Second-order Damage Tensor Research Articles

Creep modelling of masonry historic towers

This work illustrates a theoretical model developed to reproduce the behaviour of ancient masonry subjected to sustained stresses. Starting from a model recently proposed by the authors, two damage tensors have been introduced into a rheological model: the components of these tensors change both according to the intensity of the applied stress and, in case of sustained stress, to the duration of the load history. Evolution laws found in the literature for brittle materials have been employed. The principal directions of damage are meant to represent the directions of the experimental cracks; accordingly, when any damage direction is activated, it remains unchanged throughout the subsequent load history. The presence of second-order damage tensors makes it possible to describe the damage-induced anisotropy of the microcracked material. Also, since the possible increase of damage in time is accounted for, the model is able to describe creep failure and to predict the creep time to failure of the material under given stresses. The model parameters can be obtained through uniaxial creep tests on masonry samples at increasing stress levels and up to failure. The model was implemented into a finite element code, and structural analyses were carried out to assess the safety of middle-age masonry towers. The results obtained for one of these towers are briefly described and discussed.

An ‘extended’ volumetric/deviatoric formulation of anisotropic damage based on a pseudo-log rate

Following a framework of elastic degradation and damage previously proposed by the authors, an 'extended' formulation of orthotropic damage in initially isotropic materials, based on volumetric/deviatoric decomposition, is presented. The formulation is founded on the concept of energy equivalence and makes use of second-order symmetric tensor damage variables. It is characterized by fourth-order damage-effect tensors (relating nominal to effective stresses and strains) built from the underlying second-order damage tensors and decomposed in product-form in isotropic and anisotropic parts. The formulation is developed in two steps. First, secant relations are established. In the isotropic case, the model embeds a path parameter allowing to range between pure volumetric to pure deviatoric damage. With the two undamaged material constants this makes a total of three constant parameters plus an evolving scalar damage variable, giving rise to a four-parameter model with two varying isotropic material coefficients. In the anisotropic case, the model is still characterized by the same three material constants plus three evolving variables which are the principal values of a second-order damage tensor. This leads to a six-parameter restricted form of orthotropic damage. In the second step, damage evolution rules are formulated in terms of a pseudo-logarithmic rate of damage. This allows to define meaningful conjugate forces that constitute a feasible space in which loading functions and damage evolution rules can be defined. The present 'extended' formulation is closed by the derivation of the tangent stiffness.

A Formulation of Anisotropic Elastic Damage Using Compact Tensor Formalism

An anisotropic elastic-damage model for initially-isotropic materials is presented. The model is based on a pseudo-logarithmic second-order damage tensor rate. To derive the complete expression of the tangent stiffness entering the rate constitutive law, various tensor operations and derivatives of tensor functions must be developed. Such derivations have been performed in compact form. Some useful tensor derivatives and a table of tensor algebra operations are given in Appendix. This note should interest engineering researchers involved in the development of constitutive models through tensor formalism.

Dissymétrie de comportement élastique anisotrope couplé ou non à l'endommagement

A thermodynamic framework is proposed to model anisotropic elasticity different in tension and in compression. Based on Kelvin decomposition of the compliance tensor, it applies to any 3D loading. Coupling with damage is made considering fourth- and second-order damage tensors. The proposed formulation automatically satisfies the continuity of the stress tensor and of the energy release rate tensor. The particular case of initially isotropic materials is exposed.

A coupled anisotropic damage model for the inelastic response of composite materials

A coupled incremental damage and plasticity theory for rate-independent and rate-dependent composite materials is introduced here. This allows damage to be path-dependent either on the stress history or thermodynamic force conjugate to damage. This is achieved through the use of an incremental damage tensor. Damage and inelastic deformations are incorporated in the proposed model that is used for the analysis of fiber-reinforced metal–matrix composite materials. The damage is described kinematically in both the elastic and inelastic domains using the fourth-order damage effect tensor which is a function of the second-order damage tensor. A physical interpretation of the second-order damage tensor is given in this work which relates to the microcrack porosity within the unit cell. The inelastic deformation behavior with damage is viewed here within the framework of thermodynamics with internal state variables. Computational aspects of both the rate-independent and rate-dependent models are discussed in this work. The Newton–Rapson iterative scheme is used for the overall laminate system. The constitute equations of both the rate-independent and the rate-dependent plasticity coupled with damage models are additively decomposed into the elastic, inelastic and damage deformations by using the three-step split operator algorithm [J.W. Ju, Internat. J. Solids Struc. 25 (1989) 803–833]. The main framework return maping algorithm [M. Ortiz, C. Simo, Internat. J. Numer. Meth. Eng. 23 (1986) 353–366] is used for the correction of the elasto-plastic and viscoplastic states. However, for the case of the damage model these algorithms are redefined according to the governed damage constitutive relations. In order to test the validity of the model, a series of laminated systems (0 (8s)), (90 (8s)), (0/90) (4s), (−45/45) (2s) are investigated at both room and elevated temperatures of 538°C and 649°C. The results obtained from the special purpose developed computer program, DVP-CALSET (Damage and Viscoplastic Coupled Analysis of Laminate Systems at Elevated Temperatures), are then compared with the available experimental results and other existing theoretical material models obtained from the work of B.S. Majumdar, G.M. Newaz [CR-189095, NASA, 1992] and G.Z. Voyiadjis, A.R. Venson [Internat. J. Damage Mech. 4 (4) (1995) 338–361].

Thermodynamic modeling of creep damage in materials with different properties in tension and compression

A constitutive model is developed to describe the creep response of polycrystalline metals and alloys with different behavior in tension and compression. A second-order damage tensor is introduced in order to describe the creep damage under nonproportional loading in nonisothermal processes. The thermodynamic formulation of the creep equation and the damage evolution equation is used to study the creep behavior and creep damage in the initially isotropic materials. The determination of the material parameters in the proposed equations is demonstrated from a series of basic experiments outlined in this paper. The results generated from the model are compared with those obtained from experiments under uniaxial nonproportional and multiaxial nonproportional loading for both isothermal and nonisothermal processes.

The kinematics of damage for finite-strain elasto-plastic solids

In this paper the kinematics of damage for finite strain, elasto-plastic deformation is introduced using the fourth-order damage effect tensor through the concept of the effective stress within the framework of continuum damage mechanics. In the absence of the kinematic description of damage deformation leads one to adopt one of the following two different hypotheses for the small deformation problems. One uses either the hypothesis of strain equivalence or the hyphothesis of energy equivalence in order to characterize the damage of the material. The proposed approach in this work provides a general description of kinematics of damage applicable to finite strains. This is accomplished by directly considering the kinematics of the deformation field and furthermore it is not confined to small strains as in the case of the strain equivalence or the strain energy equivalence approaches. In this work, the damage is described kinematically in both the elastic domain and plastic domain using the fourth order damage effect tensor which is a function of the second-order damage tensor. The damage effect tensor is explicitly characterized in terms of a kinematic measure of damage through a second-order damage tensor. Two kinds of second-order damage tensor representations are used in this work with respect to two reference configurations. The finite elasto-plastic deformation behavior with damage is also viewed here within the framework of thermodynamics with internal state variables. Using the consistent thermodynamic formulation one introduces seperately the strain due to damage and the associated dissipation energy due to this strain.

Damage modeling of the single crystal superalloy SRR99 under monotonous creep

The evolution of material damage of single crystal superalloys depends not only on the load conditions, but also strongly on the lattice orientation. Using the theory of continuum damage mechanics, a phenomenological creep damage model for cubic single crystal superalloys is derived. In this model, a symmetric second-order damage tensor is used to describe the anisotropic nature of damage. The damage deactivation and reactivation is represented by an active damage tensor. As the effects of the current state of damage on the deformation process and on the damage development are different in nature, separate effective stresses for creep and damage are defined. Furthermore, a mapped damage active stress is introduced to reflect the influence of material symmetry on the damage growth rate by using an orientation function. Based on microscopic observations, it is assumed that only the principle tensile damage active stresses are responsible for the damage development, and that the anisotropy of the damage evolution mainly depends on the principle directions of the damage active stress and the material symmetry. Within the framework of thermodynamics, the damage evolution law is constructed by considering both the initial material anisotropy and the anisotropic influence of the current state of damage. Based on the effective stress concept, this damage model has been implemented into a three-dimensional anisotropic viscoplastic model and applied to the simulation of the creep damage behavior of the single crystal superalloy SRR99 at 760°C.

A creep damage model for initially isotropic materials with different properties in tension and compression

A continuum damage mechanics model for the dislocation creep response associated with the growth of parallel planar mesocracks in initially isotropic materials is presented. The model describes simultaneously different damage development in tension, compression and torsion, damage-induced anisotropy, as well as different creep properties in tension, compression and torsion. The proposed constitutive equation for creep and the damage growth equation contain joint invariants of the stress tensor and the second-order damage tensor. The determination of the material parameters required in the proposed equations, from a series of basic experiments, is shown. Theoretical predictions are compared with experimental data for a multiaxial stress state of various materials in the primary, secondary and tertiary creep state under proportional and non-proportional loading.

Kinematic Description of Damage

In this paper the kinematics of damage for finite elastic deformations is introduced using the fourth-order damage effect tensor through the concept of the effective stress within the framework of continuum damage mechanics. However, the absence of the kinematic description of damage deformation leads one to adopt one of the following two different hypotheses. One uses either the hypothesis of strain equivalence or the hypothesis of energy equivalence in order to characterize the damage of the material. The proposed approach in this work provides a relation between the effective strain and the damage elastic strain that is also applicable to finite strains. This is accomplished in this work by directly considering the kinematics of the deformation field and furthermore it is not confined to small strains as in the case of the strain equivalence or the strain energy equivalence approaches. The proposed approach shows that it is equivalent to the hypothesis of energy equivalence for finite strains. In this work, the damage is described kinematically in the elastic domain using the fourth-order damage effect tensor which is a function of the second-order damage tensor. The damage effect tensor is explicitly characterized in terms of a kinematic measure of damage through a second-order damage tensor. The constitutive equations of the elastic-damage behavior are derived through the kinematics of damage using the simple mapping instead of the other two hypotheses.

Anisotropic Damage Effect Tensors for the Symmetrization of the Effective Stress Tensor

Based on the concept of the effective stress and on the description of anisotropic damage deformation within the framework of continuum damage mechanics, a fourth order damage effective tensor is properly defined. For a general state of deformation and damage, it is seen that the effective stress tensor is usually asymmetric. Its symmetrization is necessary for a continuum theory to be valid in the classical sense. In order to transform the current stress tensor to a symmetric effective stress tensor, a fourth order damage effect tensor should be defined such that it follows the rules of tensor algebra and maintains a physical description of damage. Moreover, an explicit expression of the damage effect tensor is of particular importance in order to obtain the constitutive relation in the damaged material. The damage effect tensor in this work is explicitly characterized in terms of a kinematic measure of damage through a second-order damage tensor. In this work, tensorial forms are used for the derivation of such a linear transformation tensor which is then converted to a matrix form.

Plasticity and ductile fracture damage: Study of void growth in metals

The kinetics of internal deterioration by void growth around a spherical inclusion is studied as coupled with inelastic deformation. The void growth is described by the rate of a second-order damage tensor φ, whose definition supposes an appropriate averaging process over a statistically significant representative volume (the “unit cell”). The correlation between the damage tensor rate and the plastic deformation rate is established on the basis of results from auxiliary solutions of model boundary value problems for cavity growth in an infinite body. These results are properly adapted to the finite cell case. The damage tensor definition and the hypothesis on the microscopic pointwise displacement field within the cell constitute the basis of a method proposed for setting up the damage evolution equation quantifying the void growth process. The method has the advantage of being flexible enough so that the more refined approximations for a microscale displacement field can be adjusted to increase the accuracy of void growth description. Nevertheless, even starting from the simple Rice-Tracey-like microscopic velocity approximation, by applying the present method one can arrive at the tensorial damage kinetics relation which expresses the important features of the phenomena under consideration in a quantified manner. Thus, the relations obtained in this paper have the general form ·φ = ·φ(·E (p), φ, k ) and indicate the following effects: 1. (i) The damage rate as coupled with the macroscopic plastic strain rate Ė (p) depends on the actual damage state (φ); 2. (ii) it is affected by an initial inclusion size and spacing (viz. the presence of a scale factor k in the kinetics equations); 3. (iii) there is no coaxiality between strain rate and damage-rate tensors in the general case. Some exemplary situations are presented; relevant examples are developed.