## Abstract

Covariance is imposed to the second law of thermodynamics and consequences are shown for elastic-perfectly plastic bodies. In this setting, pointwise balances of standard and configurational actions, constitutive restrictions on the first Piola–Kirchhoff stress and the Eshelby one, and the structure of the dissipation are all derived from a unique source.

## 1. Introduction

The word *covariance* is used in continuum mechanics with reference to invariance properties with respect to changes in observers governed by the group of differentiable automorphisms of the geometric environment, where morphology and motions of a body are described. Covariance is considered as a tool for determining possible restrictions to constitutive structures and/or deriving balance equations [1] from an invariance requirement more stringent than the one based on the invariance of the external power alone, under isometric changes in observers. For elastic simple bodies, the evaluation of the first variation of the energy around its minimizers furnishes pointwise balances of standard and configurational actions, and constitutive restrictions [2]. The procedure can be viewed as a consequence of a covariance requirement in the same way we think of when we derive Nöther theorem in classical field theories (for such a theorem, see, e.g. [1]). The view can include non-conservative body forces when we evaluate the first variation of the energy balance [3], along the guidelines of the Marsden–Hughes theorem [1], or dissipative stresses, by considering a d'Alembert–Lagrange-type principle, as suggested in Mariano [4] within the general setting of the mechanics of complex materials. When dissipation plays the role that we recognize in standard viscosity and plasticity, a covariance approach based on the first principle of thermodynamics is no more entirely satisfactory for it does not furnish indications on the possible dissipative component of the stress (the one occurring in viscoelasticity) and, above all, the expression of the dissipation. To get it, beyond balance equations and constitutive structures, we should need to impose covariance to the second law of thermodynamics. The path is not properly straightforward. We have an inequality instead of the equality occurring in the first principle. We are then forced to formulate a covariance principle for it, as I do here.

Dissipation is determined by material mutations. They can be pictured in the reference place, which can be considered not fixed once and for all. So, in writing the mechanical dissipation inequality, we need to introduce an expression of the power of all actions involved in such mutations.

I develop here these ideas for standard finite strain elastic-perfectly plastic bodies in the isothermal setting. The results are not too much different from what we know from other paths. However, the way in which they are obtained is not standard and gives further light on their nature, at least in my opinion. What is presented here can be naturally extended to hardening materials and models of strain-gradient plasticity. Also, the procedure can be in principle used to treat non-standard situations on the basis of an efficient tool requiring a small number of assumptions. The foundational work in mechanics has also this intrinsic motivation, in fact.^{1}

## 2. Multiplicative decomposition and the local metric in the intermediate configuration

### (a) Deformations and geometric preliminaries

A connected, regularly open region in the three-dimensional Euclidean point space , a region endowed with metric^{2} *g* and provided with surface-like boundary, oriented by the normal *n* everywhere but a finite number of corners and edges, is selected as a reference place for the generic body under analysis. Actual shapes are reached from by means of *deformations*:^{3} differentiable, orientation preserving maps assigning to every point *x* in its current place in a copy of , indicated by and endowed with metric , namely
The difference between and is an isomorphism , the identification, indeed. The distinction will later render significant the standard requirement that isometric changes in observers in the *whole* ambient space leave invariant the reference place . The distinction can be accepted for it is not necessary that be occupied by the body we are thinking of along any motion. It is just a geometric environment where we measure how lengths, volumes and surfaces change under deformations, and we use it to make the comparisons defining what we can call defects.

Here, *F* indicates the spatial derivative *Du*(*x*). We call it *deformation gradient* according to tradition, although it is clear that the difference from the gradient ∇*u*(*x*) is dictated by the metric, namely^{4} ∇*u*(*x*)=*Du*(*x*)*g*^{−1}. Consider a basis {**e**_{1},**e**_{2},**e**_{3}} in a neighbourhood of and another one, , at . With respect to them, and by adopting once and for all summation over repeated indices, we have , with **e**^{A} the *A*th element of the basis {**e**^{1},**e**^{2},**e**^{3}} dual to {**e**_{1},**e**_{2},**e**_{3}}. Lower case indices refer to coordinates on while capital indices label coordinates over . We have then . By definition *F* is a linear operator mapping the tangent space to at *x*, namely , to . In short, we write . Two linear operators are naturally associated with *F*: the *formal adjoint* *F**, which maps elements of the cotangent space^{5} to elements of , and the *transpose* *F*^{T}, a linear map from to . The link between *F** and *F*^{T} is established by the metrics:^{6} . The orientation preserving condition for the deformation written formally—it is well known—. It assures then the existence of other two linear operators: the *inverse* *F*^{−1} of *F*, namely and its formal adjoint . The second-rank tensor , namely the pull-back to of the metric , with components , is the version of the right *Cauchy–Green tensor* used below.

*Motions* are time-parametrized families of deformations, namely maps of the type , with *t* the time running in some interval of the real line. We assume (sufficient) differentiability with respect to time, and we write for the velocity d*u*(*x*,*t*)/d*t*, considered as a field over , taking values in , and for the same velocity viewed now as the value in of a field over the actual place .

### (b) The ‘intermediate’ configuration

Irreversible rearrangements of the structure of the matter—slips in crystals, the exit of dislocations from crystalline grains and their collective action along intergranular boundaries, disentanglements or entanglements of polymers and so on—generate collectively the residual permanent strain evidenced along loading–unloading processes, obtained by means of assigned deformations at parts of the boundary and/or other external agencies.

A manner to represent these phenomena is the introduction of appropriate descriptors of the changes in the material morphology. This way we would enter the general model building framework of the mechanics of complex materials, developing certain classes of models as special cases. Examples exist: in plasticity such descriptors can be the slip velocity in single crystals or dislocation densities or something else. In contrast, if we accept to remain in the traditional scheme presented above, a reasonable description of plastic phenomena rests first on appropriate decompositions of the strain measure, which should feature the experimental appearance that we are not able to come back to the reference shape after an elastic–plastic process, for the collective cooperation of microscopic irreversible phenomena renders it not accessible by means of elastic unloading.

The common view—the one accepted here—is indicated by the *multiplicative decomposition* [9,10]
2.1where *F*^{e} and *F*^{p} are, respectively, the *elastic* and *plastic factors* of the deformation gradient. The following assumptions apply:
2.2and
2.3

With (2.2), we admit plastic volume changes. Moreover, *F*^{e} has positive determinant, since the transplacement is orientation preserving. With (2.3), we affirm that *F*^{p} is not the spatial derivative of any transplacement field. *F*^{p} maps the tangent space to at *x* onto another linear space that we *imagine* (just this) be the tangent space to a sort of *intermediate configuration* that we are not able to identify globally. We cannot attach to each other all linear spaces identified by *F*^{p}, varying *x* in , hoping to construct the tangent bundle to a certain configuration, because, at least in principle, these spaces could not belong to the same potential intermediate configuration. In adopting the multiplicative decomposition, we are affirming to be able to alter first the structure of the matter irreversibly, and then to apply what we imagine as a reversible (elastic, in the common sense) deformation. Such a sequence is evidently just formal because, in general, irreversible structural changes and recoverable strain are mixed and cooperate with each other.^{7}

By using (2.1), the right Cauchy–Green tensor *C* becomes
which is
once we define by
the *elastic right Cauchy–Green tensor* (or better, the fully covariant version of it). At , *C*^{e} is a metric, for it is the pull-back of the spatial metric onto the linear space determined by *F*^{p}. *C*^{e} has components , where Greek indices refer to coordinates in the linear space individuated by *F*^{p}. In that space there is another metric defined by
It is the push-forward of the material metric *g* determined by *F*^{p}. In components we have

### Proposition 2.1

*is independent of any change of frame on* *induced by diffeomorphisms of the reference space onto itself.*

### Proof.

Consider diffeomorphisms and call *x*^{A′} the image of *x*^{A}—the *A*th component of , under *h*. The spatial derivative of *h*^{−1}—call it *G*—has components . Under the action of *G* we get the following transformations:
As a consequence we have the map
and the last identity concludes this elementary proof. □

The metric allows us to define the 1-contravariant, 1-covariant version of *C*^{e} by
Here, can be interpreted as a ratio between metrics and has components

Over , a differentiable vector field
is used here to indicate the *incoming irreversible mutations* of the structure of matter leading to plastic strain. *Not necessarily* *x*↦*w* is the derivative of a time parametrized family of differentiable maps transplacing onto the intermediate configuration because such a configuration is not available in its whole. Here, *w* represents at *x* a rate determined by the incoming mutations: it tends to move the material elements in a sort of collective rearrangement of the matter. In this case, we would consider a time-parametrized family of diffeomorphisms mapping onto other possible reference places; the vector field *x*↦*w* would be exactly the infinitesimal generator of a family of reference configurations. Here, such a role can be tackled just in special cases.^{8}

## 3. Observers and three classes of their changes

An *observer* is a *representation of*—that is the assignment of frames to—*all the geometric environments necessary to describe the morphology of a body and its motion*.

These environments are here the ambient space , the reference one, , and the time axis. I do not include in the list the space where we could think of embedding the intermediate configuration—in general, a copy of —because that configuration is unknown globally, at least in the picture considered here. In principle, the linear spaces determined by *F*^{p} at two different points *x* and could be embedded in two different geometric environments, always because we do not know the intermediate configuration globally. So, the notion of observer that I use and the classes of changes listed below do not refer at all to that configuration.

The classes of changes in observers that I consider below are synchronous. Including affine changes in time would not alter the results. Moreover, more intricate changes in time would lead us towards a relativistic setting which is not considered here.

**Class 1.**Two observers and differ by time parametrized families of isometries just in the representation of the ambient space . They leave invariant the reference space and the time scale. Smooth maps and*t*↦*Q*(*t*)∈*SO*(3) describe the isometries just mentioned. If and are the velocities evaluated at*x*and*t*by and , respectively, the pull-back of into the frame defining , namely is given by 3.1where*y*_{0}is an arbitrary point in space, , and*q*is the axial vector of the skew-symmetric tensor . The relation is standard.**Class 2.**and differ by time parametrized families of isometries not only in the representation of the ambient space , but also in one of the reference spaces . The part of the change dealing with is exactly as in the previous class and (3.1) holds. Analogous behaviour is in . If*w*and*w*^{′}are the values of the material velocities measured at*x*and*t*by and , respectively, the pull back of*w*^{′}into , indicated by*w**, is given by 3.2where is, as in class 1, the translational velocity and the rotational one. In principle,*c*may be different from so as*q*from .**Class 3.**and differ by time parametrized families of diffeomorphisms not only in the representation of the ambient space but also in the one of the reference space . In other words, the two observers differ by ‘smooth’ deformations of the ambient space and the reference one. Formally, we define two time-parametrized families of maps, assumed to be smooth in time (, ), namely , with ,*h*_{0}=*identity*, and , with , . The relevant infinitesimal generators of the actions of these families are given by the vector fields and . By definition,*h*_{t}depends on*y*; so, by indicating by*v*the velocity in Euclidean representation (i.e. as a field over ), the consequent changes in observers, as evaluated by , are expressed by 3.3and 3.4However, since (Lagrangian and Eulerian representations of the velocity coincide, it is well known), we can imagine that be the value of a field defined over the reference place, so that, instead of (3.3), we can write (with a slight abuse of notation) 3.5A requirement of invariance of the structure of a basic principle, such as the balance of energy or the dissipation inequality, is what I call*covariance*in what follows.^{9}

Previous classes of changes in observers form a hierarchy becoming more and more stringent from class 1 to 3. Specific choices are structural ingredients of a model.

## 4. The relative power in the presence of bulk mutations leading to plastic strain

In 1973, Noll [25] observed that a requirement of invariance under isometric changes in observers (one observer roto-translates with respect to the other) of the external power of actions on a part of a Cauchy's body^{10} leads to integral balances of forces and torques. The result has a counterpart in the conservative setting where we assign to simple bodies a Lagrangian and prove the relevant Nöther theorem (see [1]). In both cases, we find that balances of forces and torques are related to the Killing fields of the metric in space—they are the fields along which the transport of the metric leaves the metric itself invariant. In Euclidean space such fields are just rotations and translations. The difference from what has been proposed by Noll and what emerges from the Nöther theorem has a twofold nature. In Noll's procedure, constitutive issues are not involved, while Nöther's one requires the assignment of the state variables. Moreover, from the Nöther theorem the balance of configurational actions emerge naturally, while Noll's approach does not imply them. These two views are not the sole patterns leading to the emergence of balance equations. Here, I do not discuss further the issue. I want just to remind that the approach in [25] and its subsequent generalization, including material complexity in Mariano [26] are based on an expression of the external power written thinking of a reference place fixed once and for all.

When the occurrence of material mutations suggests to consider changes in the reference shape, the one represented by *w*, we can think of writing the external power *relatively* to *w*. Moreover, since the vector field *w*(⋅) represents *material* mutations, we have to consider (i) changes in the energetic landscape and (ii) actions in power conjugated with rupture and/or formation of new material bonds.

In the presence of material mutations, we can have, in principle, energy fluxes across boundaries and consequent emergence of anisotropies in the distribution of the energy itself. Taking into account, at this stage, energy does not mean that we have to introduce constitutive classes by specifying the list of state variables. We need just to affirm that a free energy exists and changes in space and time when material mutations occur. *b*^{‡} and denote below standard bulk and contact actions, the former ones including inertial effects. The other actions mentioned above are just summarized in fields of forces *f* and couples *μ* defined over . At every point, *f* and *μ* belong to the cotangent space to . They are defined by the power that they develop on *w* and its curl, respectively. Moreover, I assume that *f* may have just a dissipative nature (so it disappears along conservative processes), while *μ* has even a conservative component appearing when the rearrangements of material elements described by the field *x*↦*w* produce anisotropy without breaking and/or reforming material bonds.

Take a part of , a subset of with the same geometric regularity of itself. The *relative power* over , indicated below by , is defined by the sum of the relative power of the standard actions, , and what I call the *power of disarrangements*, , determined by the remaining mechanisms which are peculiar to the material mutations pictured in . Formally, I write
where
with ∂_{x}*ψ* the *explicit derivative* of *ψ*(*x*,*t*,*ς*) with respect to *x*, holding fixed all the other entries of the energy.^{11} The difference is a velocity *relative* to *w*. The push-forward of *w* through *F* is motivated by the membership of and *w* to two different geometric environments.

If we impose that be invariant under changes in observers in class 2 above, namely
for any choice of *c*, *q*, and in (3.1) and (3.2), which implies
4.1we get integral balances of actions in the physical space *and* in the reference one. Precisely, the arbitrariness of *c* and *q* leads to
4.2and
4.3The standard balance of forces (4.2) brings us to Cauchy theorem and allows us to write in terms of the first Piola–Kirchhoff stress *P* (after Piola transform of Cauchy stress), with the common relation , where *n* is the normal at the point *x* of the smooth ideal cut that we make in the reference shape, and it is considered as a covector: the normalized derivative of the smooth function defining the cut itself (the procedure to determine the stress tensor is standard, indeed). Then, since is a covector (as all forces are, intrinsically) attached at *y*, the stress *P* is a linear map from the cotangent space to at *x* onto the cotangent space to at *y*. We write then , that is ; so, note, *P* is dual to *F*.

By coming back to (4.1), and taking into account the occurrence of *P*, the arbitrariness of and leads to
4.4
4.5where is Eshelby stress (or Hamilton–Eshelby stress, by taking into account its occurrence in standard issues of calculus of variations), with , Kronecker delta.

— The integral balances of forces and couples (4.2) and (4.3) are associated with the Killing fields of the metric in the ambient space.

— The integral balances of configurational actions (4.2) and (4.3) are related to the Killing fields of the metric in the reference space. The result adds further reasons to consider distinguished the ambient space and the one hosting the reference place. Moreover, there is no need to introduce

*a priori*, besides*f*and*μ*, configurational stress and bulk forces and then to identify later then in terms of free energy, Piola–Kirchhoff stress, and bulk actions (remind:*F***b*^{‡}includes also inertia actions), by means of some procedure.

Being connected with Killing fields, standard and configurational *integral* balances of actions do not appear directly when we consider changes in observers, of the type defined in class 3. Rather, we meet first pointwise balances. Moreover, we pay the stringent nature of a covariance requirement with the need to specify the list of state variables before finding the structure of balance equations, as it will be clear in what follows.

## 5. Covariance and balance equations

### (a) A structural assumption on changes in observers

The velocity determined by the change in observer in the ambient space, and naturally defined as function of *y*, has been considered as the value of a field over , since *y* depends on *x*. Now I assume that the field be differentiable and write then
Moreover, I *assume* for a multiplicative decomposition of the type
5.1The presence of *F*^{p} does not reduce the generality of , owing to the arbitrariness of *H*^{e}, a linear operator from the linear space individuated by *F*^{p} to the translation space over . However, although (5.1) is innocuous with respect to the generality of the changes in observers considered here, its consequences are crucial for the results presented below.

Let us define for the sake of notational convenience and We can write 5.2

### (b) A modified mechanical dissipation inequality and its covariance

The extended version of the mechanical dissipation inequality that I use here is the following one: 5.3

The assumptions (H1) and (H2), listed below, apply.

(H1) Perfect elastic–plastic behaviour is under scrutiny, so the free energy is of the form and satisfies the following assumptions:

(H1.a) For any linear operator with ,

(H1.b) The free energy is

*objective*^{12}in the sense that it is invariant with respect to changes of frames in the physical space, induced by the action of the orthogonal group*SO*(3), i.e. by isometries.

— H1.a implies that thanks to proposition 2.1, and the multiplicative decomposition.

^{13}— Objectivity writes formally for any

*Q*∈*SO*(3). The equality implies Since , we can write

(H2) Under diffeomorphism-based changes in observers, including both the ambient space and the reference one (class 3, then), we get where (⋅)|

_{*}indicates the pull-back in the reference place of (⋅) while (⋅)|_{↓#}is the push-forward of (⋅) in the current configuration with the additional lowering of the first index.

— (⋅)|

_{↓#}is explicitly expressed by which is in components ∂*ψ*/∂*C*^{e}|_{↓#}is then a 1-contravariant, 1-covariant tensor in the current place, an element of the dual space of .— We have also which is, in components,

— The (virtual) velocity alters the material metric

*g*, dragging it, as indicated by . So (locally, and place by place), it induces changes in the metric on the intermediate configuration, because is the push-forward of*g*induced by*F*^{p}. For this reason, instead of writing , we could consider the push-forward of through*F*^{p}, multiplying it by , which would conceptually be the same thing.— Changes in observers in the physical space of class 3 alter in principle even the spatial metric . So, since

*C*^{e}is no more than the pull-back through*F*^{e}of into the tangent space to the (globally unknown, I repeat) intermediate configuration, possible changes in the spatial metric may alter*ψ*. This one is the reason justifying the introduction of the term (∂*ψ*/∂*C*^{e})|_{↓#}in (H2).— In H2, the factor has no counterpart in the current place because

*H*^{e}is not the spatial derivative of any vector field. The remark gives reason for the presence of the factor .

(H3) Contact actions depend on the same state variables entering the energy.

^{14}

Assumptions H1 and H3 are standard and do not require additional explanations, as it is, in contrast, necessary for H2. When we look at covariance in nonlinear elasticity, a basic assumption of the Marsden–Hughes theorem is that the energy is altered ‘tensorally’ under changes in observers determined by the action of the group of diffeomorphisms from the ambient space onto itself. In that theorem, the internal energy density *e* is considered to be function of time *t*, actual placement *y* and the metric in the ambient space. Under the action of parametrized families of diffeomorphisms , the tensorality requirement reads , where indicates pull-back. The physical motivation of the requirement rests on the consciousness that deformation is a relative concept: a shape of a body is deformed *with respect to* another one that we consider ‘undeformed’. So, when we write the energy, considering only the actual place, to speak about *deformation*, we can alter the space and evaluate the energy always on the ‘original’ actual place, but taking into account a varied metric, namely —that is a varied way to measure lengths. When we want to transfer such a viewpoint in the setting that we are considering here, we should have to require ‘tensorality’ of the free energy with respect to changes of atlas in both the ambient space and the reference one. First we have to remember that *C*^{e} is a function of *F*^{e} and , while is a function of the reference metric *g* and *F*^{p}. So, by considering both and , as defined in class 3, we should require
In computing the derivative with respect to *α* (we distinguish here between the parameter in and the time *t*, for *F*^{e} and *F*^{p} depend themselves on time, then, after calculating the derivative of *ψ*, we identify *α* with *t*), the term generates the Lie derivative which is the symmetric part of . An analogous reasoning should be applied when we handle the term . However, in that case we have to be addressed by physics. In fact, owing to the decomposition (5.2), the symmetric part of would include contributions of the plastic strain which does not affect either or *C*^{e}. This is the reasoning leading to the presence of the term in H2.

### (c) Covariance principle in fully dissipative setting

For the observer , the mechanical dissipation inequality is in summary something similar to *B*≤0. The observer evaluates another inequality: say *B*^{′}≤0.

Here, *B* and *B*^{′} are the left-hand side term of (5.3), calculated over the rates and and the entries of the energy measured by the two observers, respectively. Thanks to H2 and the linearity of the relative power with respect to the velocities and *w*, the pull-back of *B*^{′} into gives rise to the inequality *B**≤0, with *B**=*B*+*B*^{†}. The addendum *B*^{†} involves the rates and , which are the infinitesimal generators of the change induced by the action of over the reference space, and the one of over the (physical) ambient. Conversely, if we push forward *B* to the frames defining the observer , we find an inequality of the type *B*^{′}+*B*^{‡}≤0 because now the change in observer is governed by the inverse of the previous maps, namely *h*^{−1} and . So, *B*^{‡} *is in principle different from* *B*^{†}.

I use here the following *covariance principle in dissipative setting*: In any change in observer of class 3, after pulling-back the mechanical dissipation inequality in the first observer, the additional term arising in the new expression of the inequality evaluated by the second observer is always non-positive.

With this principle, we are essentially affirming that the dissipative nature of a process is indifferent to changes in observers.^{15}

### (d) The covariance result

### Theorem 5.1

*If we adopt for* (5.3) *the covariance principle in dissipative setting, under assumptions H*1, *H*2 *and H*3, *the expression of the contact actions in terms of stress follows and if the fields x*↦*P* *and* *with I the identity the space of second-rank 1-contravariant, 1-covariant tensors*,^{16} *are continuous and differentiable everywhere in* *except a (fixed and free of its own energy) smooth surface* *Σ*, *oriented by the normal* *m*, *where they suffer bounded jumps, and the fields* *x*↦*b*, *x*↦*F***b*, *and* *x*↦∂_{x}*ψ* *are integrable over* *the local balance equations*
5.4
5.5
5.6
*and*
5.7*with* *Ricci's symbol with all contravariant components, namely* *hold in the bulk, while*
5.8*are valid along Σ*. *Moreover, we get*
5.9*and*
5.10*with* *and the local mechanical dissipation inequality*
5.11

### Proof.

The covariance principle in dissipative setting implies
for any choice of and (see the definition of changes in observers discussed previously). Without reducing generality, take a part across the discontinuity surface *Σ* so that the intersection is a smooth portion of *Σ* itself, with boundary a piecewise smooth curve. By taking into account the assumptions discussed above, the previous difference reduces to
If we set for a while and we consider just the equality sign, the inequality reduces to the integral balance
which can be used in the standard Cauchy theorem to get the existence of the stress, and then the relation , involving the first Piola–Kirchhoff stress after Piola transform of Cauchy stress, follows. Moreover, after some algebra, the relative power can be written as follows:
By combining the two relations, H2 and the Gauss theorem imply
The arbitrariness of and implies the validity of (5.4), (5.6) and (5.8), so that we get
5.12To continue, we note that
the last identity following from (5.2), and also
So, the inequality (5.12) changes into
The arbitrariness of , , and imply, respectively, (5.9), (5.10), (5.5) and (5.7). More precisely, we have
from which
We also get
and, by multiplying by the material metric *g* from the right, we find
which is (5.10). The derivation of (5.5) is immediate. For (5.7), consider that is the axial vector of , so that for any 3*D* vector *a* we can write
and
with defined in the statement of the theorem. Finally, the arbitrariness of *H*^{e} and the one of imply the local dissipation inequality
which is (5.11) because
and, since *H*^{e} is arbitrary, we can select *H*^{e}=*F*^{e} as a special case, getting the result. □

— The balance equations above include

*inertial effects*. To recognize them we have just to presume that bulk action*b*^{‡}admit an additive decomposition into non-inertial (objective),*b*, and inertial (non-objective),*b*^{in}, components, namely with the explicit expression of the inertial part coming from the assumption, as a first principle, of an integral balance stating that the rate of the kinetic energy is balanced by the opposite of the power developed by*b*^{in}for any choice of the velocity field, so that, eventually, we get The matter is standard and I do not reproduce it here.— Relation (5.10) has been obtained using a different procedure in [27] under the assumption that the intermediate configuration be isotropic. Here, I remove that assumption and show the validity of the formula in a more general setting. It affirms that variations of the metric in the intermediate configuration—that is the metric

*after*the structural changes summarized by the plastic strain—determine a tensor which coincides with the Eshelby one once it is pulled back in the reference place. There, such a tensor develops power in the changes of the material metric along*w*, the alterations leading to plastic strain. In this sense, the result is in agreement with a view on plastic flows pictured through time changes of the material metric (cf. [8]).^{17}— In absence of plastic phenomena, both (5.9) and (5.10) have counterparts in the Doyle–Ericksen formula [30,31] and Rosenfeld–Belinfante theorem [27]. In other words, the stress

*P*is determined by energetic variations induced by variations in the metric in the ambient space (pull-back a part), whereas is determined by analogous variations in the material metric, so by mutations in the structure of the matter.— An ingredient not appearing in the previous theorem is the flow rule for

*F*^{p}. As usual, we introduce a continuous function*f*(*P*) to define the admissible set in stress space, so that at the elastic domain*El*(*P*) is In the case that*f*be differentiable, the acceptance of the maximum dissipation principle—the stress realized along an elastic–plastic process is the one maximizing the dissipation among all elements of*El*(*P*)—gives us an acceptable flow rule for*F*^{p}from the relation and the result is with*γ*≥0 and*γ**f*(*P*)=0, and characterizes what we call associate plasticity. The issue is standard, so I do not develop details (for them see, e.g. [32]). Moreover, when*f*(⋅) is not differentiable everywhere—it means that the boundary of*El*(*P*) displays corners and edges—the flow rule has to be written in terms of subdifferential of the indicator function of*El*(*P*). Even this issue is largely investigated in literature: a basic starting point is in [33].— In what I have treated here, mass is considered constant and conserved in time. When the body is an open system (a grand-canonical ensemble from a statistical mechanics viewpoint), the mechanical dissipation inequality (5.3) has to be modified by the introduction of the chemical potential associated with the flux of mass. The subsequent analysis is rather straightforward.

— The assumption H1.b can be modified, enforced in a sense. We can accept, in fact, a different assumption.

(H1.b-bis)

*ψ*is covariant in the sense that its structure is invariant with respect to changes in the ambient space induced by diffeomorphisms, i.e. elements of the space .

— Consider . By altering by means of

*h*the ambient space, we have transformations of the type and where Formally, the requirement of covariance for the free energy reads for any*K*of the type defined above which implies, obviously that which is the same consequence of the assumption H1.b (results on constitutive restrictions induced by a covariance requirement can be found in [34]^{18}.)

It is natural to presume also that the procedure proposed here can be used in the case in which the discontinuity surface *Σ* is endowed with its own energy or there are many surfaces forming junctions (an analysis of this situation in elastic–plastic solids has been proposed in [35] from another point of view). I conjecture also that previous theorem could be extended even to different models of gradient plasticity, provided a careful evaluation of the notion of observer and the appropriate choice of its changes. Such choices are structural ingredients of a model, and not something assigned once and for all. However, this is matter of another work.

## Acknowledgements

This research is part of the activities of the research group in ‘Theoretical Mechanics’ of the ‘Centro di Ricerca Matematica Ennio De Giorgi’ of the Scuola Normale Superiore in Pisa. The support of MIUR (under the grant ‘Azioni Integrate Italia-Spagna, 2009’ and PRIN-2009 ‘Kinetic and hydrodynamic equations of complex collisional systems’) and the one of GNFM-INDAM are acknowledged.

## Footnotes

↵1 In this paper, I maintain distinguished covariant components from contravariant ones, avoiding use of the identification of and its dual, as it is commonly left unexpressed. The choice is not just a matter of personal style. On one side, the intrinsic geometric nature of some objects is specified while, on the other side, the formalism opens the way to possible extension of what is discussed here to the more intricate setting of the mechanics of complex materials.

↵2 The metric

*g*is a positive definite bilinear form over the tangent space to . For a crystalline material, it is naturally determined at a point by the optical axes of the crystal associated with that point in the continuum representation [5–7]. When the material is amorphous, the assignment of*g*is not always strictly dictated by the material texture, because it can be difficult to put in evidence a unique periodic microscopic characteristic feature. In any case, a way to interpret plastic phenomena is to consider the evolution of*g*is time, as proposed in Miehe [8].↵3 Another word used by Truesdell is

*transplacement*instead of deformation.↵4 The metric

*g*is a fully covariant tensor. Since it is positive definite by definition, its inverse*g*^{−1}exists and is a fully contravariant tensor with components indicated by*g*^{AB}.↵5 It is the space of all linear forms over the tangent space.

↵6

*F*^{T}is defined through the scalar product in the same linear space, while*F** by the duality pairing between a linear space and its dual.↵7 For an extended critical review on the use of the multiplicative decomposition in proposing constitutive structures, see [11].

↵8 Mathematical models are

*representations*of the empirical world. They are addressed by the phenomenology but, in turn, the record of data is based at least on a preliminary theoretical view on the phenomenon under scrutiny. Distinction amid different views can be motivated by predictive ability, economy in the assumptions, elegance—all factors contributing to the definition of appropriateness. There is no unique view on the plastic behaviour of solids—everybody knows that. The picture involving the intermediate configuration is just one of the possible approaches. I indicate here examples of possible alternative views. In the study of Miehe [8], the occurrence of plastic effects is pictured by considering a time-dependent material metric (an approach already mentioned). Views based on the geometrical properties of the material manifold date back to Kondo (plastic effects as topology non-preserving transformations [12,13]), Bilby and collaborators (material manifolds have torsion, but show distant parallelism [14,15]), also Noll [16] and Wang [17]. As shown in the study of Kondo [12], lattice distortions in which the Riemann–Christoffell curvature tensor cannot be made to vanish are produced physically ‘by replacing the normal lattice atoms by foreign atoms which act as centres of strain. It is shown that these may be formally replaced by distributions of dislocations, but that this replacement is not physically realistic’ (quotation from Nabarro's review of Kondo [12] on*Math. Rev*., namely MR0201121 (34#1006) available from http://www.ams.org/mathscinet/). The initial geometric view on structural changes in materials has been developed variously (Edelen & Lagoudas' [18] reviews result until 1988). For ordered solids, the*topological theory of defects*has been constructed: defects can be classified in terms of homological and/or cohomological properties of the material manifolds. In the case of crystal lattices, it is natural to define elastic invariants: ‘quantities’ which do not change under the action of diffeomorphisms over the lattice [5–7]. One of them is the dislocation density tensor. It has been used variously as an indicator of the plastic behaviour. Example is [19], where the dislocation density tensor appears as an internal variable: no peculiar actions are power-conjugated with it (although we know [20] that two neighbouring dislocations exchange an action). This picture is included in a view in which the authors ‘show that in multiplicative plasticity one can combine the reference and ‘intermediate’ configurations into a parallelizable manifold’ (p. 84). ‘In summary, instead of working with a[n] Euclidean reference manifold and an ‘intermediate’ configuration, one can assume that the material manifold is equipped with a Weitzenböck connection’ (p. 85). Besides geometrical aspects, however, and taking into account that there are well constructed and useful solutions in dislocation mechanics (such as the ones presented in [21]), a point is whether the dislocation density tensor*alone*is sufficient to describe the whole physics of the plastic behaviour, at least in metals. As regards the point, in concluding a paper on ‘Benefits and shortcomings of the continuous theory of dislocations’, Kröner wrote in 2001 [22], p. 1132 that ‘The greatest shortcoming is that the dislocation density tensor*α*, no matter whether introduced through differential geometry or in the conventional way, measures the*average*dislocation density only and, therefore, regards the internal mechanical state utmost incompletely. In principle, this shortcoming could be overcome by reorientation of dislocation theory towards a statistical theory, but only with highest expenditure of computations’. Also, for crystal lattices, a view based on the dislocation density tensor and other elastic invariants leads naturally, for purely geometrical reasons, to a multiplicative decomposition of the type [23], which reduces to the standard one, obviously, when . In contrast, a view on the elastic-perfectly plastic behaviour, free from the notion of multiplicative decomposition (or better, free from the pair reference-place/intermediate-configuration*combined*in some structures, as the ones mentioned above), is possible in a finite-strain regime if we formulate a variational principle selecting among all admissible states—admissibility defined by a certain criterion imposing constraints to the stress—as introduced in [24].↵9 In calculus of variations, when we want to evaluate the first variation of an integral functional, we can alter by means of diffeomorphisms the integration domain (which is what we call

*horizontal variations*) or we can change the target of the fields involved (in nonlinear elasticity this change allows us to derive from the elastic energy a weak form of the balance of forces in terms of Cauchy stress). In a sense, the actions of diffeomorphisms just sketched can be interpreted as changes in observers in class 3. Moreover, adopting class 3 can be also considered a step towards the description of mechanical phenomena in ambient spaces not having the ‘rigidity’ of the Euclidean one—the essential example is the relativistic setting even if, there, it could be more natural to adopt a back-to-label description for we have at disposal world-lines, not just pairs of distinct elements indicating the actual place and the relevant instant.↵10 I mean a body the morphology of which is described just by its place in space and actions over and inside it are just standard body forces and common stresses.

↵11 I have already affirmed that when we have changes in the structure of the matter occupying a part of a body, we can have in principle fluxes across boundaries, in particular, and emergence of anisotropies in the distribution of the energy. These fluxes are here summarized in the term with density (

*n*⋅*w*)*ψ*. In the special case of elastic–plastic materials treated here, such a flux can be considered as a consequence of plastic changes in volume—in crystalline materials, the flux can be directly associated with flows of dislocations which can determine changes in volume of portions of a body. In general, however, since the expression of (note: the apex*dis*means disarrangements) is not peculiar of plasticity—at least, I have this view—the energy flux can be in principle attributed to migration of microstructures (even at fixed chemical potential). The possible emergence of inhomogeneity in the energetic landscape, owing to material mutations, is accounted for by the term with density ∂_{x}*ψ*, evidently.↵12 Objectivity emerges from the isotropy of the three-dimensional Euclidean point space.

↵13 In general, the invariance requirement H1.a is interpreted as a constraint leading to a structure of the free energy of the type alone. However, evidently, such an interpretation is not the sole one, and proposition 2.1 allows us to accept the one adopted in this paper.

↵14 In presence of viscous effects such an assumption does not hold, obviously.

↵15 I have introduced this principle in a yet unpublished paper (P. M. Mariano 2013,

*Observers, relative power, and covariance in continuum mechanics*, unpublished), where I have discussed it in the case of viscoelastic and complex materials.↵16 With {

*e*_{1},*e*_{2},*e*_{3}} a vector basis in a neighbourhood of , and {*e*^{1},*e*^{2},*e*^{3}} its dual counterpart, tensor*I*is of the type .↵17 Note that relations similar to (5.10) have already been derived but not in the context of a covariance theory [28,29].

↵18 Different is the case of the combined derivation of balance equations, essential aspects of constitutive structures and the expression of the dissipation produced along elastic–plastic processes, all from a unique covariance principle. I do not know any work of others adopting a covariant principle based on the second law of thermodynamics (as I do here), rather than the first one. In fact, in the study of Yavari & Goriely [19], in a description of plasticity in which the deformation gradient from a stress-free configuration and the dislocation density tensor enter alone the energy density, the combined covariant derivation of balance equations and constitutive issues is proposed (although this is not the main focus of that paper). However, in the study of Yavari & Goriely [19], the covariance principle is imagined just for the first law of thermodynamics. So, once we recognize that the dislocation density is an elastic invariant, so it is indifferent to superposed orientation-preserving diffeomorphisms (a property proved, e.g. in [5,7], on the basis of atomic lattice description of crystalline materials), the proof in [19] is nothing but the one of Marsden–Hughes theorem, and the expression of the dissipation has to be

*added*as a second ingredient beyond the theorem itself. The choice that I make here of using the second law of thermodynamics instead of the first one allows us to derive also the dissipation. Such a character appears also when we would follow the same path for viscoelasticity.

- Received February 3, 2013.
- Accepted May 10, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.