A relaxation-based approach to damage modeling

Proc Appl Math and Mech

Mosler

2018

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A model framework for the analysis of isotropic quasi-brittle damage, was recently presented in [1]. Within this paper, the model in [1] is significantly improved. To be more precise, and in contrast to [1], the novel model: (i) eliminates unnecessary model parameters, (ii) can be better interpreted from a physics point of view, (iii) can capture a fully softened state (zero stresses) and (iv) is characterized by a very simple evolution equation which (v) can be integrated fully implicitly and (vi) the resulting time discrete evolution equation can be solved analytically providing a numerically efficient closed form solution, cf.[2].In line with [3], damage models can often be rewritten into a variational framework. To be more precise, an incremental potential can be defined whose stationary conditions define the underlying model. Following [4], the rate potentialĖ is given byĖwhere the superimposed dot denotes the material time derivative. The rate potential depends on the velocity fieldu and on the rates of internal variablesα. The quantity ρ denotes the density, b are mass-specific forces and t * prescribed tractions acting on the Neumann boundary ∂B N . The local rate potential I is defined aṡwhere D is the dissipation function and Ψ the Helmholtz free energy depending on the engineering strains ε = ∇ sym u and the internal variables. Stationary of potential (1) with respect to the velocity field yields balance of linear momentum and stationary with respect to the rates of internal variables results in Biot's equation which implies the constitutive evolution equation, cf.[5]. Prototype modelFollowing the framework of incremental energy minimization, cf.[4], the model is defined by postulating the Helmholtz free energy and the dissipation function. A common set for modeling of isotropic quasi-brittle material degradation is given byin which E is the fourth-order elasticity tensor, α an internal variable and α u an additional material parameter regulating the evolution speed of damage variable d. Stationary of potential (1) with respect toα gives the evolution equation. By considering the time discrete counterpart and the second law of thermodynamics (healing has to be excluded, i.e.α < 0) both equations can be combined to α n+1 = max(α n , Ψ 0 ), i.e., the internal variable stores the maximum Helmholtz free energy of the undamaged material. Normalized results of this prototype model are shown in Fig 1. Fig. 1a shows the normalized stressstrain-diagram, which has a typical softening behavior. Fig. 1b visualizes the normalized determinant of the corresponding tangent modulus. It can be seen that -as expected -the tangent determinant becomes negative at the onset of softening. Hence, the type of the partial differential equation changes, and thus, the boundary value problem becomes ill-posed, cf.[6].3 Rate-dependent damage modelIn this subsection the relaxation-based damage model, presented in [1], will be improved, cf. [2]. For that purpose, the local rate potential is defined as

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A model framework for the analysis of isotropic quasi-brittle damage, was recently presented in [1]. Within this paper, the model in [1] is significantly improved.

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On the rate‐depentent regularization for damage modeling

Langenfeld

Proc Appl Math and Mech

Mosler

2018

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confidence: 99%

“…This mixture, combining the advantages and preventing the disadvantages, is described by Ψ β , in which d ∈ [0, 1] is the damage variable appearing as a volume fraction between undamaged material (d = 0) and damaged material (d = 1), the variableα > 0 serves as a numerical quantity to assure nonzero driving forces for ε = 0 and d = 0, and β describes the mixture: settingα = 0 and β = 1 results in the convex Reuß energy, settingα = 0 and β → ∞ results in the non-convex Voigt energy. For more details see [1] or [2].Applying the mixture rule, the energy Ψ β combines both energy states by use of a damage function f (d) that, multiplied with the elastic constant, serves as an effective elastic tensor E eff = f (d) E 0 .This combined energy can then be used for the principle of the minimum of the dissipation potential in order to achieve the evolution equations for the internal variables, see [1]. Numerical results are shown hereinafter.…”

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confidence: 99%

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confidence: 99%

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The coupling of plasticity with a relaxation‐based approach to damage modeling

Schwarz

Proc Appl Math and Mech

Hackl

2017

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Since damage occurs in context of high stresses that are also related to the development of plastic strains, it is natural to couple damage and plasticity phenomena to achieve a more realistic model. Hereto, the new damage model presented in [2] was used and enhanced with plasticity and isotropic hardening, as first shown in [1]. Thereby, the damage model is based on a new regularization approach and provides mesh-independent results. In order to achieve mesh-independence, damage models usually take into account the non-local behavior by using a field function that couples the local damage parameter to a nonlocal level, in which differences between the local and non-local parameter as well as the gradient of the non-local parameter can be penalized [3]. In contrast, the new regularization approach no longer needs a non-local level at the finite element scale but directly provides mesh-independent results. Due to the new variational approach, we are also able to improve the calculation times and convergence behavior. Furthermore, the enhancement with plasticity and isotropic hardening allows to investigate the influences between damage and plastic strains to each other as well as the resulting influences to the cracks.The material model is based on [1] and consists of two parts: the undamaged material is described by the Helmholtz free energy Ψ 0 and the damaged material by Ψ 1 . This is realized by using different elastic constants E 0 and E 1 , whereby the damaged material is naturally much softer, thus a very small value for κ is used. Furthermore, the plastic strains are described by ε p and the isotropic hardening by the hardening variable α as well as the hardening modulusĤ.In order to achieve a mesh-independent model, a mixture between both energy states has to be chosen carefully. A linear transition, defined by the Reuß bound, provides a convex energy but does not allow a stress decrease but only a plateau instead. Whereas the Voigt bound, characterized by a sudden and complete transition from the undamaged to the damaged state, allows a stress decrease but leads to mesh-dependent results if no regularization schemes are employed. For these reasons, a mixture rule is chosen, which slightly shifts our energy from the pure Reuß bound towards the Voigt bound, whereby viscous effects are added to the model in order to prevent reductions of the mesh-independence. This mixture, combining the advantages and preventing the disadvantages, is described by Ψ β , in which d ∈ [0, 1] is the damage variable appearing as a volume fraction between undamaged material (d = 0) and damaged material (d = 1), the variableα > 0 serves as a numerical quantity to assure nonzero driving forces for ε = 0 and d = 0, and β describes the mixture: settingα = 0 and β = 1 results in the convex Reuß energy, settingα = 0 and β → ∞ results in the non-convex Voigt energy. For more details see [1] or [2].Applying the mixture rule, the energy Ψ β combines both energy states by use of a damage function f (d) that, multiplied with the elastic co...

An accurate and fast regularization approach to thermodynamic topology optimization

Jantos

Hackl

Numerical Meth Engineering

2018

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In a series of previous works, we established a novel approach to topology optimization for compliance minimization based on thermodynamic principles known from the field of material modeling. Hamilton's principle for dissipative processes directly yields a partial differential equation (referred to as the evolution equation) as an update scheme for the spatial distribution of density mass describing the topology. Consequently, no additional mathematical minimization algorithms are needed. In this article, we introduce a regularization scheme by penalization of the gradient of the spatial distribution of mass density. The parabolic evolution equation (owing to a similar structure to the transient heat-conduction equation) is solved most efficiently by an explicit time discretization. The Laplace operator is discretized via a Taylor series expansion yielding an operator matrix that is constant for the entire optimization process. This method shares some similarities to meshless methods and allows for an accurate application also on unstructured finite element meshes. The minimal size of the structure member can directly be controlled, a priori, by a numerical parameter introduced along with the regularization, similar to classical filter radii. KEYWORDSmeshless Laplacian, parabolic PDE, regularization, thermodynamic topology optimization, unstructured meshes, variational material modeling INTRODUCTIONOver recent decades, topology optimization has become a major field of interest in mechanical engineering. Approaches have been developed for different optimization objectives, constraints relating to physically feasible or manufacturable designs, or models for multiphysics problems. The most native approach is considered to be topology optimization with compliance minimization under a structure volume constraint for linear elastic materials, see, eg, the work of Bendsøe and Sigmund, 1 in which the elastic strain energy is subject to minimization and is referred to as (structural) compliance. Numerous solution approaches were developed of which the most popular are overviewed and compared in the work of Sigmund and Maute. 2 Usually, the topology has to be found within a given design space under mechanical boundary conditions, for which the mechanical problem is solved by a finite element approach. The design space contains material and void phases to identify the topology.This can be achieved either by tracing the phase borders or by introducing a density field that describes the material configuration in each point of the design space. The phase borders are subject to optimization in level-set approaches, 3,4 which Int J Numer Methods Eng. 2019;117:991-1017.wileyonlinelibrary.com/journal/nme