Material deformation

Consider a body in 3D Euclidian space. The initial or reference configuration Ω0 undergoes deformation to a deformed or current configuration, Ω, over a given time, 𝑡. The location of any point within the reference configuration is described by the material coordinates, 𝑿. Likewise, the spatial coordinates of the current configuration are defined by 𝒙. The deformation of the reference configuration to deformed configuration is described by the deformation map, 𝝌. In the interests of relevance for the analysis to come, this explanation will focus mainly on the material (Lagrangian) reference frame.

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Figure 16 – Schematic representation of the deformation field.

The displacement field 𝒖 describes the relation of the spatial and material coordinates with respect to time, such that:

For further analysis, we must not only consider the displacement of a single point, but also those immediately around it. To do this, we take the derivative of each component of the deformed vector with respect to the initial reference vector. This quantity contains nine components and is known as the deformation gradient tensor, 𝑭. This is most easily visualised in tensor notation where:

At a given time step, Equation 1 can be simplified such that:

and as such:

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where 𝑰 is the identity matrix. In cases where there is rigid body displacement, all particles move by an identical distance and direction over time, therefore being independent of 𝑿, such that 𝑭 = 𝑰. From the deformation gradient tensor we can define a number of useful quantities, which describe the process of deformation. To do this we must first decompose 𝑭 into components of pure stretch and pure rotation. Within the reference configuration this is termed the right polar decomposition and is defined such that:

where 𝑹 is the rotational tensor and 𝑼 is the right stretch tensor. The right stretch tensor measures local stretching or contraction along orthogonal eigenvectors in the material configuration. The rotational tensor captures the change in local orientation. As such, in rigid body rotation 𝑭 = 𝑹 and conversely in pure stretch 𝑭 = 𝑼.

The right Cauchy-Green tensor 𝑪 is now introduced and defined such that:

where 𝑪 is a positive, symmetric tensor. In the derivation of Equation 6 through manipulation of Equation 5, it can be shown that 𝑹^𝑇𝑹 = 𝑰. As such, rotational components are removed from right Cauchy-Green tensor, which becomes important in the calculation of strain within the material coordinates moving forward.

Volume change as a result of deformation equates to the product of the orthogonal components of the deformation gradient tensor, otherwise defined as 𝑱 = 𝑑𝑒𝑡𝑭 . When considering compressible materials, both volumetric and deviatoric components of the deformation gradient must be considered. The deviatoric deformation gradient 𝑭̃ represents pure shape change in the body. By definition, it must satisfy Equation 7:

In order to satisfy this, 𝑭̃ is calculated using Equation 8:

The deviatoric right Cauchy-Green tensor, 𝑪̃ is defined similarly to Equation 6:

The symmetric, second order, Green-Lagrange strain tensor is defined as:

𝑭 = 𝑹𝑼 ⁵

𝑪 = 𝑭^𝑇𝑭 ⁶

𝑑𝑒𝑡𝑭̃ = 1 ⁷

𝑭̃ = 𝐽⁻¹³𝑭 ⁸

𝑪̃ = 𝐽⁻²³𝑪 ⁹

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Such a definition evaluates strain, removing components of rotation and rigid body displacement.

It is noted that similar quantities can be defined in the spatial form. Material and spatial measures remain approximately equivalent at small strains.

The internal forces within the body are expressed through the quantity of stress. The Cauchy traction vector 𝒕 is defined as the force per unit area acting along the normal 𝒏, for every infinitesimal surface within the body. The similar first Piola-Kirchoff traction vector 𝑻 is defined in the reference configuration (with 𝑵 , the corresponding normal vector). Cauchy’s stress theorem proposes the existence of unique second-order tensor fields 𝝈 and 𝑷, such that:

Where 𝝈 and 𝑷 are the Cauchy stress and first Piola-Kirchoff stress tensors respectively. These tensors are further defined as:

The Cauchy stress can be decomposed into deviatoric and hydrostatic components. The hydrostatic stress is defined as:

and is equal and opposite to the pressure, p. As such, the deviatoric stress of the Cauchy stress tensor 𝝈̃ is defined:

As such, it holds that 𝑡𝑟 𝝈̃ = 0.

Hyperelasticity

Hyperelastic (Green elastic) materials have a Helmholtz free energy function 𝜓 defined per unit volume. The Helmholtz free energy is a thermodynamic measure of the energy within a closed system such that:

Where 𝑈 is the internal energy, 𝑇 is the temperature of the surrounding environment and 𝑆 is the entropy within the system. The internal energy is taken as the total energy within system at

𝑬 =1

2(𝑪 − 𝑰) ¹⁰

𝒕 = 𝝈 ∙ 𝒏 ¹¹

𝑻 = 𝑷 ∙ 𝑵 ¹²

𝝈 = 𝐽⁻¹𝑷𝑻^{𝑇 13}

𝑷 = 𝑱𝝈𝑭^{−𝑇 14}

𝝈_ℎ𝑦𝑑= −1

3𝑡𝑟 (𝝈) ¹⁵

𝝈̃ = 𝝈 − 𝝈_ℎ𝑦𝑑 = 𝝈 − 𝑝𝑰 ¹⁶

𝜓 = 𝑈 − 𝑇𝑆 ¹⁷

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constant temperature and volume. The Helmholtz energy therefore describes the energy required to create the system when the product of temperature and surrounding entropy is also taken into account.

In general, rubber-like solids and soft tissues dissipate energy through loading and un-loading, such that the stress-strain curve does not follow the same path. Figure 17 demonstrates this phenomenon, known as the Mullins effect (100).

Figure 17 – Example of a stress/strain plot depicting a region of energy loss as described by the Mullins effect.

It can be defined that the Helmholtz free energy is a function of a given strain tensor, such as 𝑭 alone. When 𝜓 = 𝜓(𝑭), it is termed the strain energy density function. In such cases where work done by stresses in the material are path independent, the material is termed hyperelastic.

Whilst hyperelasticity is theoretically impossible, it is a necessary assumption for much real-world material characterisation. The assumption of hyperelasticity and fitting of a strain energy function to experimental data is the basic premise of constitutive modelling of soft tissues.

Material isotropy exists when constitutive behaviour is independent of the material axis. Truly isotropic biological materials rarely exist. However, this assumption is often necessary, or at least convenient. In such cases 𝜓 is simply a function of the invariants of 𝑪:

where the invariants themselves are:

When it is given that 𝜓1= ^𝜕𝜓

𝜕𝐼₁, 𝜓2 =^𝜕𝜓

𝜕𝐼₂ and 𝜓3=^𝜕𝜓

𝜕𝐼₃, the second Piola-Kirchoff stress, 𝑺, can be evaluated as:

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and the Cauchy stress can now be derived in terms of the defined strain energy function and the deformation dependant invariants of 𝑪:

In practical terms, we can consider that many soft tissues are comprised of water-based fluid held within a solid scaffold. At the stresses seen in soft tissue modelling, it is generally accepted that the fluid phase can be considered incompressible. This incompressibility, or near-incompressibility is often inferred onto the entire tissue. In true near-incompressibility 𝐽 = 1 and there would be no energy change resulting from volume change. Instead, near-incompressibility is often assumed and the hyperelastic constitutive equation can be rewritten with the distortional and volumetric energy components:

Here 𝜓̃, the deviatoric strain energy density function, is derived from the deviatoric right Cauchy-Green tensor 𝜓̃ = 𝜓(𝑪̃). Different formulations of the volumetric energy component 𝑈(𝐽) are defined by different authors and hence used in different finite element implementations. One such example (101) defines:

where 𝑘 is a material parameter of compressibility; the bulk modulus. With the pressure, p given as:

The second Piola-Kirchoff stress can be reformed as Equation 27:

incorporating the discussed distortional and volumetric components.