Damage Mechanics in Metal Forming: Advanced Modeling and Numerical Simulation

By Khemais Saanouni

The aim of this book is to summarize the current most effective methods for modeling, simulating, and optimizing metal forming processes, and to present the main features of new, innovative methods currently being developed which will no doubt be the industrial tools of tomorrow. It discusses damage (or defect) prediction in virtual metal forming, using advanced multiphysical and multiscale fully coupled constitutive equations. Theoretical formulation, numerical aspects as well as application to various sheet and bulk metal forming are presented in detail.
Virtual metal forming is nowadays inescapable when looking to optimize numerically various metal forming processes in order to design advanced mechanical components. To do this, highly predictive constitutive equations accounting for the full coupling between various physical phenomena at various scales under large deformation including the ductile damage occurrence are required. In addition, fully 3D adaptive numerical methods related to time and space discretization are required in order to solve accurately the associated initial and boundary value problems. This book focuses on these two main and complementary aspects with application to a wide range of metal forming and machining processes.

Contents

1. Elements of Continuum Mechanics and Thermodynamics.
2. Thermomechanically-Consistent Modeling of the Metals Behavior with Ductile Damage.
3. Numerical Methods for Solving Metal Forming Problems.
4. Application to Virtual Metal Forming.

• Language: English
• Publisher: Wiley
• Release date: Feb 4, 2013
• ISBN: 9781118600870

Book preview

Chapter 1

Elements of Continuum Mechanics and Thermodynamics

This first chapter gives the main basic elements of mechanics and thermodynamics of the materially simple continua. A continuum is considered materially simple if the knowledge of the first transformation gradient is sufficient to define all the kinematic and state variables necessary for the characterization of the behavior of this medium. The main objective is to provide readers with the basic elements that will allow them to follow and understand without difficulty the theoretical formulations of the constitutive equations under large inelastic deformations used in virtual metal forming.

In this chapter, readers will find the basic ideas of the kinematics and dynamics of materially simple continua (section 1.1); the conservation laws or field equations (section 1.2); the thermodynamics of materially simple continua and specifically the so-called local state method in the framework of which the constitutive equations will be formulated (section 1.3); finally, we will conclude by giving an introduction to generalized continuum mechanics (GCM) by extending all kinematic and thermodynamic ideas to the context of generalized or materially non-simple continua (section 1.4). This extension allows the formulation of nonlocal constitutive equations provided at the end of Chapter 2.

For the sake of brevity, we will not recapitulate all of the mathematical details and rigorous demonstrations of all the ideas introduced. In particular, we will neither review tensor algebra and tensor analysis nor convex analysis, ideas that are indispensable for the manipulation of all mechanical quantities. For more details on these subjects, we refer the reader to the excellent book by Truesdell and Noll, first published in 1965 [TRU 65] and then republished by the same authors in a second revised and corrected edition in 1992 [TRU 92]. A third edition appeared in 2004 [TRU 04] under the aegis of the publisher Springer-Verlag and with the support of W. Noll. Directly or indirectly inspired by this work, at the origin of modern continuum mechanics, many other books have been published in which readers will find the mathematical basics and physical justifications of all basic concepts of materially simple continuum mechanics (MSCM): [CAL 60], [ERI 62], [FUN 65], [TRU 66], [ERI 67], [JAU 67], [PRI 68], [MAL 69], [KES 70], [GLA 71], [DAY 72], [SWA 72], [GER 73], [MAN 74], [SED 75], [BOW 76], [LEI 78], [KES 79], [MCL80], [GUR 81], [HUN 83], [ZIE 83], [OGD 84], [TRU 84], [MÜL85], [GER 86], [ABR 88], [SAL 88], [BOW 89], [DUV 90], [ERI 91], [DEH 93], [LAI 93], [SMI 93], [GON 94], [RAG 95], [BOU 96], [CHU 96], [COI 97], [ROU97], [DUB 98], [CHA 99], [BAS 00], [SOU 01], [LIU 02], [GAS 03], [NEM 04], [ASA 06], and [WAT 07], among many others. In the vast majority of these books, the reader will find chapters or indices dedicated to mathematical reminders on vectors and tensor analysis as well as convex analysis. However, other specialized books may be of great help to readers who wish to improve their understanding of tensor algebra and tensor analysis [LEL 63], [SOK64], [LEG 71], [FLÜ 72], [SCH 75], [WIN 79], [ABR 88], [HLA 95], [ITS 07], or of convex analysis [MOR 66], [ROC 70], [EKE 74], [DAU 84], [SEW 87]. In this book, a simple reminder of the definition and principal properties of the Legendre-Fenchel transformation are provided in Appendix 1.

1.1. Elements of kinematics and dynamics of materially simple continua

1.1.1. Homogeneous transformation and gradient of transformation

Let us consider a deformable solid occupying at time t a volume Ωt, with boundary = u ∪ F and u ∩ F=Ø. u is the portion of the boundary where displacements are imposed and F is the additional part of the boundary where forces are imposed.

1.1.1.1. Homogeneous transformation

Let us consider the description of the motion of the subdomain ℘ apart of solid Ω. Suppose that ℘ occupies at initial time t0 the initial non-deformed configuration C0. At any instant t >t0, the subdomain ℘ occupies the current deformed configuration Ct. Using a direct orthonormal Euclidian space of base , in any homogeneous transformation moving ℘ from C0 to Ct, a point P0 of coordinates in C0 transforms into Pt of coordinates in Ct by (see Figure 1.1):

[1.1]

Figure 1.1. Initial and deformed configurations of a deformable subdomain and vectors transport

ch1-fig1.1.gif

The components Xi: X1, X2, X3 of vector are the Lagrangian or material coordinates of point P0 in the reference configuration C0. The components xi: x1,x2,x3 of vector are the Eulerian or spatial coordinates of point Pt in the current configuration Ct corresponding to point P0 of C0.

The vectorial field that allows the determination at any time t of the position of point Pt is a bijection of C0 on Ct. Thus, it allows a reciprocal function , which at any point Pt of Ct is used to define in a unique manner its correspondent P0 in C0. The two vectorial functions and are continuous and continuously differentiable (except possibly on certain surfaces of discontinuity) with respect to the overall space and time variables.

If the field is expressed at any time t in the form of an affine function between the material coordinates and the spatial coordinates of the form:

[1.2]

then the transformation between C0 and Ct is considered homogeneous.

1.1.1.2. Gradient of transformation and its inverse

The gradient of the transformation defined by [1.2] is given by:

[1.3]

This is a bipoint tensor of the second-rank (or Fij) called the gradient of the homogeneous transformation between C0 and Ct. According to Figure 1.1, the homogeneous transformation is defined by:

[1.4]

where designates the displacement vector expressed in the same basis. The gradient of this homogeneous transformation is thus given by:

[1.5]

where is a non-symmetric second-rank tensor that can be broken down into symmetric and antisymmetric parts, as we will see later on. Note that in order for [1.2] and [1.4] to define correctly the motion of a continuum, we must have:

[1.6]

Since J is not zero in any point of ℘, the second-rank operator allows an inverse gradient called defined by:

[1.7]

Finally, note that for this theory of materially simple continua, the knowledge of the gradient is amply sufficient for the complete definition of the transformation kinematic of the continuum ℘ in that it allows the complete description of changes in the shape, size, and orientation of the continuum as we will see later in this chapter.

1.1.1.3. Polar decomposition of the transformation gradient

According to the well-known polar decomposition theorem, any homogeneous transformation of a subdomain ℘ can be seen as the product of a pure rotation and of a pure strain or stretch. This means that any non-singular gradient of a homogeneous transformation defined by [1.5] can be multiplicatively decomposed, in a unique manner, in the form:

[1.8]

where the symmetric and positive definite second-rank tensors and are called left and right pure strain or stretch tensors, and is the rigid body orthogonal rotation tensor . is a Lagrangian tensor defined with respect to C0, while is purely Eulerian tensor, defined with respect to Ct (see Figure 1.2).

Figure 1.2. 2D schematic illustration of the polar decomposition of the transformation gradient

ch1-fig1.2.jpg

1.1.2. Transformation of elementary vectors, surfaces and volumes

The affine nature of the relation [1.4] implies that any linear variety in the reference configuration C0 is transformed, in its transport by this homogeneous motion, into a linear variety of the same order in the current configuration Ct. This is particularly applicable to the transformation of elementary vectors, volumes, or surfaces.

1.1.2.1. Transformation of an elementary vector

We consider the set of particles occupying in C0 the segment P0Q0 as defining the Lagrangian elementary vector (Figure 1.1). Due to the affine character of the transformation [1.4], these particles occupy at time t in Ct the segment PtQt defining the Eulerian vector . Thus, and according to [1.5], the elementary vector is obtained by the transformation of the elementary vector due to the homogeneous transformation between configurations C0 and Ct:

[1.9]

1.1.2.2. Transformation of an elementary volume: the volume dilatation

Given in the configuration C0 an elementary parallelepiped constructed with the three non-coplanar vectors (Figure 1.3). Its volume in C0 is defined by:

[1.10]

where (M) is the matrix, the columns of which are the three elementary vectors. Moreover, in the current configuration Ct, the parallelepiped formed by the vectors , which are the transformation, respectively, of the vectors hasavolume dVt defined by:

[1.11]

Due to [1.9], the following relationship between the two volumes can be easily obtained:

[1.12]

Thus, J defines the volume dilatation in the homogeneous transformation between C0 and Ct. If , then the volume is preserved and the homogeneous transformation is called isochoric or incompressible (see section 1.3.3.2). It should be noted that according to [1.8], we have .

Figure 1.3. Elementary volume transformation between C0 and Ct

ch1-fig1.3.jpg

Finally, we note that it is possible to define the gradient of an isochoric or volume preserving transformation by:

[1.13]

Thus, any homogeneous transformation can be decomposed into the product of an isochoric or volume preserving transformation of gradient and of a pure dilatation of gradient , so as to have:

[1.14]

It then results, by using the polar decomposition theorem [1.8], that:

[1.15]

where and are the left and right pure stretch tensors of a purely isochoric or volume preserving transformation.

1.1.2.3. Transformation of an oriented elementary surface

Consider, in configuration C0 (see Figure 1.4), a plane elementary surface oriented by the normal vector (surrounding, for example, the point P0) of area dA0 represented by the parallelogram formed by the two coplanar vectors . The vector area of this parallelogram is defined in C0 by . This oriented plane surface, transported by the motion into the configuration Ct, is transformed into a plane surface with the normal surrounding point Pt represented by the parallelogram formed by the two vectors (respectively, transformation of the vectors by the gradient ) of vector area . By using the transformation relationships of elementary vectors as well as [1.12], the following relationship between and is obtained:

[1.16]

Called Nanson’s relation, [1.16] will subsequently be used for the definition of various forms of the stress tensor (see section 1.1.4).

Figure 1.4. Transport of elementary surface between C0 and Ct

ch1-fig1.4.jpg

1.1.3. Various definitions of stretch, strain and strain rates

We will now give the main definitions of the strain undergone by the geometry of area ℘ in the homogeneous transformation, between the reference configuration C0 and the current configuration Ct, characterized by the gradient .

1.1.3.1. On some definitions of stretches

Let us consider two non-collinear vectors in configuration C0 named with the common origin point P0; and let be their respective transformed vectors to point Pt in the current configuration Ct. The scalar product of these two vectors is given by:

[1.17]

Thus, we define in C0 the right Cauchy-Green stretch tensor , Lagrangian, symmetric and positive definite, by:

[1.18]

It is a matter of course that and, due to [1.8] the symmetry of and the orthogonality of , we have:

[1.19]

Moreover, the scalar product of the two Lagrangian vectors leads to:

[1.20]

thereby allowing the definition in Ct of the left Cauchy–Green stretch tensor , Eulerian, symmetric, and positive definite, by:

[1.21]

with . It is easy to verify, by using [1.8] and given the properties of and , that:

[1.22]

Due to the decomposition [1.14], we easily obtain the following decomposition of the Cauchy–Green stretch tensors and :

[1.23]

[1.24]

Let us finally note that extension in a given direction, for example direction , can be defined as being the ratio of the length of the transformed vector (here ) to that of the corresponding vector in C0 (here ):

[1.25]

These lengths are easily calculated by inserting and into [1.17] and [1.20] to obtain:

[1.26]

which leads to the expression of the extension in the direction , in the following forms:

[1.27]

Taking the vectors and as equal and merging them with the unit base vectors of the selected orthonormal triad, equation [1.27] permits an easy interpretation of the different diagonal components of the right Cauchy-Green tensor .

The sliding of two initially orthogonal vectors can also be defined by calculating the angle of rotation of this pair of vectors in the current configuration Ct by:

[1.28]

Knowing that , we get:

[1.29]

Taking vectors and as equal and merging them with the unit base vectors of the selected orthonormal triad, [1.29] allows for an easy interpretation of the various non-diagonal or shear components of the right Cauchy-Green tensor .

1.1.3.2. On some definitions of the strain tensors

In order to define the strain of area ℘ under the effect of a homogeneous transformation of the gradient , it is appropriate to use symmetric second-rank tensors that have no physical dimension with zero value at the origin (i.e. when ) as well as for any rigid or non-deformable body motion. A simple way to assess the material deformation of an area ℘ in the homogeneous transformation that causes it to change from the reference configuration C0 to the current configuration Ct consists of calculating the difference between the scalar products of the elementary vectors previously calculated. By using [1.17], this calculation leads to:

[1.30]

The second-rank tensor quantity can be interpreted as a convenient measure of the strain in the reference configuration C0 since it gives a measurement of the change in length of a material line during the homogeneous transformation of gradient . Accordingly, the Lagrangian Green-Lagrange strain tensor is definedinapointP0 of C0 by:

[1.31]

Now, the use of [1.20] to recalculate the difference between the scalar products leads to:

[1.32]

The second-rank tensor quantity can be interpreted as a convenient measure of the strain in the current configuration Ct since it gives a measurement of the change in length of a material line during the homogeneous transformation of gradient . The Eulerian Euler-Almansi strain tensor is defined in a point Pt of Ct by:

[1.33]

A number of other definitions (or measures) of strain can be obtained by using pure Lagrangian and pure Eulerian stretch tensors. These definitions can be rationalized in the following Lagrangian and Eulerian forms (with m being a relative integer):

[1.34]

[1.35]

For different non-zero values of the natural integer m, we find various definitions of strain tensors suggested in the literature. Table 1.1 summarizes these different tensors, which fulfill all the properties given at the beginning of this section. Particularly, it is easy to check that all of these strain measures shrinks to zero at the origin of the motion (i.e. when ) as well as for any rigid body motion.

Table 1.1. Various definitions of strain tensors in the two configurations C0 and Ct

ch1-tab1.1.gif

To illustrate these different strain measures, let us consider the one-dimensional case of a bar occupying at the initial time t0 the reference configuration defined by initial section A0 and initial length l0. Under the effect of an applied axial load, the bar deforms (or elongates) to occupy at time t the current configuration defined by the current section At and the current length lt. We call Λ(t) = (lt/l0) the ratio of lengths measuring the rate of elongation of the bar, and we calculate the various strain measurements in the reference configuration of the bar. We obtain the expressions given by:

[1.36]

The graphic representation of five different strain measures versus the elongation ratio Λ(t) is given in Figure 1.5, thus illustrating the difference between these large strain measures. In particular, all of these strain measures are zero for Λ(t) = Λ0 = 1 and they are indistinguishable in proximity to Λ0, thus resulting in what is commonly called a small strain hypothesis (SSH), as we will see later. We also note that all of these strain measures are bounded by the Green-Lagrange strain measure for the upper bound and by the Karni strain measure for the lower bound.

Figure 1.5. Comparison between various Lagrangian strain measures

ch1-fig1.5.jpg

Let us now express, for example, the Lagrangian strain tensor in terms of the first displacement gradient (see [1.5]):

[1.37]

This allows us to express the Green–Lagrange strain tensor (see [1.31]), for example, in terms of the first displacement gradient as:

[1.38]

The small strain theory mentioned above thus consists of assuming that all of the components of the tensor are very small compared to the unity, so we can reasonably disregard the term containing the double product of the displacement gradient in [1.38], and thus, obtain the definition of the small or infinitesimal strain tensor:

[1.39]

In fact, this is a kind of linearization of the strain displacement-gradient relationship around point P0 of configuration C0. In this case, all of the strain tensors presented above (see Table 1.1) are reduced to the small strain tensor defined by [1.39].

This assumption can be easily illustrated in the case of the one-dimensional bar discussed above. In fact, we define the first component of the small strain tensor at time t according to the elongation rate Λ(t) by:

[1.40]

Using [1.36], we may easily obtain, with the help if necessary of Taylor expansion up to order 2:

[1.41]

We see clearly that if we leave out the terms in (ε11(t))², all of these measurements are reduced to the small strain component ε11(t), as we can see graphically in Figure 1.5, in which all of these measurements are merged in proximity to the origin Λ(t = 0) = Λ0=1.

Finally, we note that the tensor of infinitesimal rotations (or small rotations) is given by the antisymmetric part of the displacement gradient tensor:

[1.42]

Weconfirmthat is like any second-rank tensor.

1.1.3.3. Strain rates and rotation rates (spin) tensors

The time derivative of [1.9] leads to:

[1.43]

in which the derivative of the transformation gradient with respect to time is easily obtained from [1.5] by (the material velocity vector ):

[1.44]

Thus, the derivative with respect to time of gradient is nothing but the Lagrangian gradient of the velocity vector of the material point.

Moreover, the Eulerian velocity gradient is written considering [1.44]:

[1.45]

The second-rank tensor is thus simply the Eulerian velocity gradient, which, according to the inverse of the transformation gradient [1.9], allows us to express [1.43] versus :

[1.46]

On the other hand, the derivatives with respect to time of the scalar product defined by [1.17], considering [1.31] and [1.45], leads to:

[1.47]

Thus, is the symmetric Lagrangian strain rate and is the Eulerian strain rate defined as being the symmetric part of the Eulerian velocity gradient tensor defined by [1.45]. Thus, as with any second-rank non-symmetric tensor, can be broken down into a symmetric part measuring the strain rates and an antisymmetric part measuring the rotation rates (or spin) in the current configuration Ct:

[1.48]

Taking the relationship [1.47], considering [1.9] and [1.46] and after some algebraic transformations, we easily obtain the following relationship between the Eulerian strain rate and the Lagrangian strain rate :

[1.49]

In addition, by using [1.21], [1.33] and the derivatives with respect to time of the expression , we obtain the relationship between the Eulerian strain rate of Almansi and the Eulerian strain rate :

[1.50]

Returning now to the Eulerian velocity gradient defined by [1.45], considering the polar decomposition of by [1.8] and then the orthogonality of the rotation tensor with , we get:

[1.51]

Its symmetric and antisymmetric parts, given by [1.48], then take the form:

[1.52]

[1.53]

where is the angular velocity tensor or the rigid body proper rotation rate tensor independent from dilatations, given by:

[1.54]

We note that, according to [1.53] and [1.54], the rotation of the rigid body affects the strain rate and that pure dilatation affects the material rotation rate , which is distinct from the proper rotation rate .

1.1.3.4. Volumic dilatation rate, relative extension rate and angular sliding rate

Now we will calculate the derivative of J with respect to time by using [1.12]:

[1.55]

Additionally, the derivatives with respect to time of the elementary volume dVt are given by the mixed product of the three elementary vectors as shownbelow (see [1.11]):

[1.56]

Considering [1.12], [1.45], [1.56], and [1.48], and knowing that due to the antisymmetry of , [1.55] becomes:

[1.57]

This proves that the rate of volume variation, or the variation of volume in relation to the elementary volume in current configuration Ct, is equal to the trace of the strain rate tensor , or to the divergence of the velocity vector .

The relationship [1.47] can be used to determine the elongation of a material fiber in a particular direction , with:

[1.58]

Considering the definition of relative elongation (see [1.25]) for a unit material direction defined by ( being a unit vector), we can easily get, from its temporal derivative combined with [1.58], the following quantity:

[1.59]

called the rate of instantaneous relative elongation in the material direction carried by . If is collinear, for example to , the first vector of the orthonormal basis of the Euclidian frame, we get:

[1.60]

Thus the rate of relative elongation in the direction is simply the first component of the Eulerian strain rate tensor.

The rate of angular sliding of two material directions and in the current configuration Ct can also be calculated:

[1.61]

which leads to the following relationship defining the rate of angular sliding:

[1.62]

In the specific case, where the material directions studied are initially orthogonal, the rate of angular sliding is simply deducted from [1.28] as:

[1.63]

If, for example, we take collinear to and collinear to (unit base vectors of the orthonormal Euclidian triad), then the rate of sliding is exactly twice the value of the shearing component D12 of the Eulerian strain rate tensor:

[1.64]

Applied to the three principal directions of the strain rate tensor taken two-by-two, this relationship leads to , thus allowing the following definition: the principal directions of the strain rate tensor are the orthogonal directions for which the rate of sliding is identically null.

It remains to examine the evolution of an elementary surface by using the relationship of transformation of an oriented elementary surface [1.16], which we rewrite in the following equivalent form:

[1.65]

the derivative of which, with respect to time, considering the fact that due to the equality and to [1.57], is written as:

[1.66]

1.1.4. Various stress measures

Let us consider the area ℘, initially occupying the configuration C0 and currently the configuration C, and let us examine the elementary section oriented by the normal in C0 so that , which is transformed into elementary section oriented by the normal in Ct so that (Figure 1.6). The elementary resultant force exerted at point Pt of configuration Ct on the section oriented by the normal is written as where is the elementary tension vector in this point.

The most widely used measure of the stress in a point of a continuum is the Cauchy stress (or true stress), which is defined using the measure of the elementary internal force in a point Pt of current configuration Ct. The Euler-Cauchy principle postulates that at point Pt of configuration Ct there is a symmetric second-rank tensor called the Cauchy stress tensor linked to the elementary tension vector by:

[1.67]

Figure 1.6. Representation of internal forces and definition of stress tensors

ch1-fig1.6.jpg

The resultant elementary force exerted in Pt is thus written in configuration Ct as:

[1.68]

This purely Eulerian tensor of true stress is symmetric and depends on the Eulerian coordinates of point Pt.

Other definitions of stress tensors can be presented. Let us rewrite [1.68] according to the elementary surface transformation rule or Nanson formula given by [1.16]:

[1.69]

The second-rank operator is called the Boussinesq or Piola-Lagrange stress tensor, defined by:

[1.70]

This tensor is clearly non-symmetric and, like , it is neither purely Eulerian nor purely Lagrangian. It can serve perfectly to express the equilibrium of a solid, since it can be associated with the appropriate boundary conditions on the current deformed configuration Ct.

Considering [1.69] and [1.16], let us now perform the convective inverse transformation of the elementary resultant force vector of Ct into C0 in order to have:

[1.71]

The second-rank tensor , which is symmetric due to the symmetry of and is purely Lagrangian, is called the Piola-Kirchhoff stress tensor, defined by:

[1.72]

where J = ρ0 / ρt has already been defined (see equation [1.115]).

As vector , introduced in [1.71] via inverse convective transport of the elementary resultant force vector , does not exist in a physical sense, the Piola-Kirchhoff stress tensor, unfortunately, has no physical meaning either. Specifically, it cannot express the equilibrium of a solid, since it cannot be assigned force-related boundary conditions on the boundary of reference initial configuration C0.

Finally, we introduce the Eulerian Kirchhoff stress tensor as being the correction by of the Cauchy stress tensor:

[1.73]

Like the Cauchy stress tensor is symmetric and is purely Eulerian. These various stress tensors can be easily expressed in terms of each other. Table 1.2 summarizes these different relationships.

Note that the Boussinesq stress tensor that defines current stresses in the reference configuration is often called nominal stress tensor. In fact, if the measurement of the current force in the current configuration is very easy, this is not the case for the measurement of the current deformed area, which is not a trivial task. Hence, defining a nominal stress tensor by relating current forces to the reference area holds obvious practical interest for engineers.

Table 1.2. Relationships between various stress measures

ch1-tab1.2.jpg

As in section 1.1.3.2, for the strain measures, let us illustrate the different relationships between stress measures in the simple one-dimensional case of a bar occupying at initial time t0 the reference configuration defined by the reference section A0 and the reference length l0. Subject to the effect of an axial tension force F(t) , the bar is deformed to occupy at time t the current configuration defined by current section At and current length lt. Let us consider, as in section 1.1.3.2, the relative elongation Λ(t) = (lt / l0) and note ∑(t) = (At / A0) the ratio of the areas of the bar during its deformation. The Cauchy stress tensor in the bar at a given time t has the following form:

[1.74]

By calculating the different stress tensors given in Table 1.2, we find:

[1.75]

[1.76]

[1.77]

We note that, according to the small strain theory, Λ = ∑ ≅ 1 and all of the stress measures are reduced to a single definition equal to the ratio of the current force by the current section.

1.1.5. Conjugate strain and stress measures

We have seen several definitions of strain tensors and several definitions of stress tensors. Constitutive models (see section 1.3) are simply adequate relationships between strain tensors and stress tensors. In order to be able to say which strain tensor can be in relation with which stress tensor when constructing the constitutive equations, we must express the density of massic power of the internal forces on the various configurations:

[1.78]

This stress-strain conjugacy principle shows that the Eulerian strain rate and the EulerianCauchy stress tensor or the Kirchhoff stress tensor are conjugated pairs, while the Lagrangian strain rate tensor and the Lagrangian Piola-Kirchhoff stress tensor as well as the Boussinesq stress tensor and the rate of the transformation gradient appear as convenient conjugated pairs.

1.1.6. Change of referential or configuration and the concept of objectivity

All of the mechanical quantities introduced above are expressed in a direct orthonormal Euclidian triad called ℜ. We examine now how these quantities are affected when we proceed to a change of referential from the triad ℜ to the triad ℜ' via Euclidian transformation of type:

[1.79]

where (t) is the Euclidian vector representing the translation of the triad, is an orthogonal tensor ( , and ) representing the (rigid body) rotation of the triad, and t0 is the reference time. We also suppose, to simplify the matter, that the two triads ℜ and ℜ′ overlap at the origin of time t0 = 0, which gives and .

In this change of referential, the transformation gradient given by [1.3] changes into:

[1.80]

We see clearly that in this Euclidian referential change, the transformation gradient moves as a vector, and thus it is called objective. Remember that gradient is not strictly a tensor, but is often called a bipoint tensor.

If the two triads do not overlap at the origin of the time, then the ratio above becomes:

[1.81]

Relationship [1.81] is of very limited practical interest in solid mechanics, for which an independent time reference configuration is often sought (see [SMI 93]).

It is also possible to change the reference configuration by taking a configuration at time τ different from C0. If we call the transformation field causing and Ct to correspond, and the transformation that links C0 to C0, we arrive at:

[1.82]

with .

1.1.6.1. Impact on strain and strain rates

All Lagrangian strain measures and their rates measured with respect to the Lagrangian triad are objective. To prove this, we apply [1.79] to the right Cauchy–Green strain tensor to obtain, considering [1.80]:

[1.83]

From this, we deduce immediately that:

[1.84]

This proves that the Lagrangian tensors , and are invariant through change of Euclidian referential defined by [1.79] and then objective tensors.

Concerning the left Cauchy–Green Eulerian tensor, we have:

[1.85]

This allows us to write that:

[1.86]

Thus, the Eulerian tensors , and , transforming via [1.85] and [1.86] in any change of Euclidian referential via [1.79], are referred to as objective.

It is important to examine the impact of the change of Euclidian referential [1.79] on the Eulerian velocity gradient (see [1.45]). We have:

[1.87]

Let us now look at its symmetric and antisymmetric parts (see [1.48]) while considering the orthogonality of :

[1.88]

[1.89]

Clearly, is an objective tensor; while and are neither invariant nor objective in this change of Euclidian referential via [1.79] due to the relative rotation velocity of the two triads ℜ and ℜ′ given by .

Let us look now at how the Cauchy-Green tensors and are transformed in a change of reference configuration defined by [1.82], we have:

[1.90]

[1.91]

This result shows that, unlike , which is expressed in terms of is not expressed in terms of .

Let us note, finally, that changes in reference configuration are very often useful in the practice of metal forming by large anelastic strains.

1.1.6.2. Impact on stress and stress rates

It is self-evident that the Lagrangian Piola-Kirchhoff stress tensor is invariant by the Euclidian referential change [1.79], that is . Now let us look at the impact of such a referential change on the other stress tensors listed in Table 1.2. We see that:

[1.92]

This shows that the Cauchy stress tensor is objective. The same is true for the Kirchhoff strain tensor , which is simply proportional to via the scalar J (see Table 1.2). On the other hand, the Boussinesq stress tensor is neither invariant nor objective, since:

[1.93]

We will now look at the derivatives of the objective stress tensors. Let us take a typical second-order, symmetric, and objective Eulerian tensor in any referential change defined by [1.79] i.e. . Calculating its derivative with respect to time leads to:

[1.94]

Considering the orthogonality of , we have:

[1.95]

Clearly, not every objective second-rank tensor necessarily has an objective derivative due to the relative motion between the two triads ℜ and ℜ′ (rigid body rotation) defined by:

[1.96]

This result can cause problems of objectivity when formulating nonlinear constitutive equations under finite strains, and in particular for those that are expressed in the form of tensorial relationships between stress rates and strain rates. This is the case in (visco)plasticity, where objective relationships are sought between the Cauchy stress rate tensor and the strain rate tensor . It is thus essential to use objective derivatives of any objective tensor in order to ensure the independence of the constitutive equations with regard to observer changes, that is, changes of triad, referential or configuration.

There are several objective derivatives for stress tensors, which can be formally grouped into two families:

– Convective derivatives: The transport of tensor between the two configurations is done convectively by the gradient . Among them, we can find:

  - Contravariant Lie derivative , which consists of transporting onto the reference configuration in order to derive it, and then transporting the result back onto the current configuration:

[1.97]

  - Covariant derivative :

[1.98]

  - Truesdell or Piola transport derivative :

[1.99]

– Rotational derivatives: The transport of tensor between the two configurations occurs via the rotation (t) of rotation rate with initial condition (see [1.96]). For the tensor , this derivative is written (knowing that

[1.100]

It is easy to check that these derivatives fulfill the following two properties: and , where is the trace of the tensor .

Two important specific cases may be obtained from [1.100]. The first one consists of taking the rotation rate tensor equal to the material rotation rate defined by the antisymmetric part of the velocity gradient [1.48], i.e. . In this case, the Jaumann corotational derivative of tensor is obtained.

The second case consists in taking the rotation equal to the rigid body rotation obtained via the polar breakdown of (see [1.8]) leading to . In this case, we get the Green-Naghdi proper rotation derivative of tensor .

In conclusion, derivation in a rotated frame consists of:

– Transporting the tensor under concern between the current configuration and the rotated configuration (having the same orientation as the reference configuration) by using ;

– Differentiate within this locally rotated configuration;

– Carry out the inverse transport of this derivative using (t) in order to return to the current configuration.

Thus, the calculation of the material derivative of gives:

[1.101]

And thus the following relationship between the rotational derivative and the material derivative is:

[1.102]

This rotational objective derivative performed in a locally rotated configuration is the simplest way to generalize the constitutive equations formulated under the small strain assumption, to the large strains framework, as we will see in Chapter 2.

1.1.6.3. Impact on the constitutive equations

Let us consider some scalar (α), vectorial , and second-rank tensor state variables function of which the constitutive equations will be expressed through scalar functions , vectorial functions , or tensorial functions . From the above results, the impact of the change in referential defined by [1.79] on the objectivity of these constitutive equations may be summarized as:

[1.103]

1.1.7. Strain decomposition into reversible and irreversible parts

For solids exhibiting irreversible strains ((visco)plasticity) in addition to reversible deformations (elasticity), it is essential to know how to decompose the Eulerian and/or the Lagrangian strain measures into their reversible, or elastic parts, and their irreversible, or anelastic (plastic or viscoplastic) parts. This will be the focus of Chapter 2, devoted to the modeling of thermo-elasto-(visco)plastic behavior with damage. However, it is appropriate now to describe briefly the decomposition of these strain measures into reversible and irreversible parts (see, for example, [MAU 92] for more details).

Remember that according to the small strain assumption, wherein we merge the reference and current configurations, all strain and stress measures are merged (see sections 1.1.3 and 1.1.4). In this case, the total strain tensor and its derivative can be easily decomposed into reversible and irreversible parts according to , and thus, .

In finite strains, this decomposition is generally non-trivial and depends mainly on the definition of the intermediate configurations via the unloading of the current configuration [MAU 92, BER 05].

Figure 1.7. Multiplicative decomposition of the transformation gradient with released configuration

ch1-fig1.7.gif

The most rigorous decomposition is the one that consists of breaking down the transformation gradient into its reversible part and irreversible part . Here, we will describe only the so-called Lie decomposition, which is based on the definition of an intermediate configuration via the local release of the applied load (or the elastic release of strain), called Cr in Figure 1.7. Calling the position of point Pr in Cr, we thus define, in the manner of gradient by [1.3], a reversible gradient and an irreversible gradient via:

[1.104]

Note that unlike the transformation gradient, , and are not true transformation gradients [MAU 92]. Taking from the second equation of [1.104] and from the third, and inserting them into the first, it naturally results that:

[1.105]

It results immediately from this equation that:

[1.106]

Like [1.12], which defines the volume dilatation in the homogeneous