Chapter 5
Balance Laws

5.1 Overview

The balance laws, discussed in this chapter, has to be satisfied for all bodies, irrespective of the material that they are made up of. The fundamental balance laws that would be discussed here are conservation of mass, linear momentum, and angular momentum. However, the particular form that these equations take would depend on the processes that one is interested in. Here our interest is in what is called as purely mechanical processes and hence we are not concerned about conservation of energy or electric charge or magnetic flux. These general balance equations in themselves do not suffice to determine the deformation or motion of a body subject to given loading. Also, bodies identical in geometry but made up of different material undergo different deformation or motion when subjected to the same boundary traction. Hence, to formulate a determinate problem, it is usually necessary to specify the material which the body is made. In continuum mechanics, such specification is stated by constitutive equations, which relate, for example, the stress tensor and the stretch tensor. We shall see more about these constitutive relations in the subsequent chapter while we shall focus on balance laws in this chapter.

5.2 System

We define a system as a particular collection of matter or a particular region of space. The complement of a system, i.e., the matter or region outside the system, is called the surroundings. The surface that separates the system from its surroundings is called the boundary or wall of the system.

A closed system consists of a fixed amount of matter in a region Ω in space with boundary surface ∂Ω which depends on time t. While no matter can cross the boundary of a closed system, energy in the form of work or heat can cross the boundary. The volume of a closed system is not necessarily fixed. If the energy does not cross the boundary of the system, then we say that the boundary is insulated and such a system is said to be mechanically and thermally isolated. An isolated system is an idealization for no physical system is truly isolated, for there are always electromagnetic radiation into and out of the system.

An open system (or control volume) focuses in on a region in space, Ω_c which is independent of time t. The enclosing boundary of the open system, over which both matter and energy can cross is called a control surface, which we denote by ∂Ω_c.

5.3 Conservation of Mass

At the intuitive level mass is perceived to be a measure of the amount of material contained in an arbitrary portion of the body. As such it is a non-negative scalar quantity independent of time and not generally determined by the size of the configuration occupied by the arbitrary sub-body. It is not a count of the number of material particles in the body or its sub-parts. However, the mass of a body is the sum of the masses of its parts. These statements can be formalized mathematically by characterizing mass as a set function with certain properties and we proceed on the basis of the following definitions.

Let 𝔅_t be an arbitrary configuration of a body and let 𝔄 be a set of points in 𝔅_t occupied by the particles in an arbitrary subset of . If with there is associated a non-negative real number m() having a physical dimension independent of time and distance and such that

• m(₁ ∪₂) = m(₁) + m(₂) for all pairs ₁, ₂ of disjoint subsets of , and
• m() → 0 as volume of A tends to zero,

is said to be a material body with mass function m. The mass content of 𝔄, denoted by m(𝔄), is identified with the mass m() of . Property two is a consequence of our assumption that the body is a continuum, precluding the presence of point masses. Properties (i) and (ii) imply the existence of a scalar field ρ, defined on , such that

(5.1)

ρ is called the mass density, or simply the density, of the material with which the body is made up of.

In non-relativistic mechanics mass cannot be produced or destroyed, so the mass of a body is a conserved quantity. Hence, if a body has a certain mass in the reference configuration it must stay the same during a motion. Hence, we write

(5.2)

for all times t, where 𝔅_r denotes the region occupied by the body in the reference configuration and 𝔅_t denotes the region occupied by the body in the current configuration.

In the case of a material body executing a motion {𝔅_t : t ∈ I}, the density ρ is defined as a scalar field on the configurations {𝔅_t} and the mass content m(𝔄_t) of an arbitrary region in the current configuration 𝔅_t is equal to the mass m() of . Since, m() does not depend upon t we deduce directly from (5.1) the equation of mass balance

(5.3)

ρ = (x,t), the density is assumed to be continuously differentiable jointly in the position and time variables on which it depends. Since, we are interested in the total time derivative and the current volume of the body changes with time, the differentiation and integration operations cannot be interchanged. To be able to change the order of the integration and differentiation, we have to convert it to an integral over the reference configuration. This is accomplished by using (3.75). Hence, we obtain

(5.4)

Expanding the above equation we obtain

(5.5)

Now, we compute . Towards this, using the chain rule for differentiation,

(5.6)

Using the result from equation (2.187),

(5.7)

where to obtain the last expression we have made use of the definition of a dot product of two tensors, (2.71) and the property of the trace operator, (2.68). Defining, l = grad(v), it can be seen that

(5.8)

obtained using the chain rule and by interchanging the order of the spatial and temporal derivatives. Hence, equation (5.7) reduces to,

(5.9)

on using the definition of the divergence, (2.208).

Substituting (5.9) in (5.5) we obtain

(5.10)

Again using (3.75) we obtain

(5.11)

wherein the integrand is continuous in 𝔅_t and the range of integration is an arbitrary subregion of 𝔅_t. Since, the integral vanishes in 𝔅_t and in any arbitrary subregion of 𝔅_t, the integrand should vanish, i.e.,

(5.12)

Subject to the presumed smoothness of ρ, equations (5.3) and (5.12) are equivalent expressions of the conservation of mass.

Further, since (5.4) has to hold for any arbitrary subpart of 𝔄_r, we get

(5.13)

where ρ_r is the density in the reference configuration, 𝔅_r. Note that, whenever the body occupies 𝔅_r, det(F) = 1 and hence, ρ = ρ_r, giving ρ_r as the constant value of ρ det(F) and hence the referential equation of conservation of mass

(5.14)

A body which is able to undergo only isochoric motions is said to be composed of incompressible material. Since, det(F) = 1 for isochoric motions, we see from equation (5.14) that the density does not change with time, t. Consequently, for a body made up of an incompressible material, if the density is uniform in some configuration it has to be the same uniform value in every configuration which the body can occupy. On the other hand if a body is made up of compressible material this is not so, in general.

Let ϕ be a scalar field and u a vector field representing properties of a moving material body, , and let 𝔄_t be an arbitrary material region in the current configuration of . Then show that:

5.4 Conservation of momentum

There are two kinds of forces that act in a body. They are:

• Contact forces
• Body forces

Contact forces arise due to contact between two bodies. Body forces are action at a distance forces like gravitational force. Similarly, while contact forces act per unit area of the boundary of the body, body forces act per unit mass of the body. Both these forces result in the generation of stresses.

Before deriving these conservation laws of momentum rigorously, we obtain the same using using an approximate analysis, as we did for the transformation of curves, areas and volumes. Moreover, we obtain these for a 2D body. Consider an 2D infinitesimal element in the current configuration with dimensions Δx along the x direction and Δy along the y direction, as shown in figure 5.1. We assume that the center of the infinitesimal element, O is located at (x + 0.5Δx,y + 0.5Δy).

We assume that the Cauchy stress, σ varies over the infinitesimal element in the current configuration, σ = (x,y). That is we have assumed Eulerian description for the stress. Since, we are considering only 2D state of stress, the relevant Cartesian components of the stress are σ_xx, σ_yy, σ_xy and σ_yx. Expanding the Cartesian components of the stresses using Taylor’s series up to first order we obtain:

Now, we are interested in the equilibrium of the infinitesimal element in the deformed configuration under the action of these stresses and the body force, b with Cartesian components b_x and b_y along the x and y direction respectively. Also, we shall assume that the infinitesimal element is accelerating with acceleration, a with Cartesian components a_x and a_y along the x and y direction respectively. Thus, force equilibrium along the x direction requires:

(5.23)

Similarly, writing the force equilibrium equation along the y direction:

(5.25)

Finally, we appeal to the moment equilibrium. Here we assume that there are no body couples or contact couples acting in the infinitesimal element. Since, we can take moment equilibrium about any point, for convenience, we do it about the point O, marked in figure 5.1, to obtain:

(5.27)

which in the limit Δx and Δy tending to zero, the limit that we are interested in, reduces to requiring

(5.28)

Thus, a plane Cauchy stress field should satisfy the equations (5.23), (5.25) and (5.28) for it to be admissible.

Generalizing it for 3D, the Cauchy stress field should satisfy:

(5.32)

The equations (5.29) through (5.31) are written succinctly as

(5.33)

and equations (5.32) tells us that the Cauchy stress should be symmetric.

Similarly, requiring a sector of cylindrical shell to be in equilibrium, under the action of spatially varying stresses, it can be shown that

(5.37)

has to hold.

5.4.1 Conservation of linear momentum

Let 𝔄_t be an arbitrary sub-region in the configuration 𝔅_t of a body . The total linear momentum, Γ(𝔄_t), of the material particles occupying 𝔄_t is defined by

(5.38)

Next, we find the forces acting on the body. Since, the part 𝔄_t has been isolated from its surroundings, traction t_(n)(x), introduced in the last chapter, acts on the boundary of the 𝔄_t. In addition to traction, which arises between parts of the body that are in contact, there exist forces like gravity which act on the material particles not through contact and are called body force. The body forces are denoted by b and are defined per unit mass. Hence, the resultant force, г acting on 𝔄_t is

(5.39)

Now, using Newton’s second law of motion, which states that the rate of change of linear momentum of the body must equal the resultant force that acts on the body in both magnitude and direction we obtain

(5.40)

Using (3.30), definition of acceleration, a and (5.16) the above equation can be written as

(5.41)

Next, using Cauchy’s stress theorem (4.3) and the divergence theorem (2.263) the above equation further can be reduced to

(5.42)

Since, the above equation has to hold for 𝔅_t and any subset, 𝔄_t, of it

(5.43)

Many a times, we are interested in cases for which a = o, then (5.43) becomes

(5.44)

which is referred to as Cauchy’s equation of equilibrium. Further, there arises scenarios where the body forces can be neglected and in those cases div(σ) = o. Thus, divergence free stress fields are called self equilibrated stress fields.

Equation (5.43) is in spatial form, that is it assumes that σ = (x,t), b = (x,t), a = (x,t), ρ = (x,t) are known. But on most occasions, especially in solid mechanics, we know only the material form of these fields that is σ = (X,t), b = (X,t), a = (X,t), ρ = (X,t) and in those occasions it is difficult to obtain the spatial divergence of these fields. To make it easier at these instances, we seek a statement of balance of linear momentum in material form. For this, equation (5.42a) has to be modified. Towards this, we obtain

(5.46)

It is worthwhile to emphasize that irrespective of the choice of the independent variable, the equilibrium of the forces is established only in the current configuration. In other words, whether we use material or spatial description for the stress, the forces and moments have to be equilibrated in the deformed configuration and not the reference configuration.

5.4.2 Conservation of angular momentum

Let 𝔄_t be an arbitrary sub-region in the configuration 𝔅_t of a body . The total angular momentum, Ω(𝔄_t), of the material particles occupying 𝔄_t is defined by

(5.47)

where x_o is the point about which the moment is taken and p is the intrinsic angular momentum per unit volume in the current configuration.

Next, we find the moment due to the forces acting on the body. As discussed before there are two kind of forces, surface or contact forces and body forces act on the body which contribute to the moment. Apart from moment arising due to applied forces, there could exist couples distributed per unit surface area, m and body couples c, defined per unit mass. Thus, m = (x,t,n) and c = (x,t). Hence, the resultant moment, ω acting on 𝔄_t is

(5.48)

However, in non-polar bodies that are the subject matter of the study here, there exist no body couples or couples distributed per unit surface area or intrinsic angular momentum, i.e., c = m = p = o.

The balance of angular momentum states that the rate of change of the angular momentum must equal the applied momentum in both direction and magnitude. Hence,

(5.49)

Further simplification of this balance principle involves some tensor algebra. First, we begin by calculating the left hand side of the above equation:

Next, we simplify the right hand side of the equation (5.49). Towards that, we first show that

(5.52)

where τ is the axial vector of (σ - σ^t). We begin by establishing a few vector identities which enable us to establish (5.52).

Identity - 1: If u and v are arbitrary vectors then u ∧ v is the axial vector of the skew-symmetric tensor v ⊗ u - u ⊗ v.

Proof: It is straightforward to verify that v ⊗ u - u ⊗ v is skew-symmetric. Then we observe that

(5.53)

hence, the axial vector of v ⊗ u - u ⊗ v is a scalar multiple of u ∧ v. Accordingly,

(5.54)

for any vector a. Now, we have to show that α = 1. Setting a = u in the above equation and then forming a scalar product on each side with v we have

(5.55)

Using (2.36) we find that

(5.56)

Comparing equations (5.55) and (5.56) we find that α = 1.

Identity - 2: For any vector a, b, c in the vector space

(5.57)

This identity follows immediately from the above identity.

Identity - 3: Let 𝔄_t be a regular region with boundary ∂𝔄_t, let n be the outward unit normal to ∂𝔄_t and let u be a vector field and σ a tensor field, each continuous in 𝔄_t and continuously differentiable in the interior of 𝔄_t. Then

(5.58)

Towards proving this identity, note that

(5.62)

Taking the transpose of the above identity, we obtain the required vector identity (5.58).

Now, we are in a position to prove (5.52). For this we replace b by (x - x_o) and c by σn in (5.57) and then integrating each side over the surface ∂𝔄_t we obtain

(5.63)

a being an arbitrary vector. From (5.58) it follows that

(5.67)

and hence the required identity (5.52), since a is an arbitrary vector.

Substituting (5.51) and (5.52) in (5.49) yields

(5.69)

Here we are interested only in non-polar bodies and hence, p = c = m = o. Therefore, (5.69) simplifies to requiring

(5.70)

Since, this has to hold for any arbitrary subparts of the body, 𝔅_t, τ = o. Consequently,

(5.71)

that is the Cauchy stress tensor has to be symmetric in non-polar bodies.

5.5 Summary

Since, in this course we are interested only in purely mechanical process, we studied only conservation of mass and momentum. There are other quantities like energy, charge which is also conserved. In courses on thermodynamics or electromagnetism, one would study the other conservation laws. The one equation that would be used frequently in the following chapters is the final expression for the conservation of linear momentum, also called as equilibrium equations,

(5.72)

where ρ is the density, b is the body force per unit mass, σ is the Cauchy stress tensor.

5.6 Self-Evaluation

• Determine which of the following Cauchy stress fields are possible within a body at rest assuming that there are no body forces acting on it:
  
  where the components of the stress are with respect to orthonormal Cartesian basis and (x,y,z) denote the Cartesian coordinates of a typical material particle in the current configuration of the body.
• The Cartesian components of the Cauchy stress tensor in a plate at rest is

  Find the body force that should act on the plate so that this state of stress is realizable in the body.

• A body at rest is subjected to a plane state of stress such that the non-zero Cartesian components of the Cauchy stress are σ_xx, σ_xz and σ_zz. Derive the equilibrium equations for this special case. Then, show that in the absence of body forces the equilibrium equations hold if,

  where ϕ = ϕ(x,y), for any choice of ϕ. ϕ is called the Airy’s stress potential.

• Derive form first principles and show that for the plane stress problem in cylindrical polar coordinates with the non-zero cylindrical polar components of the Cauchy stress being σ_rr, σ_rθ and σ_θθ, as shown in figure 5.2, the equilibrium equations in the absence of body forces with the body in static equilibrium are

  ---

  
  

  Figure 5.2: Figure for problem 4. Variation of the cylindrical polar components of the Cauchy stress over an infinitesimal cylindrical wedge, in plane stress state

  ---

• Is any stress field that satisfies the equilibrium equations realizable in a given body? Discuss.