elastic to distinguish it from other types of elastic behavior. If the specimen is unloaded, it returns to its original undeformed state along the same line, in much the same way as a spring. The elastic modulus in different loading modes is given different names to distinguish one from another. In tension or compression for example, it is called the Young’s modulus, defined as
where, σ is the applied tensile or compressive stress and ε is the strain. For a perfectly elastic solid, E is the same in tension and compression. The Young’s modulus is dependent on the interatomic force versus displacement curve (Chapter 2). It is one of the important mechanical properties used in design. Values for some solids used as biomaterials are given in Table 4.1. In shear and torsion, the elastic modulus is called the shear modulus or, less frequently, the rigidity modulus, whereas in bending (flexure), it is called the flexural modulus. The Young’s modulus E and the shear modulus μ are related by the equation
(4.16)
where, ν is the Poisson’s ratio, equal in magnitude to the lateral strain εy divided by the tensile strain εx, given by
(4.17)
Table 4.1 Mechanical properties of selected materials used as biomaterials.
| Material | Density (g/cm3) | Young’s modulus (GPa) | Yield strength (MPa) | Tensile strength (MPa) | Flexural strength (MPa) | Compressive strength (MPa) | Elongation to failure (%) | Fracture toughness (MPa m1/2) | Vickers hardness (GPa) |
|---|---|---|---|---|---|---|---|---|---|
| CPa Ti (F67 Grade 1−4b , c) | 4.5 | 105 | 170−485 | 240−550 | 24−15 | 100−150 | 1.5−2 | ||
| Ti6Al4V (F136 Grade 5b) | 4.4 | 115−120 | 795 | 860 | 10−15 | 50−100 | 3 | ||
|
|