Patrizio Neff

Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions

In this paper we propose a formulation of polyconvex anisotropic hyperelasticity at finite strains. The main goal is the representation of the governing constitutive equations within the framework of the invariant theory which automatically fulfill the polyconvexity condition in the sense of Ball in order to guarantee the existence of minimizers. Based on the introduction of additional argument tensors, the so-called structural tensors, the free energies and the anisotropic stress response functions are represented by scalar-valued and tensor-valued isotropic tensor functions, respectively. In order to obtain various free energies to model specific problems which permit the matching of data stemming from experiments, we assume an additive structure. A variety of isotropic and anisotropic functions for transversely isotropic material behaviour are derived, where each individual term fulfills a priori the polyconvexity condition. The tensor generators for the stresses and moduli are evaluated in detail and some representative numerical examples are presented. Furthermore, we propose an extension to orthotropic symmetry.

Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility

Abstract In this article we investigate several models contained in the literature in the case of near-incompressibility based on invariants in terms of polyconvexity and coerciveness inequality, which are sufficient to guarantee the existence of a solution. These models are due to Rivlin and Saunders, namely the generalized polynomial-type elasticity, and Arruda and Boyce. The extension to near-incompressibility is usually carried out by an additive decomposition of the strain energy into a volume-changing and a volume-preserving part, where the volume-changing part depends on the determinant of the deformation gradient and the volume-preserving part on the invariants of the unimodular right Cauchy–Green tensor. It will be shown that the Arruda–Boyce model satisfies the polyconvexity condition, whereas the polynomial-type elasticity does not. Therefore, we propose a new class of strain-energy functions depending on invariants. Moreover, we focus our attention on the structure of further isotropic strain-energy functions.

A unifying perspective: the relaxed linear micromorphic continuum

We formulate a relaxed linear elastic micromorphic continuum model with symmetric Cauchy force stresses and curvature contribution depending only on the micro-dislocation tensor. Our relaxed model is still able to fully describe rotation of the microstructure and to predict nonpolar size effects. It is intended for the homogenized description of highly heterogeneous, but nonpolar materials with microstructure liable to slip and fracture. In contrast to classical linear micromorphic models, our free energy is not uniformly pointwise positive definite in the control of the independent constitutive variables. The new relaxed micromorphic model supports well-posedness results for the dynamic and static case. There, decisive use is made of new coercive inequalities recently proved by Neff, Pauly and Witsch and by Bauer, Neff, Pauly and Starke. The new relaxed micromorphic formulation can be related to dislocation dynamics, gradient plasticity and seismic processes of earthquakes. It unifies and simplifies the understanding of the linear micromorphic models.

Existence of minimizers for a finite-strain micromorphic elastic solid

We investigate geometrically exact generalized continua of micromorphic type in the sense of Eringen. The two-field problem for the macrodeformation ϕ and the affine microdeformation ¯ P ∈ GL + (3, R) in the quasistatic, conservative load case is investigated in a variational form. Depending on material constants, two existence theorems in Sobolev spaces are given for the resulting nonlinear boundary-value problems. These results comprise existence results for the micro-incompressible case ¯ P ∈ SL(3, R) and the Cosserat micropolar case ¯ P ∈ SO(3, R). In order to treat external loads, a new condition, called bounded external work, has to be included, which overcomes the conditional coercivity of the formulation. The possible lack of coercivity is related to fracture of the micromorphic solid. The mathematical analysis uses an extended Korn first inequality. The methods of choice are the direct methods of the calculus of variations.

Existence, Uniqueness and Stability in Linear Cosserat Elasticity for Weakest Curvature Conditions

We investigate the weakest possible constitutive assumptions on the curvature energy in linear Cosserat models still providing for existence, uniqueness and stability. The assumed curvature energy is μL2 c ∥dev sym ∇axl A∥2 where axl A is the axial vector of the skewsymmetric microrotation A ∈ so(3) and dev is the orthogonal projection on the Lie-algebra sl(3) of trace free matrices. The proposed Cosserat parameter values coincide with values adopted in the experimental literature by R. S. Lakes. It is observed that unphysical stiffening for small samples is avoided in torsion and bending while size effects are still present. The number of Cosserat parameters is reduced from six to four. One Cosserat coupling parameter µc > 0 and only one length scale parameter L c > 0. Use is made of a new coercive inequality for conformal Killing vectorfields. An interesting point is that no (controversial) essential boundary conditions on the microrotations need to be specified; thus avoiding boundary layer effects. Since the curvature energy is the weakest possible consistent with non-negativity of the energy, it seems that the Cosserat couple modulus µc > 0 remains a material parameter independent of the sample size which is impossible for stronger curvature expressions.

Optimal BV estimates for a discontinuous Galerkin method for linear elasticity

We analyze a discontinuous Galerkin method for linear elasticity. The discrete formulation derives from the Hellinger-Reissner variational principle with the addition of stabilization terms analogous to those previously considered by others for the Navier-Stokes equations and a scalar Poisson equation. For our formulation, we first obtain convergence in a mesh-dependent norm and in the natural mesh-independent BD norm. We then prove a generalization of Korns second inequality which allows us to strengthen our results to an optimal, mesh-independent BV estimate for the error.

On Korn's first inequality with non-constant coefficients

In this paper we prove a Korn-type inequality with non-constant coefficients which arises from applications in elasto-plasticity at large deformations. More precisely, let Ω ⊂ R 3 be a bounded Lipschitz domain and let Γ ⊂ ∂Ω be a smooth part of the boundary with non-vanishing two-dimensional Lebesgue measure. Define and let be given with det F p ( x ) ≥ μ + > 0. Moreover, suppose that Rot . Then Clearly, this result generalizes the classical Korns first inequality which is just our result with F p = 11. With slight modifications, we are also able to treat forms of the type

NOTES ON STRAIN GRADIENT PLASTICITY: FINITE STRAIN COVARIANT MODELLING AND GLOBAL EXISTENCE IN THE INFINITESIMAL RATE-INDEPENDENT CASE

We propose a model of finite strain gradient plasticity including phenomenological Prager type linear kinematical hardening and nonlocal kinematical hardening due to dislocation interaction. Based on the multiplicative decomposition, a thermodynamically admissible flow rule for Fp is described involving as plastic gradient Curl Fp. The formulation is covariant w.r.t. superposed rigid rotations of the reference, intermediate and spatial configuration but the model is not spin-free due to the nonlocal dislocation interaction and cannot be reduced to a dependence on the plastic metric . The linearization leads to a thermodynamically admissible model of infinitesimal plasticity involving only the Curl of the nonsymmetric plastic distortion p. Linearized spatial and material covariance under constant infinitesimal rotations is satisfied. Uniqueness of strong solutions of the infinitesimal model is obtained if two non-classical boundary conditions on the plastic distortion p are introduced: on the microscopically hard boundary ΓD ⊂ ∂Ω and [Curl p] · τ = 0 on the microscopically free boundary ∂Ω\ΓD, where τ are the tangential vectors at the boundary ∂Ω. A weak reformulation of the infinitesimal model allows for a global in-time solution of the rate-independent initial boundary value problem. The method is based on a mixed variational inequality with symmetric and coercive bilinear form. We use a new Hilbert-space suitable for dislocation density dependent plasticity.

A GEOMETRICALLY EXACT PLANAR COSSERAT SHELL-MODEL WITH MICROSTRUCTURE: EXISTENCE OF MINIMIZERS FOR ZERO COSSERAT COUPLE MODULUS

The existence of minimizers to a geometrically exact Cosserat planar shell model with microstructure is proven. The membrane energy is a quadratic, uniformly Legendre–Hadamard elliptic energy in contrast to traditional membrane energies. The bending contribution is augmented by a curvature term representing the interaction of the rotational microstructure in the Cosserat theory. The model includes non-classical size effects, transverse shear resistance, drilling degrees of freedom and accounts implicitly for thickness extension and asymmetric shift of the midsurface. Upon linearization with zero Cosserat couple modulus μc = 0, one recovers the infinitesimal-displacement Reissner–Mindlin model. It is shown that the Cosserat shell formulation admits minimizers even for μc = 0, in which case the drill-energy is absent. The midsurface deformation m is found in H1(ω, ℝ3). Since the existence of energy minimizers rather than equilibrium solutions is established, the proposed analysis includes the large deformation/large rotation buckling behaviour of thin shells.

The relaxed linear micromorphic continuum: existence, uniqueness and continuous dependence in dynamics

We study well-posedness for the relaxed linear elastic micromorphic continuum model with symmetric Cauchy force-stresses and curvature contribution depending only on the micro-dislocation tensor. In contrast to classical micromorphic models our free energy is not uniformly pointwise positive definite in the control of the independent constitutive variables. Another interesting feature concerns the prescription of boundary values for the micro-distortion field: only tangential traces may be determined which are weaker than the usual strong anchoring boundary condition. There, decisive use is made of new coercive inequalities recently proved by Neff, Pauly and Witsch, and by Bauer, Neff, Pauly and Starke. The new relaxed micromorphic formulation can be related to dislocation dynamics, gradient plasticity and seismic processes of earthquakes.