A bimodular nonlinear constitutive equation for rock

A comparison of second-order constitutive theories for hyperelastic materials

This chapter deals with how to measure nonlinear viscoelastic functions and various nonlinear viscoelastic constitutive equations. Since there have been developed a number of constitutive equations which cannot be included in a single chapter, this chapter considers only a few popularly used ones, which are classified to four groups: the models based on mathematical approximations; the models obtained from the generalization of linear viscoelastic models; the models based on the simplification of polymer structure; and the Leonov model which is a class of constitutive equations based on irreversible thermodynamics.

For the class of all discrete two-velocity models of the nonlinear Boltzmann equation consistent with theH-theorem an exact representation of the corresponding nonlinear constitutive equation will be derived. Applying the fractional calculus we derive, in addition, the nonlinear fractional Boltzmann equation as the starting point for the formulation of a nonlinear fractional constitutive equation which relates the densities and fluxes through an integro-differential equation. Linearization leads to the fractional version of Cattaneo's constitutive law. The model presented here may be regarded as a first step, towards the formulation of a fractional extended irreversible thermodynamics.

A new constitutive equation has been developed for describing the nonlinear viscoelastic properties, including yield, of polymer solids. The nonlinear viscoelastic constitutive equation is the logical extension of the standard three‐dimensional linear viscoelastic constitutive equation to include the effects of deformation‐induced changes in the thermodynamic state of the polymer on the rate of viscoelastic relaxation. The rate of viscoelastic relaxation depends upon the fraction of unoccupied lattice sites (i.e., holes) in the Simha‐Somcynsky statistical thermodynamic equation of state, and the hole fraction is a known function of the temperature, pressure, and specific volume. The nonlinear viscoelastic yield and postyield behavior are a direct consequence of the deformation‐induced dilation. All material constants contained in the constitutive equation can be determined from independent elastic, linear viscoelastic, and pressure‐volume‐temperature measurements—there are no adjustable constants. The constitutive equation has been solved for model viscoelastic materials in uniaxial extension and shear deformations where the applied strain rate is constant. It is shown that an experimentally realizable “shear” deformation is only isochoric in the linear viscoelastic limit. Yield and postyield softening are predicted in both uniaxial extension and experimental shear, and the predicted yield stress exhibits the appropriate strain rate and temperature dependence. The effects of using various finite strains measures in the nonlinear viscoelastic constitutive equation are analyzed.

Considering the influence of temperature and strain variables on materials. According to the relationship of conjugate stress-strain, a complete and irreducible non-linear constitutive equation of isotropic hyperelastic materials is derived and the constitutive equations of 16 types of isotropic hyperelastic materials are given we study the transformation methods and routes of 16 kinds of constitutive equations and the study proves that transformation of two forms of constitutive equation. As an example of application, the non-linear thermo-elastic constitutive equation of isotropic hyperelastic materials is combined with the natural vulcanized rubber experimental data in the existing literature base on MATLAB, The results show that the fitting accuracy is satisfactory.

In the analysis of weak solutions relevant to evolutionary flows of incompressible fluids with non-constant viscosity or with non-linear constitutive equation, it is in general an open question whether a globally integrable pressure exists if the flows are subject to no-slip boundary conditions. Here we overcome this deficiency by considering threshold boundary conditions stating that the fluid adheres to the boundary until certain critical value for the wall shear stress is reached. Once the wall shear stress exceeds this critical value, the fluid slips. The main ingredient in our approach is to look at this type of activated, stick-slip, boundary condition as an implicit constitutive equation on the boundary.

Young’s modulus and Poisson ratio are essential parameters for rock brittleness evaluation. At present, prestack inversions of these parameters mostly adopt linear approximation, whereas the nonlinear inversion methods that are based on the exact Zoeppritz equation are seldomly conducted. Nevertheless, in the case of large incident angles and high-impedance contrast, the nonlinear equation has higher accuracy than the linear approximation. It is more in line with the nature of geophysical inversion problems, and it describes the variation of formations’ elastic parameters better than the linear approximation does. Therefore, based on the exact solution of the Zoeppritz equation, we have parameterized and derived a new nonlinear reflection coefficient equation using nonlinear Young’s modulus, Poisson ratio, and density (YPD) as variables. Subsequently, by combining the inverse operator algorithm with the YPD equation, we develop a new nonlinear amplitude-variation-with-offset inversion scheme. This approach can robustly extract Young’s modulus and Poisson ratio from seismic data and identify shale-gas sweet spot locations within the reservoir, as well as predict reservoir fluid properties. We have tested the accuracy and applicability of this approach with synthetic and real data examples.

1. The values of Young's modulus and Poisson's ratio in tension, compression, and bending, and the values of tensile and compressive strengths were obtained. It was shown that the compressive strength is about 25 times greater than the tensile strength. 2. Lithium glassceramic of composition No. 23 at room temperature practically obeys Hooke's law up to failure. 3. The forms of failure of glassceramic specimens obtained in tension, compression, and bending are analogous to failure of glass in the indicated types of loading.

The Rabotnov physically non-linear (quasi-linear) constitutive equation for non-aging elasto-viscoplastic materials with four material functions is studied analytically in order to outline the set of basic rheological phenomena which it can simulate, to clarify the material functions governing abilities, to indicate application field of the relation and to develop identification and verification techniques. General properties of the theoretic creep curves for volumetric, longitudinal and lateral strain generated by the model under uni-axial loading are investigated assuming material functions of the relation are arbitrary. Intervals of creep curves monotonicity and conditions for existence of extrema and sign changes are considered and the influence of minimal qualitative restrictions imposed on its material functions is analyzed. It is proved that the Rabotnov relation is able to simulate non-monotone behavior and sign changes of lateral strain and Poisson's ratio (lateral contraction ratio in creep). The expressions for Poisson's ratio via the strain state parameter (equal to volumetric strain divided by deviatoric strain) and via four material functions of the model are derived. The Poisso'n ratio dependence on time, stress level and material functions is examined. Assuming material functions are arbitrary, general two-sided bound for the Poisson's ratio range is obtained. Additional restrictions on material functions providing negative Poisson's ratio values are found and the criterion for its non-dependence on time is formulated. Taking into account volumetric creep (governed by two material functions of the model) is proved to affect strongly the qualitative behavior and characteristic features of longitudinal creep curves and the Poisso'n ratio evolution.

Cellular materials are widely used in various applications because of their low density and high strength. The mechanical behavior of cellular materials under various loading conditions has been investigated. Nevertheless, many of these previous studies assume that the Young’s modulus of constituting struts is the same in tension and compression. The present work first derives analytical expressions for the effective Young’s moduli of hexagonal and diamond lattices composed of struts with different tension and compression moduli under the assumption of small strain deformation. It also uses the finite element method to further investigate the mechanical responses of these lattices. The macroscopic Young’s moduli under both compressive and tensile loads are reported as a function of the ratio of compression and tension moduli of constituting struts. The numerical finite element models are implemented by a user defined material subroutine in ABAQUS. Results reveal that the effective Young’s moduli of periodic hexagonal and diamond lattices significantly decrease with decreasing ratio of compression and tension moduli of the struts. Furthermore, the mechanical behavior of hexagonal lattices composed of struts with different tension-compression moduli is dependent on the loading direction and whether they are compressed or stretched. The unique mechanical properties of bi-modulus cellular materials could find important applications in the automotive and construction industries.

Numerical simulation is one of the effective tools for the long-term stability assessment of underground structures. The reliability improvement of the numerical simulation requires precise constitutive equations to reproduce time-dependent behavior of rock such as loading-rate dependence, creep, and relaxation. These time-dependent behaviors of rock have nonlinearity between stress and strain rate and are influenced by surrounding environment. A number of constitutive equations have been proposed in previous studies; however, the equations based on the theory of linear viscoelasticity have difficulty reproducing accurately the time-dependent behavior of rock, and most of the equations based on the theory of viscoplasticity have been applied to the limited loading condition such as creep. In this article, nonlinear viscoelastic constitutive equations for rock proposed by the authors were introduced, and their exact solutions and application to experimental results were summarized. These equations can reproduce nonlinear time-dependent behavior and not only brittle but also ductile failures of rock and can be applied to the behavior under various loading conditions and experiment environment. In the equations, compliance or inelastic strain is assumed to be increased with time, but the functions and their combinations are of great variety. Hence, the differences of the solutions and applicability of the equations were detailed, and in addition future subjects were presented.

A recent review publication presented an extensive and comprehensive assessment of the phenomenological relations of Poisson's ratios (PRs) to the behavior and responses of contemporary materials under specific loading conditions. The present review and analysis paper is intended as a theoretical mechanics complement covering mathematical and physical modeling of a single original elastic and of six time and process (i.e. path and stress) dependent viscoelastic PR definitions as well as a seventh special path independent one. The implications and consequences of such models on material characterization are analyzed and summarized. Indeed, PRs based on experimentally obtained 2-D strains under distinct creep and/or relaxation processes exhibit radically different time responses for identical material specimen. These results confirm the PR's implicit path dependence in addition to their separate intrinsic time reliance. Such non-uniqueness of viscoelastic PRs renders them unsuitable as universal material descriptors. Analytical formulations and experimental measurements also examine the physical impossibility of instantaneously achieving time independent loads or strains or their rates thus making certain PR definitions based on constant state variables, while mathematically valid, physically unrealistic and unachievable. A newly developed theoretical/experimental protocol for the determination of the time when loading patterns reach stead-state conditions based on strain accelerations demonstrates the capability to measure this time from experimental data. Due to the process dependent PRs, i.e. stress and stress history paths, the non-existence of a unique viscoelastic PR and of a universal elastic-viscoelastic correspondence principle or analogy (EVCP) in terms of PRs is demonstrated. Additionally and independently, the required double convolution integral construction of linear viscoelastic constitutive relations with the inclusion of PRs is cumbersome analytically and computationally needlessly highly CPU intensive. Furthermore, there is no theoretical fundamental hint as to what loading path is required to produce a unique universal viscoelastic PR definition necessary for formulating a PR based constitutive relation or an EVCP protocol. The analysis associated with an additional Class VII viscoelastic PR establishes it as a universal representation which is loading path and strain independent while still remaining time dependent. This Class PR can be the one used if it is desired to express constitutive relations in terms of PRs, subject to the caveat applying to all PR Classes regarding the CPU intensiveness in the time space due to triple product and double convolution integral constitutive relations. However, the use PRs is unnecessary since any set of material behavior can be uniquely and completely defined in terms of only moduli and/or compliances. The mathematical model of instantaneous initial loading paths, based on Heavi-side functions, is examined in detail and shown to lead to infinite velocities and accelerations. Additionally, even if non-instantaneous gradual loading functions are employed the resulting PRs are still load and load history dependent. Consequently, they represent specialized PR responses applicable and limited to those particular load and history combinations. Although the analyses contained herein are generalized to non-homogeneous linear viscoelastic materials, the main focus is on PR time and process dependence. The non-homogeneous material results and conclusions presented herein apply equally to homogeneous viscoelasticity and per se do not influence the results or conclusions of the analytical development regarding viscoelastic PRs. In short, these PR analyses apply to all linear viscoelastic material characterization.

A theory for two-dimensional (2D) beams is presented for the case of considering new constitutive equations, wherein the linearized (infinitesimal) strain tensor is assumed to be a nonlinear function of the Cauchy stress tensor. An incremental formulation is developed to solve numerically the resultant equations. Three constitutive equations for elastic bodies are considered: a model wherein we have strain-limiting behavior, a nonlinear model for rock, and a bimodular constitutive equation for rock (which can also be considered as a nonlinear model for rock). Two boundary value problems are studied: the deformation of a cantilever beam with a point load, and a three-point beam with a point load applied on the middle. The numerical results are compared with the predictions of the classical theory for beams for the case of linearized elastic solids. The results for the stresses can be notoriously different, when comparing the predictions of the nonlinear constitutive equations with the results obtained using the linearized theory of elasticity.

This brief tutorial review first discusses the physical origin of nonlinear electric and magnetic susceptibilities of electro-magnetic media, from the audio through the microwave to the visible range of the electromagnetic spectrum. Wave propagation in non-linear media can be described by a combination of Maxwell's equations and the nonlinear constitutive equations. The Manley-Rowe relationships are verified for anisotropic lossless dielectrica. Microwave harmonic generation in ferrites and other materials, microwave modulation of light, light harmonic generation, light mixing and parametric conversion are all discussed from a general common viewpoint. The boundary conditions at the surface of a nonlinear medium lead to generalizations of the laws of reflection and refraction of light.

In the finite deformation range, the numbers of orthotropic 2n order elastic constants are studied on the basis of tensor function and of its representation theorem. On the basis of elastic constant research, the elastic orthotropic constitutive equation is derived by using the tensor method. Based on orthotropic elastic constitutive equations an in-depth study on the constitutive theory of orthotropic nonlinear thermal elasticity is carried out, and by considering the deformation produced by the coupling of temperature and load, nonlinear orthotropic thermoelastic constitutive equation is further derived with representation of the tensor invariant and scalar invariant. The constitutive equations could be used very convenient to the application in reality.