Advanced Biomechanics Practice exam questions

What is the fundamental hypothesis of continuum mechanics? How do we make use of this hypothesis in the derivation of the equilibrium equations in mechanics?

A hypothesis that any/all physical beings contain gaps between molecules, atoms, and within atoms. The fundamental hypothesis is that matter can be traced as a continuum, i.e., *space between atoms can be neglected at a reasonable scale*, and from this assumption, gradient fields can be used to describe the relation of materials to stimuli. Equilibrium equations are all derived using gradients of displacement/deformation.

What is the difference between tensor and a matrix?

A tensor relates 2 vectors and displays the relationship in the form of a matrix. A matrix is just a data organization system that can have arbitrary data and may have no physical implications.

What conditions are assumed in the linearization of strains?

1. Displacements are small 2. Displacement gradients are small 3. Rigid body motion is negligible.

What are the implications of the assumptions? 1. Frictionless contact between the compression plates and the material sample 2. Material is incompressible 3. The material exhibits LEHI behavior

1. Frictionless assumption is necessary to assume *symmetric expansion* of the material. Friction induces an opposing force that could change the amount of strain in certain ways. 2. Material being incompressible allows for us to assume total volume does not change, and with this the mass-density stays constant. A change in length of one direction --> change in another direction 3. Elastic- No energy is dissipated, all applied work Lienar- material has an intrinsic property E that relates stress/strain Homogeneous/ isotropic - Applied force field effects every point in the material in the same way.

Describe a simple experiment that allows one to draw the conclusion that a given material behaves as an elastic solid.

A uniaxial test to find the relationship between stress and strain. If loading and unloading paths of the material are equal, or the *energy applied is not somehow being dissipated*, the material can be concluded as elastic.

Why is the deformation gradient considered a fundamental measure of strain?

As shown by the graphs, deformation gradient is necessary to explain strain because as material feels constant stress, the strain verses with position. Since a constant stress should make a constant strain, the gradient is used. Because stress is constant, strain should be constant, but it's NOT. So the deformation gradient shows the varying in stress and strain.

Prior to bi-axial experimental investigations that aim at identifying the constitutive equations of soft tissue, we can perform preliminary uni-axial tests on strips or rings to get general information about the mechanical response. What type of information can such preliminary tests provide?

Can assess if it's elastic, homogenous, isotropic. Can NOT identify strain-energy function

What is the primary reason to develop constitutive models in mechanics?

Constitutive models are used because they allow findings to be compared without having to specify geometry. The primary reason to develop these models is to represent data that has been measured, and then use the representation to extrapolate data that has not been/cannot be measured.

A solid is made of an elastic and incompressible material. How do these characteristics affect the components of the Green strain tensor.

Green strain tensor E= (1/2) ( C-I ) Incompressible means III = λ1λ2λ3 = 1 A material's stretch ratios must follow this for the material to be incompressible. Meaning the strain lateral/ strain axial must follow these ratios.

What does it mean if an elastic material is isotropic? Is this valid assumption for vascular tissue? In this case, the strain energy function is fundamentally a function of which quantities?

If an elastic material is isotropic, the material response is independent of orientation. Not valid for vascular tissue due to the orthotropic nature of the tissue - dependent on axial, radial, lateral. W= W^ (I, II, III)

An incompressible elastic solid undergoes finite non-homogenous deformation. Is it possible that all stretch ratios along principle axes (i.e. the principle stretch ratios) are greater than 1?

No it is not possible. λ > 1 *implies an elongation or stretch*, and thus a gain in length. An incompressible solid has an invariance III = 1 and thus, λ1λ2λ3 = 1 which are the primary stretch ratios.

Do the dimensions of stress and strain necessarily agree? For example, does a 1D state of stress imply a 1D state of strain? Provide an example.

No, a 1D uniaxial force can cause a 1D stress but a 3D strain (like axial, radial, and circumferential directions). Hydrostatic pressure is an example.

What factors govern the passive mechanical properties of soft biological tissue? What factors govern the active mechanical properties of soft biological tissue?

Passive- Elastin and collagen content of tissue respond to load, governing the response. Active- Smooth muscle and endothelial cells respond to a load and stimulation, constricting or dilating as necessary.

What are the 3 general types of constitutive models in vascular mechanics? What are the analytical form of each type?

Phenomenological - This approach uses equations that are based on data derived from experimentations. Structure-based model- An analytical form is motivated by the composition of the material. Structure-motivated model- Means the analytical form is motivated by the structure of the tissue.

What are the physical meanings components of the Cauchy stress tensor?

Principal positions show normal stresses, or forces that acts in the direction of the force Other positions show shear stresses.

A strip of constant cross-sectional area is made of a homogenous material and is subjected to uniaxial extension. Does the strain energy function have the same value along the strip? What type of quantity does the strain energy function have the same value along the strip? What type of quantity does the strain energy function return - scalar, vector, or tensor?

Strain energy function returns a scalar value, energy (J). Strain energy function will not necessarily have the same value along the strip. For the value to remain constant along the strip, the material must be also *elastic and isotropic*. Making W = W (I, II, III) and not a function of position.

Can stress be measured directly?

Stress can not be measured directly because stress itself is mathematical construct used to normalize the physical quantity of force, and thus must be calculated for a specific point by measuring force and area.

What type of mechanical response must a material manifest to justify the existence of a strain energy function? How can the degree to which the tissue exhibits this response be quantified?

The material must manifest an elastic response. The degree of elasticity can be quantified by being how much energy is dissipated. Less energy dissipated into heat or other forms of energy means more elastic response.

How many total components are there in the Cauchy stress tensor? How many of these components are independent? What is the source of dependence among some components?

There are 9 components There are 6 independent components Source of dependence is the common assumption that the material is in *equilibrium* with EF=0 EM=0

Two arteries have different compliance at identical pressures and longitudinal stretch ratios. Does this necessarily imply that the arteries are composed of different material?

This does not necessarily imply that they are composed of different materials become compliance is a function of material properties and geometry. If the arteries are different sizes they could have different compliances with the same material makeup.