Tubes, Extension, and Torsion

arterial mechanics

Changes of mechanical properties as a result of disease could possibly be used as a marker for the disease

• e.g., “hardening of the arteries in atherosclerosis

Changes of mechanical properties may lead to additional/augmented disease states

• e.g., decreased bloodflow can lead to ischemia

Vein Graft Intimal Hyperplasia (IH)

Autologous saphenous vein remains the graft material most often used in bypass procedures.

Lumenal narrowing by IH is a primary mode of failure for autologous vein grafts, both in coronary and lower extremity positions.

physical problem with vein graft

The vein segment, normally subjected to relatively lower

and static pressure conditions, is placed in the higher-

pressure arterial system.

Urinary System

filters blood to remove waste products and extra water, turning them into urine

female urethra

3 to 4 cm long. Smooth muscle, involuntary control. Striated (skeletal) muscle, voluntary control. Urinary incontinence a significant problem in women’s health

compliance

how much deformation per unit force. Apply pressure → tube expands. treated as linear for each pair.

the measure of how easily a biological tissue, organ, or mechanical system deforms or stretches under an applied force or pressure. It is the mathematical inverse of stiffness (or elastance)

As pressure increases, compliance should decrease because of collagen

collagen

stretch and recruit more wavy fibers than once all are recruited, straighten and increase stiffness.

tubular biomechanics

• Po (ext. pressure) = 0

• Pi (int. pressure) > 0

• Incompressible

• Axisymmetric

• Linearly elastic material

• No axial variations in stress

• No shear stress

axisymmetric

same stress no matter where we go around tube → things constant with length (no axial variations)

an object, system, or physical field has symmetry around a central axis, meaning its properties do not change when rotated around that axis by any angle.

incompressibility

constant volume.

a physical property where a material's volume and density remain constant even when subjected to external pressure changes or forces

If length is constant, area is also constant

criteria for thick wall

t/R < 0.05 → Thin wall → Law of Laplace

t/R > 0.05 → Thick wall → Lamé Solutions

Classic Lamé Solutions

In solid mechanics, Lamé's solutions apply to axisymmetric boundary value problems (such as thick-walled cylinders or spheres under internal and external pressure, derived using Lamé's stress potentials and the Lamé elastic constants λ and μ)

solution to Lame equations

tangential stress distribution

in a thick-walled cylinder varies non-uniformly across the wall thickness according to Lamé's equations

Unlike thin-walled vessels where hoop stress is assumed uniform, tangential stress in thick-walled cylinders changes significantly as the radius r goes from the inner wall to the outer wall.

radial stress distribution

across a thick-walled cylinder is non-uniform and is determined using the Lamé Equations

• Inner Surface: If there is only internal pressure and zero gauge external pressure, the radial stress equals the negative of the internal pressure, acting as a compressive stress.

• Outer Surface: If external gauge pressure is zero, the radial stress drops to 0.

• Variation Across Wall: Unlike thin-walled cylinders where radial stress is assumed negligible, thick-walled cylinders experience a continuous, non-uniform variation of radial (compressive) and hoop/tangential stresses from the inner wall to the outer wall.

Lame vs Laplace

Lamé's equations apply to thick-walled solid tubes under pressure, while Laplace's law applies to thin-walled membranes, biological vessels, and capillary fluid interfaces. Both equations calculate how internal pressure relates to stress or tension in a cylindrical shape, but they solve fundamentally different physics and engineering problems.

aneurysm

using thick walled equations is good for predicting ________ rupture.

a localized, weakened bulge or "ballooning" in the wall of an artery (blood vessel that carries oxygen-rich blood from the heart to the rest of the body).

bubble inflation testing

incremental elastic modulus

Here, Δb is the change in outer radius that arises from a change in internal pressure of ΔP

the instantaneous slope of the stress-strain curve at a specific point of deformation

Unlike a standard or secant modulus (which measures overall stiffness from the origin or between distant points), this measures how stiffness changes with small additions of stress and strain during nonlinear, hyperelastic, or plastic deformation.

Stress-Strain Curves

nonlinear increase - different for each region. Proximal stretches most but achieves highest stresses.

a graphical representation in materials science and engineering that plots how much a material deforms (strain, on the x-axis) in response to an applied force intensity (stress, on the y-axis)

urethral isolation

mechanical testing

biomechanical testing

Preconditioning to remove viscoelastic effects

Testing consists of increasing intralumenal pressures from 0-20 mmHg in 2 mmHg steps

preconditioning

the process of treating, conditioning, or adjusting a raw substrate (such as polymers, concrete, food/feed blends, or thermal payloads) prior to a primary manufacturing, testing, or chemical procedure to ensure uniform processing and optimal performance

here, to remove viscoelastic effects

deformation due to extension

occurs when a stretching or tensional force is applied to an object, increasing its length.

• Hooke's Law (F = kΔ L): Within the elastic limit, the extension (Δ L) of a material is directly proportional to the applied stretching force (F), governed by the material's spring constant or stiffness (k). For LEHI materials. Axial stress is constant, applied uniformly over area.

discontinuities

determine internal forces for integrals for each section between these.

a break, jump, or non-uniformity in how a material or structure changes shape, where displacement, velocity, or the deformation gradient is not smooth or continuous across a boundary or interface

→ an applied force or moment; change in area; change in elastic modulus; etc

general solution process for deformation problems

determine all external forces applied to structure. determine all internal/reactionary forces by drawing FBDs

examples of discontinuities

1. change in area

2. change in material/elastic modulus

3. change in force

when there is one, change integrals - separate integrals at point of discontinuity

making cut

do it in the middle of a section, not directly at the discontinuity

Diaphysis

the main or central tubular shaft of a long bone (such as the femur, tibia, or humerus)

Made primarily of dense, hard cortical (compact) bone that provides high structural strength and weight-bearing support

Epiphysis

The rounded end or tip of a long bone that ossifies from a secondary center of ossification.

Composed mostly of spongy (cancellous) bone surrounded by a thin layer of compact bone, and covered at the joint surface with articular cartilage.

Metaphysis

the wide, neck-like portion of a long bone located between the central shaft (diaphysis) and the end region (epiphysis).

During childhood and adolescence, contains the epiphyseal plate (growth plate), a layer of hyaline cartilage that allows the bone to lengthen.

cortical (compact) bone

The dense, hard outer layer that forms roughly 80% of the total skeletal mass in the human body

Provides rigidity, stiffness, and high resistance to bending and torsion, allowing the skeleton to support body weight and muscle movement.

cancellous (trabecular) bone

a porous, honeycomb-like type of osseous tissue that makes up about 20% of the human skeleton

It provides structural integrity, absorbs and dampens sudden mechanical shocks (such as walking, running, or jumping), and reduces overall skeletal weight.

Wolff’s Law of bone remodeling:

collagen fibers align in direction of loading (and subsequent displacement of cells)

artificial hips

clinical relevance - 120,000 implanted annually in the US.

Femoral head subjected to compression, torsion, and bending. Impacts bone remodeling – need to consider loads in implants, transfer of load

assumptions for bone modeling problems

Model problem: axial loading of circular cylinder consisting of two LEHI materials, find forces in both

• Bone typically non-homogeneous and anisotropic

• Geometry: diaphysis region not perfect cylinder

• Unidirectional loading

statically indeterminate

when the number of unknown reaction forces or internal stresses exceeds the available equations of static equilibrium (sum F = 0) and (sum M = 0).

forces in prosthesis design

Don’t want prothesis to break → want to be really strong. Metal really stiff compared to bone. Metal carries all the load. Bad because the bone will atrophy due to disuse. Metal prosthesis designed for only slightly higher stiffness than bone.

torsion analysis

Different for circular, rectangular, etc.

Circular cross-section – twisting does not “warp” the cross-section. Not true for rectangular, elliptical, etc .Focus on circular cross-section, LEHI

torsion

the twisting of a structural member or shaft caused by an applied moment or torque (T) acting about its longitudinal axis

formulation of shear stress due to torsion

• Solid circular cylinder

• Fixed on one end, free on other

• Free end subjected to twisting moment T = M_z

• Equilibrium: Equal and opposite reaction torque, Internal torques

Shear stress due to Torsion

σ_zθ = stress on z face in θ direction

• Acts on cut face to balance applied torque T

• Acts over respective area

the internal shearing stress created in a structural member, such as a cylindrical shaft, when an external twisting force (torque) is applied

function of radius

shear stress is not constant

σ_zθ not constant: need to know σzθ as a function of radius in order to solve

Recall that for axially loaded rods, we avoided this by assuming σ_xx constant over the cross section

gamma variable

γ = Angle of circumferential motion of all material particles along a line on the cylinder

γ = γc for a line along outer radius of cylinder (r=c)

torsion equation

Only holds for small-strain LEHI behavior

Shear stress varies linearly with radial position

Failure – of LEHI material, most likely at outer circumference

polar moment of inertia (J)

measures an object's resistance to twisting (torsional deformation) around a longitudinal axis

heart

biological application of torsion.

Heart muscle twists upon ejection of blood. Unique arrangement of cardiac muscle fibers

implants

biological application of torsion.

Hip: Femur and Acetabulum

Transferred load not along implant shaft. Bending and twisting moments

different coordinate system

Recall: relate components of stress in a Coordinate

System to those in another CS

• Rewrite in Cylindrical / Polar Coordinates x → z, y →θ

General Pure Torsion: σ_zz, σ_θθ = 0

Principal Stresses and Strains in Torsion

principle stresses in torsion

Under pure torsion of a circular shaft, a small surface element experiences only shear stress Txy = Tr/J, where T is torque, r is radius, and J is the polar moment of inertia) with zero normal stress on the longitudinal and transverse planes.

Principal Stresses (σ₁, σ₂)

Using Mohr’s circle for pure torsion, the center of the circle is at the origin (0,0) and the radius equals the maximum shear stress \(\tau_{max} = \frac{Tr}{J}\).

• Major (tensile) principal stress (σ₁): σ₁ = +τ_max= +Tr/J

• Minor (compressive) principal stress (σ₂): σ₂ = -τ_max= -Tr/J

• Orientation: These principal stresses act on planes oriented at 45° (rotated by 90° on Mohr’s circle) to the longitudinal axis of the shaft. On these 45° planes, the normal stresses reach their maximum absolute values, and the shear stress is zero.

principle strains in torsion

Relating stress to strain via Hooke's law for plane stress (with shear modulus G and Poisson's ratio ν):

• Principal strains: ε1, ε2 = τ_max/2G}(1 + ν)\ = ± γ_max/2

• Strain gauges placed at 45° to the longitudinal axis align with these principal directions to directly measure principal tensile and compressive strains.

Maximum Angle of Twist in Torsion

For a circular shaft under torsion is calculated using the formula Angle of Twist Interactive Calculator

Φ = TL/JG

• T: Applied internal torque (N m(

• L: Length of the shaft segment (m)

• J: Polar moment of inertia (m⁴)

• G: Modulus of rigidity (shear modulus) of the material (Pa)