a semi-analytical solution procedure for predicting damage evolution at interfaces

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, VQL. 17, 807-819 (1993)

SHORT COMMUNICATION

A SEMI-ANALYTICAL SOLUTION PROCEDURE FOR PREDICTING DAMAGE EVOLUTION AT INTERFACES

ZHEN CHEN

New Mexico Engineering Research Institute, Unioersity of N e w Mexico, Albuquerque, N M 87131, U.S .A.

SUMMARY

A semi-analytical solution procedure is developed to predict damage evolution at the interface between two dissimilar geologic materials. The procedure consists of an analytical field solution within each finite element, and a numerical scheme for simulating structural responses. For static problems, an incremental- iterative solution strategy is constructed through the use of an initial elasticity stiffness matrix, and an evolving-localization constraint in terms of a suitable measure of damage at the most severely damaged element. For simplicity, one-dimensional problems are considered to illustrate the features of the proposed procedure, and future research is discussed based on the preliminary results obtained.

INTRODUCTION

Structure-structure, fluid-structure and fluid-fluid interactions have been considered as important factors in engineering design procedures, but the non-linear response of the interface between two dissimilar materials is still not well-understood, especially in the post-limit regime.

Recently, much research has been conducted to understand the physical nature of structural failure. With the use of both conventional and micro-experimental techniques, phenomenological and micromechanical data in the post-limit regime have been documented for some engineering materials. 1-8 It appears that materially non-linear phenomena of engineering structures arise from two distinct modes of microstructural changes: one is plastic flow and the other is the damage of material properties. In general, both plastic and damage modes are present and interacting, and a structure starts to fail when macrocracks form and propagate from the cluster of microcracks which defines a localization zone. Non-local mechanisms, in the sense of inhomogeneous interactions among material particles, are characteristic of microcracking. In other words, local constitutive models, in which the stress at a material point is related to the strain only at that point, are not representative of structural failure mechanisms, due to the lack of interactive description. Many theoretical and computational efforts have been made to remedy the deficiency of local model^.^-^' Among the continuum models proposed are non-local plasticity and damage models, rate-dependent models, Cosserat continuum models and other micromechanical models based on fracture mechanics. Preliminary results obtained for one- and two-dimensional sample problems look quite promising because, if a non-local model is used, the essential features of the post-limit response can be predicted without mesh dependency so that a non-zero energy dissipation can be obtained in the localization process.

0363-906 1/93/110807-13$11.50 0 1993 by John Wiley & Sons, Ltd.

Received 30 November 1992 Revised 22 July 1993

808 SHORT COMMUNICATION

However, non-local failure mechanisms of the interface between two dissimilar materials have not been explored thoroughly, although interface behaviours have been investigated over the last 30 years as shown by representative As a result, there exists a lack of data that exhibit the details of localized deformations adjacent to an interface. Because of the pressure dependence, quasi-brittleness and inhomogeneity, it becomes more challenging to explore the failure mechanisms of the interface between two dissimilar geologic materials than metals. To understand the essential feature of localization at the interface, torsional tests of soil-concrete interfaces were performed with photographs taken to show the change of the deformation field adjacent to the interface.I6 The post-limit configurations of the deformations near the soil-concrete interfaces are shown in Figure 1 for a circular concrete cylinder, and in Figure 2 for an elliptical concrete cylinder. It was found that the required torque rises and then falls to a residual value with the evolution of a shear band adjacent to the interface. This reduction in torque indicates that a damage effect is present. No apparent slip occurs at the interface until the

Figure I . Post-test configuration of circular soikoncrete interface16

Figure 2. Post-test configuration of elliptical soikoncrete ipterfaceI6

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evolution of the shear band stops. In the study of geogrid interaction^,^^ the evolution of a localized deformation zone was also qualitatively observed adjacent to the geogrid-soil interface. Experimental studies on the concrete/rock interface under uniaxial tensile loading indicate that characteristics of the interface failure are similar to those of material failure.36

Based on experimental data available in the literature, the following descriptions might be valid for a large class of interface problems. The interface usually initiates progressively distributed damage such as dispersed microcracks, void formation or loss of interparticle contacts in one or both of the adjacent materials. From a macroscopic point of view, large deformations can be observed adjacent to the interface. The deformation field in the softer of the two materials at contact starts to change significantly around the limit load, and is then localized into a zone which evolves as damaging progresses. It appears that no apparent slip at the interface occurs in the post-iimit regime until the evolution of the localization zone stops. When macroscopic slip occurs at the interface, the deformation field of the softer material is no longer compatible with that of the stiffer one, and the size of the localization zone is usually fixed and looks different for different materials. Thus, non-slip and damage evolution seem to be important features of non-linear interface phenomena, especially during the post-limit regime.

Within the framework of elasticity and limit design, interface phenomena have been modelled by a number of procedures. Under the assumption of compatibility between two dissimilar bodies, the following three procedures are commonly used: (a) the lumped parameter models, (b) continuum models and (c) finite element approximations. The historical development in this area can be found in representative references.I6* 2 2 ~ 2 4 * 2 6 . 3 1 - 3 3 It can be seen that Coulomb's frictional law or its variation is usually a starting point for interface modelling, for which no slip between two bodies will occur unless a critical tangential stress is reached, and the relative motion is constrained to be along the contact surface if the critical state is reached. For geologic materials such as soil, rock and concrete, however, Coulomb's law, that is a local model in nature, might not be applicable since localized deformations adjacent to the interface cannot be predicted with correct energy dissipation. In other words, the finite element or finite difference solutions using Coulomb's law are mesh-dependent so that the energy dissipation over a non-zero domain becomes trivial as the mesh is refined. With the development of theoretical and computational tools, several approaches have been proposed to advance interface modelling, such as particle modelling together with statistical analysis, special interface elements with an artificial dimension, interface modelling using the disturbed-state concept, and non-local continuum

16, 2 7 , 28 , 32 34 As a constitutive means for predicting instead of prescribing artificially the size of the localization zone, non-local models capture the essential feature of damage evolution adjacent to the interface. However, the conventional finite element or finite difference method is not robust to resolve highly localized deformation zones with correct energy dissipation because a non-local model usually involves a characteristic length factor via higher-order deformation gradients or weighted integrals of deformation fields.

As an alternative approach, a semi-analytical solution procedure is proposed here to predict the damage evolution adjacent to the interface. Based on an isotropic damage-gradient model, and on the assumption that the stress field is constant, analytical field solutions are obtained which predict the essential feature of the damage evolution. Model parameters are identified from experimental data available. For a variable stress field, a finite element mesh consisting of constant stress elements is used to discretize the whole domain of interest. An approximation of localized deformations is analytically obtained within each element. The contour enclosing the damaged elements represents the entire localization zone. For non-dynamic problems, a suitable constraint must be imposed on an incremental-iterative structural solution scheme to obtain the post-limit solutions. Based on the previous work,", 37 a robust scheme is constructed with the


use of an initial elasticity stiffness matrix, and an evolving-localization constraint in terms of a suitable measure of damage at the most severely damaged element. For simplicity, one- dimensional problems are considered to illustrate the features of the proposed procedure, and future research is discussed based on the preliminary results obtained.

AN ISOTROPIC DAMAGE-GRADIENT MODEL

To obtain an analytical field solution for the post-limit response of interface, the following assumptions are made:

1. Within the context of continuum theories, slip does not occur at the interface between two dissimilar bodies under external loading. Instead, the post-limit response is reflected through the damage evolution adjacent to the interface.

2. The imperfection at the interface reduces the strength of the softer of the two bodies at contact such that, with large enough loads, the progressively distributed damage is initiated in the region of the softer material adjacent to the interface. With this assumption, the roughness of the contact surface is simulated through the strength property of the adjacent materials rather than through a friction coefficient.

3. The non-local mechanisms of interface failure, namely, inhomogeneous interactions among material particles, are modelled by a non-local approach, that provides the means for predicting the size of the localization zone.

4. A suitable non-local model, which captures the essential feature of interface failure and makes analytical solutions feasible, is obtained if the limit stress or corresponding strain depends on the gradient of a damage parameter. The damage parameter reflects the local effect of the distributed damage, such as dispersed microcracks, void formation or loss of interparticle contacts, on the material stiffness and strength, and its gradients, hence, characterize the non-local nature of interface failure.

5. The harder of the two bodies at contact is considered to be rigid because, in engineering practice, the stiffness of the harder one is usually higher by several orders of magnitude than that of the softer one. If this disparity in stiffness is not the case, elasticity can be employed to model the harder body.

Based on the above assumptions, an isotropic damage-gradient model can be formulated through a phenomenological approach that satisfies the thermodynamic restrictions. A detailed discussion of the theories of continuum damage mechanics is not given here and, instead, several relevant papers are l i ~ t e d . ~ ~ - ~ ’ When no damage occurs, a linear isotropic elasticity relation holds for the softer of the two bodies at contact. If the limit state is reached at an initial weak point, the damage evolution is initiated. With repeated indices implying summation, and l? and $ denoting the second invariant of deviatoric strain and stress, respectively, the proposed non-local damage criterion consists of a strain-based damage surface

- - E L ( 1 - L - N ) f = E - = o

(1 - L)”

a stress-strain relation

S = 2G(1 - L)”E

and two damage variables defined by

L = W

N = aw,io,i


where EL is the value of the local strain invariant corresponding to the limit state, G the elastic shear modulus, and n and a the model parameters which are related to the post-limit response. The stress and strain tensors are material ones formulated in the unrotated configuration, and the monotonically increasing variable, o, parametrizes the damage process. The above non-local damage model becomes a local one if N , the non-local damage term involving the gradient of w, is deleted.

If the non-local term is not included, the evolution of the local damage variable, L, can easily be determined through the damage consistency condition, = 0, with the superposed dot denoting the time derivative. The damage consistency condition guarantees that the strain state remains on the damage surface during the damage process, which is a direct consequence of the damage IoadingJunloading criterion as expressed by

f < O , ci, 30, f i = o (4)

In other words, damage loading (h > 0) can occur only i f f = 0. I f f< 0, elastic unloading or no further damage occurs. When a non-local variable appears in the model, the resulting consistency condition is quite complicated because it involves not only the time derivative but an integral or a high-order differential operator on spatial variables. Hence, a numerical integration must be simultaneously performed in both time and space to find the evolution of the damage variable if an analytical approach is not available. Due to the specific forms of equations (1H3), however, the evolution of both L and N can be analytically determined under a constant stress field, as elaborated later.

If damage loading occurs, the use of equations (1) and (2) results in

s= 2GEL(1 - L - N ) = SL(I - L - N )

f = s - SL(l - L - N ) = 0

( 5 4

(5b)

or

which is a stress-based form of the damage surface with sL = 2G& being the limit strength corresponding to EL. As can be seen from equation (1) or (5b), the damage surface includes a non-local terms whether formulated in the strain space or stress space. However, the stress-strain relation, as given by equation (2), does not involve a non-local term, which simplifies the numerical part of the semi-analytical solution procedure.

It should be pointed out that the use of deviatoric field variables in formulating the non-local damage model implies the domination of mode I1 damage, i.e. shear failure mode. If mode I damage is of concern, one of the possible modelling approaches is to assess the value and direction of the maximum principal tensile stress or strain. The coupling effect between mode I and mode I1 damages is still not clear. Because of the lack of experimental data to show micromechanical mechanisms involved in interface failure, no attempt is made here to address a complete damage picture adjacent to an interface. Instead, a simple model is constructed to obtain the analytical solutions that reflect phenomenologically the damage evolution next to an interface.

ANALYTICAL SOLUTIONS

To illustrate the proposed procedure, a one-dimensional sample problem is considered first, and a generalization to other cases is discussed based on the results obtained.

Although the isotropic damage-gradient model described in the last section is applicable to large deformation analyses, the assumption of small deformations is made here for the purpose of simplicity. As a result, no distinction needs to be made among different stress and strain tensors.


Based on one-dimensional analytical approaches for a simple non-local damage model and a set of non-local plasticity models, 17, l 9 analytical solutions are developed for the damage evolution adjacent to an interface under a constant stress field. For the sample problem as shown in Figure 3, equation (5b) reduces to

(6) where S, is the normal tensile stress in the x-direction. With the harder part of the two bodies at contact being located at x < 0, the point at x = 0 denotes the interface that initiates damage. For a constant stress field, it follows that the derivative of equation (6) with respect to the x-axis vanishes, i.e.

s, = SL(l - L - N )

- sx,x = - s L(L,x + N,,) = 0 (7)

which is equivalent to the static equilibrium condition with no surface tractions or body forces acting along the x-axis. If a mesh consisting of constant stress elements, as depicted in Figure 3, is employed, equation (7) is always valid within any constant stress element whether for a static or a dynamic case. Thus, we can take the first constant stress element with element length hl as the problem domain of interest without loss of generality.

In the pre-limit regime, the elasticity relation holds and the strain distribution is constant along the x-axis up to the limit point. In the post-limit regime, the strain distribution becomes non-uniform due to the damage evolution that is governed by equation (7). With the use of definitions for the damage variables, (3), equation (7) turns out to be

LJ1 + 2aL.J = 0 ( 8 )

with two respective solutions:

L = c, X2

L = - - + c * x + c 3 4a

where C1, C 2 and C3 are integration constants. Solution (9a) gives a uniform damage distribution so that it follows from equation (6) with N = 0 that C, = 1 - Sx/SL. Because localized damage is not a constant field, equation (9a) should be disregarded. Solution (9b) predicts a damage evolution process with a length factor a. Since the damage is initiated at the interface, the boundary conditions for non-local damage are assumed to be

(10) S X S L

L(0) = 1 - and L,,(O) = 0

The integration constants in equation (9b) can then be determined and the result is

As can be seen from equation (1 l), the damage distribution is symmetrical with respect to x = 0, which is usually the case for a centre-notched specimen under tensile loading if x = 0 is considered as the centre. The selection of equation (10) for the interface damage might thus be judged based on the experimental studies on the material and interface failures.36

The model parameters involved in equation (1 1) can be identified from experimental data available. Because the damage variable L must be positive based on thermodynamic restrictions, equation (1 1 ) is valid for the domain 0 < x < x*, in which x* is the boundary of damage zone and


given by

x* = 2 J 4 1 - "1'2 SL

such that L(x*) = 0. At the beginning of the damage process, S , = sL and x* = 0, and at the end of the damage process, the stress reaches zero and x* achieves its maximum value, x*(O) = 2Ja. Since SL is a strength property and the width of a damage zone can presumably be measured experimentally, the analytical solution ( 1 1) can be compared with observed damage evolution adjacent to an interface.

The non-uniform strain distribution inside the damage zone can be found by substituting equation (1 1) into equation (2). The result is

where E x is the normal strain along the x-axis and the parameter n simulates the shape of the strain distribution. The strain field outside the zone follows the elasticity relation, i.e. Ex = SJ2G. For the one-dimensional problem, 2G = E , with E being Young's modulus. For the constant stress element, therefore, the total deflection over the element, 6, can be calculated by

Within the context of conventional displacement-based finite elements, the use of constant stress elements implies a constant strain field over an element. In this sample problem, the constant strain field is the same as the average element strain, 6/hl. Although the real strain field is not constant over the constant stress element, as shown by equation (13), there exists a one- to-one-correspondence between the real element strain and average element strain. Thus, the element stress can be found with the use of equation (14) for a given average element strain if the range of the damage zone is determined. In other words, a conventional finite element framework can still be valid here without resolving the details of the real strain field within a constant stress element. In fact, equation (14) is a strain-driven constitutive equation solver as is usual in non-linear structural analyses. As soon as the stress level is determined, the corresponding real strain distribution can be obtained analytically. Thus, a coarse mesh might be enough to resolve a highly localized deformation field adjacent to an interface, as illustrated later.

For a general case in the xyz co-ordinate system, an isotropic damage distribution of the type

can be similarly derived under a constant stress field, and the corresponding strain field can be found based on the stress-strain relation. As a result, analytical solutions for shear, uniaxial, biaxial and triaxial stress states might be recovered as special cases of equation (15), a further discussion of which is beyond the scope of the paper.

If the stress field varies with position, the above analytical approach is not applicable. With the use of constant stress elements, however, a semi-analytical structural solution procedure can be constructed as below for the isotropic damage-gradient model.


A SEMI-ANALYTICAL SOLUTION PROCEDURE

For a static problem, there exist two major difficulties in tracing the post-limit structural response, namely, the occurrence of an ill-conditioned tangent stiffness matrix around critical points and the selection of a suitable constraint on the solution path. Several approaches have been proposed to circumvent the numerical difficulties.'6* ' ' 9 3 7 3 43*44 Based on previous work,'6' '', 37 a semi-analytical structural solution procedure is designed through the use of constant stress elements, an initial elasticity stiffness matrix and an evolving-localization constraint. With this approach, the damage evolution adjacent to an interface can be predicted within a coarse mesh and without explicit singularity.

For the purpose of simplicity, the proposed procedure is outlined here within the context of proportional loading, i.e. p { 4 * } , with p being the magnitude of the external load and (q*} the reference load vector. If the load level at the end of the previous incremental step is denoted as p o { q * } , the iterative loop for a given loading increment can be written as follows with the subscript i being the iteration index:

{ P > i - l = ~ o ( q * ) - CBIT{S)i-ldV ( 164

(16b)

( 16c)

( 1 6 4

J" { S U } ~ = [ K ] - ' ( A p i { 4 * } + { p > i - l ) = Api{Sa4} + {6aP}i

{ A u } ~ = {Aa) i - l + { S U } ~

{ S } i = A ( { E } i - l > ' { A 5 } i , { L } i - l ? { N } i - l )

where the matrix [ B ] relates the strain and nodal displacement components through a differential operator and interpolation functions, and { p } is the out-of-balance force vector. To find the load increment A p indirectly through an evolving-localization constraint, the displacement increment is divided into two parts: { S d } and {Sap}. As can be seen, an LU decomposition on the initial elasticity stiffness matrix, [ K ] , needs to be performed only for the first iteration in the first incremental step. Thus, { Sa4} can be fixed during iteration. A suitable evolving-localization constraint, which is imposed on the incremental displacement vector { Aa} , can be represented by the inner product of two vectors, namely

( c ) { A a } i = A? (17)

where the localized constraint vector {c} is formulated in terms of a control element that varies in position according to the measure of damage. The sign of the constraint parameter A u is chosen to be positive for damage loading, which represents the irreversible evolution of damage. The use of equations (1 6) and (1 7) then yields

Hence, the sign change of the external load increment around a limit point is determined by the localized constraint. A detailed discussion about the formulation of the constraint condition and corresponding convergence criteria can be found in previous papers.". 37 After the load increment is found, the incremental displacement vector is updated, which yields the total strain increment { A E ) . The corresponding stress field, { S ) , is then determined with the use of the proposed non-local damage model that is symbolically represented by equation (1 6d). Specifi-


-0.50 -.-.- - 3 L -------.

cally, a forward difference scheme is adopted to calculate the local damage variable through the strain-based damage consistency condition, after which the stress level is found from equation (2). Thus, the damage distribution can be analytically solved for the given stress level. Any element, which is inside the damage zone, or the strain level of which reaches a limit state, keeps damaging until final rupture, while other elements remain in the elastic regime. The iteration loop is complete if the static equilibrium condition is satisfied.

For dynamic problems, it is not required to use a stiffness matrix and an evolving-localization constraint, but the use of constant stress elements is a necessity to obtain semi-analytical field solutions.

ILLUSTRATIONS AND DISCUSSIONS

For the one-dimensional sample problem as shown in Figure 3, model parameters are assigned with the following values which are typical of a class of geologic materials under tension: sL = 3.5 MPa and E = 35GPa. By using equations (11H13) with n = 2 for the first element, Figure 4 shows the damage distributions for different values of stress, and Figure 5 illustrates the

GlobalNodeNo.1 2 3 4

Element No. 1 2 3 ne

Elornont Longlh h, h, h,

Gkbalnode ' Coordinate '1 ' 2 X3 X4 Xn

Figure 3. Geometry and notation of a one-dimensional sample problem

3 = 0.75 SL

0.75

0.50

0.25

Figure 4. Analytical solution of the evolution of damage with decrease of stress


strain distributions corresponding to Figure 4. As can be seen, the damaged zone evolves to the right and the strain is highly localized adjacent to the interface at x = 0, with the decrease of the stress after damage occurs. Thus, the experimental observations are qualitatively replicated within a single element. In fact, the size of a localization zone is usually of the same magnitude as that of conventional finite elements used in non-linear structural analyses. However, a coarse mesh consisting of conventional elements are not able to resolve localized deformations without special treatment.

The semi-analytical solution procedure is required if the stress field varies remarkably with the position. With the use of Ja = 0.3, and 10 constant stress elements along the bar of length 1 = 1, Figure 6 illustrates the comparison between the semi-analytical solution and the corresponding analytical solution for SJS, = 0.25. The contour enclosing the step curve that represents the distribution of average element strains (6/hi) replicates the real strain distribution inside the entire localization zone. For a general problem, the three-dimensional contour enclosing the distribution of average element strains due to damage approximates the localized deformations, with the result that a coarse mesh can be used to resolve the highly localized deformations adjacent to an interface. The error introduced by the use of a coarse mesh together with analytical field solutions

5 = 0.25 SL . .-

I A.

Figure 5. Analytical solution of the strain distribution corresponding to Figure 4

Normalized Strain

- Semi-analytical

Analytical

c x

Figure 6. Comparison of the semi-analytical solution with corresponding analytical solution


1.0

within each element can be justified compared to the use of a fine mesh with approximate numerical solutions. For the same stress level in the post-limit regime, Figure 7 shows the effect of the parameter a on the size of localization zone, and Figure 8 illustrates the convergence behaviour of the semi-analytical solutions for Ja = 0-15. As can be observed, mesh-independent solutions are obtained and post-limit solution paths are easily traced by using the proposed procedure.

Since localized deformations adjacent to an interface are usually large, the small-deformation assumption might be valid only into the immediate post-limit regime. A possible extension of the porposed procedure to a case of large deformations is to combine the analytical element solutions with a material-spatial numerical scheme.45 As a result, a few material points might be enough to represent the localized damage zone with correct energy dissipation, and final rupture can be handled in a natural way. If the effect of normal stress on the interface shear resistance is of concern, a simple relation between the limit strength and the normal stress16 might be incorpor- ated into the solution procedure. Although well-defined experiments are required to understand the physics behind the interface phenomena and to verify quantitatively the model parameters, the preliminary results presented here do illustrate that the proposed semi-analytical solution procedure is a promising approach for predicting damage evolution at interfaces.

- L I _ _ _

Figure 7. Effect of the non-local model parameter, a, on the size of localization zone

1 .o

0

A

- ne = 64

b 0 0.5 1 .o 1.5

Normalized Displacement

Figure 8. Convergence study of the semi-analytical solution procedure


ACKNOWLEDGEMENTS

This research was sponsored by Sandia National Laboratories under Grant No. Sandia/12-9827 to the New Mexico Engineering Research Institute. The author is grateful to Prof. H. L. Schreyer for valuable joint discussions, and to reviewers for discerning comments on this paper. The computing assistance by Mr. S. Miao is also appreciated.

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