Derivatives and Rates of the Stretch and Rotation Tensors

General expressions for the derivatives and rates of the stretch and rotation tensors with respect to the deformation gradient are derived. They are both specialized to some of the formulas already available in the literature and used to derive some new ones, in three and two dimensions. Essential ingredients of the treatment are basic elements of differential calculus for tensor valued functions of tensors and recently derived results on the solution of the tensor equation A X + XA= H in the unknown X. This revised version was published online in August 2006 with corrections to the Cover Date.     

The Explicit Determination of the Logarithm of a Tensor and Its Derivatives

The logarithm of a tensor is often used in nonlinear constitutive relations of elastic materials. Here we show how the logarithm of an arbitrary tensor can be explicitly evaluated for any underlying space dimension n. We also present a method for the explicit evaluation of the derivatives of the logarithm of a tensor.      

Differentiability Properties of Rotationally Invariant Functions

Let f be a function on the set Lin of all tensors (= square matrices) on a vector space of arbitrary dimension. If f is rotationally invariant (with respect to the left and right multiplication by proper orthogonal tensors), it has a representation through a symmetric even function of the signed singular values of the tensor argument A∈Lin. It is shown that f is of class C ^r ,r=0,1,...,∞, if and only if is of class C ^r , and an inductive formula is given for the derivatives D ^r f. This revised version was published online in August 2006 with corrections to the Cover Date.     

A new two-point deformation tensor and its relation to the classical kinematical framework and the stress concept

Starting from the issue of what is the correct form for a Legendre transformation of the strain energy in terms of Eulerian and two-point tensor variables we introduce a new two-point deformation tensor, namely H=(F−F^−T)/2, as a possible deformation measure involving points in two distinct configurations. The Lie derivative of H is work conjugate to the first Piola–Kirchhoff stress tensor P. The deformation measure H leads to straightforward manipulations within a two-point setting such as the derivation of the virtual work equation and its linearization required for finite element implementation. The manipulations are analogous to those used for the Lagrangian and Eulerian frameworks. It is also shown that the Legendre transformation in terms of two-point tensors and spatial tensors require Lie derivatives. As an illustrative example we propose a simple Saint Venant–Kirchhoff type of a strain-energy function in terms of H. The constitutive model leads to physically meaningful results also for the large compressive strain domain, which is not the case for the classical Saint Venant–Kirchhoff material.     

Barnett-Lothe tensors and their associated tensors for monoclinic materials with the symmetry plane at x 3=0

The three Barnett-Lothe tensors S, H, L and the three associated tensors S(), H(), L() appear frequently in the real form solutions to two-dimensional anisotropic elasticity problems. Explicit expressions of the components of these tensors are derived and presented for monoclinic materials whose plane of material symmetry is at x ₃=0. We use the algebraic formalism for these tensors but the results are derived not by the straight-forward substitution of the complex matrices A and B into the formulae. Instead, we find the product –AB ^-1, whose real and imaginary parts are SL ^-1 and L ^-1, respectively. The tensors S, H, L are then determined from SL ^-1 and L ^-1. For S(), H(), L() we again avoid the direct substitution by employing an alternate approach. The new approaches require minimal algebra and, at the same time, provide simple and concise expressions for the components of these tensors. Although the new approaches can be extended, in principle, to monoclinic materials whose plane of symmetry is not at x ₃=0 and to materials of general anisotropy, the explicit expressions for these materials are too complicated. More studies are needed for these materials.     

Relation between the stress and couple-stress tensors in the microcontinuum theory of elasticity

The stress tensor is expressed in terms of an arbitrary symmetric tensor field of second rank and the couple-stress tensor. The stress and couple-stress tensors are represented by arbitrary tensor fields satisfying the homogeneous equilibrium equations. These tensors are also given in the form of the expressions satisfying the inhomogeneous equilibrium equations used in the microcontinuum theory of elasticity. The stress tensor functions are considered.     

Basis free relations for the conjugate stresses of the strains based on the right stretch tensor

In this paper, general relations between two different stress tensors T^f and T^g, respectively conjugate to strain measure tensors f(U) and g(U) are found. The strain class f(U) is based on the right stretch tensor U which includes the Seth–Hill strain tensors. The method is based on the definition of energy conjugacy and Hill’s principal axis method. The relations are derived for the cases of distinct as well as coalescent principal stretches. As a special case, conjugate stresses of the Seth–Hill strain measures are then more investigated in their general form. The relations are first obtained in the principal axes of the tensor U. Then they are used to obtain basis free tensorial equations between different conjugate stresses. These basis free equations between two conjugate stresses are obtained through the comparison of the relations between their components in the principal axes, with a possible tensor expansion relation between the stresses with unknown coefficients, the unknown coefficients to be obtained. In this regard, some relations are also obtained for T⁽⁰⁾ which is the stress conjugate to the logarithmic strain tensor lnU.     

On the Explicit Determination of the Polar Decomposition in n-Dimensional Vector Spaces

A method for the explicit determination of the polar decomposition (and the related problem of finding tensor square roots) when the underlying vector space dimension n is arbitrary (but finite), is proposed. The method uses the spectral resolution, and avoids the determination of eigenvectors when the tensor is invertible. For any given dimension n, an appropriately constructed van der Monde matrix is shown to play a key role in the construction of each of the component matrices (and their inverses) in the polar decomposition.     

Strain Rates and Material Spins

The material time rate of Lagrangean strain measures, objective corotational rates of Eulerian strain measures and their defining spin tensors are investigated from a general point of view. First, a direct and rigorous method is used to derive a simple formula for the gradient of the tensor-valued function defining a general class of strain measures. By means of this formula and the chain rule as well as Sylvester's formula for eigenprojections, explicit basis-free expressions for the material time rate of an arbitrary Lagrangean strain measure can be derived in terms of the right Cauchy–Green tensor and the material time rate of any chosen Lagrangean strain measure, e.g. Hencky's logarithmic strain measure. These results provide a new derivation of Carlson–Hoger's general gradient formula for an arbitrary generalized strain measure and supply a new, rigorous proof for Carlson–Hoger's conjecture concerning the n-dimensional case. Next, by virtue of the aforementioned gradient formula, a general fact for objective corotational rates and their defining spin tensors is disclosed: Let Ω = ϒ ( B, D, W) be any spin tensor that is continuous with respect to B, where B, D and B are the left Cauchy–Green tensor, the stretching tensor and the vorticity tensor. Then the corotational rate of an Eulerian strain measure defined by Ω is objective iff Ω = W + υ ( B, D), where Υ is isotropic. By means of this fact and certain necessary or reasonable requirements, it is further found that a single antisymmetric function of two positive real variables can be introduced to characterize a general class of spin tensors defining objective corotational rates. A general basis- free expression for all such spin tensors and accordingly a general basis-free expression for a general class of objective corotational rates of an arbitrary Eulerian strain measure are established in terms of the left Cauchy–Green tensor B and the stretching tensor B as well as the introduced antisymmetric function. By choosing several particular forms of the latter, all commonly-known spin tensors and corresponding corotational rates are shown to be incorporated into these general formulas in a natural way. In particular, with the aid of these general formulae it is proved that an objective corotational rate of the Eulerian logarithmic strain measure ln V is identical with the stretching tensor D and moreover that in all possible strain tensor measures only ln V enjoys this property. This revised version was published online in August 2006 with corrections to the Cover Date.     

Characterization of Elasticity-Tensor Symmetries Using SU(2)

A symmetry class of an elasticity tensor, c, is determined by the variance of this tensor with respect to a subgroup of the special orthogonal group, SO(3). Using the double covering of SO(3) by the special unitary group, SU(2), we determine the subgroups of SU(2) that correspond to each of the eight symmetry classes. A family of maps between C² and R³ that preserve the action of the two groups is constructed. Using one of these maps and three associated polynomials, we derive new methods for characterizing the symmetry classes of elasticity tensors. Mathematics Subject Classifications (2000) 74B05, 74E10.     

New Explicit Expression of Barnett-Lothe Tensors for Anisotropic Linear Elastic Materials

The three Barnett-Lothe tensors H, L, S appear often in the Stroth formalism of two-dimensional deformations of anisotropic elastic materials [1–3]. They also appear in certain three-dimensional problems [4, 5]. The algebraic representation of H, L, S requires computation of the eigenvalues p^v(v=1,2,3) and the normalized eigenvectors (a, b). The integral representation of H, L, S circumvents the need for computing p ^v(v=1,2,3) and (a, b), but it is not simple to integrate the integrals except for special materials. Ting and Lee [6] have recently obtained an explicit expression of H for general anisotropic materials. We present here the remaining tensors L, S using the algebraic representation. They key to our success is the obtaining of the normalization factor for (a, b) in a simple form. The derivation of L and S then makes use of (a, b) but the final result does not require computation of (a, b), which makes the result attractive to numerical computation. Even though the tensor H given in [6] is in terms of the elastic stiffnesses C^μ v while the tensors L, S presented here are in terms of the reduced elastic compliances s′ _μv , the structure of L, S is similar to that of H. Following the derivation of H, we also present alternate expressions of L, S that remain valid for the degenerate cases p ₁ p ₂ and p₁=p₂ = p ₃. One may want to compute H, L, S using either C _μv or s′ _μv v, but not both. We show how an expression in C^μ v can be converted to an expression in s′ _μv v, and vice versa. As an application of the conversion, we present explicit expressions of the extic equation for p in C^μ v and s′ _μv v. This revised version was published online in August 2006 with corrections to the Cover Date.     

Shape sensitivity analysis and the energy momentum tensor for the kinematic and static models of torsion

The aim of this paper is to bridge shape sensitivity analysis and configurational mechanics by means of a widespread use of the shape derivative concept. This technique will be applied as a systematic procedure to obtain the Eshelby’s energy momentum tensor associated to the problem under consideration. In order to highlight special features of this procedure and without loss of generality, we focus our attention in the application of shape sensitivity analysis to the problem of twisted straight bars within the framework of linear elasticity.Kinematic and static variational formulations as well as the direct method of sensitivity analysis are used to perform shape derivatives of both models. Integral expressions of first and second order shape derivatives of the total potential energy and the complementary potential energy with respect to an arbitrary transverse cross-section shape change, are achieved. These integral expressions put in evidence the relationship between shape sensitivity analysis and the first and second order Eshelby’s energy momentum tensors. Also, the null divergence property of these tensors is easily proved by comparing, in each case, the domain and boundary integral shape derivative arrived at. Finally, an example with a known exact solution, corresponding to an elastic bar with elliptical transverse cross-section submitted to twist, is presented in order to illustrate the usefulness of these tensors to compute the corresponding shape derivatives.     

A theory of homogeneous isotropic turbulence

Summary Homogeneous and isotropic turbulence has been discussed in the present paper. An attempt has been made to find the simplifying hypothesis for connecting the higher order correlation tensor with the lower ones. Starting from the Navier-Stokes equations of motion for an incompressible fluid and following the usual method of taking the averages, a differential equation in Q and X, the defining scalar of the second order correlation tensor Q _x and the defining scalar of a third order isotropic tensor X _ijk , has been derived. The tensor X _ijk stands for a tensorial expression containing the derivatives of the third and the fourth order tensors. Then the hypothesis is used that X=F(Q), where F is an unknown function. To find the forms of F, Kolmogoroff's similarity principles have been used, and thus two forms for F(Q) corresponding to two regions of the validity of these principles have been deduced.     

On the Derivative of Some Tensor-Valued Functions

An explicit expression of the derivative of the square root of a tensor is provided, by using the expressions of the derivatives of the eigenvalues and eigenvectors of a symmetric tensor. Starting from this result, the derivatives of the right and left stretch tensor U, V and of the rotation R with respect to the deformation gradient F, are calculated. Expressions for the material time derivatives of U, V and R are also given. This revised version was published online in August 2006 with corrections to the Cover Date.     

Thermal diffusion of dilute polymer solutions

In this paper diffusion of a dilute solution of elastic dumbbell model macromolecules under non-isothermal conditions is studied. Using the center of mass definition for the local polymer concentration, the diffusive flux contains a thermal diffusion dyadic d _T .  To get some idea of thermal diffusion d _T is evaluated for steady state isothermal conditions. Explicit results are presented for some homogeneous flows. It is shown that if the polymeric number density is defined via the beads (of the dumbbell) – termed n _b – then the diffusive flux j contains , where τ ^c is the intramolecular contribution to the bulk stress. Though the form of the diffusion equation for n _b thus differs from the corresponding one for n, it is shown that for essentially unbounded systems differences between n and n _b are small. Since the results involve the translational diffusion coefficient they can readily be taken over for Rouse coils. Received: 23 September 1997 Accepted: 5 June 1998     

A constitutive model for non-Newtonian materials based on the persistence-of-straining tensor

We present a constitutive equation for non-Newtonian materials which is capable of predicting, independently, steady state rheological material functions both in shear and in extension. The basic assumption is that the extra-stress tensor is a function of both the rate-of-strain tensor, D, and the persistence-of-straining tensor, -\boldsymbol{P}=\boldsymbol{D}\overline{\boldsymbol{W}}-\overline{\boldsymbol {W}}\boldsymbol{D}, introduced in Thompson and de Souza Mendes (Int. J. Eng. Sci. 43(1–2):79–105, 2005). The resulting equation falls within the category of constitutive equations of the form t=t(D,[`(W)])\boldsymbol{\tau}=\boldsymbol{\tau}(\boldsymbol {D},\overline{\boldsymbol{W}}), with the advantage of eliminating the undesirable stress jumps that may occur when [`(W)]\overline {\boldsymbol{W}} becomes locally undetermined. We also show that this formulation is not restricted to motions with constant relative principle stretch history (MWCRPSH), in contrast to what is suggested in the literature. The same basis of tensors that comes from representation theorems also arises from an elastic constitutive equation based on the difference between the Jauman and the Harnoy convected time derivatives, in the limit of small values of the Deborah number.     

Controllable infinitesimal deformations in homogeneously deformed compressible elastic materials

A controllable static deformation is a deformation that may be maintained in all materials of a given class under the action of surface forces alone. For compressible, homogeneous, isotropic elastic materials the only controllable deformations are homogeneous. However, it is known that there are solutions of the static equations of finite elasticity, linearized about a finite homogeneous deformation, which do not correspond to homogeneous deformations. These approximate solutions are investigated here. Three cases arise, depending on whether none, two, or three of the basic principal stretches are equal.Nomenclature A arbitrary vector potential - a ₁, a ₂, a ₃ bounding coordinates of body - B, B _ij left Cauchy-Green tensor - C, C _ijpq elasticity tensor - c, c ₁, c ₂, c ₃ arbitrary constants - N ₀, N ₁, N ₂ elastic response functions - n vector normal to surface of body - T ₁, T ₂, T ₃ surface tractions - t ₁, t ₂, t ₃ surface tractions - t, t _ij Cauchy stress tensor - t ⁰, t _ij ⁰ Cauchy stress corresponding to basic homogeneous deformation - u, u _i infinitesimal displacement from basic homogeneous deformation - X, X _i position vector in reference state - x, x _i position vector - arbitrary function - _ij Kronecker delta - , ₁, ₂, ₃ principal stretches - arbitrary function - arbitrary function - arbitrary function - I, II, III principal invariants of B     

Further Study on Pure Shear

It is known that the Cauchy stress tensor T is a pure shear when trT = 0. An elementary derivation is given for a coordinate system such that, when referred to this coordinate system, the diagonal elements of T vanish while the off-diagonal elements τ ₁, τ ₂, τ ₃, are the pure shears. The structure of τ _i (i = 1, 2, 3) depends on one non-dimensional parameter q = 54(detT)² / [tr(T ²)]³, 0 ≤ q ≤ 1. When q = 0, one of the three τ _i vanishes. A coordinate system can be chosen such that the remaining two have the same magnitude or one of the remaining two also vanishes. When q = 1, all three τ _i have the same magnitude. However, there is a one-parameter family of coordinate systems that gives the same three τ _i . For q ≠ 0 or 1, none of the three τ _i vanishes and the three τ _i in general have different magnitudes. Nevertheless, a coordinate system can be chosen such that two of the three τ _i have the same magnitude. ^★Professor Emeritus of University of Illinois at Chicago and Consulting Professor of Stanford University.     

On Shearing, Stretching and Spin

An analysis is presented of stretching, shearing and spin of material line elements in a continuous medium. It is shown how to determine all pairs of material line elements at a point x, at time t, which instantaneously are not subject to shearing. For a given pair not subject to shearing, a formula is presented for the determination of a third material line element such that all three form a triad not subject to shearing, instantaneously. It is seen that there is an infinity of such triads not subject to shearing. A new decomposition of the velocity gradient L is introduced. In place of the classical decomposition of Cauchy and Stokes, L=d+w, where d is the stretching tensor and w is the spin tensor, the new decomposition is L=?+ℳ, where ?, called the ldquo;modified” stretching tensor, is not symmetric, and ℳ, called the “modified” spin tensor, is skew-symmetric – the tensor ? being chosen so that it has three linearly independent real right (and left) eigenvectors. The physical interpretation of this decomposition is that the material line elements along the three linearly independent right eigenvectors of ? instantaneously form a triad not subject to shearing. They spin as a rigid body with angular velocity μ (say) associated with ℳ. Also, for each decomposition L=?+ℳ, there is a decomposition L=? ^T +, where is also skew-symmetric. The triad of material line elements along the right eigenvectors of ? ^T (the set reciprocal to the right eigenvectors of ?) is also instantaneously not subject to shearing and rotates with angular velocity (say) associated with . It is seen that the vorticity vector ω is the mean of the two angular velocities μ and , ω =(μ+)/2. For irrotational motion, ω =0, so that μ=-; any triad of material line elements suffering no shearing rotates with angular velocity equal and opposite to that of the reciprocal triad of material line elements. It is proved that provided d is not spherical, there is an infinity of choices for ? and ℳ in the decomposition L=?+ℳ. Two special types of decompositions are introduced. The first type is called “CCS-decomposition” (where CCS is an abbreviation for Central Circular Section). It is associated with the infinite family of triads (not subject to shearing) with a common edge along the normal to one plane of central circular section of an ellipsoid ? associated with the stretching tensor, and the two other edges arbitrary in the other plane of central circular section of ?. There are two such CCS-decompositions. The second type is called “triangular decomposition”, because, in a rectangular cartesian coordinate system, ? has three off-diagonal zero elements. There are six such decompositions. Received 14 November 2000 and accepted 2 August 2001