15 research outputs found
Well-posedness and long-time behavior for a class of doubly nonlinear equations
This paper addresses a doubly nonlinear parabolic inclusion of the form
. Existence of a solution is proved under suitable
monotonicity, coercivity, and structure assumptions on the operators and
, which in particular are both supposed to be subdifferentials of
functionals on . Moreover, under additional hypotheses on ,
uniqueness of the solution is proved. Finally, a characterization of
-limit sets of solutions is given and we investigate the convergence of
trajectories to limit points
Global existence for rate-independent gradient plasticity at finite strain
We provide a global existence result for the time-continuous elastoplasticity problem using the energetic formulation. For this we show that the geometric nonlinearities via the multiplicative decomposition of the strain can be controlled via polyconvexity and a priori stress bounds in terms of the energy density. While temporal oscillations are controlled via the energy dissipation the spatial compactness is obtain via the regularizing terms involving gradients of the internal variables
Free energy, free entropy, and a gradient structure for thermoplasticity
In the modeling of solids the free energy, the
energy, and the entropy play a central role. We show that the free
entropy, which is defined as the negative of the free energy divided
by the temperature, is similarly important. The derivatives of the
free energy are suitable thermodynamical driving forces for
reversible (i.e.\ Hamiltonian) parts of the dynamics, while for the
dissipative parts the derivatives of the free entropy are the
correct driving forces. This difference does not matter for
isothermal cases nor for local materials, but it is relevant in the
non-isothermal case if the densities also depend on gradients, as is
the case in gradient thermoplasticity.
Using the total entropy as a driving functional, we develop gradient
structures for quasistatic thermoplasticity, which again features
the role of the free entropy. The big advantage of the gradient
structure is the possibility of deriving time-incremental
minimization procedures, where the entropy-production potential
minus the total entropy is minimized with respect to the internal
variables and the temperature.
We also highlight that the usage of an auxiliary temperature as an
integrating factor in Yang/Stainier/Ortiz "{A} variational formulation of the coupled thermomechanical boundary-value problem for general dissipative solids" (J. Mech. Physics Solids, 54, 401-424, 2006) serves exactly the purpose
to transform the reversible driving forces, obtained from the free energy, into the needed irreversible driving forces,
which should have been derived from the free entropy. This
reconfirms the fact that only the usage of the free entropy as
driving functional for dissipative processes allows us to derive a
proper variational formulation
Linearized plasticity is the evolutionary \Gamma-limit of finite plasticity
We provide a rigorous justification of the classical linearization approach
in plasticity. By taking the small-deformations limit, we prove via
\Gamma-convergence for rate-independent processes that energetic solutions of
the quasi-static finite-strain elastoplasticity system converge to the unique
strong solution of linearized elastoplasticity.Comment: To appear on J. Eur. Math. Soc. (JEMS
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A model for the evolution of laminates in finite-strain elastoplasticity
We study the time evolution in elastoplasticity within the rate-independent framework of generalized standard materials. Our particular interest is the formation and the evolution of microstructure. Providing models where existence proofs are possible is a challenging task since the presence of microstructure comes along with a lack of convexity and, hence, compactness arguments cannot be applied to prove the existence of solutions. In order to overcome this problem, we will incorporate information on the microstructure into the internal variable, which is still compatible with generalized standard materials. More precisely, we shall allow for such microstructure that is given by simple or sequential laminates. We will consider a model for the evolution of these laminates and we will prove a theorem on the existence of solutions to any finite sequence of time-incremental minimization problems. In order to illustrate the mechanical consequences of the theory developed some numerical results, especially dealing with the rotation of laminates, are presented
A rate-independent model for the isothermal quasi-static evolution of shape-memory materials
This note addresses a three-dimensional model for isothermal stress-induced
transformation in shape-memory polycrystalline materials. We treat the problem
within the framework of the energetic formulation of rate-independent processes
and investigate existence and continuous dependence issues at both the
constitutive relation and quasi-static evolution level. Moreover, we focus on
time and space approximation as well as on regularization and parameter
asymptotics.Comment: 33 pages, 3 figure