215 research outputs found
On the characterization of drilling rotation in the 6-parameter resultant shell theory
We analyze geometrically non-linear isotropic elastic shells and prove the
existence of minimizers. In general, the model takes into account the effect of
drilling rotations in shells. For the special case of shells without drilling
rotations we present a representation theorem for the strain energy function
Counterexamples in the theory of coerciveness for linear elliptic systems related to generalizations of Korn's second inequality
We show that the following generalized version of Korn's second inequality
with nonconstant measurable matrix valued coefficients P
||DuP+(DuP)^T||_q+||u||_q >= c ||Du||_q for u in W_0^{1,q}({\Omega};R^3),
1<q<{\infty} is in general false, even if P is in SO(3), while the
Legendre-Hadamard condition and ellipticity on C^n for the quadratic form |Du
P+(DuP)^T|^2 is satisfied. Thus Garding's inequality may be violated for
formally positive quadratic forms
The geometrically nonlinear Cosserat micropolar shear-stretch energy. Part I: A general parameter reduction formula and energy-minimizing microrotations in 2D
In any geometrically nonlinear quadratic Cosserat-micropolar extended
continuum model formulated in the deformation gradient field and the microrotation field , the shear-stretch energy is necessarily of the form
\begin{equation*} W_{\mu,\mu_c}(R\,;F) := \mu\,\left\lVert{\mathrm{sym}(R^T F -
\boldsymbol{1})}\right\rVert^2 + \mu_c\,\left\lVert{\mathrm{skew}(R^T F -
\boldsymbol{1})}\right\rVert^2\;, \end{equation*} where is the Lam\'e
shear modulus and is the Cosserat couple modulus. In the present
contribution, we work towards explicit characterizations of the set of optimal
Cosserat microrotations
as a function
of and weights and . For , we prove a parameter reduction lemma which reduces the optimality
problem to two limit cases: and .
In contrast to Grioli's theorem, we derive non-classical minimizers for the
parameter range in dimension . Currently,
optimality results for are out of reach for us, but we contribute
explicit representations for which we name
and which arise for
by fixing the rotation axis a priori. Further, we compute the
associated reduced energy levels and study the non-classical optimal Cosserat
rotations for simple planar shear.Comment: 17 pages, 3 figure
The geometrically nonlinear Cosserat micropolar shear-stretch energy. Part II: Non-classical energy-minimizing microrotations in 3D and their computational validation
In any geometrically nonlinear, isotropic and quadratic Cosserat micropolar
extended continuum model formulated in the deformation gradient field and the microrotation field , the shear-stretch energy is necessarily of the form
We aim at the derivation of closed form expressions for the minimizers of
in , i.e., for the set of optimal Cosserat microrotations in
dimension , as a function of . In a previous contribution
(Part I), we have first shown that, for all , the full range of
weights and can be reduced to either a classical or a
non-classical limit case. We have then derived the associated closed form
expressions for the optimal planar rotations in and proved their global
optimality. In the present contribution (Part II), we characterize the
non-classical optimal rotations in dimension n = 3. After a lift of the
minimization problem to the unit quaternions, the Euler-Lagrange equations can
be symbolically solved by the computer algebra system Mathematica. Among the
symbolic expressions for the critical points, we single out two candidates
which we analyze and for which we can
computationally validate their global optimality by Monte Carlo statistical
sampling of . Geometrically, our proposed optimal Cosserat rotations
act in the "plane of maximal strain" and our
previously obtained explicit formulae for planar optimal Cosserat rotations in
reveal themselves as a simple special case. Further, we derive the
associated reduced energy levels of the Cosserat shear--stretch energy and
criteria for the existence of non-classical optimal rotations.Comment: 30 pages, 8 figure
Shells without drilling rotations: a representation theorem in the framework of the geometrically nonlinear 6-parameter resultant shell theory
In the framework of the geometrically nonlinear 6-parameter resultant shell
theory we give a characterization of the shells without drilling rotations.
These are shells for which the strain energy function is invariant under
the superposition of drilling rotations, i.e. is insensible to the
arbitrary local rotations about the third director . For
this type of shells we show that the strain energy density can be
represented as a function of certain combinations of the shell deformation
gradient and the surface gradient of ,
namely . For the case of
isotropic shells we present explicit forms of the strain energy function
having this property.Comment: 18 page
Existence of minimizers in the geometrically non-linear 6-parameter resultant shell theory with drilling rotations
The paper is concerned with the geometrically non-linear theory of
6-parametric elastic shells with drilling degrees of freedom. This theory
establishes a general model for shells, which is characterized by two
independent kinematic fields: the translation vector and the rotation tensor.
Thus, the kinematical structure of 6-parameter shells is identical to that of
Cosserat shells. We show the existence of global minimizers for the
geometrically non-linear 2D equations of elastic shells. The proof of the
existence theorem is based on the direct methods of the calculus of variations
using essentially the convexity of the energy in the strain and curvature
measures. Since our result is valid for general anisotropic shells, we analyze
separately the particular cases of isotropic shells, orthotropic shells, and
composite shells
On the dislocation density tensor in the Cosserat theory of elastic shells
We consider the Cosserat continuum in its finite strain setting and discuss
the dislocation density tensor as a possible alternative curvature strain
measure in three-dimensional Cosserat models and in Cosserat shell models. We
establish a close relationship (one-to-one correspondence) between the new
shell dislocation density tensor and the bending-curvature tensor of
6-parameter shells
The exponentiated Hencky-logarithmic strain energy. Part III: Coupling with idealized isotropic finite strain plasticity
We investigate an immediate application in finite strain multiplicative
plasticity of the family of isotropic volumetric-isochoric decoupled strain
energies \begin{align*} F\mapsto W_{_{\rm eH}}(F):=\hat{W}_{_{\rm
eH}}(U):=\{\begin{array}{lll} \frac{\mu}{k}\,e^{k\,\|{\rm dev}_n\log
{U}\|^2}+\frac{\kappa}{{\text{}}{2\, {\hat{k}}}}\,e^{\hat{k}\,[{\rm tr}(\log
U)]^2}&\text{if}& {\rm det}\, F>0,\\ +\infty &\text{if} &{\rm det} F\leq 0,
\end{array}.\quad \end{align*} based on the Hencky-logarithmic (true, natural)
strain tensor . Here, is the infinitesimal shear modulus,
is the infinitesimal bulk modulus with
the first Lam\'{e} constant, are dimensionless fitting
parameters, is the gradient of deformation,
is the right stretch tensor and is the deviatoric part of the strain tensor
. Based on the multiplicative decomposition , we couple
these energies with some isotropic elasto-plastic flow rules defined in
the plastic distortion , where is the subdifferential of
the indicator function of the convex elastic domain
in the mixed-variant -stress space and . While may loose ellipticity, we show that
loss of ellipticity is effectively prevented by the coupling with plasticity,
since the ellipticity domain of on the one hand, and the
elastic domain in -stress space on the other hand, are closely
related
Comparison of isotropic elasto-plastic models for the plastic metric tensor
We discuss in detail existing isotropic elasto-plastic models based on
6-dimensional flow rules for the positive definite plastic metric tensor
and highlight their properties and interconnections. We show
that seemingly different models are equivalent in the isotropic case
Loss of ellipticity for non-coaxial plastic deformations in additive logarithmic finite strain plasticity
In this paper we consider the additive logarithmic finite strain plasticity
formulation from the view point of loss of ellipticity in elastic unloading. We
prove that even if an elastic energy defined in
terms of logarithmic strain , where , is everywhere
rank-one convex as a function of , the new function need not remain rank-one convex at
some given plastic stretch (viz. ). This is in
complete contrast to multiplicative plasticity in which remains rank-one convex at every plastic distortion if
is rank-one convex. We show this disturbing feature with the
help of a recently considered family of exponentiated Hencky energies.Comment: arXiv admin note: text overlap with arXiv:1409.755
- β¦