The problem of sharp notch in microstructured solids governed by dipolar gradient elasticity

In this paper, we deal with the asymptotic problem of a body of infinite extent with a notch (re-entrant corner) under remotely applied plane-strain or anti-plane shear loadings. The problem is formulated within the framework of the Toupin-Mindlin theory of dipolar gradient elasticity. This generalized continuum theory is appropriate to model the response of materials with microstructure. A linear version of the theory results by considering a linear isotropic expression for the strain-energy density that depends on strain-gradient terms, in addition to the standard strain terms appearing in classical elasticity. Through this formulation, a microstructural material constant is introduced, in addition to the standard Lamé constants . The faces of the notch are considered to be traction-free and a boundary-layer approach is followed. The boundary value problem is attacked with the asymptotic Knein-Williams technique. Our analysis leads to an eigenvalue problem, which, along with the restriction of a bounded strain energy, provides the asymptotic fields. The cases of a crack and a half-space are analyzed in detail as limit cases of the general notch (infinite wedge) problem. The results show significant departure from the predictions of the standard fracture mechanics

Plane-Strain Problems for a Class of Gradient Elasticity Models-A Stress Function Approach

The plane strain problem is analyzed in detail for a class of isotropic, compressible, linearly elastic materials with a strain energy density function that depends on both the strain tensor epsilon and its spatial gradient a double dagger epsilon. The appropriate Airy stress-functions and double-stress-functions are identified and the corresponding boundary value problem is formulated. The problem of an annulus loaded by an internal and an external pressure is solved

Finite element modeling of trip steels

In this paper, a constitutive model that describes the mechanical behavior of steels exhibiting the "Transformation Induced Plasticity" (TRIP) phenomenon during martensitic transformation is presented. Multiphase TRIP steels are considered as composite materials with a ferritic matrix containing bainite and retained austenite, which gradually transforms into martensite. The effective properties and overall behavior of TRIP steels are determined by using homogenization techniques for non-linear composites. The developed model considers the different hardening behavior of the individual phases and estimates the apportionment of plastic strain and stress between the individual phases of the composite. A methodology for the numerical integration of the resulting elastoplastic constitutive equations in the context of the finite element method is developed and the constitutive model is implemented in a general-purpose finite element program. The prediction of the model in uniaxial tension agree well with the experimental data. The problem of necking of a bar is studied in detail

The Problem of Tension-Torsion of Pretwisted Elastic Beams Revisited

A one-dimensional technical theory for pretwisted isotropic linearly elastic beams loaded in tension and torsion was developed. The analysis was based on a kinematically admissible field written in terms of three generalized displacements, which were determined from the theorem of minimum potential energy. A general analytical solution was developed. The problem of a pretwisted beam with a built-in end and loaded in tension and torsion at the other end was analyzed. The problem also was solved by carrying out detailed three-dimensional finite-element calculations. The predictions of the technical theory agreed very well with the results of the finite-element solution. The effects of Poisson's ratio were examined, and the applicability of the model to beams of various lengths was discussed. © 2021 American Society of Civil Engineers

A non-local plasticity model for porous metals with deformation-induced anisotropy: Mathematical and computational issues

A non-local (gradient) plasticity model for porous metals that accounts for deformation-induced anisotropy is presented. The model is based on the work of Ponte Castañeda and co-workers on porous materials containing randomly distributed ellipsoidal voids. It takes into account the evolution of porosity and the evolution/development of anisotropy due to changes in the shape and the orientation of the voids during plastic deformation. A “material length” ℓ is introduced and a “non-local” porosity is defined from the solution of a modified Helmholtz equation with appropriate boundary conditions, as proposed by Geers et al. (2001); Peerlings et al. (2001). At a material point located at x, the non-local porosity f(x) can be identified with the average value of the “local” porosity floc(x) over a sphere of radius R≃3ℓ centered at x. The same approach is used to formulate a non-local version of the Gurson isotropic model. The mathematical character of the resulting incremental elastoplastic partial differential equations of the non-local model is analyzed. It is shown that the hardening modulus of the non-local model is always larger than the corresponding hardening modulus of the local model; as a consequence, the non-local incremental problem retains its elliptic character and the possibility of discontinuous solutions is eliminated. A rate-dependent version of the non-local model is also developed. An algorithm for the numerical integration of the non-local constitutive equations is developed, and the numerical implementation of the boundary value problem in a finite element environment is discussed. An analytical method for the required calculation of the eigenvectors of symmetric second-order tensors is presented. The non-local model is implemented in ABAQUS via a material “user subroutine” (UMAT or VUMAT) and the coupled thermo-mechanical solution procedure, in which temperature is identified with the non-local porosity. Several example problems are solved numerically and the effects of the non-local formulation on the solution are discussed. In particular, the problems of plastic flow localization in plane strain tension, the plane strain mode-I blunt crack tip under small-scale-yielding conditions, the cup-and-cone fracture of a round bar, and the Charpy V-notch test specimen are analyzed. © 2020 Elsevier Lt

Finite element implementation of gradient plasticity models - Part II: Gradient-dependent evolution equations

The variational formulation and finite element implementation of a class of plasticity models with gradient-dependent yield functions is covered in detail in Part I. In this sequel, attention is focussed upon the finite element formulation and implementation of a different class of plasticity models wherein spatial gradients of one or more internal variables enter the evolution equations for the state variables. Finite element solutions are obtained for the problems of localization of plastic flow in plane strain tension and of a mode-I plane strain crack. The pathological dependence of the finite element solution on the size of the elements in local plasticity models disappears when the gradient-type model is used. (C) 1998 Elsevier Science S.A. All rights reserved