Non-Smooth Thermomechanics

Frémond, Michel

doi:10.1007/978-3-662-04800-9

Cited by 455 publications

(543 citation statements)

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“…We then observe that this inequality resembles the well-known dissipation inequality of thermodynamics, stated for instance in Gurtin et al [7] or Frémond [6]. In fact, the logical status of these two inequalities are the same: we are to find a dynamical system that satisfies the inequality for all possible evolutions.…”

Section: Introductionsupporting

confidence: 68%

“…In the last term of (8), the symbol ∂ indicates the subdifferential of a convex function, see, e.g., Rockafellar [20] or, for its use in mechanics, Panagiotopoulos [18] and Frémond [6]. The convex function is in this case the indicator function of the convex set K i , i.e., a function that takes the value 0 for points inside the set and infinity for points outside the set.…”

Section: Dynamical Systemmentioning

confidence: 99%

“…see, e.g., Frémond [6]. The potential could depend on other variables thanρ i in the same way as for k i , but these act as parameters and the subdifferential is with respect toρ i .…”

Section: Dynamical Systemmentioning

confidence: 99%

“…In fact, the logical status of these two inequalities are the same: we are to find a dynamical system that satisfies the inequality for all possible evolutions. In thermodynamics, this particular modeling step is known as the Coleman-Noll procedure, [7,6]. Following this procedure we derive a general dynamical system that when specialized takes the form of a lazy zone bone remodeling theory.…”

Section: Introductionmentioning

confidence: 99%

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Lazy zone bone remodeling theory and its relation to topology optimization

Klarbring

Torstenfelt

2012

Ann. Solid Struct. Mech.

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The connection between apparent density-type bone remodeling theories and density formulations of topology optimization is well known from a large number of publications and its theoretical basis has recently been discussed by making use of a dynamical systems approach to optimization. The present paper takes this connection one step further by showing how the Coleman-Noll procedure of rational thermodynamics can be used to derive general dynamical systems, where a special case includes the lazy zone concept of bone remodeling theory. It is also shown how a numerical solution method for the dynamical system can be developed by using the sequential convex approximation idea. The method is employed to obtain a series of solutions that show the influence of modeling parameters representing elements of plasticity and viscosity in the growth process.

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Section: Introductionsupporting

confidence: 68%

Section: Dynamical Systemmentioning

confidence: 99%

“…see, e.g., Frémond [6]. The potential could depend on other variables thanρ i in the same way as for k i , but these act as parameters and the subdifferential is with respect toρ i .…”

Section: Dynamical Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Lazy zone bone remodeling theory and its relation to topology optimization

Klarbring

Torstenfelt

2012

Ann. Solid Struct. Mech.

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“…This view may appear reminiscent of those of Fried and Gurtin [2,3] and Frémond et al (see, e.g. [4,5]) in which the phase transition is based on a balance equation of microforces. The difference mainly lies in our view of the structure order as a macroscopic and observable quantity that enters a balance equation.…”

Section: Fabrizio C Giorgi and A Morromentioning

confidence: 80%

A continuum theory for first‐order phase transitions based on the balance of structure order

Fabrizio¹,

Giorgi

Morro

2007

Math Methods in App Sciences

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SUMMARYFirst-order phase transitions are modelled by a non-homogeneous, time-dependent scalar-valued order parameter or phase field. The time dependence of the order parameter is viewed as arising from a balance law of the structure order. The gross motion is disregarded and hence the body is regarded merely as a heat conductor. Compatibility of the constitutive functions with thermodynamics is exploited by expressing the second law through the classical Clausius-Duhem inequality. First, a model for conductors without memory is set up and the order parameter is shown to satisfy a maximum theorem. Next, heat conductors with memory are considered. Different evolution problems are established through a system of differential equations whose form is related to the manner in which the memory property is represented.

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The Stress Tensor in Granular Media and in other Mechanical Collections

Moreau¹

2009

Micromechanics of Granular Materials

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Instead of the pair-by-pair approach conventionally used when defining the average stress tensor of a granular sample or of a piece of masonry, a grain-by-grain construction is proposed. Its theoretical foundation lies in the assignement, to any bounded mechanical system, of the tensor moment of its internal efforts, shortly called the internal moment of the system. In classical continuous media, the Cauchy stress is nothing but the volume-density of internal moment (or its negative, depending on the sign conventions made). This approach is not limited to systems in equilibrium; furthermore, it applies to collections involving other mechanical objects than massive grains. The pertinence of the concept is demonstrated first by numerical simulations. Secondly some mathematical procedures of smoothing and homogenisation are introduced to connect the microscopic analysis with the macroscopic continuum model. Quantitatively, in usual situations, what is obtained as stress tensor differs very little from the value resulting from traditional definitions.

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Non-Smooth Thermomechanics

Cited by 455 publications

References 0 publications

Lazy zone bone remodeling theory and its relation to topology optimization

Lazy zone bone remodeling theory and its relation to topology optimization

A continuum theory for first‐order phase transitions based on the balance of structure order

The Stress Tensor in Granular Media and in other Mechanical Collections

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