Basic Equations in Mechanics
Having gained a superficial understanding of the four concepts in mechanics namely the force, stress, displacement and strain, let us look at the four equations that connect these concepts and the reasoning used to obtain them.
Equilibrium equations
Equilibrium equations are Newton’s second law which states that the rate of change of linear momentum would be equal in magnitude and direction to the net applied force. Deformable bodies are subjected to two kinds of forces, namely, contact force and body force. As the name suggest the contact force arises by virtue of the body being in contact with its surroundings. Traction arises only due to these contact force and hence so does the stress tensor. The magnitude of the contact force depends on the contact area between the body and its surroundings. On the other hand, the body forces are action at a distance forces. Examples of body force are gravitational force, electromagnetic force. The magnitude of these body forces depend on the mass of the body and hence are generally expressed as per unit mass of the body and denoted by b.
On further assuming that the Newton’s second law holds for any subpart of the body and that the stress field is continuously differentiable within the body the equilibrium equations can be written as:
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(1.6) |
where ρ is the density, a is the acceleration and the mass is assumed to be conserved. Detail derivation of the above equation is given in chapter 5. The meaning of the operator div(⋅) can be found in chapter 2.
Also, the rate of change of angular momentum must be equal to the net applied moment on the body. Assuming that the moment is generated only by the contact forces and body forces, this condition requires that the Cauchy stress tensor to be symmetric. That is in the absence of body couples, σ = σt, where the superscript (⋅)t denotes the transpose. Here again the assumptions made to obtain the force equilibrium equation (1.6) should hold. See chapter 5 for detailed derivation.
Strain-Displacement relation
The relationship that connects the displacement field with the strain is called as the strain displacement relationship. As pointed out before there is no unique definition of the strain and hence there are various strain tensors. However, all these strains are some function of the gradient of the deformation field, F; commonly called as the deformation gradient. The deformation field is a function that gives the position vector of any material particle that belongs to the body at any instance in time with the material particle identified by its location at some time to. Then, in chapter 3 we show that, the stretch ratio along a given direction A is,
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(1.7) |
where C = FtF, is called as the right Cauchy-Green deformation tensor. When the body is undeformed, F = 1 and hence, C = 1 and λ(A) = 1. Instead of looking at the deformation field, one can develop the expression for the stretch ratio, looking at the displacement field too. Now, the displacement field can be a function of the coordinates of the material particles in the reference or undeformed state or the coordinates in the current or deformed state. If the displacement is a function of the coordinates of the material particles in the reference configuration it is called as Lagrangian representation of the displacement field and the gradient of this Lagrangian displacement field is called as the Lagrangian displacement gradient and is denoted by H. On the other hand if the displacement is a function of the coordinates of the material particle in the deformed state, such a representation of the displacement field is said to be Eulerian and the gradient of this Eulerian displacement field is called as the Eulerian displacement gradient and is denoted by h. Then it can be shown that (see chapter 3),
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(1.8) |
where, 1 stands for identity tensor (see chapter 2 for its definition). Now, the right Cauchy-Green deformation tensor can be written in terms of the Lagrangian displacement gradient as,
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(1.9) |
Note that the if the body is undeformed then H = 0. Hence, if one cannot see the displacement of the body then it is likely that the components of the Lagrangian displacement gradient are going to be small, say of order 10-3. Then, the components of the tensor HtH are going to be of the order 10-6. Hence, the equation (1.9) for this case when the components of the Lagrangian displacement gradient is small can be approximately calculated as,
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(1.10) |
where
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(1.11) |
![[ ]
ϵL = 1-H + Ht ,
2](2_files/image006.webp){kind=small}
is called as the linearized Lagrangian strain. We shall see in chapter 3 that when the components of the Lagrangian displacement gradient is small, the stretch ratio (1.7) reduces to
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(1.12) |
Thus we find that ϵL contains information about changes in length along any given direction, A when the components of the Lagrangian displacement gradient are small. Hence, it is called as the linearized Lagrangian strain. We shall in chapter 3 derive the various strain tensors corresponding to the various definition of strains given in equation (1.5).
Further, since FF-1 = 1, it follows from (1.8) that
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(1.13) |
which when the components of both the Lagrangian and Eulerian displacement gradient are small can be approximated as H = h. Thus, when the components of the Lagrangian and Eulerian displacement gradients are small these displacement gradients are the same. Hence, the Eulerian linearized strain defined as,
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(1.14) |
![1-[ t]
ϵE = 2 h + h ,](2_files/image009.webp){kind=small}
and the Lagrangian linearized strain, ϵL would be the same when the components of the displacement gradients are small.
Equation (1.14) is the strain displacement relationship that we would use to solve boundary value problems in this course, as we limit ourselves to cases where the components of the Lagrangian and Eulerian displacement gradient is small.
Compatibility equation
It is evident from the definition of the linearized Lagrangian strain, (1.11) that it is a symmetric tensor. Hence, it has only 6 independent components. Now, one cannot prescribe arbitrarily these six components since a smooth differentiable displacement field should be obtainable from this six prescribed components. The restrictions placed on how this six components of the strain could vary spatially so that a smooth differentiable displacement field is obtainable is called as compatibility equation. Thus, the compatibility condition is
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(1.15) |
The derivation of this equation as well as the components of the curl(⋅) operator in Cartesian coordinates is presented in chapter 3.
It should also be mentioned that the compatibility condition in case of large deformations is yet to be obtained. That is if the components of the right Cauchy-Green deformation tensor, C is prescribed, the restrictions that have to be placed on these prescribed components so that a smooth differentiable deformation field could be obtained is unknown, except for some special cases.
Constitutive relation
Broadly constitutive relation is the equation that relates the stress (and stress rates) with the displacement gradient (and rate of displacement gradient). While the above three equations - Equilibrium equations, strain-displacement relation, compatibility equations - are independent of the material that the body is made up of and/or the process that the body is subjected to, the constitutive relation is dependent on the material and the process. Constitutive relation is required to bring in the dependance of the material in the response of the body and to have as many equations as there are unknowns, as will be shown in chapter 6.
The fidelity of the predictions, namely the likely displacement or stress for a given force depends only on the constitutive relation. This is so because the other three equations are the same irrespective of the material that the body is made up of. Consequently, a lot of research is being undertaken to arrive at better constitutive relations for materials.
It is difficult to have a constitutive relation that could describe the response of a material subjected to any process. Hence, usually constitutive relations are prescribed for a particular process that the material undergoes. The variables in the constitutive relation depends on the process that is being studied. The same material could undergo different processes depending on the stimuli; for example, the same material could respond elastically or plastically depending on say, the magnitude of the load or temperature. Hence it is only apt to qualify the process and not the material. However, it is customary to qualify the material instead of the process too. This we shall desist.
Traditionally, the constitutive relation is said to depend on whether the given material behaves like a solid or fluid and one elaborates on how to classify a given material as a solid or a fluid. A material that is not a solid is defined as a fluid. This means one has to define what a solid is. A couple of definitions of a solid are listed below:
- Solid is one which can resist sustained shear forces without continuously deforming
- Solid is one which does not take the shape of the container
Though these definitions are intuitive they are ambiguous. A class of materials called “viscoelastic solids”, neither take the shape of the container nor resist shear forces without continuously deforming. Also, the same material would behave like a solid, like a mixture of a solid and a fluid or like a fluid depending on say, the temperature and the mechanical stress it is being subjected to. These prompts us to say that a given material behaves in a solid-like or fluid-like manner. However, as we shall see, this classification of a given material as solid or fluid is immaterial. If one appeals to thermodynamics for the classification of the processes, the response of materials could be classified based on (1) Whether there is conversion of energy from one form to another during the process, and (2) Whether the process is thermodynamically equilibrated. Though, in the following section, we classify the response of materials based on thermodynamics, we also give the commonly stated definitions and discuss their shortcomings. In this course, as well as in all these classifications, it is assumed that there are no chemical changes occurring in the body and hence the composition of the body remains a constant.