Guido Borino

A symmetric nonlocal damage theory

Abstract 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.

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 residual’, which accounts for energy exchanges between different particles at the microstructural level as a consequence of the hardening/softening diffusion processes within the body; and iii) the (nonambiguous) ‘constitutive’ boundary conditions, which must be satisfied at points of the boundary surface of any (finite) region of the body where an irreversible deformation mechanism takes place (e.g. shear band). The plastic yielding laws for gradient plasticity are established with their domain and boundary equations, and their consistency with the nonlocal Clausius-Duhem inequality is assessed as well. Also, a suitable nonlocal-form maximum intrinsic dissipation theorem is provided, and the response problem of a continuous set of material particles to a given total strain rate field studied. Points of agreement and disagreement between this theory and the related literature are indicated, also via a case-study bar in uniaxial tension for which the analytical solution is worked out.

A thermodynamically consistent nonlocal formulation for damaging materials

A thermodynamically consistent nonlocal formulation for damaging materials is presented. The second principle of thermodynamics is enforced in a nonlocal form over the volume where the dissipative mechanism takes place. The nonlocal forces thermodynamically conjugated are obtained consistently from the free energy. The paper indeed extends to elastic damaging materials a formulation originally proposed by Polizzotto et al. for nonlocal plasticity. Constitutive and computational aspects of the model are discussed. The damage consistency conditions turn out to be formulated as an integral complementarity problem and, consequently, after discretization, as a linear complementarity problem. A new numerical algorithm of solution is proposed and meaningful one-dimensional and two-dimensional examples are presented.

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, in the region of active plastic yielding, some specific series representations. Finally, the structural rate response problem for assigned load rates is studied in relation to the solution uniqueness, and two variational principles are provided for this boundary value problem.

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 work is finite or not in the case of infinite duration load histories, where they then prove to be equivalent to known shakedown theorems.

Shakedown and steady-state responses of elastic-plastic solids in large displacements

Abstract Elastic-perfectly plastic solids (or structures) subjected to loads quasi-statically varying within a specified domain are addressed in the framework of large displacements and the additive strain decomposition rule. On the ground of Drucker’s principle of stability in the large, an appropriate stability requisite (called D-stability) is formulated as the positive definiteness property of a specific functional, sum of the second variation of the Helmholtz free energy with an additional term depending on higher-order geometry change effects. For a D-stable structure for which the additive strain decomposition rule is applicable, Melan’s and Koiter’s theorems of classical shake-down theory are reconsidered and reformulated for large displacements. The extended Melan and Koiter theorems so established save the essential features of the classical ones, but exhibit a greater formal complexity with consequent difficulties for engineering applications. For structures subjected to periodic loads, it is shown that—as long as the structure does not incur loss of D-stability—a long-term steady-state response (or steady cycle) occurs, which exhibits the same periodicity characteristics as in case of small displacements; that is, the (second Piola—Kirchhoff) stresses and the plastic (Green-Lagrange) strain rates become periodic as the load. A few illustrative numerical results are presented.

Consistent shakedown theorems for materials with temperature dependent yield functions

Abstract The (elastic) shakedown problem for structures subjected to loads and temperature variations is addressed in the hypothesis of elastic–plastic rate-independent associative material models with temperature-dependent yield functions. Assuming the yield functions convex in the stress/temperature space, a thermodynamically consistent small-deformation thermo-plasticity theory is provided, in which the set of state and evolutive variables includes the temperature and the plastic entropy rate. Within the latter theory the known static (Pragers) and kinematic (Konigs) shakedown theorems — which hold for yield functions convex in the stress space — are restated in an appropriate consistent format. In contrast with the above known theorems, the restated theorems provide dual lower and upper bound statements for the shakedown limit loads; additionally, the latter theorems can be expressed in terms of only dominant thermo-mechanical loads (generally the vertices of a polyhedral load domain in which the loadings are allowed to range). The shakedown limit load evaluation problem is discussed together with the related shakedown limit state of the structure. A few numerical applications are presented.

Dynamic Analysis of Prestressed Cables with Uncertain Pretension

This paper deals with finite element dynamic analysis of prestressed cables with uncertain pretension subjected to deterministic excitations. The theoretical model addressed for cable modeling is a two-dimensional finite-strain beam theory, which allows us to eliminate any restriction on the magnitude of displacements and rotations. The dynamic problem is formulated by referring the motion to the inertial frame, which leads to a simple uncoupled quadratic form for the kinetic energy. The effect of the externally applied stochastic pretension is approximately described by means of an uncertain ‘axial’ component of stress resultant, which is assumed constant along the cable in its dead load configuration. The so-called improved perturbation approach is employed to solve this stochastic problem, obtaining two coupled systems of nonlinear deterministic ordinary differential equations, governing the mean value and deviation of response. An efficient and accurate iterative procedure is proposed to obtain the solution of these equations. In order to investigate the influence of random pretension on structural response, few numerical applications are presented and results are discussed.

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 subjected to a specified infinite-duration load history. Two general bounding principles are also presented, one is non evolutive in nature and holds for repeated loads below the shakedown limit, the other is evolutive and holds for a specified load history either below and above the shakedown limit. Both principles are applicable in either statics and dynamics to construct bounds to the actual plastic deformation parameters. A continuum solid mechanics approach is used throughout, but a class of discrete models (finite elements with piecewise linear yield and saturation surfaces and plastic deformability lumped at Gauss points) are also considered. Extensions of shakedown theorems to materials with temperature dependent yield and saturation functions are also presented.

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.