Remarks on the stability of enhanced strain elements in finite elasticity and elastoplasticity

142

118

SUMMARYEnhanced strain element formulations are known to show an outstanding performance in many applications. The stability of these elements, however, cannot be guaranteed for general deformation states and arbitrarily shaped elements. In order to overcome this deÿciency, we develop an innovative control technique based on a modal analysis on element level. The control is completely automatic in the sense that no artiÿcial factors are introduced. The computational e ort is negligible. The key to the approach is the split of the element tangent matrix into constant and hourglass parts which is not possible for the classical enhanced strain concept in general. This motivates the use of a recently developed reduced integration method, which, since its stabilization part is derived on the basis of the enhanced strain method, shows the same performance and retains the crucial split. Using this formulation in combination with the new control technique, leads to a 'smart' element which is free of hourglass instabilities and generally applicable, also for strongly distorted meshes.

Section: Introductionmentioning

confidence: 94%

A stabilization technique to avoid hourglassing in finite elasticity

Reese

Wriggers

2000

142

118

“…Inserting (2) and (3) in (1), taking the scalar product of the resulting partial di erential equation in u ∈ V with a test function v ∈ V, and imposing directly the traction boundary condition t − t = 0 on @B P , we arrive at the one-ÿeld weak or variational problem VF1:…”

Section: The Continuous Problemmentioning

confidence: 99%

“…The variational problem corresponding to the three-ÿeld formulation (1), (2), and (7) may now be constructed by substituting (7) 1 in (2), and taking the scalar product of (1), (2) and (7) 2 with test functions v ∈ V;M ∈ E, and Q ∈ S, respectively; this leads to the problem of ÿnding u ∈ V, H ∈ E, and P ∈ S that satisfy VF3:…”

Section: The Continuous Problemmentioning

confidence: 99%

A new stabilization technique for finite elements in non-linear elasticity

Reese

Küssner

Reddy

1999

SUMMARYIn the present contribution, an innovative stabilization technique for two-dimensional low-order ÿnite elements is presented. The new approach results in an element formulation that is much simpler than the recently proposed enhanced strain element formulation, yet which gives results of at least the same quality. An important feature in the regime of large deformations is the stability of the element, which is addressed in detail. The main advantages of the new formulation are, besides its simplicity, its computational e ciency and robust behaviour. Only three history variables have to be stored, making this stabilization concept particularly interesting for large-scale problems.

“…In part, these improvements were triggered by papers that pointed out the potential instabilities of the original formulation. Among these were contributions by Wriggers et al [11], de Souza et al [12] and Cris"eld et al [13]. The latter authors proposed a &co-rotational formulation' [13] as a way of overcoming the di$culties.…”

Section: Introductionmentioning

confidence: 99%

A stabilised large-strain elasto-plastic Q1-P0 method

Crisfield

Norris

1999

SUMMARYThe mean dilatation method e!ectively involves bi-linear displacements and a constant pressure and is often known as the Q1-P0 formulation. Its non-linear implementation was originally derived as a three-"eld formulation which included the volume ratio via the Jacobian, J, of the deformation gradient as an additional separate variable. However, the latter term was not directly required in the numerical implementation once J was assumed constant along with the pressure. This formulation will here be termed the non-linear Q1-P0 method. It is known to give good solutions for many practical large-strain elasto-plastic problems. However, for some problems, it has been shown to be prone to severe &hour-glassing'. With a view to remedying this situation, we here re-visit the three-"eld formulation and derive a modi"ed form, which although variationally valid, is over-sti! in comparison to the original procedure (here simply called the Q1-P0 method). However, the concepts lead to a natural method for stabilising the Q1-P0 technique. The associated tangent sti!ness matrix is symmetric.