Chapter 3 Materials science

Metals and plastics are the basic principal materials used in orthotics and prosthetics. To understand recommended design and fabrication procedures, a basic knowledge of the properties of the various available materials is necessary. The practitioner must be familiar with these materials in order to cope with both standard and difficult designs and fabrication problems and have the ability to prevent structural or functional failures of device due to the material.

Selection of the correct material for a given design depends partially on understanding the elementary principles of mechanics and materials, concepts of forces, deformation and failure of structures under load, improvement in mechanical properties by heat treatment, work (strain) hardening or other means, and design of structures. For example, the choices for a knee–ankle–foot orthosis (KAFO) may include several types of steels, numerous alloys of aluminium, and titanium and its alloys. Important but minor uses of other metals include copper or brass rivets and successive platings of copper, nickel, and chromium. Plastics, fabrics, rubbers, and leathers have wide indications, and composite structures (plastic matrix with reinforcing fibers) are beginning to be used. Often complex combinations of materials are used in manners that are not appropriate from the material point of view but are appropriate for the particular clinical application. Understanding these properties not only assists with the selection, manufacture, and management of the device but extends to the management of the patient and the information that the practitioner will instill into patients. A simple example is the combination of flexible materials such as a strap and thermoplastic, using an alloy rivet.

Despite publicity for exotic materials, no single material is a panacea. One reason is that a single design frequently requires divergent mechanical properties (e.g., stiffness and flexibility required in an ankle–foot orthosis [AFO] for dorsiflexion restraint and free plantar flexion). In addition, practitioners rarely are presented with situations where they will use only one material or with single-design situations that will not require modification, customization, or variation over time.

In general, understanding by the practitioner of the mechanics and strength of materials, even if intuitive, is important during the design stage. A general understanding of stresses arising from loading of structures, particularly from the bending of beams, is needed. The practitioner then can appreciate the importance of simple methods that allow controlled deformation during fitting, provide stiffness or resiliency as prescribed, and reduce breakage whether from impact or repeated loading. A general discussion of materials and specific theory related to design, fabrication, riveting guidelines, troubleshooting, and failure considerations follows.

Consideration should be given to the international standards of terminology that are used to describe orthotics, prosthetics, properties of materials, and units of measure (whether imperial or metric) as well as the engineering principles for describing the various effects of loading upon these materials. Unless they are familiar with the particular definitions of the terms used, practitioners should generally avoid using specific terminology in favor of more objective descriptive language.

# Strength and stress

One of the practitioner’s main considerations is the strength of the material selected for fabrication of orthoses or prostheses. *Strength* is defined as the ability of a material to resist forces. When comparative studies are made of the strength of materials, the concept of stress must be introduced.

Stress relates to both the magnitude of the applied forces and the amount of the material’s internal resistance to the forces. *Stress* is defined as force per unit cross-sectional area of material and usually is expressed in pounds per square inch (psi, imperial) or megapascals (MPa, Newtons per square meter, metric). The amount of stress (σ) is computed using the equation:

Where *F* = applied force (pounds or Newtons), and *A* = cross-sectional area (square inches or square meters).

The same amount of force applied over different areas causes radically different stresses. For example, a 1-lb weight is placed on a cylindrical test bar having a cross-sectional area of 1 in^{2}. According to Equation 3-1, the compressive stress σ_{c} in the cylindrical test bar is 1 lb/in^{2} (Fig. 3-1). When the same 1-lb weight is placed on a needle having a cross-sectional area of 0.001 in^{2}, the compressive stress σ_{c} in the needle is 1000 psi (Fig. 3-2).

A force exerted on a small area always causes more stress than the same force acting on a larger area. When a woman wears high-heeled shoes, her weight is supported by the narrow heels, which have an area of only a fraction of a square inch. With flat shoes, the same weight or force is spread over a heel having a larger cross-sectional area. The stress in the heel of the shoe is much greater when high-heeled shoes are worn because less material is resisting the applied forces.

Similar problems are encountered in orthoses and prostheses. A child weighing 100 lb and wearing a weight-bearing orthosis with a 90-degree posterior stop (Fig. 3-3) can exert forces at initial contact that create stresses of thousands of pounds per square inch. If the child jumps, the force would increase with the height of the jump. The stress at the stop or on the rivet could be great enough to cause failure.

## Tensile, compressive, shear, and flexural stresses

Materials are subject to several types of stresses depending on the way that the forces are applied: tensile, compressive, shear, and flexural.

### Tensile stresses

Tensile stresses act to pull apart an object or cause it to be in tension. Tensile stresses occur parallel to the line of force but perpendicular to the area in question (Fig. 3-4). If an object is pulled at both ends, it is in tension, and sufficient force will pull it apart. Two children fighting over a fish scale and exerting opposing forces put it in tension, as shown by the indicator on the scale (Fig. 3-5).

### Compressive stresses

Compressive stresses act to squeeze or compress objects. They also occur parallel to the line of force and perpendicular to the cross-sectional area (Fig. 3-6).

A blacksmith shapes metal by hitting the material with a hammer to squeeze or compress the metal into the desired shape. In the same manner, clay yields to low compressive stress. Clay is distorted and squeezed out of shape by comparatively small forces.

### Shear stresses

Shear stresses act to scissor or shear the object, causing the planes of the material to slide over each other. Shear stresses occur parallel to the applied forces. Consider two blocks (Fig. 3-7, *A*) with their surfaces bonded together. If forces acting in opposite directions are applied to these blocks, they tend to slide over each other. If these forces are great enough, the bond between the blocks will break (Fig. 3-7, *B*). If the area of the bonded surfaces were increased, however, the effect of the forces would be distributed over a greater area. The average stress would be decreased, and there would be increased resistance to shear stress.

A common lap joint and clevis joint are examples of a shear pin used as the axis of the joint (Fig. 3-8). The lap joint has one shear area of the rivet resisting the forces applied to the lap joint (Fig. 3-8, *A*), and the rivet in the box joint (clevis) has an area resisting the applied forces that is twice as great as the area in the lap joint (assuming that the rivets in both joints are the same size; Fig. 3-8, *B*). Consequently the clevis joint will withstand twice as much shear force as the lap joint. The lap joint also has less resistance to fatigue (fluctuating stress of relatively low magnitude, which results in failure) because it is more susceptible to flexing stresses.

### Flexural stress

Flexural stress (bending) is a combination of tension and compression stresses. Beams are subject to flexural stresses. When a beam is loaded transversely, it will sag. The top fibers of a beam are in maximum compression while the bottom side is in maximum tension (Fig. 3-9). The term *fiber*, as used here, means the geometric lines that compose the prismatic beam. The exact nature of these compressive and tensile stresses are discussed later.

## Yield stress

The yield stress or yield point is the point at which the material begins to maintain a deformational change due to the load and therefore the internal stresses under which it has been exposed.

## Ultimate stress

Ultimate stress is the stress at which a material ruptures. The strength of the material before it ruptures also depends on the type of stress to which it is subjected. For example, ultimate shear stresses usually are lower than ultimate tensile stresses (i.e., less shear stress must be applied before the material ruptures than in the case of tensile or compressive stress).

## Strain

Materials subjected to any stress will deform or change their shape, even at very small levels of stress. If a material lengthens or shortens in response to stress, it is said to experience *strain*. Strain is denoted by ε and can be found by dividing the total elongation (or contraction) ΔL by the original length *L _{O}* of the structure being loaded:

Consider a change in length Δ*L* of a wire or rod caused by a change in stretching force *F* (Fig. 3-10). The amount of stretch is proportional to the original length of wire. A wire 5 inches long stretches twice as much as a wire 3 to 5 inches long, other things being equal.

## Stress–strain curve

The most widely used means of determining the mechanical properties of materials is the tension test. Much can be learned from observing the data collected from such a test. In the tension test, the dimensions of the specimen coupon are fixed by standardization so that the results can be universally understood, no matter where or by whom the test is conducted. The specimen coupon is mounted between the jaws of a tensile testing machine, which is simply a device for stretching the specimen at a controlled rate. As defined by standards, the cross-sectional area of the coupon is smaller in the center to prevent failures where the coupon is gripped. The specimen’s resistance to being stretched and the linear deformations are measured by sensitive instrumentation (Fig. 3-11).

The force of resistance divided by the cross-sectional area of the specimen is the *stress* in the specimen (Equation 3-1). The *strain* is the total deformation divided by the original length (Equation 3-2). If the stresses in the specimen are plotted as ordinates of a graph, with the accompanying strains as abscissae, a number of mechanical properties are graphically revealed. Figure 3-12 shows such a stress–strain diagram for a mild steel specimen.

The shape and magnitude of the stress–strain curve of a metal depend on its composition; heat treatment; history of plastic deformation; and strain rate, temperature, and state of stress imposed during testing. The parameters used to describe the stress–strain curve of a metal are tensile strength, yield strength or yield point, percent elongation, and reduction in area. The first two are strength parameters; the last two indicate ductility.

The general shape of the stress–strain curve (Fig. 3-12) requires further explanation. In the region from *a* to *b*, the stress is linearly proportional to strain and the strain is elastic (i.e., the stressed part returns to its original shape when the load is removed). When the applied stress exceeds the yield strength, *b* the specimen undergoes plastic deformation. If the load is subsequently reduced to zero, the part remains permanently deformed. The stress required to produce continued plastic deformation increases with increasing plastic strain (points *c, d*, and *e* on Fig. 3-12), that is, the metal strain hardens. The volume of the part remains constant during plastic deformation, and as the part elongates, its cross-sectional area decreases uniformly along its length until point *e* is reached. The ordinate of point *e* is the tensile strength of the material. After point *e*, further elongation requires less applied stress until the part ruptures at point *f* (breaking or fracture strength). Although this seems counterintuitive, it actually occurs and is best sensed when bolts are overtorqued. Correct torque settings should always be complied with, but practitioners commonly torque bolts using the “as hard as possible” technique, assuming that this method somehow secures the bolt more appropriately than the correct torque and a thread locking solution. When excessive torque has been applied, the bolt first feels like it has loosened prior to failing. This simply reflects the fact that the yield point of the material has been surpassed and the bolt is plastically deforming under a decreasing load to failure.

Stress–strain diagrams assume widely differing forms for various materials. Figure 3-13, *A* shows the stress–strain diagram for a medium-carbon structural steel. The ordinates of points *p, u*, and *b* are the yield point, tensile strength, and breaking strength, respectively. The lower curve of Fig. 3-13, *B* is for an alloy steel and the higher curve is for hard steels. Nonferrous alloys and cast iron have the form shown in Fig. 3-13, *C*. The plot shown in Fig. 3-13, *D* is typical for rubber.

For any material having a stress–strain curve of the form shown in Figs. 3-13, *A–D*, it is evident that the relation between stress and strain is linear for comparatively small values of the strain. This linear relationship between elongation and the axial force causing it was first reported by Sir Robert Hooke in 1678 and is called *Hooke*’*s law*. Expressed as an equation, Hooke‘s law becomes:

where σ = stress (psi), ε = strain (inch/inch), and *E* = constant of proportionality between stress and strain. This constant is also called *Young*’*s modulus* or the *modulus of elasticity*.

The slope of the stress–strain curve from the origin to point *p* (Figs. 3-13, *A* and *B*) is the modulus of elasticity of that particular material *E*. The region where the slope is a straight line is called the *elastic region*, where the material behaves in what we typically associate as an elastic manner, that is, it is loaded and stretched, and upon releasing the load the material returns to its original position. The ordinate of a point coincident with *p* is known as the *elastic limit* (i.e., the maximum stress that may develop during a simple tension test such that no permanent or residual deformation occurs when the load is entirely removed). Values for *E* are given in Table 3-1.

In a routine tension test (Fig. 3-14), which illustrates Hooke’s law, a bar of area *A* is placed between two jaws of a vise, and a force *F* is applied to compress the bar. Combining Equations 3-1, 3-2, and 3-3 and solving for the shortening Δ*L* gives:

Because the original length *L*_{0}, cross-sectional area *A*, and modulus of elasticity *E* are constants, the shortening Δ*L* depends solely on *F*. As *F* doubles, so does Δ*L*.

The operation of a steel spring scale is another practical illustration of Hooke’s law (Fig. 3-15). The amount of deflection of the spring for every pound of force of the load remains constant. In Fig. 3-15, *A*, the scale indicates three units (pounds, ounces, grams). With one weight added (Fig. 3-15, *B*), the scale indicates 5, or two additional units. A second weight added (Fig. 3-15, *C*) causes the scale to indicate 7, or a total of four additional units, and a third weight stretches the spring two more units (Fig. 3-15, *D*). Therefore, it is possible to make uniform gradations for every unit of force to the point beyond the range of elasticity where the spring would distort or break. Scales are manufactured with springs strong enough to bear predetermined maximum loads. A compression spring scale designed to remain within the elastic range, recording weights to about 250 lb (100 kg) and then returning back to 0, is the common type used for weighing people.

### Plastic range

Plastic range is beyond the elastic range (*b* to past *e* on the stress–strain diagram of Fig. 3-12), and the material behaves plastically. That is, the material has a set or permanent deformation when externally applied loads are removed—it has “flowed” or become plastic. In the case of the steel spring scale, if the weight did not actually break the spring, it would stretch it permanently so that the readings on the scale would be no longer accurate.

When forming orthotic bars, the practitioner must bend the bar beyond the elastic limit and into a range of plastic deformation with some associated elastic return. With experience and some basic experiments, the practitioner will be able to accurately predict the range of deformation and return for particular bends. An advisable strategy is to chart this elastic return for the regular bends and commonly used sidebars.

For most materials, the stress–strain curve has an initial linear elastic region in which deformation is reversible. Note the load σ_{2} in Fig. 3-16. This load will cause strain ε_{E}. When the load is removed, the strain disappears, that is, point *X* (σ_{2}, ε_{E}) moves linearly down the proportional portion of the curve to the origin. Similarly, when load σ_{1} is applied, strain ε_{T} results. However, when load σ_{1} is removed, point *Y* does not move back along the original curve to the origin but moves to the strain axis along a line parallel to the original linear region intersecting the strain axis at ε_{P}. Therefore, with no load, the material has a residual or permanent strain of ε_{P}. Plastic deformation is difficult to judge because of elastic and plastic deformation but can be predicted for sidebars and charted as previously mentioned. The quantity of permanent strain ε_{P} is the plastic strain, and (ε_{T} – ε_{P}) is the elastic strain ε_{E} or:

where ε_{T} = total strain under load, ε_{P} = plastic (or permanent) strain, and ε_{E} = elastic strain.

### Yield point

Yield point (point *b* on the stress–strain diagram of Fig. 3-12) refers to that point at which a marked increase in strain occurs without a corresponding increase in stress. The horizontal portion of the stress–strain curve (*b-c-d* in Fig. 3-12) indicates the yield stress corresponding to this yield point. The yield point is the “knee” in the stress–strain curve for a material and separates the elastic from the plastic portions of the curve.

### Tensile strength

The tensile strength of a material is obtained by dividing the maximum tensile force reached during the test (*e* on the stress–strain diagram in Fig. 3-12) by the original cross-sectional area of the test specimen. Practical application of the maximal tensile force is minimal because devices are never designed to be loaded to this value.

### Toughness and ductility

The area under the curve to the point of maximum stress (*a-b-c-d-e* in Fig. 3-12) indicates the *toughness* of the material, or its ability to withstand shock loads before rupturing. The supporting arms of a car bumper are an example of where toughness is of great value as a mechanical property. *Ductility* is the ability of a material to sustain large permanent deformations in tension, as drawing a rod into a wire. The distinction between ductility and toughness is that ductility deals only with the ability to deform, whereas toughness considers both the ability to deform and the stress developed during the deformation. The requirement for plastic deformation in sidebars is weighed against the ability of the sidebars to resist large rapid loads and even the forces required by the practitioner to be able to deform them.

## Thermal stress

When a material is subjected to a change in temperature, its dimensions increase or decrease as the temperature rises or falls. If the material is constrained by neighboring structures, stress is produced.

The influence of temperature change is noted through the medium of the coefficient of thermal expansion α, which is defined as the unit strain produced by a temperature change of one degree. This physical constant is a mechanical property of each material. Values of α for several materials are given in Table 3-3.

If the temperature of a bar of length *L*_{O} inches is increased Δ*T*°F (or °C, *note*: α indicates which measure of temperature it relates to), the elongation Δ*L* in inches of the unrestrained bar is given by:

If the heated rod is compressed back to its original length, then it will experience compression as given by Equation 3-4:

Combining Equations 3-6 and 3-7 and solving for stress, σ = *F/A*, gives:

Equation 3-8 allows the calculation of stress in a rod as a function of the increase in temperature Δ*T*, the modulus of elasticity *E* (Table 3-1), and the coefficient of thermal expansion α (Table 3-2). The concept of change in dimension as the result of temperature rise is illustrated in Example 1 in the Appendix.

## Centroids and center of gravity

The centroid and center of gravity of objects play important roles in their mechanical properties. The center of gravity and centroid of two identically shaped objects are the same if the density is uniform in each object. The centroid is a geometric factor, and center of gravity depends on mass.

For an object of uniform density, the term *center of gravity* is replaced by the *centroid of the area*. The *centroid of an area* is defined as the point of application of the resultant of a uniformly distributed force acting on the area. An irregularly shaped plate of material of uniform thickness *t* is shown in Fig. 3-17. Two elemental areas (*a* and *b*>) are shown with centroids (*x*_{1},*y*_{1}) and (*x*_{2},*y*_{2}), respectively. If the large, irregularly shaped plate is divided into small elemental areas, each having its own centroid, then the centroid for the irregularly shaped plate is (*x*,*y*), where:

and

The *y*-centroids for several common geometric shapes are given in Table 3-3. The general equations for the *x*– and *y*-components of the centroid are given in Example 2 in the Appendix.

## Moment of inertia

The moment of inertia of a finite area about an axis in the plane of the area is given by the summation of the moments of inertia about the same axis of all elements of the area contained in the finite area. In general, the *moment of inertia* is defined as the product of the area and the square of the distance between the area and the given axis. The moments of inertia about the centroidal axes *I*_{cc} of a few simple but important geometric shapes are determined by integral calculus and are given in Table 3-3. Although Young’s modulus is an indication of the strength of the material, the moment of inertia is an indicator of the strength of a particular shape about the aspect in which it will be loaded. This is a highly important parameter for the practitioner to know because he or she often will be able to influence the shape.

### Parallel axis theorem

When the moment of inertia has been determined with respect to a given axis, such as the centroidal axis, the moment of inertia with respect to a parallel axis can be obtained by the *parallel axis theorem*, provided one of the axes passes through the centroid of the area. The parallel axis theorem states that *the moment of inertia with respect to any axis is equal to the moment of inertia with respect to a parallel axis through the centroid added to the product of the area and the square of the distance between the two axes* (Fig. 3-18):

where *I*_{xx} = moment of inertia about *x*-axis, *I*_{cc} = moment of inertia about centroid, *A* = area, and *d* = distance between axes.

An illustration of the moment of inertia concept using the parallel axis theorem is given in Example 3 in the Appendix.

## Stresses in beams

If forces are applied to a beam as shown in Fig. 3-19, downward bending of the beam occurs. Imagine a beam is composed of an infinite number of thin longitudinal rods or fibers. Each longitudinal fiber is assumed to act independently of every other fiber (i.e., there are no lateral stresses [shear] between fibers). The beam of Fig. 3-19 will deflect downward and the fibers in the lower part of the beam undergo extension, whereas those in the upper part shorten. The changes in the lengths of the fibers set up stresses in the fibers. Those that are extended have tensile stresses acting on the fibers in the direction of the longitudinal axis of the beam, whereas those that are shortened are subject to compression stresses.

One surface in the beams always contains fibers that do not undergo any extension or compression and thus are not subject to any tensile or compressive stress. This surface is called the *neutral surface* of the beam. The intersection of the neutral surface with any cross-section of the beam perpendicular to its longitudinal axis is called the *neutral axis*. All fibers on one side of the neutral axis are in a state of tension, whereas those on the opposite side are in compression.

For any beam having a longitudinal plane of symmetry and subject to a bending torque *T* at a certain cross-section, the normal stress σ, acting on a longitudinal fiber at a distance *y* from the neutral axis of the beam (Fig. 3-20), is given by:

where *I* = moment of inertia of the cross-sectional area about the neutral or centroidal axis in inches^{4}.

These stresses vary from zero at the neutral axis of the beam (*y* = 0) to a maximum at the outer fibers (Fig. 3-20). These stresses are called *bending, flexure*, or *fiber stresses*.

### Section modulus

The value of *y* at the outer fibers of the beam is frequently denoted by *c*. At these fibers, the bending stress is a maximum and is given by:

The ratio *I/c* is called the *section modulus* and usually is denoted by the symbol *Z*. The section moduli for the shapes given in Table 3-3 are obtained by dividing the moment of inertia about the centroidal axis by the length of the centroid. For example, the moment of inertia for a rectangle about its centroidal axis is *bh*^{3}/12 and the length of the centroid is *h*/2; therefore, the section modulus is *bh*^{2/}6. Section moduli are given in Table 3-3.

### Beam torque

Most structural elements in orthoses can be represented by either a cantilever beam loaded transversely with a perpendicular force at the end (e.g., a stirrup in terminal stance; Fig. 3-21) or a beam freely supported at the ends and centrally loaded (e.g., KAFO prescribed to control valgum; Fig. 3-22).

The maximum *torque* in cantilevered (Fig. 3-21) and freely supported (Fig. 3-22) *beams* is given by:

Figure 3-23 gives the maximum torque for a few simple beams. If more than one external force acts on a beam, the bending torque is the sum of the torques caused by all the external forces acting on either side of the beam. Subsequently and not surprisingly, device failures commonly occur at the corresponding point of maximum torque (bending moment).

### Beam stress

The *stress* in a cantilevered or freely supported *beam* now can be determined by substituting Equation 3-12 or 3-13 into Equation 3-11, which gives:

If these beams have rectangular cross-sections with height *h* and base *b* (i.e., *ch*/2 and *I* = *bh*^{3}/12), then the expressions for stress can be rewritten as:

As the cross-sectional area of the beam changes shape, so does the expression for the moment of inertia *I* and the outer fiber-to-neutral axis distance *c*.

### Beam deflection

The maximum *deflection* of *beams* (sidebars, stirrups) is important to practitioners because the biomechanical objective of a prescribed device frequently depends on the ability of the device either to not deflect or to deflect a given amount. Excessive deflection (bending) of a device may either disturb alignment or prevent successful operation.

Deflection theory provides a technique of analysis for evaluating the nature and magnitude of deformations in beams. The cantilevered beam (Fig. 3-24) carries a concentrated downward load *F* at the free end. A cantilevered beam is, by definition, rigidly supported at the other end. The general expression for the downward deflection *y*, anywhere along the length (*x*-axis) of the beam, is given by:

The maximum deflection of the cantilevered beam (*y*_{max}) occurs at the free end when *x* = 0:

The general expression for the deflection of the freely supported beam with the midspan load (Fig. 3-25) is given by:

The maximum deflection of the freely supported beam (*y*_{max}) occurs at the midspan when *x* = *L/2*:

The negative sign in Equations 3-19 and 3-21 indicates that the maximum deflection is downward from the unloaded position. Example 4 in the Appendix provides an illustration of calculating KAFO stress and deflection using the concepts of moment of inertia and centroid.

# Metals

A *metal* is defined as a chemical element that is lustrous, hard, malleable, heavy, ductile, and tenacious and usually is a good conductor of heat and electricity. Of the 93 elements, 73 are classified as metals. The elements oxygen, chlorine, iodine, bromine, and hydrogen and the inert gases helium, neon, argon, krypton, xenon, and radon are considered nonmetallic. There is, however, a group of elements, such as carbon, sulfur, silicon, and phosphorus, that is intermediate between the metals and nonmetals. These elements portray the characteristics of metals under certain circumstances and the characteristics of nonmetals under other circumstances. They are referred to as *metalloids*.

The most widely used metallic elements include iron, copper, lead, zinc, aluminum (or aluminium), tin, nickel, and magnesium. Some of these elements are used extensively in the pure state, but by far the largest amount is used in the form of alloys. An *alloy* is a combination of elements that exhibits the properties of a metal. The properties of alloys differ appreciably from those of the constituent elements. Improvement of strength, ductility, hardness, wear resistance, and corrosion resistance may be obtained in an alloy by combinations of various elements. Orthotics and prosthetics typically contain alloys of aluminum and carbon steels, particularly stainless steel. Titanium also is frequently used, and, despite references to “pure titanium” (particularly in applications such as *osseointegration*), it is the alloy that is being referenced. Although these alloys (steel, aluminum, titanium) can be categorized as similar depending on the base metal and some of the contributing alloy metal, they are potentially infinitely variable.

## Crystallinity

One of the important characteristics of all metals is their crystallinity. A *crystalline substance* is one in which the atoms are arranged in definite and repeating order in a three-dimensional pattern. This regular arrangement of atoms is called a *space lattice*. Space lattices are characteristic of all crystalline materials. Most metals crystallize in one of three types of space lattices:

*Cubic system*: Three contiguous edges of equal length and at right angles—simple lattice, body-centered lattice, and face-centered lattice (Fig. 3-26)

*Tetragonal system*: Three contiguous edges, two of equal length, all at right angles—simple lattice and body-centered lattice (Fig. 3-27)

*Hexagonal system*: Three parallel sets of equal length horizontal axes at 120 degrees and a vertical axis—close-packed hexagonal (Fig. 3-28)

This orderly state also is described as balanced, unstrained, or annealed. Some metals can exist in several lattice forms, depending on the temperature. Examples of metals that normally exist in only one form are as follows:

Common iron is an example of one of many metals that may exist in more than one lattice form:

A metal in the liquid state is noncrystalline, and the atoms move freely among one another without regard to interspatial distances. The internal energy possessed by these atoms prevents them from approaching one another closely enough to come under the control of their attractive electrostatic fields. However, as the liquid cools and loses energy, the atoms move more sluggishly. At a certain temperature, for a particular pure metal, certain atoms are arranged in the proper position to form a single lattice typical of metal. The temperature at which atoms begin to arrange themselves in a regular geometric pattern (lattice) is called the *freezing point*. As heat is removed from metal, crystallization continues, and the lattices grow about each center. This growth continues at the expense of the liquid, with the lattice structure expanding in all directions until development is stopped by interference with other space lattices or with the walls of the container. If a space lattice is permitted to grow freely without interference, a single crystal is produced that has an external shape typical of the system in which it crystallizes.

Crystallization centers form at random throughout the liquid mass by the aggregation of a proper number of atoms to form a space lattice. Each of these centers of crystallization enlarges as more atoms are added, until interference is encountered. A diagrammatic representation of the process of solidification is shown in Fig. 3-29. In this diagram, the squares represent space lattices. In *A*, crystallization has begun at four centers.

As crystallization continues, more centers appear and develop with space lattices of random orientation. Successive stages in the crystallization are shown by *B, C, D, E*, and *F*. Small crystals join large ones, provided they have about the same orientation (i.e., their axes are nearly aligned). During the last stages of formation, crystals meet, but there are places at the surface of intersections where development of other space lattices is impossible. Such interference accounts for the irregular appearance of crystals in a piece of metal that is polished and etched (Fig. 3-30).

## Grain structure

During the growth process, the development of external features, such as regular faces, may be prevented by interference from the growth of other centers. In this case, each unit is called a *grain* rather than a crystal. The term *crystal* usually is applied to a group of space lattices of the same orientation that show symmetry by the development of regular faces. Each grain is essentially a single crystal. The size of the grain depends on the temperature from which the metal is cast, the cooling rate, and the nature of the metal. In general, slow cooling leads to coarse grain and rapid cooling to fine grain metals.

## Slip planes

When a force is applied to a crystal, the space lattice is distorted as evidenced by a change in the crystal’s dimensions. This distortion causes some atoms in the lattice to be closer together and others to be farther apart. The magnitude of the applied force necessary to cause the distortion depends on the forces that act between the atoms in the lattice and tends to restore it to its normal configuration. If the applied force is removed, the atomic forces return the atoms to their normal positions in the lattice. Cubic patterns (lattices) characterize the more ductile or workable materials. Hexagonal and more complex patterns tend to be more brittle or more rigid. The force required to bring about the first permanent displacement corresponds to the elastic limit. This permanent displacement, or slip, occurs in the lattice on specified planes called *slip planes*. The ability of a crystal to slip in this manner without separation is the criterion of plasticity. Practically all metals are plastic to a certain degree. During plastic deformations, the lattice undergoes distortion, thus becoming highly stressed and hardened.

Slip, or plastic deformation, can occur more easily along certain planes with a space lattice than along other planes. The planes that have the greatest population of atoms and, likewise, the greatest separation of atoms on each side of the planes under consideration are usually the planes of easiest slip. Therefore, slip takes place along these planes first when the elastic limit is exceeded. Sliding movements tend to take place at 45-degree angles to the direction of the applied load because much higher stresses are required to pull atoms directly apart or to push them straight together.

A particular characteristic of crystalline materials is that slip is not necessarily confined to one set of planes during the process of deformation. Some common planes of slip in the simple cubic system are shown in Fig. 3-31.

## Mechanical properties

The mechanical properties of metals depend on their lattice structures. In general, metals that exist with the face-centered cubic structure are ductile throughout a wide range of temperatures. Metals with the close-packed hexagonal type of lattice (Fig. 3-28) are appreciably hardened by cold working, and plastic deformation takes place most easily on planes parallel to the base of the lattice.

Of the many qualities of metals, the most significant are the related properties of elasticity and plasticity. Plasticity depends on the ability to shape and contour aluminum and stainless steel to match body contours; elasticity governs their safe and economical use as load-bearing members. The demand on the material used often is compromised depending on the consideration and prioritization of the manufacturing requirement or the clinical application.

As discussed in the section on Strength and Stress, a body is said to be *elastic* if it returns to its original shape upon removal of an external load. The *elastic limit* is the maximum stress at which the body behaves elastically. The *proportional limit* is the stress at which strain ceases to be proportional to applied stress; it is practically equal to elastic limit.

## Plasticity

*Plasticity* is the term used to express a metal’s ability to be deformed beyond the range of elasticity without fracture, resulting in permanent change in shape. Characteristically the ratio of plastic-to-elastic deformation in metals is high, on the order of 100:1 or 1000:1. Although this is rarely a consideration for commercially designed structural components because they are designed to behave within the elastic range, it is crucial to components such as sidebars, which must be plastically deformed (bent) before they are used clinically.

A simple two-dimensional representation of a cubic crystal lattice in an unstrained condition is represented by *A* in Fig. 3-32. If a shearing force within the elastic range is applied, the lattice is uniformly distorted as in *B*, with the extent of distortion proportional to the applied force.

Fig. 3-32 Deformation of a cubic crystal lattice. **A**, Unstrained condition. **B**, Elastic deformation. **C**, Plastic deformation. **D**, Permanent set as a result of slip.

When the force is removed, the lattice springs back to its original shape (*A*). However, when the force exceeds the elastic (or proportional) limit, a sudden change in the mode of deformation occurs. Without further increase in the amount of elastic strain, the lattice shears along a crystallographic plane (or slip plane). One block of the lattice makes a long glide past the other and stops (*C*). On release of load, the lattice in the two displaced blocks resumes its original shape (*D*). If the applied force is continued, slip does not continue indefinitely along the original slip plane, which on the contrary appears to acquire resistance to further motion; however, some parallel plane comes into action. Both the extent of slip per plane and the distance between active slip planes are large in comparison to the unit lattice dimensions. As slip shifts from one slip plane to another, progressively higher forces are required to accomplish it (i.e., the metal has been *work hardened*). At some stage, resistance to further slip along the primitive set of planes exceeds the resistance offered by some other set of differently directed slip planes, which then come into action. This process elaborates as plastic deformation progresses.

The actual strength of metals as ordinarily measured is but a small fraction of theoretical strength. Some significant comparisons for pure copper are as follows:

Similar relations exist for other pure metals.

Imperfections of many kinds, such as flaws in the regularity of the crystal lattice, microcracks within a grain, shrinkage voids, nonmetallic inclusions, rough surfaces, and notches of all kinds, may localize and intensify stresses. Many impurities owe their potency to a high degree of insolubility in the solid matrix coupled with high solubility in the fusion. This permits their freezing out relatively late in the solidification process, as concentrates or films between the grains, thus serving as effective internal notches. The great weakening effect of graphite flakes in cast iron is an example.

Notches act not only as stress raisers but also as stress complicators, frequently inducing stress in many directions. The deeper the notch and the sharper its root, the more effective it is in this respect. Notches are great weakeners, and practitioners do well to recognize their prevalence under many disguises (i.e., from either contouring instruments or grain boundaries). Most importantly, every care should be taken to minimize the contribution to these weakeners by not adding further notches, cracks, scratches, or rough surfaces.

# Steel and aluminum alloys

## Commercial name for metals

Before the stress–strain diagram is used as a basis for comparing the properties of various metals, it is necessary to discuss the types of steel and aluminum commercially available and used in orthotic and prosthetic applications.

The terms *surgical steel, stainless steel, tool steel*, and *heat treated* along with other general designations are freely used by manufacturers of orthotic and prosthetic components. The chemical content of these products is not identical from vendor to vendor. For example, the term *spring steel*, used by many manufacturers, refers to a group of steels ranging in chemical composition from medium- to high-carbon steel and is used to designate some alloy steels. The term *tool steel* also covers a wide variety of steels that are capable of attaining a high degree of hardness after heat treatment. More care is exercised in manufacturing tool steel to ensure maximum uniformity of desirable properties.

These general designations do not assure the orthotist or prosthetist of obtaining the exact material that is needed. Because the mechanical properties of a material and subsequent fabrication procedures depend on the material’s chemical analysis and subsequent heat treatment or working, the practice of using general descriptions for metals is seriously inadequate. In addition, reliance on these categories is not necessary because specific designations already exist for each type of steel and processing treatment. The following sections give a clearer picture of the available steel and aluminum alloys and their specific properties.

## Carbon steel

Iron as a pure metal does not possess sufficient strength or hardness to be useful for many applications. By adding as little as a fraction of 1% carbon by weight, however, the properties of the base metal are significantly altered. Iron with added carbon is called *carbon steel*. Within certain limits, the strength and hardness of carbon steel are directly proportionate to the amount of carbon added. In addition to carbon, carbon steel contains manganese and traces of sulfur and phosphorus.

## Alloy steel

To achieve desirable physical or chemical properties, other chemicals are added to carbon steel. The resultant product is known as *alloy steel*. In presenting some general characteristics distinguishing these alloys, it is necessary to define some terms commonly used to express them:

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