Forces

The International System (SI units), which is based on the metric system, is the nomenclature used by the biomedical engineering profession. The newton (N) is a direct measure of force and is recorded as intrinsic units: kg(m)/sec. As defined by Newton’s second law, force is equivalent to the product of mass and acceleration. Forces, when applied to the spine, not only consist of a magnitude but also have a directional component. The combination of a force with direction is a vector. Vectors can be displayed graphically or by trigonometric relationships. Vectors can be used to analyze biomechanical forces acting simultaneously on a biologic structure or implant material by making a free body diagram that assumes a state of equilibrium, thereby defining the forces inside the structure or implant material as dependent and proportional to those outside the structure ( Fig. 33-1 ).

Free body diagram of a lumbar vertebra with an externally applied load of 100 N at an incident angle of 60 degrees. The vertebra will have to resist the shear component as well (horizontal component—not shown in the figure—of the incident force). The resultant downward force on the vertebral body is sin (60) × 100 N = 86.7 N. The vertebra is in equilibrium with its surroundings and not moving. Therefore, a force of the same magnitude is acting on the caudal end plate.

An important principle for the spine surgeon to understand is the force-deformation relationship ( Fig. 33-2 ). When force and deformation are graphically displayed, the result is a characteristic curve. The force-deformation curve has a straight or elastic region in which materials can deform and recover to their original shape (see Fig. 33-2 , first portion of the curve). As the load increases beyond the elastic region, the deformation increases into the curved or plastic region (see Fig. 33-2 , second portion of the curve); when the specimen is unloaded, it will be permanently deformed. If deformation continues, the specimen will eventually fail (e.g., fracture; see Fig. 33-2 , third portion of the curve).

Force versus deformation curve.

The force has a straight or elastic region in which materials can deform and recover to their original shape. As load increases beyond the elastic region, the deformation increases into the plastic region; when the specimen is unloaded, it will be permanently deformed. If the deformation is limited, the specimen eventually will fail (i.e., fracture).

Assessment of the integrity of the spine is complex. The vertebral body ossifies from three primary centers, one for the centrum, which will form the major portion of the body, and the other two for neural arches. The cartilaginous growth plate is predominantly responsible for longitudinal vertebral growth. The structure of the vertebral body, therefore, provides the requirement for optimal load transfer by maximal strength with minimal weight. Bone mineral density (BMD), bone quality, microarchitecture, and material properties are the important factors that contribute to bone strength. In addition, force and displacement have been demonstrated in animal spine models. It has been demonstrated in a biomechanical cadaver study that after dorsal laminectomy and partial discectomy, the neutral zone and range of motion were not different from those in the native spine specimen. However, after pedicle screw-rod fixation, the neutral zone and range of motion of the instrumented specimen decreased significantly compared with the native specimen and the specimen after dorsal laminectomy.

Atomic Bonds, Structures, and Property Relationships

All materials are composed of molecules that interact via intermolecular forces. These bonds determine the properties of the material as a whole. If materials were composed of only one type of molecule and these molecules were perfectly consistent in their orientation, then chemistry alone would be sufficient for deriving all of the elements’ properties. However, materials typically are composed of numerous molecules of considerable diversity. Nevertheless, despite the variety of molecules in metals, certain observations can be made from their chemical composition.

Metals are created through the interaction of crystals. These crystals are formed when the electrons that surround the atoms in clouds are given up and conducted as electricity. Metal structures are polycrystalline (i.e., they are formed by a multitude of crystals). Atoms within a crystal can form one of several relationships, which define the crystal structure. They include body-centered cubic, face-centered cubic, and hexagonal close-packed arrangements ( Figs. 33-3 to 33-5 ).

The unit cell is the smallest group of atoms showing a characteristic structure. The body-centered cube is the most ductile.

The face-centered cube is moderately ductile.

The hexagonal close-packed cube is the least ductile.

In addition to variations in the unit cell of the crystal, metals have many imperfections in the crystals, consisting of line defects, point defects, missing atoms, additional atoms, and impurities with foreign atoms. Metals can be further contaminated with larger impurities from nonmetallic elements such as oxides and sulfides.

Point defects occur when a lattice site within a crystal is empty and not occupied by an atom. Point defects are present in all metals and provide a mechanism for diffusion, which is the movement of solute through a solvent.

Line defects are microscopic dislocations and are the major defect affecting a given metals mechanical properties. Line defects occur when there is an incomplete chain of atoms inside a crystal. This results in a local distortion of the structure of the crystal because of the resultant dislocation. There is considerable internal strain in the immediate vicinity of the dislocation. When a force is applied, the line defect can propagate through the crystal structure, resulting in a permanent structural change ( Figs. 33-6 and 33-7 ). This is termed plastic deformation. When a metal is plastically deformed, a permanent structural change persists after the force is removed from the metal.

During stress, individual atomic bonds are disrupted and the atoms slip along a plane.

Deformation occurs when parallel and opposite forces are applied to a structure with one side immobilized.

An example of an area defect is a grain boundary. When metal begins the solidification process, crystals form independently of one another. Each crystal grows into a crystalline structure, or grain. The size and number of grains developed by a certain amount of metal depend on the rate of nucleation, which is the initial stage of formation of a crystal. Rapid cooling usually produces smaller grains, whereas slower cooling produces larger grains. The orientation of crystal boundaries (grain boundaries) is very influential in the spread of dislocations that become cracks.

A high nucleation rate yields a high number of grains for a given amount of metal. Therefore, the grain size will be small. If the rate of crystal growth is high relative to their nucleation rate, however, fewer grains will develop, and they will be of larger size.

As a grain grows, it eventually comes in contact with another grain. The surfaces that separate grains are termed grain boundaries. Grain boundaries are the junction areas of the many metal crystals that compose an implant. The grain size has a significant effect on the mechanical properties of a metal. A higher number of grain boundaries increases strength. Grain boundaries prevent line defects from propagating from one grain to another. A higher number of grain boundaries necessitate a higher force required to induce a plastic deformation. Because more grain boundaries occur in alloys with smaller grains, smaller grains yield an increase in strength, whereas larger grains are generally associated with low strength and ductility.

The many ways in which a metal can acquire defects affecting its strength has led to the development of various strengthening mechanisms to improve the performance of a metal or alloy. All strengthening mechanisms act on the theory that impeding line defects results in increased strength.

Solid solution strengthening occurs when one or more elements are added to a metal. Atoms of the solute will take places within the crystalline lattice by substituting for a solvent (metal) atom. Alternatively, the solute atom may occupy a site not previously occupied by a solvent atom by lying in an interstitial site. Interstitial atoms usually are much smaller than the solvent, whereas substituting elements often are similar in size to the solvent. Interstitial solid solution strengthening often is more effective. The effect of solid solution strengthening is to stop line defects from spreading a dislocation by developing solute-rich regions in the area surrounding the line defect. As a result, increased force is needed to induce a plastic deformation.

Cold working deforms the metal and thus increases strength. Deformation of a metal increases the number of line defects within the metal. These dislocations then entangle with one another. The result is an increasing amount of energy that continues to move these line defects within the grain. The increase in strength from cold working comes at the expense of a decrease in ductility.

Hot working involves the use of high temperature to deform the metal. This often is used to allow a metal to form a shape while altering the microstructure of the alloy. It is possible to reduce grain size by hot working. By increasing the temperature to a level that causes a deformation, the dislocations become disentangled. The metal then recrystallizes, and new dislocation-free grains are formed.

Mechanical Properties

Knowing the dimensions of a material, when a force is applied, permits the stress or load per unit area to be determined. Stress is recorded as N/m ² (Pascal) and is a small quantity. Therefore, most materials are tested with billions of N/m ² , or gigapascals (GPa). Strain is a dimensionless unit that is the percentage of elongation (or shortening) during application of force. When both the load and the deformation are divided by the original area or length of the specimen, respectively, the result is stress and strain, which can be displayed graphically (see Fig. 33-7 ).

Spine surgeons should have a basic understanding of the typical stress-strain curve (see Fig. 33-2 ). The stress-strain curve defines the mechanical behavior of a metal under various degrees of stress and strain. The ratio of stress to strain is the modulus, the elastic modulus, or the “Young modulus.” The relationship is as follows:

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