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Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modeled as a continuum. The French mathematician Augustin Louis Cauchy was the first to formulate such models in the 19th century, but research in the area continues today.
Modeling an object as a continuum assumes that the substance of the object completely fills the space it occupies. Modeling objects in this way ignores the fact that matter is made of atoms, and so is not continuous; however, on length scales much greater than that of inter-atomic distances, such models are highly accurate. Fundamental physical laws such as the conservation of mass, the conservation of momentum, and the conservation of energy may be applied to such models to derive differential equations describing the behavior of such objects, and some information about the particular material studied is added through a constitutive relation.
Continuum mechanics deals with physical properties of solids and fluids which are independent of any particular coordinate system in which they are observed. These physical properties are then represented by tensors, which are mathematical objects that have the required property of being independent of coordinate system. These tensors can be expressed in coordinate systems for computational convenience.
Materials, such as solids, liquids and gases, are composed of molecules separated by empty space. On a macroscopic scale, materials have cracks and discontinuities. However, certain physical phenomena can be modeled assuming the materials exist as a continuum, meaning the matter in the body is continuously distributed and fills the entire region of space it occupies. A continuum is a body that can be continually sub-divided into infinitesimal elements with properties being those of the bulk material.
The validity of continuum assumption may be verified by a theoretical analysis, in which either some clear periodicity is identified or statistical homogeneity and ergodicity of the microstructure exists. More specifically, the continuum hypothesis/assumption hinges on the concepts of a Representative Volume Element (RVE) and 'separation of scales' based on the Hill-Mandel condition. [Sometimes, in place of RVE, the term Representative Elementary Volume (REV) is used.] This condition provides a link between an experimentalist's and a theoretician's viewpoint on constitutive equations (linear and nonlinear elastic/inelastic or coupled fields) as well as a way of spatial and statistical averaging of the microstructure. When the separation of scales does not hold, or when one wants to establish a continuum of a finer resolution than that of the RVE size, one employs a Statistical Volume Element (SVE), which, in turn, leads to random continuum fields. The latter then provide a micromechanics basis for stochastic finite elements (SFE). The levels of SVE and RVE link continuum mechanics to statistical mechanics. The RVE may be assessed only in a limited way via experimental testing: when the constitutive response becomes spatially homogeneous.
Continuum mechanics models begin by assigning a region in three dimensional Euclidean space to the material body modeled. The points within this region are called particles or material points. Different configurations or states of the body correspond to different regions in Euclidean space. The region corresponding to the body's configuration at time is labeled .
A particular particle within the body in a particular configuration is characterized by a position vector
where are the coordinate vectors in some frame of reference chosen for the problem (See figure 1). This vector can be expressed as a function of the particle position in some reference configuration, for example the configuration at the initial time, so that
This function needs to have various properties so that the model makes physical sense. needs to be:
For the mathematical formulation of the model, is also assumed to be twice continuously differentiable, so that differential equations describing the motion may be formulated.
A change in the configuration of a continuum body results in a displacement. The displacement of a body has two components: a rigid-body displacement and a deformation. A rigid-body displacement consist of a simultaneous translation and rotation of the body without changing its shape or size. Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration to a current or deformed configuration (Figure 2).
The motion of a continuum body is a continuous time sequence of displacements. Thus, the material body will occupy different configurations at different times so that a particle occupies a series of points in space which describe a pathline.
There is continuity during deformation or motion of a continuum body in the sense that:
It is convenient to identify a reference configuration or initial condition which all subsequent configurations are referenced from. The reference configuration need not be one that the body will ever occupy. Often, the configuration at is considered the reference configuration , . The components of the position vector of a particle, taken with respect to the reference configuration, are called the material or reference coordinates.
When analyzing the deformation or motion of solids, or the flow of fluids, it is necessary to describe the sequence or evolution of configurations throughout time. One description for motion is made in terms of the material or referential coordinates, called material description or Lagrangian description.
In the Lagrangian description the position and physical properties of the particles are described in terms of the material or referential coordinates and time. In this case the reference configuration is the configuration at . An observer standing in the referential frame of reference observes the changes in the position and physical properties as the material body moves in space as time progresses. The results obtained are independent of the choice of initial time and reference configuration, . This description is normally used in solid mechanics.
In the Lagrangian description, the motion of a continuum body is expressed by the mapping function (Figure 2),
which is a mapping of the initial configuration onto the current configuration , giving a geometrical correspondence between them, i.e. giving the position vector that a particle , with a position vector in the undeformed or reference configuration , will occupy in the current or deformed configuration at time . The components are called the spatial coordinates.
Physical and kinematic properties , i.e. thermodynamic properties and velocity, which describe or characterize features of the material body, are expressed as continuous functions of position and time, i.e. .
The material derivative of any property of a continuum, which may be a scalar, vector, or tensor, is the time rate of change of that property for a specific group of particles of the moving continuum body. The material derivative is also known as the substantial derivative, or comoving derivative, or convective derivative. It can be thought as the rate at which the property changes when measured by an observer traveling with that group of particles.
In the Lagrangian description, the material derivative of is simply the partial derivative with respect to time, and the position vector is held constant as it does not change with time. Thus, we have
The instantaneous position is a property of a particle, and its material derivative is the instantaneous velocity of the particle. Therefore, the velocity field of the continuum is given by
Similarly, the acceleration field is given by
Continuity in the Lagrangian description is expressed by the spatial and temporal continuity of the mapping from the reference configuration to the current configuration of the material points. All physical quantities characterizing the continuum are described this way. In this sense, the function and are single-valued and continuous, with continuous derivatives with respect to space and time to whatever order is required, usually to the second or third.
Continuity allows for the inverse of to trace backwards where the particle currently located at was located in the initial or referenced configuration. In this case the description of motion is made in terms of the spatial coordinates, in which case is called the spatial description or Eulerian description, i.e. the current configuration is taken as the reference configuration.
The Eulerian description, introduced by d'Alembert, focuses on the current configuration , giving attention to what is occurring at a fixed point in space as time progresses, instead of giving attention to individual particles as they move through space and time. This approach is conveniently applied in the study of fluid flow where the kinematic property of greatest interest is the rate at which change is taking place rather than the shape of the body of fluid at a reference time.
Mathematically, the motion of a continuum using the Eulerian description is expressed by the mapping function
which provides a tracing of the particle which now occupies the position in the current configuration to its original position in the initial configuration .
A necessary and sufficient condition for this inverse function to exist is that the determinant of the Jacobian should be different from zero. Thus,
In the Eulerian description, the physical properties are expressed as
where the functional form of in the Lagrangian description is not the same as the form of in the Eulerian description.
The material derivative of , using the chain rule, is then
The first term on the right-hand side of this equation gives the local rate of change of the property occurring at position . The second term of the right-hand side is the convective rate of change and expresses the contribution of the particle changing position in space (motion).
Continuity in the Eulerian description is expressed by the spatial and temporal continuity and continuous differentiability of the velocity field. All physical quantities are defined this way at each instant of time, in the current configuration, as a function of the vector position .
The vector joining the positions of a particle in the undeformed configuration and deformed configuration is called the displacement vector , in the Lagrangian description, or , in the Eulerian description.
A displacement field is a vector field of all displacement vectors for all particles in the body, which relates the deformed configuration with the undeformed configuration. It is convenient to do the analysis of deformation or motion of a continuum body in terms of the displacement field, In general, the displacement field is expressed in terms of the material coordinates as
or in terms of the spatial coordinates as
where are the direction cosines between the material and spatial coordinate systems with unit vectors and , respectively. Thus
and the relationship between and is then given by
It is common to superimpose the coordinate systems for the undeformed and deformed configurations, which results in , and the direction cosines become Kronecker deltas, i.e.
Thus, we have
or in terms of the spatial coordinates as
Continuum mechanics deals with the behavior of materials that can be approximated as continuous for certain length and time scales. The equations that govern the mechanics of such materials include the balance laws for mass, momentum, and energy. Kinematic relations and constitutive equations are needed to complete the system of governing equations. Physical restrictions on the form of the constitutive relations can be applied by requiring that the second law of thermodynamics be satisfied under all conditions. In the continuum mechanics of solids, the second law of thermodynamics is satisfied if the Clausius–Duhem form of the entropy inequality is satisfied.
The balance laws express the idea that the rate of change of a quantity (mass, momentum, energy) in a volume must arise from three causes:
Let Ω be the body (an open subset of Euclidean space) and let be its surface (the boundary of Ω).
Let the motion of material points in the body be described by the map
where is the position of a point in the initial configuration and is the location of the same point in the deformed configuration.
The deformation gradient is given by
Let be a physical quantity that is flowing through the body. Let be sources on the surface of the body and let be sources inside the body. Let be the outward unit normal to the surface . Let be the velocity of the physical particles that carry the physical quantity that is flowing. Also, let the speed at which the bounding surface is moving be un (in the direction ).
Then, balance laws can be expressed in the general form
Note that the functions , , and can be scalar valued, vector valued, or tensor valued - depending on the physical quantity that the balance equation deals with. If there are internal boundaries in the body, jump discontinuities also need to be specified in the balance laws.
If we take the Lagrangian point of view, it can be shown that the balance laws of mass, momentum, and energy for a solid can be written as
In the above equations is the mass density (current), is the material time derivative of ρ, is the particle velocity, is the material time derivative of , is the Cauchy stress tensor, is the body force density, is the internal energy per unit mass, is the material time derivative of e, is the heat flux vector, and is an energy source per unit mass.
With respect to the reference configuration, the balance laws can be written as
In the above, is the first Piola-Kirchhoff stress tensor, and ρ0 is the mass density in the reference configuration. The first Piola-Kirchhoff stress tensor is related to the Cauchy stress tensor by
We can alternatively define the nominal stress tensor which is the transpose of the first Piola-Kirchhoff stress tensor such that
Then the balance laws become
The operators in the above equations are defined as such that
where is a vector field, is a second-order tensor field, and are the components of an orthonormal basis in the current configuration. Also,
where is a vector field, is a second-order tensor field, and are the components of an orthonormal basis in the reference configuration.
The inner product is defined as
The Clausius–Duhem inequality can be used to express the second law of thermodynamics for elastic-plastic materials. This inequality is a statement concerning the irreversibility of natural processes, especially when energy dissipation is involved.
Just like in the balance laws in the previous section, we assume that there is a flux of a quantity, a source of the quantity, and an internal density of the quantity per unit mass. The quantity of interest in this case is the entropy. Thus, we assume that there is an entropy flux, an entropy source, and an internal entropy density per unit mass (η) in the region of interest.
Let Ω be such a region and let be its boundary. Then the second law of thermodynamics states that the rate of increase of η in this region is greater than or equal to the sum of that supplied to Ω (as a flux or from internal sources) and the change of the internal entropy density due to material flowing in and out of the region.
Let move with a velocity un and let particles inside Ω have velocities . Let be the unit outward normal to the surface . Let ρ be the density of matter in the region, be the entropy flux at the surface, and r be the entropy source per unit mass. Then the entropy inequality may be written as
The scalar entropy flux can be related to the vector flux at the surface by the relation . Under the assumption of incrementally isothermal conditions, we have
where is the heat flux vector, s is a energy source per unit mass, and T is the absolute temperature of a material point at at time t.
We then have the Clausius–Duhem inequality in integral form:
We can show that the entropy inequality may be written in differential form as
In terms of the Cauchy stress and the internal energy, the Clausius–Duhem inequality may be written as
|Continuum mechanics||Solid mechanics is the study of the physics of continuous solids with a defined rest shape.||Elasticity (physics) describes materials that return to their rest shape after removal of an applied force.|
|Plasticity describes materials that permanently deform (change their rest shape) after a large enough applied force.||Rheology: Given that some materials are viscoelastic (exhibiting a combination of elastic and viscous properties), the boundary between solid mechanics and fluid mechanics is blurry.|
|Fluid mechanics (including Fluid statics and Fluid dynamics) deals with the physics of fluids. An important property of fluids is viscosity, which is a fluid's resistance to a shear stress.||Non-Newtonian fluids|
On recognizing of specialists separate sections of university courses of Continuum Mechanics require revising. Many works of scientific institutes are subject to revising also.
3. Alexandr Kozachok.(2009). Paradoxes of Continuum Mechanics and contiguous fields of knowledge, Section 1. New approaches to statements and solutions of some classical Mathematical Physics problems. The Manual for Universities. VDM Verlag, ISBN-13: 978-3639135954. http://www.amazon.ca/PARADOXES-contiguous-approaches-statements-Mathematical/dp/3639135954/ref=sr_1_1?ie=UTF8&s=books&qid=1258992500&sr=1-1
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It is important for both solids and fluids (i.e., liquids and gases). The fact that all matter is made of atoms and that it commonly has some sort of heterogeneous microstructure is ignored in the simplifying approximation used in continuum mechanics.