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2015 | OriginalPaper | Chapter

4. Applications in continuum mechanics and physics of solids

Authors : Alexander Mielke, Tomàš Roubíček

Published in: Rate-Independent Systems

Publisher: Springer New York

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Abstract

The theory of rate-independent processes has a large variety of applications in continuum mechanics of solids. Rate-independent effects typically can occur inside the bulk and at the surface or along interfaces. These effects may be unidirectional, as, for example, in damage, or bidirectional. In case of a deformable continuum, one can consider the general concept of large strains or confine oneself to small strains. There might be rate-independent processes on lower-dimensional objects, typically surfaces of codimension 1 or lines (as dislocations) of dimension 1. See Table 4.1 on p. 236 for examples that will be considered in this chapter. Of course, various processes can combine with each other.

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previous chapter Rate-independent systems in Banach spaces

next chapter Beyond rate-independence

In greater detail, temperature variations on the “micro scale” are usually ultimately to be expected, but we consider them phenomenologically incorporated into “macroscopic” dissipation energies $\mathcal{D}$.

In some cases, however, there is an alternative understanding of the situation as multiple natural configurations; for this concept, see [499].

Later, on p. 267, we will, however, play with the higher-gradient contributions, too.

Minimizers or other critical points do not need a priori to be regular enough and do not need to satisfy the Euler–Lagrange equation of the type (4.1.2); cf. [39, 48]. This still emphasizes the advantages of using the energetic method, which does not rely on Euler–Lagrange equations.

The definition (4.1.3) was introduced by Ball and Murat [49], called $\mathrm{W}^{1,p}$-quasiconvexity and shown to be equivalent to the original $\mathrm{W}^{1,\infty }$-quasiconvexity introduced by Morrey [438], provided $-C \leq W(F) \leq C(1 + \vert F\vert ^{p})$ for some $C \in \mathbb{R}^{+}$; cf. [49, Proposition 2.4(i)].

Note that $\det \,F =\det \, F^{\mathsf{T}} = \sqrt{\det (F^{\mathsf{T} } F)} = \sqrt{\det \,C}$ actually depends only on C; hence (4.1.5) leads indeed to a frame-indifferent potential.

Elasticity of single crystals, which are anisotropic, involves more independent elastic moduli, depending on the symmetry system, namely 3 for cubic, 6 for tetragonal, 9 for orthorhombic, 13 for monoclinic, and 21 triclinic; cf., e.g., [478].

Instead of the Cauchy–Green stretch tensor C, the so-called right stretch tensor U: = C ^1∕2 is used and $R(F,{ \buildrel . \over F}) = R_{0}(U,{ \buildrel . \over U})$ for some function R ₀. Yet, ${ \buildrel .\over C} = U{ \buildrel . \over U} +{ \buildrel . \over U}U$, so that $R_{0}(C,{ \buildrel . \over C}) = R_{0}(U^{2},U{ \buildrel . \over U} +{ \buildrel . \over U}U)$, which reveals the form of R ₀.

Actually, $\nabla y_{\mathrm{D}}(t,y)$ is to be understand as ∂ _y y _D(t, y).

See also [378] for a similar approach using currents. In fact, [128] uses strict inequality $\det (\nabla y) > 0$ a.e. on $\varOmega$ in (4.1.20), which does not make the set of such y closed in $\mathrm{W}^{1,p}(\varOmega; \mathbb{R}^{d})$ and which is suitable only in relation to minimizing some integral functionals whose integrand $F\mapsto W(F)$ explodes to $\infty $ if $\det F \searrow 0$; cf. also Remark 4.2.12 below.

The philosophy behind arriving at (4.1.28) from (4.1.15)–(4.1.16) is to consider $\tilde{y}(x) = x +\tilde{ u}(x)$ and y _D(x) = x + u _D(x), to approximate the Green–Lagrange strain $E = \frac{1} {2}(C - \mathbb{I})$ asymptotically when writing it in terms of these new variables, i.e.,

$$\displaystyle\begin{array}{rcl} E& =& \frac{1} {2}(\mathbb{I} + \nabla \tilde{u})^{\mathsf{T}}(\mathbb{I} + \nabla u_{\mathrm{ D}}^{})^{\mathsf{T}}(\mathbb{I} + \nabla u_{\mathrm{ D}}^{})(\mathbb{I} + \nabla \tilde{u}) -\frac{1} {2}\mathbb{I} {}\\ & =& \frac{1} {2}(\nabla \tilde{u})^{\mathsf{T}} + \frac{1} {2}(\nabla u_{\mathrm{D}}^{})^{\mathsf{T}} + \frac{1} {2}\nabla u_{\mathrm{D}}^{} + \frac{1} {2}\nabla \tilde{u} + \text{h.o.t.} = e(\tilde{u} + u_{\mathrm{D}}^{}) + \text{h.o.t.}, {}\\ \end{array}$$

where “h.o.t.” abbreviates higher-order terms that are eventually neglected in (4.1.28).

This result dates back to 1906 in [313] for p = 2; see also [269] and the survey [273]. For a generalization for $\mathrm{L}^{p}$-spaces, p > 1, see [219, 445]. A generalization of (4.1.30) for p = 1 does not hold, however; cf. a counterexample in [462].

Cf. [374] for more details.

Cf. [442, Corollary 3.4] for more details.

An early idea in this direction is due to E. Aifantis [6] and O.W. Dillon, and J. Kratochvíl [161]; it has since been used widely in the engineering literature; cf. also [7, 62, 288, 290, 603]. The energetic-solution approach to gradient plasticity has been scrutinized in [243, 318, 374].

The formula (4.2.17a) uses the matrix identity $AB:C =\mathop{ \mathrm{tr}}(ABC^{\mathsf{T}}) =\mathop{ \mathrm{tr}}(A(CB^{\mathsf{T}})^{\mathsf{T}}) = A:(CB^{\mathsf{T}})$.

The formula (4.2.22a) uses the matrix identity $AB^{\mathsf{T}}:C =\mathop{ \mathrm{tr}}(AB^{\mathsf{T}}C^{\mathsf{T}}) =\mathop{ \mathrm{tr}}(A(CB)^{\mathsf{T}}) = A:(CB)$.

For many technical detials see [419].

In particular, some simplifications have been used for these simulations such as Green–Naghdi’s [232] additive decomposition instead of the multiplicative decomposition (4.2.4) and the St. Venant–Kirchhoff material (4.1.6) instead of a polyconvex material.

As pointed out by Bhattacharya et al. [71], “much remains unknown concerning the nucleation and evolution of microstructure, and the resultant hysteresis.”

Such a regularization $\mathcal{S}$ in the dissipation has been considered in [170, Remark 4.3] and [612, Sect.5] in the context of ferromagnetism, and [473, Formulas (2)–(3)] and [474, Formula (4) with (8)] for the case of damage. See also an overview and in particular [288, Formulas 26.12-13] in the context of plasticity.

This means here that $\mathcal{E}_{\varepsilon,\kappa }(0,y_{0},\lambda _{0}) \leq \mathcal{E}_{\varepsilon,\kappa }(0,y,\lambda ) + \mathcal{R}(\lambda -\lambda _{0})$ for all $(y,\lambda )\in [\mathrm{H}^{2} \cap \mathrm{W}^{1,p}](\varOmega; \mathbb{R}^{d})\times \mathrm{L}^{2}(\varOmega; \mathbb{R}^{m})$.

More precisely, the shorthand expression $\nabla ^{2}y_{\mathrm{D}}(t,y)[F,F] + \nabla y_{\mathrm{D}}(t,y)G$ in (4.2.41) means a tensor $[\sum _{n=1}^{d}\sum _{m=1}^{d}\frac{\partial ^{2}[y_{\mathrm{D}}]_{k}(t,y)} {\partial x_{m}\partial x_{n}} F_{mi}F_{nj} +\sum _{ n=1}^{d}\frac{\partial [y_{\mathrm{D}}]_{k}(t,y)} {\partial x_{n}} G_{nij}]$ indexed by (k, i, j).

We refer, e.g., to [473, Formula (2) with (9)] or [474, Formula (2) with (16)] for damage models and to [288, Formulas 26.22-23] in the context of plasticity.

Note that the pair $(y_{0},\lambda _{0})$ is then stable for every $\kappa \leq \infty $.

The idea of sharp interfaces has been considered in [372, 373] in terms of $\lambda$, involving a BV-norm of $\lambda$ into $\mathcal{E}_{\varepsilon,\kappa }$.

Such nonlocal stored energy was advocated, e.g., in [502, 510, 511] as a certain limit from the Ericksen–Timoshenko model.

We can see this from the argument that terms like $\frac{1} {2}\vert \nabla y\vert ^{2}$ and $\frac{\varepsilon }{2}\vert \nabla ^{2}y\vert ^{2}$ in (4.2.43) and (4.2.44) are physically dimensionless (before being multiplied by elastic moduli to get specific energy in J/m³), so that $\varepsilon$ has, in fact, the physical dimension m² and thus scales together with scaling the length unit.

In [416], a similar assertion was proved for the special case $\mathcal{S}$ = identity and with regularizing gradient term $\varepsilon \vert \nabla \lambda \vert ^{2}$ in the stored energy under stronger data qualification, e.g., a “nonbuckling” condition for a given specimen and given loading regime, later weakened in [328].

In particular, the Moore–Smith convergence (2.1.23) in Remark 2.1.8 uses, in fact, the nonmetrizable convex compactification of a bounded set in $\mathrm{B}([0,T]; \mathrm{L}^{p}(\varOmega; \mathbb{R}^{d\times d}))$, where deformation gradients of the approximate solutions must live in the sense of $Y _{H,\rho }^{p}(\varOmega; \mathbb{R}^{d\times d})^{[0,T]}$ with $\rho < \infty $ large enough; cf. (C.2.10).

Utilization of Young measures for numerical calculations was first implemented in [456] in a one-dimensional case. Use of (iterated) laminates has been exploited for some evolution of relaxed problems in SMA modeling in [56, 58, 65, 328, 540, 576], or for the static case, also [23, 31, 326, 327, 366] and [520, Chap. 6]. An even more sophisticated and realistic dissipation counting, also for rotation of laminates and not only for volume-fraction changes, was implemented for second-order laminates in small strains in [56].

In fact, in actual computational simulations in particular problems, if ℓ is chosen too small, an extremely slow convergence can be expected, because the oscillations of $\nabla y$ must be realized through a very fine triangulation of $\varOmega$ rather than by the laminated Young measure itself. For ℓ = 0, we are, in fact, back in the position of Section 4.2.2.1.

Shape-memory alloys mostly exhibit very regular distributions of their components, as drawn schematically in Figure 4.5(a) and are thus called intermetallics. Moreover, atoms are bonded by chemical bonds, and single crystals are thus giant molecules. Such shape-memory materials are thus very particular alloys, and not only from the purely mechanical viewpoint.

It concerns atomistic vs. continuum-mechanical models, micro- vs. meso- vs. macro-level models, small vs. large strains, static vs. rate-independent quasistatic vs. fully dynamical, isothermal vs. anisothermal, more vs. less phenomenological, single- vs. poly-crystals, etc.

Namely, $V (F):= -k_{\mathrm{B}}\theta \,\mathrm{ln}\big(\sum _{n=1}^{L}\mathrm{e}^{-V _{n}(F)/(k_{\mathrm{B}}\theta )}\big)$, where $k_{\mathrm{B}}$ is the Boltzmann constant (related per unit volume), and $\theta$ is temperature (considered here fixed). Comparing to (4.2.70), this formula does not yield the desired elastic moduli and does not keep the wells exactly on the prescribed orbits $\mathrm{SO}(3)U_{n}$, but the deviation is negligible. Cf. [337, 416, 525, 539].

See Hormann and Zimmer [274] for a construction exactly keeping the prescribed wells and elastic moduli.

In fact, V from (4.2.69)–(4.2.70) has 2N+2 minima at the orbits $\mathrm{O}(d)Q_{n}^{-1}U_{n}$, n = 1, …, N, where $\mathrm{O}(3):=\{\, Q \in \mathbb{R}^{3\times 3}\;\vert \ Q^{\mathsf{T}}Q = \mathbb{I}\,\}$ is the orthogonal group having two connected orbits. The “parasite” orbits with $\det Q < 0$ are not physical and are (usually) not seen during numerical simulations.

Considering the duality $\langle \sigma,e\rangle:=\sigma: e$ that determines $\boldsymbol{\delta }_{K}^{{\ast}}$ as the conjugate to $\boldsymbol{\delta }_{K}^{}$ with the physical dimensions Jm⁻³ and 1, the set K is in the space of dimension Jm⁻³=Nm⁻²=Pa.

Following [522, Formula (33)], this was developed and adopted in [25, 26, 337, 416, 479, 497, 524, 525, 539] and independently also in physics; see [277, 590, 614].

This phenomenology can still reflect the philosophy that in any case, the stored energy itself influences the dissipation. The common philosophy is that if the orbits $\mathrm{SO}(3)U_{n_{1}}$ and $\mathrm{SO}(3)U_{n_{2}}$ are rank-1 connected, then the dissipation within the transformation between these (phase) variants is small, or rather zero; otherwise, it is related to metastability and a stress that the material must inevitably withstand to move out of the bottoms of the wells during the phase transformation. The philosophy that the multiwell landscape of the stored energy is the only source of hysteretic response was advocated essentially in [1, 44, 45, 285, 562, 602, 605].

In [524], dealing essentially with d = 1 (see also [416] for d ≥ 2), a regularization of $\mathcal{E}_{\mathrm{rlx},\kappa }(t,y,\nu,\lambda )$ of type $\int _{\varOmega }V \bullet \nu - f(t) \cdot y +\delta \vert \nabla \lambda \vert ^{2}\,\mathrm{d}x$ with $\delta > 0$ instead of $\int _{\varOmega }V \bullet \nu - f(t) \cdot y\,\mathrm{d}x$ in (4.2.60a) was proposed, justified as a certain limit of a so-called Ericksen–Timoshenko beam model as in Ren, Rogers, and Truskinovsky [502, 511].

To show that $\mathcal{D} = \mathcal{D}_{\mathcal{R}}$ instead of (3.2.9), we use that $\mathcal{D}_{\mathcal{R}}(z_{1},z_{2}) = \mathcal{R}(z_{2}-z_{1})$, cf. (3.2.11) in Example 3.2.5, because a direct use of (3.2.9) would bring analytic troubles with ${\buildrel .\over z}$ not valued in $\mathrm{L}^{1}(\varGamma \mathrm{C})$. Then $\mathcal{D}_{\mathcal{R}}(z_{1},z_{2}) = \mathcal{R}(z_{2}-z_{1}) =\int _{\varGamma \mathrm{C}}\!a(x)\vert z_{2}(x)-z_{1}(x)\vert \,\mathrm{d}S =\int _{\varGamma _{\mathrm{D},2}\setminus \varGamma _{\mathrm{D},1}}\!a(x)\,\mathrm{d}S$.

In fact, since $\mathcal{E}(t,y,\cdot )$ is “pointwise” nondecreasing and both $\mathcal{E}(t,y,\cdot )$ and $\mathcal{R}$ are local (this argument will be more explicit in the regularized problem; cf. (4.2.93) below) and therefore there cannot be any tendency for “healing” ${\buildrel .\over z} > 0$, we could redefine $\mathcal{R}$ for ${\buildrel .\over z} > 0$ essentially arbitrarily, e.g., such that $\mathcal{R}$ will be continuous and coercive on $\mathrm{L}^{1}(\varGamma \mathrm{C})$ like $\mathcal{R}({\buildrel .\over z}):=\int _{\varGamma \mathrm{C}}a(x)\vert {\buildrel .\over z}(x)\vert \,\mathrm{d}S$. Yet such $\mathcal{R}$ would not be weakly continuous, so that there would not be any essential benefit from such a modification.

In fact, however, considering $z[[y_{\mathrm{D}}(t,\cdot ) \circ y]] = 0$ in (4.2.77d) would yield a different regularized problem (4.2.85) below.

Alternatively, the assertion follows also from Proposition 4.2.26 below. Anyhow, the direct proof is of some interest, too.

Cf. also [418, Formula (4.35)] or [373, Lemma 6.1] or [545, Formula (3.71)].

To be more specific and referring to Example 2.4.5, one can penalize the constraint r = 0 in the constraint set $A =\{\, q = (y,z,r)\;\vert \ \sqrt{z}[[y]] = r,\ r = 0\,\}$ by considering the $\mathrm{L}^{2}(\varGamma \mathrm{C})$-norm and α = 2 in (2.4.5).

Note that k in this $\varGamma \mathrm{C}$-term has, in fact, the physical dimension J/m^d+1, so that this term has indeed dimensions in Joule. The other k’s have, however, different dimensions; for simplicity, we have used the same letter k for all of them in this purely mathematical exposition.

Note, however, that the delamination-activation condition $\frac{1} {2k} \frac{\sigma ^{2}} {z^{2}} (t) + r(t) \leq a$ with $\sigma (t):= \partial _{F}^{_{}}W(\nabla y(t))\vert _{\varGamma \mathrm{C}}\nu$ in (4.2.93d) exhibits again for $k \rightarrow \infty $ the previous effect that the differential formulation no longer experiences any driving stress toward delamination. See also the arguments in Remark 4.3.54.

The linearity of W(F, ⋅ ) in (4.2.102) corresponds to a so-called 1− d model, with d having the meaning of density of microcracks or microvoids, which is very popular in engineering; in this context, we put z: = 1−d, which is occasionally used in the mathematical literature; cf. [82, 205, 206].

In case d = 3, both κ and a have the physical units Jm⁻³ = Pa.

Cf. also [418, Sect.4.2] or [258].

In contrast to BV-spaces as defined in (4.2.98) used in problems with Neumann boundary condition in Sect. 4.2.4.1, we supported the functions of BD-spaces on the closure of $\varOmega$ to have good control of traces here.

Equivalently, (4.3.12c) can also be written as $\mathcal{R}_{_{\mathrm{PR}}}({\buildrel . \over \pi }) =\int _{\bar{\varOmega }}\boldsymbol{\delta }_{S}^{{\ast}}(\mathrm{d}{\buildrel . \over \pi }/\mathrm{d}\vert {\buildrel . \over \pi }\vert )\,\mathrm{d}\vert {\buildrel . \over \pi }\vert $, where $\vert {\buildrel . \over \pi }\vert $ is the total variation of ${\buildrel . \over \pi }$, and $\mathrm{d}{\buildrel . \over \pi }/\mathrm{d}\vert {\buildrel . \over \pi }\vert $ is the Radon–Nikodym derivative of $\mathrm{d}{\buildrel . \over \pi }$ with respect to $\vert {\buildrel . \over \pi }\vert $. See [143] for further details about functions on $\mathcal{M}(\bar{\varOmega }; \mathbb{R}d\times d \mathrm{dev} )$.

In fact, suitably qualified nonvanishing f = 0 or g = 0 as a “safe load” was admitted in [143, formulas (2.17)–(2.18)].

This means that $\mathbb{B}\pi = \mathbb{C}^{-1}(\sum _{i=1}^{n}\mathbb{C}_{i}\pi _{i})$ and $\mathbb{H}\pi:\pi =\sum _{ i=1}^{n}(\mathbb{H}_{i}+\mathbb{C}_{i})\pi _{i}:\pi _{i} - \mathbb{C}\mathbb{B}\pi: (\mathbb{B}\pi )$.

In fact, since the driving force $\partial _{z}W_{\varepsilon }$ for the ansatz like (4.3.74) is always nonnegative, namely a′(z)V ₁(x, e) ≥ 0, the tendency for healing can be realized only through the grading-damage term—so up to small length effects, healing is not important.

For this, one can simply modify (5.2.84) on p. 533 straightforwardly by replacing $\bar{\xi }_{\tau }(t)$ with $\bar{\xi }_{\tau }(t) +\bar{\zeta } _{\tau }(t)$; note that r > d is not needed now.

For $z\neq 0$, we can take constant recovery sequences $z_{\varepsilon } = z$. For z = 0, the recovery sequences will depend on e: for e > 0, we choose $z_{\varepsilon } =\varepsilon$ and obtain $\mathcal{J}_{\varepsilon }(e,z_{\varepsilon }) = \vert e-1\vert $, while for e < 0, let $z_{\varepsilon } = -\varepsilon$ obtaining $\mathcal{J}_{\varepsilon }(e,z_{\varepsilon }) = \vert e+1\vert $.

More specifically, the experiments in Figures 4.16–4.18 consider Young’s modulus $E_{_{\mathrm{Young}}} = 27$ GPa, Poisson’s ratio $\nu = 0.2$, the factor of influence κ = 20 Jm⁻², and the activation threshold a = 500 Jm⁻³. The Lamé constants $\lambda$ and $\mu$ in (4.3.100) are determined by (4.1.24)

In fact, r = 2 = d was used for the calculations depicted in Figure 4.18, so that the condition r > d needed for Proposition 4.3.22 has been satisfied only “up to $\varepsilon$-tolerance.”

As we pull the springs by ever increasing load u _D(t), we do not consider a possible Signorini-type constraint u ≥ 0, nor the unidirectional constraint ${\buildrel .\over z} \leq 0$ (as considered in Sections 4.2.3, 4.3.4.2, and 4.3.4.3), which would not be active in this regime anyhow.

The other solutions take some $z(t_{_{\mathrm{M}\mathrm{D}}}) = z_{_{\mathrm{M}\mathrm{D}}}$ with $0 < z_{_{\mathrm{M}\mathrm{D}}} < 1$. To see that such solutions are also maximally dissipative, one can first exploit Example 4.3.40, and for $z_{_{\mathrm{M}\mathrm{D}}} = 1/2$, consider a ₁ = a ₂ and the scenario that only one spring breaks at time $t_{_{\mathrm{M}\mathrm{D}}}$ to see that the second spring becomes supercritical, and its break then does not contribute to a possible positive residuum that would violate (3.3.14). An analogous consideration but with more than two parallel springs clearly applies for every rational number $z_{_{\mathrm{M}\mathrm{D}}}$ from (0, 1).

For example, one can take $\xi (t) = -\partial _{z}\mathcal{E}(t,u(t), 1)$ for $t \leq t_{_{\mathrm{M}\mathrm{D}}}$ and $\xi = a$ for $t > t_{_{\mathrm{M}\mathrm{D}}}$.

Note that on considering the physical dimensions in this 1-dimensional case as [a] = J, [u] = m, $[\mathbb{K}] = [\mathbb{C}] =$ J/m², and [v ₀] = m/s, the formulas (4.3.107) and (4.3.109) give indeed a correct physical dimension of time, i.e., $[t_{_{\mathrm{E}\mathrm{S}}}] = [t_{_{\mathrm{M}\mathrm{D}}}] = [t_{_{\mathrm{V}\mathrm{V}}}] =\,$ s.

The maximum-dissipation principle is sometimes used, although rather heuristically, in engineering models of damage with plasticity and hardening; cf. [136].

More specifically, $\mathbb{C} = \mathbb{C}(\zeta )$ was affine as a function of $\zeta$ with $\mathbb{C}(1)$ as in (4.1.10) with (4.1.24) with Young’s modulus $E_{_{\mathrm{Young}}} = 27\,$ GPa and Poisson’ ratio $\nu = 0.2$, $\mathbb{C}(0) = \mathbb{C}(1)/1000$, the hardening $\mathbb{H} = \mathbb{C}(1)/4$, the elastic domain $S:=\{\,\sigma \in \mathbb{R}d\times d \mathrm{dev} \;\vert \ \vert \sigma \vert \leq \sigma _{\mathrm{y}}\,\}$ with the yield stress $\sigma _{\mathrm{y}} = 2\,$ MPa, the damage energy a = 1 kPa, and the damage length-scale coefficient κ ₁ = 10⁻⁹ J/m.

Some shortcuts were taken in comparison with the model (4.3.116), namely κ ₂ = 0 and r = 2 instead of κ ₂ > 0 and r > 2 were considered, and then, after triangulation of $\varOmega$, P1-elements were used for u and $\zeta$, while P0-elements sufficed for $\pi$ in [552]. Instead of the transformation leading to an alternating recursive quadratic and second-order cone programming, cf. Remark 3.6.15 on p. 208, to solve the discretized version of (4.3.119), a quasi-Newton iterative procedure was used; cf. [99] for details.

As in (4.1.28), we assume the prolongation $u_{\mathrm{D}}^{}(t) \in \mathrm{H}^{1}(\varOmega; \mathbb{R}^{d})$, so that in particular, $[[u_{\mathrm{D}}^{}]] \equiv 0$.

In fact, such a regularized problem was devised in [109, 310] in a position of the original problem without ambitions to pass $k \rightarrow \infty $; cf. also [373, Sect.6.2].

See [545] for more details.

More specifically, the hard cylindrical fiber with the Young modulus $E_{\mathrm{Young}}\doteq70\,$ GPa and Poisson ratio $\nu \doteq0.2$ of the diameter 15$\mu$ m in the soft cube matrix with $E_{\mathrm{Young}}\doteq2.8\,$ GPa and $\nu \doteq0.3$ of the size 30×30$\mu$ m glued by the adhesive with $\kappa _{\mathrm{n}}^{} =\kappa _{ \mathrm{t}}^{}/3 = 675$ TPa/m was used for the calculations presented in Figures 4.30 and 4.31.

We can use the damage-type construction for $\zeta$ and the quadratic trick (cf. Lemma 3.5.3) for $\pi$; cf. [542] for details.

In fact, the classical formulation (4.3.160) assumes $\varGamma \mathrm{C}$ smooth and uses the Green formula on the possibly curved surface; cf. (B.4.12) on p. 602.

As in (4.3.160), the surface Green formula (B.4.12) is used to derive the classical formulation (4.3.163).

The finite interpenetration up the depth, say, $u_{\mathrm{f}} > 0$ means that $\lim _{v\nearrow u_{\mathrm{f}}}\gamma (v) = \infty $ and $\gamma (v) = \infty $ for $v \geq u_{\mathrm{f}}$. This is a quite realistic replacement of the Signorini contact. In the context of the static friction, see [169].

We assume $\mu$ constant. For nonconstant $\mu$, the “max” in (4.3.191) distinguishes the norm of $\mu$ as an operator $v\mapsto \mu v$ on $\mathrm{H}^{1/2}(\varGamma; \mathbb{R}^{d})$ and a function in $\mathrm{L}^{\infty }(\varGamma )$, respectively. Unfortunately, the formula (4.3.191) is very implicit, and at the time of writing, there are no explicit results in any nontrivial domain $\varOmega$ except the half-space.

A discrete variant of such a maximum-dissipation principle $\int _{[0,T]\times \varGamma \mathrm{C}}\sigma _{\mathrm{t}}^{}(\bar{u}_{\tau })\sigma _{\mathrm{n}}^{}(u)[[{\buildrel .\over u}_{\tau }]]\mathrm{n}\,\mathrm{d}S\mathrm{d}t$ = $\int _{[0,T]\times \varGamma \mathrm{C}}\mu \sigma _{\mathrm{n}}^{}(\bar{u}_{\tau })\vert [[{\buildrel .\over u}_{\tau }]]\mathrm{n}\vert \,\mathrm{d}S\mathrm{d}t$ is at our disposal and allows for a passage to the limit by weak (lower semi)continuity.

At this point, we assume that the temperature in the whole ferromagnet is constant and well below the so-called Curie point. If the temperature approached the Curie point, ferromagnetism would disappear by $M\mathrm{S} \rightarrow 0$, and the material would become paramagnetic. Moreover, substantial deviations (up to 10–30%) from the constraint (4.4.1) can, in fact, be expected around the Curie point in the outer magnetic field; cf. [68, Figure 5.4]. This will be considered later, in Sect. 5.3.3.2

An alternative setting using the magnetic induction $B =\mu _{ 0}^{}(h_{\mathrm{dem}}^{} + m)$ as another nondissipative variable instead of $\phi$ while $\mathcal{Y}$ involves the constraint $\mathop{\mathrm{div}}\limits B = 0$ was implemented in [394, Sect.4.2].

Instead of magnetization, one can formulate the problem in terms of magnetic polarization $j =\mu _{ 0}^{}m$. The physical dimension of j is Tesla, while the physical dimension of the magnetization and m and of magnetic field $h_{\mathrm{ext}}$ or $h_{\mathrm{dem}}^{} = \nabla \phi$ is A/m. This also shows that $\varepsilon$ in (4.4.5) has the physical dimension Jm/A², so that the term $\frac{\varepsilon }{2}\vert \nabla m\vert ^{2}$ has the physical dimension J/m³=Pa.

More impurities or immobile dislocations make movement of domain walls harder, which further needs a bigger activation force, dissipates more energy, and leads to a wider hysteretic loop under cyclical loading by the external magnetic field $h_{\mathrm{ext}}$. Such dry-friction-type models have been given by Bergqvist [67], Jiles [286], Podio-Guidugli [481], and Visintin [610, 611], and in a nonlocal variant also [612].

The physical dimension of both $M\mathrm{S}$ and $H_{_{\mathrm{C}}}$ is the physical dimension of intensity of a magnetic field, i.e., A/m. Since $\mu _{0}^{}:= 4\pi \times 10^{-7}$ H/m, the physical dimension of $\mu _{0}^{}H_{_{\mathrm{C}}}M\mathrm{S}$ is indeed specific energy, i.e., J/m³=Pa.

If d = 3, counting that the magnetization vector can move only over the sphere $M\mathrm{S}S^{3}$ from (4.4.7), the energy in J/m³ needed (and thus dissipated) by moving a domain wall of the area 1 m² by 1 m and thereby reorienting the magnetization from m to − m is just $\mu _{0}^{}H_{_{\mathrm{C}}}M\mathrm{S}$.

See [69, Appendix A]. Alternatively, pinning effects were suggested by adding $h_{\mathrm{eff}}$ in the Landau–Lifshitz equation instead of (4.4.15) in [610], although without any mathematical justification. However, the resulting augmented equations proposed in [50] and [610] are no longer equivalent, although the original Gilbert and Landau–Lifschitz equations are equivalent to each other. The difference between the Gilbert and Landau–Lifschitz formalism has been pointed out and explained in [481].

In general, the div–curl lemma [443] says that the product of two L ²-weakly convergent sequences $(u_{k})_{k\in \mathbb{N}}^{}$ and $(v_{k})_{k\in \mathbb{N}}^{}$ converges in the sense of distributions, provided that $\{\mathop{\mathrm{div}}\limits \, u_{k}\}_{k\in \mathbb{N}}^{}$ and $\{\mathop{\mathrm{curl}}\, v_{k}\}_{k\in \mathbb{N}}^{}$ are bounded in H ⁻¹; cf., e.g., [177, Sect. 5.2]. Here it is used for the case $\mathop{\mathrm{curl}}\, v_{k} = 0$.

In fact, some shortcuts in calculations (not fully compatible with the above-presented theory) were used in [335, 537], in particular $\mathcal{S}$ in (4.4.13) considered simply the identity, and $\phi$ was evaluated only approximately. The approximation of Young measures was made a more effective adaptive activation using the Weierstrass maximum principle, cf. also [331, 332, 493].

A virgin magnetization curve arises when magnets without any domain structure begin to be magnetized in an outer field; cf. Figure 4.48.

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Title: Applications in continuum mechanics and physics of solids
Authors: Alexander Mielke
Tomàš Roubíček
Publisher: Springer New York
Book: Rate-Independent Systems
Print ISBN: 978-1-4939-2705-0

Electronic ISBN: 978-1-4939-2706-7

Copyright Year: 2015
DOI: https://doi.org/10.1007/978-1-4939-2706-7_4

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