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continuum mechanics

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definition of continuum mechanics
is a branch of mechanics that deals with the analysis of the kinematic and mechanical behavior of materials modeled as a continuum, e.g., solids and fluids (i.e., liquids and gases). A continuum is a body that can be continually sub-divided into infinitesimal small elements with properties being those of the bulk material.
What are the assumptions of linear elasticity?
"small" deformations (or strains) and linear relationships between the components of stress and strain
What is the Jacobian determinant a measure of?
The ratio of the volume in the deformed configuration to the volume in the reference configuration
Give a physical definition of an elastic materials
One that stores energy under stress but does not disspate energy then returns to its original shape when the stress is removed
What is meant by a second order tensor? How many entities/components? How about a 3rd order tensor?
It's by doing a tensor product of two vectors which gives 3^2 (3 as in 3D space, 2 independent indexes hence second order?) = 9 entities
Give a mathematical definition of an elastic material
If we can define a strain energy or elastic potential function for a material, and when differentiated that strain energy function defines stress in the material
What is meant by hyperelastic?
Nonlinear elastic material where the stiffness of the material changes under deformation
Is the Cauchy stress tensor defined in the undeformed state or deformed state?
deformed, hence energetically conjugate to Almansi-Green strain tensor
Definition of strain
the geometrical expression of deformation
What is meant by incompressible material
The volume of the tissue will not change during deformation, no matter how high the hydrostatic pressure
What are the eigenvalues and eigenvectors of stress/strain tensors are also known as ....?
Principal values and principal directions
A transformation matrix is provided in a 3 x 3 axis. If the transformation is a, what does the element a(sub12) indicates?
Given two distinct coordinate systems, the second CS can be defined given the cosine value (relationship) between every axis of the original CS and every axis of the new CS. Given the 2 CS has 3 coordinates, the transformation matrix (a) should have a size of 3 x 3, where cos (a(sub12) is the cosine value between axis 1 of original CS and axis 2 of new CS
constitutive equation
a mathematical model used to describe the relationship between stress and deformation
How is strain express in terms of length?
Change in length over length (undeformed/deformed)
Why use small strain tensor in linear elastic model?
The linear elastic model assumes that the material experience only small deformation
Given 3 mutually orthogonal UNIT vectors (e1, e2, e3), e1 . e1 = ?, e1 . e2 = ?
e1 . e1 = 1 (assume e1 is (1, 0, 0) e1 . e1 = 1i x 1i + 0j x 0j + 0k x 0k = 1. Whereas, if e2 = (0,1,0), e1 . e2 = 1i x 0i + 0j x 1j + 0k x 0k = 0.
What is the Hydrostatic pressure?
The resisting pressure
What is meant by the invariants/eigenvalues of tensors?
They have the same value in every coordinate system.
Consider a cube in 3D space with a traction force vector on each side, ...., then the vectors become the nine components of a second order stress tensor
When the cube shrunk to an infinitesimal point
Why is the linear elastic model not a suitable model for soft tissues
Most soft tissues undergo strains that qualify as large deformation. The stiffness of a soft tissue will change with deformation, unlike a linear elastic model where the stiffness is constant as long as the material is in the elastic range.
Given c indicates a component in a matrix, if there are 2 independent indices in 3 dimensional space. What is e(sub ij) representing and how many independent entities are there?
e(ij) = e11 + e12 + e13 + e21 + e22 ...... 3 ^ 2 = 9 entities
What is meant by Isotropy
The quality of a property which does not depend on the direction
What is the degree of freedom?
the number of spatial coordinates required to completely describe the motion of a system.
If A is a 3 x 3 matrix, tr (A) = ?
A11 + A22 + A33
What is the dyadic product/tensor product of u, v? What is the notation of tensor product? If C(matrix) is the tensor product, what is the value of c11 (using components of u and v)?
The product of the first vector and the transpose of the second vector.
What are the components of a strain energy function for incompressibility?
The strain energy function dependent on the deformation gradient tensor and the hydrostatic pressure (Lagrange multiplier) multiply by the incompressibility condition (J - 1)
The strain energy/elastic potential energy function is dependent on?
Deformation gradient tensor and a constant determined experimentally

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