1. Introduction

Damage in composite structures is a very complex phenomenon which can occur through a different number of failure mechanisms such as fibre breakage, fibre buckling, matrix cracking, fibre-matrix debonding and delamination either combined or individually. The increasing computational resources have allowed reliable prediction of such phenomenon with a certain degree of accuracy by using the finite element method. However, there is still a lot of work to be done in this field to better understand the physics of failure in order to improve numerical failure models for composite materials. The damage modelling in composites can be broadly divided into four different approaches:

Failure criteria approaches were initially developed for unidirectional materials and restricted to the static regime. They are divided into two categories: interactive and noninteractive criteria [1]. Noninteractive criteria assume that the failure modes are decoupled and specific expressions are used to identify each failure mechanism. In the stress based criteria, for example, each and every one of the stresses in the principal material coordinates must be less than the respective strengths; otherwise, fracture is said to have occurred. In a similar fashion, the maximum strain failure criteria states that the failure occurs if one of the strains in the principal material coordinates exceed its respective failure strain.

On the other hand, interactive criteria assume an interaction between two or more failure mechanisms and they describe the failure surface in the stress or strain space. Usually stress or strain polynomial expressions are used to describe the boundaries for the failure surface or envelope. Any point inside the envelope shows no failure in the material. Several interactive failure criteria can be found in the open literature, such as the Tsai and Wu [2] criterion, Tsai-Hill criterion, Hoffman criteria among others [1]. Nevertheless, the disadvantage in using such polynomial criteria is that they do not say anything about the damage mechanisms themselves, therefore modified versions have been used to distinguish between failure modes. Hashin [3] proposed a three-dimensional failure criteria for unidirectional composites. In his model four distinct failure modes associated with fibre failure in tension or compression and matrix cracking in tension or compression are modelled separately. Engblom and Havelka [4] proposed the use of a combination of the Hashin [3] and Lee [5] failure criteria. They used the Hashin criterion to detect in-plane failure and the Lee criteria to predict delaminations. The degradation was performed by reducing the stresses associated with each failure mechanism to zero in the composite material constitutive law.

Shivakumar et al. [6] used the Tsai-Wu failure criterion and maximum stress criterion to model low-velocity impact damage in composites. In this approach the Tsai-Wu criterion was used in order to determine whether the damage has occurred and the maximum stress criteria was used to identify the failure mode. Good agreement was obtained between computed and experimental damage areas.

F.-K. Chang and K.-Y. Chang [7, 8] proposed a progressive in-plane damage model for predicting the residual strength of notched laminated composites. Three in-plane failure modes are considered: matrix cracking, fibre-matrix shearing and fibre breakage. Their model is currently available in the material model library of the LS-DYNA3D explicit finite element code. In their model the shear stress-shear strain relation is assumed to be nonlinear and an expression proposed by Hahn and Tsain [9, 10] is used to represent composite behaviour in shear. For fibre breakage and/or fibre-matrix shearing the degree of property degradation within the damaged area depends on the size of the damage predicted by the fibre failure criterion. F.-K. Chang and K.-Y. Chang [7] proposed a property reduction model for fibre failure based on the micromechanics approach for fibre bundle failure. It is postulated that for fibre failure both and are reduced to zero, but and the shear modulus degenerate according to the Weibull distribution.

Choi et al. [11] investigated the low velocity impacts on composite plates using a line-nose impactor. They used the Chang-Chang failure criterion to detect the initiation of matrix cracking and delamination. Good agreement between experimental and numerical results was achieved.

Davies and Zhang [12] studied the low-velocity impact damage in carbon/epoxy composite plates impacted by an hemispherical impactor. The plates tested were quasi-isotropic having a symmetrical lay-up. Different impact energy levels, thickness, dimensions and boundary conditions were considered. The finite element modelling was carried out using plate elements.

The Chang-Chang failure criteria were used for the in-plane damage predictions and in general good agreement was obtained between the computed force histories and experiments.The Chang-Chang failure criteria have shown a reasonable performance for in plane damage predictions in composite structures and therefore, their applications have been widely reported in the literature [13–18].

The prediction of the onset of delamination depends on the interlaminar stress state and interlaminar strength of the laminate. Kim and Soni [19] used the distribution of interlaminar normal stress, and averaged that stress along a ply thickness distance in all cases they studied. They assumed that failure occured when the average of the normal interlaminar stress value over the fixed distance reached the interlaminar tensile strength.

Jen et al. [20] developed a model based on boundary layer theory to predict initiation and propagation of delamination in a composite laminate containing a central circular hole. The Hashin-Rotem failure criterion was adopted in their model to predict the loading and location at which the initiation of delamination occurs.

Brewer and Lagace [21] proposed a quadratic stress criterion for initiation of delamination using the approach suggested by Kim and Soni [19]. According to their criterion only out-of-plane stresses contribute to delamination and they assume that the predicted stresses should be independent of the sign of the interlaminar shear stress. A similar criteria was also proposed by Liu et al. [22] to predict delamination in composite laminates.

The disadvantage in using stress based criteria for composite materials is that the scale effects relating to the length of cracks subject to the same stress field cannot be modelled correctly [13, 23]. In the failure criteria approach either the position and size of the cracks are unknown. For these reasons the fracture mechanics approach may be more attractive. Fracture mechanics considers the strain energy at the front of a crack of a known size and compares the energy with critical quantities such as critical strain energy release rate.

The fracture mechanics approach has been used to predict compression after impact strength of composite laminates [24–27]. In such models the damaged area is replaced by an equivalent hole and the inelastic deformation associated with fibre microbuckling that develops near the hole edge is replaced with a equivalent crack loaded on its faces by a bridging traction which is linearly reduced with the crack closing displacement. The diameter of the hole is obtained from X-radiographs and/or ultrasonic C-scan images. The results showed good correlation between analytical and experimental values.

Another potential application of the fracture mechanics approach is its indirect use to predict progressive delamination in composites [28–31]. In such models stress-displacement constitutive laws describe the interfacial material behaviour and fracture mechanics concepts are used. The area defined by the constitutive relationship is equal to the fracture energy or energy release rate and once the stresses have been reduced to zero the fracture energy has been consumed and the crack propagates. Linear and quadratic interaction relationships were assumed to describe the crack propagation in mixed-mode delamination. Comparisons were made with experimental and closed-form results and good agreement was obtained.

However, the fracture mechanics approach cannot be easily incorporated into a progressive failure methodology because its application requires an initial flaw. A possible solution is to use a hybrid approach by using a stress or strain-based criterion for the failure initiation and a fracture mechanics approach for the failure propagation.

The plasticity approach is suitable for composites that exhibit ductile behaviour such as boron/aluminium, graphite/PEEK and other thermoplastic composites. Vaziri et al. [32] proposed an orthotropic plane stress material model that combines the classical flow theory of plasticity with a failure criterion. In their work the material constitutive law is assumed to be elastic-plastic and it has two stages. The first stage is the post-yield and pre-failure where an orthotropic plasticity model is used to model the nonlinear material behaviour. The second stage is the postfailure where brittle or ductile failure modes start to occur. Favourable agreement is obtained between experiment and the model results.

The damage mechanics approach has been investigated by many researchers in recent years and its application to damage modelling in composites has shown to be efficient. The method was originally developed by Kachanov [33] and Rabotnov [34] and it has the potential to predict different composite failure modes such as matrix cracking, fibre fracture and delamination.

Ladeveze and Dantec [35] proposed an in-plane model based on damage mechanics to predict matrix microcracking and fibrematrix debonding in unidirectional composites. Two internal damage variables were used to degrade the ply material properties, one of them associated with the transverse modulus and another with the in-plane shear modulus. A linear elastic-damage behaviour was assumed for tensile and compressive stresses and a plasticity model was developed to account for the inelastic strains in shear. Damage evolution laws associated with each failure mechanism were introduced which relate the damage variables to strain energy release rates in the ply. Tension, compression and cyclic shear tests were performed to determine the constants required in the damage-development laws. Comparisons with experiments were performed by the authors and a good correlation between numerical and experimental results was obtained.

Johnson [36] applied the model suggested by Ladeveze and Dantec for the prediction of the in-plane damage response of fibre reinforced composite structures during crash and impact events. The damage model was implemented for shell elements into the PAM-CRASH explicit finite element code. An experimental programme was carried out in order to validate the model. Low velocity impact tests were performed using a drop test rig. The plates were simply supported on a square steel frame and they were impacted at their centre using a hemispherical impactor with 50 mm diameter. The mass of the impactor was 21 kg. The plates were fabricated from 16 plies of carbon/epoxy with a quasi-isotropic layup. Different energy levels were considered to give a range of different failure modes from rebound to full penetration. Based on force history comparisons between experiments and numerical results the model overpredicted the peak load. According to Johnson, the discrepancy between measured and predicted peak load is explained by the neglected delamination in the model. This fact was demonstrated by the author in his subsequent work [37] where the delamination was included in the damage modelling using contact interface conditions.

Williams and Vaziri [38] implemented an in-plane damage model based on continuum damage mechanics (CDM) into LS-DYNA3D for impact damage simulation in composite laminates. The model was originally developed by Matzenmiller et al. [39]. The model considers three damage parameters: fibre failure damage parameter, matrix failure damage parameter and an extra damage parameter to account for the effect of damage in shear response. Individual stress based criteria for each failure mechanism are used for the damage initiation. The failure criterion defines certain regions in stress (or strain) space where the damage state does not change. The damage growth law adopted by Matzenmiller is a function of the strain and it assumes an exponential form. The stress/strain curve predicted by this damage function is a Weibull distribution that can be derived from statistical analysis of the probability of the failure of a bundle of fibres with initial defects. Impact simulations with different energy levels were performed and the performance of the proposed model was checked against the Chang-Chang failure criteria and experimental results. Some limitations of the model were pointed out by the authors. Firstly, the response is predicted by a single equation and, as a result the loading and postfailure responses cannot be separated, thus restricting the versatility of the model. Also, the dependence of the damage growth on the loading rate as well as mesh size.

In his recent work Williams et al. [40] proposed a plane-stress continuum damage mechanics model for composite materials. The model was implemented for shell elements into LSDYNA3D explicit finite element code. Also, the model is an extension of their previous work [41] and it deals with some issues related to the limitation of the model suggested by Matzenmiller et al. [39]. The approach adopted by the authors uses the sub-structuring concept where one integration point is used for each sub-laminate and composite laminate theory is used to obtain its effective material properties. The model assumes a bilinear damage growth law and the damage process has two phases, one associated with matrix/delamination damage and another with fibre breakage. The driving force for damage growth is assumed to be a strain potential function and the threshold values for each damage phase are experimentally determined. High-velocity and low-velocity impact simulations were performed in order to assess the performance of the model. Also, the results were compared with the model proposed by Matzenmiller et al. [39] and the existing Chang-Chang failure criteria. Pinho et al. [42] proposed a three-dimensional failure model to predict failure in composites. Their model is an extension of the plane-stress failure model proposed by Davila and Camanho [43] and it accounts for shear nonlinearities effects. The model was implemented into LS-DYNA3D finite element code and the authors obtained good correlation between numerical predictions and experimental results for static standard in-plane tests. However, neither strain rate effects nor strain localization problems associated with irregular mapped meshes were addressed by the authors.

This paper presents a formulation for a three-dimensional ply failure model for composite laminates. The proposed model is an extension of the authors previous work [44]. The new version of the model incorporates strain rate effects in shear by means of a semiempirical viscoplastic constitutive law and combines a quadratic stress based criterion with the Mohr-Coulomb failure criterion to predict interfibre-failure (IFF) without knowing a priori the orientation of the fracture plane. The new version of the model also uses the Second Piola-Kirchhoff stress tensor which potentially avoids material deformation problems. High-order damage evolution laws are proposed to avoid both numerical instabilities and artificial stress waves propagation effects commonly observed in the numerical response of Finite Element Codes based on Explicit Time Integration Schemes. These laws compared to the widely used bilinear law ensure smoothness at damage initiation and fully damaged stress onsets leading to a more stable numerical response. Iterative energy based criteria have been also added to the previous version [44] to better predict damage under multiaxial stress states. The model also incorporates a three dimensional objectivity algorithm [45] which accounts for orthotropic and crack directionality effects, making the model to be completely free of strain localization and mesh dependence problems.

2. Formulation

This section presents the failure model formulation. The formulation is based on the Continuum Damage Mechanics (CDM) approach and enables the control of the energy dissipation associated with each failure mode regardless of mesh refinement and fracture plane orientation by using a smeared cracking formulation. Internal thermodynamically irreversible damage variables were defined in order to quantify damage concentration associated with each possible failure mode and predict the gradual stiffness reduction during the fracture process. The material model has been implemented into ABAQUS explicit finite element code within brick elements as an user defined material model.

2.1. -D Orthotropic Stress-Strain Relationship

The orthotropic material law that relates second Piola-Kirchhoff stress to the Green-St. Venant strain in the local material coordinate system is written as where is the Green-St. Venant strain vector and is the transformation matrix given by where and . is the inplane rotation angle around the Z-direction which defines the orientation of the material axes with respect to the reference coordinate system. The compliance matrix is defined in terms of the material axes as where the subscripts denote the material axes, that is, Since is symmetric, The Green-St. Venant strain tensor is given by where is the deformation gradient and is the identity matrix. The Cauchy stress tensor can be determined in terms of the second Piola-Kirchhoff stress tensor as follows: where is defined as the Jacobian determinant which is the determinant of the deformation gradient . The present formulation will predict realistic material behaviour for finite displacements and rotation as long as the strains are small.

2.2. Degraded Stresses

The degraded or damaged stresses are defined as stresses transmitted across the damaged part of the cross-section in a representative volume element of the material. Based on the isotropic damage theory as originally proposed by Kachanov [33], an orthotropic relationship between local stresses acting on the damaged configuration can be written in terms of the local effective stresses in the undamaged configuration at ply level as follows:

where , , and are internal variables introduced to quantify the damage concentration within the Representative Volume Element (RVE) [46, 47]. A detailed description on the physical meaning and definition of these variables will be given in the following sections. The stress components with the superscript are the stresses degraded locally on the action fracture plane. Details about the determination of the local action plane and the local degradation procedure is given in Section 2.4.

The proposed orthotropic relationship between degraded and intact stresses ensures that the material stiffness matrix is positive defined during the degradation process. Moreover, it is physically based. By inspecting carefully (8) one can notice that transverse compression damage degrades the through-the-thickness stresses , and . From the physical point of view, this relationship incorporates the the main features observed in the experiments, such as follows:

2.3. Fibre Failure

The failure index to detect fibre failure in tension is given by

In order to detect the catastrophic failure in compression related to the total instability of the fibres, the maximum stress criteria is used to detect damage initiation: where and are the longitudinal strengths in tension and compression, respectively. When one of criteria given above is met, damage commences and grows according to the damage evolution law proposed by Donadon et al. [44]: where and are the contributions of the irreversible damage due to fibre breakage in tension and fibre kinking in compression, respectively: with where and are the failure strains in tension and compression, respectively. and are the maximum achieved strains in the strain time history in tension and compression , respectively. and are the final strains in tension and compression which are written as a function of the tensile fibre breakage and compression fibre kinking fracture toughnesses, respectively, as follows, where and are the intralaminar fracture toughnesses associated with fibre breakage in tension and compression, respectively and is characteristic element related to the size of the process zone. Experimental procedures and data reduction schemes to characterise the intralaminar fracture toughness for composites can be found in [48].

The proposed damage evolution law results in a thermodynamically consistent degradation procedure, regularizing damage and combining stress, damage mechanics and fracture mechanics based approaches within an unified way similarly to the approach proposed by Miami et al. [49]

2.4. Interfibre Failure (IFF)

The interfibre failure modes consist of transverse matrix cracking either in tension or compression. Based on the Worldwide Failure Exercise experimental results [50] Pinho et al. [42, 51] found that the failure envelope defined between the transverse stress and in-plane shear stress is accurately described by a quadratic interaction criteria. Thus, a failure index based on the interactive quadratic failure criterion given in [42, 51] has been used to predict tensile transverse matrix cracking. For tensile matrix cracking a failure index based on an interactive quadratic failure criterion written in terms of tensile and shear stresses is proposed in the following form:

Once the criterion above is met, the proposed expression for damage growth due to tensile matrix cracking is given by with where is the defined as resultant strain which is given by, with and are the damage onset resultant strain and stress, respectively, that is,

In order to account for damage irreversibility effects must be used in (16), where is the maximum achieved resultant strain in the strain time history. The derivation of the resultant failure strain associated with tensile/shear matrix cracking is based on a power law criterion, which accounts for interactions between energies per unit of volume of damaged material within the RVE subjected to tensile and shear loadings. The power law energy criterion is given in the following form, with for UD laminates. The resultant stress in the transverse direction (or matrix direction) due to combined tensile and shear loadings is given by

The tensile and resultant shear stress components can be written in terms of the resultant stress as follows: where is the angle defined between the resultant stress and tensile normal stress in the transverse direction, that is, . In a similar way the tensile and resultant shear strain components are given as follows:

For the damage evolution law given by (16) the specific fracture energies associated with tensile and shear stresses are respectively given by

where the area under the stress-strain curves defined by the proposed polynomial damage evolution laws is identical to the one defined by the widely used bilinear softening law given in [42, 44]. Substituting (25) into (21) we obtain the following expression for the final strain due to the combined tensile and shear stress state, where and are the critical specific fracture energies. These energies are related to the intralaminar fracture toughnesses. By using a smeared cracking formulation [52] and assuming that for UD laminates the values of intralaminar toughnesses associated with tensile matrix cracking and shear matrix cracking are comparable with mode I and mode II interlaminar fracture toughnesses, a relationship between specific critical fracture energies and intralaminar fracture toughnesses can be written as follows: where is the characteristic length associated with the length of the process zone for each particular failure mode. A detailed description about the characteristic length calculation will be presented in the following section.

The failure index to detect matrix cracking in compression failure is based on the criterion proposed by Puck and Schürmann [53, 54]. Their criterion is based on the Mohr-Coulomb theory and it enables the prediction of fracture planes for any given stress state related to Interfibre-Failure (IFF) Modes. This criterion is currently the state of the art to predict transverse compression response of composite laminates. The failure index to detect matrix cracking in compression based on the failure criterion proposed by Puck and Shürmann [53, 54] can be written as follows, here the subscripts , and refer to the normal and tangential directions in respect to the fracture plane direction. is the inplane shear strength and is the transverse shear strength in the potential fracture plane (Action Plane), which is given by [55], with where is the transverse compression strength. The fracture angle can be determined either experimentally or alternatively using (30), where maximises the failure criterion. Following the Mohr-Coulomb failure theory the friction coefficients can be determined as a function of the material friction angle as follows:

In the absence of experimental values an orthotropic relationship for the friction coefficients can be used [54],

The stress components acting on the potential fracture plane are written in terms of angle which defines the orientation of the fracture plane in respect to through-the-thickness direction (direction-3 in the local material coordinate system):

where and . is the rotation around the local fibre direction (direction-1 in the local material coordinate system). It is clear that in order to apply (28) must be known. Many authors have defined based on experimental results for standard compression tests in UD laminates [42, 44, 51]. This is true and has been also confirmed by Puck and Schurmann [53, 54]. However, is only valid for uniaxial compression loading. This implies that changes for different stress states and alternatives are needed in order to handle such a problem. By examining (28) it is possible to see that the criterion has also the potential of predicting transverse intralaminar shear cracking for values of different from . The transverse intralaminar shear cracking is a very important failure mode because it leads to delamination between adjacent layers. In order tackle this problem we have used an iterative procedure to compute the fracture plane orientation for a given stress state. The procedure consists of incrementally varying within the interval for a given stress state defined at ply level and check if the failure index for matrix cracking in compression being reached. Once the failure index is reached the local shear components and acting on the candidate fracture plane are degraded to zero according to the following damage evolution law: with where is the resultant shear strain on the action plane which is defined as with where and are the damage onset resultant strain and stress, respectively, that is,

The resultant final strain for transverse compression failure is defined in terms of mode II interlaminar fracture toughness as follows:

In order to account for damage irreversibility effects must be used in (34), where is the maximum achieved resultant strain on the action plane in the strain time history. After degrading the shear stresses acting on the potential fracture angle, the stresses are rotated back to the local material coordinate system using the following transformation:

2.5. In-Plane Shear Failure

The observed behaviour of glass and carbon fibres laminates generally shows marked rate dependence in matrix-dominated shear failure modes and for this reason a rate dependent constitutive model has been used to model the in-plane shear behaviour. The constitutive model formulation is based on previous work carried out by Donadon et al. [44, 56] and it accounts for shear nonlineatities, irreversible strains and damage within the RVE. The stress-strain behaviour for in-plane shear failure is defined as follows, with where is the initial shear modulus and , are material constants obtained from static/quasistatic in-plane shear tests. is the strain-rate enhancement given by the following law, where is another material constant obtained from dynamic in-plane shear tests. By decomposing the total shear-strain into inelastic and elastic components, the inelastic shear-strain can be written in terms of the elastic and total strain components as follows:

The failure index for in-plane shear failure is based on the maximum stress criterion and it is given by:

The proposed damage evolution law for in-plane shear failure is given by with where and are the total strain and total inelastic strain at failure (), respectively, that is, is written in terms of the intralaminar toughness in shear:

In the absence of experimental results for , is reasonable to assume for UD plies, where is the mode II interlaminar fracture toughness.

2.6. Objectivity Algorithm

The smeared formulation described in the previous sections relates the specific energy within a Representative Volume Element (RVE) with the fracture energy of the material for each particular failure mode. Since finite elements are volume based, mesh dependency problems will arise as a result of the mesh refinement. The correction of the postfailure softening slope according to the finite element size, as reported by Bazant [52] seems an attractive solution for the problem. However the approach has some limitations. Firstly, the crack growth direction must be parallel to one edge of the finite element, which is not the case for multidirectional composite laminates where layers can have arbitrary directions. Secondly, it cannot handle nonstructured meshes required in most of the complex finite element models with geometric discontinuities. In order to overcome such limitations and to ensure the objectivity of the model for generalized situations, a methodology originally developed by Oliver [57] has been used and extended to handle composite layers [58]. The dependence of the characteristic length on the fracture energy as well as its mathematical expression, were derived based in the work proposed by Oliver [57]. The method ensures a constant energy dissipation regardless of mesh refinement, crack growth direction and element topology so that, it is still applicable to nonstructured meshes.

2.6.1. Crack Modelling in the Continuous Medium

Imagine a singular line in a two dimensional domain as a continuous material line, across which displacements are continuous but displacement gradients are discontinuous. The condition for a point belonging to a singular line with unit normal at this point is that the determinant of the acoustic tensor in the direction be zero, that is [57], Nonpositive materials bifurcate, producing singular lines and the equation above permits their direction to be determined at each point. In the context of standard finite elements of continuity, a singular line can be modelled only by the sides of the elements, these being the only points in the mesh where displacement gradient discontinuities can be obtained. However, a crack produces not only displacement gradient discontinuities but also displacement discontinuities. This latter kind of discontinuity cannot be modelled by a finite element mesh for finite levels of discretization. However, a displacement discontinuity can be modeled as the limit of two parallel singular lines and which tend to coincide with each other. The band delimited by these lines is known as singular band, and h is its width.

By assuming an orthogonal curvilinear coordinate system () in the interior of the band, where coordinates lines are parallel to the singular lines and , and are the straight coordinates lines. Let and be the displacement vectors on and the relative displacement vector can be written as, as a vector representing the displacement “jump” between the two singular lines and If , the singular band is modelling a discontinuous displacement field as the limit of a continuous one. This allows a crack to be idealised as a limit (with mesh refinement) of a band of finite elements where, by means of some numerical mechanisms, the condition is satisfied.

2.6.2. Displacement and Traction Vectors in the Singular Band

Consider a singular band in the solid, with a width h according to Figure 1.

Figure 1: Analysis within the singular band.

Along a coordinate line , the displacement vector can be expanded from its value in the line using Taylor's series as and consenquently, for a point in the line From the (51), (53), (54) we can write where is a function to be determined, which approximates /h when .

From (51) and (56) it can be seen that The equilibrium across the singular band will be enforced by assuming the traction vector acting on the plane defined by the normal : which is constant in the direction that is

2.6.3. Energy Dissipation Within the Band

For a generic deformation process which takes place over a time the specific energy dissipation (energy per unit volume) within a closed domain (see Figure 1) is given by

For uniaxial deformation process would be, for a given point, the area under the stress-strain curve at that point. By taking the linearized geometric equations can be expressed as,

The last integrand in (61) is zero, being the product of a symmetric and antisymmetric tensor, so that where the Cauchy's equations for quasistatic processes and negligible body forces (/) have been considered. The total dissipated energy in the domain is

By applying the Gauss's theorem to (63) and using (58) we obtain,

Owing to the infitesimal width of the band, the curvilinear integral in (64) can be evaluated only on the lines and (see Figure 1): and taking into account (56) and (59) we obtain,

The first integral vanishes because the contributions on and cancel each other out. Thus, the dissipated energy on is

Now, if the case where is considered; that is, the whole band between points and in Figure 1, the total energy dissipated within the band between points and is

Equation (68) establishes that the energy dissipated within the idealized band can be written as a curvilinear integral along its length. The integrand of (68) represents the energy d