By D. Kolymbas
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Additional info for Introduction to hypoplasticity
A series of experimental data were co~municated to the participants in order to let them calibrate their models. Based on this, the participants were invited to predict some other tests, the outcomes of which were at that time kept secret. 14. FEM-implementations o f hypoplasticity 51 Such competitions appear reasonable, however none of them was conclusive. The reason is that there is no objective measure for the quality of the predictions. And, much more, there is no way to rate the calibration effort versus the quality of results.
7 Response envelopes The response envelopes introduced by GUDEHUS  to model axisymmetric response are very useful in discussing properties of constitutive models. With reference to a given axisymmetric stress state we can apply all possible axisymmetric stretchings D with ID( = 1 by setting D= ( - siria 0 0 0 -cosa/-JZ 0 0 0 - cos a/&) with 0" 5 Q 5 360". The corresponding polar representation of the obtained responses T is called the response envelope. g. for hypoelasticity) response envelopes are ellipses: For the aforementioned D-tensor a linear constitutive equation T = f(D) assumes the form which is the parametric representation of an ellipse in the Irl-62-space.
What about the unique determination of D when T is prescribed? To answer this question of unique invertibility we2 multiply the equation3 T = LD NlDl with the inverse operator L-' and obtain + A := L - ~ T =D +L-~NIDI or D =A - BID1 with B := L-lN. 1): D D = (A - BIDI) - (A - BIDI) = A A * * Noting that D - D Its solution reads i IDi2 we qp= 2 A - BID1 + B - BJDI2 . 2) is a quadratic equation for x := IDI. 2 A B f J4(A * - - B)2- 4 A 2(B - B - A(B B - 1) 1) 'private communication by J . N A D E R *The author is indebted to Dr.