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b250ac1c40ea70e64145716d279005fc7839cc8a | subsection | 9 | 80 | Mechanical stored energy | Yet, the large rotations are naturally allowed,
so we do not assume directly \nabla y\sim S which might be too
restrictive in some geophysically relevant situations.In addition to the already mentioned nonconvex \gamma -term, the
geometrically nonlinear setting of (REF ) induces additional
nonconvexity of \psi _{_{\rm ... | {
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"raw": "R. A. Toupin, Elastic materials with couple stresses, Arch. Ration. Mech. Anal., 11 (1962), pp. 385–414.",
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... | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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... | |
edc705e81e5548f426bbd60bb66d03ea048add67 | subsection | 10 | 80 | Mechanical stored energy | The exponent q in the \kappa _1-term is given and fixed to be
larger than d, which eases some points of the analysis. Note however that the
choice \kappa _1(\nabla ) |\nabla |^2 for some \kappa _1 (\nabla )\sim 1+ |\nabla |^{q-2} could be considered as well. The
gradient terms in \alpha , \phi , and \zeta are intended ... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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... | |
05af341edcdcbc3e311575d0b204048f6e54b522 | subsection | 11 | 80 | Mechanical stored energy | \ \ \ \ {Furthermore,
the
driving stress for the plastification, again involving a contribution
arising from the gradient \kappa _1-term, reads}&
_{\rm in}=\partial _{}^{}\Psi =
\sigma (\phi ) \nabla y^\top \partial _{F_{\rm el}}\psi _{_{\rm M}}(\sigma (\phi )\nabla y{\color {black}^{-1}}\!,\alpha ,\phi ,\zeta ): \part... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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... | |
ea9ac7d2ce31dc080849da93c94774c49cec06b5 | subsection | 12 | 80 | Mechanical stored energy | The
thermodynamical driving {\it pressure} for damage is }
&p_{\rm age}=
\partial _\alpha \Psi =
\partial _\alpha \psi _{_{\rm M}}(\sigma (\phi )\nabla y{\color {black}^{-1}}\!,\alpha ,\phi ,\zeta )-\kappa _2\Delta \alpha ,
{and the driving force for porosity-evolution
(a so-called effective pressure) is }
&p_{\rm eff... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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-0.008695308119058609,... | |
d1dcc84af8f06fe9de5020a6523fef3d304dde48 | subsection | 13 | 80 | Thermodynamical system | The
entropy \eta , the heat capacity c_{\rm v}, and the thermal part
\vartheta of the internal energy (per unit reference volume)
are classically recovered as&\eta =-\psi _\theta ^{\prime }=-\psi _{_{\rm T}}^{\prime }(\theta ),\ \ \ \ c_{\rm v}=-\theta \psi _{\theta \theta }^{\prime \prime }=-\theta \psi _{_{\rm T}}^{\... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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2949945bb7230b6d215b038922fac884bb174d75 | subsection | 14 | 80 | Thermodynamical system | The entropy equation reads as\theta \mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {\eta }}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {\eta }}}{{{\hspace*{1.00006pt}\text{\large .}}\over {\eta }}}{{{\hspace*{1.00006pt}\text{\large .}}\over {\eta }}}+{\rm div}\,j=\text{dissipation rate}.We assume the heat flux... | {
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"raw": "J. Cahn and J. Hilliard, Free energy of a uniform system I., Interfacial free energy, J. Chem. Phys., 28 (1958), pp. 258–267.",
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... | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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eccb8ce51d312b43fad7fcace13c8d04970503af | subsection | 15 | 80 | Thermodynamical system | This involves some relaxation time \tau _{\scriptscriptstyle \textrm {\hspace*{-3.00003pt}\rm rel}}^{}>0 and a contribution
\tau _{\scriptscriptstyle \textrm {\hspace*{-3.00003pt}\rm rel}}^{}\mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {\zeta }}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {\zeta }}}{{{\hspace... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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4684c1b7993504b42542293bfb48e0bbb0e20700 | subsection | 16 | 80 | Thermodynamical system | \\[-.1em]
p_{\rm eff}\,
\end{array}\!\!\Big )
=\Big (\!\!\begin{array}{c}0\!\! | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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-0.00... | |
d7a08f556b62461f594ccbbd7b31cd17247ca768 | subsection | 17 | 80 | Thermodynamical system | \\[-.1em]
0\!\!\end{array}\Big )
,\!\!&&\text{({\sf flow rule for damage/porosity})}
\\[-.3em]
&\mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {\zeta }}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {\zeta }}}{{{\hspace*{1.00006pt}\text{\large .}}\over {\zeta }}}{{{\hspace*{1.00006pt}\text{\large .}}\over {\zeta... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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... | |
ddcfde6d7d0027c665d719d502870ce5a34ae85c | subsection | 18 | 80 | Thermodynamical system | The need for such effective quantities stems form the fact that
driving forces are to be considered Eulerian in nature, so that a
pull-back to the reference configuration is imperative. A first choice
would then be{M}(F,\alpha ,\phi )&=(\det F)F^{-\top }\text{\large $\mathbb {M}$}(\alpha ,\phi ) F^{-1}
=(\mathrm {Cof}F... | {
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... | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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cfb8b2847d8a2b78ebe46333cc9d89ddb05d46e2 | subsection | 19 | 80 | Thermodynamical system | Indeed, the evolution is
insensitive to prior plastic deformations, for the stored energy
and the dissipation potential\widehat{\psi }_{_{\rm M}}(F,,\alpha ,\phi ,\zeta )
:=\psi _{_{\rm M}}(F_{\rm el},\alpha ,\phi ,\zeta )
\ \ \ \ \text{ and }\ \ \ \ \widehat{\mathfrak {R}}(,\alpha ,\phi ,\theta ;
\mathchoice{{{\hspace... | {
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earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0e7609fc7b86003b7d856cddd4c41eb3bf42d344 | subsection | 20 | 80 | Thermodynamical system | In particular, we can equivalently test the flow rule
() by \mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {}}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {}}}{{{\hspace*{1.00006pt}\text{\large .}}\over {}}}{{{\hspace*{1.00006pt}\text{\large .}}\over {}}}^{-1} or rewrite it as\partial _P^{}\mathfrak {R}\big (\a... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0... | |
6789bcd4fd2c5f428a44ab286d98e309e2e6418e | subsection | 21 | 80 | Thermodynamical system | As for the former
we prescribe&_{\rm el}\nu -
\mathrm {div}_{\scriptscriptstyle \textrm {\hspace*{-1.00006pt}S}}^{}\big (\kappa _0\nabla ^2y
\big )
+Ny=Ny_\flat (t),\ \ \ \ \ \ \ \kappa _0\nabla ^2y{:}(\nu \otimes \nu )=0,
\\&
M(,\alpha ,\phi )\nabla \mu ^{}{\cdot }\nu +M\mu =M\mu _\flat (t),\ \ \ \ \ \ K(,\zeta ,\thet... | {
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"raw": "V. Lyakhovsky and Y. Ben-Zion, Scaling relations of earthquakes and aseismic deformation in a damage rheology model, Geophys. J. Int.... | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0.0110... | |
30dc3eb55c0a7ec85cfc4fdcd4408de3b1aa985a | subsection | 22 | 80 | Thermodynamical system | Eventually, initial conditions read&y(0)=y_0,\ \ \ \mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {y}}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {y}}}{{{\hspace*{1.00006pt}\text{\large .}}\over {y}}}{{{\hspace*{1.00006pt}\text{\large .}}\over {y}}}(0)=v_0,\ \ \ (0)=_0,\ \ \ \alpha (0)=\alpha _0,\ \ \ \phi (0)... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0.... | |
24d78b6a5b64b1127d9ace113bd9f5c9b69f8e88 | subsection | 23 | 80 | Thermodynamical system | \frac{{\rm d}}{{\rm d}t}\bigg (\int _\Omega \frac{\varrho }{2}|\mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {y}}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {y}}}{{{\hspace*{1.00006pt}\text{\large .}}\over {y}}}{{{\hspace*{1.00006pt}\text{\large .}}\over {y}}}|^2
\,{\rm d}x
+\Psi _{_{\rm M}}(\nabla y,,\alpha ... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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-0.... | |
4477f2186168c16ab86fd72560e973fddba98783 | subsection | 24 | 80 | Thermodynamical system | Existence of
smooth solutions is however not guaranteed for y lacks time regularity due
to the possible occurrence of shock-waves in the nonlinear hyperbolic system
(REF a).
Also the power of the external mechanical load in (REF ),
i.e. y_\flat {\cdot }\mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {y}}}{{{\hsp... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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fa774d8809e0b5daa6622c3079daf9eec4a72647 | subsection | 25 | 80 | Thermodynamical system | (REF ).By adding to (REF ) the space integral of the heat equation
() we obtain the total energy balance&\hspace{-10.0pt}\frac{{\rm d}}{{\rm d}t}\bigg (
\!\!\!\begin{array}[t]{c}\begin{array}[t]{c}\underbrace{\int _\Omega \frac{\varrho }{2}|\mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {y}}}{{{\hspace*{1.00006... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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f6f5903ba744fe2fc86242d2605eacaf69613f5a | subsection | 26 | 80 | Thermodynamical system | \!From (REF ) with the heat flux j=-{K}\nabla \theta and with
the dissipation rate (=heat production rate) r
from (), one can read the entropy imbalance\frac{{\rm d}}{{\rm d}t}\int _\Omega \!\eta \,{\rm d}x
=\int _\Omega \!\frac{r+{\rm div}({K}\nabla \theta )}{\theta }\,{\rm d}x
=\int _\Omega \,\frac{r}{\theta }-{K}\na... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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74cfbd4510b18d92647f416009a0a9108bafaf20 | subsection | 27 | 80 | Thermodynamical system | In fact, we do not consider any
adiabatic-type effects, which might cause cooling.We conclude the presentation of the model with a number of
remarks and comments on modeling choices and possible extensions.Remark (Dissipation potential \mathfrak {D})
A specific form of the flow rule for the damage/porosity
() can be c... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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-0.002836158499121666,
-0.006954024080187082,
0.02873160131275654,
0.006748035550117493,
0.0... | |
3842000c3157a3502843e4a585a970059f39ddb7 | subsection | 28 | 80 | Thermodynamical system | This flow rule leads to a
dissipation
potential which is (2,\frac{n+2}{n+1})-homogeneous
in terms of the rates (\mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {\alpha }}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {\alpha }}}{{{\hspace*{1.00006pt}\text{\large .}}\over {\alpha }}}{{{\hspace*{1.00006pt}\text{\lar... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.05332248657941818,
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-0.01185029000043869,
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6803c27624b60f53ace7b0e1ca541185efd69624 | subsection | 29 | 80 | Thermodynamical system | \end{array}\right.}In the case \mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {\phi }}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {\phi }}}{{{\hspace*{1.00006pt}\text{\large .}}\over {\phi }}}{{{\hspace*{1.00006pt}\text{\large .}}\over {\phi }}}=0, the flow rule (REF )
has been used in (with \kappa _2=0) and
.... | {
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earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0.0... | |
c4d82f7caec5a6bf6ba2074ee599fef789593799 | subsection | 30 | 80 | Thermodynamical system | Such dissipation does
not admit a potential and does not control
\mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {\phi }}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {\phi }}}{{{\hspace*{1.00006pt}\text{\large .}}\over {\phi }}}{{{\hspace*{1.00006pt}\text{\large .}}\over {\phi }}}. In particular, standard existe... | {
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earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0.02985779196023941,... | |
43bb6144cd4b5d12bd1ef73ecd02a4fc9479d747 | subsection | 31 | 80 | Thermodynamical system | In this isotropic case \text{\large $\mathbb {K}$}(\theta )=k(\theta )\mathbb {I},
relation
(REF ) can also be written
by using the right Cauchy-Green tensor C as{K}(F,\theta )=\hat{{K}}(C,\theta ) =\det C^{1/2} k(\theta )C^{-1}
\quad \text{with }\ \ C=F^\top F,cf. or
.
In fact, the effective transport-coefficient tens... | {
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earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.047210078686475754,
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0.019225824624300003,
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... | |
f5e3bc41415aac019d13745cf1e679c6ffd9a278 | subsection | 32 | 80 | Existence of weak solutions | This section introduces the definition of weak solution to the
problem and brings to the statement of the our main existence result,
namely Theorem REF . Let us start by fixing some notation.We will use the standard notation C(\cdot ) for the space of continuous bounded
functions, L^p for Lebesgue spaces, and W^{k,p} f... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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bffad9e013f1370e7d7cf236d37ee4451befad39 | subsection | 33 | 80 | Existence of weak solutions | L^1(\Omega ),\ \ \, 0\le \zeta _0\le 1,\ \ \theta _0\ge 0,
\\&
g\in C(\mathbb {R}^d;\mathbb {R}^d),\ \ \ \mu _\flat \in L^2(\Sigma ),\ \ \theta _\flat \in L^1(\Sigma ),\ \ \theta _\flat \ge 0,
\\&\lambda ,G,\gamma ,m{:}\mathbb {R}^2\rightarrow \mathbb {R}^+,\ \chi ,\sigma {:}[0,1]\rightarrow \mathbb {R}^+\ \text{ Lips... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.05286213755607605,
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0.013... | |
fe4a86c7a77c907c982e7a2d163b958b2733bad7 | subsection | 34 | 80 | Existence of weak solutions | In this case, their
subdifferentials are indeed set-valued and thus
()-() are to be satisfied for any selection
from these subdifferentials. We however stick with \mathfrak {R} and
\mathfrak {D} being smooth both for the sake of simplicity and
in accord with models used in geophysical literature
, , , , , , cf. for det... | {
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"raw": "Y. Hamiel, V. Lyakhovsky, and A. Agnon, Coupled evolution of damage and porosity in poroelastic media: Theory and applications to defo... | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0.00... | |
c946bbdcff5d9d2091a35b4779c50d43f7c1321f | subsection | 35 | 80 | Existence of weak solutions | An example for \varpi considered already in
is\varpi (\det )={\left\lbrace \begin{array}{ll}\displaystyle {
\frac{\delta }{\max (1,\det )^q}
+\frac{(\det -1)^2}{2\delta }}\!\!&\text{ if }\ \det >0,\\
\qquad +\infty &\text{ if }\ \det \ge 0\,;\end{array}\right.}note that the minimum of this potential is attained just a... | {
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"raw": "T. Roubíček and U. Stefanelli, Thermoelasticity at large strains with isochoric creep or viscoplasticity under small elastic strain, Math. Mech. Solids., in print, (2018). (Preprint arXiv, no.1804.0... | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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798ce80f314ae0279dc1398a072345459d7babb1 | subsection | 36 | 80 | Existence of weak solutions | N y{\cdot }\tilde{y}\,{\rm d}S{\rm d}t=\int _Q\!
g(y){\cdot }\tilde{y}\,{\rm d}x{\rm d}t
+
\int _\Omega \!v_0{\cdot }\tilde{y}(0)\,{\rm d}x+\int _\Sigma \!
N y_\flat {\cdot }\tilde{y}\,{\rm d}S{\rm d}tholds for any \tilde{y} smooth with \tilde{y}(T)=0.
(ii)
The weak formulation of the plastic flow rule
()
in the form (... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.027949310839176178,
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0.026301642879843712,
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0.0692020058631897,
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0.01350171398371458,
0.00006376604142133147,
-... | |
d379d16b695d0949b91e400d96579ec9d9110fe3 | subsection | 37 | 80 | Existence of weak solutions | (iii)
The weak formulation of the coupled flow rule
() for (\alpha ,\phi )
holds for any \tilde{\alpha } and \tilde{\phi } smooth:&\int _Q\bigg (\partial _{F_{\rm el}}^{}\psi _{_{\rm M}}(\sigma (\phi )\nabla y\,{\color {black}^{-1}},\alpha ,\phi ,\zeta ){:}(\sigma ^{\prime }(\phi )\nabla y{\color {black}^{-1}}\tilde{\p... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.04890360310673714,
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0.0026291029062122107,
-0... | |
6ef56f891fe673770285cb1a8e5e79f89ce5ace4 | subsection | 38 | 80 | Existence of weak solutions | (v)
The weak formulation of the heat equation ()&\int _Q\!{K}(,\phi ,\zeta ,\theta )\nabla \theta {\cdot }
\nabla \tilde{\theta }-C_{\rm v}(\theta )\mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {\tilde{\theta }}}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {\tilde{\theta }}}}{{{\hspace*{1.00006pt}\text{\large ... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.003933285363018513,
0.0214251521974802,
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-0.... | |
fe8249afb2e9e6c7b7cbab67b4ae42230519de82 | subsection | 39 | 80 | Existence of weak solutions | C_{\rm v}(\theta _0)\tilde{\theta }(0)\,{\rm d}xholds for any \tilde{\theta } smooth with \tilde{\theta }(T)=0
and with C_{\rm v}(\cdot ) denoting a primitive function to c_{\rm v}(\cdot ) and with
r=r(,\alpha ,\phi ,\theta ;\mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {}}}{{{\hspace*{1.00006pt}\text{\Large .... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.028825510293245316,
0.00905661191791296,
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0.00820970069617033,
0.030641410499811172,
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-0.0... | |
b1055c534f4079afab880cdbfc88a33c9ae76030 | subsection | 40 | 80 | Existence of weak solutions | In addition{\rm div}\big (|\nabla |^{q-2}\nabla \big )\in L^2(Q;\mathbb {R}^{d\times d}),\ \ \ \ \ \ \Delta \alpha \in L^2(Q),\ \ \text{ and }\ \ \Delta \phi \in L^2(Q).Moreover, the energy conservation (REF ) holds on the time
intervals [0,t] for all t\in I in the following sense with
\vartheta (t)=C_{\rm v}(\theta (t... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.02146400697529316,
0.05531520023941994,
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-0.018229912966489792,
0.... | |
28b9ff17f76ed352f39199586419623c5d86bda0 | subsection | 41 | 80 | Existence of weak solutions | This also provides a (conceptual) algorithm that is numerically stable and
converges as the discretization and the regularization parameters h>0 and
\varepsilon >0 tend to 0. (More specifically, when successively
h\rightarrow 0 and then \varepsilon \rightarrow 0 and when \theta is reconstructed from the
rescaled temper... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0.0... | |
266017a92c73dc07e33879189c25fa4dea36d114 | subsection | 42 | 80 | Convergence of Galerkin approximations | We devote section to the proof of the existence result, namely
Theorem REF . As already mentioned,
we apply a constructive method delivering an approximation of
the problem. This results from combining a regularization in
terms of the small parameter \varepsilon and a Galerkin
approximation, described by the small para... | {
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"raw": "T. Roubíček, Nonlinearly coupled thermo-visco-elasticity, Nonlin. Diff. Eq. Appl., 20 (2013), pp. 1243–1275.",
"source_ref_id": "b824e... | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0.0005739524494856596,... | |
06a18644df8cde21b0135ffc4f176f9cd207ce05 | subsection | 43 | 80 | Convergence of Galerkin approximations | More specifically, for some given regularization parameter
\varepsilon >0, we replace these terms respectively by&r_\varepsilon =\frac{r\big (,\alpha ,\phi ,\theta ;\mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {}}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {}}}{{{\hspace*{1.00006pt}\text{\large .}}\over {}}}... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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-0.032677989453077316,
... | |
afacbb1b2eb95f10a17c4f62cac1555660b3a4f1 | subsection | 44 | 80 | Convergence of Galerkin approximations | As effect of this boundedness, we are in the
position of resorting to a L^2-theory instead of the L^1-theory for the
regularized heat problem. In addition, we perform a
regularization of the nonsmooth term \mathfrak {N}_{[0,1]}^{} in
() by means of its Yosida approximation, yielding
the mapping \mathfrak {N}_\varepsilo... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0.046354785561561584,
0.008613323792815208,
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0.010543501935899258,
-0.009803473018109798,
-0.023619884625077248,
0.002033171709626913,
-... | |
9b8c6f2666a700b15a3c007fb951cd21fb02fa67 | subsection | 45 | 80 | Convergence of Galerkin approximations | Upon
letting\mathfrak {K}(,\phi ,\zeta ,\vartheta ):=
\frac{1}{{c_{\rm v}(C_{\rm v}^{-1}(\vartheta ))}}{K}(,\phi ,\zeta ,C_{\rm v}^{-1}(\vartheta )),\ \ \ \we rewrite and regularize the system (REF ) by&\varrho \mathchoice{{{\hspace*{1.30005pt}\text{\LARGE .\hspace*{-1.30005pt}.}}\over {y}}}{{{\hspace*{1.00006pt}\text{... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.04171418398618698,
0.011229568161070347,
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-0.010909158736467361,
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0.03561115637421608,
0.00965803861618042,
0.02079606056213379,
0.011511832475662231,
-0.0028169278521090746,
0.0027597120497375727,
-0.02963019162416458,
... | |
441100bb0bf5b0c26cbaec741b9294ba6fb4217e | subsection | 46 | 80 | Convergence of Galerkin approximations | \\[-.1em]
p_{\rm eff}\,
\end{array}\!\!\Big )
=\Big (\!\!\begin{array}{c}0\!\! | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.020722869783639908,
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-0.03604375571012497,
-0.030229752883315086,
-0.018464412540197372,
0.0010758191347122192,
0.02519400417804718,
0.03961455821990967,
0.03534180298447609,
0.002819256391376257,
0.017396222800016403,
0.02166897989809513,
-0.02380535751581192,
-0.00... | |
0643d19a9aa3b15e36df20124be4545e72ef16fe | subsection | 47 | 80 | Convergence of Galerkin approximations | \\[-.1em]
0\!\!\end{array}\Big ),
\\[-.3em]
&\mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {\zeta }}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {\zeta }}}{{{\hspace*{1.00006pt}\text{\large .}}\over {\zeta }}}{{{\hspace*{1.00006pt}\text{\large .}}\over {\zeta }}}=\color {black}{\rm div}\big ({M}(,\alpha ,\phi... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.011445176787674427,
-0.016923602670431137,
-0.03882203996181488,
-0.005474609788507223,
-0.0017654185649007559,
0.003914250526577234,
0.029070749878883362,
0.03296210989356041,
0.017228806391358376,
0.004745933227241039,
-0.0003567080129869282,
0.010445632040500641,
-0.014718498103320599,... | |
f7f5031a524336515b6638f907dc45ef95d3e73c | subsection | 48 | 80 | Convergence of Galerkin approximations | Note that the
\varepsilon -regularization serves the double purpose of having a
bounded right-hand side in () as well as a smooth
nonlinearity \mathfrak {N}_\varepsilon in
(). The boundary conditions
are correspondingly modified by using (), i.e. K(,\zeta ,\theta )\nabla \theta {\cdot }\nu +K\theta =K\theta _\flat (t)
... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.008875343948602676,
0.008142727427184582,
-0.039958126842975616,
0.0053496272303164005,
-0.007726815063506365,
0.00628829188644886,
0.03134988248348236,
-0.004120968282222748,
0.03455508127808571,
0.03892025351524353,
-0.05042843893170357,
0.028144685551524162,
-0.01776595041155815,
0.0... | |
982316f012609a5b7370b77361c4ecf51720c2d5 | subsection | 49 | 80 | Convergence of Galerkin approximations | For simplicity, we assume that all initial conditions
(y_0,_0,\alpha _0,\phi _0,\zeta _0,\vartheta _{0\varepsilon })
belong to
all finite-dimensional subspaces so that no additional
approximation of such conditions is needed.The outcome of the Galerkin approximation is an
an initial-value problem for a system of ordina... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.012134763412177563,
0.018091551959514618,
-0.02275935560464859,
-0.001199366175569594,
0.024772917851805687,
-0.013622053898870945,
0.026816988363862038,
-0.00472882017493248,
0.013042391277849674,
0.007871197536587715,
-0.04390176013112068,
0.006776704918593168,
-0.02274410054087639,
0... | |
271f9e4ccc95f13c30f198b4b7b9e04487ad3980 | subsection | 50 | 80 | Convergence of Galerkin approximations | Then, the Galerkin
approximation of (REF ) with the initial/boundary conditions
(REF )—(REF ) modified by (REF ) admits a solution
on the whole time
interval I=[0,T], let us denote it by (y_{\varepsilon h},_{\varepsilon h},\alpha _{\varepsilon h},\phi _{\varepsilon h},\zeta _{\varepsilon h},\mu _{\varepsilon h},\varthe... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.006856090854853392,
0.00991214532405138,
0.007516137789934874,
-0.00837839487940073,
0.006989626679569483,
-0.027271753177046776,
0.0058641135692596436,
-0.03171275928616524,
0.03152962401509285,
0.019274886697530746,
-0.050667162984609604,
0.0065165297128260136,
-0.01597846858203411,
0... | |
c6988bb798a8682b40c53d5a2f29d98955ba09cd | subsection | 51 | 80 | Convergence of Galerkin approximations | The algebraic constraint (REF )
for
the underlying system of ordinary differential-algebraic equations
takes the more specific form\mu =
m(\alpha ,\phi )\frac{\zeta {-}\phi {-}\beta I_1}{\@root 4 \of {1{+}\epsilon I_2}}-\kappa _4\Delta \zeta +\mathfrak {N}_\varepsilon (\zeta )+\tau _{\scriptscriptstyle \textrm {\hspace... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.018254783004522324,
0.006921859923750162,
-0.009325813502073288,
-0.030648499727249146,
-0.005574882496148348,
-0.006616595666855574,
0.03550219535827637,
0.02297111041843891,
0.03950115293264389,
0.012393715791404247,
-0.040294840931892395,
-0.00687607005238533,
-0.027656912803649902,
... | |
cbb5b3b2d0cb4d3bac9e7bfbedaacfa7f1b9d8be | subsection | 52 | 80 | Convergence of Galerkin approximations | In particular, we use
\mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {y}}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {y}}}{{{\hspace*{1.00006pt}\text{\large .}}\over {y}}}{{{\hspace*{1.00006pt}\text{\large .}}\over {y}}}_{\varepsilon h}, \mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {}}}{{{\hspace*{1.... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
0.020770613104104996,
-0.01805410534143448,
-0.013712268322706223,
-0.021548939868807793,
0.01979389227926731,
-0.021945733577013016,
0.0502401627600193,
-0.013185753487050533,
0.007657351437956095,
0.020450126379728317,
-0.02527269348502159,
0.0170010756701231,
-0.024311233311891556,
-0.0... | |
7df93a2c2d092b2d5393b4e22913c55164cece25 | subsection | 53 | 80 | Convergence of Galerkin approximations | More specifically, using \mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {y}}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {y}}}{{{\hspace*{1.00006pt}\text{\large .}}\over {y}}}{{{\hspace*{1.00006pt}\text{\large .}}\over {y}}}_{\varepsilon h} as test in the
Galerkin approximation of (REF ) with
its boundary condi... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
0.018330693244934082,
-0.0052008191123604774,
0.020681172609329224,
0.008379309438169003,
0.003285330254584551,
-0.008302995003759861,
0.00039731129072606564,
-0.00415912875905633,
-0.0035562459379434586,
0.009722440503537655,
-0.019765401259064674,
0.005670151673257351,
-0.01131740957498550... | |
e079810450a69d0b397b4d87f908fe3728561b40 | subsection | 54 | 80 | Convergence of Galerkin approximations | \partial _{ \nabla y } \widehat{\psi }_{_{\rm M}}(\nabla y_{\varepsilon h},_{\varepsilon h},\alpha _{\varepsilon h},\phi _{\varepsilon h},\zeta _{\varepsilon h}){:}\nabla \mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {y}}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {y}}}{{{\hspace*{1.00006pt}\text{\large .}}\o... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
0.0015563027700409293,
0.015341788530349731,
-0.015494367107748985,
-0.01383888814598322,
0.01695149391889572,
-0.006873670499771833,
0.014266109094023705,
-0.022322265431284904,
0.003289978252723813,
0.00945225078612566,
-0.04168450087308884,
0.005916239228099585,
-0.02694539912045002,
-0... | |
67316c028b9a979bd9558f6c4c9d7f349cae6b4b | subsection | 55 | 80 | Convergence of Galerkin approximations | Such procedure leads to a (system of ordinary) differential
equation instead of the inclusion, so that conventional calculus applies.
This gives&\int _0^t\!\!\int _\Omega \!\tau _{\scriptscriptstyle \textrm {\hspace*{-3.00003pt}\rm rel}}^{}\mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {\zeta }}}{{{\hspace*{1.0... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.030599167570471764,
0.01776430383324623,
-0.03961867094039917,
0.00958417821675539,
0.0023845983669161797,
0.011385026387870312,
-0.002943548373878002,
0.015078292228281498,
0.010393033735454082,
0.004914180375635624,
-0.019244663417339325,
0.005360577255487442,
0.004132031928747892,
-0... | |
141f17959f75782fb0511f121f121fc0826c286a | subsection | 56 | 80 | Convergence of Galerkin approximations | \partial _\zeta ^{} \widehat{\psi }_{_{\rm M}}(\nabla y_{\varepsilon h}, _{\varepsilon h}, \alpha _{\varepsilon h},\phi _{\varepsilon h},\zeta _{\varepsilon h})\mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {\zeta }}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {\zeta }}}{{{\hspace*{1.00006pt}\text{\large .}}\ov... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
0.026523638516664505,
0.008714037947356701,
-0.016252977773547173,
-0.015329686924815178,
0.017183899879455566,
-0.026981469243764877,
0.03213969245553017,
-0.02260155975818634,
0.008202794007956982,
0.03842722997069359,
-0.053138844668865204,
0.012773467227816582,
-0.012773467227816582,
-... | |
b90f0adc03d67d6f02344aae7dfa9e7665498e80 | subsection | 57 | 80 | Convergence of Galerkin approximations | M\mu _{\varepsilon h}\mu _\flat \,{\rm d}S{\rm d}t
-\int _0^t\!\!\int _\Omega \!\mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {\zeta }}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {\zeta }}}{{{\hspace*{1.00006pt}\text{\large .}}\over {\zeta }}}{{{\hspace*{1.00006pt}\text{\large .}}\over {\zeta }}}_{\varepsilon... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.029008567333221436,
0.008057087659835815,
-0.031587447971105576,
-0.02270633727312088,
0.016419367864727974,
-0.007175843697041273,
0.01883039064705372,
0.007660337258130312,
0.040346477180719376,
0.01134553924202919,
-0.06183204427361488,
-0.0037214437033981085,
-0.00038935913471505046,
... | |
73d3af68361da8447e0b7c6692e90fe7ffd3665c | subsection | 58 | 80 | Convergence of Galerkin approximations | (REF ).Taking the sum of
(REF )–(REF )
and (REF ) and using the calculus& \partial _{\nabla y} \widehat{\psi }_{_{\rm M}}{:}\nabla \mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {y}}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {y}}}{{{\hspace*{1.00006pt}\text{\large .}}\over {y}}}{{{\hspace*{1.00006pt}\text{\la... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.014372438192367554,
0.030529987066984177,
-0.028424471616744995,
-0.010077489539980888,
-0.007872800342738628,
0.00006049306830391288,
0.015470967628061771,
0.015005618333816528,
0.0363125279545784,
0.021329795941710472,
-0.024060864001512527,
0.04482613876461983,
-0.027920978143811226,
... | |
950537de91eab1db27516dd6f4dbf8098ef51d8b | subsection | 59 | 80 | Convergence of Galerkin approximations | We overcome this obstruction by
by-part integration&\int _0^t\!\int _\Gamma Ny_\flat {\cdot }\mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {y}}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {y}}}{{{\hspace*{1.00006pt}\text{\large .}}\over {y}}}{{{\hspace*{1.00006pt}\text{\large .}}\over {y}}}_{\varepsilon h}\,{\... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.022927384823560715,
0.005892017390578985,
0.004873785190284252,
-0.008794169872999191,
0.0006955059361644089,
-0.014659491367638111,
-0.010311983525753021,
-0.009663671255111694,
0.051529411226511,
0.01325608603656292,
-0.050034478306770325,
0.02109685353934765,
0.01620018668472767,
0.0... | |
a3c1a0b5d9cbd7e86844b96f70388ca2deb62d4c | subsection | 60 | 80 | Convergence of Galerkin approximations | More
in detail, (REF ) follows from the
coercivity () of \mathfrak {R} so that we have also
that \mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {}}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {}}}{{{\hspace*{1.00006pt}\text{\large .}}\over {}}}{{{\hspace*{1.00006pt}\text{\large .}}\over {}}}_{\varepsilon h}^{_{... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1051/cocv:2008050",
"end": 651,
"openalex_id": "https://openalex.org/W2143607683",
"raw": "T. J. Healey and S. Krömer, Injective weak solutions in second-gradient nonlinear elasticity, ESAIM: Control, Optim. & Cal. Var., 15 (2009), pp. 8... | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.009804357774555683,
-0.0058368355967104435,
-0.02108890563249588,
0.007061426527798176,
-0.017640214413404465,
-0.005913133732974529,
0.015587784349918365,
-0.041597943753004074,
0.013726100325584412,
0.009140560403466225,
-0.03573821857571602,
0.003668050514534116,
-0.008659879676997662,... | |
792f5a25dfa18fe39ea70feed9fc5284b6f64864 | subsection | 61 | 80 | Convergence of Galerkin approximations | We obtain&\frac{{\rm d}}{{\rm d}t}\frac{1}{2}\int _\Omega \!\vartheta _{\varepsilon h}^2
\,{\rm d}x
+\int _\Omega \!
\mathfrak {K}(_{\varepsilon h},\phi _{\varepsilon h},\zeta _{\varepsilon h},\vartheta _{\varepsilon h})
\nabla \vartheta _{\varepsilon h}{\cdot }\nabla \vartheta _{\varepsilon h}\,{\rm d}x
+\int _\Gamma ... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.04152997210621834,
0.04491706192493439,
-0.0294157937169075,
-0.017988184466958046,
-0.029690422117710114,
-0.0008296077721752226,
-0.0002886939619202167,
0.004546632058918476,
0.02168041653931141,
0.010687636211514473,
-0.03554916754364967,
0.012823637574911118,
0.024258874356746674,
0... | |
d2c72f99c765c7a34b4199ab00cd64f0e73f8d6d | subsection | 62 | 80 | Convergence of Galerkin approximations | Again by comparison, using ()
with () and taking advantage of the boundedness of the term
\partial _P^{}\mathfrak {R}(\alpha _{\varepsilon h},\phi _{\varepsilon h},\theta _{\varepsilon h};\mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {}}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {}}}{{{\hspace*{1.00006pt}\te... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.04309670627117157,
0.0354357473552227,
-0.014307069592177868,
-0.01983913592994213,
-0.03241409733891487,
-0.011277785524725914,
0.022219832986593246,
0.027912138029932976,
0.00022974768944550306,
0.026279224082827568,
-0.03516105189919472,
0.012903069145977497,
-0.008744481019675732,
0... | |
5bb7e6e8cac4b6e63dd0be61199ec6f30aa1e72a | subsection | 63 | 80 | Convergence of Galerkin approximations | More precisely, we might
get here for
() the full L^2(Q)-norm upon testing
(REF ) on (\Delta \alpha ,\Delta \phi )^\top ,
which would be allowed if we assume to construct the finite
dimensional spaces starting from eigenfunctions of the Laplacian.Finally, estimate () follows
as the term
\int _0^t\int _\Omega \mathfrak ... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
0.003752525895833969,
0.021737191826105118,
0.019876183941960335,
-0.018182970583438873,
-0.0038440509233623743,
-0.0041605751030147076,
0.015505864284932613,
-0.01928127184510231,
0.013133840635418892,
0.016382979229092598,
-0.02423887699842453,
0.03477950766682625,
-0.002511217724531889,
... | |
49e52dc10eb3ee44294ce83bbccf9a13ab5a3e86 | subsection | 64 | 80 | Convergence of Galerkin approximations | Moreover, the following
a-priori estimates hold&\big \Vert {\rm div}(|\nabla _{\varepsilon }|^{q-2}\nabla _{\varepsilon })\big \Vert _{L^2(Q;\mathbb {R}^{d\times d})}\le C,
\ \ \big \Vert \Delta \alpha _{\varepsilon }\big \Vert _{L^2(Q)}\le C,\ \ \big \Vert \Delta \phi _{\varepsilon }\big \Vert _{L^2(Q)}\le C,
\\&
\big... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.033558908849954605,
0.028357278555631638,
-0.04204016178846359,
-0.003651438280940056,
-0.026709841564297676,
-0.009869370609521866,
0.06016197428107262,
-0.011127829551696777,
-0.0003513198462314904,
-0.0011135456152260303,
-0.030218273401260376,
0.006906271446496248,
-0.0125769646838307... | |
ef0be567b3ae49bee9e9ec2e4ba68eb1d3d531ef | subsection | 65 | 80 | Convergence of Galerkin approximations | Let us indicate one such weak* limit by
(y_\varepsilon ,_\varepsilon ,\alpha _\varepsilon ,\phi _\varepsilon ,\zeta _\varepsilon ,\mu _\varepsilon ,
\vartheta _\varepsilon ) and prove that it solves the regularized
problem (REF ).
Note that, the
estimates (REF ) follow from (REF i,j) which
are independent of h and h_0,... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-3-0348-0513-1",
"end": 345,
"openalex_id": "https://openalex.org/W1525219438",
"raw": "T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2nd ed., 2013.",
"source_ref_id": "925e8515... | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.04780665412545204,
-0.0010897216852754354,
-0.0005319901392795146,
0.007955064065754414,
0.0008084724540822208,
-0.022423669695854187,
0.02028808183968067,
0.023155871778726578,
0.026206711307168007,
0.01827452890574932,
-0.039477862417697906,
-0.005228376016020775,
-0.008938959799706936,... | |
f02f0a5ea586b507da49fcbff83a4318d0c5ce43 | subsection | 66 | 80 | Convergence of Galerkin approximations | Such
approximants can be constructed by projections
at the level of time derivatives.We begin by discussing the terms \mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {\alpha }}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {\alpha }}}{{{\hspace*{1.00006pt}\text{\large .}}\over {\alpha }}}{{{\hspace*{1.00006pt}\tex... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
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530dcc6593754dfd332b093ddf7fef20de58daec | subsection | 67 | 80 | Convergence of Galerkin approximations | Referring to a_\mathfrak {D}^{}>0 from the uniform monotonicity assumption
(), we can estimate&\limsup _{h\rightarrow 0}\frac{a_\mathfrak {D}^{}}{2}\Vert \mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {z}}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {z}}}{{{\hspace*{1.00006pt}\text{\large .}}\over {z}}}{{{\hspa... | {
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earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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d179735749c80733a55c1a80789bad2297fd52a5 | subsection | 68 | 80 | Convergence of Galerkin approximations | In (REF ),
we used () tested by \mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {z}}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {z}}}{{{\hspace*{1.00006pt}\text{\large .}}\over {z}}}{{{\hspace*{1.00006pt}\text{\large .}}\over {z}}}_{\varepsilon h}-\mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {\tilde{z... | {
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earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
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d623d4cde232fd19daff5b0ca3c6d38cad9e1663 | subsection | 69 | 80 | Convergence of Galerkin approximations | In (REF ), we also that
\partial _\alpha \psi _{_{\rm M}}( F_{{\rm el},\varepsilon h}, \alpha _{\varepsilon h},
\phi _{\varepsilon h},
\zeta _{\varepsilon h}), {\rm tr}\big (\sigma ^{\prime }(\phi _{\varepsilon h}) {\color {black}^{-T}_{\varepsilon h}}\nabla y_{\varepsilon h}^\top \partial _\phi \psi _{_{\rm M}}(F_{{\... | {
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earthquake modeling | [
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a055e07447157125d9790744d52b6015fc5c8674 | subsection | 70 | 80 | Convergence of Galerkin approximations | Moreover, in the last equality in (REF )
we have used
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earthquake modeling | [
"Tomas Roubicek",
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f587ff1b029dfc473a9df49c22e84aaf4d16cdcc | subsection | 71 | 80 | Convergence of Galerkin approximations | Note that, for any test function v
valued in a finite-dimensional Galerkin space we have
that& \int _Q\mathbb {M}(z_{\varepsilon h}){\color {black}^{-1}_{\varepsilon h}}\nabla \mu _{\varepsilon h}{\cdot }{\color {black}^{-1}_{\varepsilon h}}\nabla v\,{\rm d}x{\rm d}t
\\[-.4em]&\qquad \qquad \rightarrow \ \int _Q\mathbb... | {
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earthquake modeling | [
"Tomas Roubicek",
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2d6ed91c57706e5c60b254e50468e84861cac8a9 | subsection | 72 | 80 | Convergence of Galerkin approximations | In particular, by testing
the limit equation () on \mu _\varepsilon
and adding it to the limit equation
() tested on \mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {\zeta }}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {\zeta }}}{{{\hspace*{1.00006pt}\text{\large .}}\over {\zeta }}}{{{\hspace*{1.00006pt}\text{\... | {
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earthquake modeling | [
"Tomas Roubicek",
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59a3db4d55fa435f5b1e5bf22c787690e9da2585 | subsection | 73 | 80 | Convergence of Galerkin approximations | These two convergences
have to be obtained simultaneously in order to be able to
exploit a cancelation as in (REF ). | {
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} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
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2a7bc2b880dde1b874211ddb3712ad5718c0236d | subsection | 74 | 80 | Convergence of Galerkin approximations | Using (REF )
and denoting by a_\mathbb {M}>0 the positive-definiteness constant of
\mathbb {M}, we can estimate&\limsup _{h\rightarrow 0}\int _Q\tau _{\scriptscriptstyle \textrm {\hspace*{-3.00003pt}\rm rel}}^{}\big (\mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {\zeta }}}{{{\hspace*{1.00006pt}\text{\Large .}}... | {
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} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
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a0791ee4ccb598c1efa1f7dc7e601f9d1f4ffabb | subsection | 75 | 80 | Convergence of Galerkin approximations | From this, we obtain the strong convergence
() for
\nabla \mu _{\varepsilon h}\rightarrow \nabla \mu _\varepsilon . Note however that this last
convergence is not exploited in the following.The convergence of the mechanical part for h\rightarrow 0 is now
straightforward. As highest-order terms in (REF a,c-e)
are linear... | {
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earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
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f1e2866a07f7f6f6cfaefbf1bee09d91e53ff665 | subsection | 76 | 80 | Convergence of Galerkin approximations | Moreover, the following
strong convergences hold&&&
\mathchoice{{{\hspace*{1.00006pt}\text{\LARGE .}}\over {}}}{{{\hspace*{1.00006pt}\text{\Large .}}\over {}}}{{{\hspace*{1.00006pt}\text{\large .}}\over {}}}{{{\hspace*{1.00006pt}\text{\large .}}\over {}}}_{\varepsilon }^{}_{\varepsilon }^{-1}\rightarrow \mathchoice{{{\... | {
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earthquake modeling | [
"Tomas Roubicek",
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c82cb69d12c976fe407b92a9fa962d8740af8635 | subsection | 77 | 80 | Convergence of Galerkin approximations | Instead
of repeating the whole argument, we limit ourselves in pointing out
the few differences.A first difference concerns the limit passage towards the
inclusion governing \mu , due to the presence of the constraints 0\le \zeta \le 1.
From () one has that \zeta
is valued in [0,1]. To facilitate the limit passage tow... | {
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earthquake modeling | [
"Tomas Roubicek",
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b5a3818380fbb0161ca3f6c2f3b28b4ecf67a0f2 | subsection | 78 | 80 | Convergence of Galerkin approximations | This change is however immaterial
with respect to the limit passage in the mechanical
part (REF a-e). Actually, some arguments are even
simplified, for we do not need to approximate the limit into the
finite-dimensional
subspaces as we did in (REF ).
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} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
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aba1f705f5d123d4db98425f5e07400ea3199454 | subsection | 79 | 80 | Conclusion | We have addressed a model used in geophysics for poroelastic damagable rocks
with plastic-like strain, which can accomodate the large displacement
occuring during long geological time scales. The model is anisothermal,
so e.g. effects of flash heating on tectonic faults during ongoing earthquakes
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02541e7251b153556cb37c063924f908edc186ff | abstract | 0 | 127 | Abstract | The top quark mass is measured using a template method in the $t\bar{t}\to$
lepton+jets channel (lepton is $e$ or $\mu$) using ATLAS data recorded in 2012
at the LHC. The data were taken at a proton-proton centre-of-mass energy of
$\sqrt{s}=8$ TeV and correspond to an integrated luminosity of 20.2 fb$^{-1}$.
The $t\bar... | {
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} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
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c7250b1b3e4285ff622e8ac8527a687012455d08 | subsection | 1 | 127 | Introduction | The mass of the top quark is an important parameter of the Standard
Model (SM).
Precise measurements of provide crucial information for global fits of
electroweak parameters , , which help to assess
the internal consistency of the SM and probe its extensions. In addition, the
value of affects the stability of the SM Hi... | {
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} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
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b6b77bee7d28787610c0c48e7718d30c383f5652 | subsection | 2 | 127 | Introduction | After a short description of the ATLAS
detector in , the data and simulation samples are
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, followed by the description of the reconstruction of
the three observables used in the template fit in .
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} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
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05eb65298a4d70af55c4429194a687a405a40ea8 | subsection | 3 | 127 | The ATLAS experiment | The ATLAS experiment at the LHC is a multipurpose particle
detector with a forward–backward symmetric cylindrical geometry and a near
4\pi coverage in the solid angle.ATLAS uses a right-handed coordinate system with its origin at the nominal
interaction point (IP) in the centre of the detector and the z-axis along the... | {
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} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
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cb3e12cdc624d585ef4fa6825a8097891c636328 | subsection | 4 | 127 | Data and simulation samples | The analysis is based on collision data recorded by the ATLAS detector in
2012 at a of =8 .
The integrated luminosity is with an uncertainty of
.
The modelling of top quark pair () and single-top-quark signal events, as
well as most background processes, relies on MC simulations.
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} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
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"hep-ex"
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f5ab35a6ef7414668e01dea7a83c1821fb7e512c | subsection | 5 | 127 | Data and simulation samples | If the was
smaller than 0.4, the parton shower prediction was taken, while for larger
values, the matrix element prediction was used.
The sample is normalized to the inclusive NNLO
calculation . Due to the large uncertainties in the
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using data-driven... | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
-0.008696675300598145,
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0.021528085693717003,
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0.022... |
c89c47ea8898fb730448f8d7ca14720fc861dfdd | subsection | 6 | 127 | Object reconstruction, background estimation and event preselection | The reconstructed objects resulting from the top quark pair decay are electron
and muon candidates, jets and missing transverse momentum ().
In the simulated events, corrections are applied to these objects based on
detailed data-to-simulation comparisons for many different processes, so as to
match their performance i... | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
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-0.025791188701987267,
... |
eb701cf000c1a3fc22e6c370b1692440cd465ddc | subsection | 7 | 127 | Object reconstruction | Electron candidates are required to have a transverse
energy of >25 and a pseudorapidity of the corresponding EM cluster
of < 2.47 with the transition region 1.37<<1.52
between the barrel and the endcap calorimeters excluded.
Muon candidates are required to have transverse momentum
>25 and \vert \eta \vert <2.5.
To... | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
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0.03102685697376728,
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-0.010240541771054268,
-0.025456368923187256... |
509cd0bb962af6f5b6035e4277e7123960add714 | subsection | 8 | 127 | Background estimation | The contribution of events falsely reconstructed as events due to the
presence of objects misidentified as leptons (fake leptons) and NP leptons
originating from HF decays, is estimated from data using the
matrix-method .
The technique employed uses \eta - and -dependent efficiencies for
NP/fake-leptons and prompt-lept... | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
0.01094198040664196,
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0.037114217877388,
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0.03299380838871002,
-0.0013935001334175467,
0.03995... |
9bedd7fea26934740eab7c16c0941c320c792052 | subsection | 9 | 127 | Event preselection | Triggering of events is based solely on the presence of a single electron or
muon, and no information from the hadronic final state is used.
A logical OR of two triggers is used for each of the and
channels. The triggers with the lower thresholds of 24 for
electrons or muons select isolated leptons. The triggers with... | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
0.018918445333838463,
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0.003457574639469385,
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0.04595351219177246,
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0.01772841438651085,
-0.0009578416356816888,
-... |
91489a4005fe37be3995e02c79a3ebcfc3053202 | subsection | 10 | 127 | Reconstruction of the three observables | As in , a full kinematic reconstruction of the event
is done with a likelihood fit using the
package , . The algorithm
relates the measured kinematics of the reconstructed objects to the
leading-order representation of the system decay using
\rightarrow \ell \nu \,.
In this procedure, the measured jets correspond to t... | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
0.010254325345158577,
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-0.04272635653614998,
0.0133672459051013,
0.007026959676295519,
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0.03839268162846565,
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0.03366226330399513,
-0.02067650482058525,
0.05584... |
2a38320c7b8eaa1194d90486667296104fd76131 | subsection | 11 | 127 | Reconstruction of the three observables | Finally, an observable called , designed to be sensitive to the , is
computed as the scalar sum of the transverse momenta of the two jets
divided by the scalar sum of the transverse momenta of the two jets associated
with the hadronic boson decay:& = \frac{p_\mathrm {T}^{b_{\rm had}} + p_\mathrm {T}^{b_{\rm lep}}}{p_\m... | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
-0.012646781280636787,
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0.017818385735154152,
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0.04027443006634712,
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0... |
8eddc323529828f4e3a3ba8e198a23773776537c | subsection | 12 | 127 | Reconstruction of the three observables | The resulting templates have simpler shapes, which are easier to model
analytically with fewer parameters.
The preselection with these additional requirements is referred to as the standard selection to distinguish it from the boosted decision tree (BDT)
optimization for the smallest total uncertainty in , discussed in... | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
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0.013168028555810452,
0.034... |
39f9d0c19f1c0ec11fd5188005232beaabfa7b00 | subsection | 13 | 127 | Multivariate analysis and BDT event selection | For the measurement of , the event selection is refined enriching the
fraction of events with correct assignments of objects to their
counterparts which should be better measured and therefore lead to
smaller uncertainties.
The optimization of the selection is based on the multivariate BDT algorithm
implemented in the... | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
0.0036387392319738865,
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0.03606700152158737,
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0.... |
e793b8cd42123599f992b11fc2883b8baf58ead0 | subsection | 14 | 127 | Multivariate analysis and BDT event selection | In addition, reasonable agreement is found for the distributions in the
statistically independent test and training samples.
The distributions in simulation and data in REF
agree within the experimental uncertainties.
The above findings justify the application of the BDT approach to the data.
[Figure: Various classes ... | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
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0.00771014578640461,
0.01756... |
300a28afcb9c618dc0bd3d6e5906574c23fb3a34 | subsection | 15 | 127 | Multivariate analysis and BDT event selection | The sharp edge observed at 30 in REF originates from
the different selection requirements for the boson transverse mass in
the electron+jets and muon+jets final states.
[Figure: Distributions for the events passing the BDT selection.The data are shown, together with the signal-plus-background predictionnormalized to t... | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
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0.061508551239967346,
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0.... |
4b939b61fac8c61001ed38fe058209f95bd26dc1 | subsection | 16 | 127 | Template fit | This analysis uses a three-dimensional template fit technique which determines
together with the jet energy scale factors and .
The aim of the multi-dimensional fit to the data is to measure and, at the
same time, to absorb the mean differences between the jet energy scales
observed in data and MC simulated events int... | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
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0.032313697040081024,
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0.03274088352918625,
-0.02941492386162281,
... |
e915252c57a8781413284a90ec0a715f9406ef63 | subsection | 17 | 127 | Template fit | The dependence of on (not shown) is much weaker than the dependence
on .
[Figure: Template parameterizations for signal events, composed of andsingle-top-quark production events.Figures (a)–(c) show the sensitivity of to , and ,figure (d) shows the sensitivity of to andfigures (e) and (f) show the sensitivity of to an... | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
-0.029430031776428223,
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0.0426880419254303,
0.004603687208145857,
0... |
460e8a8d2f1e4a3879b1fe24b6b5795ea088f4ed | subsection | 18 | 127 | Template fit | The results of pseudo-experiments for different input values of are
obtained from statistically independent samples, while the results for
different and are obtained from statistically correlated samples
as explained above.
For each fitted quantity and each variation of input parameters, the residual,
i.e. the differen... | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
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0.0651470497250557,
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-0.... |
fee2c7ad09118972868f23586d3ba74329db1a0e | subsection | 19 | 127 | Uncertainties affecting the | This section focuses on the treatment of uncertainty sources of a systematic
nature.
The same systematic uncertainty sources as in are
investigated.
If possible, the corresponding uncertainty in is evaluated by varying the
respective quantities by \pm 1 \sigma from their default values, constructing
the corresponding e... | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
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0... |
21f2bbfbd3f7c625da7dc727e37e10744c96a3dd | subsection | 20 | 127 | Statistics and method calibration | Uncertainties related to statistical effects and the method calibration are
discussed here. | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
0.008097080513834953,
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... |
38ea5fd1666497c18c5289fe3f221f66bf8262de | subsection | 21 | 127 | Statistical: | The quoted statistical uncertainty consists of three parts: a purely
statistical component in and the contributions stemming from the
simultaneous determination of and .
The purely statistical component in is obtained from a one-dimensional
template method exploiting only the observable, while fixing the values
of and ... | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
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0.023040691390633583,
-0.0033874395303428173,
-0... |
96326d2ad6177854bcfb18f4035e79033ca9325b | subsection | 22 | 127 | Method: | The residual difference between fitted and generated when analysing a
template from a MC sample reflects the potential bias of the method.
Consequently, the largest observed fitted residual and the largest
observed statistical uncertainty in this quantity, in any of the five signal
samples with different assumed values... | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
-0.015474912710487843,
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0.039... |
cd75d7b600cb4f3f5ee565be85b02e794b722fd2 | subsection | 23 | 127 | Modelling of signal processes | The modelling of events incorporates a number of processes that have
to be accurately described, resulting in systematic effects, ranging from the
production to the hadronization of the showered objects.Thanks to the restrictive event-selection requirements, the contribution of
non- processes, comprising the single-to... | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
-0.04893593117594719,
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... |
bad52a4245a1a68c4948843a4cef5de4c5cc6477 | subsection | 24 | 127 | Signal Monte Carlo generator: | The full observed difference in fitted between the event samples produced
with the and , programs is quoted
as a systematic uncertainty.
For the renormalization and factorization scales the sample uses
the function given in , while the sample uses
=\sqrt{^2 + 0.5 (p_{{\rm T},t}^2 + p_{{\rm T}, \bar{t}}^2)}.
Both sampl... | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
0.03020063415169716,
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0.0012561342446133494,
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... |
6778525a901730fd3d1c2508b0c1fea0e6dfbc6a | subsection | 25 | 127 | Hadronization: | To cover the choice of parton shower and hadronization models, samples produced
with the program are showered with either the program
using the P2011C tune or the and programs using the ATLAS
tune.
This includes different approaches in shower modelling, such as using a
-ordered parton showering in the program or angul... | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
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... |
9d177afa8e213072069afc19b5b0352eb01c6fbf | subsection | 26 | 127 | Initial- and final-state QCD radiation (ISR/FSR): | ISR/FSR leads to a higher jet multiplicity and different jet energies than the
hard process, which affects the distributions of the three observables.
The uncertainties due to ISR/FSR modelling are estimated with samples generated
with the program interfaced to the program for which
the parameters of the generation are... | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
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e45450ec1c6d519f5bb8be093d5abdfdf2b86e1c | subsection | 27 | 127 | Underlying event: | To reduce statistical fluctuations in the evaluation of this systematic
uncertainty, the difference in underlying-event modelling is assessed by
comparing a pair of samples based on the same partonic events
generated with the CT10 PDFs.
A sample with the P2012 tune is compared with a sample with the
P2012 tune , with ... | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
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5d0b3a6dd9a8f8bada34d8e2eb120ebbaf41bb34 | subsection | 28 | 127 | Colour reconnection: | This systematic uncertainty is estimated using a pair of samples with the same
partonic events as for the underlying-event uncertainty evaluation but with the
P2012 tune and the P2012 tune for parton showering and
hadronization.
The full observed difference is assigned as a systematic uncertainty. | {
"cite_spans": []
} | 10.1140/epjc/s10052-019-6757-9 | 1810.01772 | Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets
channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous
results | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
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0.00... |
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