Stress channelling in extreme couple-stress materials Part I: Strong ellipticity,wave propagation,ellipticity, and discontinuity relations

Materials with extreme mechanical anisotropy are designed to work near a material instability threshold where they display stress channeling and strain localization, effects that can be exploited in several technologies. Extreme couple stress solids are introduced and for the first time systematically analyzed in terms of several material instability criteria: positive-definiteness of the strain energy (implying uniqueness of the mixed b.v.p.), strong ellipticity (implying uniqueness of the b.v.p. with prescribed kinematics on the whole boundary), plane wave propagation, ellipticity, and the emergence of discontinuity surfaces. Several new and unexpected features are highlighted: (i) Ellipticity is mainly dictated by the ‘Cosserat part’ of the elasticity; (ii) its failure is shown to be related to the emergence of discontinuity surfaces; and (iii) ellipticity and wave propagation are not interdependent conditions (so that it is possible for waves not to propagate when the material is still in the elliptic range and, in very special cases, for waves to propagate when ellipticity does not hold). The proof that loss of ellipticity induces stress channeling, folding and faulting of an elastic Cosserat continuum (and the related derivation of the infinite-body Green’s function under antiplane strain conditions) is deferred to Part II of this study.     

Three-dimensional Green's functions in anisotropic trimaterials

Three-dimensional elastostatic Green's functions in anisotropic trimaterials are derived, for the first time, by applying the generalized Stroh's formalism and Fourier transforms. The Green's functions are expressed as a series summation with the first term corresponding to the full-space solution and other terms to the image solutions due to the interfaces. The most remarkable feature of the present solution is that the image solutions can be expressed by a simple line integral over a finite interval [0,2π]. By partitioning the trimaterial Green's function into a full-space solution and a complementary part, the line integral involves only regular functions if the singularity is within one of the three materials, being treated analytically owning to the explicit expression of the full-space solution. When the singularity is on the interface, which occurs if the field and source points are both on the same interface, the involved singularity is handled with the interfacial Green's functions.A numerical example is presented for a trimaterial system made of two anisotropic half spaces bonded perfectly by an isotropic adhesive layer, showing clearly the effect of material layering on the Green's displacements and stresses. Furthermore, by comparing the present Green's solution to the direct (two-dimensional) 2D integral expression which is also derived in this paper, it is shown that, the computational time for the calculation of the Green's function can be substantially reduced using the present solution, instead of the direct 2D integral method.     

Discontinuous deformation gradients in plane finite elastostatics of incompressible materials

Summary This investigation is concerned with the possibility of the change of type of the differential equations governing finite plane elastostatics for incompressible elastic materials, and the related issue of the existence of equilibrium fields with discontinuous deformation gradients. Explicit necessary and sufficient conditions on the deformation invariants and the material for the ellipticity of the plane displacement equations of equilibrium are established. The issue of the existence, locally, of elastostatic shocks—elastostatic fields with continuous displacements and discontinuous deformation gradients—is then investigated. It is shown that an elastostatic shock exists only if the governing field equations suffer a loss of ellipticity at some deformation. Conversely, if the governing field equations have lost ellipicity at a given deformation at some point, an elastostatic shock can exist, locally, at that point. The results obtained are valid for an arbitrary homogeneous, isotropic, incompressible, elastic material.The results communicated in this paper were obtained in the course of an investigation supported by Contract N00014-75-C-0196 with the Office of Naval Research in Washington D.C.     

Elastic fields of dislocation loops in three-dimensional anisotropic bimaterials

By applying semi-analytical point-force Green's functions obtained via the Stroh formulism, we derive simple line integrals to calculate the elastic displacement and stress fields for a three-dimensional dislocation loop in an anisotropic bimaterial system. The solutions for the case of anisotropy are more convenient for treating an arbitrary dislocation loop compared with traditional area integration. With this new formulation, we numerically examine the displacement, stress, and energy due to the interaction between a dislocation loop and the bimaterial interface in an Al–Cu system. The interactive image energy due to the elastic moduli mismatch across the interface is then numerically evaluated. The result shows that a dislocation loop is subjected to an attractive force by the interface when it lies in the stiff material, and a repulsive force when it lies in the soft material. Moreover, the dependence of the interactive image energy of a dislocation loop on the position and size of the dislocation loop are also demonstrated and discussed. Significantly, it is found that the interactive image energy for a dislocation loop depends only on the ratio d/a, where a is the loop diameter and d is its distance to the interface. The examples studied provide benchmark solutions for anisotropic bimaterial dislocation problems.     

On the Green's Functions and Boundary Integral Formulation of Elastic Planes with Cutouts

ABSTRACT

The problem of an infinite elastic plane that contains a hole of arbitrary shape and is subjected to a concentrated unit load is considered. The Green's function (influence function) for the problem is formulated by means of two complex potential functions. This is accomplished by mapping the region that is exterior to the hole onto a unit circle. A class of closed contour hole shapes is analyzed. Green's functions for an elliptical hole and a class of triangular holes are determined. Green's functions for a class of rectangular holes are also discussed. In order to determine stress and displacement fields for the finite plane problem, Green's function is employed and an indirect boundary integral equation is formulated, with the integrand of the integral equation incorporating the effect of the hole. The contour of the hole is no longer considered a part of the boundary and only the contour of the region that is exterior to the hole is subdivided into boundary elements. Examples for elliptical and triangular holes are solved.     

Green's function solution and displacement potentials for Transformed Transversely Isotropic materials

A totally non-degenerate expression for the Green's function of infinite Transversely Isotropic (TI) materials is first deduced from the solutions given by Pan and Chou [Pan, Y.-C., Chou, T.-W., 1976. Point force solution for an infinite transversely isotropic solid. Trans. ASME, J. Appl. Mech. 43 (4), 608–612]. Then this solution and also the displacement potentials for TI materials are extended by a linear transformation to a larger family of anisotropic materials (Transformed Transversely Isotropic or TraTI materials). This family depends on 12 independent parameters and contains non-orthotropic materials and in this way a first explicit analytical solution for the Green's function for a non-orthotropic material is obtained. The TraTI materials which have orthotropic Symmetry (StraTI materials) constitute a sub-family depending on 6 independent parameters in the symmetry basis of the material. These materials present a 3D anisotropy (different stiffnesses in three orthogonal directions). General displacement potentials and the Green's function solution for STraTI materials can be deduced by a simple change and introducing one additional parameter in the well-known TI solutions.     

A note on the pure torsion of a circular cylinder for a compressible nonlinearly elastic material with nonconvex strain-energy

The large deformation torsion problem for an elastic circular cylinder subject to prescribed twisting moments at its ends is examined for a particular homogeneous isotropic compressible material, namely the Blatz-Ko material. For this material, the displacement equations of equilibrium in three-dimensional elastostatics can lose ellipticity at sufficiently large deformations. For the torsion problem, it is shown that this occurs when the prescribed torque reaches a critical value. For values of the twisting moment greater than this critical value, there is an axial core of the cylinder on which ellipticity holds, surrounded by an annular region where loss of ellipticity has occurred. The physical implications in terms of localized shear bands are briefly discussed.     

Finite plane and anti-plane elastostatic fields with discontinuous deformation gradients near the tip of a crack

In this paper the fully nonlinear theory of finite deformations of an elastic solid is used to study the elastostatic field near the tip of a crack. The special elastic materials considered are such that the differential equations governing the equilibrium fields may lose ellipticity in the presence of sufficiently severe strains.The first problem considered involves finite anti-plane shear (Mode III) deformations of a cracked incompressible solid. The analysis is based on a direct asymptotic method, in contrast to earlier approaches which have depended on hodograph procedures.The second problem treated is that of plane strain of a compressible solid containing a crack under tensile (Mode I) loading conditions. The materials is characterized by the so-called Blatz-Ko elastic potential. Again, the analysis involves only direct local considerations.for both the Mode III and Mode I problems, the loss of equilibrium ellipticity results in the appearance of curves (elastostatic shocks) issuing from the crack-tip across which displacement gradients and stresses are discontinuous.The results communicated in this paper were obtained in the course of an investigation supported by Contract N00014-75-C-0196 with the Office of Naval Research.     

On the theory of elastic shells made from a material with voids

In this paper we present a theory for porous elastic shells using the model of Cosserat surfaces. We employ the Nunziato–Cowin theory of elastic materials with voids and introduce two scalar fields to describe the porosity of the shell: one field characterizes the volume fraction variations along the middle surface, while the other accounts for the changes in volume fraction along the shell thickness. Starting from the basic principles, we first deduce the equations of the nonlinear theory of Cosserat shells with voids. Then, in the context of the linear theory, we prove the uniqueness of solution for the boundary initial value problem. In the case of an isotropic and homogeneous material, we determine the constitutive coefficients for Cosserat shells, by comparison with the results derived from the three-dimensional theory of elastic media with voids. To this aim, we solve two elastostatic problems concerning rectangular plates with voids: the pure bending problem and the extensional deformation under hydrostatic pressure.     

On the determination of elastodynamic crack tip stress fields

A linear elastic body in plane strain which contains a stationary crack and which is initially at rest and stress free is considered. It is shown that if the elastodynamic displacement field and stress intensity factor are known, as functions of crack length, for any symmetrical distribution of time-varying forces which acts on the body, subsequent to t=0, then the stress intensity factor due to any other symmetrical load system whatsoever which acts on the same body may be directly determined. The other load system may be of arbitrary spatial distribution and time variation. Further, that part of the elastodynamic displacement field due to the other load system, which arises from the presence of the crack, may also be directly determined. The results are obtained by extension of Rice's mode of derivation of the corresponding Bueckner-Rice elastostatic results to Laplace-transformed elastodynamic variables. Likewise, the existence of a universal elastodynamic “weight function” for any given cracked body is demonstrated. As an application, Freund's recent result for the stress intensity factor due to suddenly applied concentrated forces on the crack surfaces is derived directly by our method, from de Hoop's earlier solution for suddenly applied uniform pressures.     

Size effects in generalised continuum crystal plasticity for two-phase laminates

The solutions of a boundary value problem are explored for various classes of generalised crystal plasticity models including Cosserat, strain gradient and micromorphic crystal plasticity. The considered microstructure consists of a two-phase laminate containing a purely elastic and an elasto-plastic phase undergoing single or double slip. The local distributions of plastic slip, lattice rotation and stresses are derived when the microstructure is subjected to simple shear. The arising size effects are characterised by the overall extra back stress component resulting from the action of higher order stresses, a characteristic length l_c describing the size-dependent domain of material response, and by the corresponding scaling law lⁿ as a function of microstructural length scale, l. Explicit relations for these quantities are derived and compared for the different models. The conditions at the interface between the elastic and elasto-plastic phases are shown to play a major role in the solution. A range of material parameters is shown to exist for which the Cosserat and micromorphic approaches exhibit the same behaviour. The models display in general significantly different asymptotic regimes for small microstructural length scales. Scaling power laws with the exponent continuously ranging from 0 to −2 are obtained depending on the values of the material parameters. The unusual exponent value −2 is obtained for the strain gradient plasticity model, denoted “curl H^p” in this work. These results provide guidelines for the identification of higher order material parameters of crystal plasticity models from experimental data, such as precipitate size effects in precipitate strengthened alloys.     

A Variational Model for Plastic Slip and Its Regularization via Γ-Convergence

A variational model is presented able to interpret the onset of plastic deformations, here modeled as displacement jumps occurring along slip surfaces at constant yielding stress. The corresponding strain energy functional, leading to a free-discontinuity problem set in the space of SBV functions, is then approximated by a sequence of regularized elliptic functionals following the seminal work by Ambrosio and Tortorelli (Commun. Pure Appl. Math. 43, 999–1036, 1990) within the framework of Γ-convergence. Comparisons between the results obtainable with the free-discontinuity model and its regularized approximation, in terms of stability of the pure elastic phase, irreversibility of plastic slip and response under unloading, are presented, in general, for the 2-D case of antiplane shear and exemplified, in particular, for the 1-D case.     

Two-dimensional stress wave analysis in incompressible elastic solids

Two-dimensional stress waves in a general incompressible elastic solid are investigated. First, basic equations for simple waves and shock waves are presented for a general strain energy function. Then the characteristic wave speeds and the associated characteristic vectors are deduced. It is shown that there usually exist two simple waves and two shock waves. Finally, two examples are given for the case of plane strain deformation and antiplane strain deformation, respectively. It is proved that, in the case of plane strain deformation the oblique reflection problem of a plane shock is not solvable in general.     

The plane self-similar anisotropic and angularly inhomogeneous wedge under power law tractions and the asymptotic analysis of the stress field

In this study the generally anisotropic and angularly inhomogeneous wedge, under power law tractions of order n of the radial coordinate r at its external faces is considered. At first, using variable separable relations in the equilibrium equations, the strain–stress relations and the strain compatibility equation, a differential system of equations is constructed and investigated. Decoupling this system, an ordinary differential equation is derived and the stress and displacement fields may be determined. The proposed procedure is also applied to the elastostatic problem of an isotropic and angularly inhomogeneous wedge. In the sequel William's asymptotic analysis in the case of angular inhomogeneity is examined. Finally, applications for the case of an angularly inhomogeneous wedge-shape dam and for the asymptotic procedure in an isotropic wedge with angularly varying shear modulus, are made.     

Effect of yield surface vertex on crack-tip fields in mode III

An asymptotic crack-tip analysis of stress and strain fields is carried out for an antiplane shear crack (Mode III) based on a corner theory of plasticity. Because of the nonproportional loading history experienced by a material element near the crack tip in stable crack growth, classical flow theory may predict an overly stiff response of the elastic plastic solid, as is the case in plastic buckling problems. The corner theory used here accounts for this anomalous behavior. The results are compared with those of a similar analysis based on the J₂ flow theory of plasticity.     

On the equivalence between traction- and stress-based approaches for the modeling of localized failure in solids

This work investigates systematically traction- and stress-based approaches for the modeling of strong and regularized discontinuities induced by localized failure in solids. Two complementary methodologies, i.e., discontinuities localized in an elastic solid and strain localization of an inelastic softening solid, are addressed. In the former it is assumed a priori that the discontinuity forms with a continuous stress field and along the known orientation. A traction-based failure criterion is introduced to characterize the discontinuity and the orientation is determined from Mohr's maximization postulate. If the displacement jumps are retained as independent variables, the strong/regularized discontinuity approaches follow, requiring constitutive models for both the bulk and discontinuity. Elimination of the displacement jumps at the material point level results in the embedded/smeared discontinuity approaches in which an overall inelastic constitutive model fulfilling the static constraint suffices. The second methodology is then adopted to check whether the assumed strain localization can occur and identify its consequences on the resulting approaches. The kinematic constraint guaranteeing stress boundedness and continuity upon strain localization is established for general inelastic softening solids. Application to a unified stress-based elastoplastic damage model naturally yields all the ingredients of a localized model for the discontinuity (band), justifying the first methodology. Two dual but not necessarily equivalent approaches, i.e., the traction-based elastoplastic damage model and the stress-based projected discontinuity model, are identified. The former is equivalent to the embedded and smeared discontinuity approaches, whereas in the later the discontinuity orientation and associated failure criterion are determined consistently from the kinematic constraint rather than given a priori. The bi-directional connections and equivalence conditions between the traction- and stress-based approaches are classified. Closed-form results under plane stress condition are also given. A generic failure criterion of either elliptic, parabolic or hyperbolic type is analyzed in a unified manner, with the classical von Mises (J₂), Drucker–Prager, Mohr–Coulomb and many other frequently employed criteria recovered as its particular cases.     

Nonlinear Antiplane Deformation of an Elastic Body

The antiplane elastic deformation of a homogeneous isotropic prestretched cylindrical body is studied in a nonlinear formulation in actual–state variables under incompressibility conditions, the absence of volume forces, and under constant lateral loading along the generatrix. The boundary–value problem of axial displacement is obtained in Cartesian and complex variables and sufficient ellipticity conditions for this problem are indicated in terms of the elastic potential. The similarity to a plane vortex–free gas flow is established. The problem is solved for Mooney and Rivlin—Sonders materials simulating strong elastic deformations of rubber–like materials. Axisymmetric solutions are considered.     

The stress concentration near a rigid line inclusion in a prestressed, elastic material. Part II.: Implications on shear band nucleation, growth and energy release rate

The full-field and asymptotic solutions derived in Part I of this article (for a lamellar rigid inclusion, embedded in a uniformly prestressed, incompressible and orthotropic elastic sheet, subject to a far-field deformation increment) are employed to analyse shear band formation, as promoted by the near-tip stress singularity. Since these solutions involve the prestress as a parameter, stress and deformation fields can be investigated near the boundary of ellipticity loss (but still within the elliptic range). In the vicinity of this boundary, the incremental stress and displacement fields evidence localized deformations with patterns organized into shear bands, evidencing inclinations corresponding to those predicted at ellipticity loss. These localized deformation patterns are shown to explain experimental results on highly deformed soft materials containing thin, stiff inclusions. Finally, the incremental energy release rate and incremental J-integral are derived, related to a reduction (or growth) of the stiffener. It is shown that this is always positive (or negative), but tends to zero approaching the Ellipticity boundary, which implies that reduction of the lamellar inclusion dies out and, simultaneously, shear bands develop.     

Instabilities and loss of ellipticity in fiber-reinforced compressible non-linearly elastic solids under plane deformation

In a recent paper we examined the loss of ellipticity and its interpretation in terms of fiber kinking and other instability phenomena in respect of a fiber-reinforced incompressible elastic material. Here we provide a corresponding analysis for fiber-reinforced compressible elastic materials. The analysis concerns a material model which consists of an isotropic base material augmented by a reinforcement dependent on the fiber direction. The assessment of loss of ellipticity can be cast in terms of the eigenvalues of the acoustic tensors associated with the isotropic and anisotropic parts of the strain-energy function. For the anisotropic part, two different reinforcing models are examined and it is shown that, depending on the choice of model and whether the fiber is under compression or extension, loss of ellipticity may be associated with, in particular, a weak surface of discontinuity normal to or parallel to the deformed fiber direction or at an intermediate angle. Under compression the associated failure interpretations include fiber kinking and fiber splitting, while under extension fiber de-bonding and matrix failure are included.