A compatible‐incompatible decomposition of symmetric tensors in Lp with application to elasticity
Abstract: In this paper, we prove the Saint-Venant compatibility conditions in L p for p 2 .1, C1/, in a simply connected domain of any space dimension. As a consequence, alternative, simple, and direct proofs of some classical Korn inequalities in L p are provided. We also use the Helmholtz decomposition in L p to show that every symmetric tensor in a smooth domain can be decomposed in a compatible part, which is the symmetric part of a displacement gradient, and in an incompatible part, which is the incompatibility of… Show more
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Cited by 25 publications
References 23 publications
The mathematical modelling in mechanics has a long-standing history as related to geometry, and significant progresses have often been achieved by the invention of new geometrical tools. Also, it happened that the elucidation of practical issues led to the invention of new scientific concepts, and possibly new paradigms, with potential impact far beyond. One such example is Riemann's intrinsic view in geometry, that offered a radically new insight in the Physics of the early 20th century. On the other hand, the rather recent intrinsic approaches in elasticity and elasto-plasticity also share this philosophical standpoint of looking from inside, i.e., from the "manifold" point of view. Of course, this approach requires smoothness, and is thus incomplete for an analyst. Nevertheless, its first aim is to highlight the concepts of metric, curvature and torsion; these notions are addressed in the first part of this survey paper. In a second part, they are given a precise functional meaning and their properties are studied systematically. Further, a novel approach to elasto-plasticity constructed upon a model of incompatible elasticity is designed, carrying this intrinsic spirit. The main mathematical object in this theory is the incompatibility operator, i.e., a linearized version of Riemann's curvature tensor. So far, this route not only has led the authors to a new model with a solid functional foundation and proof of existence results, but also to a framework with a minimal amount of ad-hoc assumptions, and complying with both the basic principles of thermodynamics and invariance principles of Physics. The questions arising from this novel approach are complex and intriguing, but we believe that the model is now sufficiently well posed to be studied simultaneously as a problem of mathematics and of mechanics. Most of the research programme remains to be done, and this survey paper is written to present our model, with a particular care to put this approach into a historical perspective.2010 Mathematics Subject Classification. 35J48,35J58,49S05,49K20,74C05,74G99,74A05,74A15, 80A17.
The mathematical modelling in mechanics has a long-standing history as related to geometry, and significant progresses have often been achieved by the invention of new geometrical tools. Also, it happened that the elucidation of practical issues led to the invention of new scientific concepts, and possibly new paradigms, with potential impact far beyond. One such example is Riemann's intrinsic view in geometry, that offered a radically new insight in the Physics of the early 20th century. On the other hand, the rather recent intrinsic approaches in elasticity and elasto-plasticity also share this philosophical standpoint of looking from inside, i.e., from the "manifold" point of view. Of course, this approach requires smoothness, and is thus incomplete for an analyst. Nevertheless, its first aim is to highlight the concepts of metric, curvature and torsion; these notions are addressed in the first part of this survey paper. In a second part, they are given a precise functional meaning and their properties are studied systematically. Further, a novel approach to elasto-plasticity constructed upon a model of incompatible elasticity is designed, carrying this intrinsic spirit. The main mathematical object in this theory is the incompatibility operator, i.e., a linearized version of Riemann's curvature tensor. So far, this route not only has led the authors to a new model with a solid functional foundation and proof of existence results, but also to a framework with a minimal amount of ad-hoc assumptions, and complying with both the basic principles of thermodynamics and invariance principles of Physics. The questions arising from this novel approach are complex and intriguing, but we believe that the model is now sufficiently well posed to be studied simultaneously as a problem of mathematics and of mechanics. Most of the research programme remains to be done, and this survey paper is written to present our model, with a particular care to put this approach into a historical perspective.2010 Mathematics Subject Classification. 35J48,35J58,49S05,49K20,74C05,74G99,74A05,74A15, 80A17.
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 which is quadratic in the tensor inc(ε p ) and it contains isotropic hardening based on the rate of the plastic strain tensorε 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 distortionṗ to their symmetric counterpartε p = symṗ. 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. Appendix 322 Aifantis writes in [4, p. 218]: "... In conformity with established results -that the plastic strain rateεp is a state variable, rather than the strain εp itself."3 The plastic spin in the finite deformation flow theory of plasticity is defined as the skew-symmetric part of the so-called plastic distortion rate i.e., Wp := skew(ḞpF −1 p ) , while its counter-part in the small strain theory is simply skew(ṗ) , where p is the non-symmetric infinitesimal plastic distortion.
Let $$\Omega \subset \mathbb {R}^3$$ Ω ⊂ R 3 be an open and bounded set with Lipschitz boundary and outward unit normal $$\nu $$ ν . For $$1<p<\infty $$ 1 < p < ∞ we establish an improved version of the generalized $$L^p$$ L p -Korn inequality for incompatible tensor fields P in the new Banach space $$\begin{aligned}&W^{1,\,p,\,r}_0({{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}; \Omega ,\mathbb {R}^{3\times 3}) \\&\quad = \{ P \in L^p(\Omega ; \mathbb {R}^{3 \times 3}) \mid {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P \in L^r(\Omega ; \mathbb {R}^{3 \times 3}),\ {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}(P \times \nu ) = 0 \text { on }\partial \Omega \} \end{aligned}$$ W 0 1 , p , r ( dev sym Curl ; Ω , R 3 × 3 ) = { P ∈ L p ( Ω ; R 3 × 3 ) ∣ dev sym Curl P ∈ L r ( Ω ; R 3 × 3 ) , dev sym ( P × ν ) = 0 on ∂ Ω } where $$\begin{aligned} r \in [1, \infty ), \qquad \frac{1}{r} \le \frac{1}{p} + \frac{1}{3}, \qquad r >1 \quad \text {if }p = \frac{3}{2}. \end{aligned}$$ r ∈ [ 1 , ∞ ) , 1 r ≤ 1 p + 1 3 , r > 1 if p = 3 2 . Specifically, there exists a constant $$c=c(p,\Omega ,r)>0$$ c = c ( p , Ω , r ) > 0 such that the inequality $$\begin{aligned} \Vert P \Vert _{L^p(\Omega ,\mathbb {R}^{3\times 3})}\le c\,\left( \Vert {{\,\mathrm{sym}\,}}P \Vert _{L^p(\Omega ,\mathbb {R}^{3\times 3})} + \Vert {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P \Vert _{L^{r}(\Omega ,\mathbb {R}^{3\times 3})}\right) \end{aligned}$$ ‖ P ‖ L p ( Ω , R 3 × 3 ) ≤ c ‖ sym P ‖ L p ( Ω , R 3 × 3 ) + ‖ dev sym Curl P ‖ L r ( Ω , R 3 × 3 ) holds for all tensor fields $$P\in W^{1,\,p, \, r}_0({{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}; \Omega ,\mathbb {R}^{3\times 3})$$ P ∈ W 0 1 , p , r ( dev sym Curl ; Ω , R 3 × 3 ) . Here, $${{\,\mathrm{dev}\,}}X :=X -\frac{1}{3} {{\,\mathrm{tr}\,}}(X)\,{\mathbb {1}}$$ dev X : = X - 1 3 tr ( X ) 1 denotes the deviatoric (trace-free) part of a $$3 \times 3$$ 3 × 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 $$\Gamma \subset \partial \Omega $$ Γ ⊂ ∂ Ω . If no boundary conditions are imposed then the estimate holds after taking the quotient with the finite-dimensional space $$K_{S,dSC}$$ K S , d S C which is determined by the conditions $${{\,\mathrm{sym}\,}}P =0$$ sym P = 0 and $${{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P = 0$$ dev sym Curl P = 0 . In that case one can replace $$\Vert {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P \Vert _{L^r(\Omega ,\mathbb {R}^{3\times 3})} $$ ‖ dev sym Curl P ‖ L r ( Ω , R 3 × 3 ) by $$\Vert {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P \Vert _{W^{-1,p}(\Omega ,\mathbb {R}^{3\times 3})}$$ ‖ dev sym Curl P ‖ W - 1 , p ( Ω , R 3 × 3 ) . The new $$L^p$$ L p -estimate implies a classical Korn’s inequality with weak boundary conditions by choosing $$P=\mathrm {D}u$$ P = D u and a deviatoric-symmetric generalization of Poincaré’s inequality by choosing $$P=A\in {{\,\mathrm{\mathfrak {so}}\,}}(3)$$ P = A ∈ so ( 3 ) . The proof relies on a representation of the third derivatives $$\mathrm {D}^3 P$$ D 3 P in terms of $$\mathrm {D}^2 {{\,\mathrm{dev}\,}}{{\
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