Three-dimensional Maxwell stress-strain relations in viscoelasticity

Conclusion In this paper three-dimensional Maxwell stress-strain relations were deduced phenomenologically.In the first place we applied the Hamilton's principle to the viscoelastic deformation, and obtained the variational equation with respect to the elastic potential and the dissipation function.Then we assumed that the elastic potential is a function only of the stress, and the dissipation function is a function of stress and rate of stress. By the above variational equation of the virtual stress satisfying the equilibrium equation and the boundary conditions, we obtained the relations to be satisfied by the elastic potential and the dissipation function, and the conditions to be satisfied by the dissipation function.From these relations we obtained the required three-dimensional Maxwell stress-strain relations in viscoelasticity. These relations indicate that the strain is the sum of the internal elastic strain and the internal viscous strain.If a given substance is isotropic with respect to stress, the stress-strain relations are expressed by a linear Maxwell model consisting of Hookian spring in series with a Newtonian dashpot.It is the main result of this paper that the three-dimensional Maxwell stress-strain relations in viscoelasticity are deduced from physically appropriate assumptions.     

Inelastic deformations of stainless steel leaf springs-experiment and nonlinear analysis

A stainless steel leaf spring is designed and constructed followed by its performance evaluation by experiment and non-linear analysis so that an insight into the optimum use of material can be made. Cantilever beams of uniform strength, popularly termed as leaf springs, undergo much larger deflections in comparison to a beam of constant cross-section; that needs inclusions of geometric non-linearity for rigorous analysis. This study deals with such a cantilever beam, but takes into account the material non-linearity as well. Experiments were conducted for such a cantilever beam, with highly non-linear stress-strain curves. In addition to the experiment, a computer code in ‘C’ has been developed using the Runge-Kutta technique for the purpose of simulation. Effective modulus-curvature relations are obtained from the non-linear stress-strain relations for different sections of the beam and used for the analysis. It is seen that non-linear stress-strain curve governs the bending of the beam. Importantly, non-linear analysis shows the stresses are not so high as predicted by the linear theory without end-shortening. Moreover, the tensile and compressive stresses are different in magnitude and both decrease along the span. Experimental load-deflection curves are found to be initially concave upward but, non-linear and convex upward at a high load. Comparison of the numerical results with the available experimental results from another research group and theory shows excellent agreement verifying the soundness of the entire numerical simulation scheme.     

The molecular theory of viscoelasticity for thermoplastic elastomer SBS (SIS) at large deformations

A microstructure model for SBS and SIS triblock copolymers with hard domains as multifunctional reinforcing fillers is proposed. Based on this model and proposed mechanism of large deformations, the probability distribution function of the end-to-end vector for each constituent chain and the free energy of deformation for the total networks was calculated by the combination of statistical thermodynamics and kinetics. A new molecular theory of non-linear visco-elasticity for SBS and SIS at large deformations is presented. It is successful in relating the viscoelastic state to molecular constitution by three important parameters (C ₁₀₀,C ₀₂₀, andC ₂₀₀) of the networks. The relations of stress to strain for four types of deformation, the elastic modulus and the constitutive equation for the stress relaxation were derived from this theory. It provides a theoretical foundation for studying the relationships of multiphase network structures and mechanical properties at large deformations. An excellent agreement between the theoretical relationships and experimental data from the experiments and the reference was obtained.Project supported by the National Natural Foundation of China     

Plane Strain Bending of Cylindrical Sectors of Admissible Compressible Hyperelastic Materials

In the theory of nonlinear elasticity of rubber-like materials, if a homogeneous isotropic compressible material is described by a strain–energy function that is a homogeneous function of the principal stretches, then the equations of equilibrium for axisymmetric deformations reduce to a separable first-order ordinary differential equation. For a particular class of such strain–energy functions, this property is used to obtain a general parametric solution to the equilibrium equation for plane strain bending of cylindrical sectors. Specification of the arbitrary function that appears in such strain–energy functions yields some parametric solutions. In some cases, the parameter can be eliminated to yield closed-form solutions in implicit or explicit form. Other possible forms for the arbitrary constitutive function that are likely to yield such solutions are also indicated.     

The General Mathematical Theory of Plasticity and the Il’yushin Postulates of Macroscopic Definability and Isotropy

The physical laws characterizing the relation between stresses and strains are considered and analyzed in the general modern theory of elastoplastic deformations and in its postulates of macroscopic definability and isotropy for initially isotropic continuous media. The fundamentals of this theory in continuum mechanics were developed by A.A. Il’yushin in the mid-twentieth century. His theory of small elastoplastic deformations under simple loading became a generalization of Hencky’s deformation theory of flow, whereas his theory of elastoplastic processes which are close to simple loading became a generalization of the Saint-Venant–Mises flow theory to the case of hardening media. In these theories, the concepts of simple arid complex loading processes arid the concept of directing form change tensors are introduced; the Bridgman law of volume elastic change and the universal Roche–Eichinger laws of a single hardening curve under simple loading are adopted; and the Odquist hardening for plastic deformations is generalized to the case of elastoplastic hardening media for the processes of almost simple loading without consideration of a specific history of deformations for the trajectories with small arid mean curvatures. In this paper we discuss the possibility of using the isotropy postulate to estimate the effect of forming parameters in the stress-strain state appeared due to the strain-induced anisotropy during the change of the internal structures of materials. We also discuss the possibility of representing the second-rank symmetric stress and strain tensors in the form of vectors in the linear coordinate six-dimensional Euclidean space. An identity principle is proposed for tensors and vectors.     

Variational principles and generalized variational principles for nonlinear elasticity with finite displacement

In a previous paper (1979)^[1], the minimum potential energy principle and stationary complementary energy principle for nonlinear elasticity with finite displacement, together with various complete and incomplete generalized principles were studied. However, the statements and proofs of these principles were not so clearly stated about their constraint conditions and their Euler equations. In somecases, the Euler equations have been mistaken as constraint conditions. For example, the stress displacement relation should be considered as Euler equation in complementary energy principle but have been mistaken as constraint conditions in variation. That is to say, in the above mentioned paper, the number of constraint conditions exceeds the necessary requirement. Furthermore, in all these variational principles, the stress-strain relation never participate in the variation process as constraints, i.e., they may act as a constraint in the sense that, after the set of Euler equations is solved, the stress-strain relation may be used to derive the stresses from known strains, or to derive the strains from known stresses. This point was not clearly mentioned in the previous paper (1979)^[1]. In this paper, the high order Lagrange multiplier method (1983)^[2] is used to construct the corresponding generalized variational principle in more general form. Throughout this paper, V/.V. Novozhilov's results (1958)^[3] for nonlinear elasticity are used.     

Damage theory of materials with microstructure

A new form of damage theory of materials is proposed, that is valid for the case of nonconservative stresses. The partial entropy, strain and microstructure parameters are taken as the state variables. Without assuming the free energy to be a state function, the basic governing equations are derived. According to the balance of released and dissipated energy, the general form of damage evolution equation is obtained. Further, assuming the existence of independent damage mechanisms, the normality of damage evolution equation is proven. The generalized damage variables are discussed. Finally, some examples are given to show the applications of the theory. Projects Sponsered by the Joint Seismological Science Foundation.     

Basis free relations for the conjugate stresses of the strains based on the right stretch tensor

In this paper, general relations between two different stress tensors T^f and T^g, respectively conjugate to strain measure tensors f(U) and g(U) are found. The strain class f(U) is based on the right stretch tensor U which includes the Seth–Hill strain tensors. The method is based on the definition of energy conjugacy and Hill’s principal axis method. The relations are derived for the cases of distinct as well as coalescent principal stretches. As a special case, conjugate stresses of the Seth–Hill strain measures are then more investigated in their general form. The relations are first obtained in the principal axes of the tensor U. Then they are used to obtain basis free tensorial equations between different conjugate stresses. These basis free equations between two conjugate stresses are obtained through the comparison of the relations between their components in the principal axes, with a possible tensor expansion relation between the stresses with unknown coefficients, the unknown coefficients to be obtained. In this regard, some relations are also obtained for T⁽⁰⁾ which is the stress conjugate to the logarithmic strain tensor lnU.     

Variational principles in elasticity with nonlinear stress-strain relation

Since 1979, a series of papers have been published concerning the variational principles and generalized variational principles in elasticity such as [1] (1979), [6] (1980), [2,3] (1983) and [4,5] (1984). All these papers deal with the elastic body with linear stress-strain relations. In 1985, a book was published on generalized variational principles dealing with some nonlinear elastic body, but never going into detailed discussion. This paper discusses particularly variational principles and generalized variational principles for elastic body with nonlinear stress-strain relations. In these discussions, we find many interesting problems worth while to pay some attention. At the same time, these discussions are also instructive for linear elastic problems. When the strain is small, the high order terms may be neglected, the results of this paper may be simplified to the well-known principles in ordinary elasticity problems.     

Problems in determining the elastic strain energy function for rubber

Lightly crosslinked natural rubber can be stretched by 600% or more, and recovers almost completely. It is often regarded as a model highly elastic material and characterized by a strain energy function to describe its stress-strain behavior under various types of deformation. A number of such functions have been proposed; some of them appear in current finite element programs. They are usually validated by comparison with measured stress-strain relations by Treloar [7] [L.R.G. Treloar, Stress-strain data for vulcanized rubber under various types of deformation, Trans. Faraday Soc. 40 (1944) 59-70] and Jones and Treloar [15] [D.F. Jones, L.R.G. Treloar, The properties of rubber in pure homogeneous strain, J. Phys. D Appl. Phys. 8 (1975) 1285-1304]. But Treloar pointed out that the relations at high strains became markedly irreversible, and he did not assign a strain energy function for strains greater than about 300%. Rivlin's universal relation between torsional stiffness and tensile stress [14] [R.S. Rivlin, Large elastic deformations of isotropic materials. Part V1: further results in the theory of torsion, shear and flexure, Philos. Trans. R. Soc. A 243 (1949) 251-288] is applied here to show that a typical elastic solid cannot be described by any strain energy function at strains greater than about 300%. Elastic strain energy functions for higher strains, or for other rubbery materials, are thus of doubtful value unless evidence for reversibility of stress-strain relations is adduced or the applicability of a strain energy function is demonstrated.     

A damage model for crack initiation and growth

A damage accumulation model is presented for the study of the problem of crack initiation and stable growth in an elastic-plastic material. A centre-cracked specimen subjected to a uniform stress perpendicular to the crack plane is considered. A coupled stress and failure analysis is performed by using a finite element computer program based on J₂-plasticity theory in conjunction with the strain energy density theory. After initial yielding, each material element follows a different equivalent uniaxial stress-strain behavior depending on the amount of energy dissipation by permanent deformation. A host of uniaxial stress-strain curves constituting parts of the same stress-strain curve were assigned to material elements for each increment of loading. The path-dependent nature of the onset of crack initiation and growth was revealed. The proposed model predicts faster crack growth rates than those obtained on the basis of a single uniaxial stress-strain curve and is closer to experimental observation.     

The prediction of time-dependent non-linear stresses in viscoelastic materials: I. Selection of constitutive equation and strain measure

The theories for the prediction of time-dependent, non-linear stresses in viscoelastic materials such as polymers are reviewed, and it is noted that the commonly observed stress non-linearity may be ascribed either, as is usually done, to memory-function non-linearity or, alternatively, to strain-measure non-linearity. To investigate the latter alternative whilst retaining a general memory-function non-linearity, a single-integral constitutive equation of the Bird—Carreau type is employed but with an arbitrary strain measure I in place of the normally employed Finger tensor F. This model includes as special cases a large proportion of the constitutive equations previously employed for predictive purposes and in particular with a linear memory function it is shown to be indistinguishable, with the normally conducted shear experiments, from the successful BKZ model.In the new model the shear component I₁₂ of the strain measure can be found from experimental results obtained in the startup of steady shear flow, without specification or restriction of memory-function non-linearity. The form of I₁₂ found from experiment is quite non-linear in shear a for ¦a¦> 2, and hence differs from the F tensor for which F₁₂ = a. The same form for I₁₂ found for a variety of polymer solutions and a polymer melt and consequently a simple function describing I₁₂ is proposed as a new, material-independent, strain measure.     

Inversion of a full-range stress-strain relation for stainless steel alloys

This paper presents an approximate inversion of the stress-strain relation for stainless steel alloys. Using currently available stress-strain relations based on a modified Ramberg-Osgood equation, a new expression for the stress σ as an explicit function of the total strain ε is obtained. The new expression is valid over the full-range of the stress well beyond the 0.2% proof stress σ_0.2, defined as the stress level corresponding to the plastic strain value of 0.2%. The validity of the inverted expression is tested over a wide range of material parameters. The tests show that the new expression results in stress-strain curves which are both qualitatively and quantitatively consistent with the fully iterated numerical solution of the full-range stress-strain relation.     

The invariant representation of spins with applications in the theory of finite deformations

Based on the general solution given to a kind of linear tensor equations, the spin of a symmetric tensor is derived in an invariant form. The result is applied to find the spins of the left and the right stretch tensors and the relation among different rotation rate tensors has been discussed. According to work conjugacy, the relations between Cauchy stress and the stresses conjugate to Hill's generalized strains are obtained. Particularly, the logarithmic strain, its time rate and the conjugate stress have been discussed in detail. These results are important in modeling the constitutive relations for finite deformations in continuum mechanics. The project is supported by the National Natural Science Foundation of China and the Chinese Academy of Sciences (No. 87-52).     

Objective Tensor Rates and Applications in Formulation of Hyperelastic Relations

Following Ogden, a class of objective (Lagrangian and Eulerian) tensors is identified among the second-rank tensors characterizing continuum deformation, but a more general definition of objectivity than that used by Ogden is introduced. Time rates of tensors are determined using convective rates. Sufficient conditions of objectivity are obtained for convective rates of objective tensors. Objective convective rates of strain tensors are used to introduce pairs of symmetric stress and strain tensors conjugate in a generalized sense. The classical definitions of conjugate Lagrangian (after Hill) and Eulerian (after Xiao et al.) stress and strain tensors are particular cases of the definition of conjugacy of stress and strain tensors in the generalized sense used in the present paper. Pairs of objective stress and strain tensors conjugate in the generalized sense are used to formulate constitutive relations for a hyperelastic medium. A family of objective generalized strain tensors is introduced, which is broader than Hill’s family of strain tensors. The basic forms of the hyperelastic constitutive relations are obtained with the aid of pairs of Lagrangian stress and strain tensors conjugate after Hill (the strain tensors in these pairs belong to the family of generalized strain tensors). A method is presented for generating reduced forms of the constitutive relations with the aid of pairs of Lagrangian and Eulerian stress and strain tensors conjugate in the generalized sense which are obtained from pairs of Lagrangian tensors conjugate after Hill by mapping tensor fields on one configuration of a deformable body to tensor fields on another configuration.      

On solutions of the problem in stresses with the use of Maxwell stress functions

The spatial problems of elasticity are mainly solved in displacements [1, 2], i.e., the Lamé equations are taken as the initial equations. This is related to the lack of general solutions for the system of basic equations of elasticity expressed in stresses. In this connection, a new variational statement of the problem in stresses was developed in [3, 4]; this statement consists in solving six generalized equations of compatibility for six independent components of the stress tensor, while the three equilibrium equations are transferred to the set of boundary conditions. This method is more convenient for the numerical solution of problems in stresses and has been tested when solving various boundary value problems. In the present paper, analyzing the completeness of the Saint-Venant identities and using the Maxwell stress functions, we obtain a new resolving system of three differential equations of strain compatibility for the three desired stress functions φ, ξ, and ψ. This system is an alternative to the three Lamé equilibrium equations for three desired displacement components u, v, w and is simpler in structure. Moreover, both of these systems of resolving equations can be solved by the new recursive-operator method [5, 6]. In contrast to well-known methods for constructing general solutions of linear differential equations and their systems, the solutions obtained by the recursive-operator method are constructed as operator-power series acting on arbitrary analytic functions of real variables (not necessarily harmonic), and the series coefficients are determined from recursive relations (matrix in the case of systems of equations). The arbitrary functions contained in the general solution can be determined directly either from the boundary conditions (the obtained system of inhomogeneous equations with a right-hand side can also be solved by the recursive-operator method [6]) or by choosing them from various classes of analytic functions (elementary, special); a complete set of particular solutions can be obtained in the same function classes, and the coefficients of linear combinations of particular solutions can be determined by the Trefftz method, the least-squares method, and the collocation method.     

A class of universal relations in isotropic elasticity theory

A class of universal relations for isotropic elastic materials is described by the tensor equationTB = BT. This simple rule yields at most three component relations which are the generators of many known universal relations for isotropic elasticity theory, including the well-known universal rule for a simple shear. Universal relations for four families of nonhomogeneous deformations known to be controllable in every incompressible, homogeneous and isotropic elastic material are exhibited. These same universal relations may hold for special compressible materials. New universal relations for a homogeneous controllable shear, a nonhomogeneous shear, and a variable extension are derived. The general universal relation for an arbitrary isotropic tensor function of a symmetric tensor also is noted.     

Physical invariants for nonlinear orthotropic solids

Seven invariants, with immediate physical interpretation, are proposed for the strain energy function of nonlinear orthotropic elastic solids. Three of the seven invariants are the principal stretch ratios and the other four are squares of the dot product between the two preferred directions and two principal directions of the right stretch tensor. A strain energy function, expressed in terms of these invariants, has a symmetrical property almost similar to that of an isotropic elastic solid written in terms of principal stretches. Ground state and stress–strain relations are given. Using principal axes techniques, the formulation is applied, with mathematical simplicity, to several types of deformations. In simple shear, a necessary and sufficient condition is given for Poynting relation and two novel deformation-dependent universal relations are formulated. Using series expansions and the symmetrical property, the proposed general strain energy function is refined to a particular general form. A type of strain energy function, where the ground state constants are written explicitly, is proposed. Some advantages of this type of function are indicated. An experimental advantage is demonstrated by showing a simple triaxial test can vary a single invariant while keeping the remaining invariants fixed.     

Crack growth resistance in non-perfect plasticity: Isotropic versus kinematic hardening

The Strain Energy Density Theory is applied for analyzing energy dissipation and crack growth in the three-point bending specimen when the material behavior follows a multilinear strain-hardening stress-strain relationship. The problem is solved through the application of incremental theory of plasticity and finite element method.The rate of change of the strain energy density factor S with crack length a is verified to be governed by the relation . Results are obtained for isotropic and kinematic hardening. Moreover, the effects of loading step and specimen size are pointed out.     

Analysis of Strain Energy Behavior Throughout a Fatigue Process

The dissipation of strain energy density per cycle was analyzed to understand its trend through a fatigue process. The motivation behind this analysis is to improve a fatigue life prediction method, which is based on a strain energy and failure correlation. The correlation states that the same amount of strain energy is dissipated during both monotonic fracture and cyclic fatigue. This means the summation of strain energy density per cycle is equal to the total strain energy density dissipated monotonically. In order to validate this understanding, the strain energy density per cycle was analyzed at several alternating stress levels for fatigue life of Aluminum 6061-T6 (Al 6061-T6) between 10³ and 10⁵ cycles. The analysis includes the following: Alternating between high and low operating frequencies (50x magnitude difference), interruption of cyclic load during testing, and idle/zero-loading intervals of 20–40 minutes in-between cyclic loading sequences. All experimental results show a consistent trend of cyclic softening as the loading cycles approach failure; however, due to an inefficient curve fit procedure of the stress-dependent strain equation at low alternating stresses onto the experimental stress-strain data, a new approach for calculating the strain energy density per cycle is explored and shows promising results.