Kinematic Approximations

Purpose

The strain calculator ComputeLagrangianStrain converts the inverse incremental deformation gradient $var element = document.getElementById("moose-equation-b77d83ae-5cf1-4963-b58d-5c4da637b6a7");katex.render(" f^{-1}_{ij} = F^{(n)}_{iK} \\, F^{-1}_{Kj}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ into the spatial-velocity-gradient increment $var element = document.getElementById("moose-equation-b4f8b892-3f1a-45a9-adb9-163b8681239b");katex.render("\\Delta l_{ij}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ from which it derives the incremental strain $var element = document.getElementById("moose-equation-ac3b8b89-981a-4ede-b122-fd35ed2a7671");katex.render("\\Delta d_{ij} = \\tfrac{1}{2}(\\Delta l_{ij} + \\Delta l_{ji})", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ and the incremental vorticity $var element = document.getElementById("moose-equation-70053500-a8fe-4e96-9a2a-201f9c651042");katex.render("\\Delta w_{ij} = \\tfrac{1}{2}(\\Delta l_{ij} - \\Delta l_{ji})", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ . Several closed-form approximations to this conversion are available. The kinematic_approximation parameter on the strain calculator selects which approximation is used. Small kinematics (large_kinematics = false) is always linear; this parameter only affects large kinematics.

The choice of approximation has two consumers:

The strain increment $var element = document.getElementById("moose-equation-117a62fc-2001-4d36-8d5d-bf744b994b6f");katex.render("\\Delta d", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ propagated through the strain measures (and, after eigenstrain subtraction, fed to the constitutive model).
The vorticity increment $var element = document.getElementById("moose-equation-52c3ed50-f146-4620-af52-0c672ebdcd25");katex.render("\\Delta w", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ consumed by the Jaumann, Green-Naghdi, and Rashid objective rates. All four objective rate options pick up whichever $var element = document.getElementById("moose-equation-a0d51c81-de15-4563-ae92-c1dcf18f936e");katex.render("\\Delta w", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ the strain calculator produces, so any pairing of approximation and rate is consistent.

The Four Options

Let $var element = document.getElementById("moose-equation-d9df2d2e-78a1-4832-9404-637b82e79f1a");katex.render("X_{ij} = \\delta_{ij} - f^{-1}_{ij}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ throughout.

`linear` (default)

$var element = document.getElementById("moose-equation-b934b054-212c-4683-a01f-26cc9249c788");katex.render(" \\Delta l = X = I - f^{-1}, \\quad \\Delta d = \\mathrm{sym}(X), \\quad \\Delta w = \\mathrm{skew}(X).", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

This is the first-order Taylor expansion of $var element = document.getElementById("moose-equation-2e734173-652b-4f2a-af14-152de9312ef3");katex.render("-\\log f^{-1}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ in $var element = document.getElementById("moose-equation-d1e21af8-8201-4f33-b8f6-64d33ffc46cd");katex.render("X", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , and is the default.

`quadratic`

$var element = document.getElementById("moose-equation-74b5231c-6b22-41a8-85ec-e71377669308");katex.render(" \\Delta l = X + \\tfrac{1}{2} X^2.", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

One additional term in the Taylor expansion of $var element = document.getElementById("moose-equation-55fa78cd-90f8-443b-b1dc-604690378df2");katex.render("-\\log f^{-1}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ . Provided for completeness; there is no specific compatibility target.

`rashid_approximate`

Rashid's symmetric/skew split Rashid (1993). With $var element = document.getElementById("moose-equation-a4ac2e54-fdde-477f-8d10-de8fd54ac314");katex.render(" A_{ij} = X_{ik} X_{jk} - X_{ij} - X_{ji}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ the strain increment is $var element = document.getElementById("moose-equation-f0a5fec3-de84-4393-8640-b15c46b4ffc5");katex.render(" \\Delta d = -\\tfrac{1}{2} A + \\tfrac{1}{4} A^2.", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ The vorticity is built from the axial vector $var element = document.getElementById("moose-equation-9e8a8208-7cf3-40bd-beb9-341d887614a9");katex.render("\\alpha_k = \\epsilon_{kab} f^{-1}_{ab}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ : $var element = document.getElementById("moose-equation-51056d82-f5ec-4590-b96c-465449759ec0");katex.render(" \\Delta w_{ij} = -\\frac{\\theta}{2 \\sqrt{Q}}\\, \\epsilon_{ijk}\\, \\alpha_k, \\quad Q = \\tfrac{1}{4}\\, \\alpha_k \\alpha_k, \\quad \\sin \\theta = \\sqrt{Q}.", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ This option reproduces the strain and vorticity increments produced by ComputeFiniteStrain when its decomposition_method = TaylorExpansion.

`rashid_eigen`

Polar-decompose $var element = document.getElementById("moose-equation-f561a266-e569-4653-b318-cf7c9e91e2d4");katex.render("f^{-1} = r'\\, u'", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ and take the matrix logarithm of the two factors: $var element = document.getElementById("moose-equation-815fc53a-5136-419f-8060-88db1d5ac466");katex.render(" \\Delta d_{ij} = -\\log u', \\quad \\Delta w_{ij} = -\\log r'.", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ This is the "exact" $var element = document.getElementById("moose-equation-ad52cc67-2145-490c-b674-2b1ac270250c");katex.render("-\\log f^{-1}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , accurate to machine precision for any deformation. It reproduces the strain and vorticity increments produced by ComputeFiniteStrain when its decomposition_method = EigenSolution.

References

MM Rashid. Incremental kinematics for finite element applications. International Journal for Numerical Methods in Engineering, 36(23):3937–3956, 1993.

BibTeX

@article{rashid1993incremental,
    author = "Rashid, MM",
    title = "Incremental kinematics for finite element applications",
    journal = "International Journal for Numerical Methods in Engineering",
    volume = "36",
    number = "23",
    pages = "3937--3956",
    year = "1993",
    publisher = "Wiley Online Library"
}

RIS

TY  - JOUR
AU  - Rashid, MM
TI  - Incremental kinematics for finite element applications
JO  - International Journal for Numerical Methods in Engineering
PY  - 1993
VL  - 36
IS  - 23
PB  - Wiley Online Library
SP  - 3937
EP  - 3956
ER  -

Plain Text

MM Rashid. Incremental kinematics for finite element applications. International Journal for Numerical Methods in Engineering, 36(23):3937–3956, 1993.

Purpose
The Four Options
References