^{1}

^{1}

^{2}

In recent papers, Surana et al. presented internal polar non-classical Continuum theory in which velocity gradient tensor in its entirety was incorporated in the conservation and balance laws. Thus, this theory incorporated symmetric part of the velocity gradient tensor (as done in classical theories) as well as skew symmetric part representing varying internal rotation rates between material points which when resisted by deforming continua result in dissipation (and/or storage) of mechanical work. This physics referred as internal polar physics is neglected in classical continuum theories but can be quite significant for some materials. In another recent paper Surana et al. presented ordered rate constitutive theories for internal polar non-classical fluent continua without memory derived using deviatoric Cauchy stress tensor and conjugate strain rate tensors of up to orders n and Cauchy moment tensor and its conjugate symmetric part of the first convected derivative of the rotation gradient tensor. In this constitutive theory higher order convected derivatives of the symmetric part of the rotation gradient tensor are assumed not to contribute to dissipation. Secondly, the skew symmetric part of the velocity gradient tensor is used as rotation rates to determine rate of rotation gradient tensor. This is an approximation to true convected time derivatives of the rotation gradient tensor. The resulting constitutive theory: (1) is incomplete as it neglects the second and higher order convected time derivatives of the symmetric part of the rotation gradient tensor; (2) first convected derivative of the symmetric part of the rotation gradient tensor as used by Surana et al. is only approximate; (3) has inconsistent treatment of dissipation due to Cauchy moment tensor when compared with the dissipation mechanism due to deviatoric part of symmetric Cauchy stress tensor in which convected time derivatives of up to order n are considered in the theory. The purpose of this paper is to present ordered rate constitutive theories for deviatoric Cauchy strain tensor, moment tensor and heat vector for thermofluids without memory in which convected time derivatives of strain tensors up to order n are conjugate with the Cauchy stress tensor and the convected time derivatives of the symmetric part of the rotation gradient tensor up to orders 1n are conjugate with the moment tensor. Conservation and balance laws are used to determine the choice of dependent variables in the constitutive theories: Helmholtz free energy density Φ, entropy density η, Cauchy stress tensor, moment tensor and heat vector. Stress tensor is decomposed into symmetric and skew symmetric parts and the symmetric part of the stress tensor and the moment tensor are further decomposed into equilibrium and deviatoric tensors. It is established through conjugate pairs in entropy inequality that the constitutive theories only need to be derived for symmetric stress tensor, moment tensor and heat vector. Density in the current configuration, convected time derivatives of the strain tensor up to order n, convected time derivatives of the symmetric part of the rotation gradient tensor up to orders 1n, temperature gradient tensor and temperature are considered as argument tensors of all dependent variables in the constitutive theories based on entropy inequality and principle of equipresence. The constitutive theories are derived in contravariant and covariant bases as well as using Jaumann rates. The nth and 1nth order rate constitutive theories for internal polar non-classical thermofluids without memory are specialized for n = 1 and 1n = 1 to demonstrate fundamental differences in the constitutive theories presented here and those used presently for classical thermofluids without memory and those published by Surana et al. for internal polar non-classical incompressible thermofluids.

Conservation and balance laws: conservation of mass, balance of linear momenta, balance of angular momenta, balance of moments of moments (or couples), first law of thermodynamics (energy equation) and second law of thermodynamics (entropy inequality) for internal polar non-classical fluent continua were presented in references [

Another significant discussion in references [

The notations used in this paper have been used by the authors in the current literature, nonetheless some description and their use in deriving conservation and balance laws are presented in the following. Over bar is used on quantities to express quantities in the current configuration in Eulerian description, that is, all quantities with over bars are functions of current coordinates

or

If

with

In Murnaghan’s notation

in which the columns of

where

in which

tensor, and

The purpose of this paper is to present ordered rate constitutive theories for deviatoric Cauchy strain tensor, Cauchy moment tensor and heat vector for thermofluids without memory in which convected time derivatives of strain tensors up to order n are conjugate with the Cauchy stress tensor and the convected time derivatives of the symmetric part of the rotation gradient tensor up to orders ^{1}n are conjugate with the Cauchy moment tensor.

In reference [

gradient tensor. Let

(

the constitutive theories in contravariant basis, covariant basis and in Jaumann rates can be obtained. In addition to the convected derivatives of the strain tensors one must also consider convected derivatives of the rotation gradient tensor that are also basis dependent. In reference [

(), () and ()

be the convected time derivatives of the rotation gradient tensors in contravariant basis, covariant basis, and Jaumann rates. With these convected time derivatives, the

conjugate pairs are (

(

laws as well as constitutive theories. Jacobian of deformation

tors [

descriptions. Likewise

of Jacobian of deformation. Rows of

In finite deformation, a tetrahedron in the undeformed configuration with its orthogonal edges deforms into one in which the edges are non-orthogonal covariant base vectors and the vectors normal to the faces of the deformed tetrahedron are contravariant non-orthogonal base vectors that are reciprocal to the covariant base vectors. The covariant and contravariant bases are fundamental in the measures of finite deformation, rotations, etc. Consider deformed coordinates

(a) Internal rotations and rotation matrix

Consider decomposition of the Jacobian of deformation

Let

pressed as rotations about

in which

Alternatively one can also derive (15) as follows.

The sign differences between (15) and (18) are due to clockwise and counterclockwise internal rotations and will only affect sign of

The stretch tensors

1)

2)

3) Determination of

4) It suffices to note that internal rotations at a material point present in

5) The internal rotation angles

(b) Internal rotation gradient tensor and its rates using

The covariant internal rotation tensor

Let

Alternatively (16) can be written as

and then

In (22) the internal rotations

Then, one defines rotation gradient tensor

One can also define the velocity gradients as

in which

Likewise if

Remarks

1) Symmetric rotation gradient tensor in (26) is a covariant measure in Lagrangian description. It describes symmetric part of the gradients in x-frame of rotations about covariant axes expressed about the axes of the x-frame.

2) Since this measure is covariant rotation rate its work conjugate measure will be contravariant.

3) The covariant nature of this measure is intrinsic in its derivation due to

4) Convected time derivatives of

(c) Second Piola-Kirchhoff covariant rotation gradient tensor

Consider isotropic, homogeneous, compressible matter. Let

Thus, one obtains

using (36) and (37) in (34) one obtains

using (35) in (38)

hence, one obtains

and

Equations (40) and (41) are Lagrangian and Eulerian descriptions for second Piola- Kirchhoff covariant rotation gradient tensor. These are useful in deriving covariant convected time derivatives of the rotation gradient tensor

(d) Convected time derivatives of the covariant rotation gradient tensor: compressible matter

In this section derivation of convected time derivative of the covariant rotation gradient tensor

One intentionally chooses Eulerian description for Cauchy and second Piola-Kir- chhoff tensor as this is what is needed in the case of mathematical model for fluent continua. Consider material derivative of

using

in (43), factoring and regrouping, one can write

If one defines

then one obtains the following from (46)

where

In general one can write the following recursive relations that can be used to obtain the convected time derivative of any desired order k of the covariant rotation gradient tensor

For incompressible case

(a) Internal rotations and rotation matrix

Following the derivations for covariant measures, one can derive the following if one considers Jacobian of deformation

Let

in which

Alternatively one can also derive (57) as follows.

The reason for the sign difference in (57) and (60) is exactly same as for covariant measures. One notes that decomposition (53) enables explicit description of stretches (elongation per unit length and change in angles between the pair of orthogonal material lines in the undeformed configuration) and rotation tensor contained in

The stretch tensors

1)

2)

3) One notes that determination of

4) It suffices to note that internal rotations at a material point present in

5) The internal rotation angles

(b) Internal rotation gradient tensor using

The contravariant internal rotation tensor

Let

Alternatively (62) can be written as

and then

In (64) the internal rotations

Then the rotation gradient tensor

Remarks

1) Symmetric rotation gradient tensor in (67) is a contravariant measure in Eulerian description. It describes symmetric part of the gradients of rotations about contravariant axes expressed about the axes of the x-frame.

2) Since this measure is contravariant its work conjugate moment measure is expected to be covariant (see derivation of first law of thermodynamics).

3) Contravariant nature of this measure is intrinsic in its derivation, hence can not be changed. However by replacing

4) Convected time derivatives of

(c) Second Piola-Kirchhoff contravariant rotation gradient tensor

Consider isotropic, homogeneous compressible matter. Consider oblique planes of the deformed and the undeformed tetrahedra with scalar areas

Let the contravariant Cauchy rotation gradient tensor be

Substituting (71) and (72) in (70)

using

Hence, one obtains

and

Also

(d) Convected time derivatives of the contravariant rotation gradient tensor: compressible matter

Consider the material derivative of

using

and regrouping the terms one obtains

If one defines

then one can write

Here

time derivative of the contravariant Cauchy rotation gradient tensor

material derivative of (84) and follows the same steps as in case of

where

In general one can write the following recursive relation that can be used to obtain the convected time derivative up to any desired order k of the tensor

For incompressible case

It is advantageous to introduce basis independent notations so that the derivations of conservation and balance laws could be carried out independent of the basis. These can then be made basis dependent by simply replacing the basis independent quantities. Similar to Cauchy stress tensor and Cauchy moment tensor, introduce

Polar decomposition of the velocity gradient tensor is helpful in decomposing deformation into stretch rate tensor and rotation rate tensor. Whether one uses left stretch rate tensor or right stretch rate tensor, the rotation rate tensor is unique. Thus, at each location with infinitesimal volume surrounding it, the velocity gradient tensor

Let

The columns of

Then (89) holds, hence

Thus,

and following a similar procedure one can establish the following

in which

where

Explicit forms of

In reference [

At this stage the Cauchy stress tensor

The sum of work and heat added to a deforming volume of matter must result in the increase in energy of the system. Expressing this as a rate statement one can write [

where

placement, and

tributes additional rate of work due to rates of rotation in (104). Expand each of the integrals in (102)-(104). Following reference [

Using basis independent Cauchy stress tensor

Likewise using basis independent moment tensor (per unit area)

The first convected time derivative of the rotation gradient tensor,

that is conjugate to the Cauchy moment tensor

Transferring all terms to left of equality and regrouping

Using (98) (balance of linear momenta) and (99) balance of angular momenta, (110) reduces to

Since

Equation (113) is the final form of the energy equation in which

In (114) the following decomposition of

By appropriate choices of

If

Using Cauchy’s postulate for

Using (117) in (116)

One recalls that [

and

Substituting from (119) and (120) in (118) and transferring all terms to the left of inequality

Since volume

Equation (122) is called the Clausius-Duhem inequality and is the most fundamental form resulting from the second law of thermodynamics. A different form of (122) can be derived if one assumes

where

Substituting for

From energy Equation (113) (after inserting

Substituting from (126) into (125)

or

Let

Hence

Substituting from (130) into (128)

or

The entropy inequality (132) in contravariant basis, covariant bases and in Jaumann rates can be obtained by replacing

It is instructive to decompose stress tensor

where

Substituting these in the balance of linear momenta (98), balance of angular momenta (99), energy Equation (113), and entropy inequality (132) and noting that

as

one can write (137) as

Using (136)-(139) in (98), (99), (113), and (132) one can obtain

A simple calculation by expanding the terms shows that

By substituting (144) in (142) and (143) the energy equation and entropy inequality simplify.

Remarks

1) Equations (140), (141), (145), and (146) can also be expressed in contravariant basis, covariant basis and using Jaumann rates.

2) Equations (97), (140), (141), (145), and (146) constitute a complete mathematical model for internal polar fluent media in Eulerian description.

3) From (145) and (146) one can conclude that

4) This mathematical model has closure once the constitutive theories for

The choice of dependent variables in the constitutive theories must be consistent with the axiom of casualty [

Possible choices of argument tensors of dependent variables are considered, keeping in mind the principle of equipresence [

One notes that

Similarly

Secondly, since the arguments in (147) are basis dependent the heat vector is no longer

From the entropy inequality one notes that

are conjugate pairs i.e.

has no dependence on

Consider the entropy inequality (146) with the arguments of

From the continuity Equation (97) (its alternate from in

Using (151) in (150)

One notes that

Using (153) in (152)

Substituting

Regrouping terms in (155)

For (156) to hold for arbitrary but admissible

Equations (157)-(161) are fundamental relations from the entropy inequality

Remarks

1) Equation (157) implies that

2) Equation (158) implies that

3) Equation (159) implies that

4) Based on (160),

5) The last inequality is essential in the form it is stated. For example the following (or any other separation of terms)

are inappropriate due to the fact that these imply that

In view of these remarks the arguments of the dependent variables in the constitutive

theories in (149) can be modified. One can use

One notes that there are no mechanisms or conditions that permit eliminating

In order to remedy the situation discussed in remark (5), one considers decomposition of symmetric Cauchy stress tensor into equilibrium Cauchy stress tensor

in which one considers the following

That is

vanishes when

or

Since

in which

as a function of

assumes the compressive pressure to be positive, then

Inequality (170) is satisfied if

and

Inequalities (171) imply that the rate of work due to

Constitutive theories for

For incompressible matter density is constant, hence

The incompressibility condition must be enforced. Based on (174) one can add

to (167).

Using

In the case of incompressible internal polar thermofluids

Inequality (179) will hold if

and

Conditions (180) and (181) are the same for the compressible case i.e. the rate of work due to

Constitutive theories for

Remarks

1) Conditions resulting from the entropy inequality require decomposition of

2) Use of stress decomposition (164) in the conditions resulting from the entropy inequality permits determination of the constitutive theory for equilibrium stress tensor for compressible as well as incompressible internal polar thermofluids in terms of thermodynamic pressure and mechanical pressure.

3) The inequalities (170) or (179) require the rate of work due to

4) The inequality (172) or (181) can be used (shown later) to derive a simple constitutive theory for

5) The equilibrium stress

6) The rate constitutive theories for deviatoric Cauchy stress tensor, Cauchy moment tensor and heat vector are derived using theories of generators and invariants [

Consider the following (from (173))

Let

The coefficients

To determine material coefficients from (186), one considers Taylor series expansion of each

One notes that

tions of

Collecting coefficients (quantities defined in

Using (189), one can write (188) as follows

Consider (from (173))

Let

Let

The absence of unit vector in (192) is due to the fact that uniform temperature field does not contribute to

for

Then, using (193) the resulting form of (192) can be written as

Consider the following (from (173))

Let

The coefficients

To determine the material coefficients from (197), one considers Taylor series expansion of each

Then using (198) in (196) can be written as

1) The constitutive theories for

2) The configuration

3) An important point to note is that the material coefficients in the final forms of the constitutive theories are defined in a known configuration

4) Using the derivations presented in Sections 5.2-5.4 rate constitutive theories of various orders in desired basis can be derived by choosing values of n and ^{1}n, the orders of the rate theory. As the orders of the rate theory increase, the number of material constants increases significantly. Thus, the higher order rate theories necessitate elaborate experiments to calibrate them.

5) In the following rate theories of orders one (

This is the simplest possible constitutive theory for

In this case the combined generators of

and the combined invariants of

Thus, one can write

Following the general derivations in Section 5.2 for N generators and M invariants, for this specific case one can write

The definitions of material coefficients

remain the same as defined in (189). This constitutive theory requires 46 material coefficients, still too many to determine experimentally.

Consider a constitutive theory in which

In this case there are only two generators (

and the following constitutive theory (using (190) for

This constitutive theory requires 14 material coefficients and contains up to fifth degree terms in the components of

Begin with (204) and neglect those terms on the right side of (204) that are of degree higher than two in the components of

This constitutive theory requires 8 material coefficients.

If one further neglects the product terms in

This constitutive theory requires only six material coefficients. The dependence of the material coefficients on the invariants in (209) can be modified based on the assumptions used here or can be maintained as originally defined in (189).

If one neglects quadratic terms in

for

If one denotes

Material coefficients

1) One notes that the arguments of

2) Some specific remarks can be made for the simplified rate theory of order one given by (211). When one compares (211) with the similar theory for

Equation (211) implies that

Hence, one can write (211) as

That is, the linear constitutive theory of order one in (214) for deviatoric Cauchy stress tensor is basis independent.

3) Since the material coefficients

The most general constitutive theory for Cauchy moment tensor

In this case

and

The constitutive theory for

The material coefficients in (218) are defined by (198).

One begins with (218) and neglects those terms on the right side of (218) that are of degree higher than two in the components of

This constitutive theory requires eight material coefficients. If one further neglects the product terms in

This constitutive theory requires only six material coefficients. The dependence of the material coefficients on the invariants in (220) can be modified based on the assumptions used here or can be maintained as originally defined in (198).

If one denotes

The material coefficients

In reference [

gate of

be basis dependent i.e. it must be the convected time derivative of the rotation gradient tensor i.e. one must replace it with

where

Hence,

as

Thus

This is an approximation as

Recall that

or

or

Likewise

or

or

Thus for small rates of rotation gradients one can write

Thus, (228) used in reference [

In this paper ordered rate constitutive theories of orders n and ^{1}n are presented for internal polar non-classical, isotropic, homogeneous thermofluids in which the varying rates of rotations and conjugate moments in addition to usual thermofluid physics (classical) are considered in the derivations of the conservation and balance laws. The constitutive theories are presented in contravariant basis, covariant basis, and using Jaumann rates, but the derivation of the constitutive theories is carried out using basis independent stress tensor, heat vector, and moment tensor. By choosing these in the desired basis, basis dependent constitutive theories can be obtained. The theory of generators and invariants in conjunction with the conditions resulting from entropy inequality form the basis for the derivation of the constitutive theories.

The dependent variables in the constitutive theories are established by examining conservation and balance laws in conjunction with principle of casualty [

where

as argument tensors of all dependent variables (based on principle of equipresence [

and deviatoric tensors

thermodynamic pressure

and

are established using theory of generators ad invariants. General constitutive theories of up to order n for^{1}n for ^{1}n for

It is clearly shown in the paper that choice of

The work presented in this paper removes the restriction of small rotation gradient rates due to use of

The first and third authors are grateful for the support provided by their endowed professorships during the course of this research. The support and resources provided by the Computational Mechanics Laboratory (CML) of the Mechanical Engineering department of the University of Kansas is gratefully acknowledged. The financial support provided to the second author by the Department of Mechanical Engineering of the University of Kansas is greatly appreciated.

Surana, K.S., Long, S.W. and Reddy, J.N. (2016) Rate Constitutive Theories of Orders n and ^{1}n for Internal Polar Non-Classical Thermofluids without Memory. Applied Mathematics, 7, 2033- 2077. http://dx.doi.org/10.4236/am.2016.716165