1,721,107 research outputs found
A variational formulation for three-dimensional linear thermoelasticity with ‘thermal inertia’
No full textA variational model has been developed to investigate the coupled thermo-mechanical response of a three-dimensional continuum. The linear Partial Differential Equations (PDEs) of this problem are already well-known in the literature. However, in this paper, we avoid the use of the second principle of thermodynamics, basing the formulation only on a proper definition (i) of kinematic descriptors (the displacement and the entropic displacement), (ii) of the action functional (with kinetic, internal and external energy functions) and (iii) of the Rayleigh dissipation function. Thus, a Hamilton-Rayleigh variational principle is formulated, and the cited PDEs have been derived with a set of proper Boundary Conditions (BCs). Besides, the Lagrangian variational perspective has been expanded to analyze linear irreversible processes by generalizing Biot's formulation, namely, including thermal inertia in the kinetic energy definition. Specifically, this implies Cattaneo's law for heat conduction, and the well-known Lord-Shulman model for thermo-elastic anisotropic bodies is then deduced. The developed variational framework is ideal for the perspective of analyzing the thermo-mechanical problems with micromorphic and/or higher-order gradient continuum models, where the deduction of a coherent system of PDEs and BCs is, on the one hand, not straightforward and, on the other hand, natural within the presented variational deduction
Simulation results for damage with evolving microstructure and growing strain gradient moduli
No full textIn this paper, a strain gradient regularization of continuum damage mechanics is dealt with starting from a variational inequality. Full stress and strain analyses are reported for numerical tests performed with an in-house code based on the element free Galerkin method in two exemplary cases. Simulations show that the non-locality conferred to the model by the strain gradient regularization effectively solves the problem of mesh dependence exhibited by standard damage continuum models and allows to reach convergence up to deformation regimes for which equivalent negative stiffness is observed. Moreover, we will illustrate the case in which new microstructures, and therefore higher values of second gradient moduli, develop due to damage. For illustration a simple monotonic evolution rule is assumed for the growth of the second gradient moduli. Thus, for this simple rule, the simulations show development of a larger diffuse damage zone whose size can be controlled
Two-dimensional strain gradient damage modeling: a variational approach
No full textIn this paper, we formulate a linear elastic second gradient isotropic two-dimensional continuum model accounting for irreversible damage. The failure is defined as the condition in which the damage parameter reaches 1, at least in one point of the domain. The quasi-static approximation is done, i.e., the kinetic energy is assumed to be negligible. In order to deal with dissipation, a damage dissipation term is considered in the deformation energy functional. The key goal of this paper is to apply a non-standard variational procedure to exploit the damage irreversibility argument. As a result, we derive not only the equilibrium equations but, notably, also the Karush–Kuhn–Tucker conditions. Finally, numerical simulations for exemplary problems are discussed as some constitutive parameters are varying, with the inclusion of a mesh-independence evidence. Element-free Galerkin method and moving least square shape functions have been employed
Modeling and numerical investigation of damage behavior in pantographic layers using a hemivariational formulation adapted for a Hencky-type discrete model
No full textIn this study, a hemivariational formulation is presented for a Hencky-type discrete model to predict damage behavior in pantographic layers. In the discrete model, elastic behavior of pantographic layers is modeled via extensional, bending and shear springs. A damage descriptor is added for each spring type. Such a damage descriptor is non-decreasing function of time, and therefore, the standard variational formulation of the problem is generalized to a hemivariational one providing not only the Euler–Lagrange equations for the evolution of the displacements of all the standard degrees of freedom but also the Karush–Khun–Tucker condition governing the evolution of damage descriptor. The dissipation energy included in the hemivariational formulation depends upon six additional constitutive parameters (two per each spring type), and the mechanical behavior of layer is simulated with an efficient and smart strategy able to solve the nonlinear equilibrium equations coupled with the evolution of damage variables. A metallic pantographic layer which was experimentally investigated in the literature is considered to test the proposed formulation
An implicit computational approach in strain-gradient brittle fracture analysis
Full text linkWithin the context of quasi-brittle fracture mechanics analyzed by finite element approaches, the present research addresses an implicit solution scheme applied to a strain-gradient continuum damage model. The implicit scheme is based onto an iterative procedure which minimizes for each loading step the increment of both the elastic energy and the damage field between two subsequent trial solutions. The performances of the proposed scheme are compared with those of a previously developed explicit scheme. Besides a better accuracy in the static response computation, it is demonstrated that the proposed approach provides more accurate fracture propagation patterns
A strain gradient variational approach to damage: a comparison with damage gradient models and numerical results
No full textThe global response of experimental uniaxial tests cannot be homogeneous, because of the unavoidable presence of localized deformations, which is always preferential from an energetic viewpoint. Accordingly, one must introduce some characteristic lengths in order to penalize deformations that are too localized. This is what leads to the concept of nonlocal damage models. The nonlocal approach employs nonlocal terms in the internal deformation energy in order to control the size of the localization region. In phase-field models and, in general, in gradient models, dependence of the internal energy upon the first gradient of damage is assumed, while in our approach the nonlocality is given by the dependence of the internal energy upon the second gradient of the displacement field. A discussion of the advantages and challenges of using the gradient of damage and of using the second gradient of the displacement field will be addressed in the present paper. A variational inequality is formulated and partial differential equations (PDEs), boundary conditions (BCs), and Karush-Kuhn- Tucker (KKT) conditions will be derived within the framework of 2D strain gradient damage mechanics. A novel dependence of the stiffness coefficients with respect to the damage field will also be discussed. Further, an explicit derivation of the damage field evolution in loading conditions will be provided. Finally, a numerical technique based on commercial software has been introduced and discussed for a couple of standard problems
Complexity and robustness of frame structures
No full textRobustness is a prerequisite in the design of a structure and represent a modern research topic in the field of structural engineering. Many definitions have been given to this concept; many design codes focuses the attention on avoiding disproportionation between cause and effect. Lind introduced the concept of “damage tolerance” as the capacity of the system to be able to sustain some damage without failure.
Frames structures are considered as a case study of damage tolerance. The peculiarity of these schemes is the presence of a high degree of static indeterminacy, which makes the structures able to differentiate the load paths from the elevation to the foundation. In this framework, the analysis of the topology of the elements of the scheme is performed. For this purpose, a novel metric for “structural complexity” is presented. Information theory is used for the analysis of predominant mechanisms in the transfer of loads from the elevation nodes to the foundations through the set of elements of the frame. The generation of the fundamental static determinate mechanisms, similar to the load paths, is performed by means of algebraic graph theoretical concepts.
Then, the effects on the whole structure of the removal of a single element of the frame are highlighted and discussed. The removal on the element is governed by a damage parameter, which reduces the stiffness of the element. Since a general behaviour is sought, a linear elastic model is used. At the end, some general considerations on the robustness of the structures with respect to the so called “black swan events” are drawn
Energy approach to brittle fracture in strain-gradient modelling
No full textIn this paper, we exploit some results in the theory of irreversible phenomena to address the study of quasi-static brittle fracture propagation in a two-dimensional isotropic continuum. The elastic strain energy density of the body has been assumed to be geometrically nonlinear and to depend on the strain gradient. Such generalized continua often arise in the description of microstructured media. These materials possess an intrinsic length scale, which determines the size of internal boundary layers. In particular, the non-locality conferred by this internal length scale avoids the concentration of deformations, which is usually observed when dealing with local models and which leads to mesh dependency. A scalar Lagrangian damage field, ranging from zero to one, is introduced to describe the internal state of structural degradation of the material. Standard Lamé and second-gradient elastic coefficients are all assumed to decrease as damage increases and to be locally zero if the value attained by damage is one. This last situation is associated with crack formation and/or propagation. Numerical solutions of the model are provided in the case of an obliquely notched rectangular specimen subjected to monotonous tensile and shear loading tests, and brittle fracture propagation is discussed.</jats:p
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