Tensor Function Representations Research Articles

Low-Rank Tensor Function Representation for Multi-Dimensional Data Recovery.

Since higher-order tensors are naturally suitable for representing multi-dimensional data in real-world, e.g., color images and videos, low-rank tensor representation has become one of the emerging areas in machine learning and computer vision. However, classical low-rank tensor representations can solely represent multi-dimensional discrete data on meshgrid, which hinders their potential applicability in many scenarios beyond meshgrid. To break this barrier, we propose a low-rank tensor function representation (LRTFR) parameterized by multilayer perceptrons (MLPs), which can continuously represent data beyond meshgrid with powerful representation abilities. Specifically, the suggested tensor function, which maps an arbitrary coordinate to the corresponding value, can continuously represent data in an infinite real space. Parallel to discrete tensors, we develop two fundamental concepts for tensor functions, i.e., the tensor function rank and low-rank tensor function factorization, and utilize MLPs to paramterize factor functions of the tensor function factorization. We theoretically justify that both low-rank and smooth regularizations are harmoniously unified in LRTFR, which leads to high effectiveness and efficiency for data continuous representation. Extensive multi-dimensional data recovery applications arising from image processing (image inpainting and denoising), machine learning (hyperparameter optimization), and computer graphics (point cloud upsampling) substantiate the superiority and versatility of our method as compared with state-of-the-art methods. Especially, the experiments beyond the original meshgrid resolution (hyperparameter optimization) or even beyond meshgrid (point cloud upsampling) validate the favorable performances of our method for continuous representation.

ON DIFFUSE INSTABILITY OF ORTHOTROPIC VISCOPLASTIC PLATES

Elastic strain is covered by the effective medium homogenization method inside a representative volume element (RVE). It has an incremental quasi rate-independent (QRI) form obtained by the endochronic concept of thermodynamic time. The rate dependence takes place by means of stress rate dependent value of the initial yield stress. Free meso rotations and constrained micro rotations within a representative volume element (RVE) are assumed. A comparison between QRI and J2 diffuse instability equations is presented for orthotropic materials. A new QRI nonlinear evolution equation for orthotropic materials is derived by tensor function representation with Spencer-Boehler structural tensors.

Modeling the nonlinear permeability of porous composite structures with non-Newtonian fluids

This work is related to the modelling filtration flow of an incompressible non-Newtonian viscous fluid in porous composite structures. A physical mathematical model of an incompressible non-Newtonian fluid flowing in a porous composite structure has been proposed. The pore–scale description of flow in a reinforced composite is obtained using asymptotic homogenization method. Then, the nonlinear filtration law is investigated theoretically using the anisotropic tensor function representation of the tensor independent variable. The finite element method was used to calculate the local problem, and the distribution of single hole velocity, pressure and non-Newtonian viscosity was obtained. Based on the numerical results verified by the famous Darcy’s law, the nonlinear filtration law of Carreau viscosity fluid was explored and the effective permeability under different parameters was obtained.

Load-induced oriented damage and anisotropy of rock-like materials

Theoretical model for deformability of brittle rock-like materials in the presence of an oriented damage of their internal structure is formulated and verified experimentally. This model is based on the assumption that non-linearity of the stress–strain curves of these materials is a result of irreversible process of oriented damage growth. It was also assumed that a material response, represented by the strain tensor, is a function of two tensorial variables: the stress tensor and the damage effect tensor that is responsible for the current state of the internal structure of the material. The explicit form of the respective non-linear stress–strain relations that account for the appropriate damage evolution equation was obtained by employing the theory of tensor function representations and by using the results of own experiments on damage growth. Such an oriented damage that grows in the material, described by the second order symmetric damage effect tensor, results in gradual development of the material anisotropy. The validity of the constitutive equations proposed was verified by using the available experimental results for concrete subjected to the plane state of stress. The relevant experimental data for sandstone and concrete subjected to tri-axial state of stress were also used.

Modeling of large plastic deformation in crystalline polymers

When a crystalline polymer grain undergoes plastic deformation, four independent slip systems may operate with the chain direction inextensible. The inextensibility leads to texture evolution for a polycrystalline polymer under large plastic deformation. This paper develops a micromechanics model for the three-dimensional large plastic deformation of crystalline polymers. A continuous orientation distribution function is introduced to characterize the distribution of the chain axes. Restrictions on the orientation distribution imposed by material symmetry are considered. A micro–macro interaction law for deformation accounting for molecular chain inextensibility is proposed by employing tensor function representation theorems. By the equivalence between the microscopic and macroscopic work rate, an expression for the macroscopic stress is obtained. The evolution for the orientation distribution is quantified. The proposed model enables us to simulate not only the rotating process of chain axes toward the direction of the maximum stretch but also the texturing hardening under axi-symmetric straining and simple shear.

Simulation of crack tip superblunting in semi-crystalline polymers

A model for the elastic-viscoplastic flow of semi-crystalline polymers is presented. The deformation gradient is decomposed multiplicatively into elastic and plastic parts. The reference configuration for the plastic response is chosen to coincide with the intermediate configuration on which the micromechanics texture evolution is defined. A macroscopically averaged formulation of anisotropic plastic flow is derived from the microscopic description of texture proposed by Chen et al. (1996). The derivation employs tensor function representation theory and the equivalence of plastic power between the macroscopic and microscopic scales. Elasticity is included to describe the effect of elastic constraint on the development of the plastic zone. Small elastic strain is assumed. A finite element scheme is formulated within the present constitutive framework. The results of detailed numerical calculations are reported, and these results are believed to elucidate the phenomenon of crack tip superblunting. The simulated results reconfirm the essential features of crack tip superblunting and its rate dependency.

A unified invariant description of micromechanically-based effective elastic properties for two-dimensional damaged solids

This paper is devoted to an unified invariant description of the effective elastic compliance, say S, of a two-dimensional (2D) damaged solid with an isotropic matrix. A macro- and micromechanically combined approach is proposed, which consists of three major steps. First, the internal variables, i.e., the damage tensors are correlated with the microscopical geometry and distribution of the damage. It is shown that the damage due to microcracks and holes affect S through a symmetric second-order tensor, d, and an irreducible fourth-order tensor, D. Second, the general formulation of the effective compliance S as an isotropic tensor function of d and D is phenomenologically given by making use of several results in the theory of tensor function representation. The linearization and second-rank nonlinearization from the general formulation are derived, where one and six new parameters (named damage constants), respectively, are consistently introduced. It is shown that the linearization provides the most general invariant formulation for micromechanically-based non-interacting or dilute estimations of the effective compliance, and nonlinear models are accounted for the interacting effect of damage. Third, we relate these damage constants to shape and morphology factors of the damage via micromechanical analyses or computer numerical experiments for problems with simple and regular damage. The above-mentioned approach allows invariant consistent models of effective elastic compliance which are not only capable of describing complex and irregular damage, but also indicate in detail the role of the real damage. Thus, these invariant models may further constitute a solid basis of predicating damage evolution. As an application of the proposed approach, the most general second-rank interacting model for damage due to microcracks is established, which is available for any distribution of microcracks.

Anisotropic effects of quasi-isotropic composites (I) ? Modelling

Experimental observations show that quasi-isotropic materials, such as N-axial fibre-reinforced composites and woven materials, exhibit various degrees of anisotropy in elasticity and strength, and the anisotropy in strength is normally stronger than that in elasticity. In view of some available experimental data and based on the general formulation of the constitutive equations and failure criteria of quasi-isotropic materials established by using the theory of representations for tensor functions, we postulate several applicable models of the constitutive equations and strength for 3-and 4-axial quasi-isotropic materials to reveal the anisotropic effects. In a continued work (Part II), the anisotropic effect in strength of an infinitely large plate with a single elliptical hole or crack is studied, and the proposed stiffness and strength models are verified in terms of micro-mechanical analyses.

On Damage Effective Stress and Equivalence Hypothesis

The concepts of damage effective stress and damage equivalence hypothesis play an important role in the development of continuum damage mechanics. Based on a generalization of the damage equivalence hypothesis, the so-called damage isotropy principle, it is found that the effective stress as a second-order tensor-valued function of the usual stress tensor and the damage tensor(s) has to be isotropic. Particularly, this property is regardless of the initial material symmetry (isotropy or anisotropy) and the type of damage variables; and thus, it allows general invariant modeling of the effective stress by the use of theory of tensor function representations. Damage material constants are then consistently introduced to the invariant models of the effective stress and the damage effect tensors. Three new models of the damage effect tensor capable of including realistic dimensionless damage material constants are proposed. The significance of the damage material constants is examined by micromechanical analysis and computer experiments on effective elastic moduli.

A micromechanical model of texture induced orthotropy in planar crystalline polymers

This paper proposes a self-consistent micromechanical description of orthotropic texture evolution in the context of a constrained single slip model for planar crystalline polymers. A micro-macro averaging scheme accounting for molecular chain inextensibility is proposed within the framework of tensor function representation theory. The chain axes of the crystals are described by an orientation distribution function that is expressible as a convergent series of irreducible tensors. The coefficients of this expansion are termed the moment tensors. The imposed macroscopic orthotropy enables us to derive explicit and self-consistent expressions for the second moment tensors. Recurrence formulae furnish explicit expressions for the higher order moment tensors. By an orientation average, the macroscopic plastic spin is expressed in terms of the second and the fourth moment tensors. We also propose a local constraint reaction approximation that is consistent with tensor function representation theory. The proposed model simulates the process of chain axes rotation toward the direction of maximum stretch. The texture hardening is illustrated for the cases of uniaxial stretching and simple shear.

The formulation of constitutive equations for fibre-reinforced composites in plane problems: Part II

From the continuum mechanics points of view, most of commercial fibre-reinforced composites (FRCs) can be considered to be anisotropic materials with one of the five material symmetries: transverse isotropy, orthotropy, tetratropy, hexatropy and octotropy, as illustrated in the preceding paper (Part I) [1]. No properly general formulation of constitutive equations for tetratropic, hexatropic and octoctropic types of FRC has been found in the literature. In this paper, the restriction to the admissible deformation of a FRC imposed by the failure strains of the fibres is investigated. The complete and irreducible two-dimensional tensor function representations regarding tetratropy, hexatropy and octotropy derived in Part I are applied to formulate constitutive equations for FRCs in plane problems of elasticity, yielding and failure in the present work, and of heat conduction, continuum damage and asymmetric elasticity in a continued work (Part III, forthcoming).

On the Tensor Function Representations of 2nd‐Order and 4th‐Order Tensors. Part I

AbstractIn this paper, we establish the complete representations for isotropic scalar‐, 2nd‐order tensor‐ and 4th‐order tensor‐valued functions of any finite number of 2nd‐ and 4th‐order tensors in both 2‐ and 3‐dimensional spaces, and prove the irreducibility of these representations. The fourth‐order tensors under consideration are those regarded as skew‐symmetric linear transformations of both symmetric and skew‐symmetric 2nd‐order tensors in 3‐dimensional space, symmetric linear transformations of skew‐symmetric 2nd‐order tensors in 3‐dimensional space, and both symmetric and skew‐symmetric linear transformations of both symmetric and skew‐symmetric 2nd‐order tensors in 2‐dimensional space. Complete and irreducible isotropic tensor function representations in 3‐dimensional space involving 4th‐order tensors as symmetric linear transformations of symmetric 2nd‐order tensors are derived in a continued paper (Part II). These representations allow general forms of constitutive laws involving 4th‐order tensor agencies (damage tensors, internal variables) of isotropic materials to be developed.

Theory of Representations for Tensor Functions—A Unified Invariant Approach to Constitutive Equations

Representations in complete and irreducible forms for tensor functions allow general consistent invariant forms of the nonlinear constitutive equations and specify the number and type of the scalar variables involved. They have proved to be even more pertinent in attempts to model mechanical behavior of anisotropic materials, since here invariant conditions predominate and the number and type of independent scalar variables cannot be found by simple arguments. In the last few years, the theory of representations for tensor functions has been well established, including three fundamental principles, a number of essential theorems and a large amount of complete and irreducible representations for both isotropic and anisotropic tensor functions in three- and two-dimensional physical spaces. The objective of the present monograph is to summarize and recapitulate the up-to-date developments and results in the theory of representations for tensor functions for the convenience of further applications in contemporary applied mechanics. Some general topics on unified invariant formulation of constitutive laws are investigated.

Tensor function representations as applied to formulating constitutive laws for clinotropic materials

Crystals in the monoclinic system and composites reinforced with families of fibres in two non-orthogonal and mechanically non-equivalent directions are referred to as being clinotropic materials. In this paper, the complete and irreducible representations are established for clinotropic scalar-, vector-, second-order symmetric tensor- and second-order skew-symmetric tensor-valued functions of any finite number of second-order symmetric tensors, second-order skew-symmetric tensors and vectors. These representations constitute a rational basis for modelling the complex mechanical behaviour of clinotropic materials. As illustrative applications, the formulation of thermoelastic constitutive laws of clinotropic materials is developed.

Two-dimensional tensor function representation for all kinds of material symmetry

All kinds of physically possible material symmetry in two-dimensional space were investigated in a recent work of Q. -S. Zheng and J. P. Boehler. In this paper, we establish the complete and irreducible representations with respect to every kind of material symmetry for scalar-, vector-, and second-order tensor-valued functions in two-dimensional space of any finite number of vectors and second-order tensors. These representations allow general invariant forms of physical and constitutive laws of anisotropic materials to be developed in plane problems.

Tensors which characterize anisotropies

The theory of tensor function representations constitutes a rational basis for a consistent mathematical modelling of complex mechanical behaviour of anisotropic materials. The so-called structural tensors, which characterize the symmetry group of anisotropy of concern, play a key role in obtaining irreducible and coordinate-free representations for anisotropic tensor functions. In this paper, based on available properties of Kronecker products of orthogonal transformations, a simple method of determining the structural tensors with respect to any given symmetry group is developed. As its application, the structural tensors corresponding to the five transverse isotropy groups, all of their finite subgroups, and the symmetry group of the 32 crystal classes, which present the most usual and worthwhile anisotropic symmetry groups, are constructed. In particular, we also show that each of these anisotropic symmetry groups can be characterized by only one simple structural tensor.

Multiaxial Anisotropic Creep Behaviour of Rolled Sheet‐Metals

AbstractMultiaxial creep behaviour is analysed within an invariant formulation of the secondary creep constitutive equations for orthotropic solids, developed in the framework of the theory of tensor function representations. The biaxial tension method is improved in order to determine complex stress state in specimen of orthotropic rolled sheet‐metal. The shape of the specimen is taken into account with respect to the uniform stress state by applying the finite element codes.

A unified approach to constitutive equations of inelasticity based on tensor function representations

The applicability of the tensor function theory to the development of elaborate and rational constitutive equations of inelasticity is discussed to facilitate the description of the complicated inelastic response of structural components. After a brief review of the representation theory of scalar-valued as well as tensor-valued functions of a set of vector and tensor variables, previous results in this line of approach are first presented. Then, the formulation of some non-classical constitutive equations of inelasticity is discussed in some detail to elucidate the utility and the practical procedures of this theory; a yield criterion and a non-associated flow law of prestrained plastic materials, those of transversely isotropic materials, isotropic and anisotropic hardening theories of creep, and a constitutive equation of anisotropic creep damage in metals. It is shown that the present approach provides the general framework to materialize accurate constitutive equations of inelasticity.