Homogenization of plasticity equations with two-scale convergence methods

Schweizer, Ben; Veneroni, Marco

doi:10.1080/00036811.2014.896992

Cited by 20 publications

(12 citation statements)

References 29 publications

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“…The rigorous derivation of the effective two‐scale system was obtained with various approaches, covering a varying generality of the coefficient functions. A consequent use of variational aspects and the method of two‐scale convergence was exploited first by Visintin, see , and later in . Alber and Nesenenko use the method of two‐scale convergence and combine it with phase‐shift convergence to obtain rigorous results, see .…”

Section: Introductionmentioning

confidence: 99%

“…A homogenization proof based on Tartar's energy method is presented in , and general homogenization results for rate‐independent systems have been developed in . The recent analysis of is entirely based on two‐scale convergence, it allows quite general monotone flow rules and it clearifies the necessary conditions regarding x ‐ and y ‐dependence of the flow function; we refer to this work also for a further description of the literature.…”

Section: Introductionmentioning

confidence: 99%

“…To make the connection to periodic homogenization, we recall results of , where the plasticity system

\partial_{t}^{2} u^{ɛ} - \nabla \cdot σ^{ɛ} = f, C_{ɛ} σ^{ɛ} = \nabla^{s} u^{ɛ} - p^{ɛ}

\partial_{t} p^{ɛ} \in g_{ɛ} (σ^{ɛ} - B_{ɛ} p^{ɛ}),

is analyzed. We simplified here to a special case, omitting general internal variables.…”

Section: Introductionmentioning

confidence: 99%

“…In our case, the monotone multi‐valued map

g_{ɛ}

is a subdifferential,

g_{ɛ} = \partial Ψ_{ɛ}

. With the unit cell

Y : = {[0, 1 [}^{n}

, periodic coefficients are constructed as

C_{ɛ} (x) : = C (x, \frac{x}{ɛ}), B_{ɛ} (x) : = B (x, \frac{x}{ɛ}), g_{ɛ} (., x) : = g (.; x, \frac{x}{ɛ}) .

In , the homogenization of – is performed and the following limit problem is established:

\partial_{t}^{2} u - \nabla \cdot (\int_{Y} z d y) = f, C z = \nabla_{x}^{s} u + \nabla_{y}^{s} v - w,

\partial_{t} w \in g (z - B w; x, y), \nabla_{y} \cdot z = 0 .

Here,

z = z (x, y, t)

is a two‐scale stress variable,

w = w (x, y, t)

a two‐scale plastic strain variable, and

v = v (x, y, t)

stands for fine‐scale deformations; these are functions on

Q \times Y \times [0, T]

, the strong limit

u = u (x, t

…”

Section: Introductionmentioning

confidence: 99%

“…We emphasize that we deal here with the quasi‐static approximation, while in the inertia term

\partial_{t}^{2} u^{ɛ}

has been included. Nevertheless, let us formally investigate the form of the causal operator Σ in this periodic setting.…”

Section: Introductionmentioning

confidence: 99%

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Non-periodic homogenization of infinitesimal strain plasticity equations

Heida

Schweizer

2015

Z. Angew. Math. Mech.

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We consider the Prandtl-Reuss model of plasticity with kinematic hardening, aiming at a homogenization result. For a sequence of coefficient fields and corresponding solutions u ε , we ask whether we can characterize weak limits u when u ε u as ε → 0. We assume neither periodicity nor stochasticity for the coefficients, but we demand an abstract averaging property of the homogeneous system on reference volumes. Our conclusion is an effective equation on general domains with general right hand sides. The effective equation uses a causal evolution operator that maps strains to stresses; more precisely, in each spatial point x, given the evolution of the strain in the point x, the operator provides the evolution of the stress in x.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…To make the connection to periodic homogenization, we recall results of , where the plasticity system

\partial_{t}^{2} u^{ɛ} - \nabla \cdot σ^{ɛ} = f, C_{ɛ} σ^{ɛ} = \nabla^{s} u^{ɛ} - p^{ɛ}

\partial_{t} p^{ɛ} \in g_{ɛ} (σ^{ɛ} - B_{ɛ} p^{ɛ}),

is analyzed. We simplified here to a special case, omitting general internal variables.…”

Section: Introductionmentioning

confidence: 99%

“…In our case, the monotone multi‐valued map

g_{ɛ}

is a subdifferential,

g_{ɛ} = \partial Ψ_{ɛ}

. With the unit cell

Y : = {[0, 1 [}^{n}

, periodic coefficients are constructed as

C_{ɛ} (x) : = C (x, \frac{x}{ɛ}), B_{ɛ} (x) : = B (x, \frac{x}{ɛ}), g_{ɛ} (., x) : = g (.; x, \frac{x}{ɛ}) .

In , the homogenization of – is performed and the following limit problem is established:

\partial_{t}^{2} u - \nabla \cdot (\int_{Y} z d y) = f, C z = \nabla_{x}^{s} u + \nabla_{y}^{s} v - w,

\partial_{t} w \in g (z - B w; x, y), \nabla_{y} \cdot z = 0 .

Here,

z = z (x, y, t)

is a two‐scale stress variable,

w = w (x, y, t)

a two‐scale plastic strain variable, and

v = v (x, y, t)

stands for fine‐scale deformations; these are functions on

Q \times Y \times [0, T]

, the strong limit

u = u (x, t

…”

Section: Introductionmentioning

confidence: 99%

“…We emphasize that we deal here with the quasi‐static approximation, while in the inertia term

\partial_{t}^{2} u^{ɛ}

has been included. Nevertheless, let us formally investigate the form of the causal operator Σ in this periodic setting.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Non-periodic homogenization of infinitesimal strain plasticity equations

Heida

Schweizer

2015

Z. Angew. Math. Mech.

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A homogenization result in finite plasticity

Davoli,

Gavioli,

Pagliari

2024

Calc. Var.

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We carry out a variational study for integral functionals that model the stored energy of a heterogeneous material governed by finite-strain elastoplasticity with hardening. Assuming that the composite has a periodic microscopic structure, we establish the $$\Gamma $$ Γ -convergence of the energies in the limiting of vanishing periodicity. The constraint that plastic deformations belong to $$\textsf{SL}(3)$$ SL ( 3 ) poses the biggest hurdle to the analysis, and we address it by regarding $$\textsf{SL}(3)$$ SL ( 3 ) as a Finsler manifold.

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