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# Continuum mechanics

Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and mechanical behavior of materials modeled as a continuum, e.g., solids and fluids (i.e., liquids and gases). A continuum concept assumes that the substance of the body is distributed throughout — and completely fills — the space it occupies.

The continuum concept ignores the fact that matter is made of atoms, is not continuous, and that it commonly has some sort of heterogeneous microstructure, allowing the approximation of physical quantities, such as energy and momentum, at the infinitesimal limit. Differential equations can thus be employed in solving problems in continuum mechanics. Some of these differential equations are specific to the materials being investigated and are called constitutive equations, while others capture fundamental physical laws, such as conservation of mass (continuity equation), the conservation of momentum (equations of motion and equilibrium), and energy (first law of thermodynamics).

Continuum mechanics deals with physical quantities of solids and fluids which are independent of any particular coordinate system in which they are observed. These physical quantities are then represented by tensors, which are mathematical objects that are independent of coordinate system. These tensors can be expressed in coordinate systems for computational convenience.

In fluids, the Knudsen number is used to assess to what extent the approximation of continuity can be made.

## The continuum concept

Materials, such as solids, liquids and gases, are composed of molecules separated by empty space. Additionally, in a macroscopic scale, materials have cracks and discontinuities. However, certain physical phenomena can be modeled assuming materials as a continuum, i.e. the matter in the body is continuously distributed filling all the region of space it occupies. A continuum is a body that can be continually sub-divided into infinitesimal small elements with properties being those of the bulk material.

The concept of continuum is a macroscopic physical model, and its validity depends on the type of problem and the scale of the physical phenomena under consideration. A material may be assumed as a continuum when the distance between the real physical particles is very small compared to the dimension of the problem. For example, such is the case when analyzing the deformation behavior of soil deposits, i.e. settlement under a foundation, in soil mechanics. A given volume of soil is generally formed by discrete solid particles (grains) of minerals which are packed in a certain manner leaving voids between them, i.e. granular media. In this sense, soils defeat the definition of a continuum. However, in order to simplify the deformation analysis of the soil, the volume of soil can be assumed as a continuum knowing that the dimensions of particular grain particles are very small compared with the scale of the problem, i.e. the size of the foundation and the volume of the soil mass that is influenced by the foundation load (meters) is greater than the particular soil particles (millimeters).

The validity of the continuum assumption needs to be verified with experimental testing and measurements on the real material under consideration and under similar loading conditions.

## Mathematical modeling of a continuum

In continuum mechanics, a material body $mathcal B$ is a set of infinitesimal volumetric elements $X$, called particles or material points. A material body is expressed as a continuum by assuming that at any configuration, or geometrical state of the body, there is a region $R$ in a three dimensional euclidean space $mathcal E$ such that every point of that region is occupied by a material point $X$, i.e there is a one-to-one correspondence between material points and space points.

The configuration $kappa_t\left(mathcal B\right)$, or geometrical state of the material body $mathcal\left\{B\right\}$ at a particular time $t$ is characterized by the position vector $mathbf x =x_i mathbf e_i$ of all particles at that time with respect to an arbitrary frame of reference (Figure 1). Mathematically, this is expressed by the mapping function

$mathbf\left\{x\right\}=kappa_t\left(mathbf X\right)$

where $kappa_t\left(cdot\right)$ is a continuous function, i.e. uniquely invertible and differentiable as many times as necessary.

## Kinematics: deformation and motion

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 $kappa_0\left(mathcal B\right)$ to a current or deformed configuration $kappa_t\left(mathcal B\right)$ (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:

• The material points forming a closed curve at any instant will always form a closed curve at any subsequent time.
• The material points forming a closed surface at any instant will always form a closed surface at any subsequent time and the matter within the closed surface will always remain within.

It is convenient to identify a reference configuration or initial condition which all subsequent configurations are referenced from. The reference configuration need not to be one the body actually will ever occupy. Often, the configuration at $t=0$ is considered the reference configuration , $kappa_0 \left(mathcal B\right)$. The components $X_i$ of the position vector $mathbf X$ 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.

### 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 $t=0$. 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, $kappa_0\left(mathcal B\right)$. This description is normally used in solid mechanics.

In the Lagrangian description, the motion of a continuum body is expressed by the mapping function $chi\left(cdot\right)$ (Figure 2),

$mathbf x=chi\left(mathbf X, t\right)$

which is a mapping of the initial configuration $kappa_0\left(mathcal B\right)$ onto the current configuration $kappa_t\left(mathcal B\right)$, giving a geometrical correspondence between them, i.e. giving the position vector $mathbf\left\{x\right\}=x_imathbf e_i$ that a particle $X$, with a position vector $mathbf X$ in the undeformed or reference configuration $kappa_0\left(mathcal B\right)$, will occupy in the current or deformed configuration $kappa_t\left(mathcal B\right)$ at time $t$. The components $x_i$ are called the spatial coordinates.

Physical and kinematic properties $P_\left\{ijldots\right\}$, 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. $P_\left\{ijldots\right\}=P_\left\{ijldots\right\}\left(mathbf X,t\right)$.

The material derivative of any property $P_\left\{ijldots\right\}$ 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 know 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 $P_\left\{ijldots\right\}$ is simply the partial derivative with respect to time, and the position vector $mathbf X$ is held constant as it does not change with time. Thus, we have

$frac\left\{d\right\}\left\{dt\right\}\left[P_\left\{ijldots\right\}\left(mathbf X,t\right)\right]=frac\left\{partial\right\}\left\{partial t\right\}\left[P_\left\{ijldots\right\}\left(mathbf X,t\right)\right]$

The instantaneous position $mathbf x$ is a property of a particle, and its material derivative is the instantaneous velocity $mathbf v$ of the particle. Therefore, the velocity field of the continuum is given by

$mathbf v = mathbf dot x =frac\left\{dmathbf x\right\}\left\{dt\right\}=frac\left\{partial chi\left(mathbf X,t\right)\right\}\left\{partial t\right\}$

Similarly, the acceleration field is given by

$mathbf a= mathbf dot v = mathbf ddot x =frac\left\{d^2mathbf x\right\}\left\{dt^2\right\}=frac\left\{partial^2 chi\left(mathbf X,t\right)\right\}\left\{partial t^2\right\}$

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 $chi\left(cdot\right)$ and $P_\left\{ijldots\right\}\left(cdot\right)$ 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.

### Eulerian description

Continuity allows for the inverse of $chi\left(cdot\right)$ to trace backwards where the particle currently located at $mathbf x$ was located in the initial or referenced configuration$kappa_0\left(mathcal B\right)$. 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 $kappa_t\left(mathcal B\right)$, 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, i.e. fluid mechanics, as fluids do not have a previous deformed configuration, and there is no need to follow particular fluid particles. Instead, it is best to identify fixed spatial points and observe the changes through time of the different physical properties, e.g velocity, acceleration, and thermodynamic properties, that are taking place at that point in space as different material points of the continuum (fluid) pass through it.

Mathematically, the motion of a continuum using the Eulerian description is expressed by the mapping function

$mathbf X=chi^\left\{-1\right\}\left(mathbf x, t\right)$

which provides a tracing of the particle which now occupies the position $mathbf x$ in the current configuration $kappa_t\left(mathcal B\right)$ to its original position $mathbf X$ in the initial configuration $kappa_0\left(mathcal B\right)$.

A necessary and sufficient condition for this inverse function to exist is that the determinant of the Jacobian should be different from zero. Thus,

$J=left | frac\left\{partial chi_i\right\}\left\{partial X_J\right\} right |=left | frac\left\{partial x_i\right\}\left\{partial X_J\right\} right |neq0$

In the Eulerian description, the physical properties $P_\left\{ijldots\right\}$ are expressed as

$P_\left\{ij ldots\right\}=P_\left\{ijldots\right\}\left(mathbf X,t\right)=P_\left\{ijldots\right\}\left[chi^\left\{-1\right\}\left(mathbf x,t\right),t\right]=p_\left\{ijldots\right\}\left(mathbf x,t\right)$

where the functional form of $P_\left\{ij ldots\right\}$ in the Lagrangian description is not the same as the form of $p_\left\{ij ldots\right\}$ in the Eulerian description.

The material derivative of $p_\left\{ijldots\right\}\left(mathbf x,t\right)$, using the chain rule, is then

$frac\left\{d\right\}\left\{dt\right\}\left[p_\left\{ijldots\right\}\left(mathbf x,t\right)\right]=frac\left\{partial\right\}\left\{partial t\right\}\left[p_\left\{ijldots\right\}\left(mathbf x,t\right)\right]+ frac\left\{partial\right\}\left\{partial x_k\right\}\left[p_\left\{ijldots\right\}\left(mathbf x,t\right)\right]frac\left\{dx_k\right\}\left\{dt\right\}$

The first term on the right-hand side of this equation gives the local rate of change of the property $p_\left\{ijldots\right\}\left(mathbf x,t\right)$ occurring at position $mathbf x$. 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 $mathbf x$.

### Displacement Field

The vector joining the positions of a particle $P$ in the undeformed configuration and deformed configuration is called the displacement vector $mathbf u\left(mathbf X,t\right)=u_imathbf e_i$, in the Lagrangian description, or $mathbf U\left(mathbf x,t\right)=U_Jmathbf E_J$, 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

$mathbf u\left(mathbf X,t\right) = mathbf b+mathbf x\left(mathbf X,t\right) - mathbf X qquad text\left\{or\right\}qquad u_i = alpha_\left\{iJ\right\}b_J + x_i - alpha_\left\{iJ\right\}X_J$

or in terms of the spatial coordinates as

$mathbf U\left(mathbf x,t\right) = mathbf b+mathbf x - mathbf X\left(mathbf x,t\right) qquad text\left\{or\right\}qquad U_J = b_J + alpha_\left\{Ji\right\}x_i - X_J ,$

where $alpha_\left\{Ji\right\}$ are the direction cosines between the material and spatial coordinate systems with unit vectors $mathbf E_J$ and $mathbf e_i$, respectively. Thus

$mathbf E_J cdot mathbf e_i = alpha_\left\{Ji\right\}=alpha_\left\{iJ\right\}$

and the relationship between $u_i$ and $U_J$ is then given by

$u_i=alpha_\left\{iJ\right\}U_J qquad text\left\{or\right\} qquad U_J=alpha_\left\{Ji\right\}u_i$

Knowing that

$mathbf e_i = alpha_\left\{iJ\right\}mathbf E_J$
then
$mathbf u\left(mathbf X,t\right)=u_imathbf e_i=u_i\left(alpha_\left\{iJ\right\}mathbf E_J\right)=U_Jmathbf E_J=mathbf U\left(mathbf x,t\right)$

It is common to superimpose the coordinate systems for the undeformed and deformed configurations, which results in $mathbf b=0$, and the direction cosines become Kronecker deltas, i.e.

$mathbf E_J cdot mathbf e_i = delta_\left\{Ji\right\}=delta_\left\{iJ\right\}$

Thus, we have

$mathbf u\left(mathbf X,t\right) = mathbf x\left(mathbf X,t\right) - mathbf X qquad text\left\{or\right\}qquad u_i = x_i - delta_\left\{iJ\right\}X_J$

or in terms of the spatial coordinates as

$mathbf U\left(mathbf x,t\right) = mathbf x - mathbf X\left(mathbf x,t\right) qquad text\left\{or\right\}qquad U_J = delta_\left\{Ji\right\}x_i - X_J$

## Applications

 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 the force generated by a fluid in response to a velocity gradient. Non-Newtonian fluids Newtonian fluids