Material force in micromorphic plasticity

Micromorphic theory envisions a material body as a continuous collection of deformable particles with finite size and inner structure. It is considered as the most successful top-down formulation of a two-level continuum model, in which the deformation is expressed as a sum of macroscopic continuous deformation and internal microscopic deformation of the inner structure. In this work, we revisit the original micromorphic theory and further construct a mathematical theory of micromorphic plasticity with generalized strain-based return mapping algorithm. The concept of material forces, which may also be referred as Eshelbian mechanics, was first derived for micromorphic thermo-visco-elastic solid, and, now in this work, it is extended to the micromorphic plasticity. The balance law of pseudo-momentum is formulated. The detailed expressions of Eshelby stress tensor, pseudo-momentum, and material forces are derived. Following this formulation, the failure mechanisms of micromorphic thermo-visco-elastoplastic materials can be further investigated.     

The concept of material forces in phase transition problems within the level-set framework

The dynamics of a phase transition front in solids using the level set method is examined in this paper. Introducing an implicit representation of singular surfaces, a regularized version of the sharp interface model arises. The interface transforms into a thin transition layer of nonzero thickness where all quantities take inhomogeneous expressions within the body. It is proved that the existence of an inhomogeneous energy of the material predicts inhomogeneity forces that drive the singularity. The driving force is a material force entering the canonical momentum equation (pseudo-momentum) in a natural way. The evolution problem requires a kinetic relation that determines the velocity of the phase transition as a function of the driving force. Here, the kinetic relation is produced by invoking relations that can be considered as the regularized versions of the Rankine–Hugoniot jump conditions. The effectiveness of the method is illustrated in a shape memory alloy bar.     

Analytical solution for a Mode III crack in a 3D beam lattice

The brittle fracture behavior of an open cell foam is considered. The foam is modeled by an infinite lattice composed of elastic straight-line beam elements (struts) having uniform cross-sections and rigidly connected at the nodal points. The beams are parallel to the three mutually orthogonal lattice vectors thus forming a microstructure with rectangular parallelepiped cells.A semi-infinite Mode III crack is embedded in the lattice and, for the considered antiplane deformation, each node has three degrees of freedom, namely, the displacement parallel to the crack front and two rotations about the axes perpendicular to this direction. The analysis method hinges on the discrete Fourier transform, which allows to formulate the crack problem by means of the Wiener–Hopf equation. Its solution yields closed-form analytical expressions for the forces and the displacements at any cross-section, and, in particular, at the crack plane. An eigensolution for the traction-free crack faces and K-field remote loading is derived from the solution for the loaded crack using a limiting procedure. An analytical expression for the fracture toughness is derived from the eigensolution by comparing the remote stress field and the stresses in the near-tip struts. The obtained expression is found to be consistent with the known analytical and experimental results for Mode I deformation. It appears, that the dependence of the fracture toughness upon shape anisotropy ratio of the lattice material is non-monotonic. The optimal value of this parameter, which provides the maximum crack arresting ability is determined.     

J-integral and crack driving force in elastic–plastic materials

This paper discusses the crack driving force in elastic–plastic materials, with particular emphasis on incremental plasticity. Using the configurational forces approach we identify a “plasticity influence term” that describes crack tip shielding or anti-shielding due to plastic deformation in the body. Standard constitutive models for finite strain as well as small strain incremental plasticity are used to obtain explicit expressions for the plasticity influence term in a two-dimensional setting. The total dissipation in the body is related to the near-tip and far-field J-integrals and the plasticity influence term. In the special case of deformation plasticity the plasticity influence term vanishes identically whereas for rigid plasticity and elastic-ideal plasticity the crack driving force vanishes. For steady state crack growth in incremental elastic–plastic materials, the plasticity influence term is equal to the negative of the plastic work per unit crack extension and the total dissipation in the body due to crack propagation and plastic deformation is determined by the far-field J-integral. For non-steady state crack growth, the plasticity influence term can be evaluated by post-processing after a conventional finite element stress analysis. Theory and computations are applied to a stationary crack in a C(T)-specimen to examine the effects of contained, uncontained and general yielding. A novel method is proposed for evaluating J-integrals under incremental plasticity conditions through the configurational body force. The incremental plasticity near-tip and far-field J-integrals are compared to conventional deformational plasticity and experimental J-integrals.     

Body-force stress fields defined by caustic rosettes

The method of reflected caustics was extended to the study of elastic fields due to body forces. It was shown that gage perforations create caustics under the influence of body forces which are different in shape and size than those developed in usual cases. The elastic solution of a perforated strip under the influence of an external loadp and body forcesqy was developed by defining a two-term Muskhelishvili complex stress function (z). The equations of the caustics and their initial curves were established. It was shown that as the body force intensity,q, was increased relative to the external loading,p, the classical two-lobe caustic for perforated strips without body forces evoluted to an open curve and, for a further increase ofq, to a three-lobe caustic. As the body force to external force ratio was increased this third lobe was rapidly increased, relative to the two principal lobes, whereas the position of the center of the caustic was displaced along the loading axis. The maximum diametersD _t of the caustics along the loading axis of the plate yield enough information for evaluating the body force intensity, if the mechanical properties of the material and the geometry of the optical set-up are known.     

Configurational forces in crystal plasticity: an analysis of the influence of grain boundaries on crack driving forces

The theory of configurational forces is applied to standard dissipative materials. As a computational example, the method is used to analyze a crystal plastic material law, which assumes flow in the basal and prismatic slip systems of a hexagonally close packed crystal. The material data mimic titanium. Numerical examples include the driving force on a crack interacting with a grain boundary. A simple algorithm is proposed to treat straight crack growth. It is shown how the configurational force at the crack tip is influenced by the presence of the grain boundary.     

Controlling parameter of the stress intensity factors for a planar interfacial crack in three-dimensional bimaterials

Numerical solutions of singular integral equations are discussed in the analysis of a planar rectangular interfacial crack in three-dimensional bimaterials subjected to tension. The problem is formulated as a system of singular integral equations on the basis of the body force method. In the numerical analysis, unknown body force densities are approximated by the products of the fundamental density functions and power series, where the fundamental density functions are chosen to express singular behavior along the crack front of the interface crack exactly. The calculation shows that the present method gives smooth variations of stress intensity factors along the crack front for various aspect ratios. The present method gives rapidly converging numerical results and highly satisfied boundary conditions throughout the crack boundary. The stress intensity factors are given with varying the material combination and aspect ratio of the crack. It is found that the stress intensity factors K_I and K_II are determined by the bimaterial constant ε alone, independent of elastic modulus ratio and Poisson’s ratio.     

A Crack with an Electric Displacement Saturation Zone in an Electrostrictive Material

A crack with an electric displacement saturation zone in an electrostrictive material under purely electric loading is analyzed. A strip saturation model is here employed to investigate the effect of the electrical polarization saturation on electric fields and elastic fields. A closed form solution of electric fields and elastic fields for the crack with the strip saturation zone is obtained by using the complex function theory. It is found that the K _I -dominant region is very small compared to the strip saturation zone. The generalized Dugdale zone model is also employed in order to investigate the effect of the saturation zone shape on the stress intensity factor. Using the body force analogy, the stress intensity factor for the asymptotic problem of a crack with an elliptical saturation zone is evaluated numerically.     

A generalized force measure of conditions at crack tips

A tentative measure of the forces tending to cause crack growth—the apparent crack extension force—is defined within the framework of continuum mechanics. By an associated fracture criterion initiation of growth may be predicted as well as the direction of preferred growth. The theory is specialized to elastic, viscoelastic and elastic-plastic materials. Under specified conditions the apparent crack extension force may be expressed by surface integrals over the boundary of an arbitrary part of the body for quasi-static deformation and for steady-state propagation of the crack. For plane deformation and for infinitesimal deformation under plane stress conditions these surface integrals reduce to path independent line integrals which include the J integral by Rice[1] and the G integral by Sih[2] as special cases.     

Plastic flow at a crack tip. Energy failure criterion and its relationship to the J integral

Plastic flow at the tip of a crack moving in an elastic body is considered using the theory of an ideally rigid-plastic body. The material at the crack tip is treated as a body consisting of an elastic outer region and a rigid-plastic inner region. It is shown that this representation is energetically justified for small plastic regions. The distribution of the specific dissipation of the work of internal forces and deformations along the particle trajectory at the crack tip is obtained. The relationship between the specific dissipation of the work of internal forces and the J integral under plane-strain conditions is established.     

A domain independent integral expression for the crack extension force of a curved crack in three dimensions

An integral expression that is domain independent in curvilinear coordinates and compatible with zero divergence of Eshelby's (Phil. Trans. Roy. Soc. (London) 244 (1951) 87.) energy momentum tensor was obtained from the principle of virtual work. By applying Eshelby's definition of the force on a material defect a general expression of the crack extension force for a curved crack in three dimensions, here called the F-integral, was derived from the domain independent integral expression. The F-integral is given explicitly for a number of curved cracks and found to be in agreement with previously known solutions, when available. The influence of crack surface and crack front curvature upon the various forms of the F-integral is discussed. The F-integral presented in this work is a generalisation of the J-integral (Rice, J. Appl. Mech. 35 (1968) 379.) to curved cracks in orthogonal curvilinear coordinates.     

Variation of the stress intensity factor along the front of a 3-D rectangular crack subjected to mixed-mode load

Summary  The singular integral equation method is applied to the calculation of the stress intensity factor at the front of a rectangular crack subjected to mixed-mode load. The stress field induced by a body force doublet is used as a fundamental solution. The problem is formulated as a system of integral equations with r ⁻³-singularities. In solving the integral equations, unknown functions of body-force densities are approximated by the product of polynomial and fundamental densities. The fundamental densities are chosen to express two-dimensional cracks in an infinite body for the limiting cases of the aspect ratio of the rectangle. The present method yields rapidly converging numerical results and satisfies boundary conditions all over the crack boundary. A smooth distribution of the stress intensity factor along the crack front is presented for various crack shapes and different Poisson's ratio. Received 5 March 2002; accepted for publication 2 July 2002     

Finite element solutions for plane strain mode I crack with strain gradient effects

In this paper, a new phenomenological theory with strain gradient effects is proposed to account for the size dependence of plastic deformation at micro- and submicro-length scales. The theory fits within the framework of general couple stress theory and three rotational degrees of freedom ω_i are introduced in addition to the conventional three translational degrees of freedom u_i. ω_i is called micro-rotation and is the sum of material rotation plus the particles' relative rotation. While the new theory is used to analyze the crack tip field or the indentation problems, the stretch gradient is considered through a new hardening law. The key features of the theory are that the rotation gradient influences the material character through the interaction between the Cauchy stresses and the couple stresses; the term of stretch gradient is represented as an internal variable to increase the tangent modulus. In fact the present new strain gradient theory is the combination of the strain gradient theory proposed by Chen and Wang (Int. J. Plast., in press) and the hardening law given by Chen and Wang (Acta Mater. 48 (2000a) 3997). In this paper we focus on the finite element method to investigate material fracture for an elastic-power law hardening solid. With remotely imposed classical K fields, the full field solutions are obtained numerically. It is found that the size of the strain gradient dominance zone is characterized by the intrinsic material length l₁. Outside the strain gradient dominance zone, the computed stress field tends to be a classical plasticity field and then K field. The singularity of stresses ahead of the crack tip is higher than that of the classical field and tends to the square root singularity, which has important consequences for crack growth in materials by decohesion at the atomic scale.     

An electrically conducting interface crack with a contact zone in a piezoelectric bimaterial

A plane problem for an electrically conducting interface crack in a piezoelectric bimaterial is studied. The bimaterial is polarized in the direction orthogonal to the crack faces and loaded by remote tension and shear forces and an electrical field parallel to the crack faces. All fields are assumed to be independent of the coordinate co-directed with the crack front. Using special presentations of electromechanical quantities via sectionally-analytic functions, a combined Dirichlet–Riemann and Hilbert boundary value problem is formulated and solved analytically. Explicit analytical expressions for the characteristic mechanical and electrical parameters are derived. Also, a contact zone solution is obtained as a particular case. For the determination of the contact zone length, a simple transcendental equation is derived. Stress and electric field intensity factors and, also, the contact zone length are found for various material combinations and different loadings. A significant influence of the electric field on the contact zone length, stress and electric field intensity factors is observed. Electrically permeable conditions in the crack region are considered as well and matching of different crack models has been performed.     

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.     

Non-uniform warping including the effects of torsion and shear forces. Part I: A general beam theory

This two-part contribution presents a beam theory with a non-uniform warping including the effects of torsion and shear forces, and valid for any homogeneous cross-section made of isotropic elastic material. Part I is devoted to the theoretical developments and part II discusses analytical and numerical results obtained for torsion and shear-bending of cantilever beams made of different kinds of cross-section. The theory is based on a kinematics assuming that the cross-section maintains its shape and including three independent warping parameters associated to the three warping functions corresponding to torsion and shear forces. Starting from this displacement model and using the principle of virtual work, the corresponding beam theory is derived. For this theory, closed-form results are obtained for the cross-sectional constants and the three-dimensional expressions of the normal and shear stresses. Comparison with classical beam theories is carried out and additional effects due to the non-uniformity of the warping are highlighted. In particular, the contributions of primary and secondary internal forces and the effect of the non-symmetry of the cross-section on the structural behavior of the beam are specified. Simplified versions of this theory, wherein the number of degrees of freedom is reduced, are also presented. The analytical and numerical analyzes presented in part II give responses on the quality of this non-uniform beam theory and indicate also when its simplified versions could be applied.     

On the Planck scale and properties of matter

Invariant and long-lived physical properties and structures of matter are modeled by intrinsic rotations in three and four degrees of freedom. The rotations are quantized starting from the Planck scale by using a nonlinear 1/r potential and period doubling—a common property of nonlinear dynamical systems. The absolute values given by the scale-independent model fit closely with observations in a wide range of scales. A comparison is made between the values calculated from the model and the properties of the basic elementary particles, particle processes, planetary systems, and other physical phenomena. The model also shows that the perceived forces can be divided into two categories: (1) force is always attractive, like in gravitation and (2) force is attractive or repulsive, like in electrostatics. An erratum to this article can be found at     

Multiscale material modeling and its application to a dynamic crack propagation problem

Here we present a multiscale field theory for modeling and simulation of multi-grain material system which consists of several different kinds of single crystals and a large number of different kinds of discrete atoms. The theoretical construction of the multiscale field theory is briefly introduced. The interatomic forces are used to formulate the governing equations for the system. A compact tension specimen made of magnesium oxide is modeled by discrete atoms in front of the crack tip and finite elements in the far field. Results showing crack propagation through the atomic region are presented.     

The Debye Theory of Rotary Diffusion: History, Derivation, and Generalizations

Motivated by the Debye theory of rotary diffusion in a dipolar fluid, we systematically develop a continuum mechanical theory of rotary diffusion. This theory generalizes classical kinematics to include continuous rotary degrees of freedom and introduces an additional balance law associated with the rotary degrees of freedom. Various constitutive relations are proposed in accordance with standard procedures of nonlinear continuum mechanics. The resulting set of equations provides a properly invariant and thermodynamically consistent theory that allows for constitutive nonlinearities. In particular, the classical Debye theory along with the Nernst-Einstein relations are shown to follow from a special case of linear constitutive relations and an assumption of ideality in which the free energy consists only of a classical entropic contribution. Within our theory, the notion of osmotic pressure arises naturally as a consequence of accounting for forces that act conjugate to the rotary degrees of freedom and serves as the driving force for rotary diffusion. Accepted November 18, 2000?Published online April 23, 2001     

Dynamic instability and steady-state response of an elliptical cracked shaft

An elliptical front crack has been found to be more accurate and realistic for modeling the transverse surface crack in rotating machinery compared with the widely used straight front crack. When the shaft rotates, the elliptical crack opens and closes alternatively, due to gravity, and thus, a “breathing effect” occurs. This variance in shaft stiffness is time-periodic, and hence, a parametrically excited system is expected. Therefore, the dynamic instability and steady-state response of a rotating shaft containing an elliptical front crack are studied in the paper. The local flexibility due to the crack is derived, and the governing equations of the crack shaft system are established using the assumed modes method. Utilizing the Bolotin’s method and harmonic balance method, the boundaries of two typical instability regions and maximum response amplitude of the cracked shaft could be computed numerically. The elliptical crack parameters (depth, shape factor and position) and damping are, respectively, considered and discussed for their effects on the dynamic behavior of the elliptical cracked shaft. Some research results might be helpful for the crack detection in rotating machinery.