Incompatibility Tensor Research Articles

Korn–Maxwell–Sobolev inequalities for general incompatibilities

We establish a family of coercive Korn-type inequalities for generalized incompatible fields in the superlinear growth regime under sharp criteria. This extends and unifies several previously known inequalities that are pivotal to the existence theory for a multitude of models in continuum mechanics in an optimal way. Different from our preceding work [F. Gmeineder, P. Lewintan and P. Neff, Optimal incompatible Korn–Maxwell–Sobolev inequalities in all dimensions, Calc. Var. PDE 62 (2023) 182], where we focused on the case [Formula: see text] and incompatibilities governed by the matrix curl, the case [Formula: see text] considered in this paper gives us access to substantially stronger results from harmonic analysis but conversely deals with more general incompatibilities. Especially, we obtain sharp generalizations of recently proved inequalities by P. Lewintan, S. Müller and P. Neff [Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy, Calc. Var. PDE 60 (2021) 150] in the realm of incompatible Korn-type inequalities with conformally invariant dislocation energy. However, being applicable to higher-order scenarios as well, our approach equally gives the first and sharp inequalities involving Kröner’s incompability tensor inc.

The consistent coupling boundary condition for the classical micromorphic model: existence, uniqueness and interpretation of parameters

We consider the classical Mindlin–Eringen linear micromorphic model with a new strictly weaker set of displacement boundary conditions. The new consistent coupling condition aims at minimizing spurious influences from arbitrary boundary prescription for the additional microdistortion field \(\varvec{P}\). In effect, \(\varvec{P}\) is now only required to match the tangential derivative of the classical displacement \(\varvec{u}\) which is known at the Dirichlet part of the boundary. We derive the full boundary condition, in adding the missing Neumann condition on the Dirichlet part. We show existence and uniqueness of the static problem for this weaker boundary condition. These results are based on new coercive inequalities for incompatible tensor fields with prescribed tangential part. Finally, we show that compared to classical Dirichlet conditions on \(\varvec{u}\) and \(\varvec{P}\), the new boundary condition modifies the interpretation of the constitutive parameters.

Critical Roles of Incompatibility Tensor in FTMP

In FTMP (Field Theory of Multiscale Plasticity), the incompatibility tensor plays a pivotal role in many respects. This article describes the augmented usages of the incompatibility tensor in FTMP, followed by one of the up-to-date application example that tackles the limit of FTMP-based approach. The former includes (1) extension of the definition to the 4D space-time, (2) the flow-evolutionary law (working hypothesis), and (3) the interaction formalism, where (1) further leads to an important relationship that specifies the incompatibility trace rate. For the latter, we discuss the subject in the context of “micro-pillar” compression test that requires us to treat discrete sort of dislocation behaviors.

L p -versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with p-integrable exterior derivative

For $n\ge2$ and $1<p<\infty$ we prove an $L^p$-version of the generalized Korn-type inequality for incompatible, $p$-integrable tensor fields $P:\Omega \to \mathbb{R}^{n\times n}$ having $p$-integrable generalized $\underline{\operatorname{Curl}}$ and generalized vanishing tangential trace $P\,\tau_l=0$ on $\partial \Omega$, denoting by $\{\tau_l\}_{l=1,\ldots, n-1}$ a moving tangent frame on $\partial\Omega$, more precisely we have: $$\| P \|_{L^p(\Omega,\mathbb{R}^{n\times n})}\leq c\,(\| \operatorname{sym} P\|_{L^p(\Omega,\mathbb{R}^{n \times n})} + \|\underline{\operatorname{Curl}} P \|_{L^p(\Omega,(\mathfrak{so}(n))^n)} ),$$ where the generalized $\underline{\operatorname{Curl}}$ is given by $ (\underline{\operatorname{Curl}})_{ijk} :=\partial_i P_{kj}-\partial_j P_{ki}$ and $c=c(n,p,\Omega)>0$.

FTMP-based Kink Deformation and Strengthening Mechanisms for Mille-feuille Structures

This study aims at reproducing deformation-induced kink morphologies, together with the recently-reported AE measurement-based energy releasing characteristics, based on FTMP (Field Theory of Multiscale Plasticity)-incorporated crystal plasticity finite element simulation for single crystal Mg under c-axis plane strain compression condition. Two models either with the well-defined deformation twin system or a rank-1 connection-based kink system are considered, respectively, as the projection direction for the incompatibility tensor to be used in the constitutive model. The twin model yields “kink-like morphology” of growing kind, but fails to capture one of the energy releasing characteristics. The kink model, on the other hand, is demonstrated to be able to reproduce basically all the features, i.e., not only the “kink” morphology but also the AE energy-based features.&nbsp

Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy

Let Omega subset mathbb {R}^3 be an open and bounded set with Lipschitz boundary and outward unit normal nu . For 1<p<infty we establish an improved version of the generalized L^p-Korn inequality for incompatible tensor fields P in the new Banach space W01,p,r(devsymCurl;Ω,R3×3)={P∈Lp(Ω;R3×3)∣devsymCurlP∈Lr(Ω;R3×3),devsym(P×ν)=0on∂Ω}\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned}&W^{1,\\,p,\\,r}_0({{\\,\\mathrm{dev}\\,}}{{\\,\\mathrm{sym}\\,}}{{\\,\\mathrm{Curl}\\,}}; \\Omega ,\\mathbb {R}^{3\\times 3}) \\\\&\\quad = \\{ P \\in L^p(\\Omega ; \\mathbb {R}^{3 \\times 3}) \\mid {{\\,\\mathrm{dev}\\,}}{{\\,\\mathrm{sym}\\,}}{{\\,\\mathrm{Curl}\\,}}P \\in L^r(\\Omega ; \\mathbb {R}^{3 \\times 3}),\\ {{\\,\\mathrm{dev}\\,}}{{\\,\\mathrm{sym}\\,}}(P \\times \\nu ) = 0 \\text { on }\\partial \\Omega \\} \\end{aligned}$$\\end{document}where r∈[1,∞),1r≤1p+13,r>1ifp=32.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} r \\in [1, \\infty ), \\qquad \\frac{1}{r} \\le \\frac{1}{p} + \\frac{1}{3}, \\qquad r >1 \\quad \\text {if }p = \\frac{3}{2}. \\end{aligned}$$\\end{document}Specifically, there exists a constant c=c(p,Omega ,r)>0 such that the inequality ‖P‖Lp(Ω,R3×3)≤c‖symP‖Lp(Ω,R3×3)+‖devsymCurlP‖Lr(Ω,R3×3)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\Vert P \\Vert _{L^p(\\Omega ,\\mathbb {R}^{3\\times 3})}\\le c\\,\\left( \\Vert {{\\,\\mathrm{sym}\\,}}P \\Vert _{L^p(\\Omega ,\\mathbb {R}^{3\\times 3})} + \\Vert {{\\,\\mathrm{dev}\\,}}{{\\,\\mathrm{sym}\\,}}{{\\,\\mathrm{Curl}\\,}}P \\Vert _{L^{r}(\\Omega ,\\mathbb {R}^{3\\times 3})}\\right) \\end{aligned}$$\\end{document}holds for all tensor fields Pin W^{1,,p, , r}_0({{,mathrm{dev},}}{{,mathrm{sym},}}{{,mathrm{Curl},}}; Omega ,mathbb {R}^{3times 3}). Here, {{,mathrm{dev},}}X :=X -frac{1}{3} {{,mathrm{tr},}}(X),{mathbb {1}} denotes the deviatoric (trace-free) part of a 3 times 3 matrix X and the boundary condition is understood in a suitable weak sense. This estimate also holds true if the boundary condition is only satisfied on a relatively open, non-empty subset Gamma subset partial Omega . If no boundary conditions are imposed then the estimate holds after taking the quotient with the finite-dimensional space K_{S,dSC} which is determined by the conditions {{,mathrm{sym},}}P =0 and {{,mathrm{dev},}}{{,mathrm{sym},}}{{,mathrm{Curl},}}P = 0. In that case one can replace Vert {{,mathrm{dev},}}{{,mathrm{sym},}}{{,mathrm{Curl},}}P Vert _{L^r(Omega ,mathbb {R}^{3times 3})} by Vert {{,mathrm{dev},}}{{,mathrm{sym},}}{{,mathrm{Curl},}}P Vert _{W^{-1,p}(Omega ,mathbb {R}^{3times 3})}. The new L^p-estimate implies a classical Korn’s inequality with weak boundary conditions by choosing P=mathrm {D}u and a deviatoric-symmetric generalization of Poincaré’s inequality by choosing P=Ain {{,mathrm{mathfrak {so}},}}(3). The proof relies on a representation of the third derivatives mathrm {D}^3 P in terms of mathrm {D}^2 {{,mathrm{dev},}}{{,mathrm{sym},}}{{,mathrm{Curl},}}P combined with the Lions lemma and the Nečas estimate. We also discuss applications of the new inequality to the relaxed micromorphic model, to Cosserat models with the weakest form of the curvature energy, to gradient plasticity with plastic spin and to incompatible linear elasticity.

L^p-trace-free version of the generalized Korn inequality for incompatible tensor fields in arbitrary dimensions

For nge 3 and 1<p<infty , we prove an L^p-version of the generalized trace-free Korn-type inequality for incompatible, p-integrable tensor fields P:Omega rightarrow mathbb {R}^{ntimes n} having p-integrable generalized {text {Curl}}_{n} and generalized vanishing tangential trace P,tau _l=0 on partial Omega , denoting by {tau _l}_{l=1,ldots , n-1} a moving tangent frame on partial Omega . More precisely, there exists a constant c=c(n,p,Omega ) such that ‖P‖Lp(Ω,Rn×n)≤c‖devnsymP‖Lp(Ω,Rn×n)+‖CurlnP‖LpΩ,Rn×n(n-1)2,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\Vert P \\Vert _{L^p(\\Omega ,\\mathbb {R}^{n\\times n})}\\le c\\,\\left( \\Vert {\\text {dev}}_n {\\text {sym}}P \\Vert _{L^p(\\Omega ,\\mathbb {R}^{n \\times n})}+ \\Vert {\\text {Curl}}_{n} P \\Vert _{L^p\\left( \\Omega ,\\mathbb {R}^{n\\times \\frac{n(n-1)}{2}}\\right) }\\right) , \\end{aligned}$$\\end{document}where the generalized {text {Curl}}_{n} is given by ({text {Curl}}_{n} P)_{ijk} :=partial _i P_{kj}-partial _j P_{ki} and denotes the deviatoric (trace-free) part of the square matrix X. The improvement towards the three-dimensional case comes from a novel matrix representation of the generalized cross product.

Nečas–Lions lemma revisited: An Lp‐version of the generalized Korn inequality for incompatible tensor fields

For 1 &lt; p &lt; ∞, we prove an Lp‐version of the generalized Korn inequality for incompatible tensor fields P in . More precisely, let be a bounded Lipschitz domain. Then there exists a constant c = c(p, Ω) &gt; 0 such that holds for all tensor fields , that is, for all with vanishing tangential trace on ∂Ω where ν denotes the outward unit normal vector field to ∂Ω. For compatible , this recovers an Lp‐version of the classical Korn's first inequality and for skew‐symmetric an Lp‐version of the Poincaré inequality.

FTMP-based Approach toward Dislocation Dynamics of PSB

This study targets dynamic behavior of interacting dislocations associated with PSB (Persistent Slip Bands) ladder structures, ultimately responsible for fatigue crack initiations, based on FTMP (Field Theory of Multiscale Plasticity), applied to simulation results by dislocation dynamics. Focus is place on the stable/unstable of the dislocation systems in terms of FTMP-based incompatibility tensor field in particular via the attendant phase-space trajectories (PhS-Ts). The stable responses are shown to be manifested as the limit-like loci on the PhS-Ts, whereas the unstable responses mainly due to cross slip-driven vacancy-type debris loop generations are specifically represented by the incompatibility rate, i.e., the quantity well correlate the frequency of the debris loop formations. The incompatibility rate, used here as the ordinate of the PhS-Ts, can be interpreted as the creation/annihilation rate of edge dislocation pairs, in the FTMP-based context, which corroborates the correlation result.

Dislocation nucleation in the phase-field crystal model

We use the phase field crystal model to study nucleation of edge dislocations in two dimensions under an applied stress field. A dislocation dipole nucleates under the applied stress, consistent with Burgers vector conservation. The phase field correctly accounts for elastic energy storage prior to nucleation, and for dissipative relaxation during the nucleation event. We show that a lattice incompatibility field is a sensitive diagnostic of the location of the nucleation event, and of the Burgers vector and slip direction of the dislocations that will be nucleated above threshold. A direct calculation of the phase field energy accurately correlates with the nucleation event, as signaled by the lattice incompatibility field. We show that a Schmid-like criterion concerning the resolved stress at the nucleation site correctly predicts the critical nucleation stress. Finally, we present preliminary results for a three-dimensional, body-centered cubic lattice. The phase field allows a direct computation of the lattice incompatibility tensor for both dislocation lines and loops.

The geometry of incompatibility in growing soft tissues: Theory and numerical characterization

Tissues in vivo are not stress-free. As we grow, our tissues adapt to different physiological and disease conditions through growth and remodeling. This adaptation occurs at the microscopic scale, where cells control the microstructure of their immediate extracellular environment to achieve homeostasis. The local and heterogeneous nature of this process is the source of residual stresses. At the macroscopic scale, growth and remodeling can be accurately captured with the finite volume growth framework within continuum mechanics, which is akin to plasticity. The multiplicative split of the deformation gradient into growth and elastic contributions brings about the notion of incompatibility as a plausible description for the origin of residual stress. Here we define the geometric features that characterize incompatibility in biological materials. We introduce the geometric incompatibility tensor for different growth types, showing that the constraints associated with growth lead to specific patterns of the incompatibility metrics. To numerically investigate the distribution of incompatibility measures, we implement the analysis within a finite element framework. Simple, illustrative examples are shown first to explain the main concepts. Then, numerical characterization of incompatibility and residual stress is performed on three biomedical applications: brain atrophy, skin expansion, and cortical folding. Our analysis provides new insights into the role of growth in the development of tissue defects and residual stresses. Thus, we anticipate that our work will further motivate additional research to characterize residual stresses in living tissue and their role in development, disease, and clinical intervention.

Duality of the Incompatibility Tensor

Duality of the Incompatibility Tensor

Existence and asymptotic results for an intrinsic model of small-strain incompatible elasticity

A general model of incompatible small-strain elasticity is presented and analyzed, based on the linearized strain and its associated incompatibility tensor field. Strain incompatibility accounts for the presence of dislocations, whose motion is ultimately responsible for the plastic behaviour of solids. The specific functional setting is built up, on which existence results are proved. Our solution strategy is essentially based on the projection of the governing equations on appropriate subspaces in the spirit of the Leray decomposition of solenoidal square-integrable velocity fields in hydrodynamics. It is also strongly related with the Beltrami decomposition of symmetric tensor fields in the wake of previous works by the authors. Moreover a novel model parameter is introduced, the incompatibility modulus, that measures the resistance of the elastic material to incompatible deformations. An important result of our study is that classical linearized elasticity is recovered as the limit case when the incompatibility modulus goes to infinity. Several examples are provided to illustrate this property and the physical meaning of the incompatibility modulus in connection with the dissipative nature of the processes under consideration.

A fourth-order gauge-invariant gradient plasticity model for polycrystals based on Kröner’s incompatibility tensor

In this paper we derive a novel fourth-order gauge-invariant phenomenological model of infinitesimal rate-independent gradient plasticity with isotropic hardening and Kröner’s incompatibility tensor inc ( ε p ) : = Curl [ ( Curl ε p ) T ] , where ε p is the symmetric plastic strain tensor. Here, gauge-invariance denotes invariance under diffeomorphic reparametrizations of the reference configuration, suitably adapted to the geometrically linear setting. The model features a defect energy contribution that is quadratic in the tensor inc ( ε p ) and it contains isotropic hardening based on the rate of the plastic strain tensor ε · p . We motivate the new model by introducing a novel rotational invariance requirement in gradient plasticity, which we call micro-randomness, suitable for the description of polycrystalline aggregates on a mesoscopic scale and not coinciding with classical isotropy requirements. This new condition effectively reduces the increments of the non-symmetric plastic distortion p · to their symmetric counterpart ε · p = sym p · . In the polycrystalline case, this condition is a statement about insensitivity to arbitrary superposed grain rotations. We formulate a mathematical existence result for a suitably regularized non-gauge-invariant model. The regularized model is rather invariant under reparametrizations of the reference configuration including infinitesimal conformal mappings.

The deformators of high ranks and the kröner incompatibility tensors with two-dimensional structure of subscripts

In n-dimensional space (multidimensional continuum) the compatibility equations are derived for the components of the generalized deformators of rank m which are connected with the generalized displacements of rank m - 1 by analogues of the Cauchy kinematic relations (m≥1, n≥2). They may be written in form of equal to zero for all the components of the incompatibility tensor of rank m(n - 2) or for dual to it the generalized Riemann-Christoffel tensor of rank 2m. The number of independent components of these tensors coinciding with the number of compatibility equations in terms of the generalized deformations, is obtained.

High-Rank Deformators and the Kröner Incompatibility Tensors with Two-Dimensional Structure of Indices

In n-dimensional space (multidimensional continuum), the compatibility equations ( $$m \geqslant 1$$ , $$n \geqslant 2$$ ) are derived for the components of the generalized strains of rank m associated with the generalized displacements of rank m – 1 by the analogs of the Cauchy kinematic relations. The compatibility conditions may be written in the form of vanishing of all components of the incompatibility tensor of rank $$m(n - 2)$$ or of the generalized Riemann–Christoffel tensor of rank 2m, which is dual to it. The number of independent components of these tensors coinciding with the number of compatibility equations in terms of generalized strains is determined.

A canonical rate-independent model of geometrically linear isotropic gradient plasticity with isotropic hardening and plastic spin accounting for the Burgers vector

In this paper we propose a canonical variational framework for rate-independent phenomenological geometrically linear gradient plasticity with plastic spin. The model combines the additive decomposition of the total distortion into non-symmetric elastic and plastic distortions, with a defect energy contribution taking account of the Burgers vector through a dependence only on the dislocation density tensor Curl(p) giving rise to a non-symmetric nonlocal backstress, and isotropic hardening response only depending on the accumulated equivalent plastic strain. The model is fully isotropic and satisfies linearized gauge-invariance conditions, i.e., only true state-variables appear. The model satisfies also the principle of maximum dissipation which allows to show existence for the weak formulation. For this result, a recently introduced Korn's inequality for incompatible tensor fields is necessary. Uniqueness is shown in the class of strong solutions. For vanishing energetic length scale, the model reduces to classical elasto-plasticity with symmetric plastic strain sym(p) and standard isotropic hardening.

Effect of Dislocation Density and Incompatibility Tensor Fields on Misorientation Developments in Lath Block Structure based on FTMP

Effect of Dislocation Density and Incompatibility Tensor Fields on Misorientation Developments in Lath Block Structure based on FTMP

Roles of Incompatibility Tensor on Lath Block in FTMP

Roles of Incompatibility Tensor on Lath Block in FTMP

Generalized compatibility equations for tensors of high ranks in multidimensional continuum mechanics

Compatibility equations are derived for the components of generalized strains of rank m associated with generalized displacements of rank m − 1 by analogs of Cauchy kinematic relations in n-dimensional space (multi-dimensional continuous medium) (m ≥ 1, n ≥ 2). These relations can be written in the form of equating to zero all components of the incompatibility tensor of rank m(n − 2) or its dual generalized Riemann–Christoffel tensor of rank 2m. The number of independent components of these tensors is found; this number coincides with that of compatibility equations in terms of generalized strains or stresses. The inequivalence of the full system of compatibility equations to any of its weakened subsystems is discussed, together with diverse formulations of boundary value problems in generalized stresses in which the number of equations in a domain can exceed the number of unknowns.