Nonlocal continuum theory of anisotropically damaged metals

The paper deals with a consistent and systematic general framework for the development of anisotropic continuum damage in ductile metals based on thermodynamic laws and nonlocal theories. The proposed model relies on finite strain kinematics based on the consideration of damaged as well as fictitious undamaged configurations related via metric transformation tensors which allow for the interpretation of damage tensors. The formulation is accomplished by rate-independent plasticity using a nonlocal yield condition of Drucker–Prager type, anisotropic damage based on a nonlocal damage growth criterion as well as non-associated flow and damage rules. The nonlocal theory of inelastic continua is established to be able to take into account long-range microstructural interaction. The approach incorporates macroscopic interstate variables and their higher-order gradients which properly describe the change in the internal structure and investigate the size effect of statistical inhomogeneity of the heterogeneous material. The idea of bridging length-scales is made by using higher-order gradients in the evolution equations of the equivalent inelastic strain measures which leads to a system of elliptic partial differential equations which is solved using the finite difference method at each iteration of the loading step and the displacement-based finite element procedure is governed by the standard principle of virtual work. Numerical simulations of the elastic–plastic deformation behavior of damaged solids demonstrate the efficiency of the formulation. Tension tests undergoing large strains are used to investigate the damage growth in high strength steel. The influence of various model parameters on the prediction of the deformation and localization of ductile metals is discussed.     

Continuum damage models with non-conventional finite element formulations

In recent years, some research effort has been devoted to the development of non-conventional finite element models for the analysis of concrete structures. These models use continuum damage mechanics to represent the physically non-linear behavior of this quasi-brittle material. Two alternative approaches proved to be robust and computationally competitive when compared with the classical displacement finite element implementations. The first corresponds to the hybrid-mixed stress model where both the effective stress and the displacement fields are independently modeled in the domain of each finite element and the displacements are approximated along the static boundary, which is considered to include the inter-element edges. The second approach corresponds to a hybrid-displacement model. In this case, the displacements in the domain of each element and the tractions along the kinematic boundary are independently approximated. Since it is a displacement model, the inter-element boundaries are now included in the kinematic boundary. In both models, complete sets of orthonormal Legendre polynomials are used to define all approximations required, so very effective p-refinement procedures can be implemented. This paper illustrates the numerical performance of these two alternative approaches and compares their efficiency and accuracy with the classical finite element models. For this purpose, a set of numerical tests is presented and discussed.     

Nonlocal theory solution of two collinear cracks in the functionally graded materials

In this paper, the interaction of two collinear cracks in functionally graded materials subjected to a uniform anti-plane shear loading is investigated by means of nonlocal theory. The traditional concepts of the nonlocal theory are extended to solve the fracture problem of functionally graded materials. To make the analysis tractable, it is assumed that the shear modulus varies exponentially with the coordinate vertical to the crack. By use of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the triple integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularity is present near the crack tips. The nonlocal elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion in functionally graded materials. The magnitude of the finite stress field depends on the crack length, the distance between two cracks, the parameter describing the functionally graded materials and the lattice parameter of the materials.     

First-order gradient damage theory

Taking the strain tensor, the scalar damage variable, and the damage gradient as the state variables of the Helmholtz free energy, the general expressions of the firstorder gradient damage constitutive equations are derived directly from the basic law of irreversible thermodynamics with the constitutive functional expansion method at the natural state. When the damage variable is equal to zero, the expressions can be simplified to the linear elastic constitutive equations. When the damage gradient vanishes, the expressions can be simplified to the classical damage constitutive equations based on the strain equivalence hypothesis. A one-dimensional problem is presented to indicate that the damage field changes from the non-periodic solutions to the spatial periodic-like solutions with stress increment. The peak value region develops a localization band. The onset mechanism of strain localization is proposed. Damage localization emerges after damage occurs for a short time. The width of the localization band is proportional to the internal characteristic length.     

Nonlocal buckling of embedded magnetoelectroelastic sandwich nanoplate using refined zigzag theory

This paper is concerned with a buckling analysis of an embedded nanoplate integrated with magnetoelectroelastic (MEE) layers based on a nonlocal magnetoelectroelasticity theory. A surrounding elastic medium is simulated by the Pasternak foundation that considers both shear and normal loads. The sandwich nanoplate (SNP) consists of a core that is made of metal and two MEE layers on the upper and lower surfaces of the core made of BaTiO₃/CoFe₂O₄. The refined zigzag theory (RZT) is used to model the SNP subject to both external electric and magnetic potentials. Using an energy method and Hamilton’s principle, the governing motion equations are obtained, and then solved analytically. A detailed parametric study is conducted, concentrating on the combined effects of the small scale parameter, external electric and magnetic loads, thicknesses of MEE layers, mode numbers, and surrounding elastic medium. It is concluded that increasing the small scale parameter decreases the critical buckling loads.     

Nonlocal damage model using the phase field method: Theory and applications

A new nonlocal, gradient based damage model is proposed for isotropic elastic damage using the phase field method in order to show the evolution of damage in brittle materials. The general framework of the phase field model (PFM) is discussed and the order parameter is related to the damage variable in continuum damage mechanics (CDM). The time dependent Ginzburg–Landau equation which is also termed the Allen–Cahn equation is used to describe the damage evolution process. Specific length scale which addresses the interface region in which the process of changing undamaged solid to fully damaged material (microcracks) occurs is defined in order to capture the effect of the damaged localization zone. A new implicit damage variable is proposed through the phase field theory. Details of the different aspects and regularization capabilities are illustrated by means of numerical examples and the validity and usefulness of the phase field modeling approach is demonstrated.     

A symmetric nonlocal damage theory

The paper presents a thermodynamically consistent formulation for nonlocal damage models. Nonlocal models have been recognized as a theoretically clean and computationally efficient approach to overcome the shortcomings arising in continuum media with softening. The main features of the presented formulation are: (i) relations derived by the free energy potential fully complying with nonlocal thermodynamic principles; (ii) nonlocal integral operator which is self-adjoint at every point of the solid, including zones near to the solid’s boundary; (iii) capacity of regularizing the softening ill-posed continuum problem, restoring a meaningful nonlocal boundary value problem. In the present approach the nonlocal integral operator is applied consistently to the damage variable and to its thermodynamic conjugate force, i.e. nonlocality is restricted to internal variables only. The present model, when associative nonlocal damage flow rules are assumed, allows the derivation of the continuum tangent moduli tensor and the consistent tangent stiffness matrix which are symmetric. The formulation has been compared with other available nonlocal damage theories.Finally, the theory has been implemented in a finite element program and the numerical results obtained for 1-D and 2-D problems show its capability to reproduce in every circumstance a physical meaningful solution and fully mesh independent results.     

Nonlocal beam theory for nonlinear vibrations of embedded multiwalled carbon nanotubes in thermal environment

A nonlocal elastic beam model is developed to investigate the small scale effects on the large-amplitude vibration analysis of embedded multiwalled carbon nanotubes (MWCNTs) at an elevated temperature. The nested slender nanotubes are coupled with each other through the van der Waals (vdW) interlayer interaction. The curvature-dependent vdW force employed incorporates not only pairwise nearest-neighbor but also nonneighbor interactions between nested nanotubes. The incremental harmonic balance method is adopted to analytically solve the nonlinear equations that are governed by the vibrations of nested nanotubes. The influences of small scale parameter, geometrical parameters, temperature rise, and the elastic medium are fully examined.     

A stochastic theory of cumulative fatigue damage

A stochastic theory for the cumulative fatigue damage of structural components with random fatigue strength under random loading is proposed on the basis of Stratonovich-Khasminskii theorem. The analytical solutions for the probability densities of the cumulative fatigue damage and fatigue life and for the reliability function are given for steel and reinforced concrete components with constant fatigue strength subject to a narrow band stationary Gaussian stress process with zero mean. The results agree very well with those of digital simulation. It is noted that the theory can be applied, in principle, to both metallic and non-metallic materials, narrow band and wide band stress process, and adapted to a sequence ofn, stationary stress processes or quasi-stationary stress process. The scatter and degradation of fatigue strength and the inspection maintenance can also be incorporated into the theory.     

Thermodynamically consistent nonlocal theory of ductile damage

In this paper a thermodynamically consistent, weakly nonlocal theory of ductile damage is presented. The theory is based on the classical dynamical balance laws of forces and couples in the physical space and dynamical balance laws of material forces on evolving defects and on the first and second law of thermodynamics formulated for physical and material space. Assuming general constitutive equations their frame-invariant and thermodynamically admissible form is determined. It is shown that physical and material forces and stresses consist of two parts, a nondissipative part derivable from a free energy potential, and a dissipative part, which can be obtained from a dissipation pseudo-potential, if such a pseudo-potential exists.The theory can be considered as a framework with gradient elastoplasticity, isotropic and anisotropic brittle and ductile gradient damage at finite strain as special cases.     

Three-dimensional and applied formulations in the stability theory of rods made from composite materials

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 25, No. 2, pp. 7–12, February, 1989.     

Weight function theory and variational formulations for three-dimensional plane elastic cracks advancing

The weight function theory for three-dimensional elastic crack analysis received great attention after the work of Rice (1985, 1989). Several applications have been considered since then, particularly in the context of configurational stability, crack path prediction, stress intensity factor expansions, perturbation approaches. In all cases, a specific hypothesis has been made on the variation of crack shape, in order to formulate the problem in terms of Cauchy principal value. In the present note, such hypothesis is further investigated and consequences discussed. A variational statement given in Salvadori and Fantoni (2013a) is thus rephrased in terms of weight functions. Its discrete formulation shows the potential to accurate approximation of crack front propagation.     

Nonlocal Effects in Two-Dimensional Conductivity

The paper deals with the asymptotic behaviour as ε → 0 of a two-dimensional conduction problem whose matrix-valued conductivity a _ε is ε-periodic and not uniformly bounded with respect to ε. We prove that only under the assumptions of equi-coerciveness and L ¹-boundedness of the sequence a _ε , the limit problem is a conduction problem of same nature. This new result points out a fundamental difference between the two-dimensional conductivity and the three-dimensional one. Indeed, under the same assumptions of periodicity, equi-coerciveness and L ¹-boundedness, it is known that the high-conductivity regions can induce nonlocal effects in three (or greater) dimensions.     

Some results for approximate strain and rotation tensor formulations in geometrically non-linear Reissner-Mindlin plate theory

Finite element deflection and stress results are presented for four flat plate configurations and are computed using kinematically approximate (rotation tensor, strain tensor or both) non-linear Reissner-Mindlin plate models. The finite element model is based on a mixed variational principle and has both displacement and force field variables. High order interpolation of the field variables is possible through p-type discretization. Results for some of the higher order approximate models are given for what appears to be the first time. It is found that for the class of example problems examined, exact strain tensor but approximate rotation tensor theories can significantly improve the solution over approximate strain tensor models such as the von Kármán and moderate rotation models when moderate deflections/rotations are present. However, for each of the problems examined (with the exception of a postbuckling problem) the von Kármán and moderate rotation model results compared favorably with the higher order models for deflection magnitudes which could be reasonably expected in typical aeroelastic configurations.     

Finite Scale Microstructures in Nonlocal Elasticity

In this paper we develop a simple one-dimensional model accounting for the formation and growth of globally stable finite scale microstructures. We extend Ericksen's model [9] of an elastic “bar” with nonconvex energy by including both oscillation-inhibiting and oscillation-forcing terms in the energy functional. The surface energy is modeled by a conventional strain gradient term. The main new ingredient in the model is a nonlocal term which is quadratic in strains and has a negative definite kernel. This term can be interpreted as an energy associated with the long-range elastic interaction of the system with the constraining loading device. We propose a scaling of the problem allowing one to represent the global minimizer as a collection of localized interfaces with explicitly known long-range interaction. In this limit the augmented Ericksen's problem can be analyzed completely and the equilibrium spacing of the periodic microstructure can be expressed as a function of the prescribed average displacement. We then study the inertial dynamics of the system and demonstrate how the nucleation and growth of the microstructures result in the predicted stable pattern. Our results are particularly relevant for the modeling of twined martensite inside the austenitic matrix.     

A thermodynamic theory of damage in elastic inorganic and organic solids

This contribution presents the foundations of a thermodynamic theory of damage in elastic solids, developed in collaboration with the late J. Kestin and with E. Honein and T. Honein. The theory is rooted in the so-called conservative or conventional thermodynamics of irreversible processes, where the concept of a local thermodynamic state plays a prominent role. An elastic body prone to damage is regarded as a thermodynamic system characterized by a set of extensive variables that can be defined in both equilibrium and nonequilibrium states and assigned approximately the same values in both the physical space and the abstract state space (i.e., the Gibbsian phase space of constrained equilibria). The extensive variables introduced include internal parameters which describe the damaged state of the body and whose conjugate intensive variables, or affinities, constitute a generalization of Eshelby’s concept of a “force on an elastic singularity”. The local state approximation is applied by assigning to the entropy and temperature in physical space local values which can be calculated in the Gibbsian phase space by the well-established methods of equilibrium thermodynamics. This leads to an explicit expression for the entropy production. The rate equations for the damage are then postulated in such a way as to conform to the second part of the second law of thermodynamics. The resulting theory captures many features of real inorganic material behavior in which no mass loss is sustained. By contrast, damage of organic materials, such as compact bone subject to osteoporosis, is accompanied by bone mass loss. This feature can be accommodated in the theory proposed by a suitable adjustment of the expression of the Gibbs free energy.     

Pinning and Unpinning in Nonlocal Systems

We investigate pinning regions and unpinning asymptotics in nonlocal equations. We show that phenomena are related to but different from pinning in discrete and inhomogeneous media. We establish unpinning asymptotics using geometric singular perturbation theory in several examples. We also present numerical evidence for the dependence of unpinning asymptotics on regularity of the nonlocal convolution kernel.