Thermo-elastic interaction without energy dissipation in an infinite solid with distributed periodically varying heat sources

The paper deals with the thermo-elastic interactions due to distributed periodically varying heat sources in a homogeneous, isotropic, unbounded elastic medium in the context of the theory of thermo-elasticity without energy dissipation. Closed form solutions for displacement, temperature, stress and strain are derived by using Laplace transform on time and then Fourier transform on space. It reveals that the interactions consist of two coupled modified dilatational and thermal waves modified by finite thermal wave speed and thermo-elastic coupling traveling with finite speeds and without attenuation. The results are compared with previous results derived by using other generalized thermo-elasticity theories. Numerical results for a hypothetical material are presented.     

Sloshing and energy dissipation in an egg: SPH simulations and experiments

Tall structures, such as towers and bridges, can oscillate at excessive magnitudes when subjected to wind and earthquake loads. Liquid sloshing absorbers can be used to suppress these excessive oscillations by tuning the frequency of the sloshing to the critical frequency of the structure. Sloshing absorbers are simple structures consisting of a partially full container of liquid with a free surface. Tuning ensures that significant amounts of harmful energy can be extracted from the structure to the sloshing liquid. However, there needs to be a rapid means of dissipating this energy to avoid its returning back to the structure (then back to the liquid periodically).A hen׳s egg seems to have evolved to efficiently dissipate energy to protect its embryo using sloshing of its liquid content. Hence, the potential to implement the egg׳s unique properties as a sloshing absorber for structural control, is the main focus of this study. Numerical simulations, using Smoothed Particle Hydrodynamics (SPH), and experimental comparisons are presented in this paper. One objective is to demonstrate the ability of SPH to simulate complex free surface behaviour in three dimensions. Such a tool is then useful to identify different dissipation modes. Effects of fill volume and viscosity on the rate of dissipation, are also investigated.     

Inelastic deformation and energy dissipation in ceramics: A mechanism-based constitutive model

A mechanism-based constitutive model is presented for the inelastic deformation and fracture of ceramics. The model comprises four essential features: (i) micro-crack extension rates based on stress-intensity calculations and a crack growth law, (ii) the effect of the crack density on the stiffness, inclusive of crack closure, (iii) plasticity at high confining pressures, and (iv) initial flaws that scale with the grain size. Predictions of stress/strain responses for a range of stress states demonstrate that the model captures the transition from deformation by micro-cracking at low triaxiality to plastic slip at high triaxialities. Moreover, natural outcomes of the model include dilation (or bulking) upon micro-cracking, as well as the increase in the shear strength of the damaged ceramic with increasing triaxiality. Cavity expansion calculations are used to extract some key physics relevant to penetration. Three domains have been identified: (i) quasi-static, where the ceramic fails due to the outward propagation of a compression damage front, (ii) intermediate velocity, where an outward propagating compression damage front is accompanied by an inward propagating tensile (or spallation) front caused by the reflection of the elastic wave from the outer surface and (iii) high velocity, wherein plastic deformation initiates at the inner surface of the shell followed by spalling within a tensile damage front when the elastic wave reflects from the outer surface. Consistent with experimental observations, the cavity pressure is sensitive to the grain size under quasi-static conditions but relatively insensitive under dynamic loadings.     

Frictional energy dissipation in materials containing cracks

Kachanov's simplified model of microcrack interaction is applied to an investigation of the behaviour of a cracked body under predominantly compressive periodic loading, so that the cracks experience periods of closure and slip, with frictional dissipation. The model is shown to be equivalent to a discrete elastic frictional system with each crack representing one node. Theorems and algorithms from such systems are applied to determine the conditions under which the system shakes down to a state with no slip and hence no energy dissipation in friction. For conditions not too far beyond the shakedown state, the dissipation is significantly affected by the initial conditions, but with larger oscillating loads, it becomes a unique and increasing function of load amplitude. The effect of crack interaction is assessed by comparison with an uncoupled model, for which the dissipation is obtained as a summation of closed form expressions over the crack population. For small numbers of cracks, the results are significantly dependent on the randomly chosen crack locations and sizes, but with larger populations, a statistically significant decrease in dissipation is observed with increasing interaction terms.     

Heat conduction and heat wave propagation in functionally graded thick hollow cylinder base on coupled thermoelasticity without energy dissipation

In this paper, heat wave propagation and coupled thermoelasticity without energy dissipation in functionally graded thick hollow cylinder is presented based on Green–Naghdi theory. The material properties are supposed to vary as a power function of radius across the thickness of cylinder. The cylinder is considered in axisymmetry and plane strain conditions and it is divided to many sub-cylinders (layers) across the thickness. Each sub-cylinder is considered to be made of isotropic material and functionally graded property can be created by suitable arrangement of layers. The Galerkin finite element method and Newmark finite difference method are employed to solve the problem. The time history of second sounds and displacement wave propagation are obtained for various values of power function. Computed results agree well with the published data.     

Human leg impact: energy dissipation of wobbling masses

In terrestrial locomotion, the soft-tissue masses of the body undergo damped oscillations following leg impacts with the ground. Appropriate biomechanical models, therefore, describe gross soft-tissue dynamics by “wobbling masses”. We calculated mechanical energy balances of shank and thigh wobbling masses of the stance leg for the first 90 ms after touch-down in human heel-toe running. Thereto, we re-visited a data set on wobbling mass kinematics which had formerly been gained non-invasively by acquiring the motion of grids of lines painted on the skin of the corresponding muscle masses with high-speed cameras. We found frequencies ranging from 3 Hz to 55 Hz and maximum wobbling mass excursions relative to the bone ranging from 3 mm to 4 cm for the centres of mass and from 2.2° to 11.4° for the rotations. The rotational energy balance is practically neutral (±1 J). Usually, there is clearly more energy that is dissipated by wobbling mass movement in horizontal (thigh: <50 J) than in vertical direction (thigh: <15 J). There is less energy dissipated in the shank (horizontal: <10 J, vertical: <5 J). We argue that the energetic costs of separating significant wobbling masses from the skeleton may be over-compensated by avoiding metabolic costs of active impact reduction and by decreasing loads on passive skeletal structures, in particular when distal leg masses are functional, as in humans. Within reasonable biological limits, impacts are known to be even necessary for structural strengthening of bones. Beyond that, impacts might also be useful for stabilising locomotion, both by increasing basins of attraction and by providing simple mechanical signals for control.     

Temperature dependent mechanical behaviour of PVDF: Experiments and numerical modelling

The mechanical behaviour of Polyvinylidene Fluoride (PVDF) is analysed. To this end, tensile tests are performed on both smooth and notched specimens, for several values of the notch radius in order to set specific values of the stress triaxiality ratio in the net section. Tests were performed at various temperatures and at various strain rates. Experimental data together with fracture surface examinations by SEM allow the dependence of deformation and void growth processes on strain rate and temperature to be investigated. This experimental work was carried out in order to test the mechanics of porous media model. For each investigated temperature, constitutive relations take both porosity and strain rate sensitivity into account. The model is proposed for deformation leading to crazing. The material coefficients are optimised by imposing a continuous dependence on temperature.     

Biomimetic staggered composites with highly enhanced energy dissipation: Modeling, 3D printing,and testing

We investigate the damping enhancement in a class of biomimetic staggered composites via a combination of design, modeling, and experiment. In total, three kinds of staggered composites are designed by mimicking the structure of bone and nacre. These composite designs are realized by 3D printing a rigid plastic and a viscous elastomer simultaneously. Greatly-enhanced energy dissipation in the designed composites is observed from both the experimental results and theoretical prediction. The designed polymer composites have loss modulus up to ~500 MPa, higher than most of the existing polymers. In addition, their specific loss modulus (up to 0.43 km²/s²) is among the highest of damping materials. The damping enhancement is attributed to the large shear deformation of the viscous soft matrix and the large strengthening effect from the rigid inclusion phase.     

Interfacial energy and dissipation in martensitic phase transformations. Part I: Theory

A model of evolving martensitic microstructures is formulated that incorporates the interfacial energy and dissipation on three different scales corresponding to the grain boundaries attained by martensite plates, the interfaces between austenite and martensite plates, and the twin interfaces within martensite plates. Three different time scales are also considered in order to clarify the meaning of rate-independent dissipation related to instabilities at more refined temporal and spatial scales. On the slowest time scale, the process of deformation and martensitic phase transformation is investigated as being composed of segments of smooth quasi-static evolution separated by sudden jumps associated with creation or annihilation of interfaces. A general evolution rule is used in the form of minimization of the incremental energy supply to the whole compound thermodynamic system, including the rate-independent dissipation. Close relationship is shown between the evolution rule and the thermodynamic condition for stability of equilibrium, with no a priori assumption on convexity of the dissipation function. It is demonstrated that the extension of the minimum principle from the first-order rates to small but finite increments requires a separate symmetry restriction imposed on the state derivative of the dissipation function. Formulae for the dissipation associated with annihilation of interfaces are proposed that exhibit limited path-independence and satisfy that symmetry requirement. By exploiting the incremental energy minimization rule with the help of the transport theorems, local propagation conditions are derived for both planar and curved phase transformation fronts. The theory serves as a basis for the algorithm for calculation of the stress-induced evolution of martensitic microstructures along with their characteristic dimensions and related hysteresis loops in shape memory alloys; the examples are given in Part II of the paper.     

Generation and direct viscous dissipation of turbulent energy

Existing information about the generation and viscous dissipation of turbulent energy is based, as a rule, on the Laufer test data obtained for fluid flow in circular tubes at two Reynolds numbers (5 · 10⁵ and 5 · 10⁴). Computational dependences are presented herein for the generation and viscous dissipation of turbulent energy, common over the whole stream section and for the whole range of variation of the Reynolds number. The equation of the average energy balance during fluid flow in a circular tube and a flat channel is solved taking account of the equation of motion and the turbulent friction profile obtained by the author [1]. The computational dependences satisfy all the evident boundary conditions, agree with the Laufer test results [2] and yield a well-founded passage to the limit modes of average turbulent motion.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 30–36, November–December, 1973.     

Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases

Transformation plasticity in steels (i.e., the anomalous plastic flow observed during the progress of a phase transformation) is usually attributed to two distinct physical mechanisms, which have been proposed by Greenwood and Johnson and Magee. This paper proposes a theoretical approach to the problem, in the case where the Magee mechanism is negligible and the phases are ideal-plastic. An explicit expression for the transformation plastic strain rate is obtained for a steel undergoing a transformation under a small applied stress; this expression is consistent with experiments conducted on various materials. A finite element simulation provides a confirmation of the theoretical formula and allows for a detailed examination of the validity of some physical hypotheses made in the treatment. It also allows for a study of transformation plasticity under high applied stresses. Based on these results, a general (i.e., applicable for all kinds of stresses applied) model is proposed in the case of ideal-plastic phases.     

On the strain hardening and texture evolution in high manganese steels: Experiments and numerical investigation

We present a systematic investigation on the strain hardening and texture evolution in high manganese steels where twinning induced plasticity (TWIP) plays a significant role for the materials' plastic deformation. Motivated by the stress–strain behavior of typical TWIP steels with compositions of Fe, Mn, and C, we develop a mechanistic model to explain the strain-hardening in crystals where deformation twinning dominates the plastic deformation. The classical single crystal plasticity model accounting for both dislocation slip and deformation twinning are then employed to simulate the plastic deformation in polycrystalline TWIP steels. While only deformation twinning is activated for plasticity, the simulations with samples composed of voronoi grains cannot fully capture the texture evolution of the TWIP steel. By including both twinning deformation and dislocation slip, the model is able to capture both the stress–strain behaviors and the texture evolution in Fe–Mn–C TWIP steel in different boundary-value problems. Further analysis on the strain contributions by both mechanisms suggests that deformation twinning plays the dominant role at the initial stage of plasticity in TWIP steels, and dislocation slip becomes increasingly important at large strains.     

Dynamic response and energy dissipation characteristics of balsa wood: experiment and analysis

Dynamic response of a cellular sandwich core material, balsa wood, is investigated over its entire density spectrum ranging from 55 to 380 kg/m³. Specimens were compression loaded along the grain direction at a nominal strain rate of 3 × 10³ s⁻¹ using a modified Kolsky (split Hopkinson) bar. The dynamic data are discussed and compared to those of quasi-static experiments reported in a previous study (Mech. Mater. 35 (2003) 523). Results show that while the initial failure stress is very sensitive to the rate of loading, plateau (crushing) stress remains unaffected by the strain rate. As in quasi-static loading, buckling and kink band formation were identified to be two major failure modes in dynamic loading as well. However, the degree of dynamic strength enhancement was observed to be different for these two distinct modes. Kinematics of deformation of the observed failure modes and associated micro-inertial effects are modeled to explain this different behavior. Specific energy dissipation capacity of balsa wood was computed and is found to be comparable with those of fiber-reinforced polymer composites.     

Heat transfer in flow of nonlinear fluids with viscous dissipation

The rotational flow of viscoplastic fluids between concentric cylinders is examined while dissipation due to viscous effects through the energy balance. The viscosity of fluid is simultaneously dependent on shear rate and temperature. Exponential dependence of viscosity on temperature is modeled through Nahme law, and the shear dependency is modeled according to the Carreau equation. Hydrodynamically, stick boundary conditions are applied, and thermally, both constant temperature and constant heat flux on the exterior of cylinders are considered. The governing motion and energy balance equations are coupled adding complexity to the already highly correlated set of differential equations. Introduction of Nahme number has resulted in a nonlinear base flow between the cylinders. As well, the condition of constant heat flux has moved the point of maximum temperature toward the inner cylinder. Taking viscous heating into account, the effects of parameters such as Nahme and Brinkman numbers, material time and pseudoplasticity constant on the stability of the flow are investigated. Moreover, the study shows that the total entropy generation number decreases as the fluid elasticity increases. It, however, increases with increasing Nahme and Brinkman numbers.     

Damage evolution and energy dissipation of polymers with crazes

Craze damage evolution and energy dissipation of amorphous polymers with crazes have been studied. A mathematical model of a single craze (SC) is proposed by adopting the fibril creep mechanism. The viscoelastic characteristics of craze fibrils are supposed to obey the Maxwell model and the craze fibrils are assumed to be compressible. The assumption of Kausch [H.H. Kausch, The role of network orientation and microstructure in fracture initiation, J. Polym. Sci. C 32 (1971) 1–44] is adopted to describe the rupture of stressed fibril bonds. The craze damage evolution and the energy dissipation equations of a SC are derived. The equations are solved numerically and the life of a SC is computed. In a significant range of far-field stress, the dissipated energy varies linearly with the stress. Using the proposed model, the uniaxial stress-strain relation of polymers with low-density craze arrays (PLDCA) is investigated. The damage evolution equation of PLDCA is derived, which shows the mathematical relation between the damage of a SC and that of PLDCA. Based on the computed results, the variation of life of PLDCA with respect to applied stress is determined. Discussions are then given to the results and some significant conclusions are drawn.