Application of anisotropic visco-hyperelastic constitutive models for the simulation of biological tissues

(1)

Faculdade de Engenharia da Universidade do Porto

Application of anisotropic visco-hyperelastic

constitutive models for the simulation of

biological tissues

Integrated Master in Mechanical Engineering

Dissertation Thesis

Author

Maria Castro Paupério Vila Pouca

Supervisors

Prof. Dr. Marco Paulo L. Parente

Prof. Dr. Renato Natal Jorge

(2)

(3)

Application of anisotropic visco-hyperelastic

constitutive models for the simulation of

biological tissues

Maria Castro Paupério Vila Pouca

Mestrado Integrado em Engenharia Mecânica

(4)

(5)

Abstract

Computational bioengineering is experiencing major advances in the past decades, taking advantage of the latest, fast-developing computational capabilities. It employs finite element modelling to determine the effects of mechanical stresses and strains, as well as the interaction of different components and tissues, on clinical outcomes. This method has become a valuable alternative to in-vivo assessment since it has no ethical constraints.

The mechanical properties of the modelled tissues constitute a major role to achieve realistic and accurate simulations. However, tissues present an anisotropic visco-hyperelastic behaviour, which can be difficult to reproduce through numerical simulations, despite the constitutive models that have been developed over the years. Highlighting viscoelasticity, its consideration involves demanding numerical calculations since it is a time-dependent property, which consequently causes its disregard in many situations.

The aim of this work is to implement a viscoelastic constitutive model in order to account for this effect in finite element simulations involving soft tissues, using the ABAQUS®

software. A user-defined material subroutine UMAT prepared for anisotropic hyperelastic materials was previously developed by Ferreira, J., et al 2017 [1] and served as a basis for this work. Viscoelasticity was added to the developed UMAT through a recurrence update algorithm in the FORTRAN language. The constitutive model applied was the generalized Maxwell model, following the nomenclature of Holzapfel, G.A. [2].

The efficiency of the defined model is evaluated through numerical examples and general conclusions are outlined.

(6)

(7)

Resumo

A bioengenharia computacional sofreu grandes avanços ao longo das últimas décadas, acompanhando as mais recentes capacidades computacionais que têm sido desenvolvidas cada vez mais rapidamente. Envolve modelação através de elementos finitos para determinar os efeitos das tensões e deformações, assim como das interações entre os diferentes componentes e tecidos, em resultados clínicos. Este método tem-se verificado uma alternativa vantajosa a avaliações in-vivo, que se encontram frequentemente limitadas devido a restrições éticas.

A definição das propriedades mecânicas dos tecidos modelados tem um papel fundamental na obtenção de simulações exatas e realistas. No entanto, os tecidos apresentam um comportamento anisotrópico e visco-hiperelástico, o que é difícil de reproduzir, apesar dos modelos constitutivos que têm vindo a ser desenvolvidos ao longo dos anos. Evidenciando a viscoelasticidade, a sua consideração envolve cálculos numéricos exigentes dado que é uma propriedade dependente do tempo, o que faz com que seja muitas vezes desprezada, independentemente da sua importância para a simulação.

O objetivo deste trabalho é implementar um modelo constitutivo viscoelástico de forma a considerar esta propriedade em simulações numéricas envolvendo tecidos biológicos, através do software ABAQUS®_{. Uma subrotina preparada para analisar materiais}

hiperelásticos anisotrópicos (UMAT) foi desenvolvida anteriormente por Ferreira, J., et al 2017 [1] e serviu de base para este trabalho. A viscoelasticidade foi adicionada através de um algoritmo de atualização de recorrência na linguagem FORTRAN. O modelo constitutivo aplicado foi o modelo generalizado de Maxwell, seguindo a nomenclatura definida por Holzapfel, G.A. [2].

A validação e eficiência do modelo proposto é avaliada através de vários exemplos numéricos.

(8)

(9)

Agradecimentos

Ao meu orientador, Doutor Marco Parente, agradeço a oportunidade e a confiança depositada para desenvolver este trabalho, incentivando e encorajando-me em todos os momentos. Também pela disponibilidade e apoio a nível científico, e pela compreensão e amizade que sempre demonstrou.

Ao Doutor Renato Natal Jorge, agradeço novamente a oportunidade e a confiança demonstrada, assim como a sua disponibilidade e partilha de conhecimentos, que permitiu levar este trabalho ainda longe.

Ao João Ferreira, por todo o apoio e disponibilidade imediata, incentivando a minha autonomia e espírito crítico, que contribuíram grandemente para o desenvolvimento deste trabalho. Também pela amizade que demonstrou, encorajando-me nas fases mais difíceis.

À minha família: aos meus tios e primos que sempre me apoiaram e incentivaram, e em especial aos meus avós, por me mostrarem que mesmo em momentos mais difíceis, com determinação, confiança e espírito positivo, se conseguem alcançar os objetivos.

À minha irmã, que mesmo do outro lado do mundo continuou perto para me apoiar, por toda a ajuda e, claro, pelas sempre indispensáveis dicas de formatação.

Por fim aos meus pais, pela paciência e ajuda nas fases mais trabalhosas, por me ouvirem sempre, e por me incentivarem a ir mais longe.

(10)

(11)

List of Figures

Fig. 1-Schematic representation and description of motion of a continuum body. ... 16

Fig. 2- Infinitesimal volume element subjected to Cauchy stress components [7]. ... 23

Fig. 3-Body subjected to deformation. ... 23

Fig. 4- Characterization of angles Θ, Φ used for definition of fibre orientation. ... 41

Fig. 5- ABAQUS®_{flowchart ... 47}

Fig. 6- Characterization of viscoelasticity features. a) Relaxation b) Creep c) Hysteresis ... 49

Fig. 7- Maxwell Model associated with relaxation and creep behaviour. ... 51

Fig. 8- Generalized Maxwell Model to describe a viscoelastic material. ... 51

Fig. 9- Flowchart for the algorithmic steps required to compute the viscous parts of the Cauchy Stress and elasticity tensors. ... 55

Fig. 10- Undeformed configuration of the model and stress state considered. a) uniaxial stress state b) biaxial stress state ... 57

Fig. 11- Deformed configuration. a) uniaxial stress state b) biaxial stress state ... 58

Fig. 12- Cauchy Stress as a function of the stretch for the different constitutive models. a) uniaxial stress state b) biaxial stress state. ... 58

Fig. 13- Cauchy Stress as a function of the stretch for the transversally isotropic constitutive models. Uniaxial stress state. ... 59

Fig. 14- Model for the unitary cube. a) undeformed configuration b) deformed configuration ... 60

Fig. 15- Relaxation experiments: comparison of analytical results obtained from Maple with numeric results generated from ABAQUS®_{for different constitutive models and analysis of} the effects of the viscoelasticity parameter. a) NeoHooke b) Mooney-Rivlin c) Ogden d) HGO. ... 61

Fig. 16- Fiber-reinforced rubber strip model. ... 63

Fig. 17- Force Fx(t) versus time t applied to the rubber strip. ... 64

Fig. 18- Viscoelastic behaviour of a fiber-reinforced rubber strip under sinusoidal loading Fx(t). a) Time evolution of the applied force and consequent stretch. b) Applied force as a function of the stretch. c) Applied force as a function of the stretch corresponding to a period of time T=5s after the steady state is reached. ... 65

Fig. 19- Displacement distribution in the x direction (mm) on the deformed configuration of the strip after steady state is reached at the points were the strain is a) 1.240 b) 1.251 c) 1.263... 65

(14)

Fig. 20- Cauchy Stress σxx distribution (MPa) on the deformed configuration of the strip after

steady state is reached. a) peak stress b) minimum stress ... 66 Fig. 21- Representation of the Graft Model and respective dimensions. ... 67 Fig. 22- Schematic drawing of static preconditioning applied. ... 68 Fig. 23- Comparison of the relaxation curves during a static preconditioning of 2.5% strain. a) Experimental curve obtained by Graf, B. et al [29]. b) Relaxation curve obtained using the finite element model and the material and viscoelastic parameters from Table 5 and Table 6 respectively. ... 69 Fig. 24- Cauchy Stress σyy of the graft model. a) time t=0+ corresponding to peak stress b)

time t=600s ... 69 Fig. 25- Schematic drawing of loading regimens. a) Static Preconditioning with 1 min recovery time. b) Static Preconditioning with 10 min recovery time c) Cyclic Preconditioning with 1 min recovery time. ... 70 Fig. 26- Comparison of the initial relaxation behaviour (with no precondition) with the relaxation behaviour after different preconditioning conditions. ... 71 Fig. 27- Schematic of corneal structure [31]. ... 72 Fig. 28- Fibril reinforced structure of the cornea as assumed by Pandolfi, A., et al 2008 [33]. ... 74 Fig. 29-Cauchy Stress as a function of the strain. Comparison between experimental data and the proposed corneal constitutive model using the viscoelastic and material parameters from Table 7 and Table 8 respectively. ... 75 Fig. 30- Model of the corneal strip to reproduce the uniaxial tensile test in order to obtain the material parameters. a) undeformed state. b) deformed state ... 75 Fig. 31- Corneal Stress-Relaxation curves. a) experimental test Su, P. et al 2015 [31]. b) numerical simulation curve obtained using the proposed corneal constitutive model with the viscoelastic and material parameters from Table 7 and Table 8 respectively ... 76 Fig. 32- Model of the human eye. Dimensions in mm. ... 77 Fig. 33- Corneal extrusion experiment performed by Su, P. et al 2015 [31]. ... 77 Fig. 34- Relationship curve between the extrusion force and the displacement obtained experimentally by Su, P. et al 2015 [31]. ... 77 Fig. 35- Model of the eyeball used in the simulations with the respective discretization. ... 78 Fig. 36- Representation of the fiber directions obtained for the first family (red vectors) and the second family (blue vectors). ... 79 Fig. 37- Relationship between the extrusion force and displacement obtained in the numerical simulation and comparison with the experimental data from Ref [31]. ... 79

(15)

Fig. 38- Distribution of the maximum Principal Stress (MPa) of the eye extrusion simulation. Cut in the xy plane corresponding to a) 0 mm extrusion displacement b) 1.09 mm extrusion displacement (point A) c) 3.13 mm extrusion displacement (point B) d) final extrusion displacement 4mm. ... 80 Fig. 39- Distribution of the maximum principal stresses (MPa) of the eye extrusion simulation. Visualization of the complete eye without displaying the plate corresponding to a) 1.09 mm extrusion displacement (point A) b) 3.13 mm extrusion displacement (point B) c) final extrusion displacement 4mm. ... 80 Fig. 40- Relationship between the extrusion force and displacement obtained in the numerical simulation for different levels of IOP. ... 81 Fig. 41- Stress relaxation curve for viscoelastic parameters of the pelvic floor muscles ... 83 Fig. 42-Finite element model used to define the viscoelastic parameters. a) undeformed configuration b) deformed configuration. ... 83 Fig. 43- Stress-stretch curve of the uniaxial tensile test. visco: viscoelastic element with the properties of pelvic floor muscles. without visco: element with the properties of the pelvic floor muscles without considering viscous effects. ... 84 Fig. 44-Finite element model of the mother showing the pelvic floor muscles in red, the supporting structures in brown and the surface to provide contact between the foetus head and the pelvic bones in grey. ... 85 Fig. 45-Finite element model of the pelvic floor muscles with identification of the fixed nodes. ... 85 Fig. 46-Finite element model of the foetus with the identification of the reference nodes that control its movement. ... 86 Fig. 47-Vertical displacement of the foetus head as a function of the simulation time for the precipitous (30min), normal (2.5h) and prolonged (5.5h) labours. ... 87 Fig. 48- Antero-posterior reaction forces in the pelvic floor muscles measured in fixed nodes, identified in Fig. 45, during the vertical displacement of the foetus head for a normal labour with simpler viscoelastic parameters and a labour without viscous effects. ... 88 Fig. 49- Antero-posterior reaction forces in the pelvic floor muscles measured in fixed nodes, identified in Fig. 45, during the vertical displacement of the foetus head for a normal labour and a labour without viscous effects. ... 88 Fig. 50- Antero-posterior reaction forces in the pelvic floor muscles measured in fixed nodes, identified in Fig. 45, during the vertical displacement of the foetus head for a normal labour with controlled time increments and a labour without viscous effects. ... 89 Fig. 51- Antero-posterior reaction forces in the pelvic floor muscles measured in fixed nodes, identified in Fig. 45, during the vertical displacement of the foetus head for a

(16)

precipitous, normal and prolonged labour. For comparison purposes, the result of a labour without viscous effects is included. ... 90 Fig. 52-Model of the mother and curve used to evaluate the stresses. ... 91 Fig. 53- Maximum principal Cauchy stresses (MPa) calculated along the normalized path at the most inferior portion of the pelvic floor (see Fig. 52) for precipitous, normal, prolonged and without viscous effects labours. a) peak stresses instant; b) end of the simulation. ... 92 Fig. 54- Distribution of the maximum principal Cauchy stresses (MPa) on the model of the mother and the foetus head. a) peak stresses instant. b)end of the simulation ... 92 Fig. 55-Vertical displacement of the foetus head as a function of the simulation time for a precipitous labour with resting stages. ... 93 Fig. 56- Antero-posterior reaction forces in the pelvic floor muscles measured in fixed nodes, identified in Fig. 45, during the vertical displacement of the foetus head for a precipitous labour and a precipitous labour with three resting stages. ... 94 Fig. 57- Distribution of the maximum principal stresses (MPa) on the model of the mother and the foetus head at the peak stresses instant a) Precipitous labour with three resting stages of three minutes at a vertical displacement of 30, 57 and 70mm. b) Precipitous labour. ... 94

(17)

List of Tables

Table 1- Material parameters selected for each constitutive model (consistent system of

units) ... 58

Table 2- Material parameters selected for each constitutive model (consistent system of units). ... 60

Table 3- Maximum and minimum relative error of the Cauchy stress for the different constitutive models and viscoelastic parameters analysed. ... 63

Table 4- Material, structural and viscoelastic parameters selected for the rubber strip ... 64

Table 5- Patellar tendon material parameters [26]. ... 67

Table 6- Patellar tendon viscoelastic parameters ... 69

Table 7-Cornea viscoelastic parameters ... 73

Table 8- Material parameters selected for the cornea ... 75

Table 9- Constitutive material parameters for the pelvic floor muscle tissue ... 83

Table 10- Constitutive viscoelastic parameters for the pelvic floor muscle tissue ... 84

Table 11- Viscoelastic parameters applied in the initial test for the pelvic floor muscle tissue ... 87

(18)

(19)

List of Symbols

Ω0 Reference configuration

Ω Deformed configuration

x _{Spatial coordinates}

X Material or initial coordinates

u Displacement field t time F Deformation Gradient I Unit tensor J Jacobian U Right-Stretch tensor V Left-Stretch tensor R Rotation matrix

C Right Cauchy-Green deformation tensor B Left Cauchy-Green deformation tensor 𝐚𝐚0 unit vector in the reference configuration

a Unit vector in the deformed configuration

λ Stretch

𝛌𝛌a0 Stretch vector

�𝐍𝐍�1,2,3� Principal stretch directions in the reference configuration

�λ1,2,3� Principal Stretches

�𝐧𝐧�1,2,3� Principal stretch directions in the deformed configuration

E Lagrangian strain tensor e Eulerian strain tensor

v Velocity field

L Velocity gradient tensor

𝐅𝐅̇ Time derivative of the gradient tensor

D Stretch rate

W Spin tensor

𝐄𝐄̇ Material strain rate tensor

𝐂𝐂̇ Time derivative of the right Cauchy-Green deformation tensor

t Traction vector

σ Cauchy stress tensor

τ Kirchhoff stress tensor

T Nominal of first Piola-Kirchhoff traction vector P First Piola-Kirchhoff stress tensor

S Second Piola-Kirchhoff stress tensor

ρ Mass density in the deformed configuration

(20)

f Body force vector

V Volume of the body in the current configuration ρ0 Mass density in the reference configuration

V0 Volume of the body in the reference configuration

Ѱ Strain-energy function



Lagrangian or Material elasticity tensor

 Eulerian or spatial elasticity tensor

I1,2,3 Invariants of the Cauchy-Green deformation tensors

p Indeterminate Lagrange multiplier

𝐅𝐅� Isochoric part of the deformation gradient

𝐂𝐂� Isochoric part of the right Cauchy-Green deformation tensor �λ�1,2,3� Isochoric part of the principal stretches

Ѱvol Volumetric part of the strain-energy function

Ѱiso Isochoric part of the strain-energy function



Fourth-order unit tensor



Fourth-order projection tensor in the lagrangian or material configuration 𝐒𝐒� Isochoric part of the second Piola-Kirchhoff stress tensor

𝐁𝐁� Isochoric part of the left Cauchy-Green deformation tensor

 Projection tensor in the eulerian or spatial configuration 𝛔𝛔� Isochoric part of the Cauchy stress tensor

I1,2,3

�� Invariants of isochoric part of the Cauchy-Green deformation tensors I4,5 Pseudo-invariants arising directly from anisotropy

Ѱmat Ground matrix part of the strain-energy function

Ѱfib Fiber part of the strain-energy function

D1 Penalty parameter of the penalty function to ensure incompressibility

C10 Parameter of the Neohooke and Mooney-Rivlin constitutive models

C01 Parameter of the Mooney-Rivlin constitutive model

μp, αp Parameters of the Ogden constitutive model

C, b Parameters of the ground matrix part of the Martins constitutive model A, a Parameters of the fiber part of the Martins constitutive model

ѰfibPE Passive elastic part of the fiber contribution of the strain-energy function

Ѱ_fibSE _{Active part of the fiber contribution of the strain-energy function}

λ�f Stretch ratio of the muscle fibers

T0m Maximum muscle tension at resting length

δ Activation variable

ki1, ki2 Parameters of the i-th family of fibers of the HGO constitutive model

Φ, Θ Angles that define the direction of the fiber

ρi Normalized probabilistic distribution function of the i-th fiber orientation

ϖ Unit sphere

erfi Imaginary error function

(21)

erf Error function

κ Dispersion parameter

H Generalized structure tensor

M Arbitrary unit vector located in the three dimensional eulerian space

δW Virtual Work

δu Virtual displacement field Fint Internal equivalent nodal forces

Fext External equivalent nodal forces

Rf Residual force

K Stiffness Matrix

γ Dissipative Potential

Г Internal Variables

𝚪𝚪� Isochoric part of the internal variables α Number of viscoelastic processes

β Free-energy parameter

τ Relaxation time

𝐒𝐒vol∞ Volumetric elastic part of the second Piola-Kirchhoff stress tensor

𝐒𝐒iso∞ Isochoric elastic part of the second Piola-Kirchhoff stress tensor

Qα Isochoric non-equilibrium stress tensor

𝐒𝐒mat∞ Ground matrix elastic part of the second Piola-Kirchhoff stress tensor

𝐒𝐒_fib∞ _{Fiber elastic part of the second Piola-Kirchhoff stress tensor}

Qmat α Ground matrix part of the non-equilibrium stress tensor

Qfib α Fiber part of the non-equilibrium stress tensor

H History Term

vol ∞

 Volumetric elastic part of the material elasticity tensor

mat ∞

 Ground matrix elastic part of the material elasticity tensor

fib ∞

 Fiber elastic part of the material elasticity tensor

mat α

 Ground matrix viscous part of the material elasticity tensor

fib α

 Fiber viscous part of the material elasticity tensor

List of Operators

∇ Gradient ⋅ Scalar product : Double product ⨂ Tensor product 9

(22)

(23)

Chapter 1 Introduction

1.1 Context

Bioengineering and biomechanics can be defined as the biological or medical application of engineering principles. It includes the study of motion, material deformations and load bearing, tissue engineering and flow of bodily fluids. The biomedical problems are thus solved based on a myriad of laws of physics, thermodynamics and biology. The applications are diverse and over the past decades the advances in bioengineering have contributed to numerous innovations that have resulted in significant improvement in our quality of life.

Computational bioengineering is a more recent field that apply the latest computational capabilities to deal with biomedical problems. Employing the finite element method, it is possible to model and simulate biological components in order to analyse and determine the effects of mechanical stresses and strains as well as interactions of different components and tissues on clinical outcomes. It offers predictive capabilities to assess new surgical concepts, medical devices and postoperative surgical outcomes that otherwise must be studied through in-vivo assessments. Since it is not restricted to ethical constraints, it become a powerful and valuable tool that is currently experiencing major advances.

To achieve accurate and realistic simulations it is necessary to obtain not only a geometrical approximate model but also to understand the mechanical properties of each modelled tissues and components. The physical properties of materials are specified by constitutive equations. Since tissues are known to present an intricate behaviour, being anisotropic and visco-hyperelastic, the constitutive equations were unknown until recently and are still being developed so that more specific particularities can be considered. The first constitutive models were defined by Mooney and Rivlin in the 1940s, who developed the Mooney-Rivlin and the Neohooke material models. These material models are relatively simple, and they were initially used to represent rubber [3]. Even nowadays these models are frequently applied since they do not need a great number of constant material parameters, facilitating the application in numerical simulations. Throughout the years, a great number of constitutive models were developed, such as the Ogden model in the 1970s [2].

A constitutive model developed by Holzapfel was presented in the 2000 [2], to account for anisotropy due to the frequently existence of collagen fibers in the tissues. This particular model is still in development since it initially considered a perfect alignment for

(24)

each family of fibers and the consideration of fiber dispersion following specific distribution functions is being studied [3], [4]. Viscoelasticity is also a feature presented in living tissues and this behaviour can be likewise captured by mechanical models that were developed to correlate experimental data on real materials, such us the Maxwell, the Kelvin-Voigt and the generalized Maxwell model. Since viscoelasticity is a time dependent property, the introduction of this feature in finite element simulations involves the application of a recurrence update procedure in which demanding numerical calculations are preceded. Consequently, despite the importance of this property in the analysis of biological tissues, this feature is often disregarded in biomechanical models.

The numerical simulations are developed employing the finite element method using proper software’s such as ABAQUS®_{. The complete ABAQUS}®_{environment allows to model}

the problem, manage, monitor and visualize the results. Although the software is prepared to consider a wide range of materials, specially the most everyday materials in structural engineering, it does not include most of the constitutive models necessary for biological tissues. Therefore, it is necessary to apply a user defined material, defining the constitutive equations and the relation between stress and strain in a subroutine that must be developed in the FORTRAN language. The user subroutine is designated as UMAT in ABAQUS®

language and documentation.

In this work, it is intended to introduce viscoelasticity as a feature of tissues in numerical simulations using the software ABAQUS®_{. The basis of the work is a user material}

subroutine UMAT developed by Ferreira, J., et al 2017 [1] that is able to consider hyperelastic materials using the Neohooke, Mooney-Rivlin or the Ogden constitutive models and also anisotropy, following the constitutive model presented by Holzapfel, G.A. [2].

This work involves therefore the following set of objectives:

 _{Extend the existing UMAT introducing a recurrence update algorithm for} viscoelasticity in the FORTRAN language following the generalized Maxwel model as it is presented by Holzapfel, G.A. [2];

 _{Analyse and validate the material behaviour through an introductory and simple} numerical example;

 _{Analyse the possible applications in biomechanical models developing a set of} numerical examples;

 _{Verify the influence and the effects of viscoelasticity through the results of the} numerical simulations;

(25)

1.2 Thesis Organisation

Chapter 2 of this thesis provides an overview over the principles of Continuum Mechanics and the Finite Element software, which provides the basic tools to define the constitutive equations using the software ABAQUS®_{. Starting with kinematics, motion and}

descriptions are defined so as the most applied deformation tensors. The concept of stretch and principal stretches followed by the essential stress and strain measures are also exposed. Then, equations of motion and equilibrium were established and the constitutive equations for isotropic and anisotropic hyperelastic materials are presented so as some forms of strain energy function. Finally, the basic of the finite element method is outlined so as the overall structure of the software ABAQUS®_{, in order to contribute to a better}

understanding of its functioning.

Chapter 3 focus essentially on viscoelasticity, providing in the first place the basis of its fundaments. Afterwards, viscoelastic materials are defined and the assumptions necessary for viscoelasticity implementation are presented. Ultimately, some considerations about the developed user defined material subroutine UMAT are discussed.

The Chapter 4 includes the numerical examples, providing insights related to the materials approached and its properties, more specifically viscoelasticity.

Finally, the ultimate conclusions and some suggestions for future work are delineated in Chapter 5.

(26)

(27)

Chapter 2 Continuum Mechanics and Finite Element Method

2.1 Standard Continuum Mechanics

Continuum Mechanics studies the response of materials to different loading conditions. Its study can be developed through general and well known principles such as the principle of conservation of mass, for example, or through constitutive equations defining idealized materials. Since the mechanical behaviour of real materials varies not only from material to material but also with different loading conditions, there are many constitutive equations defining the many different aspects of material behaviour [5].

Continuum mechanics theory allows the study of phenomena regarding the behaviour of materials, developing models that neglect their structure on a small scale. Therefore, matter is considered indefinitely divisible and a particle is the designation given to an infinitesimal volume of the material. The validity of this theory depends of the given situation and is justified by experimental results and observations over the years [5].

2.2 Kinematics

Kinematics can be described as the study of motion and deformation without reference to the cause. Therefore, it provides and defines fundamental quantities and constitutive descriptions that allows the characterization of motion and deformation.

2.2.1 Description of Motion

A continuum is defined by an infinite amount of particles. Thus, a vector with a set of coordinates is used to identify the position of a given particle at some reference time. The set of vectors, each one of them defining one particle, defines the configuration of the body. Naturally, a body can have multiple configurations representing different states of deformation that are usually compared with its reference configuration. The base configuration is the configuration defined as the origin of the displacements. Although in most situations the base and reference configurations represent the same state, it will actually depend on the kinematically description that it will be used, as it will be introduced afterwards. In Fig. 1 we can observe a body where Ω0 represents the set of vectors in the

reference configuration and Ω in the deformed configuration. The vectors that define the position of each particle can be represented by:

i= (X ,X ,X , ), with (X ,X ,X , ) Xi 1 2 3 t i 1 2 3 t0 = i

x x x _(2.1)

(28)

Fig. 1-Schematic representation and description of motion of a continuum body.

Equation (2.1) defines the motion of a continuum through the description of the pathlines of each particle. X1, X2 e X3 are known as the material or initial coordinates, while

x1, x2 e x3 are known as the spatial coordinates.

The displacement of a particle from a reference position to a position at an instance t is the given by:

= ( , )−

u x Xt X _(2.2)

Therefore, if the pathline x(X, t) of a particle is known, its displacement field is also known.

In conclusion, the motion of a continuum can be described through the pathlines of every particle, also called kinematic equations of motion (eq. (2.1)), or through the displacement field, defined by eq. (2.2) [5], [6].

2.2.2 Kinematics Descriptions

When a continuum is in motion, its temperature, velocity, and stress tensor (to be defined afterwards) may change with time. If these changes are described by following the particles identified by the material coordinates (X1, X2, X3) and time, it is assumed that the

Lagrangian or Material description is being applied. On the other hand, if the changes are observed at fixed locations, meaning that particles are identified by the spatial coordinates (x1, x2, x3), the description is known as Eulerian or spatial Description. The spatial

coordinates are related to the material coordinates by eq. (2.1), therefore, it is also possible to obtain one description from the other. It is important to observe that in the Eulerian Description, what is described or measured is the change of quantities at a fixed location as a function of time. This means that direct information regarding changes in particle properties as they move is not provided using this description [5].

2.2.3 Deformation Gradient

Considering a material element dX at a reference position that is transformed through motion into a material element dx, the relationship between dX and dx is given by:

P(t0) P(t) e1 e2 e2 X x

Ω

0

Ω

16

(29)

dx=x(X+d , )X t −x( , ) (X t = ∇x)dX _(2.3)

Defining the deformation gradient tensor as F=

∇x

, we obtain:

dx=F Xd (2.4)

Considering the displacement field defined by eq. (2.2), it is possible to obtain the deformation gradient through the following equation:

F I= + ∇u _(2.5)

The deformation gradient F contains all the required information regarding the local changes in lengths, volumes, and angles due to the deformation exhibited by every material point in the reference configuration.

If F is independent of the material coordinates, the deformation is called homogeneous. On the other hand, if F varies according to the position X inside the material body, the deformation is non-homogeneous.

One particularity of the deformation gradient is the fact that its determinant represent the local rate of change of the deformed configuration volume v with respect to the reference configuration volume V. We can also designate by Jacobian this measure of the volume variation caused by a deformation, obtaining the following definition:

= = d

J det dV

F v _(2.6)

For the matrix F to adequately describe physical deformations preserving the material orientation and to ensure the material won’t diminish to zero volume, the condition detF>0 is required.

If the material is incompressible, the deformation is called isochoric, implying that detF=1. On the other hand, if the mass density of particle of the material in the initial configuration is ρ0, the density of that particle in the deformed configuration will be ρ= ρ0/J

[5]–[7].

2.2.4 Polar Decomposition Theorem

Considering a body in which its reference configuration is subjected to a generic deformation, it is verified that the motion that leads to the final configuration can be separated into stretching and rotation. It is also known that the rotation part of the deformation does not contribute to the strain. Therefore, it is important to decompose the deformation gradient into two components, one defining the pure stretch of the deformation, and the other defining the rotation.

Knowing that any real tensor with a nonzero determinant can be decomposed into the product of a proper orthogonal tensor and a symmetric tensor, we can write:

(30)

F = RU= VR (2.7) U and V are positive definite symmetric matrices that define pure stretch deformations and are called the right-stretch tensor and the left-stretch tensor, respectively. R is a rotation matrix.

The decomposition is unique since there is only one R, one U and one V satisfying eq. (2.7).

It is possible to obtain the tensors U, R and V only through the deformation gradient, attending the properties of symmetric and orthogonal tensors [5], [6]. With eq. (2.7) we can write: T ( ) ( ) T T T T F F = RU RU U R RU U U UU= = = _(2.8) Therefore, 2₌ T U F F (2.9) 1 − = R FU (2.10) T T = = V FR RUR (2.11)

2.2.5 Right-Cauchy Green and Left Cauchy-Green Deformation Tensor

The deformation gradient F presents some shortcomings that are not useful for the development of constitutive models such as the fact that is not symmetrical and is not insensible to a pure body rotation. Therefore, the following deformation tensors are introduced.

The Right-Cauchy Green deformation tensor can be defined through eq. (2.12).

2 T

C U F F= = _(2.12)

In a similar approach, the Left-Cauchy Green deformation tensor can be defined by eq. (2.13).

2 T

B V= =FF (2.13)

2.2.6 Stretch and Principal Stretches

Considering a unit vector a0,a0 =1 in the reference configuration which describes the direction of the material line element (which can be imagined as a fiber) is it possible to define the stretch vector λ as: a₀

0( , ) 0

a

λ Xt =F a (2.14)

The length, defined by λ =λa0 is simply known as stretch and it is a measure of how much the unit vector a0 has stretched. If λ<1 the element was compressed, if λ=1, the

element maintained the same length, and if λ>1, the element was extended.

(31)

It is possible to obtain the square of the stretch using the Right-Cauchy Green deformation tensor according to the definition (2.12):

2 T

0 0 0 0 0 0

F a F a a F F a a Ca

λ = ⋅ = ⋅ = ⋅ (2.15)

In the deformed configuration, the unit vector a, a =1, is the spatial element characterizing the direction of the fiber. Therefore, the relationship between a0 and a can

be written as:

0

a

λ = λa _(2.16)

There are two sets of principal stretch directions, associated with the deformed and the undeformed solids. The principal stretch directions in the undeformed solid are denoted by

{ }

Nˆ ,_a a 1,2,3= _{and are the mutually orthogonal and normalized eigenvectors of U or C. The}

principal stretches can be denoted by

{ }

λa , a 1,2,3= and correspond to the eigenvalues of U as:

a a a a a=1, 2, 3

ˆ = λ ˆ _with ˆ =_{1 ,}

UN N N (2.17)

Furthermore, combining (2.12) with (2.17) it is possible to obtain the eigenvalue problem for C:

2 2

a a a a , a=1, 2, 3

ˆ ˆ ˆ

CN U N= = λ N _(2.18)

Therefore, although C and B present the same orthonormal eigenvectors

{ }

ˆN , the a positive real eigenvectors differ, and for the tensor C, they represent the squares of the principal stretches: 2

a λ

Following the same reasoning, it is possible to obtain the eigenvalue problem for the tensor V. Recalling equation (2.11) and noticing that RTR=I, we can obtain the following

equation from (2.17): T a a a a a=1, 2, 3 ˆ ˆ ˆ ( ) ( ) ( ) , V RN =RUR RN = λ RN _(2.19)

Regarding the tensor B, if equation (2.13) is combined with (2.19) the eigenvalue problem develop as:

2 2

a a a a a=1, 2, 3

ˆ ˆ ˆ

( ) ( ) ( ) ,

B RN =V RN = λ RN _(2.20)

Equations (2.19) and (2.20) allow to understand that the two tensors V and B have the same eigenvectors RN , while their positive and real eigenvalues are ˆa λ and a λ 2a respectively. The eigenvectors are those of U and C rotated with R. Therefore, it is possible to define a new set of principal directions denoted by

{ }

ˆn that are associated with the a deformed solid and represent the eigenvectors of V and B.

(32)

a ˆa a a=1, 2, 3

ˆ with ˆ 1 ,

n =RN n = _(2.21)

Finally, the four introduced symmetric strain tensors can be summarized as: 3 2 2 a a a a 1 ˆ ˆ = = U C N N = λ ⊗

∑

(2.22) 3 2 2 a a a a 1 ˆ ˆ = = V B n n = λ ⊗

∑

_(2.23)

It is possible and it will be more convenient in some cases to express the material behaviour in terms of the principal directions. Therefore, it is important to define the deformation gradient using these concepts, as:

3 2 a a a a 1 ˆ ˆ = F n N = λ ⊗

∑

(2.24)

Since F is non-symmetric, λa in the previous equation may not be interpreted as the

eigenvalues of F.

2.2.7 Strain Measures

In order to quantify the relative distance change between two material points, that is, to characterize the straining, it is necessary to define strain measures. Since the strains are not a measurable but a conceptual quantity, there are numerous strain tensors defined in the literature. In this section, the Lagrangian and Eulerian strain tensors are defined, since they are both suitable to analyse bodies submitted to large deformations.

The Lagrangian Strain Tensor can be defined from the deformation gradient as:

=1₍ T − ₎ 2

E F F I (2.25)

Considering the equations (2.12) and (2.5), that is, considering the definition of the right Cauchy-Green deformation tensor and the displacement field, respectively, equation (2.25) can be rewritten:

=1₍ − =₎ 1_∇ + ∇₍ ₎T + ∇₍ ₎T∇ _

2 2

E C I u u u u _(2.26)

Analysing the previous equation and remembering that C=U2_{, it is verified that the}

Lagrangian strain tensor will only be function of the component of the deformation gradient regarding stretch, U. Being independent of the rotation, this tensor will represent a valid approach in order to analyse the strain of a body submitted to large displacements, large rotations and large deformations. It can be also noted that for a rigid-body displacement, C=I, implying that the tensor will also be zero in that case. The Lagrangian strain tensor is a strain measure usually applicable for materials that exhibit large strains and rotations, such as rubber and biological tissue [5]–[7].

The Eulerian Strain Tensor is defined as:

(33)

− − −

=1₍ − 1₎=1₍ − T 1₎

2 2

e I B I F F (2.27)

Its physical significance is similar to the Lagrange strain tensor and it is also applied when it is intended to analyse materials that suffer large deformations. However this tensor enables the computation of the strain of an infinitesimal line element from its orientation after deformation.

2.2.8 Velocity and Material Time Derivatives

Nonlinear processes are often time-dependent, therefore, it is necessary to consider velocity and material time derivatives of various quantities. Even if the process is not rate-dependent, it is convenient to establish the equilibrium equations in terms of virtual velocities associated virtual-time dependent quantities [7], [8].

Recalling the equations of motion presented previously, x=x( , )X t , the velocity of a

particle can be defined as:

( , ) ( , )=∂ ∂ X v X t t t x _(2.28)

Despite being written in terms of the material coordinates, the velocity field is a spatial vector. Therefore, the derivative with respect to the spatial coordinates defines the velocity gradient tensor L as:

( , ) ∂ = = ∇ ∂ v L x t v x (2.29)

The time derivative of the gradient tensor F can be defined as:

d d v F X X X ∂ ∂ ∂ ∂     = _ _= _ _= ∂ ∂ ∂ ∂      x x t t (2.30)

Introducing the concept of the velocity gradient tensor, it is possible to obtain:

∂ ∂ ∂ = = = ∂ ∂ ∂ v v F LF X X  x x (2.31)

And therefore, the velocity gradient tensor can also be obtained from the previous equation as [7], [8]:

1

L FF_{= } −

(2.32) It is possible to define the stretch rate, which quantifies the rate of stretching of a material fiber in the deformed solid. It can be defined as:

T T 1₍ ₎ 1 2 2 _∂ __∂ _  = + =  +_ _  ∂ ∂      v v D L L x x (2.33) 21

(34)

It is also possible to measure the average angular velocity of all material fibers passing through a material point through the spin tensor, defined as follows:

T

1( )

2

= −

W L L _(2.34)

The sum of the stretch rate and the spin tensor represent another way to obtain the velocity gradient defined previously in eq.(2.32).

L D W= + _(2.35)

It is possible to define another rate tensor obtained from the Lagrangian strain tensor that provides the rate of stretching in terms of the initial elemental vectors. It is called the material strain rate tensor and can be defined recalling equations (2.25) and (2.26) as:

T T

1 1₍ ₎

2 2

E= C= F F F F +  _(2.36)

It is possible to relate both the stretch rate D and the material strain rate tensor E through the gradient tensor F, since it represents a relation between the spatial and material coordinates. Therefore, the following equations can be stated [7], [8]:

=  T E F DF _(2.37) − − = T 1 D F EF _(2.38)

2.2.9 Stress Measures

Motion and deformation give rise to interactions between the material and neighbouring material in the interior part of the body, which leads to stress. Since the notion of stress is responsible for the deformation of materials, it is crucial in continuum mechanics, and there are several alternative forms to measure this concept.

In the first place, it is important to define the traction vector at a point on the plane, which represents the force acting on the surface per unit area of the deformed solid as follows:

→ = dA 0 d lim dA n P t _(2.39)

where dA represents an element of area on a plane with normal vector n subjected to a force dP. The traction vector t depends on the orientation of the plane, being a function of its outward normal vector n [5], [6]:

( , , ) =

t t x t n _(2.40)

The relationship between the traction vector and the normal vector n can be defined by the following linear transformation:

T

( , , )= ( , ).

t x nt σ xt n _(2.41)

(35)

Where σ represents the Cauchy stress Tensor [6]. It is important to indicate that the Cauchy Stress is always symmetric, therefore, the previous definition can be presented as the transpose.

The Cauchy Stress Tensor completely characterizes the internal forces acting in a deformed solid and the physical significance of each component can be seen in Fig. 2. σij

represents the component of traction in the ej direction acting on the plane with normal in

the ei direction. Since the Cauchy Stress represents force per unit area of the deformed solid,

it is commonly called “true stress” [7].

Fig. 2- Infinitesimal volume element subjected to Cauchy stress components [7].

There are some situations, for instance when a body is subjected to large deformations, where it is more convenient to formulate equations of motion with respect to the reference configuration rather than the current configuration. Therefore, stress tensors based on the undeformed area were defined and are designated as the first and second Piola-Kirchhoff stress tensors. To define such tensors it is necessary to know both the actual and the initial configuration of the solid. Reminding that deformation can be described through the deformation gradient F and that J represents its Jacobian defined by J=det F, it is possible to define the Kirchhoff stress from the Cauchy stress as follows:

= J

τ σ _(2.42)

Unlike the Cauchy Stress, the Kirchhoff stress has no obvious physical significance [7]. In order to define the nominal or the first Piola-Kirchhoff stress, we can imagine a body being deformed as shown in Fig. 3.

Fig. 3-Body subjected to deformation.

Whereas the Cauchy traction vector t is the actual physical force per area on the deformed configuration, the nominal or First Piola-Kirchhoff traction vector T(X,t,N) is a fictitious quantity and does not describe the actual intensity. It represents the force acting

N T

dS Reference

configuration configuration Current

n t

ds

df = tds = TdS

(36)

on an element in the current configuration divided by the area of the corresponding element in the reference configuration:

= d dS

f

T (2.43)

The relationship between the traction vector T and the nominal or first Piola-Kirchhoff stress tensor P can be defined as follows.

( , , )= ( , )

T X Nt P X Nt _(2.44)

Finally, it is possible to obtain a relationship between the Cauchy stress tensor and the first Piola-Kirchhoff stress tensor:

T J P σF₌ − (2.45) 1 ij ik jk P J F_{= σ} −

The Cauchy stress is symmetric, but since the deformation gradient is not, the first Piola-Kirchhoff stress tensor will also not be symmetric. This particularity restricts its use and it actually led to the definition of the material or Second Piola-Kirchhoff stress tensor. It can be defined from the Cauchy stress tensor or from the first Piola-Kirchhoff stress as it follows:

1 T 1 J S F σF₌ − − ₌F P− (2.46) ik1 1 ik1 ij kl jl kj S ₌JF−_σ F− ₌F P−

The second Piola-Kirchhoff stress tensor is a symmetric tensor and can be visualized as force per unit undeformed area, except that forces are regarded as acting within the undeformed solid rather than on the deformed solid.

In practice it is best not to try to attach too much physical significance to these stress measures. They are best regarded to as generalized forces, which are work conjugate to particular strain measures. Although the Cauchy stress and the Kirchhoff stress are not conjugated to any strain measure, the first and second Piola-Kirchhoff stress tensor are conjugated to the deformation gradient and to the Lagrange strain tensor respectively [7], [9], [2].

2.3 Equations of Motion and Equilibrium for Deformable Solids

In this section, Newton’s Laws of motion (conservation of linear and angular momentum) is generalized to a deformable solid.

2.3.1 Linear Momentum Balance in terms of Cauchy Stress

The principle of conservation of linear momentum states that the rate of change of the total linear momentum of a continuum medium equals the vector sum of all external forces

(37)

acting on the body. Therefore, this principle leads to the following equation of motion, that also express Newton’s third law of motion:

2 2 σ u f ∂ ∂ + = ρ ∂x ∂t (2.47)

where ρ represents the mass density of the deformed solid, f is the body force vector (per unit volume) and σ the Cauchy stress distribution. These equations must be satisfied for any continuum in motion and are also called Cauchy’s Equations of Motion. If the acceleration vanishes, we obtain the static equilibrium equation:

0

σ f

∂ _{+ =}

∂x (2.48)

If the previous equations are used to solve problems it is common to say that the strong formulation is being applied. However, there are situations where the differential form of the equations of motion complicates the process, which led to the development of the weak formulation, obtained through the principle of virtual work [10].

2.3.2 Principle of Virtual Work

To understand the principle of virtual work consider a deformable body subjected to loading that induces a displacement field u(X) and a velocity field v(X). The loading consists of a prescribed displacement on part of the surface of the body, denoted by Su and by a

traction, t (per unit of current area), applied to the rest of the boundary denoted by St. Let

δv be a kinematically admissible virtual velocity field in the body, which is arbitrary except that δv=0 on Su. Let V be a volume occupied by a part of a body in the current configuration

and f the body force (per unit of current volume). Additionally, the virtual velocity gradient and the virtual stretch rate can be defined respectively by:

∂δ δ = = ∇δ ∂ v L v x (2.49) T T 1₍ ₎ 1 2 2 _∂δ __∂δ _  δ = δ + δ =  _∂ +_ _∂ _        v v D L L x x (2.50)

The principle of virtual work states that the stress, body force and traction are in equilibrium if and only if the rate of work done by the external forces subjected to any virtual velocity field equals the rate of work done by equilibrating stresses on the rate of deformation of the same virtual velocity field. Therefore, the virtual work equation becomes: δ + ρ δ − ⋅ δ − ⋅ δ =

∫

t V V V S d : dV dV dV dA 0 dt v σ D v f v t v _(2.51) 25

(38)

To prove this result, the divergence theorem and also some statements regarding the properties of the Cauchy Stress can be used.

The divergence theorem can be defined as an equality relationship between surface integrals and volume integrals, with the divergence of an arbitrary vector field involved denoted by ( ):

( )

V S . dV . dS ∂   ₌ _∂   

∫

n x (2.52)

Properties regarding the symmetry of the Cauchy stress allow us to observe that:

j j i i i ij ij ij ji ij ji j i j i j v v v v v 1 1 D 2 2 ∂δ ∂δ   ∂δ ∂δ  ∂δ σ δ = σ _ + _= _σ + σ _= σ ∂ ∂ ∂ ∂ ∂  x x   x x  x

It is also possible to note that:

(

)

ji i ji ji i i j j j v _v ∂σ _v ∂ ∂ σ = σ δ − δ ∂x ∂x ∂x

Applying these concepts to part of the virtual work equation (2.51), it is obtained:

V V :δ dV= ∂δ dV= ∂

∫

σ D

∫

σ v x (2.53) V V ( _{) dV} _dV ∂ δ ∂ = − δ ∂ ∂

∫

σ v_x

∫

σ v_x

Applying now the divergence theorem to the first integral of equation (2.53), we can rewrite it as: i St V (σ n v) dA ∂σ v dV = ⋅ δ − δ ∂

∫

_x (2.54)

and replacing the previous statement into the virtual work equation (2.51), we obtain:

t i St V V V S d ( ) dA dV dV dV dA 0 d σ v σ n v⋅ δ − ∂ δv + ρ δv − ⋅δf v − t v⋅δ = ∂

∫

_x

∫

_t

∫

(2.55)

In order to rewrite the previous statement, recall from eq. (2.41)that: t σ n= ⋅ _{. Therefore,}

the following expression is obtained:

V V V d dV dV dV d σ v f v⋅δ + ∂ δv = ρ δv ∂

∫

_x

∫

_t (2.56)

Finally, for every virtual velocity field and also considering that the volume is arbitrary, equation (2.56) recovers the equilibrium equation in the volume of the body stated in the previous section: 2 2 ∂ _{+ = ρ}∂ ∂ ∂ σ _f u t x (2.47) 26

(39)

which validates the principal of virtual work.

The principal of virtual work can also be seen as a different way of rewriting partial differential equation for linear momentum balance, in an equivalent integral form that is more suited for computer solution. As referred in the previous section, the principle of virtual work is designated as the weak form of the equilibrium equations and is used as the basic equilibrium statement for the finite element formulation [7], [11], [12].

It is often convenient to implement the virtual work equation using the different stress measures presented previously. To do so, it is necessary to define the virtual rate of change of the deformation gradient and the virtual rate of change of Lagrange Strain, which are presented in eq. (2.57) and (2.58) respectively.

∂δ ∂ ∂δ δ = = ∂ ∂ ∂ v v F X X  x x (2.57) T T 1 δ = (δ δ ) 2 E F F F F +  _(2.58)

Consequently, it becomes possible to express the virtual work equation in terms of the Kirchhoff stress: 0 0 0 t 0 0 0 0 V V V S d dV dV dV dA 0 dt v τ D⋅ δ + ρ δv − f v⋅ δ − t v⋅ δ =

∫

(2.59)

In terms of the First Piola- Kirckhoff Stress:

0 0 0 t 0 0 0 0 V V V S d dV dV dV dA 0 d ⋅ δ + ρ δ − ⋅ δ − ⋅ δ =

∫

P F

∫

v v

∫

f v

∫

t v t (2.60)

And finally in terms of the Second Piola- Kirckhoff Stress:

0 0 0 t 0 0 0 0 V V V S d dV dV dV dA 0 d ⋅ δ + ρ δ − ⋅ δ − ⋅ δ =

∫

S E

∫

v v

∫

f v

∫

t v t (2.61)

It is important to realize that all volume integrals are now taken over the undeformed solid, which is convenient for computer applications since the shape of the undeformed solid is known. However, the area integral is still evaluated over the deformed solid and can become an inconvenient when solving some problems [7].

2.4 Constitutive Equations

The constitutive model for a material is a set of equations relating stress to strain with the aim of representing the real behaviour of matter. They depend on the type of material under consideration and can be dependent or independent of time. These equations must satisfy certain physics laws but they are generally fit to experimental measurements since they cannot be calculated using fundamental physics laws [7].

(40)

The constitutive equations must obey the thermodynamic laws and satisfy the condition of objectivity or material frame indifference. In addition, it is important to ensure that the material satisfies the Drucker stability criterion.

The first thermodynamic law states that the work done by stresses must either be stored as recoverable energy in the solid or be dissipated as heat (or a combination of both). The second law requires that if a sample of material is subjected to a cycle of deformation that starts and ends with identical strain and internal energy (at constant temperature or without heat exchange with the surroundings), the total work done must be positive or zero.

The term objectivity can be defined as the condition that the tensor valued functions relating stress to deformation must transform correctly under a change of basis and change of origin for the coordinate system.

In this section, the constitutive equations for the non-linear regime will be presented, considering hyperelastic materials.

2.4.1 Hyperelasticity

Materials for which the constitutive behaviour is only a function of the current state of deformation are defined as elastic. In the particular case when the work done by stresses during a deformation is dependent only on the initial state at time t0 and the final

configuration at time t, the behaviour of the material is said to be path-independent and the material is called hyperelastic. As a consequence of this particular behaviour it is possible to stablish a stored energy function, also known as strain-energy function or elastic potential, Ѱ, per unit of undeformed value, which represents the work done by the stresses from the initial to the current position:

0 d ( ) : d ; : d F P F Ψ P F Ψ =

_∫

t  =  t t t (2.62)

P represents the first Piola-Kirchhoff stress tensor defined by eq. (2.45) and F the time derivative of the gradient tensor defined by eq. (2.30).

The rate of work done per unit of undeformed value can also be written as: ij ij F d dt F ∂ Ψ ∂Ψ = ∂ ∂t (2.63)

Comparing the previous definition with eq.(2.62), it is possible to obtain an expression for the first Piola-Kirchhoff stress tensor as a function of the strain-energy function:

ji ij ( )_{, P} F F P F ∂Ψ ∂Ψ = = ∂ ∂ (2.64) 28

(41)

It is presumed that Ѱ can be obtained from physical experiments, which defines a given material.

The strain-energy function can be represented in equivalent forms. It is known that Ѱ must remain invariant under rigid body rotations, which means that it is independent of the rotational part R of the deformation gradient F=RU. As a consequence, it is possible to write the strain energy as a function of only the stretch tensor U. However, it is more common to write this function in order to the right Cauchy-Green tensor C, since C=U2_{. Recalling also}

equation (2.36) which defines the time derivative of the Lagrange strain tensor:  1=  2

E C and

observing that this tensor is the work conjugate of to the second Piola-Kirchhoff stress, S, it is possible to obtain a totally Lagrangian constitutive equation:

d _: 1 _: d C C 2S C Ψ ₌∂Ψ ₌ ∂   t (2.65) ∂Ψ ∂Ψ = = ∂ ∂ ( ) ( ) 2 C E S C E (2.66)

Finally, since the Cauchy stress can be defined from eq. (2.45) as: _σ₌_J−1_PFT

it can also be expressed in terms of the strain-energy function Ѱ substituting P with the definitions stated previously, obtaining: − ∂Ψ − ∂Ψ = = ∂ ∂













T 1 ( ) 1 ( ) T J F J F σ F F F F − ∂Ψ = ∂ 1 ( ) T 2J C σ F F C (2.67)

and in terms of the right Cauchy-Green tensor C [2], [8]:

− ∂Ψ = ∂ 1 ( ) T 2J C σ F F C (2.68)

2.4.2 Elasticity Tensors

In order to obtain solutions for the nonlinear problems in computational finite elasticity and inelasticity, iterative solutions techniques are applied to solve a sequence of linearized problems. This strategy requires the linearization of the constitutive equations, which is basically differentiation.

In order to linearize the relationship between S and C or E, the chain rule can be applied and it is possible to obtain a linear relationship between the directional derivative of S and the linearized strain DuE in component form as:

IJ IJ KL 0 d D S S (E [ ]) d u u ∈= = φ+ ∈ = ∈ 3 IJ KL K ,L 1 _KL ₀ S d _{E [} _] E d u = _∈= ∂ = φ+ ∈ = ∂ ∈

∑

29

(42)

3 IJ KL K ,L 1 KL S D E E = ∂ = ∂

∑

u (2.69)

The previous equation can be more concisely expressed introducing a new forth order tensor



as:

DuS=



:DuE _(2.70)

where



represents the Lagrangian elasticity tensor and can be defined as:

IJKL IJ KL S ( ) 2 or 2 C C S C C ∂ ∂ = = ∂ ∂  (2.71)

Or in terms of the Lagrange Strain tensor E: I IJK J KL L S ( ) or E

C

S E E ∂ ∂ = = ∂ ∂



(2.72)

The quantity



measures the change in stress which results from a change in strain. It is always symmetric in its first and second slots, therefore it is said that possesses the minor symmetries:

IJKL JIKL IJLK

C

=

C

=

C

_(2.73)

Recalling both eq. (2.66), where S can be derived from Ѱ, and equation (2.71), it is possible to obtain the following relation for the elasticities in the material description:

IJKL 2 2 IJ KL ( ) 4 or 4 C C

C

C C C ∂ Ψ ∂ Ψ = = ∂ ∂ ∂ ∂



(2.74)

with the symmetries:

T

IJKL KLIJ or

C

= =

 

(2.75)

Therefore, it is also said that



possesses the major symmetries. The condition (2.75) it is a necessary and sufficient condition for a material to be hyperelastic [2], [8].

The elasticity tensor in the spatial or Eulerian description, denoted by  is defined as the Piola push-forward operation of



, and after some manipulation, it can be obtained as:

1

ijkl iI jJ kK lL IJKL

c ₌J F F F F−

C

(2.76) With the minor symmetries: cijkl=cjikl=cijlk, and also for hyperelastic materials, with the major symmetries:.cijkl=cklij

2.4.3 Isotropic Hyperelasticity

Isotropy is a property that is based on the physical idea that the response of the material, when studied in a stress-strain experiment, is the same in all directions. Therefore, it is possible to restrict the strain-energy function for materials, such as rubber, that present this particular behaviour [2].

Application of anisotropic visco-hyperelastic constitutive models for the simulation of biological tissues

Faculdade de Engenharia da Universidade do Porto

Application of anisotropic visco-hyperelastic

constitutive models for the simulation of

biological tissues

Integrated Master in Mechanical Engineering

Dissertation Thesis

Author

Maria Castro Paupério Vila Pouca

Supervisors

Prof. Dr. Marco Paulo L. Parente

Prof. Dr. Renato Natal Jorge

Application of anisotropic visco-hyperelastic

constitutive models for the simulation of

biological tissues

Maria Castro Paupério Vila Pouca

Mestrado Integrado em Engenharia Mecânica

Abstract

Resumo

Agradecimentos

Contents

List of Figures

List of Tables

List of Symbols







List of Operators

Chapter 1

Introduction

1.1 Context

1.2 Thesis Organisation

Chapter 2

Continuum Mechanics and Finite Element Method

2.1 Standard Continuum Mechanics

2.2 Kinematics

2.2.1 Description of Motion

2.2.2 Kinematics Descriptions

2.2.3 Deformation Gradient

Ω

Ω

∇x

2.2.4 Polar Decomposition Theorem

2.2.5 Right-Cauchy Green and Left Cauchy-Green Deformation Tensor

2.2.6 Stretch and Principal Stretches

{ }

{ }

{ }

{ }

∑

∑

∑

2.2.7 Strain Measures

2.2.8 Velocity and Material Time Derivatives

2.2.9 Stress Measures

2.3 Equations of Motion and Equilibrium for Deformable Solids

2.3.1 Linear Momentum Balance in terms of Cauchy Stress

2.3.2 Principle of Virtual Work

∫

∫

∫

∫

( )

( )

∫

∫

(

)

∫

∫

∫

∫

∫

∫

∫

∫

∫

∫

∫

∫

_∫