## Balance Equations: Mass Balance

Let and represent the reference and the deformed configurations, respectively, of a body embedded in . Let represent the image of at a certain point in time under the bijective smooth mapping . Notice that the set which is the range of changes as time changes. The velocity vector is described by the vector valued function . and are the deformation and the velocity gradients, respectively, while . The density distribution of the continuum body in the reference configuration is given by the function . As the body deforms, the density at each material point changes with time and thus can be described by a function of both time and material points . The spatial density distribution of the continuum body in the deformed configuration is a function of the position and of time and is described by the function . The difference between and is merely a change of coordinates from to :

In this section, we will use the physical law of mass balance to establish the relationship between the mass densities , and . If and represent an infinitesimal volume in the deformed and reference configurations, respectively, then, the relationship between the volume in the deformed versus the reference configurations is given by:

If is the actual mass of both infinitesimal volumes and assuming that **mass is preserved** then the relationship between the densities is given by:

This relationship can also be obtained using integrals over the whole continuum body as follows. The total mass of the reference configuration can be calculated by the integral:

The total mass in the deformed configuration can be calculated by the integral:

The volume integral over can be replaced by the volume integral over by using the relationship :

Assuming that **the total mass** is conserved, i.e., , and since is arbitrary, the relationship between the densities is retrieved:

### The Continuity Equation

The continuity equation describes the rates of change of the density functions. We first recall the relationship between the velocity gradient and obtained in the velocity gradient section:

Assuming **preservation of mass**, i.e., the rate of change of is equal to zero leads to the following equivalent forms of the continuity equation:

#### Lagrangian Formulation

The continuity equation as a function of is given by:

Alternatively, the continuity equation as a function of is given by:

Since we reach the following form of the continuity equation:

#### Eulerian Formulation

The continuity equation as a function of is given by:

Since we reach the following form of the continuity equation:

where, according to the results in the vector calculus section:

#### Continuity Equation of Fluids

In fluids, the Eulerian formulation is adopted along with the approximation that the fluid is incompressible, i.e., and . Therefore, the continuity equation reduces to:

#### Continuity Equation for Growth or Mass Transfer

In problems where there is growth or mass transfer, the density in the reference configuration is assumed to vary in time and its rate of change is equal to the source or the sink of the local mass change and the continuity equation is then derived from the equation:

#### Example

The position function of a 2units by 2units plate has the form:

where units. The density of the material in the reference configuration at is given by the function:

The following tool draws the contour plots of and in the reference configuration and in the deformed configuration as a function of time. Vary the time to see the effect on the deformation and the effect on the density plots. Note that does not change when time evolves. The computations are based on first calculating the deformation gradient as follows:

thus, the determinant of is given by:

The density as time evolves is equal to:

Finding requires solving a nonlinear set of equations to express in terms of .

#### Problem

The two dimensional displacement field of a 2 units by 3 units plate is given by:

If the density in the reference configuration is given by the function units of density, find the expressions for and . For , draw the contour plots of , , and .

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