Tendon and Ligament Viscoelasticity: Hysteresis, Creep, and Load Relaxation

Stress is internal resistance per unit area (σ = F/A), measured in N/m² or N/mm². Types include direct stress (applied along axis), shear stress (parallel to cross-section, parabolic behavior), and torsion stress (linear behavior). True stress uses actual cross-sectional area after deformation, while engineering stress uses original area. The stress-strain curve plots stress on Y-axis and strain on X-axis. The linear elastic region (O to A) follows Hooke's Law where stress is proportional to strain. The proportional limit (Point A) marks the maximum stress where this linear relationship holds. The elastic limit (Point B) marks the maximum stress where the material returns to its original shape. Beyond this point, permanent deformation occurs.

Solid Mechanics studies how solid materials behave under applied forces. Stress is the internal resistance force per unit area (Force/Area), while strain is the change in length per unit original length (dimensionless). The stress-strain curve for mild steel shows key regions: the linear elastic region where stress is proportional to strain (Hooke's Law), the proportional limit where linearity ends, the elastic limit where permanent deformation begins, the yield point where plastic deformation starts, the ultimate stress point representing maximum load-carrying capacity, and the fracture point where the material completely breaks apart.

Stress is force per unit area (σ = F/A), with normal stress acting perpendicular to surfaces and shear stress acting parallel. Normal stress causes dimensional changes (tensile elongation or compressive shortening), while shear stress causes angular changes. Strain (ε = ΔL/L₀) is dimensionless. Both stress and strain are tensor quantities requiring matrix representation. Units include Pascal (N/m²) and PSI (pound-force per square inch), with 1 atm ≈ 101.3 kPa ≈ 14.7 PSI.

This section covers the foundational concepts of solid mechanics. Stress is defined as restoring force per unit area, while strain is the ratio of change in dimension to original dimension. Three types of stress exist: tensile (stretching), compressive (squeezing), and shearing (sliding). Hooke's Law states that stress is directly proportional to strain for many materials, with the modulus of elasticity as the constant of proportionality. Three moduli exist: Young's modulus (for tensile/compressive stress), Shear modulus (for shearing stress), and Bulk modulus (for fluids). The elastic region on a stress-strain curve represents where Hooke's Law holds and the material returns to its original shape.

Stress is defined as the restoring force per unit area, not simply force divided by area. Strain is defined as the change in dimension divided by the original dimension. These definitions are fundamental to understanding material behavior under load. The stress-strain curve is a graphical representation of a material's mechanical properties, showing how a material responds to applied stress. In stress-strain curves, strain is plotted on the x-axis and stress on the y-axis because stress is the cause (independent variable) and strain is the effect (dependent variable). The curve reveals important information about the material's elasticity, strength, ductility, and failure characteristics. This curve is fundamental to understanding material behavior and is used for material classification and selection in engineering design.

Viscoelastic fluids exhibit both elastic and viscous behavior simultaneously, common in polymers and plastics. Hooke's Law describes elastic behavior (force proportional to deformation), while Newton's Law describes viscous behavior. Unlike elastic materials that recover their shape, viscoelastic materials show partial recovery. The Maxwell model combines these behaviors mathematically.

Hooke's Law states that the force required to stretch or compress an elastic material is directly proportional to the extension or compression distance (F = kx). This law applies only to elastic materials that return to their original shape after deformation, distinguishing them from plastic materials that undergo permanent deformation. The spring constant k characterizes each material's stiffness—the greater the k, the more force needed for a given extension. The elastic limit marks the boundary beyond which permanent deformation occurs. At the atomic level, elastic behavior results from reversible stretching of atomic bonds, while exceeding the elastic limit causes bond breaking or permanent reorganization.

Rheology studies flow and deformation behavior of materials. Newton's Law of Viscosity states shear stress is proportional to velocity gradient (τ = μdu/dy). Newtonian fluids (water, air, petrol) follow this linear relationship. Non-Newtonian fluids (toothpaste, ketchup) have viscosity that changes with applied stress. Viscoelastic materials combine viscous and elastic properties. Yield stress is the minimum force to initiate flow. Rheology considers stress, strain, and time variables to characterize material behavior.

Materials exhibit two fundamental mechanical behaviors: elasticity and viscosity. Elastic solids follow Hooke's Law (F = kx), where force is proportional to deformation, allowing them to return to original shape after force removal. The spring constant determines deformation resistance. In contrast, Newtonian fluids exhibit viscous behavior where shear stress depends on deformation rate rather than magnitude—faster deformation produces greater resistance. This distinction between time-independent elastic response and time-dependent viscous response forms the foundation for understanding all material deformation mechanisms.

Elastic solids follow Hooke's law (ε = σ/G) and can be modeled as springs, exhibiting immediate and complete recovery upon stress removal. Newtonian fluids follow Newton's law (dε/dt = σ/η) and are modeled as dashpots, showing continuous deformation under constant stress. These fundamental models provide the basis for understanding more complex viscoelastic behaviors. The spring represents stored energy and rigidity, while the dashpot represents dissipated energy through viscous flow.

Biomechanics studies mechanical forces acting on living organisms, combining 'bio' (living bodies) with 'mechanics' (forces and movements). Three primary forces interact during movement: gravity pulling downward, muscle forces generating movement, and ground reaction forces creating equal upward pressure. Human tissues are adaptable and deform under mechanical loads—temporary during movement, lasting with prolonged loading. For example, sitting 10+ hours causes hip extension difficulty and back pain. This understanding enables teachers to explain why certain movements become challenging and how mechanical forces shape body function over time.

Biological tissues experience six main types of mechanical loads: (1) Compression - axial force pushing molecules together, causing shortening and widening (bones during standing); (2) Tension - axial force pulling molecules apart, causing micro tears and elongation (muscles, ligaments, tendons); (3) Shear - frictional sliding force parallel to the surface, causing molecules to slide past each other (skin blisters, ligament tension from bone sliding); (4) Torsion - twisting along the axis, causing spiral fractures; (5) Bending - composite loading creating tension on one side and compression on the opposite side; (6) Shear - frictional sliding force parallel to the plane. Each load type affects tissues differently and can lead to specific injury patterns.

Mechanical loading refers to the various ways forces act upon biological tissues, including tension (pulling forces that stretch structures), compression (pushing forces that shorten structures), torsion (twisting forces like wringing a towel), shear (offset compressive forces causing sliding deformation), and bending (forces perpendicular to the long axis creating both tension and compression simultaneously on opposite sides). Bones are much stronger in compression than tension, and bending creates tension on one side and compression on the other, which is clinically significant because stress fractures typically occur on the compression side where healing is more effective.

This comprehensive section establishes core principles governing how mechanical forces interact with biological tissues. Key concepts include: (1) Progressive force application causes gradual changes over time, as demonstrated by sustained mechanical loading; (2) Cellular migration serves as a fundamental mechanism for tissue repositioning during development; (3) Tension applied to one area produces corresponding effects on opposite areas, creating balanced physiological responses; (4) Optimum force thresholds exist where minimum effective force produces desired biological responses without causing damage; (5) Fibroblasts actively respond to mechanical forces by altering their behavior and contributing to tissue remodeling; (6) Blood pressure changes represent systemic physiological responses to force application. These foundational principles provide the theoretical basis for understanding how mechanical forces influence biological tissues at multiple organizational levels.

Tension is the internal force that tissue exerts to oppose an external deforming force. Stress is the internal force that tissue supports to resist the deforming force applied to it. Deformation is the amount of change produced by a force. When a force is applied, it generates deformation, which in turn generates tension. This relationship between force, deformation, and tension is fundamental to understanding how tissues respond to mechanical loads and how they may develop pathology under repeated stress.