A lower bound estimate of the critical load for compressible elastic solids

On the Stability of Incompressible Elastic Cylinders in Uniaxial Extension

“…Equivalently, one can derive estimates on the energy that ensure that the second variation of E is strictly positive at the deformation u h λ and, consequently, that u h λ is both linearization stable and a strict weak relative minimizer. For a compressible cylindrical solid in tension this technique was used by Spector [27] and, more recently, by Del Piero and Rizzoni [7] and Fosdick, Foti, Fraddosio, and Piccioni [10] (see, also, Del Piero [6]) to obtain estimates upon the values of λ where bifurcations cannot occur. Additionally, in [7] and [10] similar estimates are derived for compression (λ ≤ 1); incompressible materials are also considered in [7].…”

Geometric numerical integrators based on the Magnus expansion in bifurcation problems for non-linear elastic solids

“…A more detailed discussions on such a choice, together with an exhaustive analysis of problems closely related to the one here presented, are developed in [17], [18]; in particular, in [17] one may also find stability issues based on [19] and partly on [20]. This allows us to skip here the description of certain analytical developments, for which we refer to the works above mentioned.…”

Shear driven planar Couette and Taylor-like instabilities for a class of compressible isotropic elastic solids

“…In fact, there are some choices of material parameters for which C(α)[H] · H looses its positive definiteness property in Lin 1 at a certain value of α > 0, but inequality (4.2) continues to remain continuously valid for larger values of α and the simple shear statẽ χ is Hadamard stable for these larger shear strains. In order to report something meaningful regarding the Hadamard stability ofχ in such situations, we develop in [9] a lower bound α lower for the critical load by adapting to the present case, and improving upon, a lower bound result of Del Piero [4]. Specifically, in [9] we first write grad u = E + W, where E ∈ Sym, W ∈ Skew, and re-write the Hadamard integral in where β * (α) is structured in accordance with the first integrand of the integral in (4.8) and is defined by The function m(α, η), defined for each η ∈ J :=]−(t 2 (α) + t 3 (α)), ∞[, is structured by the form of the second and third integrands of the integral in (4.8) and the ordering of the principal stresses of T(α) as…”

“…Introducing the Korn constant K (independent of α becauseχ maps Ω ∪ ∂ 1 Ω ∪ ∂ 2 Ω into itself), we argue in [9] that m(α, η) has the form…”

“…In[9], our arguments for estimating critical loads apply to more general deformations than homogeneous pure shearthe focus of the present paper 6. Restricted in order to be compatible with x 1 -independence expressed in the definition of the set A in (4.1).…”