Isothermal Elasticity Research Articles

Comparison of Isothermal and Adiabatic Elasticity Characteristics of the Single Crystal Nickel-Based Superalloy CMSX-4 in the Temperature Range Between Room Temperature and 1300°C

Based on the results of performed thermophysical measurements and available experimental data for adiabatic elastic stiffnesses, isothermal elastic characteristics of the single crystal nickel-based superalloy CMSX-4 have been calculated for a wide temperature range, from room temperature to 1300°C. According to the results obtained, the adiabatic and isothermal values of such elastic characteristics as elastic stiffnesses {{c}_{{11}}}, {{c}_{{12}}} and bulk modulus of elasticity B differ significantly at high temperatures. The reasons for this are significant changes in the thermophysical properties of the alloy with temperature and a temperature increase of its Poisson’s ratio approaching the limiting value for cubic crystals, equal to 0.5. It is shown that the use of adiabatic elastic constants instead of isothermal ones in engineering calculations affects the relationship between the volumetric strain and hydrostatic stress, and this effect is similar to introducing a field of thermal dilatation into the analyzed object. At low temperatures, this effect is small, but at high temperatures, typical for the service conditions of the blade material of aircraft gas turbine engines, it increases many times.

An entropy-stable updated reference Lagrangian smoothed particle hydrodynamics algorithm for thermo-elasticity and thermo-visco-plasticity

This paper introduces a novel upwind Updated Reference Lagrangian Smoothed Particle Hydrodynamics (SPH) algorithm for the numerical simulation of large strain thermo-elasticity and thermo-visco-plasticity. The deformation process is described via a system of first-order hyperbolic conservation laws expressed in referential description, chosen to be an intermediate configuration of the deformation. The linear momentum, the three incremental geometric strains measures (between referential and spatial domains), and the entropy density of the system are treated as conservation variables of this mixed coupled approach, thus extending the previous work of the authors in the context of isothermal elasticity and elasto-plasticity. To guarantee stability from the SPH discretisation standpoint, appropriate entropy-stable upwinding stabilisation is suitably designed and presented. This is demonstrated via the use of the Ballistic free energy of the coupled system (also known as Lyapunov function), to ensure the satisfaction of numerical entropy production. An extensive set of numerical examples is examined in order to assess the applicability and performance of the algorithm. It is shown that the overall algorithm eliminates the appearance of spurious modes (such as hour-glassing and non-physical pressure fluctuations) in the solution, typical limitations observed in the classical Updated Lagrangian SPH framework.

Behavior of the isothermal Elasticity operator with non-linear friction in a thin domain

This paper deals with the asymptotic behavior of a boundary value problem in a three dimensional thin domain Ω ε with non-linear friction of Coulomb type. We will establish a variational formulation for the problem and prove the existence and uniqueness of the weak solution. We then study the asymptotic behavior when one dimension of the domain tends to zero. In which case, the uniqueness result of the displacement for the limit problem is also proved.

A New Updated Reference Lagrangian Smooth Particle Hydrodynamics algorithm for isothermal elasticity and elasto-plasticity

This paper presents a new Updated Reference Lagrangian Smooth Particle Hydrodynamics (SPH) algorithm for the analysis of large deformation isothermal elasticity and elasto-plasticity. Taking as point of departure a Total Lagrangian setting and considering as referential configuration an intermediate configuration of the deformation process, the equation of conservation of linear momentum and three geometric conservation laws (for the deformation gradient, its cofactor and its determinant) are re-written leading to a very generic system of first order conservation laws. The key feature of the formulation is a suitable multiplicative decomposition of the conservation variables, leading to a very simple final set of equations with striking similarities to the conventional Total Lagrangian system albeit re-written in terms of alternative Referential Updated conservation variables which are evolved in time. Taking advantage of this new Updated Reference Lagrangian formalism, a second order entropy-stable SPH upwinding stabilisation method will be introduced. With respect to previous publications by the group, a new three-stage Runge–Kutta time integration method is implemented in order to increase the CFL stability restriction. Finally, and to demonstrate the robustness and applicability of the methodology, a wide spectrum of challenging problems will be presented and compared, including some benchmark three-dimensional large deformation elasto-plasticity problems. To facilitate its ease of implementation, the paper explores the use of a series of novel expressions for the evaluation of kernels and the gradients of kernels to provide the SPH user the freedom to choose amongst various options, all leading to equally convincingly robust results.

An entropy-stable Smooth Particle Hydrodynamics algorithm for large strain thermo-elasticity

This paper presents a novel Smooth Particle Hydrodynamics computational framework for the simulation of large strain fast solid dynamics in thermo-elasticity. The formulation is based on the Total Lagrangian description of a system of first order conservation laws written in terms of the linear momentum, the triplet of deformation measures (also known as minors of the deformation gradient tensor) and the total energy of the system, extending thus the previous work carried out by some of the authors in the context of isothermal elasticity and elasto-plasticity (Lee et al., 2016; Lee et al., 2017; Lee et al., 2019). To ensure the stability (i.e. hyperbolicity) of the formulation from the continuum point of view, the internal energy density is expressed as a polyconvex combination of the triplet of deformation measures and the entropy density. Moreover, and to guarantee stability from the spatial discretisation point of view, consistently derived Riemann-based numerical dissipation is carefully introduced where local numerical entropy production is demonstrated via a novel technique in terms of the time rate of the so-called ballistic free energy of the system. For completeness, an alternative and equally competitive formulation (in the case of smooth solutions), expressed in terms of the entropy density, is also implemented and compared. A series of numerical examples is presented in order to assess the applicability and robustness of the proposed formulations, where the Smooth Particle Hydrodynamics scheme is benchmarked against an alternative in-house Finite Volume Vertex Centred implementation.

Structure and intrinsic properties of dispersion relation in hyperbolic thermoelasticity

ABSTRACTFor a system of field equations of hyperbolic thermoelasticity, we derive a propagation condition for a thermoelastic disturbance in a form of homogeneous plane wave in deformation and temperature. The corresponding dispersion relation is given in an explicit form, together with the dependence of characteristic coefficients on the principal invariants of the tensor of isothermal elasticity, heat conductivity, and thermoelasticity. Discussion of different types of homogeneous thermoelastic plane waves is given as well. Derived methodology is applied in the analysis of Green–Naghdi model.

Variational Theorem for the Dynamical Problem of Coupled Mechanothermodiffusion in Inhomogeneous Anisotropic Shells with Distortions

The variational principles are extensively used in the construction of new mathematical models and approximate methods for the solution of boundary-value problems of the elasticity theory [2, 10]. In the works by Biot [1], the variational principles of isothermal elasticity were generalized to the theory of coupled thermoelasticity for three-dimensional problems [6, 7, 9] and the problems of the theory of shells [4, 5, 12]. However, in the variational statement, the dynamical problems were formulated in these works without initial conditions. The approaches to the construction of variational principles proposed by Gurtin [8] made it possible to pose the boundary-value dynamical problems of thermoelasticity in the variational form as problems with initial conditions [11]. However, in the theory of shells, these approaches have not been widely applied yet. In the present work, we propose a variational statement of the dynamical problem of mechanothermodiffusion with initial conditions for inhomogeneous anisotropic shells with distortions. Consider a shell of constant thickness 2h . We refer the points of the shell in the three-dimensional space to a normal curvilinear coordinate system x = {x α ,z} , α =1,2 , where z = 0 is the middle surface of the shell G bounded by the contour g . It is assumed that the material of the shell is an inhomogeneous anisotropic two-component solid solution with a single symmetry plane at any point (for the physicomechanical properties of the shell). Suppose that, at the initial time, the shell is in the natural state at a temperature T 0 . Starting from the time τ > 0 , the shell suffers the action of an external surface force load q = {q α ,q 3 } ,

A first order hyperbolic framework for large strain computational solid dynamics. Part I: Total Lagrangian isothermal elasticity

This paper introduces a new computational framework for the analysis of large strain fast solid dynamics. The paper builds upon previous work published by the authors (Gil et al., 2014) [48], where a first order system of hyperbolic equations is introduced for the simulation of isothermal elastic materials in terms of the linear momentum, the deformation gradient and its Jacobian as unknown variables. In this work, the formulation is further enhanced with four key novelties. First, the use of a new geometric conservation law for the co-factor of the deformation leads to an enhanced mixed formulation, advantageous in those scenarios where the co-factor plays a dominant role. Second, the use of polyconvex strain energy functionals enables the definition of generalised convex entropy functions and associated entropy fluxes for solid dynamics problems. Moreover, the introduction of suitable conjugate entropy variables enables the derivation of a symmetric system of hyperbolic equations, dual of that expressed in terms of conservation variables. Third, the new use of a tensor cross product [61] greatly facilitates the algebraic manipulations of expressions involving the co-factor of the deformation. Fourth, the development of a stabilised Petrov–Galerkin framework is presented for both systems of hyperbolic equations, that is, when expressed in terms of either conservation or entropy variables. As an example, a polyconvex Mooney–Rivlin material is used and, for completeness, the eigen-structure of the resulting system of equations is studied to guarantee the existence of real wave speeds. Finally, a series of numerical examples is presented in order to assess the robustness and accuracy of the new mixed methodology, benchmarking it against an ample spectrum of alternative numerical strategies, including implicit multi-field Fraeijs de Veubeke–Hu–Washizu variational type approaches and explicit cell and vertex centred Finite Volume schemes.

Isothermal stress and elasticity tensors for ions and point dipoles using Ewald summations

The isothermal stress tensor and isothermal elasticity tensor for systems of point charges and of nonpolarizable point dipoles are derived from the strain derivatives of the free energy. For the case of point dipoles, it is shown that the angular dependence of the interaction potential gives rise to additional contributions to the stress and elasticity tensors not recognized previously.

Three-dimensional thermoelastic problems of interface cracks in periodic two-layered composites

This paper outlines the potential function method for solving three-dimensional interface crack problems of a two-layered periodic space, treated within the framework of linear stationary thennoelasticity with microlocal parameters. By constructing the appropriate harmonic functions, the resulting boundary-value problems regarding an interface crack are reduced to corresponding ordinary problems dealing with mechanical loading in homogeneous isothermal elasticity. This result—known as an analogy between the thermal crack problems and their mechanical counterparts—may be exploited to yield closed-form solutions following directly from that obtained in the isotropic case. In connection with the theory of fracture mechanics, the thermal stress singularities as well as the effects of the geometrical and mechanical parameters of the composite structure on them are discussed.

A Continuum Theory for Materials With Directional Porosity

A theory is developed for porous materials with directionally dependent lineal porosity. The assumed form of the directional dependence leads to an idealization of the porous material as a continuum with ordinary and dipolar displacements. The field equations for a linearized isothermal elasticity theory are derived for isotropic materials with nondirectional lineal porosity in the reference state.

The behavior of elastic heat conductors with second-order response functions

Second-order forms for the material response functions of an elastic heat conductor are derived by approximating the response functions by polynomials in the appropriate invariants. Solutions based upon these forms of the response functions are exact for special materials and approximate for general materials. The second-order dependence on temperature of isothermal elasticity solutions is found, and the results are shown to agree well with experimental data taken on rubber. Within the second-order theory for incompressible and isotropic bodies, the problems of biaxial stretching of a plate with transverse heat flow and the simultaneous extension and shear of a cylindrical annulus with radial heat flow are solved.

Successive approximations in nonlinear thermoelasticity

Starting with the basic equations of finite thermoelasticity for an incompressible, isotropic, homogeneous body, we generalize the usual method of successive approximations by expanding the relevant fields in terms of two parameters. These parameters separately characterize the mechanical and thermal loading on the body. The successive equations are derived, and due to the incompressibility and isotropy assumptions, it is seen that at each order they are not only linear but also completely uncoupled. The displacement is determined by the equations of linear isothermal elasticity, while the temperature field is determined by the equation of linear heat conduction. As an example, the solution, through second-order terms, is found for the problem of an annular cylinder with axial mechanical loading and radial thermal loading.

Biaxial stretching of heat‐softened plastic sheets: Experiments and results

AbstractA second‐generation apparatus was built and used to inflate heat‐softened plastic sheets to shapes from hemispheres to large spheroidal bubbles. Three sheet materials, polystyrene, high‐impact polystyrene, and cellulose acetate butyrate, were successfully formed beyond the hemispherical shape at temperatures somewhat below industrial levels. Four others, rigid poly‐(vinyl chloride), an acrylic‐modified PVC, cast poly(methyl methacrylate), and polycarbonate, could not be deformed beyond the hemispherical shape without rupture. Measurements included bubble profiles, thickness distributions and local extension ratios for the final bubbles; two plots of profile growth measured from high‐speed motion pictures; and a continuous record of bubble temperatures and blowing pressure from which the meridional temperature distributions were plotted.Fifteen bubbles were measured and analyzed. Profiles and extension ratios were compared with those computed from the equations of isotropic, isothermal elasticity, using several different strain‐energy functions with constant material parameters. Agreement over all fifteen bubbles on all properties was about ±10.8 percent (absolute) for the best models considered. Bubble growth occurred over periods of 5 to 8 seconds, with about 90 percent of the final area of the larger bubbles being generated in the final 1.5 seconds. The maximum areal elongations obtained were 18,600 percent for clear polystyrene at 238°F, 14,300 percent for high‐impact polystyrene at 224°F, and 1,010 percent for cellulose acetate butyrate at 280°F.

Note on a paper by Mehta and Chaudhry

Mehta and Chaudhry [1] have dealt with the problem of determining the thermoelastic stresses in a rotating circular solid cylinder with axially symmetric temperature distribution, according to the finite deformation theory. In this note, it is shown that their paper is in error. After the error is corrected, their problem reduces to the corresponding one in isothermal elasticity, whose solution is well known.

On a Variational Theorem in Coupled Thermoelasticity

Washizu’s principle for stresses, displacements, and strains in isothermal elasticity is extended to the field of coupled thermoelasticity. All field equations, including solid and thermal strain-displacement relations, are deduced from the extended functional. The relation to other known theorems is discussed.

On some steady state thermoelastic stress distributions in a slab

The calculation of the steady state thermal stresses in an isotropic elastic half space or slab with traction free faces has been the subject of several investigations. Steinberg and McDowell (1), using an extension of the Bousinesq-Papkowitch method of isothermal elasticity, first derived the now well-known result that in such a body which contains no heat sources there exists a plane state of stress parallel to the boundary planes. Sneddon and Lockett (2) approached this class of problems by direct solution of the equations of thermoelasticity using a double Fourier integral transform method, the results being transformed to Hankel type integrals in the case of axial symmetry. A further approach due to Nowinski (3) exploits the fact that in steady state thermoelasticity each component of the displacement vector is a biharmonic function which can be expressed as a combination of harmonics. However, possibly the most economical method of solution of this type of problem is that of Williams (4) who expressed the displacement vector in terms of two scalar potential functions, one of which is directly related to the temperature field. The same principle has also been used by Fox (5) in treating thermoelastic distributions in a slab containing a spherical cavity.

On variational principles in thermoelasticity and heat conduction

The variational principles for displacements, for stresses and for both displacements and stresses in isothermal elasticity are extended to the coupled processes of thermoelasticity and heat conduction in a three-dimensional, anisotropic body. The character of these principles is examined and it is established that in a stable system one is concerned with a minimum, a maximum and a stationary value problem, respectively.