The quasiconvex envelope of the Saint Venant–Kirchhoff stored energy function

1995 ◽  
Vol 125 (6) ◽  
pp. 1179-1192 ◽  
Author(s):  
Hervé Le Dret ◽  
Annie Raoult
Keyword(s):  
Explicit Expression ◽  
Energy Function ◽  
Singular Values ◽  
Convex Envelope ◽  
External Loading ◽  
Stored Energy ◽  
Quasiconvex Envelope

We give an explicit expression for the quasiconvex envelope of the Saint Venant–Kirchhoff stored energy function in terms of the singular values. This envelope is also the convex, polyconvex and rank 1 convex envelope of the Saint Venant–Kirchhoff stored energy function. Moreover, it coincides with the Saint Venant–Kirchhoff stored energy function itself on, and only on, the set of matrices whose singular values arranged in increasing order are located outside an ellipsoid. It vanishes on, and only on, the set of matrices whose singular values are less than 1. Consequently, a Saint Venant–Kirchhoff material can be compressed under zero external loading.

On the global stability of two-dimensional, incompressible, elastic bars in uniaxial extension

2009 ◽  
Vol 466 (2116) ◽  
pp. 1167-1176 ◽  
Author(s):  
Jeyabal Sivaloganathan ◽  
Scott J. Spector
Keyword(s):  
Elastic Energy ◽  
Energy Function ◽  
Shear Bands ◽  
Uniaxial Extension ◽  
Two Dimensional ◽  
Stored Energy ◽  
Absolute Minimizer ◽  
Local Bifurcations ◽  
Maximum Value ◽  
Incompressible Hyperelastic Material

When a rectangular bar is subjected to uniaxial tension, the bar usually deforms (approximately) homogeneously and isoaxially until a critical load is reached. A bifurcation, such as the formation of shear bands or a neck, may then be observed. One approach is to model such an experiment as the in-plane extension of a two-dimensional, homogeneous, isotropic, incompressible, hyperelastic material in which the length of the bar is prescribed, the ends of the bar are assumed to be free of shear and the sides are left completely free. It is shown that standard constitutive hypotheses on the stored-energy function imply that no such bifurcation is possible in this model due to the fact that the homogeneous isoaxial deformation is the unique absolute minimizer of the elastic energy. Thus, in order for a bifurcation to occur either the material must cease to be elastic or the stored-energy function must violate the standard hypotheses. The fact that no local bifurcations can occur under the assumptions used herein was known previously, since these assumptions prohibit the load on the bar from reaching a maximum value. However, the fact that the homogeneous deformation is the absolute minimizer of the energy appears to be a new result.

scholarly journals The Quasiconvex Envelope of Conformally Invariant Planar Energy Functions in Isotropic Hyperelasticity

2020 ◽  
Vol 30 (6) ◽  
pp. 2885-2923
Author(s):  
Robert J. Martin ◽  
Jendrik Voss ◽  
Ionel-Dumitrel Ghiba ◽  
Oliver Sander ◽  
Patrizio Neff
Keyword(s):  
Singular Values ◽  
Elliptic Partial Differential Equations ◽  
Singular Value ◽  
Energy Functions ◽  
Conformally Invariant ◽  
Princeton University ◽  
Common Representation ◽  
Special Cases ◽  
Quasiconvex Envelope ◽  
Positive Determinant

Abstract We consider conformally invariant energies W on the group $${{\,\mathrm{GL}\,}}^{\!+}(2)$$ GL + ( 2 ) of $$2\times 2$$ 2 × 2 -matrices with positive determinant, i.e., $$W:{{\,\mathrm{GL}\,}}^{\!+}(2)\rightarrow {\mathbb {R}}$$ W : GL + ( 2 ) → R such that $$\begin{aligned} W(A\, F\, B) = W(F) \quad \text {for all }\; A,B\in \{a\, R\in {{\,\mathrm{GL}\,}}^{\!+}(2) \,|\,a\in (0,\infty ),\; R\in {{\,\mathrm{SO}\,}}(2)\}, \end{aligned}$$ W ( A F B ) = W ( F ) for all A , B ∈ { a R ∈ GL + ( 2 ) | a ∈ ( 0 , ∞ ) , R ∈ SO ( 2 ) } , where $${{\,\mathrm{SO}\,}}(2)$$ SO ( 2 ) denotes the special orthogonal group and provides an explicit formula for the (notoriously difficult to compute) quasiconvex envelope of these functions. Our results, which are based on the representation $$W(F)=h\bigl (\frac{\lambda _1}{\lambda _2}\bigr )$$ W ( F ) = h ( λ 1 λ 2 ) of W in terms of the singular values $$\lambda _1,\lambda _2$$ λ 1 , λ 2 of F, are applied to a number of example energies in order to demonstrate the convenience of the singular-value-based expression compared to the more common representation in terms of the distortion $${\mathbb {K}}:=\frac{1}{2}\frac{\Vert F \Vert ^2}{\det F}$$ K : = 1 2 ‖ F ‖ 2 det F . Applying our results, we answer a conjecture by Adamowicz (in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni, vol 18(2), pp 163, 2007) and discuss a connection between polyconvexity and the Grötzsch free boundary value problem. Special cases of our results can also be obtained from earlier works by Astala et al. (Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton University Press, Princeton, 2008) and Yan (Trans Am Math Soc 355(12):4755–4765, 2003). Since the restricted domain of the energy functions in question poses additional difficulties with respect to the notion of quasiconvexity compared to the case of globally defined real-valued functions, we also discuss more general properties related to the $$W^{1,p}$$ W 1 , p -quasiconvex envelope on the domain $${{\,\mathrm{GL}\,}}^{\!+}(n)$$ GL + ( n ) which, in particular, ensure that a stricter version of Dacorogna’s formula is applicable to conformally invariant energies on $${{\,\mathrm{GL}\,}}^{\!+}(2)$$ GL + ( 2 ) .

Dynamical modelling of phase transitions by means of viscoelasticity in many dimensions

1992 ◽  
Vol 121 (1-2) ◽  
pp. 101-138 ◽  
Author(s):  
Piotr Rybka
Keyword(s):  
Phase Transitions ◽  
Energy Function ◽  
Boundary Data ◽  
Initial Conditions ◽  
Stored Energy ◽  
Dynamical Modelling

SynopsisWe study the equations of viscoelasticity in a multidimensional setting for the ‘no-traction’ boundary data. For the sake of modelling phase transitions we do not assume elliptieity of the stored energy function W. We construct dynamics in W1,2(Ωℝn) globally in time. Next, we study the question of stability for a class of equilibria. Moreover, we show a certain kind of decay in time of solutions for arbitrary initial conditions.

Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber

1951 ◽  
Vol 243 (865) ◽  
pp. 251-288 ◽  
Keyword(s):  
Energy Function ◽  
Pure Shear ◽  
Thin Sheet ◽  
The Other ◽  
Homogeneous Deformation ◽  
Simple Extension ◽  
Stored Energy ◽  
Elastic Deformations ◽  
Isotropic Materials ◽  
Deformation Curves

It is shown in this part how the theory of large elastic deformations of incompressible isotropic materials, developed in previous parts, can be used to interpret the load-deformation curves obtained for certain simple types of deformation of vulcanized rubber test-pieces in terms of a single stored-energy function. The types of experiment described are: (i) the pure homogeneous deformation of a thin sheet of rubber in which the deformation is varied in such a manner that one of the invariants of the strain, I 1 or I 2 , is maintained constant; (ii) pure shear of a thin sheet of rubber (i.e. pure homogeneous deformation in which one of the extension ratios in the plane of the sheet is maintained at unity, while the other is varied); (iii) simultaneous simple extension and pure shear of a thin sheet (i.e. pure homogeneous deformation in which one of the extension ratios in the plane of the sheet is maintained constant at a value less than unity, while the other is varied); (iv) simple extension of a strip of rubber; (v) simple compression (i.e. simple extension in which the extension ratio is less than unity); (vi) simple torsion of a right-circular cylinder; (vii) superposed axial extension and torsion of a right-circular cylindrical rod. It is shown that the load-deformation curves in all these cases can be interpreted on the basis of the theory in terms of a stored-energy function W which is such that δ W /δ I 1 is independent of I 1 and I 2 and the ratio (δ W /δ I 2 ) (δ W /δ I 1 ) is independent of I 1 and falls, as I 2 increases, from about 0*25 at I 2 = 3.

A Simple w′(λ) Function for The Valanis-Landel Form of Stored Energy Function

1998 ◽  
Vol 71 (2) ◽  
pp. 234-243 ◽  
Author(s):  
Robert F. Landel
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Long Range ◽  
Energy Function ◽  
Good Accuracy ◽  
Linear Range ◽  
Stored Energy ◽  
Element Analysis ◽  
Strong Curvature

Abstract In the Valanis-Landel formulation of the stored energy function W, stresses depend on the function w′(λ)(=dw/dλ). This function exhibits strong curvature, making it difficult to represent analytically with good accuracy. It is found for both SBR and NR that the function λw′(λ) is not only far less curved, it is essentially linear in λ for the range of about 0.4 < λ < 2.0. The long range of simple proportionality to strain invites examination of molecular theories of rubberlike elasticity. Above the linear range the response can be approximated by kλn. These simplifications should make it easier to convert w′(λ) to the W1 and W2 functions employed in finite element analysis.

Sign in / Sign up

Export Citation Format

Share Document