CONTINUUM MECHANICS

It is often appropriate to model living tissues as continuous media. For example, at the tissue level, the arterial wall can be modeled as a continuum. This assumption breaks down when the length scales of interest approach the order of the micro structural details of the material.

The basic postulates of continuum mechanics are :
1.) Conservation of linear and angular momentum,
2.) Conservation of mass, conservation of energy, and
3.) Entropy inequality.
Solids are usually modeled using "reference" or "Lagrangian" coordinates, whereas fluids are often modeled using "spatial" or "Eulerian" coordinates. Using these postulates and some assumptions regarding the particular problem at hand, a set of equilibrium equations can be established. The kinematics and constitutive relations are also needed to model a continuum.

Second and fourth order tensors are crucial in representing many quantities in electromechanical. In practice, however, the full tensor form of a fourth-order constitutive matrix is rarely used. Instead, simplifications such as isotropy, transverse isotropy, and incompressibility reduce the number of independent components. Commonly-used second-order tensors include the Cauchy stress tensor, the second Viola-Kirchhoff stress tensor, the deformation gradient tensor, and the Green strain tensor. A reader of the mechanic's literature would be well-advised to note precisely the definitions of the various tensors which are being used in a particular work.

DEFINITION:
1.)Continuum mechanics
Continuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter, including both solids and fluids (i.e., liquids and gases).

2.)Solid Mechanics
Solid mechanics is the study of the physics of continuous solids with a defined rest shape.

3.) Fluid Mechanics
Fluid mechanics (including Fluid statics and Fluid dynamics) deals with the physics of fluids. An important property of fluids is viscosity, which is the force generated by a fluid in response to a velocity gradient.

4.) Length Scale
Length scale is a particular length or distance determined with the precision of one order (or a few orders) of magnitude.

5.)Linear Momentum
Momentum (SI unit kg m/s) is the product of the mass and velocity of an object.The law of conservation of momentum is a fundamental law of nature, and it states that the total momentum of a closed system of objects (which has no interactions with external agents) is constant.

6.)Conservation of Mass
The law of conservation of mass/matter, also known as law of mass/matter conservation (or the Lomonosov-Lavoisier law), states that the mass of a closed system of substances will remain constant, regardless of the processes acting inside the system.

7.)Conservation of Energy
The conservation of energy states that the total amount of energy in an isolated system remains constant, although it may change forms, e.g. friction turns kinetic energy into thermal energy. In thermodynamics, the first law of thermodynamics is a statement of the conservation of energy for thermodynamic systems, and is the more encompassing version of the conservation of energy. In short, the law of conservation of energy states that energy can not be created or destroyed, it can only be changed from one form to another, such as when electrical energy is changed into heat energy.

8.)Entropy
Entropy is a measure of the uniformity of the distribution of energy.

9.)Lagrangian and Eulerian coordinates
In fluid dynamics and finite-deformation plasticity the Lagrangian reference frame is a way of looking at fluid motion where the observer follows individual fluid particles as they move through space and time. Plotting the position of an individual particle through time gives the pathline of the particle. This can be visualized by sitting in a boat drifting down a river.

The Eulerian reference frame is a way of looking at fluid motion that focuses on specific points in the space through which the fluid moves. This can be visualized by sitting on the bank of a river and watching the water pass your location. Values about the fluid flow are determined as vectors at discrete locations.

They are related by the Convective derivative or Lagrangian derivative (sometimes called the material derivative):


This tell us the rate of change of F whilst moving with the fluid at velocity u.

10.)Kinematics
Kinematics is a branch of mechanics which describes the motion of objects without the consideration of the masses or forces that bring about the motion. By contrast, dynamics is concerned with the forces and interactions that produce or affect the motion.

Kinematics studies how the position of an object changes with time. Position is measured with respect to a set of coordinates. Velocity is the rate of change of position. Acceleration is the rate of change of velocity. Velocity and Acceleration are the two principal quantities which describe how position changes.

The simplest application of kinematics is to point particle motion (translational kinematics or linear kinematics). The description of rotation (rotational kinematics or angular kinematics) is more complicated. The state of a generic rigid body may be described by combining both translational and rotational kinematics (rigid-body kinematics). A more complicated case is the kinematics of a system of rigid bodies, possibly linked together by mechanical joints. The kinematic description of fluid flow is even more complicated, and not generally thought of in the context of kinematics.