Analysis of nonlinear vibration for symmetric angle-ply laminated viscoelastic plates with damage

The behavior of nonlinear vibration for symmetric angle-ply laminated plates including the material viscoelasticity and damage evolution is investigated. By employing the von Karman's nonlinear theory, strain energy equivalence principle and Boltzmann superposition principle, a set of governing equations of nonlinear integro-differential type are derived. By applying the finite difference method, Newmark method and iterative procedure, the governing equations are solved. The effects of loading amplitudes, exciting frequencies and different ply orientations on the critical time to failure initiation and nonlinear vibration amplitudes of the structures are discussed. Numerical results are presented for the different parameters and compared with the available data. The project supported by the National Natural Science Foundation of China (10272042) and the Special Science Fund of the Doctoral Discipline of the Ministry of Education, China (20020532018) The English text was polished by Ron Marshall.     

NDE of metal damage: ultrasonics with a damage mechanics model

This research describes a nondestructive method for the quantitative estimation of property variations due to damage in metal materials. The method employs a damage mechanics model, which accounts for stiffness degradation and damage evolution of a metal medium with a measurement of ultrasonic velocity. In order to describe the progressive deterioration of materials prior to the initiation of macrocracks, we have developed a new damage mechanics model. Thereafter, a finite element model valid for numerically describing such damage process has been developed by ABAQUS/Standard code, and correlations between damage state, elastic stiffness and plastic strain could be found by the results of the finite element simulation. The property variations due to damage evolution are calculated based on the Mori–Tanaka theory, and then the ultrasonic velocity can be predicted by Christoffel’s equation. When the measured velocity is coupled with the theoretically predicted velocity, the unknown damage variable is solved, from which other residual properties are determined by the predictions of damage model. The proposed technique is performed on type 304 stainless steel bars. The numerical results obtained by the simulation were compared with experimental ones in order to verify the validity of the proposed finite element model and good agreement was found. It is shown that the damaged properties of metals can be estimated accurately by the proposed method.     

Investigation of the damage variable basic issues in continuum damage and healing mechanics

Consistent mathematics and mechanics are used here to properly interpret the damage variable within the confines of the concept of reduced area due to damage. In this work basic issues are investigated for the damage variable in conjunction with continuum damage and healing mechanics. First, the issue of the additive decomposition of the damage variable into damage due to voids and damage due to cracks in continuum damage mechanics is discussed. The accurate decomposition is shown to be non-additive and involves a term due to the interaction of cracks and voids. It is shown also that the additive decomposition can only be used for the special case of small damage. Furthermore, a new decomposition is derived for the evolution of the damage variable. The second issue to be discussed is the new concept of independent and dependent damage processes. For this purpose, exact expressions for the two types of damage processes are presented. The third issue addressed is the concept of healing processes occurring in series and in parallel. In this regard, systematically and consistently, the equations of healing processes occurring either consecutively or simultaneously are discussed. This is followed by introducing the new concept of small healing in damaged materials. Simplified equations that apply when healing effects are small are shown. Finally, some interesting and special damage processes using a systematic and original formulation are presented.     

Thermodynamics of polycrystalline materials treated by the theory of mixtures with continuous diversity

The physics of polycrystalline materials is described via microscopic processes such as grain boundary migration, grain growth, grain rotation, polygonization (the bending and breaking of crystallites) and evolution of dislocation density. The importance of taking these processes into account lies in their influence on the macroscopic mechanical behaviour of the material. Constitutive equations to describe such phenomena have been proposed in the literature. The main result of this paper is to give a general and thermodynamically consistent approach for such constitutive equations. The framework of the Theory of Mixtures with Continuous Diversity (TMCD) is used. The inclusion of both orientation and grain-size distributions is presented in this paper for the first time. Their introduction requires the formulation of a new and general constitutive theory that is, therefore, given. The method of Lagrange multipliers used in the context of the entropy principle (Liu, Arch. Rat. Mech. Anal. 46, 131–148 (1972)) provides the restrictions of the second law of thermodynamics on the constitutive equations. The success of this work is that all the main results present in the literature can be incorporated in this framework.     

On the thermodynamic consistency of the equivalence principle in continuum damage mechanics

We consider theories of continuum damage mechanics involving damage effect variables of different tensorial ranks. It turns out that orthotropic damage together with the use of Lemaitre's equivalence principle for the elastic part does not allow thermodynamic potentials such as the free enthalpy to exist. As the existence of these potentials is, however, a strict thermodynamic requirement, a theory employing orthotropic damage in this way is inconsistent. We show that the use of a rank-4 damage effect variable allows a consistent use of the equivalence principle.     

A continuum damage model for piezoelectric materials

In this paper, a constitutive model is proposed for piezoelectric material solids containing distributed cracks. The model is formulated in a framework of continuum damage mechanics using second rank tensors as internal variables. The Helrnhotlz free energy of piezoelectric mate- rials with damage is then expressed as a polynomial including the transformed strains, the electric field vector and the tensorial damage variables by using the integrity bases restricted by the initial orthotropic symmetry of the material. By using the Talreja＇s tensor valued internal state damage variables as well as the Helrnhotlz free energy of the piezoelectric material, the constitutive relations of piezoelectric materials with damage are derived. The model is applied to a special case of piezoelectric plate with transverse matrix cracks. With the Kirchhoff hypothesis of plate, the free vibration equations of the piezoelectric rectangular plate considering damage is established. By using Galerkin method, the equations are solved. Numerical results show the effect of the damage on the free vibration of the piezoelectric plate under the close-circuit condition, and the present results are compared with those of the three-dimensional theory.     

A note on viscoelastic materials that can age

Materials can stress relax for a variety of reasons, for example stress relaxation in classical viscoelastic bodies and that due to aging or damage. Here, within the context of a specific model we show that the stress relaxation that is characteristic of viscoelasticity is distinctly different from that of aging.     

Prediction of the softening and damage effects with permanent set in fibrous biological materials

Deformation induced softening is an inelastic phenomenon frequently accompanying mechanical response of soft biological tissues. Inelastic phenomena which occur in mechanical testing of biological tissues are very likely to be associated with alterations in the internal structure of these materials.In this study, a novel structural constitutive model is formulated to describe the inelastic effects in soft biological tissues such as Mullins type behavior, damage and permanent set as a result of residual strains after unloading. Anisotropic softening is considered by evolution of internal variables governing the anisotropic properties of the material. We consider two weight factors w_i (softening) and s_k (discontinuous damage) as internal variables characterizing the structural state of the material. Numerical simulations of several soft tissues are used to demonstrate the performance of the model in reproducing the inelastic behavior of soft biological tissues.     

Effect of thickness and boundary conditions on the behavior of viscoelastic layers in sliding contact with wavy profiles

In this work, the sliding contact of viscoelastic layers of finite thickness on rigid sinusoidal substrates is investigated within the framework of Green's functions approach. The periodic Green's functions are determined by means of a novel formalism, which can be applied, in general, to either 2D and 3D viscoelastic periodic contacts, regardless of the contact geometry and boundary conditions.Specifically, two different configurations are considered here: a free layer with a uniform pressure applied on the top, and a layer rigidly confined on the upper boundary. It is shown that the thickness affects the contact behavior differently, depending on the boundary conditions. In particular, the confined layer exhibits increasing contact stiffness when the thickness is reduced, leading to higher loads for complete contact to occur. The free layer, instead, becomes more and more compliant as thickness is reduced.We find that, in partial contact, the layer thickness and the boundary conditions significantly affect the frictional behavior. In fact, at low contact penetrations, the confined layer shows higher friction coefficients compared to the free layer case; whereas, the scenario is reversed at large contact penetrations. Furthermore, for confined layers, the sliding speed related to the friction coefficient peak is shifted as the contact penetration increases. However, once full contact is established, the friction coefficient shows a unique behavior regardless of the layer thickness and boundary conditions.     

Formulation of a damage internal state variable model for amorphous glassy polymers

The following article proposes a damage model that is implemented into a glassy, amorphous thermoplastic thermomechanical inelastic internal state variable framework. Internal state variable evolution equations are defined through thermodynamics, kinematics, and kinetics for isotropic damage arising from two different inclusion types: pores and particles. The damage arising from the particles and crazing is accounted for by three processes of damage: nucleation, growth, and coalescence. Nucleation is defined as the number density of voids/crazes with an associated internal state variable rate equation and is a function of stress state, molecular weight, fracture toughness, particle size, particle volume fraction, temperature, and strain rate. The damage growth is based upon a single void growing as an internal state variable rate equation that is a function of stress state, rate sensitivity, and strain rate. The coalescence internal state variable rate equation is an interactive term between voids and crazes and is a function of the nearest neighbor distance of voids/crazes and size of voids/crazes, temperature, and strain rate. The damage arising from the pre-existing voids employs the Cocks–Ashby void growth rule. The total damage progression is a summation of the damage volume fraction arising from particles and pores and subsequent crazing. The modeling results compare well to experimental findings garnered from the literature. Finally, this formulation can be readily implemented into a finite element analysis.     

Elastic and viscoelastic indentation of flat surfaces by pyramid indentors

The present work presents the results of frictionless and adhesionless contact of flat surfaces by pyramid indentors. The materials of the contacting solids were modelled as homogeneous and isotropic, linear elastic, as well as linear viscoelastic. The theoretical analysis is complemented by experiments and numerical calculations. The results include explicit relations between the normal applied load and the depth of penetration, details of the contact area shapes, the surface stresses and the contact pressure distributions. The standard shapes of the Vickers, Berkovich and Knoop pyramids were examined in particular. Certain aspects of geometrical imperfections, transverse isotropy and adhesion were considered. When elasticity or viscoelasticity provide adequate models of material behavior, micro- or nano-indentation by sharp pyramid indentors can be very useful and perhaps the only possible test in probing mechanical properties of small volumes of materials. The results can be particularly useful in using instrumented indentation for assessing mechanical properties of materials at cryogenic temperatures, of bio-materials and of micro-electro-mechanical components.     

A viscoelastic constitutive model of rubber under small oscillatory load superimposed on large static deformation considering the Payne effect

The viscoelastic behavior of carbon-black-filled rubber under small oscillatory loads superimposed on large static deformation is dealt with. In this class of problems, as the strain amplitudes of the load increase, the dynamic stiffness decreases, and this phenomenon is known as the Payne effect. Besides the effects of the static deformation and the frequencies of the superimposed dynamic load, the Payne effect is considered in this study. Influence factors are introduced in this model in order to consider the influence of static predeformation, the dynamic-strain-dependent properties, and frequency-dependent properties. For simplicity, separation of the three dominant variables, frequency, prestatic deformation, and dynamic amplitude of strain, is assumed. The Kraus model is used for describing the Payne effect. Dynamic tension tests are executed to obtain the model parameters and also for the verification of the proposed model. The suggested constitutive equation shows reasonable agreement with test data.     

Evolution of fabric tensors in damage mechanics of solids with micro-cracks: Part I – Theory and fundamental concepts

The evolution of fabric tensors based on micro-crack distributions is formulated based on sound thermodynamic principles. In Part I of this work, the exact definition of fabric tensors based on micro-crack distributions is presented. This definition is seen to incorporate both the orientation and length of a micro-crack. In this regard, the micro-crack distribution is assumed to be radially symmetric, i.e. symmetric about a line through the origin.The equations of thermodynamics are employed in order to derive the exact evolution equations of the fabric tensors defined in the first part. In this regard, a thermodynamic force that is associated with the fabric tensor is defined and utilized in the derivation of the evolution equations. The application of the theory to the case of uniaxial tension is derived in Part II (companion paper) of this work.     

Zur Aufstellung von Stoffgleichungen in der Kriechmechanik anisotroper Körper

Zusammenfassung Der Aufsatz vermittelt einen kurzen Überblick über mögliche Darstellungsformen von Stoffgleichungen unter Einbeziehung der Anisotropie und etwaiger Kriechschäden. Bei der Aufstellung von Stoffgleichungen werden Darstellungstheoreme von Tensorfunktionen zugrunde gelegt. Für praktische Anwendungen wird eine vereinfachte Theorie skizziert.

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Summary The paper will give a short survey about some possible representations of constitutive equations involving (initial) anisotropy of the material (e.g. from rolling) and creep-damage. The formulation of such equations is based upon theorems regarding scalarvalued and tensor-valued tensor functions. In order to attain to constitutive equations for more practical use, a simplified theory is proposed.

Mit 1 Abbildung     

Bi-variable damage model for fatigue life prediction of metal components

Based on the theory of continuum damage mechanics,a bi-variable damage mechanics model is developed,which,according to thermodynamics,is accessible to derivation of damage driving force,damage evolution equation and damage evolution criteria. Furthermore,damage evolution equations of time rate are established by the generalized Drucker’s postulate. The damage evolution equation of cycle rate is obtained by integrating the time damage evolution equations,and the fatigue life prediction method for smooth specimens under repeated loading with constant strain amplitude is constructed. Likewise,for notched specimens under the repeated loading with constant strain amplitude,the fatigue life prediction method is obtained on the ground of the theory of conservative integral in damage mechanics. Thus,the material parameters in the damage evolution equation can be obtained by reference to the fatigue test results of standard specimens with stress concentration factor equal to 1,2 and 3.     

A constitutive theory for creep behavior of initially isotropic materials sustaining unilateral damage

The goal of the present work is to modify structure of the creep constitutive equations existing in the literature, and simultaneously to incorporate both damage induced anisotropy and unilateral damage into the constitutive model. The proposed nonlinear-tensor constitutive equation for creep together with the damage evolution equation take into account the secondary and tertiary creep of the initially isotropic materials. The material parameters of the model are determined using basic experiments. It is shown that the creep model is capable of describing available experimental data for the lateral creep responses under uniaxial compression.