Terms and definitions

• Stress-Strain

Stress is defined as a pressure or a tension exerted on an object. It is calculated dividing the force by its area of application. Strain, in turn, is the amount of deformation, and is calculated dividing the change of length by the initial length.

See equations 3.3 an 3.4. Hence, a tissue’s material properties may be obtained from force-elongation test data dividing the recorded force by the original cross-sectional area to give stress, and by dividing the variation of the specimen length by its original length to give strain. It allows constructing a stress-strain dia-gram that approximates the material’s stress-strain behavior independently of the sample dimensions. Stress-strain, also called force-elongation, curves are typ-ically described in terms of four regions. These four regions are illustrated in the ligament test of figure 3.26.

σ = F~

A (3.3)

ε= ∆l l0

(3.4)

• Structural vs. Material Properties

A biological tissue is often described in terms of its structural and material prop-erties. Structural properties characterize the tissue in its intact form. Important

Chapter 3. Deformation Model

structural properties are represented by a relationship between force and defor-mation, and must be understood in order to predict how a tissue will behave in-vivo. Material properties characterize the behavior of the material comprising the tissue and to a first approximation are independent of the size of the tissue.

The material properties are usually expressed in terms of the stress-strain rela-tionship of the material. Structural and material properties curves are similar in appearance, differing only by a scaling factor.

• Stiffness (Elasticity)

The stiffness of a material represents the materials ability to resist to deformation.

Stiffness is commonly characterized by the slope of the linear region of a stress-strain curve, also referred to as Young’s Modulus. To describe the slope of other regions of the stress-strain curve a Tangent Modulus is often defined. If a Tangent Modulus is defined it should have associated with it a strain value or a range of strains. There can be different modules depending on the loading conditions (e.g.

shear modulus, compression modulus). The larger the stiffness, the greater the force required to cause a given deformation. If the stress in a material is directly proportional to the strain for all strains, the material is called aHookean material.

• Anisotropy and Non-homogeneity

Ideal materials are isotropic and homogeneous. A material is called isotropic when its properties are the same in each of three coordinate axes (x,y,z). Tensile and compressive properties may be different, but each respective property must be the same in three directions. A material is said to be homogeneous if it is made of the same material throughout. Biological tissues are, instead, anisotropic and non-homogeneous.

• Viscoelastic Properties

Biological tissues are viscoelastic materials. It means that their behavior is time and history dependent. A number of different behaviors characterizes a material as viscoelastic. They are: stress-relaxation, creep, strain rate sensitivity, and hysteresis.

Force-relaxation(or stress-relaxation) is a phenomenon that occurs in a tissue stretched and held at a fixed length. Over time the force present within the tissue continually declines. Force-relaxation is strain rate sensitive. In general, the higher the strain rate, the larger the peak force and subsequently the greater the magnitude of the force-relaxation.

In contrast to stress-relaxation, that occurs when a tissue’s length is held fixed, creepoccurs when a constant force is applied across the tissue. When subjected to a constant tensile force, a tissue elongates with time. The general shape of the displacement-time curve depends on the past loading history (e.g. peak force, loading rate).

Another time-dependent property is strain rate sensitivity. Different tissues show different sensitivities to strain rate. For example, there may be little

differ-3.3. Bio-tissues properties

ence in the stress-strain behavior of ligaments subjected to tensile tests varying in strain rate over three decades while bone properties may change considerably.

Besides, theloading andunloading curves obtained from a force-deformation test of biological tissues, for example, do not follow the same path. The difference in the calculated area under the loading and unloading curves is termed the area of hysteresis and represents the energy lost due to internal friction in the material.

The amount of energy liberated or absorbed during a tensile test is defined as the integral of the force and the displacement.

• Viscosity

The viscosity of a fluid is a measure of the fluid’s resistance to flow. Viscosity of water is used as reference to calculate other fluids viscosity, and is considered to be 1. The capsule of diarthrodial joints is normally filled with a fluid of viscosity 10 called synovial fluid. This fluid helps to reduce friction and wear of articulating surfaces. Just for comparison, the viscosity of olive oil, for example, is 84 [Hawkins, 2002].

• Coefficient of Friction

The coefficient of friction is that fraction of the force transmitted across two bearing surfaces that must be used to initiate movement ( s -static friction) or keep the surfaces moving ( d -dynamic friction). The static coefficient of friction between two surfaces is always greater than the dynamic coefficient of friction.

Joints of the human body are well designed to reduce the coefficient of friction between articulating surfaces. See the section 4 that is dedicated to cartilage.

• Testing procedures

Structural and material properties of biological tissues are usually determined through some form of mechanical testing (e.g. tensile tests, compressive tests, bending and torsion tests). Customized workstations utilizing force transducers, clamps, and an actuator to control the distance between clamps are common-place. Commercial systems are also available and vary in design depending on the type of tissue being studied (e.g. macroscopic vs. microscopic, hard tissue vs. soft tissue etc.) and the type of loading rates required. Instron [Instron]

and MTS [MTS] are the two most common suppliers of mechanical testing sys-tems. Most systems allow either force control or length control. See pictures in figure 3.27. Mechanical testing of tissue in-vivo is very difficult and hence not commonly performed. Some of the techniques that have been utilized include: 1) buckle transducers to monitor tendon and ligament forces, 2) telemetered pres-sure sensors to meapres-sure joint contact prespres-sure, and 3) strain gauges to quantify bone and ligament strain. More general procedures include the works of Nava [Nava et al., 2003] and Valtorta [Valtorta and Mazza, 2004] which used respec-tively aspiration and torsional resonance tests in-vivo to determine bio-tissues properties. Some non-invasive approaches have also been employed. Ultrasound techniques have been used to detect changes in the speed of sound in different tissues and these changes have been correlated with the tissue’s elastic properties.

Chapter 3. Deformation Model

Various imaging techniques have also been used to quantify tissue geometry and deformation [Hawkins, 2002].