Читать книгу Materials for Biomedical Engineering - Mohamed N. Rahaman - Страница 119
4.2.2 Elastic Modulus
ОглавлениеFor 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 | ||
| Stainless steel (316L) | 7.8 | 200 | 240−1000 | 550−1240 | 12−55 | 50−150 | 2−3 | ||
| CoCr (F75b; as‐cast/annealed) | 8.3−8.5 | 210 | 448−517 | 655−889 | 10−20 | 50−100 | 3−4 | ||
| CoCr (F799b; hot forged) | 8.3−8.5 | 210 | 896−1200 | 1399−1586 | 5−15 | 50−100 | 3−4 | ||
| Al2O3 | 3.98 | 350−400 | 360−400 | 2000−3000 | <0.1 | 4−5 | 18−20 | ||
| ZrO2‐toughened Al2O3 (ZTA) | 4.40 | 300−350 | 700−1000 | 4000−4500 | <0.1 | 6−10 | 16−20 | ||
| YSZ | 6.05 | 200−220 | 1000−1200 | <0.1 | 6−12 | 12−13 | |||
| Si3N4 | 3.2 | 290−310 | 600−1000 | <0.1 | 6−10 | 14−15 | |||
| Hydroxyapatite | 3.16 | 90−120 | 80−200 | 100−200 | 500−1000 | <0.2 | ~1.0 | 5−6 | |
| β‐TCP | 3.07 | 35−90 | 100−150 | 460−700 | |||||
| UHMWPE | 0.93−0.94 | 0.5−1.3 | 20−30 | 30−40 | 130−400 | 2−3 | 0.06−0.09 | ||
| PMMA | 1.2 | 2.5−3.2 | 60−80 | 4−5 | 1−1.5 | 0.1−0.2 | |||
| Cortical bone (porosity 5−10%) | 1.7−2.0 | 10−20 | 125−150(//) | 135−190 | 130−180 | 1.5−3 | 2−12 | 0.7−0.8(//) | |
| 50(⊥) | 0.5−0.6(⊥) | ||||||||
| Trabecular bone (porosity 50−90%) | 0.1−5 | 1−5 | 10−20 | 2−12 | 0.1−0.8 |
These data correspond to those for the dense (pore‐free) metals and polymers and for dense or nearly dense ceramics (<5% porosity) at room temperature (approximately 25 °C). The mechanical properties of metals are dependent on their microstructure and heat treatment whereas for ceramics, they are dependent on the amount of residual pores and grain size.
YSZ, yttria‐stabilized zirconia (3 mol% Y2O3); ZTA, alumina matrix composite containing 15−20 vol% tetragonal ZrO2 particles; UHMWPE, Ultrahigh molecular weight polyethylene; PMMA, polymethyl methacrylate.
a CP, commercial purity.
b ASTM designation.
c Strength increases but elongation to failure decreases from Grade 1 to Grade 4.
//, parallel to long direction.
⊥, perpendicular to long direction.
The negative sign is included because most materials contract in the direction perpendicular to a tensile stress, that is, the strain in the lateral direction is opposite in sign to the strain in the tensile direction. Consequently, ν is positive for most materials. Values for the Poisson’s ratio for the majority of materials are in the range ~0.25–0.40.
There are other forms of elastic behavior, although the stress versus strain curve is not a straight line or the unloading part of the curve does not follow the same path as the loading part. Polymers show elastic behavior in that they recover completely when the stress is removed but the recovery is slow and the stress–strain curve can be linear or nonlinear. Materials such as synthetic and natural rubbers, and the natural material elastin undergo large deformations and when unloaded, they return to their original shape. In spite of this, their stress–strain curve is nonlinear.
