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Development of a curvilinear viscoelastic constitutive relationship for time dependent materials. Part A: Theoretical discussion.


Materials such as polymers (plastics) are complicated viscoelastic materials, the stress and strain of which are not easily related to each other. Much has been done relating stress and strain in a small deformation context. Many formulations have also been proposed to solve problems involving finite deformations and viscoelasticity. Huilgol and PhanThien (1) published a significant work outlining many contributions in this field.

Many viscoelastic relationships involve use of the time derivative of a tensor. The derivative chosen is typically the convected time derivative, the Jaumann rate derivative, or a variation of the Jaumann rate derivative in order to retain objectivity with respect to a given fixed coordinate system. These derivatives can, unfortunately, be quite cumbersome to work with and integrate to calculate a solution.

This paper proposes a viscoelastic model involving the use of a convected or material coordinate system where the coordinates are embedded in the body and rotate and deform with the material. This type of system has the advantage that quantities associated with the same point of material at different times can be added as the sum, making calculation of the influence of the history of the deformation relatively straightforward.

The constitutive model proposed in this paper is formulated using the generalized curvilinear form of Hooke's Law. Extensive use of components of the Eucidean metric tensor enabled formulation in a material coordinate system. The proposed model is for simple viscoelastic fluids in that the stress is a function of the history of the deformation gradient.

The constitutive model is developed and formulated here in Part A. The second portion of this work, Part B, illustrates example problems solved using this model.


This section provides a description of the deformation of a continuum in terms of curvilinear coordinates. More general treatments can be found in several texts (2, 3). Let x represent a fixed Cartesian reference frame with coordinates [x.sub.t], and let [zeta] denote a curvilinear material (convected) reference frame with coordinates [[zeta].sup.t]. Assume that the two reference frames coincide at time [t.sub.o] as shown in Fig. 1. Furthermore, assume that the mapping from x to [zeta] is unique and invertible at any time t within the domain of interest. Then one can write

x = x([zeta], t) (1a)

to determine the location of a particular material point [zeta] at time t. Alternatively one can write

[zeta] = [zeta](x, t) (1b)

to establish the material point that is x positioned at at time t.

Let us define unit base vectors [e.sub.i] in the Cartesian coordinate system. Sets of covariant base vectors [g.sub.i] and contravariant base vectors [g.sup.i], illustrated in Fig. 2, are obtained at any material point [zeta] at any time t from the relationships:

[g.sub.i](t) = [partial][x.sub.j]/[partial][[zeta].sup.i] [e.sub.j](t) (2a)

[g.sup.i](t) = [partial][[zeta].sup.i]/[partial][x.sub.j] [e.sub.j](t) (2a)

In Eqs 1 and 2 and throughout the remainder of this paper, the obvious additional dependence of field quantities on [zeta] is omitted. The usual index notation is employed with summations over repeated indices. The components of the covariant ([g.sub.ij]), contravariant ([g.sup.ij]), and the mixed ([g.sup.j.sub.i]) metric tensors can now be defined by the following:

[g.sub.ij](t) = [g.sub.i](t) * [g.sub.j](t) (3a)

[g.sup.ij](t) = [g.sup.i](t) * [g.sup.j](t) (3b)

[g.sup.j.sub.i](t) = [g.sub.i](t) * [g.sup.j](t) = [[delta].sup.j.sub.i](t) (3c)

The mixed tensor components in Eq 3c reduces to the mixed Kronecker delta tensor components ([[delta].sub.i.sup.j]) since the covariant and contravariant base vectors are biorthogonal, as Illustrated in Fig. 2.

The metric tensors describe the local deformation field and consequently can be used to define strain. In particular, the length ds of an infinitesimal length is written

d[s.sup.2](t) = [g.sub.ij](t)d[[zeta].sup.i] d[[zeta].sup.i] (4)

The Green strain tensor (E) may then be defined in terms of its covariant components as

[E.sub.ij](t) = 1/2([g.sub.ij](t) - [g.sub.ij]([t.sub.o])) (5)


E(t) = [E.sub.ij](t)[g.sup.i](t)[g.sup.j](t) (6)

In writing the strain as E(t), its dependence on [t.sub.o] is not stated explicitly In order to simplify the notation. Meanwhile, the covariant components of the displacement vector u(t) are

[u.sub.i](t) = [x.sub.i](t) - [x.sub.i]([t.sub.o]) (7)

where, again, the dependence of u on time [t.sub.o] is understood.


Let [tau](t) represent the Cauchy stress tensor at time t for a material point [zeta]. Thus, in terms of contravariant components:

[tau](t) = [[tau].sup.ij](t)[g.sub.i](t)[g.sub.j](t) (8)

The traction vector T on any surface having outer unit normal v can be expressed as:

[T.sup.j](t) = [[tau].sup.ij](t)[v.sub.t](t) (9a)


[T.sup.j](t) = [[tau].sup.j.sub.i](t)[v.sup.i](t) (9b)

In the absence of dynamic effects, the balance of linear momentum reduces to the requirements for local equilibrium:

[[tau].sup.ij][\.sub.i](t) + [rho][F.sup.j](t) = 0 (10)

where [rho] is the mass density, F is the body force vector per unit mass, and the first term implies covariant differentiation.

Sokolnikoff (4) developed a virtual work expression in general curvilinear coordinates. The derivation is accomplished by taking the dot product of the integrand of the stress equilibrium equation (Eq 10) with a virtual displacement vector and then integrating the resulting expression over the volume.

The overall scalar product expression takes the form [Ag.sup.r] * [Bg.sub.s] = [Abg.sup.r.sub.s] = AB[[delta].sup.r.sub.s]. Therefore the scalar product result is:

[[integral].sub.v] {([[tau].sup.ij][\.sub.i](t) + [rho][F.sup.j](t))[delta][u.sub.j](t)} dV = 0 (11)

Use of the product rule for covariant differentiation and application of the divergence theorem gives:

[[integral].sub.V] [[tau].sup.ij][delta][u.sub.j][\.sub.i](t)dV = [[integral].sub.s] [[tau].sup.j.sub.i][delta][u.sub.j][v.sup.i](t)dS + [[integral].sub.v][rho][F.sup.j][delta][u.sub.j](t)dV (12)

The surface traction vector components will arise from the surface integral term in Eq 12 via Eq 9b. Equation 12 is easily rewritten with the contravariant stress and virtual covariant strain components ([delta][[epsilon].sub.ij]) in the more familiar form:

[[integral].sub.v] [[tau].sup.ij][delta][[epsilon].sub.ij](t)dV = [[integral].sub.s] [T.sup.j] [delta][u.sub.j] (t) dS + [[integral].sub.v] [rho][F.sup.j] [delta][u.sub.j](t)dV (13)

Equation 13 provides an elegant general formulation of the equation of virtual work in curvilinear coordinates. At this stage, the virtual displacements need not be compatible with the essential boundary conditions defined on S.


Sokolnikoff (4) and McConnell (5) developed a generalized Hooke's law in curvilinear coordinates. The derivation is summarized below with all quantities evaluated at the current time t.

The generalized Hooke's law for small deformations is written

[[tau].sub.ij] = [C.sub.ijkl][E.sub.kl] (14)

where for isotropic materials the quality [C.sub.ijkl] is a component of the constitutive tensor defined as

[C.sub.ijkl] = [lambda][[delta].sub.ij][[delta].sub.kl] + G[[delta]][[delta].sub.jk] + G[[delta].sub.ik][[delta].sub.jl] (15)

In curvilinear coordinates:

[[tau].sup.i.sub.j] = [C.sup.ik.sub.jl] [E.sup.l.sub.k] (16)


[C.sup.ik.sub.jl] = [lambda][g.sup.i.sub.j][g.sup.k.sub.l] + [Gg.sup.k.sub.j] [g.sup.i.sub.1] + [Gg.sub.jl][g.sup.ik] (17)

Then, the generalized Hooke's law for isotropic materials as provided in Sokolnikoff (4) in curvilinear coordinates becomes:

[[tau].sup.i.sub.j] = [lambda][E.sup.k.sub.k][[delta].sup.i.sub.j] + 2[GE.sup.i.sub.j] (18)

The contravariant stress is related to the covariant strain using:

[[tau].sup.ij] = [C.sup.ijkl][E.sub.kl] (19)


[C.sub.ijkl] = [lambda][g.sup.ij][g.sup.kl] + [][g.sup.jk] + [Gg.sup.ik][g.sup.jl] (20)

Inclusion of the effect of thermal expansion in the generalized Hooke's Law in curvilinear coordinates can be derived by use of the mixed thermal expansion coefficient tensor. The generalized Hooke's Law can therefore be written as:

[[tau].sup.i.sub.j] = [lambda][E.sup.k.sub.k][[delta].sup.i.sub.j] + 2[GE.sup.i.sub.j] - ([lambda][[alpha].sup.k.sub.k][[delta].sup.i.sub.j] + 2G[[alpha].sup.i.sub.j]) [DELTA]T (21a)

where [DELTA]T represents the temperature change from a stress free datum. For an isotropic materials, the mixed thermal expansion coefficient tensor component, [[alpha].sup.i.sub.j], is simply [[alpha].sup.i.sub.j] = [alpha][[delta].sup.i.sub.j] where [alpha] is the coefficient of thermal expansion. With symmetry in [E.sub.kl] and [[alpha].sub.kl], Eq 21a is rewritten as:

[[tau].sub.ij] = ([lambda][g.sup.ij][g.sup.kl] + 2[][g.sup.jk])([E.sub.kl] - [alpha][g.sub.kl][DELTA]T) (21b)

In Eqs 21a and b, the actual strain (deformation) is simply defined by Eq 5.


A viscoelastic constitutive model must retain objectivity for each time step with respect to a specified coordinate system. IN many large strain inelastic applications, the Updated Lagrangian Jaumann (U.L.J.) formulation is adopted (6, 7) in which the Jaumann

Stress rate (D[[tau].sub.ij]/Dt) is used to express the relationship between stress and strain:

D[[tau].sub.ij]/Dt = [C.sub.ijrs][] (22)

where [C.sub.ijrs] is a component of the constitutive tensor and [] are the components of the rate of deformation tensor. The Jaumann rate derivative is simply

D[[tau].sub.ij]/Dt = D[[tau].sub.ij]/Dt - [[tau].sub.ip][[OMEGA].sub.pj] - [[tau]][[OMEGA].sub.pi] (23)

where [[OMEGA].sub.ij] is a component of the anti-symmetric spin tensor and D/Dt represents the substantial (material) derivative operator.

Prager (8) illustrates that the Jaumann rate derivative is a very practical rate of change of stress and does not contain a dependence on the fixed reference coordinate system one happens to be using. In summary, the substantial rate derivative ensures objectivity with respect to rigid body translation, while the Jaumann rate derivative ensures objectivity with respect to both rigid body translation and rotation.

The Jaumann stress rate is invariant under a rigid body rotation and translation, which lends itself nicely to the U.L.J. formulation. This approach works well for small deformations. Unfortunately, there are problems associated with the traditional Jaumann rate approach in a finite deformation context (9). The main problem of the U.L.J. formulation is that it requires integration of the Jaumann rate derivative in its implementation, which can lead to severe problems, as discussed by Dienes (9) and Atluri (10). A particular problem is that the Jaumann stress rate is a very low order approximation to a rotational transformation. This makes it very difficult to integrate accurately over finite rotation and stretch increments. Another problem encountered is that physically unrealistic results can arise for some finite deformation problems. For example, in finite shear deformation (Fig. 3), the stress oscillates with time in the following manner (9-11):

[[tau].sub.11] = - [[tau].sub.22] = G[1 - cos(2[omega]t)] (24a)

[[tau].sub.12] = G sin (2[omega]t) (24b)

Atluri (10) explores the Jaumann and other objective rate derivatives and shows that some of these derivatives give physically unrealistic results. The oscillatory results indicate that the spin tensor (angular velocity tensor) alone does not accurately calculate the rate of rigid body rotation for pure finite shear deformation when the rotation is measured with respect to the initial configuration. Consequently, it could be argued that the angular spin is not the only contribution to rigid body rotation. Rolph and Bathe (12) show how the displacements in the principal directions produce additional rotation increments due to changes in the principal directions.

Sedov (13-15) and Bondar (16) derive and describe several alternative convective stress rates. Peric (17) and Szabo and Balla (18) review and derive additional rates. All of the fundamental stress rates described seek to overcome the problems of the Jaumann stress rate. Szabo et al. also solve the problem of simple shear using several stress rates in addition to the Jaumann rate and finds that other stress rates can also give oscillating solutions. The main problem with using stress rates is that it is difficult to retain objectivity when integrating the rate to obtain a solution for the stress and strain. A way some try to retain objectivity is to: i) transform the stress and/or strain tensor from the current instantaneous configuration back to either the initial or an alternative configuration; ii) differentiate with respect to time; iii) re-transform the tensor in the opposite direction as described in (i). Needless to say, this process is not trivial. Peric describes how some will simply take advantage of a hyperelastic constitutive model to avoid the time derivative altogether by employing logarithmic strains in a material coordinate system to ensure objectivity.


Several simple elastic examples show how use of a material coordinate system can retain objectivity with respect to both translation and rotation.

Using Fig. 3, the [x.sub.1] and [x.sub.2] coordinates at time t for simple shear deformation are

[x.sub.1](t) = [x.sub.1](0) + 2[omega][tx.sub.2](0) (25a)

[x.sub.2](t) = [x.sub.2](0) (25b)

where [t.sub.o] = 0. The metric tensors were calculated for two dimensions using Eqs 2 and 3 where [[zeta].sup.1] and [[zeta].sup.2] are the material coordinates and d[x.sub.i]/d[[zeta].sup.i](0) = 1 in a material (convected) coordinate system. The material coordinates and the covariant base vectors [g.sub.1] and [g.sub.2] are shown in Fig. 3.

Physical components of stress [sigma] and strain [epsilon] tensors can then be determined. The results illustrating the variation of [[epsilon].sub.22], [[sigma].sup.11], [[sigma].sup.12], and [[sigma].sup.22] with [[epsilon].sub.12] are shown in Figs. 4 and 5. In these plots a value of 0.30 was assumed for the Poisson's ratio ([mu]). Observe that both [[sigma].sup.11] and [[sigma].sup.22] are negative. Negative results for these two normal stresses were also found by Green and Zerna (19). Figure 6 shows that the original lines [AB.sub.o], and [AC.sub.o] (heavy dashed lines) compress to AB and AC (heavy solid lines), respectively, in the directions of the contravariant vectors and the corresponding stresses are compressive.

A second example involving simple extension is shown in Fig. 7. In this case, the object is being pulled in the [x.sub.1] direction and contracts in the [x.sub.2] direction. The [x.sub.1] and [x.sub.2] coordinates are:

[x.sub.1](t) = f(t)[x.sub.1](0) + [x.sub.1](0) f(t) [greater than or equal to] 0 (26a)

[x.sub.2](t) = h(t)[x.sub.2](0) + [x.sub.2](0) h(t) [less than or equal to] 0 (26b)

with the function h(t) chosen such that [[tau].sup.22] = 0 for all t. This example then has some interesting characteristics. The relationship between the physical components of stress and strain are identical to that seen in small deformation theory i.e.:

[[epsilon].sub.11] = [[epsilon].sub.22](1 - [mu]/[mu]) (27a)

[[sigma].sup.11] = [lambda]([[epsilon].sub.11] + [[epsilon].sub.22]) + 2 G[[epsilon].sub.11] (27b)

An example of simple rotation is shown in Fig. 8. The [x.sub.1] and [x.sub.2] coordinates are:

[x.sub.1] = [[[([x.sub.1](0)).sup.2] + [([x.sub.2](0)).sup.2]].sup.1/2] cos

[[theta] + asin {[x.sub.2](0)/[[[([x.sub.1](0)).sup.2] + [([x.sub.2](0)).sup.2]].sup.1/2]} - [[theta].sub.o]] (28a)

[x.sub.2] = [[[([x.sub.1](0)).sup.2] + [([x.sub.2](0)).sup.2]].sup.1/2] sin

[[theta] + asin {[x.sub.2](0)/[[[([x.sub.1](0)).sup.2] + [([x.sub.2](0)).sup.2]].sup.1/2]} - [[theta].sub.o]] (28b)

where [theta] and [[theta].sub.o] are the final and original angles of rotation from the [x.sub.1] axis, respectively.

The metric tensor components are simply [g.sub.ij] = [[delta].sub.ij] and [g.sup.ij] = [[delta].sup.ij]. Consequently, no deformation takes place when a body undergoes simple rotation using this metric tensor approach. This underscores the very important result that objectivity is maintained with a metric tensor approach using material coordinates in that no stress or deformation arose solely as a result of rotation.

Furthermore, the [x.sub.1] and [x.sub.2] coordinates for the case of simple shear with rotation and simple extension with rotation are found by simply substituting [x.sub.1] and [x.sub.2] of Eqs 25 and 26 into [x.sub.1](0) and [x.sub.2](0) of Eqs 28. After application of some trigonometric identities and the fact that a convected coordinate system was used, one can show that the Eucidean metric tensor components for these cases will be exactly the same as in the two cases without rotation. Thus, objectivity is maintained with the present metric tensor formulation.


The theory of deformation and flow of homogeneous materials was discussed in detail by Oldroyd (20-22) with an excellent summary of his work published posthumously in 1984 (22). In these papers, Oldroyd discussed several properties that a constitutive model should possess i.e. that the stress and strain usually depend on all the previous rheological states and that the stress and strain are i) independent of any frame of reference, ii) independent of the position and the translational and rotational motion and iii) independent of the states of neighboring material elements. In summary, position and motion must be completely irrelevant to a rheological problem.

Rheological equations of state are usually expressed in a convected or a material coordinate system where the coordinates are constant and deform with the material. The material coordinates and the current time t are taken as the independent variables. In the material coordinate system, the axes are embedded in the body and rotate with the material. The material (convected) coordinate system has the significant advantage that similar quantities associated with the same material at different times can be added as the sum (22).

Many times the constitutive equations of state are to be solved simultaneously with the equations of motion and continuity with prescribed boundary conditions. These equations are typically written with respect to a fixed coordinate system (frame). Consequently, one needs to transform the constitutive model into a new form where all tensor quantities are referred to the fixed coordinate system, which is usually Cartesian. This transformation can make solving the equations of motion and continuity extremely difficult. The operations of integration and differentiation with respect to time and the addition of quantities associated with different times becomes fairly complicated when one has to relate quantities to a fixed coordinate system. Integration and addition are usually accomplished by transforming the material (convected) components of a tensor quantity such as stress or strain and then performing the addition (20). Unfortunately, however, convected differentiation is generally much more cumbersome, as noted earlier.

Use of the Jaumann derivative to formulate co-rotational rheological models is discussed in detail by Bird et al. (23) and Tadmor and Gogos (24). A big advantage of co-rotational constitutive equations is that only Cartesian tensors are used. Consequently, these models are more easily used In the governing equations of motion and continuity using the Jaumann rate derivative to retain objectivity. Goddard and Miller outlined techniques to integrate the Jaumann rate derivatives (25) within a rheological context. Goddard further expressed functions of the stress and strain using co-rotational expansion models (26).

The results presented in the previous section are very important in that they illustrate the objectivity of the metric tensor approach using material (convected) coordinates. In particular, no extra stress or deformation was induced as a result of any rotation. This point is emphasized in great detail by Lodge (27-30). As particles of a material pass through different places in space, all the changes in shape of the deforming body are described by the Euclidean metric tensor, which does not change in a rigid motion. This metric tensor approach is essentially the convected or material approach of Oldroyd and can be regarded as a coordinate-free version of the Oldroyd formulations (27, 28).

Lodge utilizes the Boltzmann superposition principle as opposed to the Jaumann rate derivative to derive an integral form of a viscoelastic constitutive equation. The strain equation (Eq 5) has the important property of additivity in material coordinates, which is not restricted to the size or orientation of the strains (22, 29). The Boltzmann superposition principle, unfortunately, has the annoying restriction to small strains. Lodge makes use of the property of additivity of a material (convected) coordinate system to formulate his model in a finite deformation mode.

The stress is tantamount to a mapping of body metric tensor components (material coordinate components) onto the material stress tensor components. This mapping can include all tensorially admissible operations such as addition, differentiation, and integration. Everything is evaluated at one particle of material. All of the material constants are treated as scalars (27, 28).

Lodge formulated his rubber-like equation in an Eulerian context for large deformations where the final configuration was used as the reference and the strain was measured as the difference in the contravariant metric tensor components at time t' and t:

[[tau].sup.ij] = [[integral].sup.t.sub.-[infinity]] M(t - t') ([g.sup.ij](t') - [g.sup.ij](t))dt' (29)

Equation 29 is a fairly easy to use constitutive equation since it does not involve the complicated derivatives and is formulated using a material coordinate system. Lodge refers to Oldroyd's convected system as a body metric coordinate system. Bird et al. (31) rewrite this equation for a Cartesian coordinate system. Lodge shows that the rubber-like liquid equation satisfies Oldroyd's rules of rheological invariance and is considered "admissible."

In Eq 29, M(t - t') is called the memory function. For many cases, the memory function can be formulated using linear viscous elements (dashpots) and linear spring elements (springs), thus providing a physically intuitive description of viscoelastic materials. Each spring and dashpot placed in series represents a Maxwell element, which in turn can be arranged in parallel to represent a generalized Maxwell fluid. The memory function of a Maxwell fluid is written as:

M(t - t') = [summation over (N/p=1)] [G.sub.p]/[[xi].sub.p] exp (- (t - t')/[[xi].sub.p]) (30)

where [G.sub.p] and [[xi].sub.p] are the shear modulus and relaxation time, respectively, of each Maxwell element. These ideas and the formulation of the memory function are discussed in great detail by both Bird et al. (31) and Rosen (32). The exponential decay term indicates the fading memory property that viscoelastic materials possess (33, 34). The stress and strain of Eq 29 depend only on the separation of the particles at a particular instant and contain no reference to any other configuration in a convected coordinate system. This is an important point that will be used in the development of our visco-elastic model.

Coleman and Nail (33) and Flugge (35) give detailed descriptions of the use of the superposition principle to derive a visco-elastic model in the limit of small strains. The stress-strain relationship is simply

[[tau].sub.ij](t) = [E.sub.ij](t)Y(0) - [[integral].sup.t.sub.o] M(t - t') [E.sub.ij](t')dt' (31)

where [[tau].sub.ij] the stress at time t, [E.sub.ij] is the strain, M(t - t') is a memory function, and Y(0) is the relaxation modulus at t = 0 which is usually the elastic constants, i.e., the Young's and shear modulus or the Lame constants. Generally, the memory function has the same form as Eq 30, with the Lame constants or the Young's modulus sometimes used in place of the shear modulus. The integral in Eq 31 is usually referred to as the hereditary integral, representing the fact that a simple viscoelastic material "remembers" its strain history (33, 34). All of the previous strains are added; however, the material "remembers" the most recent past better than behavior that occurred at the most distant past time.

Both Coleman and Noll (32, 33) and Lodge (30) derive a viscoelastic law for finite strains by taking an Eulerian approach where the present configuration is taken as the reference configuration. This is typically done for a fluid that has no preferred configuration. The Eulerian viewpoint asks, "Here I am now; where have I been?" The solid mechanics viewpoint, which will be developed here, asks, "Here I am at time zero; now where do I go from here?" This approach is good for materials where one needs to ascertain the deformation from a preferred or reference configuration.

An equation relating stress and strain needs to be rewritten for a viscoelastic material to take into account that while the stress is evaluated at the current time, it depends on the deformation that occurred at the past time and space. The concept of a two-point (double) tensor field is required to handle this issue. Double-tensor fields are functions of two points in space (36). A double-tensor field is defined as [[tau].sup.Ij] ([zeta](t'), [zeta](t)), which depends on the set of variables [zeta](t') and [zeta](t), all of which are evaluated at the past time t' and the current time t. Upper case letters are used to denote the past time (t'). Lower case letters denote the current time (t). The metric tensor components are calculated similar to Eqs 3 as:

[g.sub.Ii]([zeta](t'), [zeta](t)) = [g.sub.I](t') * [g.sub.i](t) (32a)

[g.sup.Ii]([zeta](t'), [zeta](t)) = [g.sup.I](t') * [g.sup.i](t) (32b)

[g.sup.I.sub.i]([zeta](t'), [zeta](t)) = [g.sup.I](t') * [g.sub.i](t) (32c)

[g.sup.i.sub.I]([zeta](t'), [zeta](t)) = [g.sub.I](t') * [g.sup.i](t) (32d)

While the covariant and the contravariant metric tensor components have the property of symmetry, the mixed metric tensor components are not necessarily symmetric, nor do they necessarily reduce to the Kronecker delta tensor components.

The viscoelastic model is now formulated using Eqs 21b and 31:

[[tau].sup.ij](t) = {([summation over (N/p=1)] [[lambda].sub.p]) [g.sup.ij](t)[g.sup.kl](t) + 2 ([summation over (N/p=1)] [G.sub.p]) [g.sup.ij](t)[g.sup.jk](t)}

([E.sub.kl] - [[alpha].sub.kl][DELTA]T)(t) - [[integral].sup.t.sub.0] [M.sub.1](t - t') [g.sup.ij](t)[g.sup.KL](t')([E.sub.KL] - [[alpha].sub.KL][DELTA]T)

(t')dt' - [[integral].sup.t.sub.0] 2[M.sub.2](t - t') [g.sup.iL](t, t') [g.sup.jK](t, t')([E.sub.KL] - [[alpha].sub.KL][DELTA]T)(t')dt' (33)

The variable N is the number of Maxwell elements. The functions [M.sub.1] and [M.sub.2] are the memory functions associated with the Lame constants, represented as

[M.sub.1](t - t') = [summation over (N/p=1)] [[[lambda].sub.p]/[[xi].sub.p]] exp(-(t - t')/[[xi].sub.p]) (34a)

[M.sub.2](t- t') = [summation over (N/p=1)] [[G.sub.p]/[[xi].sub.p]] exp(-(t - t')/[[xi].sub.p]) (34b)

In Eqs 34, [[lambda].sub.p] and [G.sub.p] are the Lame constants of the pth Maxwell element and [[xi].sub.p] is the corresponding relaxation time. The relaxation time of each element is assumed the same for both equations in Eqs 34 because the viscosities of each corresponding element would relate exactly as the Lame constants themselves. The relaxation time is the ratio of the shear viscosity to the shear modulus, and is assumed to be identical for the Lame constant [lambda].

In the case of a Maxwell solid, the dashpot of the Nth element does not exist. The summation in Eqs 34 is replaced by a summation as p goes from 1 to N-1, while the summation in Eq 33 remains unchanged.

Use of the material (convected) coordinate system has the advantage that quantities at different times can be added as the sum. Integration can be regarded as a special case of the addition of quantities associated with a material point at different times (27). Transformation of the deformation to a fixed reference system is not required. The memory functions defined in Eqs 34 are all scalar quantities. The relationships between the stress and deformation described in Eq 33 are all functions of the scalar Lame constants and components of the Euclideen metric and associated metric tensors. These tensor components are formed by the simple scalar product of either the covariant or contravariant base vectors, respectively. The result is an integration of a scalar function over time making the hereditary integral easy to compute. Integration of the Jaumann rate and convected time derivatives are not necessary in this formulation.

The metric tensor components used to calculate the stresses of Eq 33 were all evaluated at a specific material particle at various times ranging from the original time zero to the present time t. The operations of integration with time and multiplication of the metric tensor components are all evaluated at a particle without any influence from any neighboring particles. Consequently, material objectivity is maintained in this formulation. This extremely important result cannot be overemphasized. The objectivity tests of the previous section show that rotation does not induce any stresses or strains in the material. The concept of a two-point tensor field ensures proper evaluation of the stress at the current time through proper evaluation of the metric tensor components (Eq 33) and the past strains. Position and motion are completely irrelevant to the rheological relationship being proposed. All of Oldroyd's criteria are satisfied.


The generalized Hooke's law in a material (convected, body metric) coordinate system yielded physically realistic solutions to several simple problems of shear deformation, extension, and rotation. The application of rotation along with either extension or shear illustrated the important property of objectivity in the metric tensor approach taken here. This is a key point in satisfying Oldroyd's rules of admissibility of a rheological model.

The effect of thermal expansion was easily incorporated into the generalized Hooke's law using a material (convected) coordinate system.

The use of the material coordinate system was key in the model development. Tensorially admissible operations such as addition and integration can be performed at a material point without influence from neighboring material elements. The important property of additivity was crucial to formulate and compute the hereditary integral in a finite deformation context. Complicated integration of either the Jaumann rate or convected time derivative was not necessary. This formulation provides a model that is fairly straightforward to implement and that could be used to simulate many practical processes involving use of viscoelastic materials.

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M.J. STEPHENSON (1)(*) and G.F. DARGUSH (2)

(*.) Corresponding author.

(1.) Research and Development Group Aristech Acrylics LLC Florence, KY 41042

(2.) Department of Civil Engineering State University of New York at Buffalo Buffalo, NY 14260
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Date:Mar 1, 2002
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