Classification of the Response of Materials - Civil Engineering (CE) PDF Download

Classification of the Response of Materials 

First, it should be clarified that one should not get confused with the real body and its mathematical idealization. Modeling is all about idealizations that lead to predictions that are close to observations. To illustrate, the earth and the sun are assumed as point masses when one is interested in planetary motion. The same earth is assumed as a rigid sphere if one is interested in studying the eclipse. These assumptions are made to make the resulting problem tractable without losing on the required accuracy. In the same sprit, the all material responses, some amount of mechanical energy is converted into other forms of energy. However, in some cases, this loss in the mechanical energy is small that it can be idealized as having no loss, i.e., a non-dissipative process.

Non-dissipative response 

A response is said to be non-dissipative if there is no conversion of mechanical energy to other forms of energy, namely heat energy. Commonly, a material responding in this fashion is said to be elastic. The common definitions of elastic response,
1. If the body’s original size and shape can be recovered on unloading, the loading process is said to be elastic.
2. Processes in which the state of stress depends only on the current strain, is said to be elastic.

The first definition is of little use, because it requires one to do a complimentary process (unloading) to decide on whether the process that needs to be classified as being elastic. The second definition, though useful for deciding on the variables in the constitutive relation, it also requires one to do a complimentary process (unload and load again) to decide on whether the first process is elastic. The definition based on thermodynamics does not suffer from this drawback.  However, many processes (approximately) satisfy all the three definitions.

This class of processes also proceeds through thermodynamically equilibrated states. That is, if the body is isolated at any instant of loading (or displacement) then the stress, displacement, internal energy, entropy do not change with time.

Ideal gas, a fluid is the best example of a material that responds in a non-dissipative manner. Metals up to a certain stress level, called the yield stress, are also idealized as responding in a non dissipative manner. Thus, the notion that only solids respond in a non-dissipative manner is not correct.

Thus, for these non-dissipative, thermodynamically equilibrated processes the Cauchy stress and the deformation gradient can in general be related through an implicit function. That is, for isotropic materials (see chapter 6 for when a material is said to be isotropic), f(σ, F) = 0. However, in classical elasticity it is customary to assume that Cauchy stress in a isotropic material is a function of the deformation gradient, σ = f(F). On requiring the restriction6 due to objectivity and second law of thermodynamics to hold, it can be shown that if

                   (1.16)

where (J₁, J₂, J₃) is the Helmoltz free energy defined per unit volume in the reference configuration, also called as the stored energy, B = FF^t and J₁ = tr(B), J₂ = tr(B⁻¹ ), J₃ =  .    When the components of the displacement gradient is small, then (1.16) reduces to,

σ = tr(∈)λ1 + 2µ∈,                                     (1.17)

on neglecting the higher powers of the Lagrangian displacement gradient and where λ and µ are called as the Lam`e constants. The equation (1.17) is the famous Hooke’s law for isotropic materials. In this course Hooke’s law is the constitutive equation that we shall be using to solve boundary value problems.

Before concluding this section, another misnomer needs to be clarified. As can be seen from equation (1.16) the relationship between Cauchy stress and the displacement gradient can be nonlinear when the response is nondissipative. Only sometimes as in the case of the material obeying Hooke’s law is this relationship linear. It is also true that if the response is dissipative, the relationship between the stress and the displacement gradient is always nonlinear. However, nonlinear relationship between the stress and the displacement gradient does not mean that the response is dissipative. That is, nonlinear relationship between the stress and the displacement gradient is only a necessary condition for the response to be dissipative but not a sufficient condition.

Dissipative response 

A response is said to be dissipative if there is conversion of mechanical energy to other forms of energy. A material responding in this fashion is popularly said to be inelastic. There are three types of dissipative response, which we shall see in some detail.

Plastic response 

A material is said to deform plastically if the deformation process proceeds through thermodynamically equilibrated states but is dissipative. That is, if the body is isolated at any instant of loading (or displacement) then the stress, displacement, internal energy, entropy do not change with time. By virtue of the process being dissipative, the stress at an instant would depend on the history of the deformation. However, the stress does not depend on the rate of loading or displacement by virtue of the process proceeding through thermodynamically equilibrated states.
For plastic response, the classical constitutive relation is assumed to be of the form,

σ = f(F, F^p , q₁, q₂),                          (1.18)

where F_p , q₁, q₂ are internal variables whose values could change with deformation and/or stress. For illustration, we have used two scalar internal variables and one second order tensor internal variable while there can be any number of tensor or scalar internal variables. In some theories the internal variables are given a physical interpretation but in general, these variable need not have any meaning and are proposed for mathematical modeling purpose only.

Thus, when a material deforms plastically, it does not return back to its original shape when unloaded; there would be a permanent deformation. Hence, the process is irreversible. The response does not depend on the rate of loading (or displacement). Metals like steel at room temperature respond plastically when stressed above a particular limit, called the yield stress.

Viscoelastic response 

If the dissipative process proceeds through states that are not in thermodynamic equilibrium⁷ , then it is said to be viscoelastic. Therefore, if a body is isolated at some instant of loading (or displacement) then the displacement (or the stress) continues to change with time. A viscoelastic material when subjected to constant stress would result in a deformation that changes with time which is called as creep. Also, when a viscoelastic material is subjected to a constant deformation field, its stress changes with time and this is called as stress relaxation. This is in contrary to a elastic or plastic material which when subjected to a constant stress would have a constant strain.
The constitutive relation for a viscoelastic response is of the form,
                             (1.19)

where  denotes the time derivative of stress and  time derivative of the deformation gradient. Though here we have truncated to first order time derivatives, the general theory allows for higher order time derivatives too.
Thus, the response of a viscoelastic material depends on the rate at which it is loaded (or displaced) apart from the history of the loading (or displacement). The response of a viscoelastic material changes depending on whether load is controlled or displacement is controlled. This process too is irreversible and there would be unrecovered deformation immediately on removal of the load. The magnitude of unrecovered deformation after a long time (asymptotically) would tend to zero or remain the same constant value that it is immediately after the removal of load.

Constitutive relations of the form,

        (1.20)

which is a special case of the viscoelastic constitutive relation (1.19), is that of a viscous fluid.
In some treatments of the subject, a viscoelastic material would be said to be a combination of a viscous fluid and an elastic solid and the viscoelastic models are obtained by combining springs and dashpots. There are several philosophical problems associated with this viewpoint about which we cannot elaborate here.

Viscoplastic response 

This process too is dissipative and proceeds through states that are not in thermodynamic equilibrium. However, in order to model this class of response the constitutive relation has to be of the form,

                          (1.21)

where F^p , q₁, q₂ are the internal variables whose values could change with deformation and/or stress. Their significance is same as that discussed for plastic response. As can be easily seen the constitutive relation form for the viscoplastic response (1.21) encompasses viscoelastic, plastic and elastic response as a special case.

In this case, constant load causes a deformation that changes with time. Also, a constant deformation causes applied load to change with time. The response of the material depends on the rate of loading or displacement. The process is irreversible and there would be unrecovered deformation on removal of load. The magnitude of this unrecovered deformation varies with rate of loading, time and would tend to a value which is not zero. This dependance of the constant value that the unrecovered deformation tends on the rate of loading, could be taken as the characteristic of viscoplastic response.

Figure 1.3 shows the typical variation in the strain for various responses when the material is loaded, held at a constant load and unloaded, as discussed above. This kind of loading is called as the creep and recovery loading, helps one to distinguish various kinds of responses.

As mentioned already, in this course we shall focus on the elastic or nondissipative response only.
 

Figure 1,3: Schematic of the variation in the strain with time for various responses when the material is loaded and unloaded.