Scientific Research

An Academic Publisher

Rate Constitutive Theories of Orders n and 1n for Internal Polar Non-Classical Thermofluids without Memory

**Author(s)**Leave a comment

KEYWORDS

1. Introduction

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 and time t. is the density of the fluid in the current configuration and is a function of and, , and denote the Helmholtz free-energy density, temperature, and entropy density, respectively in the current configuration and are also functions of. is the Cauchy stress tensor (in Eulerian description in contravariant basis). The superscript “0” is used to signify that it is rate of order zero and the lowercase parenthesis destinguish it from the second Piola-Kirchhoff stress tensor used in Lagrangian description. Dot on any quantity refers to the material derivative. As explained above undeformed and deformed configurations can be used in the derivatives as long as the final equations from the conservation and balance laws contain and t and do not have displacements and strains in them as these are not available for fluent continua. In the following a brief explanation of notations is necessary as some of the notations are new. and denote the position coordinates of a material point in the reference and current configurations, respectively, in a fixed frame (x-frame)

(1)

or

(2)

If and are the components of length and in the reference and current configurations, and if one neglects the infinitesimals of orders two and higher in both configurations, then one obtains

(3)

(4)

with

(5)

In Murnaghan’s notation

(6)

in which the columns of are covariant base vectors, whereas the rows of are contravariant base vectors [4] . and are Jacobians of deformation in covariant and contravariant bases. Furthermore is Lagrangian description while is Eulerian description. The basis defined by is reciprocal to the basis defined by. The following relations are useful in the paper:

(7)

(8)

where

(9)

in which stands for material time derivative, is the spatial velocity gradient

tensor, and are velocity components of a material point in the current configuration in the x-frame. Over bar on all dependent quantities refers to their Eulerian description, i.e. they are functions of and t whereas the quantities without over bar are their Lagrangian description i.e. they are functions of and t. Thus and are Eulerian and Lagrangian description of a quantity Q in the current configuration.

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.

2. Rotation Gradients, Their Convected Time Derivatives and Conservation and Balance Laws

, symmetric part of the velocity

gradient tensor. Let define Cauchy stress tensor, Cauchy moment tensor, and convected time derivatives of the conjugate strain tensor in a chosen basis. Derivations of the constitutive theories is presented using this notation so that the resulting derivations are basis independent. By replacing

with (),

(), and (),

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 [3] the authors show that Cauchy moment tensor and symmetric part of the gradient of the rate of rotation tensor are conjugate. In reference [3] , the authors considered symmetric part of the gradients of rates of rotation obtained using skew symmetric part of the velocity gradient tensor. One notes that the Cauchy moment tensor is basis dependent: being moment tensors in contravariant basis, covariant basis, and in Jaumann rates. Thus, the convected time derivatives of the symmetric part of the rotation gradient tensor in general must also be basis dependent. Let

(), () 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 (), () and

() in contravariant and covariant bases and in Jaumann rates. Covariant and contravariant bases are important in conservation and balance

laws as well as constitutive theories. Jacobian of deformation or is Lagrangian description. Columns of are covariant base vec-

tors [4] . Thus, quantities derived using are in covariant basis and are Lagrangian

descriptions. Likewise or is Eulerian description

of Jacobian of deformation. Rows of are contravariant base vectors. Hence, quantities derived using are in contravariant basis and are Eulerian descriptions. The convected time derivatives of the rotation gradient tensors in covariant and contravariant bases must be derived using rotation gradient tensor obtained using and. Details are presented in the following.

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 of a material point in the current configuration with undeformed coordinates in the reference configuration. Then

(10)

2.1. Covariant Basis: Internal Rotations, Rotation Matrix, Rotation Gradient Tensor and Their Convected Time Derivatives

(a) Internal rotations and rotation matrix

Consider decomposition of the Jacobian of deformation into symmetric and skew-symmetric tensors.

(11)

(12)

(13)

Let be the components of the rotations ex-

pressed as rotations about, , and axes of the x-frame, then one can write

(14)

in which

(15)

Alternatively one can also derive (15) as follows.

(16)

(17)

(18)

is the permutation tensor.

The sign differences between (15) and (18) are due to clockwise and counterclockwise internal rotations and will only affect sign of term in the balance of angular momenta. If one uses (15) as the definition of rotations then the term containing in the balance of angular momenta must have negative sign. If the rotations in (18) are defined as, , and then the term containing in the balance of angular momenta must have positive sign. Regardless, the resulting equations and the following derivations are not affected. One notes that decomposition in (11) enables explicit description of stretches and rotations contained in due to deformation of solid matter. The stretch tensor and the rotation tensor can also be obtained using polar decomposition of into right stretch tensor or left stretch tensor and pure rotation tensor [1] [2] [3] [4] .

(19)

The stretch tensors and are symmetric and positive-definite and the rotation tensor is orthogonal. Since in (19) and in (15) are both obtained from the same deformation in, these contain details of the same internal rotation physics but in different forms. One may make the following remarks.

1) is rotation matrix, hence relates undeformed orthogonal frame to a new orthogonal rotated frame (due to deformation).

2) on the other hand contains rotation angles due to deformation about the axes of the x-frame.

3) Determination of from or determination of from is not necessary. Two different mathematical forms of rotation physics in and is sufficient. However, this process of obtaining from or from in general is not unique and may not even be possible without some approximation [5] [6] [7] [8] .

4) It suffices to note that internal rotations at a material point present in can be expressed either in or in. Both forms contain mathematical description of same physics, hence either can be used as deemed suitable, but determination of from or from is not necessary.

5) The internal rotation angles are present at every material point and are a result of deformation. Between two neighboring material points the variation of is perhaps small otherwise there may be permanent damage or separation between them. Regardless of the magnitude of, these are strictly deterministic from, , or the polar decomposition.

(b) Internal rotation gradient tensor and its rates using

The covariant internal rotation tensor is a tensor of rank two, hence alternatively one can write

(20)

Let be the internal rotation gradient tensor, a tensor of rank three. Using (20) one can define

(21)

Alternatively (16) can be written as

(22)

and then

(23)

In (22) the internal rotations are expressed as a tensor of rank one (i.e., , as a vector), hence its gradient in (23) appears as a tensor of rank two. The representation (22) is more appealing for matrix and vector forms given in the following. Let

(24)

Then, one defines rotation gradient tensor and its decomposition into symmetric and skew-symmetric tensors and.

(25)

(26)

(27)

One can also define the velocity gradients as

(28)

in which

(29)

(30)

Likewise if or is the rotation rate then its gradients are given by

(31)

(32)

(33)

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, hence can not be changed. However, by replacing with these measures can be converted to Eulerian description.

4) Convected time derivatives of of orders up to are defined as.

(c) Second Piola-Kirchhoff covariant rotation gradient tensor

Consider isotropic, homogeneous, compressible matter. Let and be the scalar and and be the vector areas of the oblique planes of the deformed and the undeformed tetrahedra. Let the resultant rotation gradient vector acting on these areas be and and the average rotation gradient vectors on these areas be and. Let the covariant Cauchy rotation gradient tensor be acting on the deformed tetrahedron and let be the corresponding second Piola-Kirchhoff covariant rotation gradient tensor acting on the undeformed tetrahedron. Consider the following correspondence rule [4] .

(34)

(35)

Thus, one obtains

(36)

(37)

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

(38)

using (35) in (38)

(39)

hence, one obtains

(40)

and

(41)

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. For incompressible fluent continua, hence for this case (40) and (41) can be modified.

(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 for compressible matter is presented. Consider

(42)

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

(43)

using

(44)

(45)

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

(46)

If one defines

(47)

(48)

then one obtains the following from (46)

(49)

is the first convected time derivative of the covariant rotation gradient tensor for compressible matter. To obtain the second convected time derivative of the covariant rotation gradient tensor one can take material derivative of (49), and following the same steps as in case of one obtains:

(50)

where

(51)

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 compressible continua.

(52)

For incompressible case and in (52).

2.2. Contravariant Basis: Internal Rotations, Rotation Matrix, Rotation Gradient Tensor and Their Convected Time Derivatives

(a) Internal rotations and rotation matrix

Following the derivations for covariant measures, one can derive the following if one considers Jacobian of deformation in contravariant basis. Rows of are con- travariant base vectors. Consider decomposition of into symmetric and skew- symmetric tensors.

(53)

(54)

(55)

Let be the components of the rotations about, , and axes of the x-frame, then one can write

(56)

in which

(57)

Alternatively one can also derive (57) as follows.

(58)

(59)

(60)

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 and the rotation tensor can also be obtained using polar decomposition of into right stretch tensor or left stretch tensor and rotation tensor [1] [2] [3] [4] .

(61)

The stretch tensors and are symmetric and positive-definite and the rotation tensor is orthogonal. Since in (61) and in (57) are both obtained from the same deformation in, these contain details of the same internal rotation physics but in different forms. The following remarks parallel to those for covariant measures can be made.

1) is rotation matrix due to deformation, hence relates two orthogonal frames.

2) on the other hand contains rotation angles due to deformation about the axes of the x-frame due to rotations about contravariant axes.

3) One notes that determination of from or determination of from is not necessary. Two different mathematical forms of rotation physics is sufficient in derivation of the conservation and balance laws. However, this process of obtaining from or from in general is not unique and may not even be possible without some approximation [5] [6] [7] [8] .

4) It suffices to note that internal rotations at a material point present in can be expressed either in or in. Both forms contain mathematical description of same physics, hence either can be used as deemed suitable, but determination of from or from is not necessary.

5) The internal rotation angles are present at every material point and are a result of deformation. Between two neighboring material points the variation of is perhaps small otherwise there may be permanent damage or separation between them. Regardless of the magnitude of, these are strictly deterministic from, , or the polar decomposition.

(b) Internal rotation gradient tensor using

The contravariant internal rotation tensor is a tensor of rank two, hence alternatively one can define

(62)

Let be the internal rotation gradient tensor, a tensor of rank three. Using (62) one can define

(63)

Alternatively (62) can be written as

(64)

and then

(65)

In (64) the internal rotations are expressed as a tensor of rank one (i.e., , as a vector), hence its gradient in (65) appears as a tensor of rank two. The representation (64) is more appealing for matrix and vector representations given in the following. Let

(66)

Then the rotation gradient tensor and its decomposition into symmetric and skew-symmetric tensors and are defined as:

(67)

(68)

(69)

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 with, these measures will become Lagrangian descriptions.

4) Convected time derivatives of of orders up to in contravariant basis are defined as.

(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 and and vector areas and. Let the resultant rotation gradient vector acting on these areas be and and the average rotation gradient vectors on these areas be and.

(70)

Let the contravariant Cauchy rotation gradient tensor be acting on the deformed tetrahedron and let be the corresponding second Piola-Kirchhoff covariant rotation gradient tensor acting on the faces of the undeformed tetrahedron derived from using correspondence rule (70). Then one can write

(71)

(72)

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

(73)

using

(74)

(75)

Hence, one obtains

(76)

and

(77)

Also in (77) can be replaced by if so desired. Equations (76) and (77) define contravariant second Piola-Kirchhoff rotation gradient tensor in Lagrangian and Eulerian descriptions. For incompressible fluent continua, hence (76) and (77) can be modified for this case.

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

Consider the material derivative of, the second Piola-Kirchhoff rotation gradient tensor (Equation (77)) derived using contravariant Cauchy rotation gradient tensor.

(78)

using

(79)

(80)

and regrouping the terms one obtains

(81)

If one defines

(82)

(83)

then one can write

(84)

Here is the first convected time derivative of the contravariant Cauchy rotation gradient tensor for compressible matter. To obtain the second convected

time derivative of the contravariant Cauchy rotation gradient tensor, one takes

material derivative of (84) and follows the same steps as in case of, then one obtains the following:

(85)

where

(86)

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 compressible matter.

(87)

For incompressible case and, hence the expressions for the convected time derivatives can be modified for this 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 as basis independent convected time derivatives of the rotation gradient tensor. By choosing to be or or one can obtain convected time derivatives of the rotation gradient tensor in contravariant basis, covariant basis and in Jaumann rates.

2.3. Polar Decomposition of Velocity Gradient Tensor and Consideration of Local Rotation Rates

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 can be decomposed into pure rates of rotation and right or left stretch rate tensors and. is orthogonal and and are symmetric and positive definite. The rotation rate tensor can equivalently be obtained due to rotation rates at each location in the flow domain. Thus, at each location in the flow domain the rotation rate matrix can be viewed as being due to. If varying rotation rates at varying locations in the flow domain are resisted by the constitution of the fluent continua then this must result in additional dissipation that requires existence of energy conjugate moments in the deforming matter. Thus, at the onset and its conjugate are considered in the derivation of the polar continuum theory for the fluent continua. Details of polar decomposition of and rotation rates are given in the following. Let

(88)

Let be the eigenvalues of in which, then

(89)

The columns of are eigenvectors and is a diagonal matrix of. If one chooses

(90)

Then (89) holds, hence can be defined using (90). can now be determined using (88)

(91)

Thus, and are established in polar decomposition (88). Using

(92)

and following a similar procedure one can establish the following

(93)

(94)

in which are eigenpairs of. defined by (91) or (94) is unique. The rate of rotation matrix can equivalently be obtained due to rotation rates at each location. Thus, at each location can be viewed as being due to rates of rotations. Rate of energy dissipation due to requires coexistence of moments (per unit area) on the oblique surface of the tetrahedron in the deforming matter. Thus

(95)

where

(96)

Explicit forms of i.e., , and or, , and are defined in terms of velocity gradients. These rotation rates on the oblique plane of the tetrahedron are conjugate with the resultant moment tensor, hence result in rate of work. At this stage these are not basis dependent. When is converted to moment tensor or or, then there is basis dependency in. In the energy equation and likewise in entropy inequality their derivations are continued with until at a later stage when gradients of are needed, convected time derivatives of the rotation gradient tensor in the appropriate basis are introduced.

2.4. Conservation and Balance Laws

In reference [1] [2] conservation and balance laws were derived for internal polar fluent continua. These derivations were presented using, and as basis independent constitutive tensors which were then given appropriate definitions of, ,;, ,; and, , depending on whether the basis of choice is contravariant, covariant or Jaumann rates. Additionally were used as argument tensors that are basis independent convected time derivatives of strain tensor that could be or or depending upon the contravariant or covariant basis or Jaumann rates. The constitutive theory for thermofluids without memory in reference [3] utilizes and as conjugate pairs in the derivation. Since is basis dependent, it’s conjugate pair(s) must also be basis dependent. Choice of the rotation rate gradient resulting from the velocity gradient tensor as conjugate to limits the applicability of the resulting theory to small strain rates and small rotation rates and its gradients. Furthermore, this choice can not be extended to higher order time derivatives of the rotation gradient tensor as it is not the convected time derivative of rotation gradient tensor in Eulerian description. The work presented in this paper proposes and replaces with the true convected time derivatives of the rotation gradient tensor that can be considered in a chosen basis. Use of correct convected time derivatives of the rotation gradient tensor requires the rederivation of energy equation and entropy inequality using correct measures. Using as measures of Cauchy stress tensor, Cauchy moment tensor, and heat vector, as convected time derivatives of the appropriate strain tensors and as measures of the convected time derivatives of the rotation gradient tensor (all will eventually be basis dependent) one can write the following for conservation of mass, balance of linear momenta, balance of angular momenta and balance of moments of moments (or couples).

(97)

(98)

(99)

(100)

At this stage the Cauchy stress tensor is nonsymmetric but the Cauchy moment tensor is symmetric (due to moment of moments Equation (100)). In (98) - (100) basis independent and have been used as opposed to their contravariant measures, but the details of the derivations remain similar. Since the energy equation and the entropy inequality require rate of work due to in addition to rate of work due to, their derivations in reference [3] does not hold here as the conjugate pair to is no longer gradient of rotation rate tensor resulting from the velocity gradient tensor, instead it is convected time derivative of the rotation gradient tensor in appropriate basis depending upon the choice of basis for the Cauchy moment tensor.

2.4.1. First Law of Thermodynamics: Energy Equation

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 [4] [9] [10] [11]

(101)

, , and are total energy, heat added, and work done. These can be written as

(102)

(103)

(104)

where is specific internal energy, is body force per unit mass, are dis-

placement, and is rate of heat. Note the additional term in con-

tributes additional rate of work due to rates of rotation in (104). Expand each of the integrals in (102)-(104). Following reference [4] , it is straight forward to show that:

(105)

(106)

(107)

Using basis independent Cauchy stress tensor, Cauchy principle, and following the details in reference [4] one can write

(108)

Likewise using basis independent moment tensor (per unit area), Cauchy principle, and following the details similar to these used in deriving (108), one can write

(109)

The first convected time derivative of the rotation gradient tensor,

that is conjugate to the Cauchy moment tensor has been used in (109). Using (105)-(109) in (101)

(110)

Transferring all terms to left of equality and regrouping

(111)

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

(112)

Since is arbitrary, (112) implies that

(113)

Equation (113) is the final form of the energy equation in which is a nonsymmetric Cauchy stress tensor and is a symmetric Cauchy moment tensor. Thus in (113) one can use

(114)

In (114) the following decomposition of into symmetric and antisymmetric tensors has been used.

(115)

By appropriate choices of, the explicit form of the energy equation in any desired basis can be obtained.

2.4.2. Second Law of Thermodynamics: Entropy Inequality

If is the entropy density in volume, is the entropy flux between and the volume of matter surrounding it and is the source of entropy in due to non-contacting bodies, then the rate of increase in entropy in volume is at least equal to that supplied to from all contacting and non-contacting sources [4] . Thus

(116)

Using Cauchy’s postulate for i.e.

(117)

Using (117) in (116)

(118)

One recalls that [4]

(119)

and

(120)

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

(121)

Since volume is arbitrary, (121) implies

(122)

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

(123)

where is absolute temperature, is heat vector, and is a suitable potential. Using (123)

(124)

Substituting for from (123) and for from (124) into (122) and multiplying by.

(125)

From energy Equation (113) (after inserting term) in basis independent form one can write the following.

(126)

Substituting from (126) into (125)

(127)

or

(128)

Let be Helmholtz free energy density (specific Helmholtz free energy) defined by

(129)

Hence

(130)

Substituting from (130) into (128)

(131)

or

(132)

is symmetric but is not symmetric. Since is symmetric, one can use the following in (132).

(133)

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

3. Stress Decomposition and Balance Laws

It is instructive to decompose stress tensor into symmetric and antisymmetric tensors

(134)

where

(135)

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

(136)

(137)

as

(138)

one can write (137) as

(139)

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

(140)

(141)

(142)

(143)

A simple calculation by expanding the terms shows that

(144)

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

(145)

(146)

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 and are con- jugate pairs, hence are responsible for conversion of mechanical energy into heat or entropy. The conjugate pairs are instrumental in deciding the dependent variables in the constitutive theories and some of their argument tensors. These conjugate pairs suggest that can be expressed as a function of and as a function of. One notes that and are also conjugate, thus can be expressed as a function of. These details of the constitutive theories are presented in the following sections.

4) This mathematical model has closure once the constitutive theories for, and (total of 15 equations) are added to the already existing eight equations for the conservation and balance laws (conservation of mass, balance of linear momenta, balance of angular momenta and energy equation) giving rise to a total of 23 equations in 23 variables,.

4. Dependent Variables in the Constitutive Theories

The choice of dependent variables in the constitutive theories must be consistent with the axiom of casualty [4] [9] [10] . The self observable quantities and those that can be derived from them by simple differentiation and/or integration can not be considered as dependent variables in the constitutive theories. Thus velocities, temperatures, temperature gradients, etc. are ruled out as choices of dependent variables in the constitutive theories. From the entropy inequality one notes that are possible choices of dependent variables in the constitutive theories. The choice of, , and as dependent variables in the constitutive theories is also supported by balance of linear momenta, balance of angular momenta, and the energy equation. can not be chosen as dependent variables in the constitutive theories as these are deterministic from the balance of angular momenta. Choice of or is a matter of preference as these are related through. In the present work are chosen, hence need not be considered as a dependent variable in the constitutive theories. Thus, and are the possible dependent variables in the constitutive theories. At a later stage of the derivation, some of these may be ruled out as dependent variables in the constitutive theories if so warranted by some other considerations.

Possible choices of argument tensors of dependent variables are considered, keeping in mind the principle of equipresence [4] [9] [10] , i.e. at the onset all dependent variables in the constitutive theories possibly must contain the same argument tensors. For compressible fluent media, density is certainly an argument tensor. is a natural choice as an argument tensor. The choice of as an argument tensor is necessitated due to the dependent variable in the constitutive theory and the physics of heat conduction. The choice of and as argument tensors is also clear as these are conjugate to and. From conservation of mass in Lagrangian description i.e. compressibility is due to, hence it is fitting to consider as an argument tensor in Eulerian description as opposed to for the dependent variables in the constitutive theories. At a later stage dependence on can be replaced by dependence on by using calculus. Thus, based on the principle of equipresence [4] [9] [10] all dependent variables have the same argument tensors.

(147)

One notes that is the first convected time derivative of the strain tensor (Almansi tensor or Green’s tensor or Jaumann rates) and is a fundamental kinematic tensor. In addition to, , are also fundamental kinematic tensors up to order n that are convected time derivatives of orders of strain tensor in a chosen basis. With the choice of, the first convected time derivative of the strain tensor only in (147) the resulting constitutive theories would be rate constitutive theories of order one. can be replaced with as these all are fundamental kinematic tensors to generalize the derivation for the rate constitutive theories to up to order n.

Similarly is the symmetric part of the first convected derivative of the rotation gradient tensor and is a fundamental kinematic tensor. In addition to, one also has as fundamental kinematic tensors that are symmetric part of the convected time derivatives of the rotation gradient tensor. With the use of only in the constitutive theory, the resulting constitutive theory will be of order one in rotation gradient rates. One replaces by to generalize the derivation of the rate constitutive theories up to order.

Secondly, since the arguments in (147) are basis dependent the heat vector is no longer, but instead is indicating that it could be depending upon the choice of the basis. One can write the following for the dependent variables in the constitutive theories and their arguments tensors.

(148)

From the entropy inequality one notes that and

are conjugate pairs i.e. has no dependence on and likewise

has no dependence on. Thus one can modify the argument tensors of and in (148).

(149)

5. Entropy Inequality and Constitutive Theories

Consider the entropy inequality (146) with the arguments of defined in (149). Now

can be obtained which is needed in the entropy inequality.

(150)

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

(151)

Using (151) in (150)

(152)

One notes that

(153)

Using (153) in (152)

(154)

Substituting from (154) in the entropy inequality (146)

(155)

Regrouping terms in (155)

(156)

For (156) to hold for arbitrary but admissible, , and, the following must hold

(157)

(158)

(159)

(160)

(161)

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

Remarks

1) Equation (157) implies that is not a function of.

2) Equation (158) implies that is not a function of.

3) Equation (159) implies that is not a function of.

4) Based on (160), is not a dependent variable in the constitutive theories as

, hence is deterministic from.

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

(162)

are inappropriate due to the fact that these imply that is not a function of, as is not a function of these. This is contrary to (149). Inequality (161) in this form is unable to provide one with further details regarding the derivation of the constitutive theories.

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

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

(163)

One notes that there are no mechanisms or conditions that permit eliminating and from the argument list of, hence one must keep them as in (163). Based on (160) and remark (4), is no longer a dependent variable in the constitutive theories. With (161) and (163) one has no further mechanisms to proceed with the derivation of the constitutive theories.

5.1. Decomposition of Stress Tensor

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

(164)

in which one considers the following

(165)

That is is not a function of and and

vanishes when and are zero. Substituting (164) into (161)

(166)

or

(167)

5.1.1. Constitutive Theory for Equilibrium Stress: Compressible Internal Polar Thermofluids

Since is not a function of and and neither is (due to (165)), then the constitutive theory for must be derivable from

(168)

in which

(169)

is called thermodynamic pressure for compressible internal polar thermofluids and is generally referred to as an equation of state [4] [9] in which is expressed

as a function of and or and, where is specific volume. If one

assumes the compressive pressure to be positive, then in (168) can be replaced by. Using (168), inequality (167) reduces to

(170)

Inequality (170) is satisfied if

(171)

and

(172)

Inequalities (171) imply that the rate of work due to i.e. and due to i.e. must be positive. In view of (168) one can write the following for compressible internal polar thermofluids.

(173)

Constitutive theories for, , and must satisfy (171) and (172).

5.1.2. Constitutive Theory for Equilibrium Stress: Incompressible Matter

For incompressible matter density is constant, hence, thus for this case

, hence the constitutive theory for the incompressible case must consider

as. The incompressibility condition given by (174) must be incorporated in the entropy inequality.

(174)

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

(175)

to (167). is arbitrary Lagrange multiplier.

(176)

Using in (176) and regrouping terms.

(177)

In the case of incompressible internal polar thermofluids is a function of only, hence from (177) one obtains

(178)

is called mechanical pressure. Since is an arbitrary Lagrange multi- plier, is not deterministic from the deformation field. In view of (178), (177) reduces to

(179)

Inequality (179) will hold if

(180)

and

(181)

Conditions (180) and (181) are the same for the compressible case i.e. the rate of work due to and must be positive and the constitutive theory for must satisfy (181). In view of (178) one can write the following for incompressible internal polar thermofluids.

(182)

Constitutive theories for, , and must satisfy (180) and (181).

Remarks

1) Conditions resulting from the entropy inequality require decomposition of into equilibrium and deviatoric stresses and (164) to proceed fur- ther.

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 and be positive but provide no mechanisms for deriving constitutive theories for and.

4) The inequality (172) or (181) can be used (shown later) to derive a simple constitutive theory for (Fourier heat conduction law), but better constitutive theories are possible for (shown in subsequent sections)

5) The equilibrium stress is independent of the basis for compressible as well as incompressible polar thermofluids due to the fact that is basis independent. This implies that

(183)

6) The rate constitutive theories for deviatoric Cauchy stress tensor, Cauchy moment tensor and heat vector are derived using theories of generators and invariants [4] [9] [12] - [27] .

5.2. Rate Constitutive Theories of up to Order n for Deviatoric Symmetric Cauchy Stress Tensor: Compressible

Consider the following (from (173))

(184)

Let be the combined generators of the argument tensors

, and that are symmetric tensors of rank two [4] and

be the combined invariants of the same argument tensors [4] . Then, one can express as a linear combination of and iden- tity tensor.

(185)

The coefficients are functions of, and in the current configuration

(186)

To determine material coefficients from (186), one considers Taylor series expansion of each in about a known configuration and retains only up to linear terms in and the invariants (for simplicity).

(187)

One notes that are func-

tions of, and whereas are functions of the same quantities but in the current configuration (186). When (187) is substituted in (185), one obtains the final expression for the most general rate constitutive theory of up to order n for for compressible internal polar thermofluids. The final expression defines the material coefficients in the known configuration. Details are given in the following. First, substitute (187) in (185),

(188)

Collecting coefficients (quantities defined in) of the terms in (188) that are defined in the current configuration and also grouping those terms that are completely defined in the known configuration. Let

(189)

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

(190)

and are material coefficients defined in known configura- tions. This constitutive theory requires material coefficients. The material coefficients defined in (190) are functions of, and. This constitutive theory is based on integrity, hence it is com- plete.

5.3. Rate Constitutive Theories of up to Order n and ^{1}n for Heat Vector: Compressible

Consider (from (173))

(191)

Let be the combined generators of the argument tensors

, , and that are tensors of rank one.

Let be the combined invariants of the same argument tensors. Then, one can express as a linear combination of.

(192)

The absence of unit vector in (192) is due to the fact that uniform temperature field does not contribute to. The negative sign in (192) is because a positive in the direction of the exterior unit normal to the surface of the volume of matter results in heat removal from the volume of matter. The coefficients are functions of, and in the current configuration. To determine the material coefficients from (that are defined in the current configuration) in (192), one considers Taylor series expansion of each about a known configuration in and and retains only up to linear terms in and the invariants and then substitutes these back in (192). If one defines the following in the final expression for

(193)

for and.

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

(194)

, , and are material coefficients defined in known configuration. This constitutive theory defined by (194) requires material coefficients. The material coefficients are functions of, and. This theory is based on integrity, hence is complete.

5.4. Constitutive Theory for Cauchy Moment Tensor: Compressible

Consider the following (from (173))

(195)

Let be the combined generators of the argument tensors of that are symmetric tensors of rank two and be the combined invariants of the same argument tensors of. One can express as a linear combination of and identity tensor.

(196)

The coefficients are functions of and in the current configuration i.e.

(197)

To determine the material coefficients from (197), one considers Taylor series expansion of each in, about a known configuration and retains only up to linear terms in and the invariants (for simplicity) and then substitutes these back in (196). Define the following

(198)

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

(199)

and are material coefficients defined in known configurations. This constitutive theory requires material coefficients. The material coefficients defined in (198) are functions of, and. This constitutive theory is based on integrity, hence is complete.

5.5. Remarks

1) The constitutive theories for and can be made basis specific by choosing these and and specific to the basis of choice.

2) The configuration can be chosen to be reference configuration (undeformed configuration before the commencement of the evolution) in which case the material coefficients will be independent of the deformation. If one chooses to be a known deformed configuration, then the deformation dependent material coefficients are possible in the constitutive theories. Dependence of the material coefficients on the invariants of the argument tensor in a known configuration permits complex description of material coefficients.

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, whereas the constitutive equations hold in the current configuration for which the deformation field is yet to be determined. This of course is a consequence of the Taylor series expansion of the coefficients in the linear combination (using combined generators) about a known configuration. In the currently used constitutive models in the published works [28] for variable material coefficients, the coefficients are expressed as a function of the unknown deformation field in the current configuration. This is obviously not supported by the derivations presented in Sections 5.2-5.4.

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 () and their further simplifications are considered to present rather simple theories that could be used in model problem studies.

5.6. Rate Constitutive Theories of Order One () for: Compressible

This is the simplest possible constitutive theory for in which there is interaction between and. Consider

(200)

In this case the combined generators of and that are symmetric ten- sors of rank two are ()

(201)

and the combined invariants of, are ()

(202)

Thus, one can write

(203)

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

(204)

The definitions of material coefficients and as well as

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

5.6.1. Simplified Rate Constitutive Theory of Order One () for: Compressible

Consider a constitutive theory in which is not dependent on i.e.

(205)

In this case there are only two generators () and three invariants ()

(206)

and the following constitutive theory (using (190) for and) is obtained.

(207)

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

5.6.2. Simplified Rate Constitutive Theory of Order One () for That Is Quadratic in the Components of: Compressible

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

(208)

This constitutive theory requires 8 material coefficients.

If one further neglects the product terms in (last two terms on the right side of (208)) in (208) one obtains

(209)

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).

5.6.3. Simplified Rate Constitutive Theory of Order One () for That Is Linear in the Components of: Compressible

If one neglects quadratic terms in in (209), then one obtains a constitutive theory

for that is linear in.

(210)

If one denotes and, then one can write (210) as

(211)

Material coefficients and can be functions of, and invariants of in the known configuration. The constitutive theory (211) is the simplest possible constitutive theory for deviatoric symmetric Cauchy stress tensor.

5.7. Remarks on Constitutive Theories for: Compressible

1) One notes that the arguments of are same as those of, deviatoric Cauchy stress tensor for classical thermofluids [4] . Thus the constitutive theories for for internal polar thermofluids are the same as those for for classical thermofluids. The fundamental difference is that even though the constitutive theories are the same, they are for different stress measures. is the deviatoric part of the total Cauchy stress tensor, whereas is the deviatoric part of the symmetric part of the Cauchy stress tensor.

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, one notes that and are similar to first and second viscosities and is thermal mo- dulus. Since

(212)

Equation (211) implies that

(213)

Hence, one can write (211) as

(214)

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

3) Since the material coefficients and are functions of, and invariants of in the known configuration, they can be defined using power law, Carreau-Yasuda model, Sutherland law, etc. similar to classical thermofluids (see reference [28] ). In case of incompressible fluid in (214).

5.8. Simplified Constitutive Theories for: Compressible

The most general constitutive theory for Cauchy moment tensor has been presented in Section 5.4. Unfortunately this constitutive theory for requires forty seven material coefficients. In this section simplified constitutive theories are presented that are derived using the general constitutive theory presented in Section 5.4.

5.8.1. Constitutive Theory for without as Argument Tensor: Compressible Case

In this case

(215)

and and. The generators and invariants are

(216)

(217)

The constitutive theory for based on the theory of generators and invariants is given by (199) with and. This constitutive theory requires fourteen material coefficients, still too many for practical applications. Explicit form is given by the following after Taylor series expansion of the coefficients in the linear combination about a known configuration.

(218)

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

5.8.2. Constitutive Theory for That Is Quadratic in But Independent of: Compressible

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.

(219)

This constitutive theory requires eight material coefficients. If one further neglects the product terms in (last two terms on the right side of (219)) in (219), then one obtains

(220)

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).

5.8.3. Constitutive Theory for That Is Linear in But Independent of: Compressible

(221)

If one denotes and, then one can write (221) as

(222)

The material coefficients and can be functions of, and in- variants of in the known configuration. The constitutive theory (222) is the simplest possible constitutive theory for Cauchy moment tensor but permits deformation dependent material coefficients. Thus, here also one can use concepts similar to power law, Carreau-Yasuda model, Sutherland law etc. that are used for the material coefficients in the constitutive theory for.

5.9. Remarks

In reference [3] instead of, the symmetric part of the convected time derivatives of the rotation gradient tensor, only was used as conju-

gate of. Details are presented in the following. Recall from the derivation of the energy equation. Since is basis dependent must also

be basis dependent i.e. it must be the convected time derivative of the rotation gradient tensor i.e. one must replace it with. This treatment is consistent. Authors in [3] use the following.

(223)

is the gradients of the rotation rates, not the convected time derivatives of the rotation gradient tensor.

(224)

where

(225)

Hence,

(226)

as

(227)

Thus

(228)

This is an approximation as is an approximation to the symmetric part of the convected time derivative of the rotation gradient tensor. Use of is justified only when the rates of rotation gradients are very small in which case

(229)

Recall that

(230)

or

(231)

or

(232)

Likewise

(233)

or

(234)

or

(235)

Thus for small rates of rotation gradients one can write

(236)

Thus, (228) used in reference [3] is an approximation to the symmetric part of the convected time derivative of the rotation gradient tensor. Use of in the constitutive theory for is consistent and removes the restriction of small rates of rotation gradients.

6. Summary and Conclusions

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 [4] [9] . By examining the conditions resulting from entropy inequality and after introducing stress decomposition one finally arrives at as dependent variables in the constitutive theory. Back superscript zero implies that these quantities are basis dependent. From entropy inequality the conjugate pairs are established:

.

where and are convected time derivatives of appropriate basis dependent strain tensor and rotation gradient tensor. In reference [3] , symmetric part of the gradients of rotation rates is used in place of. This only holds when the rates of rotation gradients are small. This is a fundamental difference between the rate theories presented here and those in [3] . Following [3] one may generalize the derivation of the constitutive theory by replacing with. However with the use of conjugate to such generalization was not possible in [3] . This is the second major difference in the constitutive theories presented here and those in [3] . is also replaced with at the onset of the derivation of energy equation and subsequently is replaced with. This gives

as argument tensors of all dependent variables (based on principle of equipresence [4] [9] ) in the constitutive theories for thermofluids. Using the conjugate pairs in entropy inequality it is straightforward to eliminate as arguments of and as arguments of. is decomposed into equilibrium

and deviatoric tensors. The constitutive theories for as

thermodynamic pressure for compressible matter and mechanical pressure for incompressible matter are established using entropy inequality and incompressibility condition. Constitutive theories for

,

and

are established using theory of generators ad invariants. General constitutive theories of up to order n for, of up to order ^{1}n for and of up to orders n and ^{1}n for are derived. The simplified forms for and are also given. Further simplification of the theories of orders and that are linear in and are also derived.

It is clearly shown in the paper that choice of conjugate to is justified when the rates of rotation gradients are small. For finite rates of rotation gradients convected time derivatives of the rotation the gradient tensor (in the basis of choice) must be considered conjugate to. The derivations of the convected time derivatives of the rotation gradient tensor in contravariant basis and covariant basis in both Lagrangian and Eulerian descriptions are given in the paper. It is shown that use of conjugate to is not only limiting in terms of magnitude of the rates of rotation gradients but it also inhibits extension of the constitutive theories for and to higher rates of rotation gradient tensor. In the derivation of the constitutive theory for as well as are retained as its argument tensors as there are no conditions or mechanism that suggest their removal from the argument list of. The constitutive theory for heat vector as presented in this paper is naturally basis dependent.

The work presented in this paper removes the restriction of small rotation gradient rates due to use of in [3] and presents a general constitutive theory for and that are functions of the true convected time derivatives of the rotation gradient tensor up to orders in the basis of choice.

Acknowledgments

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.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

*Applied Mathematics*,

**7**, 2033-2077. doi: 10.4236/am.2016.716165.

[1] | Surana, K.S., Reddy, J.N. and Powell, M. (2015) A Polar Continuum Theory for Uent Continua. International Journal of Engineering Research and Industrial Applications, 8, 107-146. |

[2] | Surana, K.S., Powell, M. and Reddy, J.N. (2015) A More Complete Thermodynamic Framework for Uent Continua. Journal of Thermal Engineering, 1, 1-16. |

[3] | Surana, K.S., Powell, M. and Reddy, J.N. (2015) Ordered Rate Constitutive Theory for Internal Polar Thermouids. International J. of Math. Sci. & Engg. Appls., 9, 51-116. |

[4] | Surana, K.S. (2015) Advanced Mechanics of Continua. CRC/Taylor & Francis Group, Boca Raton. |

[5] |
Steinmann, P. (1994) A Micropolar Theory of Finite Deformation and Finite Rotation Multiplicative Elastoplasticity. International Journal of Solids and Structures, 31, 1063-1084.
http://dx.doi.org/10.1016/0020-7683(94)90164-3 |

[6] |
Srinivasa, A.R. and Reddy, J.N. (2013) A Model for a Constrained, Finitely Deforming, Elastic Solid with Rotation Gradient Dependent Strain Energy, and Its Specialization to von Kármán Plates and Beams. Journal of the Mechanics and Physics of Solids, 61, 873-885.
http://dx.doi.org/10.1016/j.jmps.2012.10.008 |

[7] | Segerstad, P.H., Toll, S. and Larsson, R. (2009) A Micropolar Theory for the Finite Elasticity of Open-Cell Cellular Solids. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 465, 843-865. |

[8] |
Shield, R.T. (1973) The Rotation Associated with Large Strains. SIAM Journal on Applied Mathematics, 25, 483-491. http://dx.doi.org/10.1137/0125048 |

[9] | Eringen, A.C. (1967) Mechanics of Continua. John Wiley and Sons, Hoboken. |

[10] | Eringen, A.C. (1962) Nonlinear Theory of Continuous Media. McGraw-Hill, Boston. |

[11] | Reddy, J.N. (2013) An Introduction to Continuum Mechanics. Cambridge University Press, Cambridge. |

[12] |
Smith, G.F. (1960) On the Minimality of Integrity Bases for Symmetric 3 × 3 Matrices. Archive for Rational Mechanics and Analysis, 5, 382-389.
http://dx.doi.org/10.1007/BF00252916 |

[13] |
Smith, G.F. (1965) On Isotropic Integrity Bases. Archive for Rational Mechanics and Analysis, 18, 282-292. http://dx.doi.org/10.1007/BF00251667 |

[14] | Smith, G.F. (1970) On a Fundamental Error in Two Papers of CC Wang on Representations for Isotropic Functions, Part I and II. Arch. Ratl. Mech. Anal, 36. |

[15] |
Smith, G.F. (1971) On Isotropic Functions of Symmetric Tensors, Skew-Symmetric Tensors and Vectors. International Journal of Engineering Science, 9, 899-916.
http://dx.doi.org/10.1016/0020-7225(71)90023-1 |

[16] |
Spencer, A.J.M. (1961) The Invariants of Six Symmetric 3 × 3 Matrices. Archive for Rational Mechanics and Analysis, 7, 64-77. http://dx.doi.org/10.1007/BF00250750 |

[17] |
Spencer, A.J.M. (1965) Isotropic Integrity Bases for Vectors and Second-Order Tensors. Archive for Rational Mechanics and Analysis, 18, 51-82.
http://dx.doi.org/10.1007/BF00253982 |

[18] |
Spencer, A.J.M. and Rivlin, R.S. (1958) Finite Integrity Bases for Five or Fewer Symmetric 3 × 3 Matrices. Archive for Rational Mechanics and Analysis, 2, 435-446.
http://dx.doi.org/10.1007/BF00277941 |

[19] |
Spencer, A.J.M. and Rivlin, R.S. (1958) The Theory of Matrix Polynomials and Its Application to the Mechanics of Isotropic Continua. Archive for Rational Mechanics and Analysis, 2, 309-336. http://dx.doi.org/10.1007/BF00277933 |

[20] |
Spencer, A.J.M. and Rivlin, R.S. (1959) Further Results in the Theory of Matrix Polynomials. Archive for Rational Mechanics and Analysis, 4, 214-230.
http://dx.doi.org/10.1007/BF00281388 |

[21] |
Spencer, A.J.M. and Rivlin, R.S. (1962) Isotropic Integrity Bases for Vectors and Second-Order Tensors. Archive for Rational Mechanics and Analysis, 9, 45-63.
http://dx.doi.org/10.1007/BF00253332 |

[22] |
Wang, C.C. (1969) On Representations for Isotropic Functions. Archive for Rational Mechanics and Analysis, 33, 249-267. http://dx.doi.org/10.1007/BF00281278 |

[23] |
Wang, C.C. (1969) On Representations for Isotropic Functions. Archive for Rational Mechanics and Analysis, 33, 268-287. http://dx.doi.org/10.1007/BF00281279 |

[24] |
Wang, C.C. (1970) A New Representation Theorem for Isotropic Functions: An Answer to Professor GF Smith’s Criticism of My Papers on Representations for Isotropic Functions. Archive for Rational Mechanics and Analysis, 36, 166-197.
http://dx.doi.org/10.1007/BF00272241 |

[25] |
Wang, C.C. (1970) A New Representation Theorem for Isotropic Functions: An Answer to Professor G. F. Smith’s Criticism of My Papers on Representations for Isotropic Functions. Archive for Rational Mechanics and Analysis, 36, 198-223.
http://dx.doi.org/10.1007/BF00272242 |

[26] |
Wang, C.C. (1971) Corrigendum to My Recent Papers on Representations for Isotropic Functions. Archive for Rational Mechanics and Analysis, 43, 392-395.
http://dx.doi.org/10.1007/BF00252004 |

[27] |
Pennisi, S. and Trovato, M. (1987) On the Irreducibility of Professor GF Smith’s Representations for Isotropic Functions. International Journal of Engineering Science, 25, 1059-1065. http://dx.doi.org/10.1016/0020-7225(87)90097-8 |

[28] | White, F.M. (1991) Viscous Fluid Flow. McGraw-Hill, Inc., Boston. |

Copyright © 2018 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.