**2. Biomechanical basic concepts**

#### **2.1 Strain**

Strain is a local deformation parameter expressed as units of length per length, usually expressed as a percentage, and is therefore dimensionless when the bone is loaded with different force vectors.

The mathematical definition (Eq. (1)) of strain is the change in length divided by the original length.

Formula for strain calculus:

$$\text{Strain} \left( \% \right) = \frac{\text{Change in length (mm)}}{\text{Original length (mm)}} \tag{1}$$

Due to the dimensionless characteristic of this parameter, strain provides a clinically useful scaled measure of the displacement of bone fragments and can compare strain values in bones of different lengths. For example, a 1 mm fracture gap displacement is more significant in a 10 mm rat (strain 10%) femur than in a 300-mm dog femur (strain 0.3%) [1, 2].

#### **2.2 Stress**

One of the main functions of the appendicular skeleton is to support the body weight in rest or during movement, and consequently, bone is the tissue that supports more mechanical loads. When a force is applied to a bone, this will cause a stress situation [1, 2].

By definition, like pressure, stress is a local force expressed in units of force per unit area (Eq. (2)). The SI unit of force is the Newton, and force is often expressed as N/m2 or Pascal (Pa) [1, 2].

Formula for stress calculus:

$$\text{Stress} \left( \frac{\text{N}}{\text{mm}^2} \text{or} \text{Pa} \right) = \frac{\text{Load (N)}}{\text{Cross} - \text{sectional area } (\text{mm}^2)} \tag{2}$$

The damage that the load will cause depends on the area over which it is being distributed: a large force applied over a small area will result in greater stress, the contrary not being true. For instance, that is what happens when a skeletally immature dog falls from a height and supports all the weight on the hind limb. In these cases, the load will be equally distributed proximally and distally to the knee; however, due to the minor dimension of the distal part (tibial crest) of the tibial tuberosity, the fracture is normally located in this point due to stress concentration in a small area [3].

Bone tissue is constantly submitted to mechanical loads, comprehending forces/ loads of compression (axial), torsion, tension, bending, and shearing. Usually, these forces act in combination; however, they can be more predominant in an isolated way in certain locations of bones. Conceptually, it is considered that the deformation occurs in the bone tissue when small animals move and will vary between 0.04 and 0.3%, hardly exceeding 0.1%. This interval characterizes the elastic deformation of the bone, which is conceptually an initial response to the establishment of a load in the bone. In this scenario, the deformation/length of the bone returns to the initial dimensions/shape once the load is removed. It is a natural and important process for the homeostasis of bone tissue. An interesting characteristic of bone tissue is related to bone deformation according to the mechanical load that is applied. There are materials conceptually defined as isotropic, which respond to mechanical load regardless of the orientation of the material. By contrast, a material is considered anisotropic if the response to mechanical load varies with orientation. Bone is an example of an anisotropic material with mechanical properties that depend on the orientation of the bone lamellae. Thus, the mechanical properties are not equal in all directions and depend on the direction of the load applied. Long bones are stronger in longitudinal orientation than in tangential or radial orientation, since osteons have a longitudinal orientation in cortical bone. The maximum load that the bone will support is directly related to the direction in which the force is being applied. An illustrative example of this statement is that the appendicular bone supports a greater axial (compressive) load if compared to the transverse load. This difference between maximum strength in different directions emphasizes the anisotropic characteristic of bone. An example of an isotropic material is the stainless-Steel 316L (metallic alloy commonly used in the production of plates, which presents a similar behavior regardless of the direction of the load that is applied, with a similar resistance [3].

From a mechanical point of view, bone is also considered a viscoelastic tissue. Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Practically, this represents the ability of the bone to resist deformation without loss of definitive structural integrity. Viscoelasticity of bone is dependent on several factors including water, mineral, and collagen type I content [4].

This concept justifies the resistance of bone to sudden loads/impacts like jumps or falls: if there is some degree of deformation, there is later a return to the original form.

However, bone tissue is not always capable of withstanding all the loads that are applied to it, and this failure occurs when the imposed force exceeds the elastic deformation capacity, which may cause a complete or incomplete fracture. In this last situation, permanent deformation occurs in the tissue even after the load is removed, triggering microfractures and trabecular disruption and preceding macroscopic rupture [3].

*Biomechanical Basis of Bone Fracture and Fracture Osteosynthesis in Small Animals DOI: http://dx.doi.org/10.5772/intechopen.112777*

**Figure 1.**

*Graphic representation of the stress/strain curve, and the biomechanical concepts of young modulus, yield, and fracture point.*

When a supraphysiological force is applied, the bone will be deformed. The mechanical properties of bone are well represented by the load/stress-strain curve (**Figure 1**). The load/stress response of the bone directly depends on the length, thickness, density, shape, and type of bone (cortical vs. cancellous), so the stressstrain curve represents the structural properties of the object. Bone is characterized by a stress-strain curve with initially a linear response, the so-called elastic deformation. From the point at which the response ceases to be linear to the load and starts to express a curve (physiological limit point), plastic deformation appears and microfractures occur at the structural level, which can culminate in macroscopic fractures if the load is not interrupted or maximum force is reached (fracture point) [3].

#### **2.3 Strength and stiffness**

Strength denotes the ultimate load a material can withstand before a catastrophic failure, which is also designated as the fracture point (**Figure 1**) when this concept is applied to bone [1–3].

The stiffness of a biomaterial or bone tissue is the mechanical property that characterizes and quantifies the changes in the original shape when a force vector or a load is applied to it. A graphic that represents stiffness is called a load/displacement curve, and the relationship between stress and strain for materials, or load and displacement for structures, can help us understand these properties better. The slope of the straight part of the curve that ascends represents the elastic modulus or stiffness. The steeper the slope of this part of the curve, the stiffer the material. The strain (or change in shape) in this part of the curve is elastic, which means the material can return to its original shape after the force is removed. There is a point on the curve, called Y, known as the yield point or yield load where the curve stops being nonlinear (**Figure 1**). At this point, the strain exceeds the material's ability to recover from its

original shape, and the material gets a permanent change in shape if the load is removed. This point shows where the material changes from elastic to plastic deformation. This point is important for clinical reasons because it means that the bone acquires a different shape from its original [2].

This permanent change in shape, called plastic (instead of elastic) deformation, occurs when covalent bonds break at a molecular level. The point on the curve where the material breaks or fails is called U, or ultimate failure/fracture point (**Figure 1**). The curve also shows how much energy the material can absorb during the loading process [2]. This is called toughness, and it is represented by the area under the curve [2].

#### **2.4 Elastic modulus of young**

The elastic modulus, also called the modulus of Young (Y), represents the relationship between stress and strain and is one of the most useful parameters for mechanically comparing biomaterials. It is calculated by applying the formula represented by Eq. (3), where the slope of the stress is plotted versus the strain curve (**Figure 1**). It has the same units as stress (N/mm2 ) because strain does not have any units [3]. It quantifies the relationship between tensile/compressive stress. Young's moduli values are normally so large that this parameter must be expressed in gigapascals (GPa) instead of in Pascals. The components of the formula for Young's modulus calculation are: σ (force per unit area) and axial strain, ε (proportional deformation) in the linear elastic region of a material and is determined using the formula (Eq. (3)):

Young modulus formula:

$$E = \frac{\sigma}{\varepsilon} \tag{3}$$

The elastic modulus of Young is a measure of the linear elasticity of a material. This parameter allows grading materials in two categories: flexible and rigid. Materials with higher Y values are considered rigid. The bone tissue, for example, is included in the rigid category with a value of 15 GPa but is less rigid when compared to materials used in the manufacture of orthopedic implants. The elastic modulus of stainless-steel implants is usually 188 GPa, pure titanium is 116 GPa, and titanium alloy (Ti-6Al-4 V) is 113 GPa. A single value of Y assumes a linear relationship, which is true for metals (until their yield point) [3].

#### **2.5 Area moment of inertia (AMI)**

The area moment of inertia is a geometric parameter to be considered when the mechanics of implants are studied. The AMI is a measure of the resistance of materials exclusively related to flexion loads. This parameter is only influenced by the geometry and not by the composition of materials. Implants manufactured with bigger AMI have the least probability of structural collapse when submitted to higher flexion loads (higher stiffness to flexion loads). AMI does not take into account material properties, and for that reason, AMI must be only used to compare different constructs of the same material. AMI is a geometric parameter that is calculated based on the dimensions of the structure in the direction of bending. For a circular implant (e.g., a pin or interlocking nail), the direction is not relevant, and the formula used for this particular type of implant is the following (Eq. (4)):

AMI formula for circular implant:

$$\frac{1}{4.p.r4} \tag{4}$$

In Eq. (4), the radius is raised to the fourth power, so a small increase in the diameter of a pin or other circular implant has a large impact on its bending stiffness.

This concept can be illustrated by the following example: if the AMI of a 2.4 mm pin (3/32 inch) is 1.6 mm<sup>4</sup> and for a 3.2-mm (1/8 inch) pin is 5.1 mm<sup>4</sup> , an increase of 33.3% of the pin diameter results in an increased AMI value, 3 times larger than an original pin.

For solid rectangular structures, AMI is calculated using a different formula (Eq. (5)):

AMI formula for solid rectangular structures:

$$\frac{1}{3.b.h3} \tag{5}$$

In the AMI formula, *b* is the width, and h is the height in the direction of bending. Plate thickness is an important parameter because this dimension is cubed. However, for bone plates, the presence of the screw holes adds complexity to this calculus. At the screw holes, the AMI is usually less than half the value that would be calculated from its external dimensions [5].

#### *2.5.1 The impact of plate orientation on AMI*

If the direction of the bending force of a fracture is known or the vector force can be simplified to a craniocaudal direction, the orthopedic surgeon can also use this concept of AMI to consider alternative plate locations. The classic example of this concept is for distal radius fractures. In this type of fractures, the primary direction of bending is considered to be in the craniocaudal plane; if a 2.7-mm LC-DCP (Limited Contact-Dynamic Compression Plate) is placed on the medial aspect, a higher AMI value (solid section of approximately 111 mm<sup>4</sup> ) will be measured when compared to a 3.5 mm LC-DCP placed on the cranial aspect (AMI of 30 mm4 ), because the height of the 2.7 mm plate in the direction of bending is 8 mm (almost 3 times greater), compared with 3.3 mm for the 3.5 mm plate [2].

Another variant that influences the AMI is the position of the implant regarding the neutral axis of the bone, which is represented by the medullary canal of the bone. If the implant is positioned more distant from the neutral axis, the implant is less efficient to resist the bending forces. For the mentioned reason, the interlocking nail is the most mechanically favored implant to resist bending forces [3].

The use of AMI helps the decision-making process for choosing the osteosynthesis method but is not exclusively based on this parameter. Every long bone has a tension side and a compression side when axial loading is applied to the bone that will cause deformation, promoting bending. When the axial loading is applied and the bone bends, one side will experience tension and the cortices suffer traction. At the same time, the opposite side of the bone and the bone cortices will experience compression. Every long bone has a neutral axis that corresponds to the medullary cavity and that does not suffer compression or traction forces, and also has a tension and compression sides [1–3].

By using the AMI and tension/compression sides defined for each bone, in the decision-making process of orthopedic surgery, two premises must be fulfilled to succeed:

