By D. Kolymbas

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The method itself consists in establishing equilibrium not at every point of a continuous body but only at some selected points, the so-called nodes. This approach can be derived from the field equations of equilibrium, daij/drcj pgi = 0, by multiplying them with U , , the field of virtual displacements, and setting the integral over the body equal to zero. The resulting equation is known as the weak form of the equilibrium conditions. Applying the theorem of GAUSSand using appropriate virtual displacements leads to the aforementioned equilibrium conditions for the nodes.

Chapter 4. Hypoplasticity 48 A more general representation of the cyclic behaviour in hypoplasticity requires an additional state variable such as a structure tensor E351 which involves the recent deformation history. A ‘memory function’ [4] or an ‘intergranular strain’ [51] have been proposed for this purpose. 6 was published in 1985 [33]: 1 T = C1-(TD + D T ) 2 + Cztr(TD)l+ C3T+ C4- 1m trT2 T -I L . In order to remove some deficiencies, WU WEI [76] modified the tensorial terms and obtained the aforementioned relation 3 % h m + cT* ,--m.

If we deform a rate-independent material twice as fast, then the stress rate will also be doubled. e. constitutive equations of the type T = h(T,D)), rate-independence means that the stress rate T is positively homogeneous of the first degree with respect to D: h(T,XD) = Xh(T,D) for X > 0 . Note that this homogeneity does by no means imply linearity (cf. the relation g = Jzl, which is homogeneous in the above sense, but not linear). Soils are not exactly rateindependent. Clays are more pronouncedly rate dependent than sands, but also sands exhibit rate dependence [12].

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