Search Results (12)

Boundary element methods provide powerful techniques for the analysis of problems involving coupled multi-physical response. This paper presents the integral equation formulation for the size-dependent thermoelastic response of solids under steady-state conditions in three dimensions. The formulation is based upon consistent couple stress theory, which features a skew-symmetric couple-stress pseudo-tensor. For general anisotropic thermoelastic material, there is not only thermal strain deformation, but also thermal mean curvature deformation. Interestingly, in this size-dependent multi-physics model, the thermal governing equation is independent of the deformation. However, the mechanical governing equations depend on the temperature field. First, thermal and mechanical weak forms and reciprocal theorems are developed for this theory. Then, an integral equation formulation for three-dimensional size-dependent thermoelastic isotropic materials is derived, along with the corresponding singular infinite-space fundamental solutions or kernel functions. For isotropic materials, there is no thermal mean curvature deformation, and the thermoelastic effect is solely the result of thermal strain deformation. As a result, the size-dependent behavior is specified entirely by a single characteristic length scale parameter $l$, while the thermal coupling is defined in terms of the thermal expansion coefficient $\alpha$, as in the classical theory of steady-state isotropic thermoelasticity. Full article

We develop a general approach for the computation of the effective properties of nonlinear viscoelastic composites. For this purpose, we employ the asymptotic homogenisation technique to decouple the equilibrium equation into a set of local problems. The theoretical framework is then specialised to the case of a strain energy density of the Saint-Venant type, with the second Piola–Kirchhoff stress tensor also featuring a memory contribution. Within this setting, we frame our mathematical model in the case of infinitesimal displacements and employ the correspondence principle which results from the use of the Laplace transform. In doing this, we obtain the classical cell problems in asymptotic homogenisation theory for linear viscoelastic composites and look for analytical solutions of the associated anti-plane cell problems for fibre-reinforced composites. Finally, we compute the effective coefficients by specifying different types of constitutive laws for the memory terms and compare our results with available data in the scientific literature. Full article

The hard rock in the protective coal seam of the Pingdingshan Mine in China is a typical quasi-brittle material exhibiting complex mechanical characteristics. According to available research on the mechanical property, the inelastic deformation and development of damage are considered related with crack initiation and propagation, which are main causes of the material degradation. In the present study, an original experimental investigation on the rock sample of the Pingdingshan coal mine is firstly carried out to obtain the basic mechanical responses in a conventional triaxial compression test. Based on the homogenization method and thermodynamic theory, a damage–friction coupled model is proposed to simulate the non-linear mechanical behavior. In the framework of micromechanics, the hard rock in a protective coal seam is viewed as a heterogeneous material composed of a homogeneous solid matrix and a large number of randomly distributed microcracks, leading to a Representative Elementary Volume (REV), i.e., the matrix–cracks system. By the use of the Mori–Tanaka homogenization scheme, the effective elastic properties of cracked material are obtained within the framework of micromechanics. The expression of free energy on the characteristic unitary is derived by homogenization methods and the pairwise thermodynamic forces associated with the inelastic strain and damage variables. The local stress tensor is decomposed to hydrostatic and deviatoric parts, and the effective tangent stiffness tensor is derived by considering both the plastic yield law and a specific damage criterion. The associated generalized Coulomb friction criterion and damage criterion are introduced to describe the evolution of inelastic strain and damage, respectively. Prepeak and postpeak triaxial response analysis is carried out by coupled damage–friction analysis to obtain analytical expressions for rock strength and to clarify the basic characteristics of the damage resistance function. Finally, by the use of the returning mapping procedure, the proposed damage–friction constitutive model is applied to simulate the deformation of Pingdingshan hard rock in triaxial compression with respect to different confining pressures. It is observed that the numerical results are in good agreement with the experimental data, which can verify the accuracy and show the obvious advantages of the micromechanic-based model. Full article

The present review paper consists of two main parts, which are not connected. The first part is devoted to a general axisymmetric elastic–plastic plane stress solution, assuming polar anisotropy. Strains are infinitesimal. The principal stress trajectories coincide with the principal axes of anisotropy. No restrictions are imposed on the yield criterion other than the conventional restrictions imposed on the yield criteria in plasticity. The plastic portion of the strain rate tensor is determined from the associated flow rule. A simple example illustrates the general solution. The second part is devoted to the stationary ideal flow theory for anisotropic materials under axial symmetry. The elastic portion of the strain tensor is neglected. A piece-wise linear yield criterion is adopted. This criterion generalizes Tresca’s yield criterion. The existence of ideal flow is proven. It is also shown that the available solutions for Tresca’s yield criterion can be used for deriving solutions for the yield criterion under consideration. Miscellaneous topics are shortly discussed in the third part of the paper. Full article

We used the recently introduced stress tensor trajectory U_σ space construction within the framework of next-generation quantum theory of atoms in molecules (NG-QTAIM) for a chirality investigation of alanine when subjected to a non-structurally distorting electric field. The resultant sliding of the axial-bond critical point (BCP) responded significantly, up to twice as much, in the presence of the applied electric field in comparison to its absence. The bond flexing, a measure of bond strain, was always lower by up to a factor of four in the presence of the electric field, depending on its direction and magnitude. An achiral character of up to 7% was found for alanine in the presence of the applied electric field. The achiral character was entirely absent in the presence of the lowest value of the applied electric field. Future applications, including molecular devices using left and right circularly polarized laser pulses, are briefly discussed. Full article

The mechanical properties of anisotropic materials are generally characterized based on the orthotropy or transverse isotropy. However, the two-dimensional plane stress problems cannot comprehensively characterize the anisotropy of materials. In this study, based on the theory of elasticity and the transformation of the three-dimensional space coordinate system, combined with the projection relationship of the Cauchy stress tensor of an arbitrary section, the transformation relationship of the elastic modulus, shear modulus, and stress–strain between the orthogonal and load coordinate systems are obtained. The orthotropic Johnson-Cook (JC) constitutive model of AA7050-T7451 aluminum alloy is modified by fitting, and the constitutive relationship at any spatial angle is theoretically calculated by combining the obtained spatial coordinate transformation matrix. The generated spatial constitutive model is verified and modified through experiments. The results demonstrate that the theoretical mechanical properties and the modified spatial constitutive model can accurately reflect the effect of the spatial angle on the material stress distribution. Full article

Molecular dynamics (MD) and experiments indicate that the high-speed dislocations dominate the plasticity properties of crystal materials under high strain rate. New physical features arise accompanied with the increase in dislocation speed, such as the “Lorentz contraction” effect of moving screw dislocation, anomalous nucleation, and annihilation in dislocation interaction. The static description of the dislocation is no longer applicable. The elastodynamics fields of non-uniformly moving dislocation are significantly temporal and spatially coupled. The corresponding mathematical formulas of the stress fields of three-dimensional (3D) and two-dimensional (2D) dislocations look quite different. To clarify these differences, we disclose the physical origin of their connections, which is inherently associated with different temporal and spatial decoupling strategies through the 2D and 3D elastodynamics Green tensor. In this work, the fundamental relationship between 2D and 3D dislocation elastodynamics is established, which has enlightening significance for establishing general high-speed dislocation theory, developing a numerical calculation method based on dislocation elastodynamics, and revealing more influences of dislocation on the macroscopic properties of materials. Full article

In the paper, a thermodynamically consistent model of elastic damaged material in the framework of small strain theory is formulated, describing the process of deterioration in quasibrittle materials, concrete in particular. The main goal is to appropriately depict the distinction between material responses in tension and compression. A novel Helmholtz energy and a dissipation potential including three damage parameters are introduced. The Helmholtz function has a continuous first derivative with respect to strain tensor. Based on the assumed functions, the strain–stress relationship, the damage condition, the evolution laws, and the tangent stiffness tensor are derived. The model’s predictions for uniaxial tension, uniaxial compression, uniaxial cyclic compression–tension, and pure shear tests are calculated using Wolfram Mathematica in order to identify the main features of the model and to grasp the physical meaning of an isotropic damage parameter, a tensile damage parameter, and a compressive damage parameter. Their values can be directly bound to changes of secant stiffness and generalized Poisson’s ratio. An interpretation of damage parameters in association with three mechanisms of damage is given. The considered dissipation potential allows a flexible choice of a damage condition. The influence of material parameters included in dissipation function on damage mode interaction is discussed. Full article

In this study, a dynamic Mindlin–Reissner-type plate is developed based on a simplified version of Mindlin’s form-II first-strain gradient elasticity theory. The governing equations of motion and the corresponding boundary conditions are derived using the general virtual work variational principle. The presented model contains, apart from the two classical Lame constants, one additional microstructure material parameter g for the static case and one micro-inertia parameter h for the dynamic case. The formal reduction of this model to a Kirchhoff-type plate model is also presented. Upon diminishing the microstructure parameters g and h, the classical Mindlin–Reissner and Kirchhoff plate theories are derived. Three points distinguish the present work from other similar published in the literature. First, the plane stress assumption, fundamental for the development of plate theories, is expressed by the vanishing of the z-component of the generalized true traction vector and not merely by the zz-component of the Cauchy stress tensor. Second, micro-inertia terms are included in the expression of the kinetic energy of the model. Finally, the detailed structure of classical and non-classical boundary conditions is presented for both Mindlin–Reissner and Kirchhoff micro-plates. An example of a simply supported rectangular plate is used to illustrate the proposed model and to compare it with results from the literature. The numerical results reveal the significance of the strain gradient effect on the bending and free vibration response of the micro-plate, when the plate thickness is at the micron-scale; in comparison to the classical theories for Mindlin–Reissner and Kirchhoff plates, the deflections, the rotations, and the shear-thickness frequencies are smaller, while the fundamental flexural frequency is higher. It is also observed that the micro-inertia effect should not be ignored in estimating the fundamental frequencies of micro-plates, primarily for thick plates, when plate thickness is at the micron scale (strain gradient effect). Full article

Studies have suggested that brittle fractures occur in steel because microcracks in the brittle layer at grain boundaries propagate as a result of the increase in piled-up dislocations. Therefore, prestraining can approach the limits of a material, which could lead to a decrease in fracture toughness. However, strains are tensors comprising multiple components, so the effect of prestrain on fracture toughness is not simple. Additionally, the mechanism of change in critical stress due to prestrain has not been thoroughly investigated. For the lifetime evaluation of steel structures with a complicated load history, it is important to generalize the effect of complicated prestrain on the decrease in fracture toughness. In this paper, a single prestrain was applied in a direction different from the crack opening direction. A general three-point bending test was employed for fracture evaluation. Numerical analyses using the strain gradient plasticity (SGP) theory, which is a method based on the finite element method (FEM) are carried out; conventional macroscopic material damage rules are considered as well. Using these FEM analyses, the critical stress is calculated. Finally, the change in critical stress can be expressed by the yield point increase and dislocation density and formulated based on the identified micromechanisms. Full article

This paper considers non-classical continuum theory for thermoviscous fluids without memory incorporating internal rotation rates resulting from the antisymmetric part of the velocity gradient tensor to derive ordered rate constitutive theories for the Cauchy stress and the Cauchy moment tensor based on entropy inequality and representation theorem. Using the generalization of the conjugate pairs in the entropy inequality, the ordered rate constitutive theory for Cauchy stress tensor considers convected time derivatives of the Green’s strain tensor (or Almansi strain tensor) of up to orders $n_{\epsilon }$ as its argument tensors and the ordered rate constitutive theory for the Cauchy moment tensor considers convected time derivatives of the symmetric part of the rotation gradient tensor up to orders $n_{\mathsf{\Theta }}$ . While the convected time derivatives of the strain tensors are well known the convected time derivatives of higher orders of the symmetric part of the rotation gradient tensor need to be derived and are presented in this paper. Complete and general constitutive theories based on integrity using conjugate pairs in the entropy inequality and the generalization of the argument tensors of the constitutive variables and the representation theorem are derived and the material coefficients are established. It is shown that for the type of non-classical thermofluids considered in this paper the dissipation mechanism is an ordered rate mechanism due to convected time derivatives of the strain tensor as well as the convected time derivatives of the symmetric part of the rotation gradient tensor. The derivations of the constitutive theories presented in the paper is basis independent but can be made basis specific depending upon the choice of the specific basis for the constitutive variables and the argument tensors. Simplified linear theories are also presented as subset of the general constitutive theories and are compared with published works. Full article

In the Newtonian approach to mechanics, the concepts of objective tensors of various ranks and types are introduced. The tough classification of objective tensors is given, including tensors of material and spatial types. The diagrams are constructed for non-degenerate (“analogous”) relations between tensors of one and the same (any) rank, and of various types of objectivity. Mappings expressing dependence between objective tensor processes of various ranks and types are considered. The fundamental concept of frame-independence of such mappings is introduced as being inherent to constitutive relations of various physical and mechanical properties in the Newtonian approach. The criteria are established for such frame-independence. The mathematical restrictions imposed on the frame-independent mappings by the objectivity types of connected tensors are simultaneously revealed. The absence of such restrictions is established exclusively for mappings and equations linking tensors of material types. Using this, a generalizing concept of objective differentiation of tensor processes in time, and a new concept of objective integration, are introduced. The axiomatic construction of the generalized theory of stress and strain tensors in continuum mechanics is given, which leads to the emergence of continuum classes and families of new tensor measures. The axioms are proposed and a variant of the general theory of constitutive relations of mechanical properties of continuous media is constructed, generalizing the known approaches by Ilyushin and Noll, taking into account the possible presence of internal kinematic constraints and internal body-forces in the body. The concepts of the process image and the properties of the five-dimensional Ilyushin’s isotropy are generalized on the range of finite strains. Full article