6.1 Stress, strain, and elasticity

Stress, strain, and elasticity are key concepts in understanding how biological tissues respond to forces. These principles explain why muscles stretch, bones resist compression, and tendons store energy during movement. They're crucial for grasping tissue function and injury mechanisms.

Tissues exhibit complex stress-strain relationships due to their unique structures. Bone is stiff and strong, while tendons are elastic and tough. Cartilage is viscoelastic, and skin is highly non-linear. Understanding these properties helps us analyze movement and design better treatments for injuries.

Stress and Strain in Tissues

Fundamental Concepts of Stress and Strain

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• Stress develops within a material as internal force per unit area in response to applied external force
• Strain represents material deformation or shape change relative to original dimensions when subjected to stress
• Interconnected concepts describe how tissues respond to mechanical loads
• Non-linear and time-dependent relationship in tissues due to complex structure and composition
• Various stress types (tensile, compressive, shear) applied to biological tissues result in different strain responses
• Stress-strain relationship in biological tissues influenced by tissue type, age, hydration, and previous loading history

Tissue-Specific Stress-Strain Characteristics

• Bone tissue exhibits high stiffness and strength, with relatively low strain before failure (brittle behavior)
• Tendons and ligaments display high tensile strength and significant elasticity, allowing for energy storage and return
• Cartilage demonstrates viscoelastic properties, with time-dependent deformation under load
• Skin shows highly non-linear behavior, with initial low stiffness followed by rapid stiffening at higher strains
• Blood vessels combine elastic and viscous properties to accommodate pulsatile blood flow
• Muscle tissue exhibits active and passive mechanical properties, changing stiffness with contraction state

Elasticity in Tissue Mechanics

Fundamentals of Tissue Elasticity

• Elasticity enables material to return to original shape and size after force removal
• Crucial property determines tissue response to mechanical loads and deformations
• Elastic modulus (Young's modulus) quantifies deformation resistance
  • Calculated as stress-to-strain ratio in linear elastic region
• Viscoelastic behavior in biological tissues combines elastic and viscous properties
  • Affects time-dependent mechanical response
• Essential for understanding tissue function, injury mechanisms, and biomaterial design
• Elasticity varies widely among tissues (highly elastic tendons vs. less elastic bone)
• Elastic properties change due to aging, disease, or injury
  • Impacts mechanical function and damage susceptibility

Elastic Modulus and Tissue Function

• Elastic modulus serves as a measure of tissue stiffness
• Higher elastic modulus indicates greater resistance to deformation (bone, ~20 GPa)
• Lower elastic modulus allows for more flexibility and energy storage (tendons, ~1 GPa)
• Mismatch in elastic moduli between adjacent tissues can lead to stress concentrations
  • Potential sites for injury or failure (bone-tendon interface)
• Gradients in elastic moduli within tissues optimize force transmission (muscle-tendon junction)
• Changes in elastic modulus can indicate pathological conditions
  • Increased liver stiffness in cirrhosis
  • Decreased bone mineral density in osteoporosis

Linear vs Non-linear Elasticity

Linear Elastic Behavior

• Characterized by direct proportional relationship between stress and strain
• Follows Hooke's law: $\sigma = E\varepsilon$
  • $\sigma$ represents stress
  • $E$ represents elastic modulus
  • $\varepsilon$ represents strain
• Rarely observed in biological tissues over large strain ranges
• Approximated in some tissues for small deformations (cortical bone under low loads)
• Simplifies mathematical modeling and analysis of tissue mechanics
• Useful for comparing mechanical properties between different materials or tissues

Non-linear Elastic Behavior in Tissues

• Exhibits variable relationship between stress and strain
• Increasing stiffness at higher strains common in soft tissues
• Most biological tissues display non-linear elasticity due to complex structure
• Initial toe region in stress-strain curve represents non-linear behavior at low strains
  • Often due to straightening of collagen fibers (tendons, ligaments)
• Linear region represents tissue behavior under normal physiological loads
  • Used to calculate elastic modulus
• Transition from linear to non-linear behavior crucial for understanding functional limits
• Varying degrees of non-linearity observed in different tissue types
  • Highly non-linear (skin, blood vessels)
  • Moderately non-linear (tendons, ligaments)
  • Less non-linear (cartilage, bone)

Stress-Strain Curve Analysis

Key Features of Stress-Strain Curves

• Graphically represent relationship between applied stress and resulting strain
• Curve shape provides information on mechanical properties
  • Stiffness, strength, energy absorption capacity
• Key points define tissue's mechanical limits
  • Yield point marks onset of plastic deformation
  • Ultimate strength indicates maximum stress before failure
  • Failure point represents complete tissue rupture
• Area under curve represents energy absorbed during deformation (toughness)
• Slope of linear region corresponds to elastic modulus (tissue stiffness measure)
• Hysteresis in loading-unloading cycles indicates energy dissipation and viscoelasticity

Interpreting Stress-Strain Curves for Different Tissues

• Tendon stress-strain curve exhibits distinct regions
  • Toe region (initial non-linear portion)
  • Linear region (physiological loading range)
  • Yield region (microdamage accumulation)
  • Failure region (complete rupture)
• Bone stress-strain curve shows relatively linear behavior until yield point
  • Followed by brief plastic deformation before failure
• Cartilage stress-strain curve demonstrates time-dependent behavior
  • Creep (increasing strain under constant stress)
  • Stress relaxation (decreasing stress under constant strain)
• Comparing curves reveals functional and pathological differences
  • Age-related changes in tendon elasticity
  • Disease-induced alterations in bone strength
  • Effects of exercise on muscle mechanical properties