GATE Biomedical Engineering Biomechanics Syllabus

Differentiate normal stress from shear stress.

Normal stress $\sigma = \frac{F}{A}$ acts perpendicular to the surface (tensile or compressive). Shear stress $\tau = \frac{F}{A}$ acts parallel (tangential) to the surface.

State Hooke's law for a linear elastic material.

Within the elastic limit, stress is proportional to strain: $$\sigma = E \varepsilon$$ where $E$ is Young's modulus (modulus of elasticity).

Define Young's modulus (modulus of elasticity).

The ratio of normal stress to normal strain in the linear elastic region: $$E = \frac{\sigma}{\varepsilon}$$ It measures material stiffness; SI unit pascal.

Define shear modulus (modulus of rigidity).

The ratio of shear stress to shear strain: $$G = \frac{\tau}{\gamma}$$ where $\gamma$ is the shear strain (angular deformation).

Define Poisson's ratio.

The negative ratio of transverse (lateral) strain to axial (longitudinal) strain under uniaxial loading: $$\nu = -\frac{\varepsilon_{lateral}}{\varepsilon_{axial}}$$ For most biological/engineering materials $0 \leq \nu \leq 0.5$.

Give the relationship between $E$, $G$, and $\nu$ for an isotropic material.

$$G = \frac{E}{2(1+\nu)}$$

Distinguish elastic deformation from plastic deformation.

Elastic deformation is fully recoverable when load is removed (material returns to original shape). Plastic deformation is permanent and remains after unloading, occurring beyond the yield point.

Define the yield point (yield strength) on a stress–strain curve.

The stress at which a material begins to deform plastically; beyond it, deformation is no longer fully recoverable. It marks the end of the elastic (linear) region.

Define ultimate tensile strength.

The maximum stress a material can withstand before failure, corresponding to the highest point on the stress–strain curve.

What is toughness of a material in terms of the stress–strain curve?

The total energy absorbed per unit volume up to fracture, equal to the area under the entire stress–strain curve: $$U = \int_0^{\varepsilon_f} \sigma \, d\varepsilon$$

What is the modulus of resilience?

The strain energy absorbed per unit volume up to the elastic (yield) limit, equal to the area under the elastic portion of the stress–strain curve: $$U_r = \frac{\sigma_y^2}{2E}$$

Distinguish ductile from brittle material behavior.

Ductile materials undergo large plastic deformation before fracture (e.g., trabecular bone is more ductile). Brittle materials fracture with little plastic deformation (e.g., cortical bone behaves more brittly), showing little necking.

Compare cortical and cancellous bone by structure.

Cortical (compact) bone is dense, solid, with low porosity (about 5–10%), forming the outer shell of bones, organized into osteons (Haversian systems). Cancellous (trabecular/spongy) bone is porous (about 50–90%), made of a lattice of trabeculae, found at the ends of long bones and in vertebrae.

Compare the mechanical properties of cortical versus cancellous bone.

Cortical bone is stiffer and stronger (Young's modulus roughly $15$–$20\,\text{GPa}$), bearing high compressive/bending loads. Cancellous bone is more compliant and weaker (modulus roughly $0.1$–$2\,\text{GPa}$), absorbing energy and distributing loads near joints.

What is the functional significance of cancellous bone's porous structure?

Its trabecular lattice provides light weight with high energy absorption, distributes loads at joint ends, houses bone marrow, and aligns trabeculae along principal stress trajectories (Wolff's law) for efficient load transfer.

State Wolff's law of bone remodeling.

Bone adapts its structure to the mechanical loads placed upon it: bone is deposited where stress is high and resorbed where stress is low, so trabecular architecture aligns with principal stress directions.

What type of mechanical material is bone (in terms of directionality)?

Bone is anisotropic (specifically orthotropic) and viscoelastic — its mechanical properties depend on the direction and rate of loading; it is stronger in compression than in tension or shear.

Rank bone's strength under compression, tension, and shear.

Bone is strongest in compression, intermediate in tension, and weakest in shear: compression $>$ tension $>$ shear.

Define a viscoelastic material.

A material exhibiting both viscous (fluid-like, time/rate-dependent) and elastic (solid-like, recoverable) behavior. Its stress–strain response depends on the rate of loading and exhibits time-dependent phenomena such as creep, stress relaxation, and hysteresis.

Define creep in a viscoelastic material.

The progressive increase in strain (deformation) over time under a constant applied stress.

Define stress relaxation in a viscoelastic material.

The gradual decrease in stress over time when a viscoelastic material is held at constant strain.

What is hysteresis in viscoelastic loading?

The phenomenon in which the loading and unloading stress–strain curves differ, forming a loop whose enclosed area represents the energy dissipated (lost as heat) per cycle.

How does loading rate affect a viscoelastic material like bone?

At higher strain rates the material behaves stiffer and stronger (higher modulus and failure stress); at lower rates it is more compliant. This rate-dependence is a hallmark of viscoelasticity.

Describe the Maxwell model of viscoelasticity.

A spring (elastic, modulus $E$) and a dashpot (viscous, viscosity $\eta$) connected in series. It represents a viscoelastic fluid: it models stress relaxation well but predicts unbounded creep.

Write the constitutive (governing) equation of the Maxwell model.

$$\frac{d\varepsilon}{dt} = \frac{1}{E}\frac{d\sigma}{dt} + \frac{\sigma}{\eta}$$ The total strain rate is the sum of the spring and dashpot strain rates.

What does the Maxwell model predict for stress relaxation at constant strain?

Stress decays exponentially: $$\sigma(t) = \sigma_0 \, e^{-t/\tau}, \quad \tau = \frac{\eta}{E}$$ where $\tau$ is the relaxation time.

Describe the Kelvin–Voigt model of viscoelasticity.

A spring (modulus $E$) and a dashpot (viscosity $\eta$) connected in parallel. It represents a viscoelastic solid: it models creep well but cannot represent instantaneous elastic deformation or stress relaxation.

Write the constitutive equation of the Kelvin–Voigt model.

$$\sigma = E\varepsilon + \eta \frac{d\varepsilon}{dt}$$ Total stress is the sum of the spring (elastic) and dashpot (viscous) stresses, which share the same strain.

What does the Kelvin–Voigt model predict for creep under constant stress $\sigma_0$?

Strain rises asymptotically toward its equilibrium value: $$\varepsilon(t) = \frac{\sigma_0}{E}\left(1 - e^{-t/\tau}\right), \quad \tau = \frac{\eta}{E}$$

Contrast the Maxwell and Kelvin–Voigt models in their elements and behaviors.

Maxwell: spring and dashpot in series; models a viscoelastic fluid; captures stress relaxation but gives unbounded (unrealistic) creep. Kelvin–Voigt: spring and dashpot in parallel; models a viscoelastic solid; captures creep and recovery but no instantaneous elasticity or stress relaxation.

Define viscosity and give its SI unit.

Viscosity $\eta$ is a fluid's resistance to shear flow, defined by $\tau = \eta \frac{du}{dy}$ (Newton's law of viscosity). SI unit is pascal-second ($\text{Pa}\cdot\text{s}$); the poise (CGS) is also used.

Why is blood a non-Newtonian fluid?

Because its apparent viscosity is not constant but decreases with increasing shear rate (shear-thinning behavior), owing to red blood cell aggregation (rouleaux) at low shear and cell deformation/alignment at high shear.

What is the Fåhraeus–Lindqvist effect?

The phenomenon in which the apparent viscosity of blood decreases as the diameter of the tube/vessel decreases (in vessels roughly $10$–$300\,\mu\text{m}$), because red cells migrate to the center, leaving a cell-free plasma layer near the wall that lubricates flow.

Define hematocrit and state its effect on blood viscosity.

Hematocrit is the volume fraction of red blood cells in blood (normally about 40–45%). Blood viscosity increases markedly (nonlinearly) with rising hematocrit.

State the Hagen–Poiseuille equation for laminar flow of blood through a cylindrical vessel.

$$Q = \frac{\pi r^{4} \Delta P}{8 \eta L}$$ where $Q$ is volumetric flow rate, $r$ the radius, $\Delta P$ the pressure drop, $\eta$ the viscosity, and $L$ the vessel length.

From Poiseuille's law, how does vascular resistance depend on vessel radius?

Resistance $R = \frac{8\eta L}{\pi r^{4}}$ varies inversely with the fourth power of the radius, so small changes in radius (vasoconstriction/dilation) produce large changes in flow and resistance.

Define the Reynolds number and its significance for blood flow.

$$Re = \frac{\rho v D}{\eta}$$ a dimensionless ratio of inertial to viscous forces. Low $Re$ indicates laminar flow; flow becomes turbulent above a critical value (about 2000–2300 in tubes), which can occur at heart valves or stenoses.

What is wall shear stress on a blood vessel and why is it physiologically important?

The tangential frictional force per unit area exerted by flowing blood on the endothelium; for Poiseuille flow $\tau_w = \frac{4\eta Q}{\pi r^{3}}$. It regulates endothelial function, vasodilation (nitric oxide release), and is implicated in atherosclerosis where shear is low/oscillatory.

State the law of Laplace for a thin-walled cylindrical vessel.

$$T = P \cdot r$$ wall tension equals transmural pressure times radius (for a cylinder; wall stress $\sigma = \frac{P r}{t}$ for wall thickness $t$). It explains why large-radius/thin vessels are prone to rupture (e.g., aneurysms).

What is plug flow (blunted velocity profile) of blood in small vessels?

In narrow vessels, red cells concentrate in the central core moving as a 'plug', flattening the otherwise parabolic velocity profile, due to axial migration of cells and the cell-free plasma layer at the wall.