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Material objects move and deform under the action of forces and moments. While statics and dynamics describe the motion of rigid objects in response to systems of forces, solid mechanics describes the force-induced deformation of objects or materials. A few basic concepts of the mechanics of materials are presented in the following section.

Load and displacement

When a sample of any kind of material in the shape of a bar is subjected to a force F in the longitudinal direction the bar stretches, meaning that it deforms in the longitudinal direction (Figure 1.a,b). As the intensity of the forces increases the central section of the bar narrows (Figure 1.c). A further increase leads to the fracture of the bar in correspondence to the narrowed region (Figure 1.d).

Figure 1: Deformation of a bar under tensile loading

The tensile forces applied and the instantaneous length of the bar can be measured during deformation. If the load is plotted against the displacement of the surface it is applied on, the load-displacement curve obtained depends not only on the properties of the material but also on the size and shape of the sample tested. The notions of stress and strain, which are independent from size and shape of the sample, are then introduced.

Stress and strain

When a tensile or compressive force F is applied to a sample the normal stress σ is defined as the ratio of the normal force to the cross-sectional area A: σ = F/A. Stress has the dimensions of force per unit area, N/m² or Pascal (Pa).
The normal strain ε is defined as the ratio of lenght to the initial length: ε = ΔL/L₀, where the deformation is expressed as the difference between the actual length L and the initial length L₀: ΔL = L-L₀. Strain ε is a pure number. It can also be expressed as a percentage, multiplying the above value of ε by 100. Strain is positive in tension and negative in compression. Shear stress and strain are defined in case of shear loading.
The above definitions are referred to simple loading conditions and they do not take into account the real tensorial nature of stress and strain.
Figure 2 shows the tensile stress-strain curve for the same steel whose load displacement curve is shown above.

Figure 2: Tensile stress-strain curve for steel

A number of characteristic points may be identified on a stress-strain curve, revealing important properties of the material.
Point A identifies the proportional limit. Below the proportional limit, stress is a linear function of strain. In this region the elastic modulus is defined as the ratio of stress and strain, i.e. the slope of the initial linear portion of the stress-strain curve.
The longitudinal elastic modulus in tension and compression is referred to as the Young’s modulus and it is represented with the letter E: E = σ/ε.

Point B identifies the elastic limit or yield point. The yield stress σ_y and the yield strain ε_y are the maximum stress and strain for which the material shows an elastic behaviour. Elastic deformation is a deformation which is temporarily and reversible. On removal of load the sample returns to its original size and shape. When a material shows an elastic behaviour the same stress-strain path is followed during loading and unloading (Figure 3).

Figure 3: Elastic behaviour

Beyond the yield point plastic deformation occurs. Plastic deformation is a non-reversible deformation: after releasing the load the sample doesn't return to its original size and shape. When a material behaves plastically the loading and unloading path are different (Figure 4.a). When unloading is completed the plastic deformation ε_p is maintained. When a new loading takes place the previous unloading path is followed (Figure 4.b).

Figure 4: Plastic behaviour

Since the exact point at which yield occurs may not be easily determined a conventional yield stress σ_0.2% is often used, that is the stress correspondent to 0.2% irreversible strain.
In correspondence to point C the ultimate stress σ_m is reached: this is the maximum stress that occurs in the material before fracture. The sample breaks in correspondence to point D, where the stress at failure σ_f and the correspondent strain at failure ε_f are reached. The drop in stress magnitude between point C and point D is due to necking, which is a localized decrease of the cross-sectional area.

The application of a tensile or compressive force to a specimen provokes not only a longitudinal deformation but also a transverse deformation and thus a change in the cross-sectional area (Figure 5).

Figure 5: Longitudinal and transversal deformation for compressive loading

The Poisson’s ratio ν is defined as ν = - ε₂/ε₁ which represents the ratio between the transverse strain ε₂ and the longitudinal strain ε₁. The Poisson's ratio normally has a value included between 0, for a material in which no transverse deformation occurs, and 0.5, for a material that maintains constant volume during deformation. Most materials have a Poisson’s ratio ranging between 0.2 and 0.5.

 

A description of the mechanical properties of bone and titanium can be found following the links:

• Mechanical properties of bone
• Mechanical properties of titanium