Optimum design for work-hardening adaptation

Abstract The finite element-linear programming approach and the work-hardening adaptation criterion are used to formulate a general theory of optimum design of rigid-work-hardening structures subjected to loads which vary statically within given limits. Self-weight, as well as some technological constraints, can be introduced into the framework of the optimization problem. The optimality conditions are discussed with the aid of geometrical descriptions as well, and a comparison is made with the standard limit design. Numerical applications are given for a plane truss and a plane frame with axial force-bending moment interaction.

Interfacial energy effects within the framework of strain gradient plasticity

AbstractIn the framework of strain gradient plasticity, a solid body with boundary surface playing the role of a dissipative boundary layer endowed with surface tension and surface energy, is addressed. Using the so-called residual-based gradient plasticity theory, the state equations and the higher order boundary conditions are derived quite naturally for both the bulk material and the boundary layer. A phenomenological constitutive model is envisioned, in which the bulk material and the boundary layer obey (rate independent associative) coupled plasticity evolution laws, with kinematic hardening laws of differential nature for the bulk material, but of nondifferential nature for the layer…

Optimum plastic design for multiple sets of loads

We study optimum plastic design of structures made up, or conceived as assemblies of finite elements, each having an elemental piece-wise linear rigid-plastic behaviour. Since cost function linearly dependent on design variables are considered, optimization problems in linear programming are encountered. Allowance is made for design dependent mass forces, and for some technological constraints. The design growing process is studied in the case of various sets of alternative applied loads, and the optimality conditions are written in a proper geometrical form which leads to a generalization of the concept of Foulkes mechanism.

Dynamic shakedown of structures under repeated seismic loads

Elastic, perfectly plastic structures are considered under the action of repeated short-duration exitations of seismic type acting in an unknown time sequence, but belonging to a given polyhedral excitation domain. The basic excitations (vertices of the polyhedron) are chosen as discrete-spectrum waves each with frequencies coincident with the first natural frequencies of the structure, and amplitudes related to the ground features and earthquake intensity (according to the Kanai and Tajimi filter model) in such a way that every admissible excitation-obtained as a linear convex combination of the basic ones-has a maximum power not exceeding a given value. In the framework of unrestricted dy…

A thermodynamic approach to nonlocal plasticity and related variational principles

Elastic-plastic rate-independent materials with isotropic hardening/softening of nonlocal nature are considered in the context of small displacements and strains. A suitable thermodynamic framework is envisaged as a basis of a nonlocal associative plasticity theory in which the plastic yielding laws comply with a (nonlocal) maximum intrinsic dissipation theorem. Additionally, the rate response problem for a (continuous) set of (macroscopic) material particles, subjected to a given total strain rate field, is discussed and shown to be characterized by a minimum principle in terms of plastic coefficient. This coefficient and the relevant continuum tangent stiffness matrix are shown to admit, …

BIEM-based variational principles for elastoplasticity with unilateral contact boundary conditions

The structural step problem for elastic-plastic internal-variable materials is addressed in the presence of frictionless unilateral contact conditions. Basing on the BIEM (boundary integral equation method) and making use of deformation-theory plasticity (through the backward-difference method of computational plasticity), two variational principles are shown to characterize the solution to the step problem: one is a stationarity principle having as unknowns all the problem variables, the other is a saddle-point principle having as unknowns the increments of the boundary tractions and displacements, along with the plastic strain increments in the domain. The discretization by boundary and i…

Strain gradient elasticity within the SGBEM

Dynamic shakedown by modal analysis

Dynamic shakedown of discrete elastic-perfectly plastic structures under a specified load history is studied using the dynamic characteristics of the structure provided by modal analysis. Several statical and kinematical theorems are presented, including lower and upper bound theorems for the minimum adaptation time of the structure. In the formulation of the kinematical theorems a crucial role is played by the appropriate definition of ≪admissible plastic strain cycle≫.

Size effects on the plastic collapse limit load of thin foils in bending and thin wires in torsion

Abstract Following a previous paper by the author [Strain gradient plasticity, strengthening effects and plastic limit analysis, Int. J. Solids Struct. 47 (2010) 100–112], a nonconventional plastic limit analysis for a particular class of micron scale structures as, typically, thin foils in bending and thin wires in torsion, is here addressed. An idealized rigid-perfectly plastic material is considered, which is featured by a strengthening potential degree-one homogeneous function of the effective plastic strain and its spatial gradient. The nonlocal (gradient) nature of the material resides in the inherent strengthening law, whereby the yield strength is related to the effective plastic st…

A unifying variational framework for stress gradient and strain gradient elasticity theories

Abstract Stress gradient elasticity and strain gradient elasticity do constitute distinct continuum theories exhibiting mutual complementary features. This is probed by a few variational principles herein presented and discussed, which include: i) For stress gradient elasticity, a (novel) principle of minimum complementary energy and an (improved-form) principle of stationarity of the Hellinger–Reissner type; ii) For strain gradient elasticity, a (known) principle of minimum total potential energy and a (novel) principle of stationarity of the Hu–Washizu type. Additionally, the higher order boundary conditions for stress gradient elasticity, previously derived by the author (Polizzotto, Int…

Internal-variable constitutive model for rate-independent plasticity with hardening saturation surface

An elastic-plastic material model with internal variables and thermodynamic potential, not admitting hardening states out of a saturation surface, is presented. The existence of such a saturation surface in the internal variables space — a consequence of the boundedness of the energy that can be stored in the material's internal micro-structure — encompasses, in case of general kinematic/isotropic hardening, a one-parameter family of envelope surfaces in the stress space, which in turn is enveloped by a limit surface. In contrast to a multi-surface model, noad hoc rules are required to avoid the intersection between the yield and bounding/envelope surface. The flow laws of the proposed mode…

Interaction Diagram of a Circular Bar in Torsion and Extension

For a circular bar of perfectly plastic material and subjected to a cyclically variable torque and a constant axial force, the interaction (or generalized Bree) diagram is derived by a direct method in which Melan’s theorem is used to locate the nonratchetting load boundary.

A unified approach to quasi-static shakedown problems for elastic-plastic solids with piecewise linear yield surface

The paper concerns shakedown analysis of elastic-plastic bodies subjected to quasi-statically varying loads within a given domain. Using a perturbation method, a general inequality is given, from which, by simply specializing the perturbing terms, the generalized Melan theorem as well as bounds on various deformation parameters (such as displacements or plastic strain intensities) are derived. The solution of the «perturbed» shakedown problem in finite or holonimic terms permits the bound to be the most stringent and expressible in «local» terms instead of integral terms. A simple application concludes the paper.

From the Euler–Bernoulli beam to the Timoshenko one through a sequence of Reddy-type shear deformable beam models of increasing order

Abstract A sequence of elastic Reddy-type shear deformable beams of increasing (odd) order is envisioned, which starts with the Euler–Bernoulli beam (first order) and terminates with the Timoshenko beam (infinite order). The kinematics of the generic beam, including the warping mode of the cross sections, is specified in terms of three deformation variables (two curvatures, one shear angle), work-conjugate of as many stress resultants (two bending moments, one shear force). The principle of virtual power is used to determine the (static) equilibrium equations and the boundary conditions. The equations relating the bending moments and shear force to the curvatures and shear angle are also re…

Strain-gradient elastic-plastic material models and assessment of the higher order boundary conditions

Abstract A gradient elastic material model exhibiting gradient kinematic and isotropic hardening is addressed within a thermodynamic framework suitable to cope with nonlocal-type continua. The Clausius–Duhem inequality is used, in conjunction with the concepts of energy residual, insulation condition and locality recovery condition, to derive all the pertinent restrictions upon the constitutive equations, including the PDEs and the related higher order (HO) boundary conditions that govern the gradient material behaviour. Through a suitable limiting procedure, the HO boundary conditions are shown to interpret the action, upon the body's boundary surface, of idealized extra HO constraints cap…

A class of shear deformable isotropic elastic plates with parametrically variable warping shapes

A homogeneous shear deformable isotropic elastic plate model is addressed in which the normal transverse fibers are allowed to rotate and to warp in a physically consistent manner specified by a fixed value of a real non-negative warping parameter ω. On letting ω vary continuously (at fixed load and boundary conditions), a continuous family of shear deformable plates Pω is generated, which spans from the Kirchhoff plate at the lower limit ω=0, to the Mindlin plate at the upper limit ω=∞; for ω=2, Pω identifies with the third-order Reddy plate. The boundary-value problem for the generic plate Pω is addressed in the case of quasi-static loads, for which a principle of minimum total potential …

An energy residual-based approach to gradient effects within the mechanics of generalized continua

AbstractGeneralized continua exhibiting gradient effects are addressed through a method grounded on the energy residual (ER)-based gradient theory by the first author and coworkers. A main tool of this theory is the Clausius-Duhem inequality cast in a form differing from the classical one only by a nonstandard extra term, the (nonlocality) ER, required to satisfy the insulation condition (its global value has to vanish or to take a known value). The ER carries in the nonlocality features of the mechanical problem through a strain-like rate field, being the specific nonlocality source, and a concomitant higher-order long-range stress (or microstress) field. The thermodynamic restrictions on …

A second strain gradient elasticity theory with second velocity gradient inertia – Part II: Dynamic behavior

Abstract This paper is the sequel of a companion Part I paper devoted to the constitutive equations and to the quasi-static behavior of a second strain gradient material model with second velocity gradient inertia. In the present Part II paper, a multi-cell homogenization procedure (developed in the Part I paper) is applied to a nonhomogeneous body modelled as a simple material cell system, in conjunction with the principle of virtual work (PVW) for inertial actions (i.e. momenta and inertia forces), which at the macro-scale level takes on the typical format as for a second velocity gradient inertia material model. The latter (macro-scale) PVW is used to determine the equilibrium equations …

Shakedown Under Thermomechanical Loads

A shakedown theory for elastic–perfectly plastic structures subjected to thermomechanical loads varying within a given range is outlined under the assumption of temperature-dependent yield stress, but temperature-independent elastic moduli and thermal expansion coefficient are considered. Inertia and creep effects, along with thermal coupling phenomena, are considered negligible. A nonstandard constitutive model is used in which a central role is played by the yield function-assumed convex in the stress–temperature space. The inherent flow mechanism obeys the normality rule and includes, beside the standard plastic strain rates, an extra scalar variable work conjugate of the temperature, co…

Nonlocal Elastic-Damage Models

A theory of nonlocal isotropic damage for elastic quasi-brittle materials is presented under the assumption of isothermal conditions and small deformations. Key ingredients of this theory are a self-adjoint (regularization) operator which transforms a local field into a related nonlocal one while preserves uniform fields and a free energy which depends on the strain and (linearly) on the nonlocal damage variable, as well as on an (scalar) internal variable accounting for the damage hardening. The relevant thermodynamic restrictions on the constitutive equations are obtained by means of two alternative procedures, one based on the principle of virtual power and the other on the concept of “n…

Surface effects, boundary conditions and evolution laws within second strain gradient plasticity

Abstract The principle of the virtual power (PVP) is used in conjunction with the concepts of “energy residual” and “insulation condition” to address second strain gradient plasticity. The energy residual with its typical divergence format is an extra stress power playing the role of basic state variable to describe the gradient effects, whereas the insulation condition constitutes a global energy characterization of the body as part of the body/environment system. The microstructure of a second strain gradient material (but not of a first strain gradient one) is shown to exhibit surface effects with the formation of a thin boundary layer. This boundary layer is in local (and global) equili…

A method to transform a nonlocal model into a gradient one within elasticity and plasticity

Abstract A method based on the principle of the virtual power (PVP) is presented, by which a mechanical problem of nonlocal elasticity, or plasticity, is transformed into one of gradient nature. Different Taylor series expansion techniques are applied to the driving local strain fields of the nonlocal problem, either full spatial expansion within the bulk volume, or uni-directional expansion along the normal to the thin boundary layer. This, at the limit when the boundary layer thickness tends to zero, makes the PVP of the nonlocal model transform itself into one featuring a counterpart gradient model. Also, for a class of “associated” nonlocal and gradient elasticity models (i.e. the kerne…

A Consistent Boundary/Interior Element Method for Evolutive Elastic Plastic Structural Analysis

A symmetric/sign-definite formulation of the BEM to address the evolutive elastic plastic analysis of structures is presented. A wide class of material models with internal variables and thermodynamic potential is considered. Different energy methods—namely the boundary min-max principle, the Helmholtz free energy and the maximum intrinsic dissipation theorem—axe employed in order to provide the discretization operations by boundary elements and cell elements with inherent variational consistency. The resulting space-discretized equations can be solved by a step-by-step procedure and a predictor/corrector iteration scheme, with corrections operated locally cell-by-cell, just as with the FEM…

A STRAIN-DIFFERENCE BASED NONLOCAL ELASTICITY THEORY FOR SMALL-SCALE SHEAR-DEFORMABLE BEAMS WITH PARAMETRIC WARPING

A strain-difference-based nonlocal elasticity model

Abstract A two-component local/nonlocal constitutive model for (macroscopically) inhomogeneous linear elastic materials (but constant internal length) is proposed, in which the stress is the sum of the local stress and a nonlocal-type stress expressed in terms of the strain difference field, hence identically vanishing in the case of uniform strain. Attention is focused upon the particular case of piecewise homogeneous material. The proposed model is thermodynamically consistent with a suitable free energy potential. It constitutes an improved form of the Vermeer and Brinkgreve [A new effective nonlocal strain measure for softening plasticity. In: Chambon, R., Desrues, J., Vardulakis, I. (E…

Elastic-Viscoplastic Solids Subjected to Thermal and Loading Cycles

— A class of elastic-viscoplastic materials with dual internal variables, thermodynamic potential and temperature-dependent plastic and creep data is considered. For solids (or structures) of such materials, subjected to cyclic loads and temperature variations, the existence of a steady-state response is ascertained and its periodicity characteristics established. Particular steady-state responses, like, elastic and inelastic shakedown, are addressed. By means of a sensitivity analysis of the steady cycle with respect to the load parameter changes, a number of basic features of inelastic shakedown (the viscoplastic counterpart of plastic shakedown) are also addressed.

Shakedown theorems for elastic–plastic solids in the framework of gradient plasticity

Abstract Static and kinematic shakedown theorems are given for a class of generalized standard materials endowed with a hardening saturation surface in the framework of strain gradient plasticity. The so-called residual-based gradient plasticity theory is employed. The hardening law admits a hardening potential, which is a C 1 -continuous function of a set of kinematic internal variables and of their spatial gradients, and is required to satisfy a global sign restriction (but not to be necessarily convex). The totally produced, the accumulated and the freely moving dislocations per unit volume, distinguished as statistically stored and geometrically necessary ones, are in this way accounted…

Boundary/Field Variational Principles for the Elastic Plastic Rate Problem

An elastic-plastic continuous solid body under quasi-statically variable external actions is herein addressed in the hypoteses of rate-independent material model with dual internal variables and of infinitesimal displacements and strains. The related analysis problem for assigned rate actions is first formulated through a boundary/field integral equation approach, then is shown to be characterized by two variational principles, one of which is a stationarity theorem, the other a min-max one.

Bounds to internal forces for elastic-plastic solids subjected to variable loads

Considering an elastic-plastic workhardening solid with piecewise linear yield surfaces and a piecewise linear workhardening law, we give a method for constructing bounds to the internal forces and to the (hardened) yield stresses produced by the action of variable loads at any point of the body and at any time. The loading history is supposed to be unknown, but the loads range within a given domain.

Gradient elasticity and nonstandard boundary conditions

Abstract Gradient elasticity for a second gradient model is addressed within a suitable thermodynamic framework apt to account for nonlocality. The pertinent thermodynamic restrictions upon the gradient constitutive equations are derived, which are shown to include, besides the field (differential) stress–strain laws, a set of nonstandard boundary conditions. Consistently with the latter thermodynamic requirements, a surface layer with membrane stresses is envisioned in the strained body, which together with the above nonstandard boundary conditions make the body constitutively insulated (i.e. no long distance energy flows out of the boundary surface due to nonlocality). The total strain en…

Shakedown of Structures Accounting for Damage Effects

Shakedown theory for elastic-plastic-damage materials is exposed. Two kinds of shakedown are considered: i) Enlarged shakedown (or simply shakedown), in which both plastic deformations and damage eventually cease, after which the structural response is purely elastic; ii) Weak-form shakedown, in which plastic deformations eventually cease together with their consequences (including ductile damage), not necessarily damage from other sources (which are however escluded by assumption). An (enlarged) shakedown static-type theorem is given for a class of D-stable structures. Sufficient theorems of weak-form shakedown are provided, i.e. a static-type one (quite similar to that of Hachemi and Weic…

An alternative formulation of the boundary element method

Abstract The paper suggests an alternative formulation of the Boundary Element Method, in which singular solutions generated by unit dislocations are required and moreover the stresses at the interior points of the body are directly computed from the boundary quantities, without passing through the displacements. Relationships between the singular solutions for unit dislocation and unit force are derived.

Thermodynamics-based gradient plasticity theories with an application to interface models

AbstractIn the framework of small deformations, the so-called residual-based gradient plasticity theory is reconsidered and improved. Using the notion of moving geometrically necessary dislocations (GNDs), suitable micromechanics interpretations are heuristically given for the higher order boundary conditions and the long distance particle interactions. Also, a comparison is made between this theory and the analogous virtual work principle (VWP)-based one, whereby their respective conceptual and methodological features are pointed out. The conditions under which the two theories lead to a same constitutive model are investigated, showing that, correspondingly, a certain indeterminacy exhibi…

Unified thermodynamic framework for nonlocal/gradient continuum theories

Abstract A thermodynamic framework, equipped with the concept of nonlocality (energy) residual, is utilized to address nonlocal/gradient internal variable material models. A unified procedure is provided for either nonlocal and gradient materials, which makes it possible to determine the thermodynamic restrictions upon the constitutive equations, and in particular the pertinent state equations, the consistent form of the dissipation power and the constitutive expression of the nonlocality residual. Additionally, for gradient models, the associated nonstandard boundary conditions are derived, pointing out their basically constitutive nature and their substantial differences from the standard…

Variational Formulations for Coupled BE/FE Methods in Elastostatics

Ein gekoppeltes BEM/FEM-Problem aus der Elastostatik, ein typisches Substrukturproblem, wird im Rahmen der symmetrisch-definiten BEM behandelt. Es werden vier verschiedene Variationsformulierungen vorgestellt, in deren jeder die Transmissionsbedingungen gegenuber der Trennflache zwischen FE-Unterregion und BE-Unterregion die Rolle naturlicher Randbedingungen spielen. Zwei der oben erwahnten Formulierungen sind Stationaritatsprinzipien in gemischter Form, die anderen beiden sind Sattelpunkt-Prinzipien, d. h. Kombinationen des Rand-min-max-Prinzips entweder mit dem Prinzip der minimalen Gesamtpotentialenergie oder mit dem Prinzip der minimalen Gesamtkomplementaritatsenergie. Jedes der oben an…

Nonlocal elasticity and related variational principles

Abstract The Eringen model of nonlocal elasticity is considered and its implications in solid mechanics studied. The model is refined by assuming an attenuation function depending on the `geodetical distance' between material particles, such that in the diffusion processes of the nonlocality effects certain obstacles as holes or cracks existing in the domain can be circumvented. A suitable thermodynamic framework with nonlocality is also envisaged as a firm basis of the model. The nonlocal elasticity boundary-value problem for infinitesimal displacements and quasi-static loads is addressed and the conditions for the solution uniqueness are established. Three variational principles, nonlocal…

Strain gradient plasticity, strengthening effects and plastic limit analysis

Abstract Within the framework of isotropic strain gradient plasticity, a rate-independent constitutive model exhibiting size dependent hardening is formulated and discussed with particular concern to its strengthening behavior. The latter is modelled as a (fictitious) isotropic hardening featured by a potential which is a positively degree-one homogeneous function of the effective plastic strain and its gradient. This potential leads to a strengthening law in which the strengthening stress, i.e. the increase of the plastically undeformed material initial yield stress, is related to the effective plastic strain through a second order PDE and related higher order boundary conditions. The plas…

Mathematical Programming Methods for the Evaluation of Dynamic Plastic Deformations

Dynamic plastic deformation can be evaluated with two accuracy levels, nemely either by a full analysis making use of a step-by-step procedure, or by a simplified analysis making use of a bounding technique. Both procedures can be achieved by means a unified mathematical programming approach here presented. It is shown that for a full analysis both the direct and indirect methods of linear dynamics coupled with mathematical programming methods can be successfully applied, whereas for a simplified analysis a convergent bounding principle, holding both below and above the shakedown limit, can be utilized to produce an efficient linear programming-based algorithm.

Strain gradient elasticity within the symmetric BEM formulation

The symmetric Galerkin Boundary Element Method is used to address a class of strain gradient elastic materials featured by a free energy function of the (classical) strain and of its (first) gradient. With respect to the classical elasticity, additional response variables intervene, such as the normal derivative of the displacements on the boundary, and the work-coniugate double tractions. The fundamental solutions - featuring a fourth order partial differential equations (PDEs) system - exhibit singularities which in 2D may be of the order 1/ r 4 . New techniques are developed, which allow the elimination of most of the latter singularities. The present paper has to be intended as a resear…

Integral and differential approaches to Eringen's nonlocal elasticity models accounting for boundary effects with applications to beams in bending

A nonlocal strain gradient plasticity theory for finite deformations

Abstract Strain gradient plasticity for finite deformations is addressed within the framework of nonlocal continuum thermodynamics, featured by the concepts of (nonlocality) energy residual and globally simple material. The plastic strain gradient is assumed to be physically meaningful in the domain of particle isoclinic configurations (with the director vector triad constant both in space and time), whereas the objective notion of corotational gradient makes it possible to compute the plastic strain gradient in any domain of particle intermediate configurations. A phenomenological elastic–plastic constitutive model is presented, with mixed kinematic/isotropic hardening laws in the form of …

A second strain gradient elasticity theory with second velocity gradient inertia – Part I: Constitutive equations and quasi-static behavior

Abstract A multi-cell homogenization procedure with four geometrically different groups of cell elements (respectively for the bulk, the boundary surface, the edge lines and the corner points of a body) is envisioned, which is able not only to extract the effective constitutive properties of a material, but also to assess the “surface effects” produced by the boundary surface on the near bulk material. Applied to an unbounded material in combination with the thermodynamics energy balance principles, this procedure leads to an equivalent continuum constitutively characterized by (ordinary, double and triple) generalized stresses and momenta. Also, applying this procedure to a (finite) body s…

A hierarchy of simplified constitutive models within isotropic strain gradient elasticity

Abstract Simplified isotropic models of strain gradient elasticity are presented, based on the mutual relationship between the inherent (dual) gradient directions (i.e. the gradient direction of any strain gradient source and the lever arm direction of the promoted double stress). A class of gradient-symmetric materials featured by gradient directions obeying a reciprocity relation and by 4 independent h.o. (higher order) coefficients is envisioned, along with the sub-classes of hemi-collinear materials (3 h.o. coefficients, gradient directions in part coincident), collinear materials (2 h.o. coefficients, equal gradient directions) and micro-affine materials (1 h.o. coefficient, behavioral…

A thermodynamically consistent formulation of nonlocal and gradient plasticity

Stress gradient versus strain gradient constitutive models within elasticity

Abstract A stress gradient elasticity theory is developed which is based on the Eringen method to address nonlocal elasticity by means of differential equations. By suitable thermodynamics arguments (involving the free enthalpy instead of the free internal energy), the restrictions on the related constitutive equations are determined, which include the well-known Eringen stress gradient constitutive equations, as well as the associated (so far uncertain) boundary conditions. The proposed theory exhibits complementary characters with respect to the analogous strain gradient elasticity theory. The associated boundary-value problem is shown to admit a unique solution characterized by a Helling…

A note on the higher order strain and stress tensors within deformation gradient elasticity theories: Physical interpretations and comparisons

Abstract Higher order strain and stress tensors encompassed within gradient elasticity theories are discussed with a particular concern to the physical meaning of double and triple stresses. A single rule is shown to hold for the physical interpretation of the indices of a higher order stress tensor both within distortion gradient and strain gradient theories, whereas the analogous Mindlin’s rule holds only within distortion gradient theories. Double and triple stresses are discussed separately with the aid of simple illustrative examples. A corrigendum to a previous paper by the author (IJSS 50 (2013) 3749–3765) is also presented.

A symmetric Galerkin boundary/domain element method for finite elastic deformations

Abstract The Symmetric Galerkin Boundary Element Method (SGBEM) is reformulated for problems of finite elasticity with hyperelastic material and incompressibility, using fundamental solutions related to a (fictitious) homogeneous isotropic and compressible linear elastic material. The proposed formulation contains, besides the standard boundary integrals, domain integrals which account for the problem's nonlinearities through some (fictitious) initial strain and stress fields required to satisfy appropriate “consistency” equations. The boundary/domain integral equation problem so obtained is shown to admit a stationarity principle (a consequence of the Hu-Washizu one), which covers a number…

On the Conditions to Prevent Plastic Shakedown of Structures: Part I—Theory

For a structure of elastic perfectly plastic material subjected to a given cyclic (mechanical and/or kinematical) load and to a steady (mechanical) load, the conditions are established in which plastic shakedown cannot occur whatever the steady load, and thus the structure is safe against the alternating plasticity collapse. Static and kinematic theorems, analogous to those of classical shakedown theory, are presented.

Size effects of small-scale beams in bending addressed with a strain-difference based nonlocal elasticity theory

Abstract A strain-difference based nonlocal elasticity model devised by the authors elsewhere (Polizzotto et al., Int. J. Solids Struct. 25 (2006) 308–333) is applied to small-scale homogeneous beam models in bending under static loads in the purpose to describe the inherent size effects. With this theory —belonging to the strain-integral nonlocal model family, but exempt from anomalies typical of the Eringen nonlocal theory— the relevant beam problem is reduced to a set of three mutually independent Fredholm integral equations of the second kind (each independent of the beam’s ordinary boundary conditions, only one depends on the given load), which can be routinely solved numerically. Appl…

A Study on Plastic Shakedown of Structures: Part I—Basic Properties

For a continuous elastic-perfectly plastic solid body subjected to a combination of cyclic (mechanical and/or kinematical) load and of a steady (mechanical) load such as to produce plastic shakedown (i.e., alternating plasticity), a number of characterizing properties are established and discussed. The conditions for the body’s transition from plastic shakedown to ratchetting are also addressed.

On shakedown of elastic plastic solids

Making reference to elastic perfectly plastic solids subjected to cyclic loads, the problem of the shakedown load factor is considered and the relevant Euler-Lagrange equations are discussed. It is proved that the solution to these equations describes the gradient, with respect to the load multiplier, of the steady-state response of the solid body to the cyclic loads at the shakedown limit, and that it thus enables one to predict the nature of the impending collapse. These results are then extended to the more general case of loads varying within a given load domain.

Variational formulations and extra boundary conditions within stress gradient elasticity theory with extensions to beam and plate models

Abstract The principle of minimum total potential energy and the primary principle of virtual power for stress gradient elasticity are presented as kinematic type constructs dual of analogous static type principles from the literature (Polizzotto, 2014; Polizzotto, 2015a). The extra gradient-induced boundary conditions are formulated as “boundary congruence conditions” on the microstructure’s deformation relative to the continuum, which ultimately require that the normal derivative of the stresses must vanish at the boundary surface. Two forms of the governing PDEs for the relevant boundary-value problem are presented and their computational aspects are discussed. The Timoshenko beam and th…

Anisotropy in strain gradient elasticity: Simplified models with different forms of internal length and moduli tensors

Abstract Anisotropy of centro-symmetric (first) strain gradient elastic materials is addressed and the role there played by the dual gradient directions (i.e. directions of strain gradient and of double stress lever arm) is investigated. Anisotropy manifests itself not only through the classical fourth-rank elasticity tensor C (21 independent constants) in the form of moduli anisotropy, but also through a sixth-rank elasticity tensor B (171 independent constants) in a unified non-separable form as compound internal length/moduli anisotropy. Depending on the microstructure properties, compound anisotropy may also manifest itself in a twofold separable form through a decoupled tensor B = L C …

Shakedown Analysis Within the Framework of Strain Gradient Plasticity

A class of rate-independent material models is addressed within the framework of isotropic strain gradient plasticity. These models exhibit a size dependence through the strengthening effects (Hall–Petch effects), whereby the yield stress is related to the effective plastic strain by a suitable second-order partial differential equation with related boundary conditions. For a perfectly plastic material with strengthening effects, the classical concepts of plastic and shakedown limit analysis do hold, which lead to size dependent plastic and shakedown limit loads according to the dictum: smaller is stronger. In the perspective of a development of direct methods for applications to small-scal…

Bounding principles for elastic-plastic-creeping solids loaded below and above the shakedown limit

Solids of elastic-perfectly plastic creeping material subjected to variable loads are considered within the infinitesimal displacement framework and a bounding principle is presented which holds below and above the shakedown limit. Through the choice of some free parameters, this principle generates a number of deformation bounds with practical meanings, some of wich coincide with known results for creeping and noncreeping material, while others constitute new results or generalizations of known results. The topic will be further studied in a subsequent paper [35].

Constitutive equations for no-tension materials

For a material which is incapable of sustaining tensile stresses (no-tension material, NTM), the local stability postulate is utilized in order to derive the appropriate equations which relate, within general 3D situations, cracking strain states and stress states to each other. Several alternative forms of these equations are discussed, either in terms of stress and strain components, or in terms of stress and strain invariants. The results obtained improve known results regarding the NTM's.

Minimum theorems for displacement and plastic strain rate histories in structural elastoplasticity

The finite element method approach is used to obtain formulations of analysis problems relative to elastic-plastic structures when subjected to prescribed programmes of loads, and under the restrictive hypotheses:a) the yielding surfaces are piecewise linearized, andb) the plastic flow-laws are supposed to be of holonomic type within a single “finite” time interval. For mulations are given as linear complementarity problems and quadratic programming problems: one pair of formulations in terms of velocity and plastic multiplier rate histories, and another pair in terms of plastic multiplier rate histories only. The solutions are shown to be characterized by two minimum principles for displac…

Workhardening adaptation of rigid-plastic structures

The paper considers discrete rigid-plastic structures which are subjected to the action of loads varying quasi-statically within given limits. It studies the conditions for workhoardening adaptation, that is the conditions to ensure that the structure, after an initial rigid-plastic phase, shows a purely rigid behavior. The safety factor against the workhardening inadaptation is defined by two dual optimization problems. Some characteristic features of the yielding surface at failure are pointed out, using also a proper geometric description. Static and kinematic theorems, which are similar to those of shakedown theory, are given. A simple application concludes the paper.

A link between the residual-based gradient plasticity theory and the analogous theories based on the virtual work principle

A link is shown to exist between the so-called residual-based strain gradient plasticity theory and the analogous theories based on the (extended) virtual work principle (VWP). To this aim, the former theory is reformulated and cast in a residual-free form, whereby the insulation condition and the (nonlocal) Clausius–Duhem inequality, on which the theory is grounded, are substituted with equivalent residual-free ingredients, namely the energy balance condition and the residual-free form of the Clausius–Duhem inequality. The equivalence of the residual-free formulation to the original one is shown, also in their ability to cope with energetic size effects and interfacial energy ones. It emer…

Limit analysis of arch-beam structures by dynamic programming

We study one-dimensional structures like arch-beams in the limit state of plastic collapse, on the ground of a two-dimensional yielding surface (bending moment and normal generalized stress). The proposed method, which is able to give a numerical solution of the problem of finding the limit load, rests on the upper bound theorem of limit analysis and uses dynamic programming. We examine also some questions linked with numerical procedures. A future work devoted to applications will complete the treatment.

A micromorphic approach to stress gradient elasticity theory with an assessment of the boundary conditions and size effects

Thermodynamically consistent residual-based gradient plasticity theory and comparison

A gradient plasticity theory for small deformations is presented within the framework of nonlocal continuum thermodynamics. The second principle (Clausius–Duhem inequality), enriched by an additional term named energy residual, is employed in conjunction with the concepts of insulation condition and locality recovery condition, in order to derive all the pertinent restrictions upon the constitutive equations. These include the expressions of the energy residual and of the plastic dissipation density, as well as the PDEs governing the gradient kinematic and isotropic hardening of the material, together with the related higher-order boundary conditions for both the fixed and the moving bounda…

A gradient elasticity theory for second-grade materials and higher order inertia

Abstract Second-grade elastic materials featured by a free energy depending on the strain and the strain gradient, and a kinetic energy depending on the velocity and the velocity gradient, are addressed. An inertial energy balance principle and a virtual work principle for inertial actions are envisioned to enrich the set of traditional theoretical tools of thermodynamics and continuum mechanics. The state variables include the body momentum and the surface momentum, related to the velocity in a nonstandard way, as well as the concomitant mass-accelerations and inertial forces, which do intervene into the motion equations and into the force boundary conditions. The boundary traction is the …

Shakedown of Structures Subjected to Dynamic External Actions and Related Bounding Techniques

The shakedown theory for dynamic external actions is expounded considering elastic-plastic internal-variable material models endowed with hardening saturation surface and assuming small displacements and strains as long with negligible effects of temperature variations on material data. Two sorts of dynamic shakedown theories are presented, i.e.: i) Unrestricted dynamic shakedown, in which the structure is subjected to (unknown) sequences of short-duration excitations belonging to a known excitation domain, with no-load no-motion time periods in between and for which a unified framework with quasi-static shakedown is presented; and ii) Restricted dynamic shakedown, in which the structure is…

Shakedown of elastic—plastic solids with frictionless unilateral contact boundary conditions

Abstract Elastic perfectly plastic solids (or structures) in frictionless unilateral contact with a rigid obstacle and subjected to quasi-statically variable loads within a given domain are considered. In the hypothesis that the structure undergoes small displacements and complies with a d -stability requisite herein introduced, a Melantype shakedown theorem is presented. This theorem is conceptually similar to the classical one; namely, it requires that the unilateral-contact elastic stress response to the loads and to some initial plastic strains be plastically admissible everywhere in the body and for all load conditions. A method for evaluating the shakedown load boundary is also discus…

A thermodynamics-based formulation of gradient-dependent plasticity

Abstract A nonlocal thermodynamic theoretical framework is provided as a basis for a consistent formulation of gradient-dependent plasticity in which a scalar internal variable measuring the material isotropic hardening/softening state is the only nonlocal variable. The main concepts of this formulation are: i) the ‘regularization operator’, of differential nature, which governs the relation between the above nonlocal variable and a related local variable (scalar measure of plastic strain) and confers a unified character to the proposed formulation (this transforms into a formulation for nonlocal plasticity if the regularization operator has an integral nature); ii) the ‘nonlocality residua…

A Linear Programming Method for Bounding Plastic Deformations

A method for providing upper and lower bounds to plastic deformations is presented, which has the feature of being applicable both below and above the structure shakedown limit. The bounds provided are expressed in terms of some fictitious plastic strains obeying relaxed yielding laws, whose evaluation is made by means of a suitable LP-based algorithm.

Symmetric Galerkin Boundary Element Methods

This review article concerns a methodology for solving numerically, for engineering purposes, boundary and initial-boundary value problems by a peculiar approach characterized by the following features: the continuous formulation is centered on integral equations based on the combined use of single-layer and double-layer sources, so that the integral operator turns out to be symmetric with respect to a suitable bilinear form. The discretization is performed either on a variational basis or by a Galerkin weighted residual procedure, the interpolation and weight functions being chosen so that the variables in the approximate formulation are generalized variables in Prager’s sense. As main con…

A Study on Plastic Shakedown of Structures: Part II—Theorems

For a continuous elastic-perfectly plastic solid body subjected to a combination of cyclic (mechanical and/or kinematical) load and of a steady (mechanical) load, two theorems of plastic shakedown are presented, one stating a necessary condition, another stating a sufficient condition. The problem of the direct determination of the plastic shakedown boundary is also briefly addressed.

Shear Effects in Elastic Nanobeams

Small-scale, shear deformable nanobeams, subjected to quasi-static loads, are analyzed by a nonlocal (integral) elasticity model with the main goal to evaluate the influence of shear deformation on size effects. To this aim a warping parametric model is considered in order to obtain a continuous family of shear deformable beam models which span from the Euler-Bernoulli to the Thimoshenko beam model, passing from the Reddy model. The strain difference based nonlocal elasticity theory is applied under the hypotheses of small displacements and isotropic material. The results, obtained by analysing a cantilever nonlocal nanobeam, indicate that shear deformation has a considerable influence upon…

Variational methods for the steady state response of elastic–plastic solids subjected to cyclic loads

Abstract Solids (or structures) of elastic–plastic internal variable material models and subjected to cyclic loads are considered. A minimum net resistant power theorem, direct consequence of the classical maximum intrinsic dissipation theorem of plasticity theory, is envisioned which describes the material behavior by determining the plastic flow mechanism (if any) corresponding to a given stress/hardening state. A maximum principle is provided which characterizes the optimal initial stress/hardening state of a cyclically loaded structure as the one such that the plastic strain and kinematic internal variable increments produced over a cycle are kinematically admissible. A steady cycle min…

A nonhomogeneous nonlocal elasticity model

Nonlocal elasticity with nonhomogeneous elastic moduli and internal length is addressed within a thermodynamic framework suitable to cope with continuum nonlocality. The Clausius–Duhem inequality, enriched by the energy residual, is used to derive the state equations and all other thermodynamic restrictions upon the constitutive equations. A phenomenological nonhomogeneous nonlocal (strain difference-dependent) elasticity model is proposed, in which the stress is the sum of two contributions, local and nonlocal, respectively governed by the standard elastic moduli tensor and the (symmetric positive-definite) nonlocal stiffness tensor. The inhomogeneities of the elastic moduli and of the int…

Shakedown Analysis by Elastic Simulation

Shakedown analysis of elastic plastic structures is widely credited as a valuable analytical/numerical tool for design purposes. For complex structures and loading conditions, e. g. for fast breeder nuclear reactor plants, full inelastic analysis is rarely performed, practically never within the early stages of the design advancement and the inherent decision process. The essential information therein needed can in fact be obtained, at moderate computational costs, by application of the shakedown methods and rules, at least within some limits related to the present developments of shakedown theory and its applicability to practical engineering problems, see e. g. Ponter et al. (1990), Carte…

A unified residual-based thermodynamic framework for strain gradient theories of plasticity

Abstract A unified thermodynamic framework for gradient plasticity theories in small deformations is provided, which is able to accommodate (almost) all existing strain gradient plasticity theories. The concept of energy residual (the long range power density transferred to the generic particle from the surrounding material and locally spent to sustain some extra plastic power) plays a crucial role. An energy balance principle for the extra plastic power leads to a representation formula of the energy residual in terms of a long range stress, typically of the third order, a macroscopic counterpart of the micro-forces acting on the GNDs (Geometrically Necessary Dislocations). The insulation …

Theorems of restricted dynamic shakedown

Abstract Dynamic shakedown for a rate-independent material with internal variables is addressed in the hypothesis that the load values are restricted to those of a specified load history of finite or even infinite duration, thus ruling out the possibility—typical of classical shakedown theory—of indefinite load repetitions. Instead of the usual approach to dynamic shakedown, based on the bounded plastic work criterion, another approach is adopted here, based on the adaptation time criterion. Static, kinematic and mixed-form theorems are presented, which characterize the minimum adaptation time (MAT), a feature of the structure-load system, but which are also able to assess whether plastic w…

Dynamic shakedown of structures with variable appended masses and subjected to repeated excitations

Elastic shakedown for discrete, or finite-element discretized, structures subjected to combinations of static and time-variable loads is addressed in the hypothesis of elastic-perfectly plastic material behavior. The static load is conceived as the weight of an additional mass appended to the structure, whereas the time-variable load is conceived as an unknown sequence of excitations belonging to a specified domain, with intervals between subsequent excitations during which the structure is considered as being motionless. It is shown that, in the plane of the static and time-variable load parameters, the structure's dynamic shakedown domain is nonconvex and that its boundary curve generally…

Thermodynamics and continuum fracture mechanics for nonlocal-elastic plastic materials

Nonlocal elasticity is used as an improved elasticity model which engenders no crack-tip stress singularities and thus makes applicable the classical stress-based failure criteria. Considering nonlocal-elastic plastic materials exposed to softening by particle decohesion in a process surface and to subsequent surface separation by fracture, fracture mechanics is addressed within the framework of irreversible internal-variable thermodynamics in the hypothesis of small strains and arbitrary (but sufficiently regular) fracture surface (crack surface plus process surface). The state equations and the energy dissipation densities are derived for the bulk material and for the process surface, for…

On the Conditions to Prevent Plastic Shakedown of Structures: Part II—The Plastic Shakedown Limit Load

Following the results of a companion paper, the concept of plastic shakedown limit load is introduced for an elastic-perfectly plastic material structure subjected to combined cyclic (mechanical and/or kinematical) loads and steady (mechanical) load. Static and kinematic approaches are available for the computation of this load, in perfect analogy with the classic (elastic) shakedown limit load. The plastic shakedown limit state of the structure being in an impending alternating plasticity collapse is studied and a number of interesting features of it are pointed out.

Shakedown analysis for a class of strengthening materials within the framework of gradient plasticity

Abstract The classical shakedown theory is extended to a class of perfectly plastic materials with strengthening effects (Hall–Petch effects). To this aim, a strain gradient plasticity model previously advanced by Polizzotto (2010) is used, whereby a featuring strengthening law provides the strengthening stress, i.e. the increase of the yield strength produced by plastic deformation, as a degree-zero homogeneous second-order differential form in the accumulated plastic strain with associated higher order boundary conditions. The extended static (Melan) and kinematic (Koiter) shakedown theorems are proved together with the related lower bound and upper bound theorems. The shakedown limit loa…

The shakedown load boundary of an elastic-perfectly plastic structure

In the hypothesis of small displacements and combined time-variable/steady loads, the geometrical-mechanical properties of the shakedown load boundary are investigated. It is shown that, in the load space, the shakedown load boundary plays the role of yield surface, and that a certain plastic strain accumulation vector—characterizing some impending inadaptation collapse mechanism—obeys the normality rule, whereas a specific form of the maximum plastic work theorem constitutes an effective tool for the evaluation of the shakedown limit load corresponding to a specified inadaptation collapse mode. The equations governing the state of the structure at the shakedown limit are provided and the r…