Lines Matching refs:bm

16 …am below, where $\bm \sigma$ and $\bm \epsilon$ are stress and strain, respectively, in the small …
20   {\overbrace{\bm S(\bm E)}^{\text{Finite Strain Hyperelastic}}}
22   {\overbrace{\bm S = \mathsf C \bm E}^{\text{St. Venant-Kirchoff}}} \\
23 …@V{\text{geometric}}V{\begin{smallmatrix}\bm E \to \bm \epsilon \\ \bm S \to \bm \sigma \end{small…
24 …@V{\begin{smallmatrix}\bm E \to \bm \epsilon \\ \bm S \to \bm \sigma \end{smallmatrix}}V{\text{geo…
25   {\underbrace{\bm \sigma(\bm \epsilon)}_\text{Small Strain Hyperelastic}}
27   {\underbrace{\bm \sigma = \mathsf C \bm \epsilon}_\text{Linear Elastic}}
47 \nabla \cdot \bm{\sigma} + \bm{g} = \bm{0}
50 where $\bm{\sigma}$ and $\bm{g}$ are stress and forcing functions, respectively.
51 …}`lin-elas` by a test function $\bm v$ and integrate the divergence term by parts to arrive at the…
54 \int_{\Omega}{ \nabla \bm{v} \tcolon \bm{\sigma}} \, dV
55 - \int_{\partial \Omega}{\bm{v} \cdot \left(\bm{\sigma} \cdot \hat{\bm{n}}\right)} \, dS
56 - \int_{\Omega}{\bm{v} \cdot \bm{g}} \, dV
57 = 0, \quad \forall \bm v \in \mathcal V,
60 where $\bm{\sigma} \cdot \hat{\bm{n}}|_{\partial \Omega}$ is replaced by an applied force/traction …
65 In their most general form, constitutive models define $\bm \sigma$ in terms of state variables.
66 …ariables are constituted by the vector displacement field $\bm u$, and its gradient $\nabla \bm u$.
70 \bm{\epsilon} = \dfrac{1}{2}\left(\nabla \bm{u} + \nabla \bm{u}^T \right).
73 This constitutive model $\bm \sigma(\bm \epsilon)$ is a linear tensor-valued function of a tensor-v…
78 Q \bm \sigma(\bm \epsilon) Q^T = \bm \sigma(Q \bm \epsilon Q^T),
81 which means that we can change our reference frame before or after computing $\bm \sigma$, and get …
82 Constitutive relations in which $\bm \sigma$ is uniquely determined by $\bm \epsilon$ while satisfy…
83 Here, we define a strain energy density functional $\Phi(\bm \epsilon) \in \mathbb R$ and obtain th…
86 \bm \sigma(\bm \epsilon) = \frac{\partial \Phi}{\partial \bm \epsilon}.
90 The strain energy density functional cannot be an arbitrary function $\Phi(\bm \epsilon)$; it can o…
93 \gamma(\bm \epsilon) = \gamma(Q \bm \epsilon Q^T)
102 \bm{\Phi} = \frac{\lambda}{2} (\operatorname{trace} \bm{\epsilon})^2 + \mu \bm{\epsilon} : \bm{\eps…
108 \bm\sigma = \lambda (\operatorname{trace} \bm\epsilon) \bm I_3 + 2 \mu \bm\epsilon,
111 where $\bm I_3$ is the $3 \times 3$ identity matrix, the colon represents a double contraction (ove…
120 \bm{\sigma} = \mathsf{C} \!:\! \bm{\epsilon}.
123 For notational convenience, we express the symmetric second order tensors $\bm \sigma$ and $\bm \ep…
147 \bm{\Phi} = \lambda (1 + \operatorname{trace} \bm{\epsilon}) (\log(1 + \operatorname{trace} \bm\eps…
153 \bm{\sigma} = \lambda \log(1 + \operatorname{trace} \bm\epsilon) \bm{I}_3 + 2\mu \bm{\epsilon}
156 where $\bm{\epsilon}$ is defined as in {eq}`small-strain`.
164 \diff \bm{\sigma} = \dfrac{\partial \bm{\sigma}}{\partial \bm{\epsilon}} \tcolon \diff \bm{\epsilon}
170 \diff \bm{\epsilon} = \dfrac{1}{2}\left( \nabla \diff \bm{u} + \nabla \diff \bm{u}^T \right)
176 \diff \nabla \bm{u} = \nabla \diff \bm{u} .
182 \diff \bm{\sigma}  = \bar{\lambda} \cdot \operatorname{trace} \diff \bm{\epsilon} \cdot \bm{I}_3 + …
227 In this formulation, we solve for displacement $\bm u(\bm X)$ in the reference frame $\bm X$.
234 - \nabla_X \cdot \bm{P} - \rho_0 \bm{g} = \bm{0}
238 $\bm{P}$ and $\bm{g}$ are the *first Piola-Kirchhoff stress* tensor and the prescribed forcing func…
240 The tensor $\bm P$ is not symmetric, living in the current configuration on the left and the initia…
242 $\bm{P}$ can be decomposed as
245 \bm{P} = \bm{F} \, \bm{S},
248 …bm S$ is the *second Piola-Kirchhoff stress* tensor, a symmetric tensor defined entirely in the in…
249 Different constitutive models can define $\bm S$.
256 \bm C = \bm F^T \bm F
262 \bm E = \frac 1 2 (\bm C - \bm I_3) = \frac 1 2 \Big( \nabla_X \bm u + (\nabla_X \bm u)^T + (\nabla…
265 the latter of which converges to the linear strain tensor $\bm \epsilon$ in the small-deformation l…
266 …bm S$ as a function of $\bm E$, similar to the linear case, shown in equation  {eq}`linear-stress-…
269 We will assume without loss of generality that $\bm E$ is diagonal and take its set of eigenvalues …
270 …y alternate choices, such as $\operatorname{trace}(\bm E), \operatorname{trace}(\bm E^2), \lvert \…
271 …mon in the literature for invariants to be taken from $\bm C = \bm I_3 + 2 \bm E$ instead of $\bm …
277 \Phi(\bm E) &= \frac{\lambda}{2}(\log J)^2 - \mu \log J + \frac \mu 2 (\operatorname{trace} \bm C -…
278   &= \frac{\lambda}{2}(\log J)^2 - \mu \log J + \mu \operatorname{trace} \bm E,
282 where $J = \lvert \bm F \rvert = \sqrt{\lvert \bm C \rvert}$ is the determinant of deformation (i.e…
287 …partial \bm E} = \frac{\partial \sqrt{\lvert \bm C \rvert}}{\partial \bm E} = \lvert \bm C \rvert^…
290 where the factor of $\frac 1 2$ has been absorbed due to $\bm C = \bm I_3 + 2 \bm E.$
294 \bm S = \lambda \log J \bm C^{-1} + \mu (\bm I_3 - \bm C^{-1}).
301 \bm S = \lambda \log J \bm C^{-1} + 2 \mu \bm C^{-1} \bm E,
304 which is more numerically stable for small $\bm E$, and thus preferred for computation.
305 Note that the product $\bm C^{-1} \bm E$ is also symmetric, and that $\bm E$ should be computed usi…
308 To sketch this idea, suppose we have the $2\times 2$ non-symmetric matrix $\bm{F} = \left( \begin{s…
316 For example, if $u_{i,j} \sim 10^{-8}$, then naive computation of $\bm I_3 - \bm C^{-1}$ and $\log …
321 …\bm C = 3 + 2\trace \bm E$ and $J$, Mooney-Rivlin models depend on the additional invariant, $\mat…
332 \bm S &=  \lambda \log J \bm{C}^{-1} - (\mu_1 + 2\mu_2) \bm{C}^{-1} + \mu_1\bm I_3 + \mu_2(\mathbb{…
333 &= (\lambda \log J - \mu_1 - 2\mu_2) \bm C^{-1} + (\mu_1 + \mu_2 \mathbb I_1) \bm I_3 - \mu_2 \bm C,
341 …\bm E} &= 2 \bm I_3, & \frac{\partial \mathbb{I_2}}{\partial \bm E} &= 2 \mathbb I_1 \bm I_3 - 2 \…
391 …-hookean-stress` around $\bm E = 0$, for which $\bm C = \bm I_3 + 2 \bm E \to \bm I_3$ and $J \to …
394 \bm S = \lambda (\trace \bm E) \bm I_3 + 2 \mu \bm E,
401 …nearities, $\bm E \to \bm \epsilon$ and $\bm C \to \bm I_3$, while retaining the nonlinear depende…
406 We multiply {eq}`sblFinS` by a test function $\bm v$ and integrate by parts to obtain the weak form…
407 find $\bm u \in \mathcal V \subset H^1(\Omega_0)$ such that
410 \int_{\Omega_0}{\nabla_X \bm{v} \tcolon \bm{P}} \, dV
411  - \int_{\Omega_0}{\bm{v} \cdot \rho_0 \bm{g}} \, dV
412  - \int_{\partial \Omega_0}{\bm{v} \cdot (\bm{P} \cdot \hat{\bm{N}})} \, dS
413  = 0, \quad \forall \bm v \in \mathcal V,
416 where $\bm{P} \cdot \hat{\bm{N}}|_{\partial\Omega}$ is replaced by any prescribed force/traction bo…
417 … in defining $\bm S(\bm E)$, as well as geometric nonlinearities through $\bm P = \bm F\, \bm S$, …
426 …bm P = \frac{\partial \bm P}{\partial \bm F} \!:\! \diff \bm F = \diff \bm F\, \bm S + \bm F \unde…
432 \diff \bm E = \frac{\partial \bm E}{\partial \bm F} \!:\! \diff \bm F = \frac 1 2 \Big( \diff \bm F…
435 and $\diff\bm F = \nabla_X\diff\bm u$.
436 The quantity ${\partial \bm S} / {\partial \bm E}$ is known as the incremental elasticity tensor, a…
437 We now evaluate $\diff \bm S$ for the Neo-Hookean model {eq}`neo-hookean-stress`,
440 \diff\bm S = \frac{\partial \bm S}{\partial \bm E} \!:\! \diff \bm E
441 = \lambda (\bm C^{-1} \!:\! \diff\bm E) \bm C^{-1}
442   + 2 (\mu - \lambda \log J) \bm C^{-1} \diff\bm E \, \bm C^{-1},
448 \diff \bm C^{-1} = \frac{\partial \bm C^{-1}}{\partial \bm E} \!:\! \diff\bm E = -2 \bm C^{-1} \dif…
452 In the small-strain limit, $\bm C \to \bm I_3$ and $\log J \to 0$, thereby reducing {eq}`eq-neo-hoo…
460 \diff\bm S &= \lambda (\bm C^{-1} \tcolon \diff\bm E) \bm C^{-1} \\
461 &\quad + 2(\mu_1 + 2\mu_2 - \lambda \log J) \bm C^{-1} \diff\bm E \bm C^{-1} \\
462 &\quad + 2 \mu_2 \Big[ \trace (\diff\bm E) \bm I_3 - \diff\bm E\Big] .
475 \diff \bm P &= \diff \bm F\, \bm S
476 …+ \lambda (\bm C^{-1} : \diff \bm E) \bm F^{-T} + 2(\mu - \lambda \log J) \bm F^{-T} \diff\bm E \,…
477 &= \diff \bm F\, \bm S
478 …+ \lambda (\bm F^{-T} : \diff \bm F) \bm F^{-T} + (\mu - \lambda \log J) \bm F^{-T} (\bm F^T \diff…
479 &= \diff \bm F\, \bm S
480 …+ \lambda (\bm F^{-T} : \diff \bm F) \bm F^{-T} + (\mu - \lambda \log J) \Big( \diff \bm F\, \bm C…
484 where we have exploited $\bm F \bm C^{-1} = \bm F^{-T}$ and
487 …bm C^{-1} \!:\! \diff \bm E = \bm C_{IJ}^{-1} \diff \bm E_{IJ} &= \frac 1 2 \bm F_{Ik}^{-1} \bm F_…
490 …ive, requiring access to (non-symmetric) $\bm F^{-1}$ in addition to (symmetric) $\bm C^{-1} = \bm…
493 :::{dropdown} $\diff\bm S$ in index notation
498 \diff\bm S_{IJ} &= \frac{\partial \bm S_{IJ}}{\partial \bm E_{KL}} \diff \bm E_{KL} \\
499 …&= \lambda (\bm C^{-1}_{KL} \diff\bm E_{KL}) \bm C^{-1}_{IJ} + 2 (\mu - \lambda \log J) \bm C^{-1}…
500 …Big( \lambda \bm C^{-1}_{IJ} \bm C^{-1}_{KL} + 2 (\mu - \lambda \log J) \bm C^{-1}_{IK} \bm C^{-1}…
506 …point $\bm F$ and solution increment $\diff \bm F = \nabla_X (\diff \bm u)$ (which we are solving …
508 1. recover $\bm C^{-1}$ and $\log J$ (either stored at quadrature points or recomputed),
509 …d line of {eq}`eq-neo-hookean-incremental-stress-index` to compute $\diff \bm S$ while avoiding co…
510 3. conclude by {eq}`eq-diff-P`, where $\bm S$ is either stored or recomputed from its definition ex…
513 …m-initial` may be written as a weak form for linear operators: find $\diff\bm u \in \mathcal V_0$ …
516 \int_{\Omega_0} \nabla_X \bm v \!:\! \diff\bm P dV = \text{rhs}, \quad \forall \bm v \in \mathcal V…
519 where $\diff \bm P$ is defined by {eq}`eq-diff-P` and {eq}`eq-neo-hookean-incremental-stress`, and …
522 The decision of whether to recompute or store functions of the current state $\bm F$ depends on a r…
526 Along with 6 entries for $\bm S$, this totals 27 entries of overhead compared to computing everythi…
527 This compares with 13 entries of overhead for direct storage of $\{ \bm S, \bm C^{-1}, \log J \}$, …