In the present paper, an evaluation of the damage behaviour of quasi-brittle composites exposed to mechanical, thermal, and other loads is studied by means of viscoelastic and/or viscoplastic material models, applying some non-local regularisation techniques to the initiation and development of damages. The methods above are presented as a strong tool for a deeper understanding of material structures in miscellaneous engineering disciplines like civil, mechanical, and many others. Nevertheless, all of the software packages reflect certain compromises between the need for effective computational tools, with parameters obtained from inexpensive experiments, within the possibilities and the complexity of both physical and geometrical descriptions of structure deformation within processes. The article is devoted to the mathematical aspects regarding a considerably wide class of computational modelling problems, emphasising the following ones: (i) the existence and the uniqueness of solutions of engineering problems formulated in terms of the deterministic initial and boundary value problems of partial differential equations theory; (ii) the problems of convergence of computational algorithms applied to (i). Both aspects have numerous references to possible generalisations and investigations connected with open problems.

Let Grk,m be the subgroup of the jet group Grk formed by elements projectable to Grm ? Grm ? {jr 0 idRk?m }. We define a partition V on reg Jr0 (Rk,Rm)0 of Gr m-orbits with respect to the left action defined by the jet composition. Elements of reg Jr0 (Rk,Rm)0 are considered as vectors from Rk with the standard inner product over them. The r-th order Grassmannian Gr(r, k,m) is defined as the basis of the principal bundle ?p# : reg Jr0 (Rk,Rm)0 ? V identified with the reduction of the principal bundle ? p : Gr k ? Gr k,m \Gr k to the structure group Gr m ? Gr m ? {jr 0 idRk?m}. Adding the claim of the first-order orthonormality and modifying V to VOrt we obtain the geometrical structure over Gr(r, k,m) in the form of the so-called orthonormal Grassmann bundle ?p# Ort. We construct an atlas on Gr(r, k,m) from a finite system of local sections of ? p# Ort and define the r-th order Grassmannian bundle functor with standard fiber Gr(r, k,m) on the category Mfm of m-dimensional manifolds and local diffeomorphism. For the jet algebra Dr k, a Weil algebra A = Dr k/I and the projection homomorphism pA : Dr k ? A we define the partition VA on reg TA 0 Rm formed by orbits of the left action of the group of all TA 0 h; h ? Diff0 Rm, which is by [29] identified with Gr m. We prove that the local sections of ?p# Ort above satisfy some kind of TA-respecting property and determine an atlas on VA. We define the Weil Grassmannian Gr(A,m) = VA and the bundle functor GrA defined onMfm with Gr(A,m) as its standard fiber. We prove the coincidence of Gr(A,m) with the quotient of V by the map [epA,Rm ] : V ? VA induced by pA. We define the principle bundle ?p# A : reg Jr0 (Rk,Rm)0 ? Gr(A,m) with the structure group Gr m. We define a partition VA on reg TA 0 Rm coarser then VA and some auxiliary partitions within reg TA 0 Rm. It is proved that the factorization [?p#A] : reg TA0 Rm ? VA of ?p#A to reg TA 0 Rm can be considered as the disjoint union of bundles with standard fiber identified with some sublagebra of A of widthA = m.SA nema

Let M be an m-dimensional manifold and A = D_k^r/I = R⊕N_A a Weil algebra of height r. We prove that any A-covelocity T_x^Af ∈ T_x^A*M, x ∈ M is determined by its values over arbitrary max{width A,m} regular and under the first jet projection linearly independent elements of T_x^AM. Further, we prove the rigidity of the so-called universally reparametrizable Weil algebras. Applying essentially those partial results we give the proof of the general rigidity result T^A*M ≃ T^r*M without coordinate computations, which improves and generalizes the partial result obtained in Tomáš (2009) from m ≥ k to all cases of m. We also introduce the space J^A(M,N) of A-jets and prove its rigidity in the sense of its coincidence with the classical jet space J^r(M,N).