Existence of long‐time solutions to dynamic problems of viscoelasticity with rate‐and‐state friction

Pipping, Elias

doi:10.1002/zamm.201800263

Cited by 7 publications

(18 citation statements)

References 18 publications

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“…Despite the success of this empirical constitutive law to represent experimental data, it does not have a sound thermodynamical basis allowing for a proper balance of energies through external powers and dissipation (Pipping, 2019 ). This mathematical drawback can be alleviated by proper regularization.…”

Section: Theorymentioning

confidence: 99%

“…This mathematical drawback can be alleviated by proper regularization. Pipping ( 2019 ) adopted two possible regularizations that, in the frame of the aging law for the state evolution, can lead to a well defined boundary value problem with unique solution in space and time. According to this author, the friction coefficient can be given using any of the two following expressions:

in the above expressions,

is a positive coefficient depending on the material,

and:

where

is a positive coefficient depending on the material and

and

are reference values for the friction coefficient and the slip velocity, respectively (also dependent on the material).…”

Section: Theorymentioning

confidence: 99%

“…According to this author, the friction coefficient can be given using any of the two following expressions:

in the above expressions,

is a positive coefficient depending on the material,

and:

where

is a positive coefficient depending on the material and

and

are reference values for the friction coefficient and the slip velocity, respectively (also dependent on the material). It can be shown that (Pipping, 2019 ):

where we set

, with

. The state variable,

, satisfies the general evolution law:

where

is a non‐decreasing function and

is Lipschitz.…”

Section: Theorymentioning

confidence: 99%

“…where 𝐴𝐴 𝐴𝐴 is a positive coefficient depending on the material and 𝐴𝐴 𝐴𝐴0 and 𝐴𝐴 𝐴𝐴0 are reference values for the friction coefficient and the slip velocity, respectively (also dependent on the material). It can be shown that (Pipping, 2019): , with 𝐴𝐴 𝐴𝐴𝑐𝑐 being a characteristic slip distance (Scholz, 2019). By integrating with respect to time Equation ( 24) and using the non-decreasing property of 𝐴𝐴 𝐴𝐴(𝜓𝜓) , we obtain…”

Section: Rate-and-state Frictionmentioning

confidence: 99%

See 3 more Smart Citations

Preventing Instabilities and Inducing Controlled, Slow‐Slip in Frictionally Unstable Systems

Stefanou

Tzortzopoulos

2022

JGR Solid Earth

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We propose a theory for preventing instabilities and inducing controlled, slow‐slip in frictionally unstable systems, such as the Generalized‐Burridge‐Knopoff (GBK) model and seismic fault models. We exploit the dependence of friction on pressure and use it as a backdoor for altering the dynamics of the underlying dynamical system. We use the mathematical Theory of Control and, for the first time, we manage to (a) stabilize and restrict chaos in this kind of systems, (b) guarantee slow frictional dissipation and (c) tune the system toward desirable global asymptotic equilibria of lower energy. Our control approach is robust and does not require exact knowledge of the frictional or elastic behavior of the system. Numerical examples of control are given for a Burridge‐Knopoff system and a strike‐slip fault model obeying rate‐and‐state friction. GBK models are known to present Self‐Organized Critical (SOC) behavior. Therefore, the presented methodology shows an additional example of SOC Control. Even though further developments are necessary before any practical application, we expect our methodology to inspire earthquake mitigation strategies regarding anthropogenic and/or natural seismicity.

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Section: Theorymentioning

confidence: 99%

in the above expressions,

is a positive coefficient depending on the material,

and:

where

is a positive coefficient depending on the material and

and

are reference values for the friction coefficient and the slip velocity, respectively (also dependent on the material).…”

Section: Theorymentioning

confidence: 99%

“…According to this author, the friction coefficient can be given using any of the two following expressions:

in the above expressions,

is a positive coefficient depending on the material,

and:

where

is a positive coefficient depending on the material and

and

are reference values for the friction coefficient and the slip velocity, respectively (also dependent on the material). It can be shown that (Pipping, 2019 ):

where we set

, with

. The state variable,

, satisfies the general evolution law:

where

is a non‐decreasing function and

is Lipschitz.…”

Section: Theorymentioning

confidence: 99%

Section: Rate-and-state Frictionmentioning

confidence: 99%

See 2 more Smart Citations

Preventing Instabilities and Inducing Controlled, Slow‐Slip in Frictionally Unstable Systems

Stefanou

Tzortzopoulos

2022

JGR Solid Earth

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show abstract

“…To our knowledge, existence and uniqueness of solutions of Problem 2.2 is widely open. In case of unilateral frictional contact with a rigid foundation and Dieterich's law (16), long-time existence of solutions was established by Pipping [28].…”

Section: I and Introduce The Closed Linear Subspacementioning

confidence: 99%

Numerical simulation of multiscale fault systems with rate- and state-dependent friction

Gräser¹,

Kornhuber²,

Podlesny³

2021

Preprint

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We consider the deformation of a geological structure with non-intersecting faults that can be represented by a layered system of viscoelastic bodies satisfying rate-and state-depending friction conditions along the common interfaces. We derive a mathematical model that contains classical Dieterich-and Ruina-type friction as special cases and accounts for possibly large tangential displacements. Semi-discretization in time by a Newmark scheme leads to a coupled system of non-smooth, convex minimization problems for rate and state to be solved in each time step. Additional spatial discretization by a mortar method and piecewise constant finite elements allows for the decoupling of rate and state by a fixed point iteration and efficient algebraic solution of the rate problem by truncated non-smooth Newton methods. Numerical experiments with a spring slider and a layered multiscale system illustrate the behavior of our model as well as the efficiency and reliability of the numerical solver.This research has been funded by Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1114 "Scaling Cascades in Complex Systems", Project Number 235221301, Project B01 "Fault networks and scaling properties of deformation accumulation".

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Dynamic contact problem with moderate displacement and time‐dependent normal

Wu¹

2022

Z Angew Math Mech

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In the publications on existence of solutions to the dynamic contact problems, small displacement was basically assumed. To extend the qualitative study for more general applications with moderate displacement and general shape of surfaces, the normal of contact surface is treated as time‐dependent variable in the contact mechanics model. The frictional dynamic contact problem with unilateral constraint is formulated in a variational inequality. The penalized variational equation is derived with two penalty terms, verifying contact condition and applying constraint on the normal velocity, respectively. For given penalty parameter the penalized variational equation is proved to have solution, which converges to the solution of the variational inequality. The true contact boundary, varying in time, is determined in the process and the constraint condition is satisfied. The regularity of the solution is similar to the results by others under the assumption of small displacement.

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Existence of long‐time solutions to dynamic problems of viscoelasticity with rate‐and‐state friction

Cited by 7 publications

References 18 publications

Preventing Instabilities and Inducing Controlled, Slow‐Slip in Frictionally Unstable Systems

Preventing Instabilities and Inducing Controlled, Slow‐Slip in Frictionally Unstable Systems

Numerical simulation of multiscale fault systems with rate- and state-dependent friction

Dynamic contact problem with moderate displacement and time‐dependent normal

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