Local micromorphic non-affine anisotropy for materials incorporating elastically bonded fibers
LLocal micromorphic non-affine anisotropy formaterials incorporating elastically bonded fibers
Sebastian Skatulla , Carlo Sansour , and Georges Limbert Department of Civil Engineering, University of Cape Town, SouthAfrica Center for Research in Computational and Applied Mechanics,University of Cape Town, South Africa Department of Mathematics, Bethlehem University, Bethlehem,Palestine National Centre for Advanced Tribology at Southampton(nCATS) and Bioengineering Science Research Group, Departmentof Mechanical Engineering, Faculty of Engineering and PhysicalSciences, University of Southampton, Southampton SO17 1BJ, UK Laboratory of Biomechanics and Mechanobiology, Division ofBiomedical Engineering, Department of Human Biology, Faculty ofHealth Sciences, University of Cape Town, Observatory 7935, CapeTown, South AfricaMarch 4, 2021
Abstract
There has been increasing experimental evidence of non-affine elas-tic deformation mechanisms in biological soft tissues. These observationscall for novel constitutive models which are able to describe the dom-inant underlying micro-structural kinematics aspects, in particular rel-ative motion characteristics of different phases. This paper proposes aflexible and modular framework based on a micromorphic continuum en-compassing matrix and fiber phases. It features in addition to the dis-placement field so-called director fields which can independently deformand intrinsically carry orientational information. Accordingly, the fibrousconstituents can be naturally associated with the micromorphic directorsand their non-affine motion within the bulk material can be efficientlycaptured. Furthermore, constitutive relations can be formulated basedon kinematics quantities specifically linked to the material response ofthe matrix, the fibres and their mutual interactions. Associated stressquantities are naturally derived from a micromorphic variational princi-ple featuring dedicated governing equations for displacement and director a r X i v : . [ phy s i c s . c o m p - ph ] M a r elds. This aspect of the framework is crucial for the truly non-affineelastic deformation description.In contrast to conventional micromorphic approaches, any non-local higher-order material behaviour is excluded, thus significantly reducing the num-ber of material parameters to a range typically found in related classicalapproaches.In the context of biological soft tissue modeling, the potential and appli-cability of the formulation is studied for a number of academic examplesfeaturing anisotropic fiber-reinforced composite material composition toelucidate the micromorphic material response as compared with the oneobtained using a classical continuum mechanics approach. A large number of physical in and ex vivo experiments on biological tissuesand cells have demonstrated that mechanical forces are central to many biolog-ical processes from morphogenesis [2], development and ageing to disease andhealing by controlling cell behaviour and biochemical signalling pathways [25].Mechanobiology, the science of how mechanics and biology affect each other,has emerged as one of the most active and promising branch of biophysics. Itholds the keys for fundamental insights and understanding of human biology andphysiology in health and disease [30] whilst opening up a wide array of practi-cal applications including tissue engineering [36] and regenerative medicine [43],diagnosis and treatment of diseases such as cancer, osteoarthritis and cardio-vascular disorders.It is therefore essential to provide robust knowledge bases that shed light onthe fundamental force transmission and deformation mechanisms of biologicalstructural assemblies, from cell through tissue to organ level. An importantaspect of biological soft tissues is the fibrous nature of their extracellular matrixcomponent which takes the form of collagen and elastic fibres networks withvarious degrees of structural order and symmetry. Of particular significance,the structural characteristics of these fibre networks are intrinsically linked tothe physiological and biophysical functions of tissues [22].In the continuum-based constitutive modelling of biological soft tissues it istypically assumed that a tissue can be represented as a composite material madeof one or several families of (oriented) collagen fibres embedded in a highly com-pliant isotropic solid matrix composed mainly of proteoglycans [56, 28, 37]. Moreoften than not, in these types of approaches, it is assumed that fibres deformaffinely such that the macroscopic and microscopic principal strains of the fibrephase are coaxial. In other words, fibre and matrix kinematics are propagatedacross length scales. While this could be considered a reasonable assumption,various experimental studies have evidenced the existence of non-affinity of de-formation mechanisms in biological soft tissues [35, 26, 6] and polymer hydrogels[10, 57].In their experimental study to quantify fibre kinematics of porcine aor-tic valves leaflets and bovine pericardium under biaxial stretch,
Billiar and acks [6] observed via in situ imaging significant local reorientation of collagenfibres in pericardium specimens resulting in an almost uniform fibre alignmentwith the maximum principal strain. As highlighted by these authors this in-dicated non-affinity of deformations and a generally high rotational mobilityof fibres optimising the reinforcement of the material [13]. In a comparableexperimental study, Tower et al. [54] characterised heterogeneous collagenfibre re-alignment in native porcine aortic valve leaflets and tissue equivalentsunder uniaxial extension. It was demonstrated by
Screen et al. [47] thatinter-fibre sliding occurs in uncrimped rat tail tendon fascicles during uniaxialtraction and that this constitutes the dominant mechanism responsible for fas-cicle extension. Significant non-affine microstructural deformation mechanismswere evidenced in the hexagon-like microstructure of lung alveoli under tensionby
Brewer et al. [8].
Krasny et al. [35] showed that adventitial collagenfibres in carotid arteries from New Zealand White rabbit exhibited non-affinityby reorienting along the load direction to a degree that cannot be accounted forby affine kinematics alone.It is clear from this selected list of characterisation studies on biological softtissues that complex microstructural deformation mechanisms are responsiblefor a variety of non-affine behaviours.
Zarei et al. [59] showed with a discretefibre-network model of axons embedded in an extra-cellular matrix consisting ofcollagen fibres that, depending on the degree of anisotropy and mode of loading,the overall maximum principal strain can both, increase or decrease. However,true multiscale computational models explicitly accounting for individual mi-crostructural components of tissues, e.g. [59], remain unpractical because ofthe high computational cost and challenges in mechanically characterising thesemicro-constituents. Therefore, microstructurally-based continuum constitutivemodels capable of accounting for non-affine kinematics of biological fibres offeran attractive prospect in the quest for robust predictive models in biophysicalsciences.The majority of continuum-based approaches dealing with relative fibre-matrix deformation and fibre reorientation and thus, non-affine kinematics rela-tions, are of an inelastic dissipative nature. Generally, the continuum approachimplies that matrix and fibre constituents occupy the same position in space.The dissipative relative fibre motion can then be determined through energeticconsiderations controlled via the entropy inequality [33] or linked to a principleof coaxiality with the principal strain directions [16, 27]. This principle of coax-iality can also be utilized within an elastic fibre reorientation framework [44].However, in contrast to a dissipative approach introducing the fibre deformationas an independent internal variable, the elastic approach lacks this independentvariable. Consequently, albeit this type of elastic fibre deformation is not di-rectly linked via the deformation gradient tensor, the relative matrix-fibre de-formation is still affine and no truly independent relative motion can take place.Equally, multiscale homogenization approaches consider an independent defor-mation behaviour within the RVE, but its volume-averaged response must becompatible with the macroscopic deformation, e.g. [11, 52, 41, 38]. In contrast,if elastic relative fibre-matrix deformation is to be considered which is governed3y the stationarity principle of the total energy with no prior explicit assump-tions on the constituents’ relative motion, e.g. the alignment with the maximumprincipal strain direction, then the fibre reorientation must be described by anindependent field and an associated dedicated equilibrium equation.The linking of macro-kinematics and micro-kinematics is a core aspect ofgeneralised continuum approaches, such as the milestone works of the
Cosserat continuum by
Toupin [53] with its thermodynamic implications discussed by
Capriz [9] or the micromorphic continuum by
Mindlin [39] and
Eringen [18].Generalised continuum theories possess the benefit of additional degrees of free-dom and/or higher-order strain and stress quantities which can be linked tomicro-kinematics features but require a corresponding extended list of materialparameters and/or characteristic length scale parameters to be identified (foran overview see e.g. [20]). The internal length scale parameters can be usedto define size of the representative volume element (RVE) in homogenisationapproaches controlling the non-local material behaviour [1, 50, 29, 45, 7]. Ingranular mechanics, the micromorphic continuum formulation has been shownto be able to address scale-dependent material behaviour as linked to an inde-pendent micro-kinematics of interacting particles [17, 24, 58]. It is also capableto deal with mass flux in growth and remodelling of biological material [31].The number of material parameters can be reduced by restrictions posed onthe higher-order quantities associated with the non-local deformation response,e.g. by considering only micro-dilatation [51, 21], rigid body motion as in the
Cosserat continuum [14], the micropolar continuum [32] and in a relaxed micro-morphic continuum [42] by selectively disregarding specific deformation modes[48], or by formulating the constitutive response within an extended continuumas a function of a generalized strain tensor [46, 49]. It also can be shown thatthe
Euler-Lagrange equations of linear micropolar and micromorphic elasticitycan be recovered from an internal variable approach based on thermodynamicconsiderations [4, 5].From the above, one notices that especially the micromorphic continuumoffers great flexibility, as the additional degrees of freedom can be interpreted inmany different ways. The micromorphic continuum as conceived by
Mindlin [39] and
Eringen [18] considered the extra degrees of freedom in vectorialform, as so-called directors. These directors can deform independently of thedisplacement field in terms of change of length and orientation. Accordingly, anadditional independent (micro-)deformation tensor can be defined which intrin-sically carries orientational information of a material point. The latter has beenexploited in the context of linear elastic anisotropy where the micromorphic con-tinuum has recently found application in scale-dependent wave propagation anddispersion of crystalline material [15, 40], micro-mechanics of cellular or latticemetamaterials [34], special anisotropic constitutive relations constraining themicro-kinematics [3].The micromorphic constitutive framework proposed in this paper is moti-vated by the aforementioned uniaxial and biaxial tension experiments [6, 54, 35]exhibiting elastic collagen fibre re-alignment in load direction. The approachequips the fibrous constituents with the ability to elastically reorient themselves4ltering the material response depending on the loading experienced as sug-gested by
Chew et al. [13]. As a preliminary step we will lay down thebasic theoretical foundations that will enable the flexible integration of specificdeformation modes into non-affine models of biological soft tissues [12].There are six unique characteristics of this approach:1. The fibres are elastically linked to the matrix material such that non-dissipative relative (non-affine) fiber-matrix deformation can be captured.This is made possible by the introduction of independent vector fields, themicromorphic directors, which describe the initial orientation and subse-quent deformation of the fibres. This is in contrast to conventional fiberreorientation approaches which are either affine-linked to the displacementgradient or non-affine but dissipative;2. As the non-affine deforming micromorphic directors are identified with thepreferred material directions, the resulting anisotropic material behaviourcan become either more pronounced than conventional anisotropy duringloading or less pronounced. In the limit case, the medium can even becomemechanically isotropic, depending on the elastic fibre-matrix bond stiffnessand the stiffness of the directors themselves;3. The fibre-matrix linkage differentiates between elastically constrained rel-ative axial and rotational/shear motion and can therefore be associatedwith dedicated physically motivated elastic material parameters, namelyan axial bond stiffness and a rotational/shear bond stiffness;4. No specific a priori choice is made with regards to the relative fiber motion(e.g. linked to maximum principal strain direction) other than the stiff-ness parameters of the matrix, directors and elastic bonds. This aspectis particularly beneficial for more complex fibre composition hierarchiessuch as those found in biological soft tissues where for arbitrary loadingconditions the specific mode of relative fibre-matrix deformation is notnecessarily known a priori. Consequently, the fibre orientation must bea truly independent field which directly and independently responds tothe loading via additional
Euler-Lagrange equations. By identifying eachfibre family with a micromorphic director field, this condition is naturallymet, as each director is governed by a dedicated equilibrium equation andcorresponding boundary conditions;5. The equilibrium equations derived from the micromorphic variational prin-ciple intrinsically account for the interaction between matrix and fibre byexplicitly incorporating additional stress-like quantities naturally arisingfrom the non-affine anisotropic micromorphic constitutive framework;6. The chosen micromorphic continuum description specifically excludes non-local material behaviour considering the characteristic length scale param-eter vanishing in the mathematical limit, i.e. l C →
0, and no homogeni-sation process is required. In this way, the number of additional material5arameters can be limited to that required to describe the elastic materialbehaviour of the fibres themselves and that of the matrix-fibre linkage.The paper is organised as follows: Sec. 2 introduces the micromorphic non-affine anisotropy framework in terms of its kinematics, constitutive relationsand variational principle with corresponding governing equations and boundaryconditions. Sec. 3 illustrates the main features and characteristics of the ap-proach by means of academic examples considering fibrous materials. Finally,in Sec. 4, the constitutive framework is discussed and its potential for futurebiomechanical applications is highlighted.
Making use of the micromorphic approach introduced by
Sansour et al. [46]and von Hoegen et al. [55], a generalised continuum can be constructedfrom a matrix-continuum,
B ⊂ E (3), representing the bulk material and a one-dimensional fibre-continuum, S , representing a fibre embedded in the bulk ma-terial.Figure 1: Schematic description of the generalized configuration space.Here, we assume that the placement vector ˜ x of a material point P is ofan additive nature, namely the sum of its position in the matrix-continuum, x ∈ B t , and in the fibre-continuum, ξ ∈ S t , both at time t ∈ R , as follows˜ x ( X k , ζ, t ) = x ( X k , t ) + ξ ( X k , ζ, t ) . (1)According to this additive structure, the configuration of the generalised contin-uum is defined by the Cartesian product G := B × S ⊂ E (3 + 1) and the integra-tion over the matrix and the fibre-continuum can be performed separately. Eachfibre-continuum represents a specific fibre family. The matrix-continuum, B , isparameterized by the Cartesian coordinates, X i , and the fibre-continuum, S , bythe Cartesian coordinate, ζ . Here, and in what follows, Latin indices take the6alues 1 , ,
3. As shown in Fig. 1 the matrix-placement vector, x , defines theorigin of the fibre co-ordinate system such that the fibre placement, ξ , within S is assumed to be relative to the matrix-placement. For convenience but withoutloss of generality we identify G with the un-deformed reference configuration ata fixed time, t , in what follows.The definition of the generalised continuum, and so the extra degrees offreedom, depends directly on the choices to be made for the fibre-deformation ξ ( X k , ζ, t ). The theory is based on the fact that the dependency on the fibre-coordinate, ζ , must be determined a priori. The simplest case is provided bythe linear ansatz ˜ x = x ( X k , t ) + ζ a f ( X k , ζ, t ) . (2)The vector function a f ( X k , t ) is a so-called director with its associated fibre-coordinate, ζ , which is related to its equivalent in the reference configuration, A f , via a f = A f ( X k , ζ ) + w f ( X k , t ) , (3)with w f ( X k , t ) denoting the increment/change of the director and its threevector components being the extra degrees of freedom. The orientation of thefibre-continuum in the undeformed configuration, S , is specified by the unde-formed director, A f , with | A f | = 1 and the size of S is given by definition spaceof the fibre-coordinate, ζ = [ − l f ; l f ], where l f denotes the characteristic lengthof fibre-continuum. Accordingly, for the purpose of micromorphic non-affinefibre-matrix mechanics, we identify the fibre with the director at time t = 0 inthe undeformed/unloaded configuration: A f = ˜ V f = ˜ V F . (4)where the directional index referring to the fibre is chosen to be uppercase as( • ) F when linked to the matrix kinematics and lowercase as ( • ) f when linkedto the fibre kinematics. In the most general form the undeformed director, A f , depends on the matrix-coordinates, X k , as well as the fibre-coordinate, ζ , and is not uniform throughout B and S . This is, for instance, the casefor most biological materials due to fibre orientation variations within the bulkmaterial and fibre dispersion effects linked to micro-continuum, see e.g. [19, 23].The change of the director, w f , however, is assumed constant in S and so itscomponents which are the extra degrees of freedom.As a starting point we assume A f = A f ( X k ) disregarding any dispersion ofits orientation in the fibre-continuum. Taking the derivatives of ˜ x with respectto the matrix-coordinates, X i ,˜ x ,i = ∂ ˜ x ∂X i = x ,i + ζ a f,i (5)as well as with respect to the fibre co-ordinate, ζ ,˜ x ,f = ∂ ˜ x ∂ζ = a f (6)7he generalized deformation gradient tensor can then be expressed as follows˜ F = ˜ F (0) + ˜ F (1) = ( x ,i + ζ a f,i ) ⊗ ˜ G i + a f ⊗ ˜ V f . (7)The operator ⊗ denotes the dyadic product of two vectors and the tangent space T G in the reference configuration is defined by the pair ( ˜ G i × ˜ V f ) given by˜ G i = ∂ ˜ X ∂X i and ˜ V f = ∂ ˜ X ∂ζ = A f , (8)where the corresponding dual contra-variant vectors are denoted by ˜ G i and ˜ V f ,respectively. Due to the chosen Cartesian coordinate spaces for both, B and S ,respectively, it holds ˜ G i = ˜ G i and ˜ V f = ˜ V f . From Eq. (8) it is clear thatthe director, A f ∈ T G , is of differential nature and equally, its change, w f .This implies that the components of w f , the extra degree of freedom, are effec-tively strain-like quantities based on the specific definition of the micromorphicplacement vector (Eq. (2).In the following, we only want to disregard non-local scale-dependent effectswith respect to the matrix, relative matrix-fibre and fibre deformation. Thelatter implies that the characteristic length of the fibre-continuum l f → ζ = 0. In this way, the higher-order scale-dependent part of ˜ F (0) (Eq. (7)) is removed which thus becomes the classical deformation gradient ten-sor given as ˜ F (0) = x ,i ⊗ ˜ G i . Accordingly, the local micromorphic deformationgradient tensor is expressed in the deformed and undeformed configurations as˜ F = x ,i ⊗ ˜ G i + a f ⊗ ˜ V f and ˜ F ref = ˜ G i ⊗ ˜ G i + ˜ V f ⊗ ˜ V f , (9)respectively. As such, we can define two local micromorphic right Cauchy-Green deformation tensors as follows:˜ C (0) = ( ˜ F (0) ) T ˜ F (0) = x ,k · x ,l ˜ G k ⊗ ˜ G l = C (0) (10)˜ C (1) = ( ˜ F (1) ) T ˜ F (1) = a f · a f ˜ V f ⊗ ˜ V f . (11)˜ C (0) quantifies strain only relating to the matrix-space, whereas ˜ C (1) is a purefibre-strain, that is the squared director stretch. Disregarding the higher-ordercontributions of the micromorphic kinematics is a clear point of departure fromconventional micromorphic theories (e.g. [46]) which has the advantage that noknowledge of the material underlying characteristic lengths is required for thepurpose homogenization. This is also means that this micromorphic continuumframework is effectively defined on the macroscale.The proposed micromorphic continuum provides a direct means to describeanisotropic material behaviour on kinematics level. There is, however, a signif-icant difference to a classical approach. Considering Eqs. (4) and (9) we findthat a fibre deforming affine with matrix is described by˜ v F = ˜ F (0) ˜ V F (12)8igure 2: Illustration of the independent non-affine deformation mapping ofmatrix and director tangent spaces in fibre direction, respectively.and a fibre identified with a corresponding micromorphic director deformingnon-affine to the matrix by a f = ˜ F (1) ˜ V f . (13)This highlights the fact that in the undeformed state ˜ V F = ˜ V f = A f but indeformed state ˜ v F (cid:54) = a f as illustrated in Fig. 2. Accordingly, the micromorphicanisotropy approach inherently features a non-affine matrix-fibre deformationdescription through the independently deforming micromorphic director. Forlater use, with Eqs. (9), (10) and (11) two micromorphic Green strain tensorsare defined as ˜ E (0) = 12 (cid:16) ˜ C (0) − (cid:17) (14)˜ E (1) = 12 (cid:16) ˜ C (1) − ˜ V f ⊗ ˜ V f (cid:17) . (15) Now, let us expand on the previous considering two distinct fibre families andcorresponding directors, a f and a s , respectively, which are in the undeformedconfiguration perpendicular to each other given as V f and V s , respectively. This9eans six extra degrees of freedom arise from the two director change vectors, w f and w s , respectively. Accordingly, we re-define the director deformationgradient part in Eq. (9)) as˜ F (1) = a f ⊗ ˜ V f + a s ⊗ ˜ V s and ˜ F (1) ref = ˜ V f ⊗ ˜ V f + ˜ V s ⊗ ˜ V s . (16)For later use, we need to introduce two micromorphic Almansi strain tensorsobtained from Eqs. (14) and (15) via push-forward operations˜ e (0) = ( ˜ F (0) ) − T ˜ E (0) ( ˜ F (0) ) − (17)˜ e (1) = ( ˜ F (1) ) − T ˜ E (1) ( ˜ F (1) ) − (18)which can also be related to two corresponding micromorphic Cauchy deforma-tion tensors given by ˜ c (0) = ( ˜ F (0) ref ) − T ( ˜ F (0) ref ) − − e (0) (19)˜ c (1) = ( ˜ F (1) ref ) − T ( ˜ F (1) ref ) − − e (1) . (20)Then, let us consider the micromorphic non-affine anisotropy strain energy asthe sum of contributions from the matrix alone, ˜ ψ m , elastic axial motion and change of angle between matrix- and fibre, ˜ ψ mf , as well as the stretch of thedirector itself, ˜ ψ f , which can be expressed in terms of ˜ E (0) , ˜ E (1) , a f and a s asfollows: ˜ ψ = ˜ ψ m ( ˜ E (0) ) + ˜ ψ mf ( ˜ E (0) , a f , a s ) + ˜ ψ f ( ˜ E (1) ) . (21)For the isotropic matrix strain energy, we consider the following standard linearelastic approach: ˜ ψ m = 12 a (cid:16) tr ˜ E (0) (cid:17) + a tr (cid:16) ˜ E (0) (cid:17) (22)with a and a denoting the elastic material constants.Now, we proceed with the non-affine anisotropic material description whichis achieved by identifying the preferred directions as the non-affine deformingfibres, a f and a s , respectively. The external loading and the internal responseto it is primarily associated with the matrix. As such the reaction of the ma-trix provokes in turn a reaction of the fibres. However, the directors generallydeform to a certain degree independent of the matrix which could even includethe possibility that the matrix stretches along the director-direction whereasthe director itself contracts evading the loading. Accordingly, the interactionbetween matrix and fibre has to be suitably incorporated in the material descrip-tion to achieve a physically reasonable deformation behaviour of both materialconstituents. In the most general case, the linkage between matrix and fibreneeds to account for relative axial motion and change of angle , α , between thedirector and the matrix as depicted in Fig. 3.In order to allow for reversible non-affine fibre motion relative to the matrix,the matrix-fibre bond is of elastic nature and is described in terms of suitable10igure 3: Schematic description of the elastic structural matrix-fibre interactions interms of axial fibre motion (left) and rotational fibre motion (right) where the elasticbond is idealised by springs indicated in blue. pseudo strain invariants combining matrix and fibre kinematics relations. Forthe relative axial strain of both fibres we define in the deformed configurationusing the matrix
Cauchy deformation tensor (Eq. 19) as projected onto thedirectors I f = ˜ c (0) : ( a f ⊗ a f ) (23) I s = ˜ c (0) : ( a s ⊗ a s ) (24)and for the relative shear strains we have I fs = ˜ c (0) : 12 ( a f ⊗ a s + a s ⊗ a f ) . (25)The fibres reinforce the matrix lending it additional axial stiffness in direction a f and a s , respectively, which is associated with corresponding pseudo straininvariants of the fibres L f = ˜ e (1) : ( a f ⊗ a f ) = ˜ E (1) : ( ˜ V f ⊗ ˜ V f ) (26) L s = ˜ e (1) : ( a s ⊗ a s ) = ˜ E (1) : ( ˜ V s ⊗ ˜ V s ) (27)where we made use of Eqs. (13) and (18). The pseudo strain invariants I f , I s , I fs , L f and L s have each a clear physical meaning, as they provide the meansto separately address the relative stretch and shear of the matrix with respectto the non-affine deforming fibres as well as the stretch of the fibres themselves.11 emark: Due to the non-affine deforming fibres there are several implica-tions which highlight the inherent differences of the micromorphic non-affineanisotropic constitutive framework from conventional ones:1. Albeit, A i = ˜ V i , i = 1 ,
2, the usual pull-back operation applied to pseudoinvariants does not apply considering Eqs. (12) and (13): I (˜ e (0) )4 i = ˜ e (0) : ( a i ⊗ a i ) (cid:54) = ˜ E (0) : ( ˜ V i ⊗ ˜ V i ) . (28)Instead, we find with Eqs. (13) and (17) I (˜ e (0) )4 i = ˜ e (0) : ( a i ⊗ a i )= ( ˜ F (1) ) T ( ˜ F (0) ) − T ˜ E (0) ( ˜ F (0) ) − ˜ F (1) : ( ˜ V i ⊗ ˜ V i ) . (29)2. Formulating the pseudo-invariants quantifying the relative matrix-fibremotion directly in terms of the micromorphic Almansi strain tensors isnot suitable, because the
Hessians relating to the matrix-fibre interaction, ∂ ˜ S (0) ∂ a f and ∂ ˜ S (0) ∂ a s , respectively, vanish.3. There is no direct pull-back operation for the pseudo-invariants formu-lated in terms of the Cauchy deformation tensor, ˜ c (0) (Eqs. (23)-(25)).Accordingly, for a Lagrangian formulation considering Eqs. (17) and (19)the change of those invariants with respect to ˜ E (0) , a f , a s and ˜ F (0) needsto be evaluated.Finally, making use of the previously defined pseudo invariants (Eqs. (23)-(27)) we can complete the elastic non-affine anisotropic material descriptionbased on two preferred material directions represented by a f and a s as follows:˜ ψ m = 12 a (cid:16) tr ˜ E (0) (cid:17) + a tr (cid:16) ˜ E (0) (cid:17) (30)˜ ψ mf = b f ( I f −
1) + b s ( I s −
1) + b fs I fs + p f ( I f −
1) + p s ( I s −
1) (31)˜ ψ f = c f L f + c s L s (32)where the extra constants p f and p s relating to the linear term in Eq. (31)are to be determined as a function of the other material parameters such thatall stress quantities defined further below vanish at the reference configuration.This marks a clear point of departure from classical anisotropy, as in contrast,affine anisotropic material behaviour would be formulated in terms of affinepseudo-invariants, e.g. I F = ˜ e (0) : ( v F ⊗ v F ) = ˜ E (0) : ( ˜ V F ⊗ ˜ V F ). Furthermore,the degree of non-affine fibre deformation of the preferred material directions iscontrolled by the material parameters b f , b s and b fs . The required magnitudeof these parameters depends on the relative stiffness difference between thematrix given by the material constants a and a , and the fibres given by c f and c s , respectively. The additional stiffness contribution of the preferred material12irections is therefore the result of the combination of fibre stiffness and matrix-fibre linkage.Due to the two micromorphic strain measures we also have two micromorphicsecond Piola-Kirchhoff -type stress tensors˜ S (0) = ∂ ˜ ψ m ∂ ˜ E (0) + ∂ ˜ ψ mf ∂I i ∂I i ∂ ˜ c (0) ∂ ˜ c (0) ∂ ˜ e (0) ∂ ˜ e (0) ∂ ˜ E (0) (33)˜ S (1) = ∂ ˜ ψ f ∂L i ∂L i ∂ ˜ E (1) . (34)As previously remarked, from the non-affine anisotropy description via thepseudo invariants representing the matrix-fibre interface strains (Eqs. (23)-(25))three additional stress-like quantities arise:˜ z f = ∂ ˜ ψ mf ∂I i ∂I i ∂ a f (35)˜ z s = ∂ ˜ ψ mf ∂I i ∂I i ∂ a s (36)˜ Y = ∂ ˜ ψ mf ∂I i ∂I i ∂ ˜ c (0) ∂ ˜ c (0) ∂ ˜ e (0) ∂ ˜ e (0) ∂ ( ˜ F (0) ) − ∂ ( ˜ F (0) ) − ∂ ˜ F (0) . (37)In this sense, interaction between matrix and fibre is quantified via ˜ Y , ˜ z f , ˜ z s and the second part of ˜ S (0) . Lastly, corresponding micromorphic stress tensorsof the first Piola-Kirchhoff -type can be considered:˜ P (0) = ˜ F (0) ˜ S (0) (38)˜ P (1) = ˜ F (1) ˜ S (1) . (39)Obviously, with Eq. (37) the total first Piola-Kirchhoff of is given by ˜ P (0) + ˜ Y . Let us now consider a non-linear boundary value problem in the domain B withthe boundary ∂ B . In the following we choose that the external forces are nota function of the fibre-coordinate and define the external virtual work in theLagrangian form as follows W ext ( u ) = (cid:90) B b · δ u dV + (cid:90) ∂ B N t ( n ) · δ u dA (40)with b denoting the external body forces and t ( n ) the external traction . Fur-thermore, dV is a volume element of the matrix domain B , whereas dA is asurface element of its corresponding boundary ∂ B .Furthermore, we assume that the body under consideration B is hyperelasticand possesses an elastic potential Ψ represented by the stored strain energy per13nit volume ˜ ψ ( ˜ E (0) , ˜ E (1) , a f , a s ). The first law of thermodynamics providesthen the following variational principleΨ ( u , w ) = (cid:90) B (cid:110) ˜ S (0) : δ ˜ E (0) + ˜ S (1) : δ ˜ E (1) (cid:111) dV + (cid:90) B (cid:110) ˜ z f : δ w f + ˜ z s : δ w s + ˜ Y : δ ˜ F (0) (cid:111) dV − W ext = 0 (41)which separately accounts for the internal powers referring to the two differentmicromorphic strain tensors plus the additional non-affine anisotropy contribu-tions. The micromorphic variational formulation is supplemented with Dirichletboundary conditions for the displacement and change of director fields, respec-tively: u = ˆ u on ∂ B uD (42) w f = ˆ w f on ∂ B wD (43) w s = ˆ w s on ∂ B wD . (44)As mentioned before, the fibre-space is considered in the mathematical limit with l f →
0. Consequently, the integration of the micromorphic variational principle(Eq. (41)) over S is not required anymore, as it defined in B which is contrast tothe similar but non-local approach by von Hoegen et al. [55]. Furthermore,corresponding to the independent displacement and change of director fields wecan identify the following equilibrium equations:Div ˜ P (0) + Div ˜ Y + b = in B (45)˜ P (1) ˜ V f + ˜ z f = in B (46)˜ P (1) ˜ V s + ˜ z s = in B (47)as well as a Neumann boundary condition corresponding to Eq. (45):˜ P (0) n + Yn − t ( n ) = on ∂ B N . (48)The governing equations for the two change of director fields (Eqs. (46) and (47))establish the equilibrium between the fibre stress ˜ P (1) and the correspondinginteraction stress vectors, ˜ z i , i = 1 , a f and a s (Eqs. (46)-(47)), respectively,is crucial such that the latter can be directly solved for from first principles forany chosen constitutive behaviour, in particular in case of an elastic response.Furthermore, external loading enters through the governing equation for thedisplacement field and corresponding Neumann boundary condition, Eqs. (45)and (48), respectively. This elucidates that the external loading is primarilyassociated with the matrix which has been previously mentioned in Sec. 2.2.14 Numerical examples
The numerical examples in the following aim at elucidating the qualitative be-haviour of the local micromorphic non-affine anisotropy framework introducedin the previous sections. Based on the results its potential to model biologicalsoft tissue is discussed in the subsequent section, Sec. 4.To better understand what type anisotropic behaviour can be expected foreach example and to highlight the differences to an affine approach, an equivalentconventional anisotropic strain energy function based on the classical
Green strain tensor, E = ˜ E (0) , is used as well. The latter makes use of the followingpseudo strain invariants I F = ˜ E (0) : ( ˜ V F ⊗ ˜ V F ) (49) I S = ˜ E (0) : ( ˜ V S ⊗ ˜ V S ) (50) I F S = ˜ E (0) : 12 (cid:16) ˜ V F ⊗ ˜ V S + ˜ V S ⊗ ˜ V F (cid:17) (51)so that a classical anisotropic strain energy function can then be expressed as ψ = 12 ¯ a (cid:16) tr ˜ E (0) (cid:17) + ¯ a tr (cid:16) ˜ E (0) (cid:17) ¯ a F I F + ¯ a S I S + ¯ a F S I F S (52)with its parameter values listed in Tab. 1. For the micromorphic non-affineTable 1: Overview of the material parameters used for the classical anisotropymodel (Eq. (52)).Set ¯ a ¯ a ¯ a F ¯ a S ¯ a F S . × . × × Table 2: Overview of the material parameters used for the micromorphic non-affine anisotropy model (Eq. (21)).Set a a b f b s b fs c f c s . × . × . × . × . × . × × . × . × . × . × . × . × anisotropy model (Eq. (21)) three parameter sets are considered as listed inTab. 2. Set 1 considers only axial coupling of the fibres with the matrix throughparameters b f and b s , respectively, but no shear coupling setting b fs = 0whereas Set 2 includes shear coupling as well setting b fs = 5 × . Set 3 con-siders a very soft linkage of matrix and fibres. For all three sets, the parameterchoice facilitates that the preferred material direction, ˜ V f , is by one order ofmagnitude stiffer than the other one, ˜ V s . As only a qualitative comparison15etween the conventional and micromorphic model is undertaken, the param-eters of both models have not been calibrated with each other. The matrixparameters of the micromorphic model, a and a , respectively, and the fibrestiffness parameters, c f and c s , respectively, are chosen to be identical to thecorresponding parameters of the classical model, ¯ a , ¯ a , ¯ a F and ¯ a S , respec-tively. However, as previously mentioned, the effect of fibre stiffness on the bulkmaterial response depends on bond stiffness parameters.As the classical and micromorphic models are not calibrated together, it isnot possible to quantify the actual difference between affine deforming fibres ofthe classical approach and the non-affine deforming fibres in the micromorphicapproach magnitude. However, the discrepancy of between affine and non-affinefibre deformation (Eqs. (12) and (13)) can be determined for the micromorphicapproach alone, that is the relative motion between ˜ v F and a f as well as ˜ v S and a s . The axial motion between matrix and fibres is given by the differencein stretch using Eqs. (26), (27), (49) and (50) as follows J F f = I F − L f (53) J Ss = I S − L s . (54)For the change of angle , α , between matrix and fibre, we make use of the scalarproduct of affine and non-affine deformed fibres and normalise the result toexclusively obtain the change of angle: J F f = 1 − cos α F f = 1 − (˜ v F · a f ) (2 I F + 1)(2 L f + 1) (55) J Ss = 1 − cos α Ss = 1 − (˜ v S · a s ) (2 I S + 1)(2 L s + 1) . (56)Consequently, α F f = sin − (cid:112) J F f and α Ss = sin − √ J Ss . Note, J F f , J F f , J F f and J Ss could be associated with penalty parameters to establish thetransition to affine anisotropy, namely a f → ˜ v F and a s → ˜ v S , respectively. Inorder to evaluate the matrix stretch along the non-affine deforming fibres forthe micromorphic model, we introduce the following quantities Z F = ˜ E (0) : ( ˜ Z F ⊗ ˜ Z F ) (57) Z S = ˜ E (0) : ( ˜ Z S ⊗ ˜ Z S ) , (58)where the normalised pull-back of the non-affine deforming preferred materialdirections are given as˜ Z F = ( ˜ F (0) ) − a f | ˜ F (0) ) − a f | and ˜ Z S = ( ˜ F (0) ) − a s | ˜ F (0) ) − a s | (59)respectively.Three case studies are investigated in the following subsections: (1) a plateunder uniaxial tension, (2) a plate under biaxial tension with fixed transverse16ontraction, and (3) a plate with a circular hole under biaxial tension. Forthe numerical simulations, both, the classical and the micrormorphic approach,are implemented in an in-house C++ code using standard linear hexahedralfinite elements. As the examples are effectively two-dimensional problems, thedisplacement degree of freedom, u = 0 and the change of director degrees offreedom, w f = 0 and w s = 0, respectively, have been enforced throughout theproblem domain for all three examples. Furthermore, the discretisation acrossthe thickness dimension of the plates consists of only one element. This meansthat the normal strain in plate thickness direction is zero and the two directorscan only deform parallel to the plate plane. Figure 4:
Fibre-reinforced plate subjected to uniaxial tension.
The first example is a fibre-reinforced plate with a thickness of 0.1 subjectedto uniaxial tension in horizontal direction as shown in Fig. 4. The plate issimply supported on the left edge preventing horizontal displacement but notconstraining the transverse contraction of the plate. There are two uniformlydistributed fibre families, ˜ V f and ˜ V s , under 30 ° and − ° to the horizontal direc-tion, respectively. The uniaxial tension is applied via a displacement boundarycondition, ¯ u = 1 .
3, along the right edge.If also an isotropic case were considered, this problem would warrant a homo-geneous deformation response in terms uniaxial horizontal stretch of the plate.Choosing two preferred material directions with one significantly stiffer thanthe other one and both being neither parallel nor perpendicular to the horizon-tal direction results in a downwards deflection and a generally highly nonlineardeformation response for the classical formulation as shown in the contour plotsof the matrix strain in fibre direction Fig. 5. In contrast, the micromorphicnon-affine anisotropy provides a strictly linear deformation behaviour with asignificantly lower maximum magnitude of the matrix strain as both non-affinedeforming fibre families are free to uniformly re-align themselves relative to theimposed horizontal matrix stretch as illustrated in Fig. 6. When comparing the17igure 5:
Contour plots of the matrix stretch along the fibre directions, classicanisotropy with Set 0 (left), micromorphic non-affine anisotropy with Set 1 facilitatinga strong bond axially (right). The first row of tiles depicts I F (Eq. (49)) and Z F (Eq. (57)) and the second row of tiles I S (Eq. (50)) and Z S (Eq. (58)), respectively. orientation of both fibre families with the principal direction of the Almansi strain tensor shown in Fig. 7, it can be clearly seen that both fibre families areparallel to the longitudinal principal strain direction which is not the case forthe classical model.If the fibre orientations are fixed on both ends of the plate, we need to applycorresponding change of director boundary conditions, w f = w s = . Theresulting deformation response for the micromorphic model then becomes highlynon-linear as shown in Fig. 8, because the free motion capability of the fibresrelative to the matrix is significantly constrained at the regions close to boundarycondition application. Further inwards, however, both fibre families are againperfectly aligned with the principal strain direction along the longitudinal axisof the plate.Inclusion of matrix-fibre shear interaction using Set 2 in Tab. 2 also leadsto homogeneous deformation response, but the deformed fibre directions do notrealign themselves as strongly with the longitudinal plate direction as depicted18n Fig. 9. As such the plate remains stiffer in transverse direction so that thetransverse contraction is not as large as for Set 1. The additional shear couplingdoes not only affect relative rotational matrix fibre motion but also betweenthe directors themselves and thus, constrains the rotation of both directorsindependent from each other. Even though, the fibre orientation differencebetween classical and micromorphic model is in average not as significant ascompared with Set 1, the homogenisation effect of the micromorphic approachis retained. This implies that the freedom of relative axial matrix-fibre motionis here the dominant aspect with regards to the aforementioned homogenisationproperty of the micromorphic anisotropy model.Considering a weak matrix-fibre bond in the micromorphic model using Set3 in Tab. 2 results in less fibre realignment as shown in Fig. 10 as comparedwith that using a strong bond depicted in Fig. 6. A strong matrix-fibre bondgives with J F f = 0 . J Ss = 0 . J F f = 0 . J Ss = 0 . V f amounts to two orders of magnitude. The second example is a fibre-reinforced plate with a thickness of 37.5 subjectedto biaxial tension in horizontal and vertical directions as shown in Fig. 11. Thistime the two uniformly distributed fibre families, ˜ V f and ˜ V s , have angles of − ° and 60 ° to the horizontal direction, respectively. The biaxial tension isinduced via displacement boundary conditions, ¯ u = 300, along the two verticaledges whereas the height is kept constant applying along the horizontal edges¯ u = 0.The deformation response of the biaxially stressed plate is inhomogeneousand non-linear for the classical anisotropy model using material parameter Set0 and also the micromorphic non-affine anisotropy model using Set 1 as shownin Fig. 12 illustrating the matrix strain field along the two fibre directions inboth cases. There are highly localised strain magnitudes in the plate’s cornersfor classical model but to a much lesser degree for the micromorphic model.The strain localisation effects are mirrored by the fibre orientation distributionsdepicted in Fig. 13 which are perturbed in the plate’s corners but again consid-erably less for the micromorphic model. The orientation of both fibre familiesgenerally tend to align themselves with the horizontal maximum principal straindirection but to a significantly larger degree for the micromorphic model as il-lustrated in Fig. 13. Accordingly, fibre family ˜ V F / ˜ V f rotates counter-clockwiseand ˜ V S / ˜ V s clockwise as both stretch and the resulting vector fields representingthe change of length and orientation have a downwards and upwards compo-nent, respectively, as shown in Fig. 14. Even though the strain magnitudes arenot directly comparable as the classical and micromorphic model have not been19alibrated with each other, clearly, the non-affine fibre stretch and realignmentexhibited for the micromorphic anisotropy model has a homogenising effect.A strong bond stiffness exceeding the fibre stiffness in the micromorphicmodel results in the fibre stretch being significantly larger than the stretch ofthe matrix along fibre direction as shown in Fig. 15 which effectively increasesthe capacity of the former to align with the horizontal maximum principal straindirection.The incorporation of anisotropic shear stiffness, ¯ a fs , in the classical anisotropymodel (Eq. (52)) further increases strain localisation as illustrated in Fig. 16,whereas the use of b fs in the non-affine micromorphic anisotropy model (Eq. (21))decreases the stretch and the rotation of the fibres when compared with Fig. 13.Choosing relatively small values for the matrix-fibre interaction parameters b f , b s and b fs , respectively, in the micromorphic model using Set 3 in Tab. 2significantly reduces the fibre’s horizontal realignment and stretching with themaximum principal strain direction as shown in Fig. 17. Furthermore, thediscrepancy in terms of rotational deformation between affine and non-affinedeforming fibres quantified via Eqs. (55) and (56), respectively is less as depictedin Fig. 18. In particular, ˜ V s rotates considerably more for the stiffer matrix-fibrebond as shown in the bottom right tile in Fig. 18. The third example is a fibre-reinforced plate with a centered circular hole and athickness of 37.5 subjected to biaxial tension in horizontal and vertical directionsas shown in Fig. 19. The two uniformly distributed fibre families, ˜ V f and ˜ V s ,are now oriented 45 ° and 135 ° to the horizontal direction, respectively. Thetension is applied via outward displacement boundary condition, ¯ u = 2, alongall edges.This example has been specifically chosen as the hole introduces a strongnon-linear deformation response under biaxial tension loading which can beclearly seen for the matrix strain field along the two fibre families as displayedin Fig. 20 for the classical approach and the non-affine micromorphic approachwith Set 1. As for the previous example, the micromorphic approach exhibitsa homogenising property distributing and reducing the localised strain maximaoccurring at the hole. This is also reflected by the deformed fibre distributionswhere the non-affine deformation leads to a better ”force flow” around the holedue to the stronger realignment of the fibres to become tangential with thehole’s circumference as shown in Fig. 21. As a result, the maximum principaldirection of the Almansi strain tensor follows the circumference of the hole andthe minimum principal strain direction is strictly radial as displayed in Fig. 22.For the classical approach, in contrast, the principal strain directions tend toremain aligned with the undeformed preferred material directions, especially inthe corners of the plate.Lastly, we consider a weak matrix-fibre bond in the micromorphic modelassigning small values to parameters b f , b s and b fs , respectively, which is Set3 in Tab. 2. For this case, the tangential alignment of the fibres is not as signif-20cant as for the strong bond when comparing Fig. 23 with the previously shownFig. 21. This characteristic can also be verified by the location and magnitudeof non-affine rotational deformation. In order to obtain a fibre orientation whichis tangentially aligned with the hole, the most significant change of direction hasto occur at the four diagonal points. This is where the relative rotational defor-mation has maxima in the corresponding contour plots shown in the two righttiles in Fig. 24. For the weaker bond, the magnitudes of non-affine rotationalmotion is one order of magnitude less and occurs not at the diagonal points. In biological soft tissue, collagen fibres play an important structural load bearingrole and have been found to exhibit non-affine elastic reorientation in directionof the maximum principal strain, e.g.
Billiar and Sacks [6] and
Krasny etal. [35]. In order to model this type of material behaviour, a microstructurally-based continuum mechanics framework needs to provide the following:1. composite kinematics describing independently the motion of fibres andbulk material such that relative motion between both constituents can beunambiguously captured;2. elastic constitutive relations separately describing the response of the bulkmaterial, the fibres and their interaction;3. dedicated governing equations for the fields representing the motion ofbulk material and fibres, respectively.The proposed local micromorphic non-affine anisotropy framework in Sec. 2meets these requirements. It features to independent primary fields, the dis-placement field representing the kinematics of the bulk material (the matrix)and the director fields representing the kinematics of the fibres. The preferredmaterial directions of the anisotropic constitutive laws are linked to the non-affine deforming fibres and the degree of relative matrix-fibre deformation iscontrolled by the dedicated bond material parameters, b f , b s and b fs , inEq. (31) as linked to axial and shear motion, respectively. It has been shownfor all three numerical examples in the previous section that, besides the fibrestiffness constants in Eq. (32), c f and c s , respectively, the magnitude of thebond parameters determine how much load the fibres attract and thus, directlyrelate to amount of interaction experienced by matrix and fibres, respectively. Inparticular, under uniaxial tension the discrepancy between affine and non-affinedeforming fibre is largest for the stronger fibre family.The micromorphic variational principle (Eq. (41)) features besides the vir-tual power of matrix and fibres additional contributions relating to the non-affinematrix-fibre interaction. As a consequence, the governing equations (Eqs. (45)-(47)) also include the linkage of the stress response referring to matrix, fibreand their bond. In particular, each director field is associated with a dedicated21overning equation establishing force equilibrium between fibre and matrix-fibreinteraction stresses (Eqs. (46) and (47), respectively). The governing equationof the displacement, on the other hand, ensures that stress fluctuation of thematrix translate into stress response of the fibres giving rise to a strong ho-mogenising effect of the approach which has been clearly demonstrated in allthree numerical studies. As the two primary fields are independent from eachother, a unique solution for both is the result of the stationarity principle of thetotal energy which ultimately governs the resulting fibre reorientation distribu-tion.Specially, it has been shown that both fibre families undergo significant non-affine deformation in terms of relative axial and rotational matrix-fibre motionsuch that they seem to align themselves with the maximum principal strain di-rection. This behaviour was also observed for the classical anisotropy approachbut not to such an extent because of the affine nature of its underlying kine-matics. It was found that especially the non-affine axial motion has a dominantinfluence on the degree of fibre reorientation. Large magnitudes of the axialinteraction parameters, b f and b s , respectively, translated the imposed matrixstrain into a fibre stretch significantly exceeding the matrix stretch resulting ina better alignment of the fibres with the maximum principal strain direction.This can be explained considering that the micromorphic director, a f , is ba-sically a differential line element deforming within a corresponding differentialarea element, dS , via ˜ F (1) (Eq. 13), as shown in Fig. 25 for horizontal stretch.For the director to become horizontally aligned, a large stretch would be neces-sary which would lead to an equally large and unrealistically matrix deformationfor an affine approach. The second numerical study relates to the biaxial ten-sion experiment of pericardial tissue by Billiar and Sacks in terms of similarloading and geometry. The question therefore arises whether the almost perfectalignment of the collagen fibres with the maximum principal strain directionin the experiment is due to their uncrimping. In this case, the kinematics ofthe collagen fibre and the micromorphic director exhibiting significantly largestretch than the matrix could be indeed assumed similar in the mathematicallimit.Contrary to the isotropic case, the principal strain and stress directions arenot coaxial for the general anisotropic case. From the examples, it was observedthat the fibres have a tendency to align themselves with the direction of theprincipal strain and reduce the degree of anisotropic material behaviour. Thisseems to imply that principal stress and strain directions are driven to becomecoaxial in the limit. However, this warrants further investigation.Incorporating explicitly a shear bond between matrix and fibre via parameter b fs in Eq. (31) decreases the magnitude of non-affine stretch and the rotationexperienced by the fibres which is explained by the consideration that this shearparameter also establishes a direct coupling between the two fibre families whichis otherwise only indirectly given via the matrix. Also, the consideration ofboundary conditions fixing the director orientations at both ends of the platesubjected to uniaxial tension significantly reduce the fibre reorientation but onlynear the boundary. 22astly, we note that the initial derivation of the micromorphic approachproposed here also includes higher-order contributions which would be neededto address non-local scale-dependent phenomena of matrix-fibre interaction, e.g.inter-fibre sliding of tendon fascicles [47], as well as fibre dispersion phenomena. The local micromorphic non-affine anisotropy framework introduced in this pa-per was shown to naturally provide the flexibility to deal with characteristickinematics aspects of fibrous composite material concerning the relative elas-tic motion of fibres within the bulk material. The approach is complementedwith suitable constitutive relations making use of physically motivated param-eters linked to matrix-fibre bond and fibre stiffness. An extended variationalprinciple provides the means to solve for the complex composite deformationresponse.In contrast to other models, the approach considers truly non-affine deform-ing primary fields within an exclusively elastic setting which is a behaviouralso experienced by biological soft tissue. In particular, the non-affine deform-ing fibres representing the preferred material directions profoundly influence theanisotropic material response for larger strains which optimises and homogenisesthe resulting deformation. The latter can be expected from biological tissues.The application of the framework to soft tissue, however, will require to castthe constitutive relation into a Fung-type exponential form.In the absence of quantifiable evidence of a non-local microstructural mate-rial response, all higher-order contributions of conventional micromorphic theo-ries have been disregarded. If such information becomes available, e.g. withregards to scale-dependent matrix-fibre bond or fibre dispersion effects, theframework provides the flexibility to consider the needed higher-order non-localcontributions as well.
Acknowledgement
This research has been supported by the National Research Foundation of SouthAfrica (Grant Numbers 104839 and 105858). Opinions expressed and conclu-sions arrived at, are those of the author and are not necessarily to be attributedto the NRF.
References [1] M Agoras, Oscar Lopez-Pamies, and P Ponte Casta˜neda. A general hyper-elastic model for incompressible fiber-reinforced elastomers.
Journal of theMechanics and Physics of Solids , 57(2):268–286, 2009.232] Farid Alisafaei, Xingyu Chen, Thomas Leahy, Paul A Janmey, and Vivek BShenoy. Long-range mechanical signaling in biological systems.
Soft matter ,2020.[3] Gabriele Barbagallo, Angela Madeo, Marco Valerio d’Agostino, RafaelAbreu, Ionel-Dumitrel Ghiba, and Patrizio Neff. Transparent anisotropyfor the relaxed micromorphic model: macroscopic consistency conditionsand long wave length asymptotics.
International Journal of Solids andStructures , 120:7–30, 2017.[4] Arkadi Berezovski, J¨uri Engelbrecht, and G´erard A Maugin. Generalizedthermomechanics with dual internal variables.
Archive of Applied Mechan-ics , 81(2):229–240, 2011.[5] Arkadi Berezovski, M Erden Yildizdag, and Daria Scerrato. On the wavedispersion in microstructured solids.
Continuum Mechanics and Thermo-dynamics , 32(3):569–588, 2020.[6] KL Billiar and MS Sacks. A method to quantify the fiber kinematics ofplanar tissues under biaxial stretch.
Journal of biomechanics , 30(7):753–756, 1997.[7] Raja Biswas, Leong Hien Poh, and Amit Subhash Shedbale. A micro-morphic computational homogenization framework for auxetic tetra-chiralstructures.
Journal of the Mechanics and Physics of Solids , 135:103801,2020.[8] Kelly K Brewer, Hiroaki Sakai, Adriano M Alencar, Arnab Majumdar,Stephen P Arold, Kenneth R Lutchen, Edward P Ingenito, and B´ela Suki.Lung and alveolar wall elastic and hysteretic behavior in rats: effects of invivo elastase treatment.
Journal of Applied Physiology , 95(5):1926–1936,2003.[9] Gianfranco Capriz. Continua with latent microstructure.
Archive for Ra-tional Mechanics and Analysis , 90:43–56, 1985.[10] Preethi L Chandran and Victor H Barocas. Affine versus non-affine fibrilkinematics in collagen networks: theoretical studies of network behavior.2006.[11] Huan Chen, Yi Liu, Xuefeng Zhao, Yoram Lanir, and Ghassan S Kassab.A micromechanics finite-strain constitutive model of fibrous tissue.
Journalof the Mechanics and Physics of Solids , 59(9):1823–1837, 2011.[12] Huan Chen, Xuefeng Zhao, Xiao Lu, and Ghassan Kassab. Non-linear mi-cromechanics of soft tissues.
International journal of non-linear mechanics ,56:79–85, 2013. 2413] Paul H Chew, Frank CP Yin, and Scott L Zeger. Biaxial stress-strain prop-erties of canine pericardium.
Journal of molecular and cellular cardiology ,18(6):567–578, 1986.[14] Eugene Cosserat and Fran¸cois Cosserat.
Th´eorie des corps d´eformables . A.Hermann et fils, 1909.[15] Fabrizio Dav´ı. Wave propagation in micromorphic anisotropic continuawith an application to tetragonal crystals.
Mathematics and Mechanics ofSolids , page 1081286520971840, 2020.[16] Niels JB Driessen, Martijn AJ Cox, Carlijn VC Bouten, and Frank PTBaaijens. Remodelling of the angular collagen fiber distribution in cardio-vascular tissues.
Biomechanics and modeling in mechanobiology , 7(2):93,2008.[17] Wolfgang Ehlers and Sami Bidier. From particle mechanics to micromor-phic media. part i: Homogenisation of discrete interactions towards stressquantities.
International Journal of Solids and Structures , 187:23–37, 2020.[18] A Cemal Eringen and ES Suhubi. Nonlinear theory of simple micro-elasticsolids—i.
International Journal of Engineering Science , 2(2):189–203, 1964.[19] Helen M Finlay, Peter Whittaker, and Peter B Canham. Collagen orga-nization in the branching region of human brain arteries.
Stroke , 29(8):1595–1601, 1998.[20] Samuel Forest. Nonlinear regularization operators as derived from themicromorphic approach to gradient elasticity, viscoplasticity and damage.
Proceedings of the Royal Society A: Mathematical, Physical and Engineer-ing Sciences , 472(2188):20150755, 2016.[21] Samuel Forest and Rainer Sievert. Nonlinear microstrain theories.
Inter-national Journal of Solids and Structures , 43(24):7224–7245, 2006.[22] YC Fung and Richard Skalak. Biomechanics: mechanical properties ofliving tissues. 1981.[23] T Christian Gasser, Ray W Ogden, and Gerhard A Holzapfel. Hyperelas-tic modelling of arterial layers with distributed collagen fibre orientations.
Journal of the royal society interface , 3(6):15–35, 2006.[24] Ivan Giorgio, Francesco dell’Isola, and Anil Misra. Chirality in 2d cosseratmedia related to stretch-micro-rotation coupling with links to granular mi-cromechanics.
International Journal of Solids and Structures , 2020.[25] Chin-Lin Guo, Nolan C Harris, Sithara S Wijeratne, Eric W Frey, andChing-Hwa Kiang. Multiscale mechanobiology: mechanics at the molecular,cellular, and tissue levels.
Cell & bioscience , 3(1):25, 2013.2526] Yu Long Han, Pierre Ronceray, Guoqiang Xu, Andrea Malandrino, Roger DKamm, Martin Lenz, Chase P Broedersz, and Ming Guo. Cell contractioninduces long-ranged stress stiffening in the extracellular matrix.
Proceedingsof the National Academy of Sciences , 115(16):4075–4080, 2018.[27] G Himpel, A Menzel, E Kuhl, and P Steinmann. Time-dependent fibrereorientation of transversely isotropic continua—finite element formulationand consistent linearization.
International journal for numerical methodsin engineering , 73(10):1413–1433, 2008.[28] Gerhard A Holzapfel and Thomas C Gasser. A viscoelastic model for fiber-reinforced composites at finite strains: Continuum basis, computationalaspects and applications.
Computer methods in applied mechanics andengineering , 190(34):4379–4403, 2001.[29] Geralf H¨utter. Homogenization of a cauchy continuum towards a micro-morphic continuum.
Journal of the Mechanics and Physics of Solids , 99:394–408, 2017.[30] Christopher R Jacobs, Sara Temiyasathit, and Alesha B Castillo. Osteocytemechanobiology and pericellular mechanics.
Annual review of biomedicalengineering , 12:369–400, 2010.[31] Mohammadjavad Javadi, Marcelo Epstein, and Mohsen Asghari. Thermo-mechanics of material growth and remodeling in uniform bodies based onthe micromorphic theory.
Journal of the Mechanics and Physics of Solids ,138:103904, 2020.[32] CB Kafadar and A Cemal Eringen. Micropolar media—i the classical the-ory.
International Journal of Engineering Science , 9(3):271–305, 1971.[33] Igor Karˇsaj, Carlo Sansour, and Jurica Sori´c. The modelling of fibre reori-entation in soft tissue.
Biomechanics and modeling in mechanobiology , 8(5):359–370, 2009.[34] Sergei Khakalo and Jarkko Niiranen. Anisotropic strain gradient thermoe-lasticity for cellular structures: Plate models, homogenization and isoge-ometric analysis.
Journal of the Mechanics and Physics of Solids , 134:103728, 2020.[35] Witold Krasny, Claire Morin, H´el`ene Magoariec, and St´ephane Avril. Acomprehensive study of layer-specific morphological changes in the mi-crostructure of carotid arteries under uniaxial load.
Acta Biomaterialia ,57:342–351, 2017. ISSN 18787568. doi: 10.1016/j.actbio.2017.04.033. URL http://dx.doi.org/10.1016/j.actbio.2017.04.033 .[36] Hugo Krynauw, Rodaina Omar, Josepha Koehne, Georges Limbert, Neil HDavies, Deon Bezuidenhout, and Thomas Franz. Electrospun polyester-urethane scaffold preserves mechanical properties and exhibits strain stiff-ening during in situ tissue ingrowth and degradation.
SN Applied Sciences ,2(5):1–12, 2020. 2637] Georges Limbert and Mark Taylor. On the constitutive modeling of bio-logical soft connective tissues: a general theoretical framework and explicitforms of the tensors of elasticity for strongly anisotropic continuum fiber-reinforced composites at finite strain.
International Journal of Solids andStructures , 39(8):2343–2358, 2002.[38] Michele Marino and Peter Wriggers. Micro–macro constitutive modelingand finite element analytical-based formulations for fibrous materials: Amultiscale structural approach for crimped fibers.
Computer Methods inApplied Mechanics and Engineering , 344:938–969, 2019.[39] RD Mindlin. Micro-structure in linear elasticity.
Rational Mechanics andAnalysis , 6:51–78, 1964.[40] H Moosavian and HM Shodja. Mindlin–eringen anisotropic micromorphicelasticity and lattice dynamics representation.
Philosophical Magazine , 100(2):157–193, 2020.[41] Claire Morin, St´ephane Avril, and Christian Hellmich. Non-affine fiberkinematics in arterial mechanics: a continuum micromechanical investiga-tion.
ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschriftf¨ur Angewandte Mathematik und Mechanik , 98(12):2101–2121, 2018.[42] Patrizio Neff, Bernhard Eidel, Marco Valerio d’Agostino, and AngelaMadeo. Identification of scale-independent material parameters in the re-laxed micromorphic model through model-adapted first order homogeniza-tion.
Journal of Elasticity , pages 1–30, 2019.[43] Dame Julia Polak. Regenerative medicine. opportunities and challenges:a brief overview.
Journal of the Royal Society Interface , 7(suppl 6):S777–S781, 2010.[44] Arun Raina and Christian Linder. A homogenization approach for nonwo-ven materials based on fiber undulations and reorientation.
Journal of theMechanics and Physics of Solids , 65:12–34, 2014.[45] O Rokoˇs, MM Ameen, RHJ Peerlings, and MGD Geers. Extended mi-cromorphic computational homogenization for mechanical metamaterialsexhibiting multiple geometric pattern transformations.
Extreme mechanicsletters , page 100708, 2020.[46] Carlo Sansour, Sebastian Skatulla, and H Zbib. A formulation for themicromorphic continuum at finite inelastic strains.
International Journalof Solids and Structures , 47(11-12):1546–1554, 2010.[47] HRC Screen, DA Lee, DL Bader, and JC Shelton. An investigation into theeffects of the hierarchical structure of tendon fascicles on micromechanicalproperties.
Proceedings of the Institution of Mechanical Engineers, Part H:Journal of Engineering in Medicine , 218(2):109–119, 2004.2748] Mohamed Shaat. A reduced micromorphic model for multiscale materialsand its applications in wave propagation.
Composite Structures , 201:446–454, 2018.[49] S Skatulla and C Sansour. A formulation of a cosserat-like continuum withmultiple scale effects.
Computational materials science , 67:113–122, 2013.[50] Ashwin Sridhar, Varvara G Kouznetsova, and Marc GD Geers. Homog-enization of locally resonant acoustic metamaterials towards an emergentenriched continuum.
Computational mechanics , 57(3):423–435, 2016.[51] Holger Steeb and Stefan Diebels. Modeling thin films applying an ex-tended continuum theory based on a scalar-valued order parameter.: Parti: isothermal case.
International Journal of Solids and Structures , 41(18-19):5071–5085, 2004.[52] Triantafyllos Stylianopoulos and Victor H Barocas. Multiscale, structure-based modeling for the elastic mechanical behavior of arterial walls.
Journalof Biomechanical Engineering , 129:611–618, 2007.[53] Richard A Toupin. Theories of elasticity with couple-stress.
Archive forRational Mechanics and Analysis , 17:85–112, 1964.[54] Theodore T Tower, Michael R Neidert, and Robert T Tranquillo. Fiberalignment imaging during mechanical testing of soft tissues.
Annals ofbiomedical engineering , 30(10):1221–1233, 2002.[55] Markus von Hoegen, Sebastian Skatulla, and J¨org Schr¨oder. A generalizedmicromorphic approach accounting for variation and dispersion of preferredmaterial directions.
Computers & Structures , 232:105888, 2020.[56] Jeffrey A Weiss, Bradley N Maker, and Sanjay Govindjee. Finite ele-ment implementation of incompressible, transversely isotropic hyperelas-ticity.
Computer methods in applied mechanics and engineering , 135(1-2):107–128, 1996.[57] Qi Wen, Anindita Basu, Paul A Janmey, and Arjun G Yodh. Non-affinedeformations in polymer hydrogels.
Soft matter , 8(31):8039–8049, 2012.[58] Chenxi Xiu, Xihua Chu, Jiao Wang, Wenping Wu, and Qinglin Duan.A micromechanics-based micromorphic model for granular materials andprediction on dispersion behaviors.
Granular Matter , 22(4):1–22, 2020.[59] Vahhab Zarei, Sijia Zhang, Beth A Winkelstein, and Victor H Barocas. Tis-sue loading and microstructure regulate the deformation of embedded nervefibres: predictions from single-scale and multiscale simulations.
Journal ofThe Royal Society Interface , 14(135):20170326, 2017.28igure 6:
Vector plots of fibre/director fields, classic anisotropy with Set 0 (left), mi-cromorphic non-affine anisotropy with Set 1 facilitating a strong bond axially (right).The first row of tiles illustrates in the deformed configuration the fibre field ˜ v F anddirector field a f , respectively, and the second row of tiles the fibre field ˜ v S and di-rector field a s , respectively. V f , V s , A f and A s indicate the orientation of thefibres/directors in their undeformed state. Vector plots of the in-plane longitudinal principal direction field of the ma-trix
Almansi strain tensor (Eq. (17)), classic anisotropy with Set 0 (left), micromorphicnon-affine anisotropy with Set 1 (right).
Figure 8:
Vector plots of the deformed director fields, a f (left) and a s (right), re-spectively, of micromorphic non-affine anisotropy using Set 1 with both director fieldsfixed on either end of the plate. Vector plots of fibre/director fields, classic anisotropy with Set 0 (left),micromorphic non-affine anisotropy with Set 2 (right) facilitating a strong bond axiallyand rotationally. The first row of tiles illustrates in the deformed configuration thefibre field ˜ v F and director field a f , respectively, and the second row of tiles the fibrefield ˜ v S and director field a s , respectively. Vector plots of the deformed director fields, a f (left) and a s (right),respectively, of micromorphic non-affine anisotropy with Set 3 facilitating a weak bondaxially. Figure 11:
Fibre-reinforced plate subjected to biaxial tension.
Contour plots of the matrix stretch along the fibre directions, classicanisotropy with Set 0 (left), micromorphic non-affine anisotropy with Set 1 facilitatinga strong bond axially (right). The first row of tiles depicts I F (Eq. (49)) and Z F (Eq. (57)) and the second row of tiles I S (Eq. (50)) and Z S (Eq. (58)), respectively. Vector plots of fibre/director fields, classic anisotropy with Set 0 (left), mi-cromorphic non-affine anisotropy with Set 1 facilitating a strong bond axially (right).The first row of tiles illustrates in the deformed configuration the fibre field ˜ v F anddirector field a f , respectively, and the second row of tiles the fibre field ˜ v S and di-rector field a s , respectively. V f , V s , A f and A s indicate the orientation of thefibres/directors in their undeformed state. Vector plots of the change of fibre/director fields, classic anisotropy withSet 0 (left), micromorphic non-affine anisotropy with Set 1 facilitating a strong bondaxially (right). The first row of tiles illustrates in the deformed configuration the fibrefield change ˜ v F − ˜ V F and director field change w f , respectively, and the second rowof tiles ˜ v S − ˜ V S and w s , respectively. Contour plots of matrix stretch along fibre-direction, Z F (Eq. (57)) and Z S (Eq. (58)), respectively (left) and fibre stretch, L f (Eq. (26)) and L s (Eq. (27)),respectively (right) using micromorphic non-affine anisotropy with Set 1 facilitating astrong bond axially. Vector plots of fibre/director fields, classic anisotropy with Set 0 (left),micromorphic non-affine anisotropy with Set 2 facilitating a strong bond axially androtationally (right). The first row of tiles illustrates in the deformed configuration thefibre field ˜ v F and director field a f , respectively, and the second row of tiles the fibrefield ˜ v S and director field a s , respectively. Vector plots of the deformed director fields, a f and a s , respectively (left)and change of director fields, w f and w s , respectively (right) using micromorphicnon-affine anisotropy with Set 3 facilitating a weak bond axially. Contour plots of relative rotational matrix-fibre deformation using micro-morphic non-affine anisotropy with weak bond stiffness Set 3 (left) and with strongbond stiffness Set 1 (right). The first row of tiles shows J Ff (Eq. (55)) and the secondrow J Ss (56)). Fibre-reinforced plate with a hole subjected to biaxial tension.
Contour plots of the matrix stretch along the two fibre directions, classicanisotropy with Set 0 (left), micromorphic non-affine anisotropy with Set 1 facilitatinga strong bond axially (right). The first row of tiles depicts I F (Eq. (49)) and Z F (Eq. (57)) and the second row I S (Eq. (50)) and Z S (Eq. (58)), respectively. Vector plots of fibre/director fields, classic anisotropy with Set 0 (left),micromorphic non-affine anisotropy with Set 1 facilitating a strong bond axially (right).The first row illustrates in the deformed configuration the fibre field ˜ v F and directorfield a f , respectively, and the second row the fibre field ˜ v S and director field a s ,respectively. V f , V s , A f and A s indicate the orientation of the fibres/directors intheir undeformed state. Vector plots of the in-plane principal directions of the matrix
Almansi strain tensor (Eq. (17)), classic anisotropy with Set 0 (left), micromorphic non-affineanisotropy with Set 1 facilitating a strong bond axially (right).
Vector plots of the deformed director fields, a f (left) and a s (right),respectively, using micromorphic non-affine anisotropy with Set 3 facilitating a weakbond axially. Contour plots of relative rotational matrix-fibre deformation using micro-morphic non-affine anisotropy with weak bond stiffness Set 3 (left) and with strongbond stiffness Set 1 (right). The first row of tiles shows J Ff (Eq. (55)) and the secondrow J Ss (56)). Schematic deformation mapping of the director, a f , within a differentialarea element, ds , in terms of stretch through ˜ F (1) (Eq. (16)) alone.(Eq. (16)) alone.