Strain localization in plane strain micropolar elasticity

Summary Strain localization of nonlinear micropolar elastic material in plane strain deformation is discussed. Attention is focused on the case when the micro-rotation is linearly related to the macro-rotation and when the material is centro-symmetric. The strain localization is regarded as a perturbation on the otherwise homogeneous deformation field. General formulations for the initiation of the localization are given. Numerical results for a special material are presented. Since the effects of microstructures of materials are considered, the analysis gives not only the critical stretches for localized banding, but also the thickness of the band.

---

Lokalisieru g der Formänderungen bei mikropolar elastischem, ebenem Formänderungszustand

Übersicht Untersucht wird die Lokalisierung der Verformungen von nichtlinearen, mikropolar elastischen Werkstoffen im ebenen Formänderungszustand. Hauptaugenmerk liegt dabei auf der linearen Abhängigkeit der Mikrodrehung von der Makrodrehung im Falle eines zentralsym-metrischen Werkstoffes. Die Lokalisierung wird als eine Störung des ansonsten homogenen Verformungsfeldes betrachtet. Algemeine Bedingungen für die Initiierung der Lokalisierung werden formuliert und für einen ausgewählten Werkstoff numerisch ausgewertet. Dank der Betrachtung der Mikrostruktureinflüsse liefert die dargestellte Analyse nicht nur die Werte der kritischen Streckungen für lokalisierte Bänder, sondern auch deren Dicke.

---

---

This work was prepared when the Author was visiting University of Kaiserslautern in Germany. The author appreciates very much the fellowship awarded him by the A. v. Humboldt Foundation and the hospitality of Prof. Dr.-Ing. U. Wittek. Later part of the work was also supported by National Natural Science Foundation of China through grant 19002014.     

Cosserat (micropolar) elasticity in Stroh form

The theory of defects in Cosserat continua is sketched out in strict analogy to the theory of line defects in anisotropic elasticity (Stroh theory). This rewrite of the second order equilibrium equations of elasticity in a 3-dimensional space as first order equations in a 6-dimensional space is analogous to replacing the Laplace equation by the Riemann–Cauchy equations. For generalized plane strain of anisotropic micropolar (Cosserat) elasticity one obtains a 14-dimensional coupled linear system of differential equations of first order and for plane strain of anisotropic micropolar (Cosserat) elasticity we obtain a 6-dimensional coupled linear system of differential equations of first order.     

Unified theory of structures based on micropolar elasticity

Meccanica - This paper intends to establish a unified theory of structures based on the micropolar elasticity (ME). ME allows taking into consideration the microstructure of the material, through...     

Plane strain dynamics of elastic solids with intrinsic boundary elasticity, with application to surface wave propagation

In this paper, in a development of the static theory derived by Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853), we establish the equations of motion for a non-linearly elastic body in plane strain with an elastic surface coating on part or all of its boundary. The equations of (linearized) incremental motions superposed on a finite static deformation are then obtained and applied to the problem of (time-harmonic) surface wave propagation on a pre-stressed incompressible isotropic elastic half-space with a thin coating on its plane boundary. The secular equation for (dispersive) wave speeds is then obtained in respect of a general form of incompressible isotropic elastic strain-energy function for the bulk material and a general energy function for the coating material. Specialization of the form of strain-energy function enables the secular equation to be cast as a quartic equation and we therefore focus on this for illustrative purposes. An explicit form for the secular equation is thereby obtained. This involves a number of material parameters, including residual stress and moment in the properties of the coating. It is shown how this equation relates to previous work on waves in a half-space with an overlying thin layer set in the classical theory of isotropic elasticity and, in particular, the significant effect of omission of the rotatory inertia term, even at small wave numbers, is emphasized. Corresponding results for a membrane-type coating, for which the bending moment, inertia and residual moment terms are absent, are also obtained. Asymptotic formulas for the wave speed at large wave number (high frequency) are derived and it is shown how these results influence the character of the wave speed throughout the range of wave number values. A bifurcation criterion is obtained from the secular equation by setting the wave speed to zero, thereby generalizing the bifurcation results of Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853) to the situation in which residual stress and moment are present in the coating. Numerical results which show the dependence of the wave speed on the various material parameters and the finite deformation are then described graphically. In particular, features which differ from those arising in the classical theory are highlighted.     

Soliton-like solutions based on geometrically nonlinear Cosserat micropolar elasticity

The Cosserat model generalises an elastic material taking into account the possible microstructure of the elements of the material continuum. In particular, within the Cosserat model the structured material point is rigid and can only experience microrotations, which is also known as micropolar elasticity. We present the geometrically nonlinear theory taking into account all possible interaction terms between the elastic and microelastic structures. This is achieved by considering the irreducible pieces of the deformation gradient and of the dislocation curvature tensor. In addition we also consider the so-called Cosserat coupling term. In this setting we seek soliton type solutions assuming small elastic displacements, however, we allow the material points to experience full rotations which are not assumed to be small. By choosing a particular ansatz we are able to reduce the system of equations to a sine–Gordon type equation which is known to have soliton solutions.     

Plane waves in a fractional order micropolar magneto-thermoelastic half-space

This paper deals with the problem of magneto-thermoelastic interactions in an unbounded, perfectly conducting half-space whose surface suffers a time harmonic thermal source in the context of micropolar generalized thermoelasticity with fractional heat transfer allowing the second sound effects. The medium is assumed to be unstrained and unstressed initially and has uniform temperature. The Laplace–Fourier double transform technique has been used to solve the resulting non-dimensional coupled field equations. Expressions for displacements, stresses and temperature in the physical domain are obtained using a numerical inversion technique. The effects of fractional parameter, magnetic field and micropolarity on the physical fields are noticed and depicted graphically. For a particular model, these fields are found to be significantly affected by the above mentioned parameters. Some particular cases of interest have been deduced from the present problem. Numerical results predict finite speed of propagation for thermoelastic waves.     

Singular solutions and Green's method in micropolar theory of elasticity

The matrices of fundamental solutions are constructed for a concentrated force as well as a concentrated couple varying harmonically in time and acting in an unbounded micropolar elastic continuum. These solutions are then used to obtain solutions for some other loading singularities. Integral representations, for the displacement and the rotation vectors are obtained by making use of the basic singular solutions. The formal solutions to two fundamental boundary value problems are expressed in terms of integrals which include given surface and body data and Green's functions.     

Invariant characteristic representations for classical and micropolar anisotropic elasticity tensors

By extending and developing the characteristic notion of the classical linear elasticity initiated by Lord Kelvin, a new type of representation for classical and micropolar anisotropic elasticity tensors is introduced. The new representation provides general expressions for characteristic forms of the two kinds of elasticity tensors under the material symmetry restriction and has many properties of physical and mathematical significance. For all types of material symmetries of interest, such new representations are constructed explicitly in terms of certain invariant constants and unit vectors in directions of material symmetry axes and hence they furnish invariants which can completely characterize the classical and micropolar linear elasticities. The results given are shown to be useful. In the case of classical elasticity, the spectral properties disclosed by our results are consistent with those given by similar established results.     

A general solution to some plane problems of micropolar elasticity

We obtain a general solution to the field equations of plane micropolar elasticity for materials characterized by a hexagonal or equilateral triangular structure. These materials exhibit 3-fold symmetry in the plane and the elastic response is isotropic. Utilizing two displacement potential functions, the solution is obtained in terms of two analytic functions and a third function satisfying the modified homogeneous Helmholtz equation. Expressions for the two-dimensional components of displacement, stress, and couple stress, along with the resultant force on a contour, are presented. We observe that micropolar effects are most significant in material regions subjected to large deformation gradients. Specific results are presented for the classical crack problem, the half plane loaded uniformly on the surface, Flamant's problem, and the circular cylinder compressed by equal and opposite concentrated forces.     

Some problems of linear elasticity for cylinders in micropolar orthotropic material

Some special problems for axisymmetric solids made of linearly elastic orthotropic micropolar material with central symmetry are dealt with. The first one is a hollow circular cylinder of unlimited length, subjected to internal and external uniform pressure. The second one is a hollow or solid circular cylinder of finite length, subjected to a relative rotation of the bases about its axis. In both cases, one of the axes of elastic symmetry is parallel to the cylinder axis; the other two are arbitrarily oriented in the plane of any cross-section of the solid. The elastic properties are invariant along the cylinder axis. It is shown that the two problems are governed by formally similar sets of ordinary differential equations in the kinematic fields (in-plane displacements and microrotations). In the general case, numerical solutions are derived. The solution for the cylinder subjected to radial pressure does not significantly differ from that obtained in classical elasticity, at least in terms of radial and hoop force stresses. In the case of a cylinder subjected to torsion the difference between the micropolar and the classical solutions is more pronounced. The torque induces twisting couple stresses about the cylinder axis of variable sign. Finally, size effects in terms of torsional inertia are pointed out.     

Complementary energy release rates and dual conservation integrals in micropolar elasticity

The complementary energy momentum tensor, expressed in terms of the spatial gradients of stress and couple-stress, is used to construct the and conservation integrals of infinitesimal micropolar elasticity. The derived integrals are related to the release rates of the complementary potential energy associated with a defect translation or rotation. A nonconserved integral is also derived and related to the energy release rate that is associated with a self-similar cavity expansion. The results are compared to those obtained on the basis of the classical energy momentum tensor, expressed in terms of the spatial gradients of displacement and rotation, and the release rates of the potential energy. It is shown that the evaluation of the complementary conservation integrals is of similar complexity to that of the classical conservation integrals, so that either can be effectively used in the energetic analysis of the mechanics of defects. The two-dimensional versions of the dual conservation integrals are then derived and applied to an out-of-plane shearing of a long cracked slab.     

Plane strain problems in second-order elasticity theory

The equations of second-order elasticity are developed in polar coordinates R, θ for plane strain deformations of incompressible isotropic elastic materials. By considering a ‘displacement function’ the second-order problem is reduced to the solution of an equation of the form ⁴ψ = g(R, Θ) where ² is Laplace's differential operator and g(R, Θ) depends only on the first-order solution. The displacement function technique is then applied to obtain a second-order solution to the problem of an elastic body contained between two concentric rigid circular boundaries, when the outer boundary is held fixed and the inner is subjected to a rigid body translation.     

Construction of eigentensor columns in the linear micropolar theory of elasticity

The internal structure of the block tensor matrix of the elastic modulus tensors is studied for the case of micropolar theory. In particular, the problem of finding the eigenvalues and eigentensor columns of block tensor matrices is considered. The complete orthonormal system of eigentensor columns for a block tensor matrix is constructed. A number of definitions and theorems are formulated. Several newly introduced terms are used to propose various representations of the specific strain energy and the corresponding constitutive relations.     

Axisymmetric problem for a half-space in the micropolar theory of elasticity

The general solution of the equilibrium equations is obtained for a half-space subjected to arbitrary normal pressure. Four particular cases, including a concentrated force, are considered in detail. The singular parts of the displacements, rotations and stresses are obtained for the case of a concentrated force. The corresponding classical results have been derived. The graphs for various physical and geometrical quantities have been drawn to illustrate the micropolar effects.

---

Zusammenfassung Die ellgemeine Lösung für Gleichgewicht eines Halbraums unter beliebigem Normaldruck wird gefunden. Vier Spezialfälle darunter eine konzentrieste Kraft werden genauer untersucht. Für diesen Fall werden die Verschiebungen, Drehungen und Spannungen gefunden. Die entsprechenden klassischen Lösungen wurden erhalten. Zur Illustration der Mikropolareffekte wurden Diagramme der verschiedenen physikalischen und geometrischen Grössen gezeichnet.

     

Plane elasticity problem for a multi-wedge system with a thin wedge

The paper presents a method for studying a system of elastic wedges containing a thin wedge with the angle Θ₀, which may be arbitrary small. An analysis shows that the considered problem, involving 2-D vectors of tractions and displacements, cannot be solved by straight-forward extension of the method previously worked out by the authors for analogous scalar problems. The difficulty arises because of the disclosed feature of the dependences between the Mellin transformed displacements and tractions at the boundaries of a thin wedge: they are linearly dependent when their Taylor’s expansions in Θ₀ are represented by the first terms only. The difficulty is removed by using the consequences of the linear dependence and by an appropriate re-arrangement of variables. Then simple physical models, simulating the influence of a thin wedge on a multi-wedge system, become available. The models cover the cases of a very rigid and very compliant thin wedge and also intermediate cases. The ranges of the models applicability are studied analytically and illustrated by numerical results.     

Plane waves and vibrations in the theory of micropolar thermoelasticity for materials with voids

The present paper concerns with the linear theory of micropolar thermoelasticity for materials with voids. Some basic properties of wave numbers of the longitudinal and transverse plane harmonic waves are treated. The existence theorems of non-trivial solutions and eigenfrequencies of the interior homogeneous boundary value problems of steady vibrations are proved. The connection between plane harmonic waves and eigenfrequencies of the aforementioned problems is established.     

Compatibility conditions and equations of motion in the linear micropolar theory of elasticity

An analog of Cesàro’s formula and several compatibility conditions are given for the three-dimensional and two-dimensional linear micropolar theory of elasticity in the form different from that used in the literature. A number of formulas are obtained to determine the antisymmetric part of the strain (stress) tensor in terms of the symmetric part of the strain tensor and the symmetric part of the bending-torsion (stress and couple-stress) tensor and to determine the antisymmetric part of the bending-torsion (couple-stress) tensor in terms of the symmetric part of the bending-torsion (couplestress) tensor. Some integro-differential equations of motion expressed in terms of the symmetric parts of the stress and couple-stress tensors are proposed for the micropolar theory of elasticity.     

Cavitation in finite elasticity with surface energy effects

Cavity formation in incompressible as well as compressible isotropic hyperelastic materials under spherically symmetric loading is examined by accounting for the effect of surface energy. Equilibrium solutions describing cavity formation in an initially intact sphere are obtained explicitly for incompressible as well as slightly compressible neo-Hookean solids. The cavitating response is shown to depend on the asymptotic value of surface energy at unbounded cavity surface stretch. The energetically favorable equilibrium is identified for an incompressible neo-Hookean sphere in the case of prescribed dead-load traction, and for a slightly compressible neo-Hookean sphere in the case of prescribed surface displacement as well as prescribed dead-load traction. In the presence of surface energy effects, it becomes possible that the energetically favorable equilibrium jumps from an intact state to a cavitated state with a finite cavity radius, as the prescribed loading parameter passes a critical level. Such discontinuous cavitation characteristics are found to be highly dependent on the relative magnitude of the surface energy to the bulk strain energy.