A note on material forces in finite inelasticity

In this contribution we aim to elaborate material forces in the context of multiplicative elasto-plasticity, which is considered as a representative and general framework for finite inelasticity. The comparison of different representations of the balance of linear momentum enables us to identify relevant Eshelbian stress tensors and corresponding volume forces. These material, or rather configurational, forces incorporate dislocation density tensors due to the general incompatibility of the underlying intermediate configuration. As an interesting application, the celebrated Peach–Koehler force, driving single dislocations in the context of finite-deformation inelasticity, allows representation in terms of the derived configurational volume forces.     

The Eshelby stress tensor,angular momentum tensor and dilatation flux in gradient elasticity

The Eshelby (static energy momentum) stress tensor, the angular momentum tensor and the dilatation flux are derived for anisotropic linear gradient elasticity in non-homogeneous materials. The divergence of these tensors gives the configurational forces, moments and work terms in gradient elasticity. There are several types of configurational forces, acting on the dislocation density and its gradient, on the inhomogeneities, proportional to the distortion, and linear and quadratic in the distortion gradient, and on the body force.     

Configurational forces: are they needed?

It is shown that standard forces alone are unable to account for the body’s response to the evolution of material structures with time. It is also shown what configurational forces, both at a distance and at contact, are needed to give a complete picture of the body’s response. Moreover, representations of these configurational forces are given in terms of standard constructs, such as the first and second deformation gradients, the Piola stress, and the free-energy density.     

Configurational forces and couples in fracture mechanics accounting for microstructures and dissipation

Configurational forces and couples acting on a dynamically evolving fracture process region as well as their balance are studied with special emphasis to microstructure and dissipation. To be able to investigate fracture process regions preceding cracks of mode I, II and III we choose as underlying continuum model the polar and micropolar, respectively, continuum with dislocation motion on the microlevel. As point of departure balance of macroforces, balance of couples and balance of microforces acting on dislocations are postulated. Taking into account results of the second law of thermodynamics the stress power principle for dissipative processes is derived.Applying this principle to a fracture process region evolving dynamically in the reference configuration with variable rotational and crystallographic structure, the configurational forces and couples are derived generalizing the well-known Eshelby tensor. It is shown that the balance law of configurational forces and couples reflects the structure of the postulated balance laws on macro- and microlevel: the balance law of configurational forces and configurational couples are coupled by field variable, while the balance laws of configurational macro- and microforces are coupled only by the form of the free energy. They can be decoupled by corresponding constitutive assumption.Finally, it is shown that the second law of thermodynamics leads to the result that the generalized Eshelby tensor for micropolar continua with dislocation motion consists of a non-dissipative part, derivable from free and kinetic energy, and a dissipative part, derivable from a dissipation pseudo-potential.     

A non-ordinary state-based peridynamic method to model solid material deformation and fracture

In this paper, we develop a new non-ordinary state-based peridynamic method to solve transient dynamic solid mechanics problems. This new peridynamic method has advantages over the previously developed bond-based and ordinary state-based peridynamic methods in that its bonds are not restricted to central forces, nor is it restricted to a Poisson’s ratio of 1/4 as with the bond-based method. First, we obtain non-local nodal deformation gradients that are used to define nodal strain tensors. The deformation gradient tensors are used with the nodal strain tensors to obtain rate of deformation tensors in the deformed configuration. The polar decomposition of the deformation gradient tensors are then used to obtain the nodal rotation tensors which are used to rotate the rate of deformation tensors and previous Cauchy stress tensors into an unrotated configuration. These are then used with conventional Cauchy stress constitutive models in the unrotated state where the unrotated Cauchy stress rate is objective. We then obtain the unrotated Cauchy nodal stress tensors and rotate them back into the deformed configuration where they are used to define the forces in the nodal connecting bonds. As a first example we quasi-statically stretch a bar, hold it, and then rotate it ninety degrees to illustrate the methods finite rotation capabilities. Next, we verify our new method by comparing small strain results from a bar fixed at one end and subjected to an initial velocity gradient with results obtained from the corresponding one-dimensional small strain analytical solution. As a last example, we show the fracture capabilities of the method using both a notched and un-notched bar.     

Mechanical modeling of growth considering domain variation—Part II: Volumetric and surface growth involving Eshelby tensors

The growth of biological tissues is here described at the continuum scale of tissue elements. Relying on a previous work in Ganghoffer and Haussy (2005), the rephrasing of the balance laws for tissue elements under growth in terms of suitable Eshelby tensors is done in the present contribution, considering successively volumetric and surface growth. Balance laws for volumetric growth are written in both compatible and incompatible configurations, highlighting the material forces for growth associated to Eshelby tensors. Evolution laws for growth are written from the expression of the local dissipation in terms of a relation linking the growth velocity gradient to a growth-like Eshelby stress, in the spirit of configurational mechanics. Surface growth is next envisaged in terms of phenomena taking place in a varying reference configuration, relying on the setting up of a surface potential depending upon the surface transformation gradient and to the normal to the growing surface. The balance laws resulting from the stationnarity of the potential energy are expressed, involving surface Eshelby tensors associated to growth. Simulations of surface growth in both cases of fixed and moving generating surfaces evidence the interplay between diffusion of nutrients and the mechanical driving forces for growth.     

On the representation of Prandtl-Reuss tensors within the framework of multiplicative elastoplasticity

This article presents some aspects of the formulation of finite strain elastoplasticity based on the multiplicative decomposition of the deformation gradient. A “canonical” structure of multiplicative elastoplasticity is discussed characterized by a geometrical setting relative to the intermediate configuration in terms of mixed-variant tensors and the exploitation of fundamental dissipation principles. The symmetric fourth-order elastoplastic moduli (so-called ‘Prandtl-Reuss-Tensors’ of the associative theory) appear as a consequence of the assumed metric-dependence of the flow criterion function in a characteristic structure which seems to be typical for large strain multiplicative elastoplasticity. Particular representations of “Prandtl-Reuss tensors” are outlined for isotropic response as well as for decoupled volumetric-isochoric stress response.     

On a formulation for anisotropic elastoplasticity at finite strains invariant with respect to the intermediate configuration

The paper is concerned with a formulation of anisotropic finite strain inelasticity based on the multiplicative decomposition of the deformation gradient F=F_eF_p. A major feature of the theory is its invariance with respect to rotations superimposed on the inelastic part of the deformation gradient. The paper motivates and shows how such an invariance can be achieved. At the heart of the formulation is the mixed-variant transformation of the structural tensor, defined as the tensor product of the privileged directions of the material as given in a reference configuration, under the action of F_p. Issues related to the plastic material spin are discussed in detail. It is shown that, in contrast to the isotropic case, any flow function formulated purely in terms of stress quantities, necessarily exhibits a non-vanishing plastic material spin. The possible construction of spin-free rates is discussed as well, where it is shown that the flow rule must then depend not only on the stress but on the strain as well.     

关于铁电铁弹材料的自然构形

铁电换弹材料常在极化下工作,其性能受材料的微结构直接影响。铁电材料在自极化过程中自变形和自极化方向的不唯一性,以及多晶材料中存在各种不同方向晶界的不可避免性,使得其自极化稳定构形具有极其复杂的结构。根据铁电铁材料单晶特征,提出在自极化过程中该材料能量密度是变形梯度和电位移向量的非凸函数。并且从能量角度出发,导出角电铁弹材料的自极化稳定构形所应满足的必要条件;得出在自然条件下（即无外力和无外加电场作     

A screw dislocation in a functionally graded material using the translation gauge theory of dislocations

The aim of this paper is to provide new results and insights for a screw dislocation in functionally graded media within the gauge theory of dislocations. We present the equations of motion for dislocations in inhomogeneous media. We specify the equations of motion for a screw dislocation in a functionally graded material. The material properties are assumed to vary exponentially along the x and y-directions. In the present work we give the analytical gauge field theoretic solution to the problem of a screw dislocation in inhomogeneous media. Using the dislocation gauge approach, rigorous analytical expressions for the elastic distortions, the force stresses, the dislocation density and the pseudomoment stresses are obtained depending on the moduli of gradation and an effective intrinsic length scale characteristic for the functionally graded material under consideration.     

A numerical spectral approach for solving elasto-static field dislocation and g-disclination mechanics

A spectral approach is developed to solve the elasto-static equations of field dislocation and g-disclination mechanics in periodic media. Given the spatial distribution of Nye’s dislocation density and/or g-disclination density tensors in heterogeneous or homogenous linear elastic media, the incompatible and compatible elastic distortions are respectively obtained from the solutions of Poisson and Navier-type equations in the Fourier space. Intrinsic discrete Fourier transforms solved by the Fast Fourier Transform (FFT) method, which are consistent with the pixel grid for the calculation of first and second order spatial derivatives, are preferred and compared to the classical discrete approximation of continuous Fourier transforms when deriving elastic fields of defects. Numerical examples are provided for homogeneous linear elastic isotropic solids. For various defects, a regularized defect density in the core is considered and smooth elastic fields without Gibbs oscillations are obtained, when using intrinsic discrete Fourier transforms. The results include the elastic fields of single screw and edge dislocations, standard wedge disclinations and associated dipoles, as well as “twinning g-disclinations”. In order to validate the present spectral approach, comparisons are made with analytical solutions using the Riemann–Graves integral operator and with numerical simulations using the finite element approximation.     

Kinematics of finite-strain elastic-inelastic deformation

Polar decomposition tensors are constructed for slightly disturbed kinematic elastic, inelastic, and thermal strain tensors. Provided that the inelastic and thermal site gradients are pure deformations without rotations, relations are obtained between inelastic small strains and small rotations and between thermal small strains and small rotations which transform an intermediate configuration to a close current configuration. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 1, pp. 165–172, January–February, 2008.     

A thermodynamic based higher-order gradient theory for size dependent plasticity

A physically motivated and thermodynamically consistent formulation of small strain higher-order gradient plasticity theory is presented. Based on dislocation mechanics interpretations, gradients of variables associated with kinematic and isotropic hardenings are introduced. This framework is a two non-local parameter framework that takes into consideration large variations in the plastic strain tensor and large variations in the plasticity history variable; the equivalent (effective) plastic strain. The presence of plastic strain gradients is motivated by the evolution of dislocation density tensor that results from non-vanishing net Burgers vector and, hence, incorporating additional kinematic hardening (anisotropy) effects through lattice incompatibility. The presence of gradients in the effective (scalar) plastic strain is motivated by the accumulation of geometrically necessary dislocations and, hence, incorporating additional isotropic hardening effects (i.e. strengthening). It is demonstrated that the non-local yield condition, flow rule, and non-zero microscopic boundary conditions can be derived directly from the principle of virtual power. It is also shown that the local Clausius–Duhem inequality does not hold for gradient-dependent material and, therefore, a non-local form should be adopted. The non-local Clausius–Duhem inequality has an additional term that results from microstructural long-range energy interchanges between the material points within the body. A detailed discussion on the physics and the application of proper microscopic boundary conditions, either on free surfaces, clamped surfaces, or intermediate constrained surfaces, is presented. It is shown that there is a close connection between interface/surface energy of an interface or free surface and the microscopic boundary conditions in terms of microtraction stresses. Some generalities and utility of this theory are discussed and comparisons with other gradient theories are given. Applications of the proposed theory for size effects in thin films are presented.     

EFFECTS OF DISLOCATION CONFIGURATION, CRACK BLUNTING AND FREE SURFACES ON THE TRIGGERING LOAD OF DISLOCATION SOURCES

The effects of dislocation configuration,crack blunting and free surfaces on the triggering load of dislocation sources in the vicinity of a crack or a wedge tip subjected to a tensile load in the far field are investigated.An appropriate triggering criterion for dislocation sources is proposed by considering the configurational forces acting on each dislocation.The triggering behaviors of dislocation sources near the tips of a crack and a wedge are compared.It is also found that the blunting of crack tip and the presence of free surfaces near the crack or the wedge have considerable influences on the triggering load of dislocation sources.This study might be of significance to gaining a deeper understanding of the brittle-to-ductile transition of materials.     

An elasto-plastic theory of dislocation and disclination fields

A linear theory of the elasto-plasticity of crystalline solids based on a continuous representation of crystal defects – dislocations and disclinations – is presented. The model accounts for the translational and rotational aspects of lattice incompatibility, respectively associated with the presence of dislocations and disclinations. The defects content relates to the incompatible plastic strain and curvature tensors. The stress state is described by using the conjugate variables to strain and curvature, i.e., the stress and couple-stress tensors. Defect motion is described by two transport equations. A dynamic interplay between dislocations and disclinations results from a disclination-induced source term in the transport of dislocations. Thermodynamic guidance provides the driving forces conjugate to dislocation and disclination velocity in a continuous context, as well as admissible constitutive relations for the latter. When dislocation and disclination velocity vanish, the model reduces to deWit’s elasto-static theory of crystal defects. It also reduces to Acharya’s linear elasto-plastic theory for dislocation fields when the disclination density is ignored. The theory is intended for use in instances where rotational defects matter, such as grain boundaries. To illustrate its applicability, a finite high-angle tilt boundary is modeled using a disclination dipole and its behavior under tensile loading normal to the boundary is shown.