Crack occurrence in bodies with gradient polyconvex energies
aa r X i v : . [ m a t h . A P ] F e b Crack occurrence in bodies with gradientpolyconvex energies
Martin Kruˇz´ık*, Paolo Maria Mariano**, Domenico Mucci*** *Czech Academy of Sciences, Institute of Information Theory and AutomationPod Vod´arenskou v˘eˇz´ı 4, CZ-182 00 Prague 8, Czechiae-mail: [email protected]**DICEA, Universit`a di Firenzevia Santa Marta 3, I-50139 Firenze, Italye-mail: paolomaria.mariano@unifi.it, paolo.mariano@unifi.it***DSMFI, Universit`a di ParmaParco Area delle Scienze 53/A, I-43134 Parma, Italye-mail: [email protected]
Abstract
Energy minimality selects among possible configurations of a continuous body withand without cracks those compatible with assigned boundary conditions of Dirichlet-type. Crack paths are described in terms of curvature varifolds so that we considerboth “phase” (cracked or non-cracked) and crack orientation. The energy consideredis gradient polyconvex: it accounts for relative variations of second-neighbor surfacesand pressure-confinement effects. We prove existence of minimizers for such an en-ergy. They are pairs of deformations and varifolds. The former ones are taken to be
SBV maps satisfying an impenetrability condition. Their jump set is constrainedto be in the varifold support.
Key words:
Fracture, Varifolds, Ground States, Shells, Microstructures, Calculusof Variations
Deformation-induced material effects involving interactions beyond those offirst-neighbor-type can be accounted for by considering, among the fieldsdefining states, higher-order deformation gradients. In short, we can say thatthese effects emerge from latent microstructures, intending those which donot strictly require to be represented by independent (observable) variablesaccounting for small-spatial-scale degrees of freedom. Rather they are suchhat ‘though its effects are felt in the balance equations, all relevant quanti-ties can be expressed in terms of geometric quantities pertaining to apparentplacements’ [9, p. 49]. A classical example is the one of Korteweg’s fluid: thepresence of menisci in capillarity phenomena implies curvature influence onthe overall motion; it is (say) measured by second gradients [25] (see also[13] for pertinent generalizations). In solids length scale effects appear to benon-negligible for sufficiently small test specimens in various geometries andloading programs; in particular, when plasticity occurs in poly-crystalline ma-terials, such effects are associated with grain size and accumulation of bothrandomly stored and geometrically necessary dislocations [17], [16], [24].These higher-order effects influence possible nucleation and growth of cracksbecause the corresponding hyperstresses enter the expression of Hamilton-Eshelby’s configurational stress [34]. Here we refer to this kind of influence.We look at energy minimization and consider a variational description of cracknucleation in a body with second-gradient energy dependence. We do not referto higher order theories in abstract sense (see [13] for a general setting, [9] fora physical explanations in terms of microstructural effects, [34] for a general-ization of [13] to higher-order complex bodies), rather we consider a specificenergy, in which we account for the gradient of surface variations and confine-ment effects due to the spatial variation of volumetric strain. Specifically, theenergy we consider reads as F ( y, V ; B ) := Z B ˆ W (cid:16) ∇ y ( x ) , ∇ [cof ∇ y ( x )] , ∇ [det ∇ y ( x )] (cid:17) dx + ¯ aµ V ( B ) + Z G ( B ) a k A k p dV + a k ∂V k , (1.1)with B a fit region in the three-dimensional real space, ¯ a , a , and a positiveconstants, y : B −→ ˜ R , a special bounded variation map, a deformation thatpreserves the local orientation and is such that its jump set is contained inthe support over B of a two-dimensional varifold V with boundary ∂V andgeneralized curvature tensor A . Such a support is a 2-rectifiable subset of B with measure µ V ( B ). We identify such a set with a possible crack path, andthe terms ¯ aµ V ( B ) + Z G ( B ) a k A k p dV + a k ∂V k represent a modification of the traditional Griffith energy [23], which is just¯ aµ V ( B ) (i.e., it is proportional to the crack lateral surface area), so they havea configurational nature. The energy density ˆ W is assumed to be gradientpolyconvex, according to the definition introduced in reference [7].We presume that a minimality requirement for F ( y, V ; B ) selects amongcracked and free-of-crack configurations. To this aim we prove an existencetheorem for such minima under Dirichlet-type boundary conditions; minimiz-ing deformations satisfy also a condition allowing contact of distant body2oundary pieces but avoiding self-penetration. This is the main result of thispaper. ∇ [cof( · )] : a significant case The choice of allowing a dependence of the energy density ˆ W on ∇ [cof ∇ y ] hasphysical ground: we consider an effect due to relative variations of neighboringsurfaces. Such a situation occurs, for example, in gradient plasticity. We donot tackle directly its analysis here, but in this section we explain just itsgeometric reasons.In periodic and quasi-periodic crystals, plastic strain emerges from dislocationmotion through the lattice [36], such phenomenon includes meta-dislocationsand their approximants in quasi-periodic lattices [15], [35]. In polycristallinematerials, dislocations cluster at granular interstices obstructing or favoringthe re-organization of matter. In amorphous materials other microstructuralrearrangements determining plastic (irreversible) strain occur. Examples arecreation of voids, entanglement and disentanglement of polymers.At macroscopic scale, the one of large wavelength approximation, a traditionalway to account indirectly for the cooperative effects of irreversible microscopicmutations is to accept a multiplicative decomposition of the deformation gra-dient, commonly indicated by F , into so-called “elastic”, F e , and “plastic”, F p factors [26], [29], namely F = F e F p , which we commonly name the Kr¨oner-Lee decomposition . The plastic factor F p describes rearrangements of matterat a low scale, while F e accounts for macroscopic strain and rotation.In such a view, the plastic factor F p indicates through its time-variation justhow much (locally) the material goes far from thermodynamic equilibriumtransiting from an energetic well to another, along a path in which the matterrearranges irreversibly. In the presence of quasi-periodic atomic arrangements,as in quasicrystals, such a viewpoint requires extension to the phason fieldgradient [30], [32].Here, we restrict the view to cases in which just F and its decomposition playa significant role: they include periodic crystals, polycrystals, even amorphousmaterials like cement or polymeric bodies, when we neglect at a first glancedirect representation of the material microstructure in terms of appropriatemorphological descriptors to be involved in Laundau-type descriptions coupledwith strain. 3ith B a reference configuration for the body under scrutiny, at every itspoint x , the plastic factor F p maps the tangent space of B at x into a linearspace not otherwise specified, except assigning a metric g L to it—indicate sucha space by L F p . Then, F e transforms such a space into the tangent space ofthe deformed configuration.In general, the plastic factor F p allows us to describe an incompatible strain, soits curl does not vanish, i.e., curl F p = 0, unless we consider just a single crystalin which irrecoverable strain emerges from slips along crystalline planes. So,the condition curl F p = 0, which may hold notwithstanding curl F = 0, doesnot allow us to sew up one another the linear spaces L F p , varying x ∈ B , sowe cannot reconstruct an intermediate configuration, with the exception of asingle crystal behaving as a deck of cards, parts of which can move along slipplanes. Of course, curl F p = 0 when F e reduces to the identity.In modeling elastic-perfectly-plastic materials in large strain regime, we usu-ally assume that the free energy density ψ has a functional dependence onstate variables of the type ψ := ˜ ψ ( x, F, F p ). Further assumptions are listedbelow. • Plastic indifference , which is invariance under changes in the reference shape,leaving unaltered the material structure ( material isomorphims ); formallyit reads as ˜ ψ ( x, F, F p ) = ˜ ψ ( x, F G, F p G ) , for any orientation preserving unimodular second rank tensor G mapping atevery x the tangent space T x B of B at x onto itself (the requirement det G =1 ensures mass conservation along changes in reference configuration). • Objectivity : invariance with respect to the action of SO (3) on the physicalspace; it formally reads ˜ ψ ( x, F, F p ) = ˜ ψ ( x, QF, F p ) , for any Q ∈ SO (3).Plastic indifference implies ˜ ψ ( x, F, F p ) = ˆ ψ ( x, F e ). Then, objectivity requiresˆ ψ ( x, F e ) = ˆ ψ ( x, ˜ C e ), with ˜ C e the right Cauchy-Green tensor ˜ C e = F e T F e ,where ˜ C e = g − L C e , with C e := F e ∗ ˜ gF e the pull-back in L F p of the metric in B a (the asterisk denotes formal adjoint, which coincides with the transposewhen the metrics involved are flat). However, plastic indifference implies also˜ ψ ( x, F, F p ) = ˆ ψ ( x, F e , ¯ g ), where ¯ g := F p −∗ gF p − is at each x push-forward ofthe material metric g onto the pertinent intermediate space L F p through F p .Since by the action of G over the reference space g becomes G ∗ gG , we get¯ g = F p −∗ gF p − G −→ ( F p −∗ G −∗ ) G ∗ gG ( G − F p − ) = ¯ g .To account for second-neighborhood effects, we commonly accept the free en-ergy density to be like ˆ ψ ( x, F e , D α F e ) or ˆ ψ ( x, F e , ¯ g, D α F e ), with α indicating4hat the derivative is computed with respect to coordinates over L F p ( x ) .We claim here that this choice—i.e., the presence of D α F e in the list of statevariables—is related to the possibility of assigning energy to oriented area vari-ations of neighboring staking faults when det F p = 1.To prove the statement, first consider that the second-rank minors of F p , col-lected in cof F p , govern at each point x the variations of oriented areas fromthe reference shape to the linear intermediate space associated with the samepoint. Neighboring staking faults determine such variations in the microstruc-tural arrangements collected in what we call plastic flows. Since det F p > F p = (det F p ) F p −∗ , where −∗ indicates adjointof F p − .Consequently, assigning energy to area variations due to first-neighbor stakingfaults, we may take a structure for the free energy as ψ := ˜ ψ ( x, F, F p , q D cof F p ) , where D indicates the spatial derivative with respect to x , and the apex q indicates minor left adjoint operation of the first two indexes of a third ordertensor (it corresponds to the minor left transposition when the metric is flat orthe first two tensor components are both covariant or contravariant). At leastin the case of volume-preserving crystal slips over planes (det F p = 1), we havecof F p = F p −∗ , whence we can write in operational form D cof F p = F p −∗ ⊗ D so that q D cof F p = F p − ⊗ D . Under the action of G , describing a changein the reference shape, as above, we have F p −∗ ⊗ D G −→ (( GF p − ) ∗ ⊗ D ) G .Consequently, for volume-preserving plastic flows, the requirement of plasticinvariance reads˜ ψ ( x, F, F p , F p − ⊗ D ) = ˜ ψ ( x, F G, F p G, (( G − F p − )) ⊗ DG )for any choice of G with det G = 1. The condition implies˜ ψ ( x, F, F p , F p −∗ ⊗ D ) = ˜ ψ ( x, F F p − , ¯ g, (( F F p − ) ⊗ D ) F p − )= ˜ ψ ( x, F F p − , ¯ g, ( DF e ) F p − ) = ˆ ψ ( x, F e , ¯ g, D α F e ) , which concludes the proof.Alternatively, if we choose ψ := ˜ ψ ( x, F, F p , D cof F p ) , with the same argument as above we get˜ ψ ( x, F, F p , D cof F p ) = ˆ ψ ( x, F e , ¯ g, D α F e ∗ ) .
5n our analysis here the density ˆ W is less intricate than ˜ ψ ( x, F, F p , D cof F p ),however, the analysis of its structure indicates a fruitful path for dealing withmore complex situations.Finally, from now on we just assume flat metrics so that we write ∇ insteadof D , which appears to indicate the weak derivative of special bounded varia-tion functions, a measure indeed. Also, we refer just to F and do not considerthe plasticity setting depicted by the multiplicative decomposition. Despitethis, our choice of considering the gradient of cof F among the entries of ˜ W is intended as an indicator of relative surface variation effects. Also, as al-ready mentioned, the dependence of ˜ W on ∇ det F is a way of accounting forconfinement effects due to non-homogeneous volume variations (see [8] for apertinent analysis in small strain regime).Explanations a part are necessary for justifying the representation of cracksin terms of varifold, which are special vector-valued measures. Take a reference configuration B of a body that can be cracked, and a set ofits infinitely many copies differing one another just by a possible crack path,each a H -rectifiable set. In this reference picture, each crack path can beconsidered fictitious, i.e., the projection over B of the real crack occurring inthe deformed shape; in other words, it can be considered as a shadow over awall. Assigned boundary conditions, a question can be whether a crack mayoccur so that the deformed configuration is in one-to-one correspondence withat least one of the infinitely many reference configurations just depicted.We may imagine of giving an answer by taking an expression of the energyincluding both bulk and crack components, asking its minimality as a crite-rion of selecting among configurations with or without cracks. This is whathas been proposed in reference [18] taking Griffith’s energy [23] as the appro-priate functional. This minimality criterion is also a first step to approximatea cracking process [18]. To this aim we may select a finite partition of the timeinterval presuming to go from the state at instant k to the one at k + 1 byminimizing the energy. In principle, the subsequent step should be computingthe limit as partition interval goes to zero. This path rests on De Giorgi’snotion of minimizing movements [12].In the minimum problem, deformation and crack paths are the unknowns. Anon-trivial difficulty emerges: in three dimensions we cannot control minimiz-ing sequences of surfaces. A way of overcoming the difficulty is to consider asunknown just the deformation taken, however, in the space of those specialfunctions with bounded variations, which are orientation preserving. We give6heir formal definition in the next section. Here, we just need to know thatthey admit a jump set with non-zero H measure. Once found minima ofsuch a type, we identify the crack path with the deformation jump set [11].Although such a view is source of nontrivial analytical problems and pertinentresults [11], it does not cover cases in which portions of the crack margins arein contact but material bonds across them are broken. To account for thesephenomena, we need to recover the original proposal in reference [18], takingonce again separately deformations and crack paths. However, the problemof controlling minimizing sequences of surfaces or more irregular crack pathsreappears. A way of overcoming it is to select minimizing sequences withbounded curvature because this restriction would avoid surface blow up. Thisis the idea leading to the representation of cracks in terms of varifolds.Take x ∈ B , the question to be considered is not only whether x belongs toa potential crack path or not but also, in the affirmative case, what is thetangent (even in approximate sense) of the crack there, among all planes Πcrossing x . Each pair ( x, Π) can be viewed as a typical point of a fiber bundle G k ( B ), k = 1 ,
2, with natural projector π : G k ( B ) −→ B and typical fiber π − ( x ) = G k, the Grassmanian of 2D-planes or straight lines associated with B . A k -varifold over B is a non-negative Radon measure V over the bundle G k ( B ) [3], [1], [2], [31]. For the sake of simplicity, here we consider just G ( B ),avoiding one-dimensional crack in a 3 D -body. The generalization to include1 D cracks is straightforward. Itself, V has a projection π V over B , whichis a Radon measure over B , indicated for short by µ V . Specifically, we mayconsider varifolds supported by H -rectifiable subsets of B , i.e., by potentialcrack paths. We look at those varifolds admitting a certain notion of gener-alized curvature (its formal definition is in the next section) and parametrizethrough them the set of infinitely many reference configurations describedabove. Rather than sequences of cracks, we consider sequences of varifolds.The choice allows us to avoid the problem of controlling sequences of surfacesbut forces us to include the varifold and its curvature in the energy, leading(at least in the simplest case) to a variant of Griffith’s energy augmented by Z G ( B ) a k A k p dV + a k ∂V k with respect to the traditional term just proportional to the surface crack area,namely ¯ aµ V ( B ). Such a view point has been introduced first in references [20]and [33] (see also [19]).The discussion in this section justifies a choice of a energy functional like F ( y, V ; B ), indicated above, which we analyze in the next sections.7 Background analytical material
For G : R n → R N a linear map, where n ≥ N ≥
1, we indicate also by G = ( G ji ), j = 1 , . . . , N , i = 1 , . . . n , the ( N × n )-matrix representing G oncewe have assigned bases ( e , . . . , e n ) and ( ǫ , . . . , ǫ N ) in R n and R N , respectively.For any ordered multi-indices α in { , . . . , n } and β in { , . . . , N } with length | α | = n − k and | β | = k , we denote by G βα the ( k × k )-submatrix of G withrows β = ( β , . . . , β k ) and columns α = ( α , . . . , α k ), where α is the elementwhich complements α in { , . . . , n } , and 0 ≤ k ≤ n := min { n, N } . We alsodenote by M βα ( G ) := det G βα the determinant of G βα , and set M ( G ) := 1. Also, the Jacobian | M ( G ) | ofthe graph map x ( Id ⊲⊳ G )( x ) := ( x, G ( x )) from R n into R n × R N satisfies | M ( G ) | := X | α | + | β | = n M βα ( G ) . (3.1) Let Ω ⊂ R n be a bounded domain, with L n the pertinent Lebesgue measure.For u : Ω → R N an L n -a.e. approximately differentiable map, we denote by ∇ u ( x ) ∈ R N × n its approximate gradient at a.e. x ∈ Ω. The map u has a Lusinrepresentative on the subset e Ω of Lebesgue points pertaining to both u and ∇ u . Also, we have L n (Ω \ e Ω) = 0.In this setting, we write u ∈ A (Ω , R N ) if • ∇ u ∈ L (Ω , M × ) and • M βα ( ∇ u ) ∈ L (Ω) for any ordered multi-indices α and β with | α | + | β | = n .The graph G u of a map u ∈ A (Ω , R N ) is defined by G u := (cid:26) ( x, y ) ∈ Ω × R N | x ∈ e Ω , y = e u ( x ) (cid:27) , where e u ( x ) is the Lebesgue value of u . It turns out that G u is a countably n -rectifiable set of Ω × R N , with H n ( G u ) < ∞ . The approximate tangent n -plane at ( x, e u ( x )) is generated by the vectors t i ( x ) = ( e i , ∂ i u ( x )) ∈ R n + N ,for i = 1 , . . . , n , where the partial derivatives are the column vectors of thegradient matrix ∇ u , and we take ∇ u ( x ) as the Lebesgue value of ∇ u at x ∈ e Ω.8he unit n -vector ξ ( x ) := t ( x ) ∧ t ( x ) ∧ · · · ∧ t n ( x ) | t ( x ) ∧ t ( x ) ∧ · · · ∧ t n ( x ) | provides an orientation to the graph G u .For D k (Ω × R N ) the vector space of compactly supported smooth k -forms in Ω × R N , and H k the k -dimensional Hausdorff measure, one defines the current G u carried by the graph of u through the integration of n -form on G u , namely h G u , ω i := Z G u h ω, ξ i d H n , ω ∈ D n (Ω × R N ) , where h , i indicates the duality pairing. Consequently, by definition G u is anelement of the (strong) dual of the space D n (Ω × R N ). Write D n (Ω × R N ) forsuch a dual space. Any element of it is called a current .By writing U for a open set in R n + N , we define mass of T ∈ D k ( U ) the number M ( T ) := sup {h T, ω i | ω ∈ D k ( U ) , k ω k ≤ } and call a boundary of T the ( k − ∂T defined by h ∂T, η i := h T, dη i , η ∈ D k − ( U ) , where dη is the differential of η .A weak convergence T h ⇀ T in the sense of currents in D k ( U ) is definedthrough the formulalim h →∞ h T h , ω i = h T, ω i ∀ ω ∈ D k ( U ) . If T h ⇀ T , by lower semicontinuity we also have M ( T ) ≤ lim inf h →∞ M ( T h ) . With these notions in mind, we say that G u is an integer multiplicity (in shorti.m.) rectifiable current in R n (Ω × R N ), with finite mass M ( G u ) equal to thearea H n ( G u ) of the u -graph. According to (3.1), since the Jacobian | M ( ∇ u ) | of the graph map x ( Id ⊲⊳ u )( x ) = ( x, u ( x )) is equal to | t ( x ) ∧ t ( x ) ∧ · · · ∧ t n ( x ) | , by the area formula h G u , ω i = Z Ω ( Id ⊲⊳ u ) ω = Z Ω h ω ( x, u ( x )) , M ( ∇ u ( x )) i dx for any ω ∈ D n (Ω × R N ), so that M ( G u ) = H n ( G u ) = Z Ω | M ( ∇ u ) | dx < ∞ . u is of class C , the Stokes theorem implies h ∂G u , η i = h G u , dη i = Z G u dη = Z ∂ G u η = 0for every η ∈ D n − (Ω × R N ), i.e., the null-boundary condition( ∂G u ) Ω × R N = 0 . (3.2)Such a property (3.2) holds true also for Sobolev maps u ∈ W ,n (Ω , R N ), byapproximation. However, in general, the boundary ∂G u does not vanish andmay not have finite mass in Ω × R N . On the other hand, if ∂G u has finitemass, the boundary rectifiability theorem states that ∂G u is an i.m. rectifiablecurrent in R n − (Ω × R N ). An extended treatment of currents is in the two-volume treatise [22]. Let { u h } be a sequence in A (Ω , R N ).Take N = 1, i.e., consider real-valued maps u . Suppose also to have in handssequences { u h } and {∇ u h } such that u h → u strongly in L (Ω) and ∇ u h ⇀ v weakly in L (Ω , R n ), where u ∈ L (Ω) is an a.e. approximately differentiablemap and v ∈ L (Ω , R n ). In general, we cannot conclude that v = ∇ u a.e.in Ω. The question has a positive answer provided that { u h } is a sequencein W , (Ω). Notice that, when N = 1, the membership of a function u ∈ A (Ω , R ) to the Sobolev space W , (Ω) is equivalent to the null-boundarycondition (3.2).When N ≥
2, assume that u h → u strongly in L (Ω , R N ), with u some a.e.approximately differentiable L (Ω , R N ) map. Presume also that M βα ( ∇ u h ) ⇀v βα weakly in L (Ω), with v βα ∈ L (Ω), for every multi-indices α and β , with | α | + | β | = n . A sufficient condition ensuring that v βα = M βα ( ∇ u ) a.e. is againthe validity of equation (3.2) for each u h .We can weaken such a condition by requiring a mass control on G u h boundariesof the type sup h M (( ∂G u h ) Ω × R N ) < ∞ , (3.3)as stated by Federer-Fleming’s closure theorem [14], which refers to sequencesof graphs G u h which have equi-bounded masses, sup h M ( G u h ) < ∞ and satisfythe condition (3.3) [22, Vol. I, Sec. 3.3.2]. Theorem 3.1 (Closure theorem).
Let { u h } be a sequence in A (Ω , R N ) such that u h → u strongly in L (Ω , R N ) to an a.e. approximately differentiable ap u ∈ L (Ω , R N ) . For any multi-indices α and β with | α | + | β | = n , assume M βα ( ∇ u h ) ⇀ v βα weakly in L (Ω) , with v βα ∈ L (Ω) . If the bound (3.3) holds, the inclusion u ∈ A (Ω , R N ) holdsand, for every α and β , v βα ( x ) = M βα ( ∇ u ( x )) L n -a.e in Ω . (3.4) Moreover, we find G u h ⇀ G u weakly in D n (Ω × R N ) , and also M ( G u ) ≤ lim inf h →∞ M ( G u h ) < ∞ M (( ∂G u ) Ω × R N ) ≤ lim inf h →∞ M (( ∂G u h ) Ω × R N ) < ∞ . A summable function u ∈ L (Ω) is said to be of bounded variation if thedistributional derivative Du is a finite measure in Ω. Such a function u isapproximately differentiable L n -a.e. in Ω. Its approximate gradient ∇ u agreeswith the Radon-Nikodym derivative density of Du with respect to L n . Then,the decomposition Du = ∇ u L n + D s u holds true, where the component D s u issingular with respect to L n . Also, the jump set S ( u ) of u is a countably ( n − H n − -essentially with the complement of u Lebesgue’s set. If, in addition, the singular component D s u is concentratedon the jump set S ( u ), we say that u is a special function of bounded variation ,and write in short u ∈ SBV (Ω).A vector valued function u : Ω → R N belongs to the class SBV (Ω , R N ) if allits components u j are in SBV (Ω). In this case, Du = ∇ u L n + D s u , wherethe approximate gradient ∇ u belongs to L (Ω , R N × n ), and the jump set S ( u )is defined component-wise as in the scalar case, so that D s u = ( u + − u − ) ⊗ ν H n − S ( u ), where ν is an unit normal to S ( u ) and u ± are the one-sidedlimits at x ∈ S ( u ). Therefore, for each Borel set B ⊂ Ω we get | Du | ( B ) = Z B |∇ u | dx + Z B ∩ S ( u ) | u + − u − | d H n − . Compactness and lower semicontinuity results hold in
SBV . The treatise [6]offers an accurate analysis of
SBV landscape. Here, we just recall that thecompactness theorem in [4] relies on a generalization of the following charac-terization of
SBV functions with H n − -rectifiable jump sets.According to reference [5], we denote by T (Ω × R ) the class of C -functions11 ( x, y ) such that | ϕ | + | Dϕ | is bounded and the support of ϕ is contained in K × R for some compact set K ⊂ Ω. Proposition 3.1
Take u ∈ BV (Ω) . Then, u ∈ SBV (Ω) , with H n − ( S ( u )) < ∞ , if and only if for every i = 1 , . . . , n there exists a Radon measure µ i in Ω × R such that Z Ω (cid:18) ∂ϕ∂x i ( x, u ( x )) + ∂ϕ∂y ( x, u ( x )) ∂ i u ( x ) (cid:19) dx = Z Ω × R ϕ dµ i for any ϕ ∈ T (Ω × R ) . In this case, we have µ i = − ( Id ⊲⊳ u + ) ( ν i H n − S ( u )) + ( Id ⊲⊳ u − ) ( ν i H n − S ( u )) . As a consequence, we infer that if a sequence { u h } ∈ A (Ω , R N ) satisfiessup h (cid:18) k u h k ∞ + Z Ω | M ( ∇ u h ) | p dx (cid:19) < ∞ , p > { u h } ∈ SBV (Ω , R N ) andthe SBV compactness theorem hold. In fact, by Proposition 3.1 we get H n − S ( u h ) ≤ π | ∂G u h | ( B ) ∀ h where π : Ω × R N → Ω is the projection onto the first n coordinates, and | · | the total variation, so that π | ∂G u | ( B ) = | ∂G u | ( B × R N ) for each Borel set B ⊂ Ω. When the bound sup h k u h k ∞ < ∞ fails, the SBV compactness theorem cannotbe applied. This happens, e.g., if u h = ∇ y h for some sequence { y h } ⊂ W ,p (Ω).When such sequences play a role in the problems analyzed, we find it conve-nient to call upon generalized special functions of bounded variation , the classof which is commonly denoted by GSBV .To define them, first write
SBV loc (Ω) for functions v : Ω → R such that v | K ∈ SBV ( K ) for every compact set K ⊂ Ω. Definition 3.1
A function u : Ω → R N belongs to the class GSBV (Ω , R N ) if φ ◦ u ∈ SBV loc (Ω) for every φ ∈ C ( R N ) with the support of ∇ φ compact. The following compactness theorem holds.12 heorem 3.2
Let { u h } ⊂ GSBV (Ω , R N ) be such that sup h (cid:18)Z Ω (cid:16) | u h | p + |∇ u h | p (cid:17) dx + H n − ( S u h ) (cid:19) < ∞ for some real exponent p > . Then, there exists a function u ∈ GSBV (Ω , R N ) and a (not relabeled) subsequence of { u h } such that u h → u in L p (Ω , R N ) , ∇ u h ⇀ ∇ u weakly in L p (Ω , R N × n ) , and H n − S ( u h ) weakly converges in Ω to a measure µ greater than H n − S ( u ) .3.6 Curvature varifolds with boundary We now turn to the physical dimension n = 3 and denote by B a connectedbounded domain in R with surface-like boundary that can be oriented bythe outward unit normal to within a finite number of corners and edges. Inthis setting, we take the deformation as a map y : B −→ ˜ R , where ˜ R isa isomorphic copy of R , the isomorphism given by the identification. Sucha distinction is necessary for example when we consider changes in observers(which are frames on the entire ambient space) leaving invariant the referenceconfiguration, which is B in this case. Definition 3.2
A general 2-varifold in B is a non-negative Radon measureon the trivial bundle G ( B ) := B × G , , where G , is the Grassmanian man-ifold of -planes Π through the origin in R . If C is a 2-rectifiable subset of B , for H C a.e. x ∈ B there exists theapproximate tangent 2-space T x C to C at x . We thus denote by Π( x ) the 3 × R onto T x C and define V C ,θ ( ϕ ) := Z G ( B ) ϕ ( x, Π) dV C ,θ ( x, Π) := Z C θ ( x ) ϕ ( x, Π( x )) d H ( x ) (3.5)for any ϕ ∈ C c ( G ( B )), where θ ∈ L ( C , H ) is a nonnegative density func-tion. If θ is integer valued, then V = V C ,θ is said to be the integer rectifiablevarifold associated with ( C , θ, H ).The weight measure of V is the Radon measure in B given by µ V := π V ,where π : G ( B ) → B is the canonical projection. Then, we have µ V = θ H C and call k V k := V ( G ( B )) = µ V ( B ) = Z C θ d H a mass of V . Definition 3.3
An integer rectifiable -varifold V = V C ,θ is called a curvature2-varifold with boundary if there exist a function A ∈ L ( G ( B ) , R ∗ ⊗ R ⊗ ∗ ) , A = ( A ℓij ) , and a R -valued measure ∂V in G ( B ) with finite mass k ∂V k ,such that Z G ( B ) (Π D x ϕ + AD Π ϕ + ϕ t tr ( AI )) dV ( x, Π) = − Z G ( B ) ϕ d∂V ( x, Π) for every ϕ ∈ C ∞ c ( G ( B )) . Moreover, for some real exponent p > , the sub-class of curvature -varifolds with boundary such that | A | ∈ L p ( G ( B )) isindicated by CV p ( B ) . Varifolds in CV p ( B ) have generalized curvature in L p [31]. Therefore, Allard’scompactness theorem applies (see [1], [2], but also [3]): Theorem 3.3
For < p < ∞ , let { V ( h ) } ⊂ CV p ( B ) be a sequence of cur-vature -varifolds V ( h ) = V C h ,θ h with boundary. The corresponding curvaturesand boundaries are indicated by A ( h ) and ∂V ( h ) , respectively. Assume that thereexists a real constant c > such that for every hµ V ( h ) ( B ) + k ∂V ( h ) k + Z G ( B ) | A ( h ) | p dV ( h ) ≤ c. Then, there exists a (not relabeled) subsequence of { V ( h ) } and a -varifold V = V C ,θ ∈ CV p ( B ) , with curvature A and boundary ∂V , such that V ( h ) ⇀ V, A ( h ) dV ( h ) ⇀ A dV, ∂V ( h ) ⇀ ∂V, in the sense of measures. Moreover, for any convex and lower semicontinuousfunction f : R ∗ ⊗ R ⊗ R ∗ → [0 , + ∞ ] , we get Z G ( B ) f ( A ) dV ≤ lim inf h →∞ Z G ( B ) f ( A ( h ) ) dV ( h ) . According to references [7,27,28], we take a continuous functionˆ W : R × × R × × × R → ( −∞ , + ∞ ] , and we set ˆ W = ˆ W ( G, ∆ , ∆ ). We assume also existence of four real expo-nents p, q, r, s satisfying the inequalities p > , q ≥ pp − , r > , s > c such that for every ( G, ∆ , ∆ ) ∈ R × × R × × × R the following estimates holds:ˆ W ( G, ∆ , ∆ ) ≥ c (cid:16) | G | p + | cof G | q + (det G ) r + (det G ) − s + | ∆ | q + | ∆ | r (cid:17)
14f det
G >
0, and ˆ W ( G, ∆ , ∆ ) = + ∞ if det G ≤ Definition 3.4
With B ⊂ R the domain already described, consider thefunctional J ( F ; B ) := Z B ˆ W (cid:16) F ( x ) , ∇ [cof F ( x )] , ∇ [det F ( x )] (cid:17) dx defined on the class of integrable functions F : B → R × for which the ap-proximate derivatives ∇ [cof F ( x )] , ∇ [det F ( x )] exist for L -a.e. x ∈ B andare both integrable functions in B . Then, J ( F ; B ) is called gradient polycon-vex if the integrand ˆ W ( G, · , · ) is convex in R × × × R for every G ∈ R × . To assign the Dirichlet condition, we assume that Γ ∪ Γ is an H -measurablepartition of the B boundary such that H (Γ ) >
0. For some given measur-able function y : Γ → R , we consider the classˆ A p,q,r,s := { y ∈ W ,p ( B , R ) | cof ∇ y ∈ W ,q ( B , R × ) , det ∇ y ∈ W ,r ( B ) , det ∇ y > B , (det ∇ y ) − ∈ L s ( B ) , y = y on Γ } , where p, q, r, s satisfy the inequalities (3.6).The following existence result has been proven in reference [7] (see also [27]). Theorem 3.4
Under the previous assumptions, if the class ˆ A p,q,r,s is non-empty and inf { J ( ∇ y ; B ) | y ∈ ˆ A p,q,r,s } < ∞ , the functional y J ( ∇ y ; B ) attains a minimum in A p,q,r,s . We now look at an energy modified by the introduction of a varifold, throughwhich we parametrize possible fractured configurations with respect to thereference one. Specifically, we consider a curvature varifold with boundary: V ∈ CV p ( B ). The choice implies a fracture energy modified with respect tothe Griffith one. In fact, the latter is just proportional to the crack area, whichimplies considering material bonds of spring-like type. The additional presencein our case of the generalized curvature tensor implies, instead, consideringbeam-like material bonds for which bending effects play a role. In a certainsense, the energy we propose is a regularization of the Griffith one, since werequire that the coefficient in front of the curvature tensor square modulusdoes not vanish.In this setting, we look for minimizing deformations that are bounded and mayadmit a jump set contained in the varifold support. We cannot assume the de-15ormation y to be a Sobolev map. More generally we require y ∈ SBV ( B , R ).The main issue in proving existence is recovering the weak convergence of mi-nors. To achieve it we look at the approximate gradient and exploit Federer-Fleming’s closure theorem as in Theorem 3.1. On the other hand, since someproperties as the bound k cof ∇ y k ∞ < ∞ fails to hold, we assume cof ∇ y to bein the class GSBV , with jump set controlled by the varifold support. In thisway we recover the weak continuity of the approximate gradients ∇ [cof ∇ y h ]along minimizing sequences.Our existence result below could be generalized to the case in which the crackpath is described by a stratified family of varifolds in the sense introduced inreferences [20] and [33] (see also [19]). This choice had been made we could haveassigned additional curvature-type energy to the crack tip, taking possiblyinto account energy concentrations at tip corners, when the tip is not smooth.Also, we could describe the formation of defects with codimension 2 in frontof the crack tip, specifically dislocations nucleating in front of the tip (see[19]). However, for the sake of simplicity, we restrict ourselves to the choice ofa single varifold, avoiding to foresee an additional tip energy and also cornerenergies.Consequently, we consider the energy functional F ( y, V ; B ) := J ( ∇ y ; B ) + E ( V ; B ) , where F J ( F ; B ) is the functional in Definition 3.4, and E ( V ; B ) := ¯ aµ V ( B ) + Z G ( B ) a k A k p dV + a k ∂V k , with ¯ a , a , and a positive constants.The couples deformation-varifold are in the class A p,p,q,r,s,K,C defined below. Definition 4.1
Let p > and p, q, r, s real exponents satisfying (3.6) , let K, C be two positive constants, and let y : Γ → R be a given measurable function,where Γ ∪ Γ is an H -measurable partition of the boundary of B . We saythat a couple ( y, V ) belongs to the class A p,p,q,r,s,K,C if the following propertieshold:(1) V = V C ,θ is a curvature -varifold with boundary in CV p ( B ) ;(2) y ∈ A ( B , R ) , with k y k ∞ ≤ K and y = y on Γ ;(3) π | ∂G y | ≤ C · µ V ;(4) the approximate gradient ∇ y ∈ L p ( B , R × ) , cof ∇ y ∈ L q ( B , R × ) , and det ∇ y ∈ L r ( B ) ;(5) det ∇ y > a.e. in B , and (det ∇ y ) − ∈ L s ( B ) ;(6) cof ∇ y ∈ GSBV ( B , R × ) , with |∇ [cof ∇ y ] | ∈ L q (Ω) ;(7) det ∇ y ∈ GSBV ( B , R ) , with ∇ [det ∇ y ] ∈ L r (Ω) ; H n − S (cof ∇ y ) ≤ µ V and H n − S (det ∇ y ) ≤ µ V . Assumptions (2) and (3) imply y ∈ SBV ( B , R ), with jump set containedin the varifold support, namely H n − S ( y ) ≤ µ V . Moreover, if y ∈ ˆ A p,q,r,s ,the graph current G y has null boundary ( ∂G y ) B × R = 0, see [22, Vol. I,Sec. 3.2.4]. Therefore, taking V = 0, i.e., in the absence of fractures, it turnsout that the couple ( y,
0) belongs to the class A p,p,q,r,s,K,C ( B ), provided that k y k ∞ ≤ K , independently from the choice of p and C . Theorem 4.1
Under previous assumptions, if the class A := A p,p,q,r,s,K,C ofadmissible couples ( y, V ) is non-empty and inf { F ( y, V ; B ) | ( y, V ) ∈ A } < ∞ , the functional ( y, V ) F ( y, V ; B ) attains a minimum in A . Proof.
Let { ( y h , V ( h ) ) } be a minimizing sequence in A . By Theorem 3.3, sincesup h E ( V ( h ) ; B ) < ∞ we can find a (not relabeled) subsequence of { V ( h ) } anda 2-varifold V = V C ,θ ∈ CV p ( B ), with curvature A and boundary ∂V , suchthat V ( h ) ⇀ V , A ( h ) dV ( h ) ⇀ A dV , and ∂V ( h ) ⇀ ∂V in the sense of measures,so that by lower semicontinuity E ( V ; B ) ≤ lim inf h →∞ E ( V ( h ) ; B ) < ∞ . The domain B being bounded, in terms of a (not relabeled) subsequence { y h } ⊂ A ( B , R ), we find an a.e. approximately differentiable map y ∈ L ( B , R ) such that y h → y strongly in L ( B , R ) and for any multi-indices α and β , with | α | + | β | = 3, functions v βα ∈ L ( B ) such that M βα ( ∇ y h ( x )) ⇀ v βα ( x ) weakly in L ( B ) . Moreover, we get the bound sup h M ( G y h ) < ∞ on the mass of the i.m. rectifi-able currents G y h in R ( B × R ) carried by the y h graphs, whereas the inequal-ities π | ∂G y h | ≤ C · µ V ( h ) imply the bound sup h M (( ∂G y h ) B × R ) < ∞ onthe boundary current masses. Therefore, Theorem 3.1 yields y ∈ A ( B , R )and v βα ( x ) = M βα ( ∇ y ( x )) a.e in B , for every α and β , whereas G y h ⇀ G y weakly in D ( B × R ); the current G y is i.m. rectifiable in R ( B × R ), andthe inequality π | ∂G y | ≤ C · µ V holds true.By taking into account that H n − S ( y h ) ≤ µ V ( h ) and sup h k y h k ∞ ≤ K , thecompactness theorem in SBV applies to the sequence { y h } ⊂ SBV ( B , R ),yielding the convergence Dy h ⇀ Dy as measures, whereas H n − S ( y ) ≤ µ V and k y k ∞ ≤ K , by lower semicontinuity, and clearly y = y on Γ .By using the uniform boundsup h Z B (cid:16) |∇ y h | p + | cof ∇ y h | q + | det ∇ y h | r (cid:17) dx < ∞ , W of the func-tional F J ( F ; B ), we get ∇ y h ⇀ ∇ y in L p ( B , R × ), cof ∇ y h ⇀ cof ∇ y in L q ( B , R × ), and det ∇ y h ⇀ det ∇ y in L r ( B ).Also, the inequalities H n − S (cof ∇ y h ) ≤ µ V ( h ) and the lower bound on ˆ W imply that the sequence { cof ∇ y h } ⊂ GSBV ( B , R × ) satisfies the inequalitysup h (cid:18)Z B (cid:16) | cof ∇ y h | q + |∇ [cof ∇ y h ] | q (cid:17) dx + H n − ( S (cof ∇ y h )) (cid:19) < ∞ . Therefore, by Theorem 3.2 we infer that • cof ∇ y ∈ GSBV ( B , R × ) , • cof ∇ y h → cof ∇ y in L q ( B , R × ) , • ∇ [cof ∇ y h ] ⇀ ∇ [cof ∇ y ] weakly in L q ( B , R × × ), and • H n − S (cof ∇ y ) ≤ µ V .Similarly, the inequalities H n − S (det ∇ y h ) ≤ µ V ( h ) and the lower bound onˆ W imply that the sequence { det ∇ y h } ⊂ GSBV ( B ) satisfies the inequalitysup h (cid:18)Z B (cid:16) | det ∇ y h | r + |∇ [det ∇ y h ] | r (cid:17) dx + H n − ( S (det ∇ y h )) (cid:19) < ∞ , so that Theorem 3.2 entails that • det ∇ y ∈ GSBV ( B ) , • det ∇ y h → det ∇ y in L r ( B ) , • ∇ [det ∇ y h ] ⇀ ∇ [det ∇ y ] weakly in L r ( B , R ), and • H n − S (det ∇ y ) ≤ µ V .Arguing as in the proof of Theorem 3.4, reported in reference [27], we obtaindet ∇ y > B , and (det ∇ y ) − ∈ L s ( B ), whence we get ( y, V ) ∈ A = A p,p,q,r,s,K,C .Finally, on account of the previous convergences, the gradient polyconvexityassumption implies the lower semicontinuity inequality J ( ∇ y ; B ) ≤ lim inf h →∞ J ( ∇ y h ; B ) . Then, F ( y, V ) ≤ lim inf h →∞ F ( y h , V ( h ) ) , which is the last step in the proof. 18 .1 By avoiding self-penetration The restriction imposed to det ∇ y ( x ) ensures that the deformation locally pre-serves orientation. However, we have also to allow possible self-contact betweendistant portions of the boundary preventing at the same time self-penetrationof the matter. To this aim, in 1987 P. Ciarlet and J. Neˇcas proposed theintroduction of an additional constraint, namely Z B ′ det ∇ y ( x ) dx ≤ L ( e y ( f B ′ ))for any sub-domain B ′ of B , where f B ′ is intersection of B ′ with the domain f B of Lebesgue’s representative e y of y [10].We adopt here a weaker constraint, introduced in 1989 by M. Giaquinta, G.Modica, and J. Souˇcek [21] (see also [22, Vol. II, Sec. 2.3.2]). It reads Z B f ( x, u ( x )) det ∇ y ( x ) dx ≤ Z R sup x ∈ B f ( x, y ) dy , for every compactly supported smooth function f : B × R → [0 , + ∞ ).We thus denote by f A p,p,q,r,s,K,C the set of couples ( y, V ) ∈ A p,p,q,r,s,K,C suchthat the deformation map y satisfies the previous inequality.Since that constraint is preserved by the weak convergence as currents G y h ⇀G y along minimizing sequences, arguing as in Theorem 4.1 we readily obtainthe following existence result. Corollary 4.2
Under the previous assumptions, if the class f A := f A p,p,q,r,s,K,C of admissible couples ( y, V ) is non-empty and inf { F ( y, V ) | ( y, V ) ∈ f A } < ∞ , then the minimum of the functional ( y, V ) F ( y, V ) is attained in f A . Acknowledgements . This work has been developed within the activitiesof the research group in “Theoretical Mechanics” of the “Centro di RicercaMatematica Ennio De Giorgi” of the Scuola Normale Superiore in Pisa. PMMwishes to thank the Czech Academy of Sciences for hosting him in Pragueduring February 2020 as a visiting professor. We acknowledge also the supportof GA ˇCR-FWF project 19-29646L (to MK), GNFM-INDAM (to PMM), andGNAMPA-INDAM (to DM). 19 eferences [1] Allard W. K. (1972), On the first variation of a varifold,
Ann. of Math. , ,417-491.[2] Allard W. K. (1975), On the first variation of a varifold: boundary behavior, Ann. of Math. , , 418-446.[3] Almgren F. J. Jr. (1965), Theory of varifolds , mimeographed notes, Princeton(1965).[4] Ambrosio L. (1995), A new proof of the
SBV compactness theorem,
Calc. Var.Partial Differential Equations , 127-137.[5] Ambrosio L., Braides A., Garroni A. (1998), Special functions with boundedVariation and with weakly differentiable traces on the jump set, NoDEANonlinear Differential Equations Appl. , , 219-243.[6] Ambrosio L., Fusco N., Pallara D. (2000), Functions with Bounded Variationand Free Discontinuity Problems , Oxford University Press, Oxford.[7] Beneˇsov´a B., Kruˇz´ık M., Schl¨omerkemper A. (2018), A note on locking materialsand gradient polyconvexity,
Math. Mod. Methods Appl. Sci. , , 2367-2401.[8] Bisconti L., Mariano P. M., Markenscoff X. (2019), A model of isotropic damagewith strain-gradient effects: existence and uniqueness of weak solutions forprogressive damage processes, Math. Mech. Solids , , 2726-2741.[9] Capriz G. (1985), Continua with latent microstructure, Arch. Rational Mech.Anal. , , 43-56.[10] Ciarlet P. G., Neˇcas J. (1987), Unilateral problems in nonlinear three-dimensional elasticity, Arch. Rat. Mech. Anal. , , 171-188.[11] Dal Maso G., Toader R. (2002), A model for the quasi-static growth of brittlefractures: Existence and approximation results, Arch. Rational Mech. Anal. , , 101-135.[12] De Giorgi E. (1993), New problems on minimizing movements, in Ennio DeGiorgi - Selected Papers , L. Ambrosio, G Dal Maso, M. Forti, M. Miranda, S.Spagnolo Edt.s, pp. 699-713, Springer Verlag, 2006.[13] Dunn J. E. and Serrin J. (1985), On the thermomechanics of intertistitialworking,
Arch. Rational Mech. Anal. , , 95-133.[14] Federer H., Fleming W. (1960), Normal and integral currents, Ann. of Math. , , 458-520.[15] Feuerbacher M., Heggen M. (2011), Metadislocations in complex metallic alloysand their relation to dislocations in icosahedral quasicrystals, Israel J. Chem. , , 1235-1245.
16] Fleck N. A., Hutchinson J. W. (1993), A phenomenological theory for straingradient effects in plasticity,
J. Mech. Phys. Solids , , 1825-1857.[17] Fleck N. A., Muller G. M., Ashby M. F., Hutchinson J. W., (1994), Straingradient plasticity: theory and experiment, Acta Metall. Mater. , , 475-487.[18] Francfort G. A., Marigo J. J. (1998), Revisiting brittle fracture as an energyminimization problem, J. Mech. Phys. Solids , , 1319-1342.[19] Giaquinta M., Mariano P. M., Modica G. (2010), A variational problem in themechanics of complex materials, Disc. Cont. Dyn. Syst. A , , 519-537.[20] Giaquinta M., Mariano P. M., Modica G., Mucci D. (2010), Ground statesof simple bodies that may undergo brittle fracture, Physica D - NonlinearPhenomena , , 1485-1502.[21] Giaquinta M., Modica G., Souˇcek J. (1989), Cartesian currents, weakdiffeomorphisms and existence theorems in nonlinear elasticity, Arch. RationalMech. Anal. , , 97-159. Erratum and addendum, Arch. Rational Mech. Anal. ,(1990) , 385-392.[22] Giaquinta M., Modica G., Souˇcek J. (1998),
Cartesian Currents in the Clculusof Variations , voll. I and II, Springer-Verlag, Berlin.[23] Griffith A. A. (1920), The phenomena of rupture and flow in solids,
Phil. Trans.Royal Soc. A , CCXXI , 163-198.[24] Gudmundson P. (2004), A unified treatment of strain gradient plasticity,
J.Mech. Phys. Solids , Arch.N´eerl. Sci. Exactes Nat. Ser. II , , 1-24.[26] Kr¨oner E. (1960), Allgemeine Kontinuumstheorie der Versetzungen undEigenspannungen, Arch. Rational Mech. Anal. , , 273-334.[27] Kruˇz´ık M., Pelech P., Schl¨omerkemper A. (2020), Gradient polyconvexity inevolutionary models of shape-memory alloys, J. Opt. Theory Appl. , , 5-20.[28] Kruˇz´ık M., Roub´ıˇcek, T. (2019), Mathematical Methods in ContinuumMechanics of Solids , Springer, Switzerland.[29] Lee E. H. (1969), Elastic-plastic deformations at finite strains,
J. Appl. Mech. , , 1-6.[30] Lubensky T. C., Ramaswamy S., Toner, J. (1985), Hydrodynamics oficosahedral quasicrystals, Phys. Rev. B , , 7444-7452.[31] Mantegazza C. (1996), Curvature varifolds with boundary, J. DifferentialGeom. , , 807-843.
32] Mariano P. M. (2006), Mechanics of quasi-periodic alloys,
J. Nonlinear Sci. , ,45-77.[33] Mariano P. M. (2010), Physical significance of the curvature varifold-baseddescription of crack nucleation, Rendiconti Lincei , , 215-233.[34] Mariano P. M. (2017), Second-neighbor interactions in classical field theories:invariance of the relative power and covariance, Math. Meth. Appl. Sci. , ,1316-1332.[35] Mariano P. M. (2019), Mechanics of dislocations and metadislocations inquasicrystals and their approximants: power invariance and balance, Cont.Mech. Thermodyn. , , 373-399.[36] Phillips R. (2001), Crystals, Defects and Microstructures , Cambridge UniversityPress, Cambridge., Cambridge UniversityPress, Cambridge.