On Finite Shear

If a pair of material line elements, passing through a typical particle P in a body, subtend an angle Θ before deformation, and Θ+γ after deformation, the pair of material elements is said to be sheared by the amount γ. Here all pairs of material elements at P are considered for arbitrary deformations. Two main problems are addressed and solved. The first is the determination of all pairs of material line elements at P which are unsheared. The second is the determination of that pair of material line elements at P which suffers the maximum shear. All unsheared pairs of material elements in a given plane π(S) with normal S passing through P are considered. Provided π(S) is not a plane of central circular section of the C-ellipsoid at P (where C is the right Cauchy-Green strain tensor), it is seen that corresponding to any material element in π(S) there is, in general, one companion material element in π(S) such that the element and its companion are unsheared. There are, however, two elements in π(S) which have no companions. We call their corresponding directions \textit{limiting directions.} Equally inclined to the direction of least stretch in the plane π(S), the limiting directions play a central role. It is seen that, in a given plane π(S), the pair of material line elements which suffer the maximum shear lie along the limiting directions in π(S). If Θ _L is the acute angle subtended by the limitig directions in π(S) before deformation, then this angle is sheared into its supplement π−Θ _L so that the maximum shear γ^*;(S) is γ^*=π− 2 Θ _L . If S is given and C is known, then Θ _L may be determined immediately. Its calculation does not involve knowing the eigenvectors or eigenvalues of C. When all possible planes through P are considered, it is seen that the global maximum shear γ^* _G occurs for material elements lying along the limiting directions in the plane spanned by the eigenvectors of C corresponding to the greatest principal stretch λ₃ and the least λ₁. The limiting directions in this principal plane of C subtend the angle and . Generally the maximum shear does not occur for a pair of material elements which are originally orthogonal. For a given material element along the unit vector N, there is, in general, in each plane π(S passing through N at P, a companion vector M such that material elements along N and M are unsheared. A formula, originally due to Joly (1905), is presented for M in terms of N and S. Given an unsheared pair π(S), the limiting directions in π(S) are seen to be easily determined, either analytically or geometrically. Planar shear, the change in the angle between the normals of a pair of material planar elements at X, is also considered. The theory of planar shear runs parallel to the theory of shear of material line elements. Corresponding results are presented. Finally, another concept of shear used in the geology literature, and apparently due to Jaeger, is considered. The connection is shown between Cauchy shear, the change in the angle of a pair of material elements, and the Jaeger shear, the change in the angle between the normal N to a planar element and a material element along the normal N. Although Jaeger's shear is described in terms of one direction N, it is seen to implicitly include a second material line element orthogonal to N. Accepted: May 25, 1999     

In General There Are No Unsheared Tetrads

Suppose the principal stretches are all different at a point P in a deformed body. In this case, it has been shown [1] that generally there is an infinity of non coplanar infinitesimal material line elements at P which remain unsheared following the deformation – that is, the angle between the arms of each pair of material line elements forming the triad remains unchanged. Here it is shown that in this case when all three principal stretches at P are different, there is no set of four infinitesimal material line elements, no three of which are coplanar, and such that the angle between each pair of the six pairs of material line elements is unchanged following the deformation. It is only when all three principal stretches at P are equal to each other, that there are unsheared tetrads at P, and in that case all tetrads are unsheared. This revised version was published online in June 2006 with corrections to the Cover Date.     

Extended Polar Decompositions for Plane Strain

In a finite deformation at a particle of a continuous body, a triad of infinitesimal material line elements is said to be “unsheared” when the angles between the three pairs of line elements of the triad suffer no change. In a previous paper, it has been shown that there is an infinity of unsheared oblique triads. With each oblique unsheared triad may be associated an “extended polar decomposition” F = QG = HQ of the deformation gradient F, in which Q is a rotation tensor, and G, H are not symmetric. Both G and H have the same real eigenvalues which are the stretches of the elements of the triad. In this paper, a detailed analysis of extended polar decompositions is presented in the case when the finite deformation is that of plane strain. Then, we may deal with a 2 × 2 deformation gradient F′ = Q′G′ = H′Q′ instead of the full 3 × 3 tensor F. In this case, the extended polar decompositions are associated with “unsheared pairs,” i.e., pairs of infinitesimal material line elements in the plane of strain which suffer no change in angle in the deformation. If one arm of an unsheared pair is chosen in the plane of strain, then, in general, its companion in the plane is determined. It follows that all possible extended polar decompositions may then be described in terms of a single parameter, the angle that the chosen arm makes with a coordinate axis in the plane. Explicit expressions for G′ and H′ are obtained, and various special cases are discussed. In particular, we note that the expressions for G′ and H′ remain valid even when the chosen arm is along a “limiting direction,” that is the direction of a line element which has no companion element in the plane forming an unsheared pair with it. The results are illustrated by considering the cases of simple shear and of pure shear.Dedicated to Professor Piero Villaggio as a symbol of our friendship and esteem.     

On Infinitesimal Shear

The setting for this note is the theory of infinitesimal strain in the context of classical linearized elasticity. As a body is subjected to a deformation the angle between a pair of material line elements through a typical point P is changed. The decrease in angle is called the shear of this pair of elements. Here, we determine all pairs of material line elements at P which are unsheared in a deformation. It is seen, in general, that corresponding to any given material line element in a given plane through P, there is one corresponding “companion” material line element such that the given element and its conjugate are unsheared in the deformation. There are two exceptions. If the plane through P is a plane of central circular section of the strain ellipsoid, then every material line element through P in this plane has an infinity of companion elements in this plane – all pairs of material line elements in the plane(s) of central circular section of the strain ellipsoid are unsheared. If the plane through P is not a plane of central circular section of the strain ellipsoid, then there are two exceptional material line elements through P such that neither of them has a companion material line element forming an unsheared pair with it. The directions of these exceptional elements in the plane are called “limiting directions”. It is seen that it is the pair of elements along the limiting directions in a plane which suffer the maximum shear in that plane. A geometrical construction is presented for the determination of the extensional strains along the pairs of elements which are unsheared. Also, it is shown that knowing one unsheared pair in a plane and their extensions is sufficient to determine the principal extensions and the principal axes in this plane. Expressions for all unsheared pairs in a given plane are given in terms of the normals to the planes of central circular sections of the strain ellipsoid. Finally, for a given material line element, a formula is derived for the determination of all other material line elements which form an unsheared pair with the given element.     

Determination of the Infinitesimal Strain Tensor from Shears and Elongations

It is assumed that at a point P in a body, the longitudinal strains (elongations) along three non-coplanar directions are known from observation and that the shears of the three pairs of infinitesimal material line elements along the three non-coplanar directions are also known. With this information the strain tensor e at P is determined explicitly. The strain tensor e takes a simpler form in the special case when the three shears are zero. This simpler form is precisely the form obtained by Boulanger and Hayes in their study (Boulanger and Hayes, Proc R Ir Acad 103A:113–141, 2003) of the consequences of writing the displacement gradient at P as the sum of a skew symmetric tensor and a tensor with three real eigenvalues. The special case when the three elongations are zero is also considered. Dedicated to Franz Ziegler on the occasion of his seventieth birthday.     

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     

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     

A Note on Maximum Shear

The problem of the determination at any point P in a body of that pair of infinitesimal material line elements which suffers the maximum shear in a deformation has been solved [1]. Here that problem is revisited and a short proof, of geometrical type, of the result is presented. This revised version was published online in July 2006 with corrections to the Cover Date.     

An Eulerian formulation for large deformations of anisotropic elastic and viscoelastic solids and viscous fluids

An Eulerian formulation has been developed for the constitutive response of a group of materials that includes anisotropic elastic and viscoelastic solids and viscous fluids. The material is considered to be a composite of an elastic solid and a viscous fluid. Evolution equations are proposed for a triad of vectors m _i that represent the stretches and orientations of material line elements in the solid component. Evolution equations for an orthonormal triad of vectors s _i are also proposed to characterize anisotropy of the fluid component. In particular, for an elastic solid it is shown that the material response is totally characterized by the functional form of the strain energy and by the current values of m _i , which are measurable in the current state of the material. Moreover, it is shown that the proposed Eulerian formulation removes unphysical arbitrariness of the choice of the reference configuration in the standard formulation of constitutive equations for anisotropic elastic solids.     

Infinitesimal elastic stability of homogeneous deformations and the zero moment condition

We investigate the relationships between the infinitesimal elastic stability of homogeneous deformations and the zero moment condition. Under dead loading, for physically reasonable constitutive assumptions, we find that if the infinitesimal deformation satisfies the zero moment condition, it is stable under a very weak condition, one which includes an all-round compressive state. We show further that for a given stretching D the deformation L with the zero moment condition is the minimum (maximum) stable deformation in the state 53-1. Here 53-2 and t _a, a=1, 2, 3, are the principal Biot and Cauchy stresses, respectively. Finally, we examine stability when the prescribed traction rate is controlled such that the zero moment condition is satisfied for any deformation.     

Minimum distance and optimal transformations on SO(N)

The group of special (or proper) orthogonal matrices, SO(N), is used throughout engineering mechanics in the analysis and representation of mechanical systems. In this paper, a solution is presented for the optimal transformation between two elements of SO(N). The transformation is assumed to occur during a specified finite time, and a cost function that penalizes the transformation rates is utilized. The optimal transformation is found as a constant-rate rotation in each of the principal planes relating the two elements. Although the kinematics of SO(N) are nonlinear and governed by Poisson’s equation, the solution is found to be a linear function of the generalized principal angles. This is made possible by the extension of principal-rotation kinematics from three-dimensional rotations to the general SO(N) group. This extension relates the N-dimensional angular velocity to the derivatives of the principal angles. The cost of the optimal transformation, the square root of the sum of the principal angles squared, also provides a useful measure for the angular distance between two elements of SO(N).     

Second-Order Shell Kinematics Implied by Rotation Constraint-Equation

The paper presents a general methodology of introducing the shell-type variables which is based on the rotation constraint-equation (RC-equation). The RC-equation is proven to be equivalent to the polar decomposition of the deformation gradient formula, and the rotations which it yields are interpreted in terms of rotations of vectors of an ortho-normal basis. The deformation function and rotations are assumed as polynomials of the thickness coordinate ζ, and in this form used in the RC-equation. Solving this equation, we can express the coefficients of the quadratic deformation function in terms of the following shell-type variables: (a) the mid-surface position x ₀, (b) the constant rotation Q ₀, (c) the rotation vector ψ ^* for the ζ-dependent rotations, and (d) the normal components U ₃₃ ⁰ and U ₃₃ ¹ of the right stretching tensor. This new methodology (i) ensures that all shell kinematical variables are consistent with the RC-equation, which is justified on 3D grounds, (ii) provides a general framework from which various Reissner-type hypotheses can be obtained by suitable assumptions. As an example, two generalized Reissner hypotheses are derived: one with two normal stretches, and the other with the in-plane twist and the bubble-like warping parameters. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

A New Approach to the Existence of Weak Solutions of the Steady Navier–Stokes System with Inhomogeneous Boundary Data in Domains with Noncompact Boundaries

We prove the existence of a weak solution to the steady Navier–Stokes problem in a three dimensional domain Ω, whose boundary ∂Ω consists of M unbounded components Γ¹, . . . , Γ ^M and N − M bounded components Γ ^M+1, . . . , Γ ^N . We use the inhomogeneous Dirichlet boundary condition on ∂Ω. The prescribed velocity profile α on ∂Ω is assumed to have an L ³-extension to Ω with the gradient in L ²(Ω)^3×3. We assume that the fluxes of α through the bounded components Γ ^M+1, . . . , Γ ^N of ∂Ω are “sufficiently small”, but we impose no restriction on the size of fluxes through the unbounded components Γ¹, . . . , Γ ^M .     

Initial stresses in elastic solids: Constitutive laws and acoustoelasticity

On the basis of the nonlinear theory of elasticity, the general constitutive equation for an isotropic hyperelastic solid in the presence of initial stress is derived. This derivation involves invariants that couple the deformation with the initial stress and in general, for a compressible material, it requires 10 invariants, reducing to 9 for an incompressible material. Expressions for the Cauchy and nominal stress tensors in a finitely deformed configuration are given along with the elasticity tensor and its specialization to the initially stressed undeformed configuration. The equations governing infinitesimal motions superimposed on a finite deformation are then used to study the combined effects of initial stress and finite deformation on the propagation of homogeneous plane waves in a homogeneously deformed and initially stressed solid of infinite extent. This general framework allows for various different specializations, which make contact with earlier works. In particular, connections with results derived within Biot's classical theory are highlighted. The general results are also specialized to the case of a small initial stress and a small pre-deformation, i.e. to the evaluation of the acoustoelastic effect. Here the formulas derived for the wave speeds cover the case of a second-order elastic solid without initial stress and subject to a uniaxial tension [Hughes and Kelly, Phys. Rev. 92 (1953) 1145] and are consistent with results for an undeformed solid subject to a residual stress [Man and Lu, J. Elasticity 17 (1987) 159]. These formulas provide a basis for acoustic evaluation of the second- and third-order elasticity constants and of the residual stresses. The results are further illustrated in respect of a prototype model of nonlinear elasticity with initial stress, allowing for both finite deformation and nonlinear dependence on the initial stress.     

On Determinants having Complex Conjugate Columns or Rows

It is proved that the determinant, det D, of an N × N matrix D having n ≤ N/2 pairs of complex conjugate columns (or rows), while all other elements are real-valued, is given by det D =(−2i)ⁿdet S,n = 0,1,2,... in which S is a certain residual matrix having real-valued elements. Thus, det D is either real-valued or pure imaginary according as n is even (including n = 0 ) or odd, respectively. The general theorem is illustrated in an example. This revised version was published online in August 2006 with corrections to the Cover Date.     

Mode III stress intensity factors in an interfacial crack in dissimilar bonded materials

The antiplane shear deformation problem of two edge-bonded dissimilar isotropic wedges is considered. In the case when the sum of the two apex angles is equal to 2π, the problem reduces to that of two edge-bonded dissimilar materials with an interfacial crack subjected to concentrated antiplane shear tractions on the crack faces. An explicit expression is extracted for the stress intensity factor at the crack tip. In the special cases of different combinations of the apex angles, the obtained expression for the stress intensity factor may be simplified and relations of a simpler form are given for the stress intensity factor. It is shown that the stress intensity factor is dependent on the material properties as well as the geometry and loading. However, in special cases of equal apex angles as well as the case of similar materials the dependency of the stress intensity factor on the material properties disappears.     

Stability of an elastic cube under dead loading: Two equal forces

A cube of incompressible neo-Hookean material undergoes a pure homogeneous deformation and is held in equilibrium by three specified pairs of equal and opposite forces, two of which are the same, applied normally to its faces and uniformly distributed over them. The possible equilibrium states are determined and the stability of each is studied with respect to arbitrary superposed infinitesimal deformations. The stability limits are found to be different from those obtained when only infinitesimal deformations having the same principal directions as those of the basic equilibrium state are considered. The differences arise from rotational and shearing types of instabilities that may occur in the general case. A critical inference is drawn concerning the nature of the dead loading conditions employed.     

Analytical solutions for tapering quadratic and cubic rat-holes in highly frictional granular solids

New exact analytical solutions are presented for both stress and velocity fields for a Coulomb–Mohr granular solid assuming non-dilatant double-shearing theory. The solutions determined apply to highly frictional materials for which the angle of internal friction φ is assumed equal to 90°. This major assumption is made primarily to facilitate exact analytical solutions, and it is discussed at length in the Introduction, both in the context of real materials which exhibit large angles of internal friction, and in the context of using the solutions derived here as the leading term in a regular perturbation solution involving powers of 1−sinφ. The analytical velocity fields so obtained are illustrated graphically by showing the direction of the principal stress as compared to the streamlines. The stress solutions are also exploited to determine the static stress distribution for a granular material contained within vertical boundaries and a horizontal base, which is assumed to have an infinitesimal central outlet through which material flows until a rat-hole of parabolic or cubic profile is obtained, and no further flow takes place. A rat-hole is a stable structure that may form in storage hoppers and stock-piles, preventing any further flow of material. Here we consider the important problems of two-dimensional parabolic rat-holes of profile y=ax², and three-dimensional cubic rat-holes of profile z=ar³, which are both physically realistic in practice. Analytical solutions are presented for both two and three-dimensional rat-holes for the case of a highly frictional granular solid, which is stored at rest between vertical walls and a horizontal rigid plane, and which has an infinitesimal central outlet. These solutions are bona fide exact solutions of the governing equations for a Coulomb–Mohr granular solid, and satisfy exactly the free surface condition along the rat-hole surface, but approximate frictional conditions along the containing boundaries. The analytical solutions presented here constitute the only known solutions for any realistic rat-hole geometry, other than the classical solution which applies to a perfectly vertical cylindrical cavity.     

Influence of roughness on attachment line boundary layer transition in hypersonic flow

 Attachment line boundary layer transition on swept cylinders is studied in a low enthalpy hypersonic wind tunnel at M _∞=7.14. Sweep angles of 60° and 70° are used and transition is detected by means of heat flux measurements. The influence on attachment line transition of single 2D-roughness elements, in the form of tripwires or slots, as well as 3D obstacles is determined and the results are analyzed with respect to Poll’s criterion. Received: 16 January 1996 / Accepted: 12 July 1996