Extensions of Kronecker product algebra with applications in continuum and computational mechanics

Summary Kronecker product algebra is widely applied in control theory. However, it does not appear to have been commonly applied to continuum and computational mechanics (CCM). In broad terms the goal of the current investigation is to extend Kronecker product algebra so that it can be broadly applied to CCM. Many CCM quantities, such as the tangent compliance tensor in finite strain plasticity, are very elaborate or difficult to derive when expressed in terms of tensor indicial or conventional matrix notation. However, as shown in the current article, with some extensions Kronecker product algebra can be used to derive compact expressions for such quantities. In the following, Kronecker product algebra is reviewed and there are given several extensions, and applications of the extensions are presented in continuum mechanics, computational mechanics and dynamics. In particular, Kronecker counterparts of quadratic products and of tensor outer products are presented. Kronecker operations on block matrices are introduced. Kronecker product algebra is extended to third and fourth order tensors. The tensorial nature of Kronecker products of tensors is established. A compact expression is given for the differential of an isotropic function of a second-order tensor. The extensions are used to derive compact expressions in continuum mechanics, for example the transformation relating the tangent compliance tensor in finite strain plasticity in undeformed to that in deformed coordinates. A compact expression is obtained in the nonlinear finite element method for the tangent stiffness matrix in undeformed coordinates, including the effect of boundary conditions prescribed in the current configuration. The aforementioned differential is used to derive the tangent modulus tensor in hyperelastic materials whose strain energy density is a function of stretch ratios. Finally, block operations are used to derive a simple asymptotic stability criterion for a damped linear mechanical system in which the constituent matrices appear explicitly.Appendix: Notation A, Â, a_ij matrix, second-order tensor - a, a_i vector, VEC (A) - a vector - â scalar - B, b_ij matrix, second-order tensor - b, b_i VEC (B) - b vector - block permutation matrix - C, C, c_ijkl fourth-order tensor - right Cauchy Green strain tensor - c VEC () - c_i eigenvalues of - C _a,C _b third-order tensors - C ₁,C ₂,C ₃ outer product functions - D deformation rate tensor - D damping matrix - d VEC (D) - E boundary stiffness matrix - e VEC () - Eulerian strain - F isotropic tensor-valued function ofA - deformation gradient tensor - f VEC - f scalar valued counterpart ofF - G coordinate transformation tensor - G strain-displacement matrix - g VEC (G) - g consistent force vector - H dynamic system matrix - h_n lowest eigenvalue ofH - I,I _n,I ₉ identity matrix/tensor - i VEC (I) - I₁ TRACE () - I index - i index - J determinant of - J matrix relating d to da - J index - j index - J matrix relating d to da - K,K _T,K _b,K stiffness matrices - K index - k index - L velocity gradient tensor - L index - l index - M mass matrix - strain-displacement matrix - M matrix arising from Ogden model - M index - m index - N shape function matrix - N matrix arising from Ogden model - n, n₀ exterior normal vectors - n n² - N index - n index - P, dynamic system matrix - p VEC (P) - p_n eigenvalue ofP - Q rotation tensor - R,r _ij tensor used with outer products - r rank, index - r VEC (R) - S,s _ij tensor used with outer products - s VEC (S) - S, S₀ surface area - S matrix diagonalizingA - s index - T unitary matrix - t₀, t traction - t VEC () - VEC ( ) - t time - T time interval - U _n ,U ₉,U _M permutation matrices - u displacement vector - V matrix appearing in linear system stability criterion - V projection matrix - W,W _i multipliers d - w, w₁ vectors - strain energy function - X undeformed position - x deformed position - Y,Y coordinate system - y, y_j vectors - z, z_j vectors - _j eigenvalue ofA - _j eigenvalue ofB - , ₁, ₂ nodal displacement vectors - _j eigenvalue of - matrix - , ₁, ₂, _a, _b diagonal matrices - _ij entries of the Kronecker tensor (I) - Lagrangian strain - _ijk permutation tensor - _i coefficient of Ogden model - parameter in linear system stability criterion - _i eigenvalue - Lamé coefficient - matrix - surface area factor - Lamé coefficient - _j eigenvalue ofA - _i coefficient of Ogden model - matrix - second Piola-Kirchhoff stress - Cauchy stress - Truesdell stress flux - matrix/tensor - matrix/tensor - matrix/tensor - TEN22 (C) - d rotation vector - d rotation tensor - TRACE(.) trace of a matrix - left Kronecker function - right Kronecker function - VEC(.) vectorization operator - VECB(.) block vectorization operator - TEN22(.) tensor operator - TEN12(.) tensor operator - TEN21(.) tensor operator - _x, _y divergence operator - d(.) differential operator - (.) variational operator - (right) Kronecker product - Kronecker sum - Kronecker difference - block Kronecker product - left Kronecker product - AsB block Kronecker sum ofA andB - AdB block Kronecker difference ofA and      

New models of continuum mechanics

Possible ways of transition from traditional models of continuum mechanics to new-generation models, using a model of a multicomponent model as an example, are considered.     

Rate formulations in nonlinear continuum mechanics

In solving problems of geometrically nonlinear structural mechanics, a prominent role is played by formulation of rate equilibrium conditions. In the computational machinery, the evaluation of the stiffness operator provides the trial incremental displacement field as fixed point of an iterative algorithm. The issue is investigated by a new geometric approach to continuum mechanics. Kinematics is described by the motion along a trajectory manifold embedded in the affine four-dimensional space-time. Variational conditions of equilibrium and rate equilibrium are formulated in terms of natural time rates of stress and stretching. The rate elastostatic problem is formulated in the full nonlinear context by adopting a newly contributed rate-elastic constitutive model. The geometric stiffness and forcing operators are expressed in terms of an arbitrary linear spatial connection. It is shown that the adoption of a Levi- Civita connection provides a linear expression of the geometric stiffness involving a curvature term. For bodies in motion in the flat Euclid space with parallel transport by translation, a symmetric expression of the geometric stiffness is obtained, thus extending the standard formula to bodies of any dimensionality.     

Micropolar continuum mechanics of fractal media

This paper builds on the recently begun extension of continuum thermomechanics to fractal media which are specified by a fractional mass scaling law of the resolution length scale R. The focus is on pre-fractal media (i.e., those with lower and upper cut-offs) through a technique based on a dimensional regularization, in which the fractal dimension D is also the order of fractional integrals employed to state global balance laws. In effect, the governing equations are cast in forms involving conventional (integer-order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving D and R, as well as a surface fractal dimension d. While the original formulation was based on a Riesz measure—and thus more suited to isotropic media—the new model is based on a product measure capable of describing local material anisotropy. This measure allows one to grasp the anisotropy of fractal dimensions on a mesoscale and the ensuing lack of symmetry of the Cauchy stress. This naturally leads to micropolar continuum mechanics of fractal media. Thereafter, the reciprocity, uniqueness and variational theorems are established.     

Phenomenological evolution equations for the backstress and spin tensors

Summary A formulation of a constitutive model involving a new kinematic hardening rule is presented in the Eulerian reference system. The corotational stress and backstress rates involving spin tensors are discussed and incorporated in the evolution equations. The backstress evolution model is compared with other models and is found to be decomposable into a model that consists of two backstresses whose evolution is independent of each other.The elasto-plastic stiffness tensor is derived for all the models considered. It is shown in the proposed backstress evolution model we obtain an implicit system of differential equation in the stress and backstress rates. The theory is applied to two yield functions: one of the von-Mises type and the other of the anisotropic type. It is shown that for both these yield criteria the stress evolution is independent of the stress rate. As an example, the torsion of a cylindrical bar with fixed ends is investigated.     

Concerning the fictitious continuum in damage mechanics

Consistent theories to describe damage processes are generally presented within the framework of effective stress and internal parameters. It is well known that damage is concerned with the progressive deterioration of elastic properties due to microscopic defects, such microvoids or microcracks. In the framework of Continuum Mechanics, damage is related to irreversible changes (on the microlevel) of small vicinities surrounding material points in the body. So a convenient definition of these small vicinities, named “representative material element”, will be recalled in Part 1, and application will be made to elastoplasticity in Part 2. In the subsequent parts, a fictitious suitable undamaged elastoplastic body accompanying the real damaged one is introduced in order to define the effective stress in the framework of large strains and its use in the construction of damaged elasticity law. Finally application is made to infinitesimal strains that concern most of the examples in literature. Due to limitation of place, plasticity coupled with damage is not considered in this paper.     

Reduction of mesh sensitivity in continuum damage mechanics

Summary Continuum damage theories can be applied to simulate the failure behaviour of engineering constructions. In the constitutive equations of the material a damage parameter is incorporated. A damage criterion and a damage evolution law are postulated and quantified based on experimental data. The elaboration of the mathematical formulation is performed by common finite element techniques. Without special precautions the numerical results appear to be unacceptably dependent on the measure of the spatial discretization. It is shown that a simple but effective procedure leads to the conservation of objectivity.     

Interpretation of the Rice integral in continuum mechanics

The author starts by defining, thanks to the virtual work principle, the notion of “succion force” which represents in an continuous mediums the influence of the medium itself on each of its molecules. Mathematically this quantity appears as dual to the displacement of the molecule with regard to the medium; in Lagrange variables, it is analogue to a mass force expressing the influence of the extorior. This force is shown to act only on the points where the medium is elastically inhomogeneous (inclusion, crack, plastified zone…). The study is made for any particular position, eventually a very deformed one. The vectorial sum of these forces in a given volume is calculated with the help of an integral stretched to the surface which limits this volume. This integral generalises the Rice integral which had only been written in the case of the approximation of small deformations.     

The archetype-genome exemplar in molecular dynamics and continuum mechanics

We argue that mechanics and physics of solids rely on a fundamental exemplar: the apparent properties of a system depend on the building blocks that comprise it. Building blocks are referred to as archetypes and apparent system properties as the system genome. Three entities are of importance: the archetype properties, the conformation of archetypes, and the properties of interactions activated by that conformation. The combination of these entities into the system genome is called assembly. To show the utility of the archetype-genome exemplar, this work presents the mathematical ingredients and computational implementation of theories in solid mechanics that are (1) molecular and (2) continuum manifestations of the assembly process. Both coarse-grained molecular dynamics (CGMD) and the archetype-blending continuum (ABC) theories are formulated then applied to polymer nanocomposites (PNCs) to demonstrate the impact the components of the assembly triplet have on a material genome. CGMD simulations demonstrate the sensitivity of nanocomposite viscosities and diffusion coefficients to polymer chain types (archetype), polymer–nanoparticle interaction potentials (interaction), and the structural configuration (conformation) of dispersed nanoparticles. ABC simulations show the contributions of bulk polymer (archetype) properties, occluded region of bound rubber (interaction) properties, and microstructural binary images (conformation) to predictions of linear damping properties, the Payne effect, and localization/size effects in the same class of PNC material. The paper is light on mathematics. Instead, the focus is on the usefulness of the archetype-genome exemplar to predict system behavior inaccessible to classical theories by transitioning mechanics away from heuristic laws to mechanism-based ones. There are two core contributions of this research: (1) presentation of a fundamental axiom—the archetype-genome exemplar—to guide theory development in computational mechanics, and (2) demonstrations of its utility in modern theoretical realms: CGMD, and generalized continuum mechanics.     

Modelling DNA loops using continuum and statistical mechanics

The classical Kirchhoff elastic-rod model applied to DNA is extended to account for sequence-dependent intrinsic twist and curvature, anisotropic bending rigidity, electrostatic force interactions, and overdamped Brownian motion in a solvent. The zero-temperature equilibrium rod model is then applied to study the structural basis of the function of the lac repressor protein in the lac operon of Escherichia coli. The structure of a DNA loop induced by the clamping of two distant DNA operator sites by lac repressor is investigated and the optimal geometries for the loop of length 76 bp are predicted. Further, the mimicked binding of catabolite gene activator protein (CAP) inside the loop provides solutions that might explain the experimentally observed synergy in DNA binding between the two proteins. Finally, a combined Monte Carlo and Brownian dynamics solver for a worm-like chain model is described and a preliminary analysis of DNA loop-formation kinetics is presented.     

Classification of the models of materials in continuum mechanics

The classification of the models of materials in continuum mechanics proposed by the author on the basis of the general theory of Noll constitutive relations is developed by using the methods of rational continuum mechanics. __________ Translated from Problemy Prochnosti, No. 5, pp. 79–89, September–October, 2006. Report on International Conference “Dynamics, Strength, and Life of Machines and Structures” (1–4 November 2006, Kiev, Ukraine).     

On a generalized system of equations in continuum mechanics

Summary A system of equations of which the equations of elasticity and the Stokes equations of hydrodynamics are particular cases, is examined. Galerkin-type representations are constructed for this system with the aid of a matrix inversion technique. These representations give rise to the fundamental singular solution which together with a derived reciprocal relationship yield integral representations for the unknown parameters in the given system of equations. The integral representations lead in a natural way to the introduction of surface potentials whose properties are stated. Some well-known cases are deduced from our general results.     

Finite elements on the basis of continuum mechanics

The formulation of finite element models on the basis of different variational principles is reviewed. The degrees-of-freedom of the elements are defined in an abstract way without the help of nodal points. In this manner it is possible to describe elements of arbitrary shape and accuracy. The formulation is confined to linear elasto-statics. For two-dimensional structures two hybrid element models are developed using Legendre polynomials on the element boundaries. Examples of plane stress problems are used to test the generation of convergence by increasing the accuracy of the elements vs. by increasing the number of elements.     

On Hill's stress rate in the continuum mechanics of polycrystals

Summary The lattice corotational stress rate, suggested byHill [1], is contrasted with the material co-rotational (Jaumann-Zaremba) stress rate and shown to be preferred in phenomenological continuum models of crystal behavior. A simple application to a crystal deforming via quasi-static single slip is included.

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Zur Hillschen Spannungsgeschwindigkeit in der Kontinuumsmechanik von Vielkristallen

Zusammenfassung Die vonHill [1] vorgeschlagene, mit dem Gitter mitrotierende Spannungsgeschwindigkeit wird der mit der Materie mitrotierenden (Jaumann-Zaremba) gegenübergestellt. Es wird gezeigt, daß erstere in einem phänomenologischen Kontinuumsmodell des Kristallverhaltens vorzuziehen ist. Eine einfache Anwendung auf den durch quasistatisches Gleiten sich verformenden Kristall wird angegeben.

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This work was supported by the Office of Naval Research and by the Advanced Research Projects Agency of the Department of Defense under Contract No. N00014-68-A-0187.     

On the mathematical modelling of material behavior in continuum mechanics

Summary The classical theories of continuum mechanics — linear elasticity, viscoelasticity, plasticity and hydrodynamics — are defined by special constitutive equations. These can be understood to be asymptotic approximations of a quite general constitutive model, valid under restrictive assumptions for the stress functional or the input processes. The general theory of material behavior develops systematic methods to represent material properties in a context of physical evidence and mathematical consistency. According to experimental observations material behavior may be rate independent or rate dependent with or without equilibrium hysteresis. This motivates four different constitutive theories, namely elasticity, plasticity, viscoelasticity and viscoplasticity. Constitutive equations can be formulated explicitly as functionals. Then, the particular constitutive models correspond to continuity properties of these functionals, related to convenient function spaces. On the other hand, a system of differential equations may lead to an implicit definition of a stress functional. In this case additional variables are introduced, which are called internal variables. For these variables additional evolution equations must be formulated, specifying the rate of change of the internal variables in dependence on their present values and the strain (or stress) input. In the context of different models of inelastic material behavior the evolution equations have different mathematical characteristics. These concern the existence of equilibrium solutions and their stability properties. Rate independent material behavior is modelled by means of evolution equations, which are related to an arclength instead of the time as independent variable. It can be shown that the rate independent constitutive equations of elastoplasticity are the asymptotic limit of rate dependent viscoplasticity for slow deformation processes.This paper is an extended version of a lecture held at the First Conference of the GAMM working group on material theory in Stuttgart, Germany, February 28, 1992. The author thanks Prof. Dr. F. Ziegler for the opportunity to participate in this conference.     

Probabilistic analysis of continuum damage mechanics of brittle materials

Uncertainty of material properties in solution of engineering problems is often a fundamental question. Statistical methods give a powerful tool for analysis of uncertainty. Monte Carlo simulations together with Gumbel distribution are used as a possible way to study influence of data dispersion on assessment of damage of brittle materials.     

Numerical prediction of slant fracture with continuum damage mechanics

Ductile specimens always exhibit an inclined fracture surface with an angle relative to the loading axis. This paper reports a numerical study on the cup-cone fracture mode in round bar tensile tests and the slant fracture in plane-strain specimens based on continuum damage mechanics. A combined implicit-explicit numerical scheme is first developed within ABAQUS through user defined material subroutines, in which the implicit solver: Standard, and the explicit solver: Explicit, are sequentially used to predict one single damage/fracture process. It is demonstrated that this numerical approach is able to significantly reduce computational cost for the simulation of fracture tests under quasi-static or low-rate loading. Comparison with various tensile tests on 2024-T351 aluminum alloy is made showing good correlations in terms of the load-displacement response and the fracture patterns. However, some differences exist in the prediction of the critical displacement to fracture.     

An anisotropic theory of elasticity for continuum damage mechanics

This paper presents the development of an anisotropic elastic damage theory. This is achieved by deriving a modified damage effect tensor M(D) for the effective stress equations capable of including the effect of anisotropic material damage. The modified tensor removes the restriction of a priori knowledge of the directions of principal stresses imposed by a damage effect tensor developed earlier and can now be made for general practical engineering applications of failure analysis. Reduction of the proposed tensor to a scalar for isotropic damage is shown to be possible when it is expressed not only in the principal directions but also in any arbitrary coordinate system, a necessary condition to verify the validity of the proposed tensor. Uniaxial tension and pure torsion are chosen to illustrate the application of the theory as well as associated damage variables that may be experimentally determined using laboratory size specimens. The measured damage variables confirm the presence of anisotropic damage from an initially isotropic material specimen and the magnitude is more pronounced at higher stresses and strains.

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Résumé On présente un développement d'une théorie sur l'endommagement élastique anisotrope en déduisant un tenseur modifié décrivant l'effet de l'endommagement pour un système d'équations de contraintes effectives susceptible d'inclure l'effet d'un endommagement dans un matériau anisotrope. Le tenseur modifié supprime la restriction de la connaissance a priori des directions des contraintes principales imposées par un tenseur d'effet d'endommagement développé précédemment; il peut à présent entrer dans les applications pratiques en construction de l'analyse des ruptures.On montre qu'il est possible de réduire le tenseur proposé à une valeur scalaire dans le cas d'un dommage isotrope, dès lors qu'il est exprimé non seulement suivant les directions principales, mais dans un système de coordonnées arbitraires, ce qui est une condition nécessaire pour en vérifier la validité.On choisit une traction multiaxiale et une torsion pure pour illustrer l'application de la théorie ainsi que des variables d'endommagement associées, susceptibles d'être déterminées expérimentalement à l'aide d'éprouvettes de laboratoire.Les variables d'endommagement mesurées confirment la présence d'un dommage anisotrope dans le cas d'une éprouvette d'un matériau initialement isotrope; son amplitude est plus prononcée à des contraintes ou des déformations plus importantes.