Conjugate Stress/Strain Pairs


The governing tensors for stress and the rate of deformation can be encoded into a set of independent conjugate base pairs wherein each stress/strain conjugate pair is a scalar pair. Individual constitutive equations govern the response of each base pair, which may be elastic, viscoelastic, etc., as determined from experiment. After the separate constitutive equations for all base pairs have been solved, their constituents can be decoded to update the tensorial states of stress and the rate of deformation. An admissible encoding/decoding algorithm is one-to-one [1].


In 2003 Dr. K. R. Rajagopal [2] discussed implicit elastic constitutive models that Helmholtz free energy can be written as a function of temperature, stress and strain. This paper also provides answers either stress causes to strain or strain causes to stress question like most commonly used stress strain relation Hooke’s law \sigma= E \epsilon.

In 2012 Dr. A. R. Srinivasa [3] explained benefit of upper triangular decomposition for Green elastic materials. With these decomposition researchers don’t need to do tensor invariant calculations to derive constitutive relations. Deformation gradient can be decomposed physically apparent, meaningful and simple elements.

Material Models

Modeling biological fibers

Getting inspiration from these papers, Dr. Freed proposed 1d fiber model [4] which can represents strain limiting behavior. Constitute model can be seen below where E^E is elastin elastic modulus, E^C is collegen elastic modulus and \beta=1/ \epsilon^{crimp}_{max}.


Material parameters are pretty easy to measure by using stress strain curve asymptotes. This method was introduced by Babu [5].

Planar Analysis of Biological Tissues

After upper triangular (QR) factorization, deformation gradient decomposed into rotation tensor and upper triangular matrix.  \mathrm{F} = \mathrm{Q}\mathrm{\~F} where \mathrm{Q} is orthogonal matrix  and  \mathrm{\~F} is upper triangular matrix.

 [ \tilde{\mathbf{F}} ] = \begin{bmatrix}a & a \gamma \\ 0 & b \end{bmatrix}

Let’s consider unit square deform into a parallelogram. Extensions a and b and shear \gamma =\tan \phi can be directly observed unlike right stretch tensor (U) which comes from polar decomposition.

Conjugate pair stress/strain approach gives us an opportunity to decompose \tilde{\mathbf{F}} into product of  three fundamental modes of deformation which are dilation, squeeze and shear in material frame [1] .


Anisotropic Fiber Reinforced Composite Model

Conjugate pair approach is expanded upon to include anisotropic materials. This model has some advantages over  widely used fiber model of  Spencer’s [5]. A new parameter n is introduced to quantify the strength of anisotropy. It arises in the decoding/encoding maps, and not in the constitutive equations themselves, which remain isotropic [6]. Differences and similarities between the two models can be seen in the table below.


1) Freed, Alan, Veysel Erel, and Michael Moreno. “Conjugate stress/strain base pairs for planar analysis of biological tissues.” Journal of Mechanics of Materials and Structures 12.2 (2016): 219-247.

2) Rajagopal, Kumbakonam R. “On implicit constitutive theories.” Applications of Mathematics 48.4 (2003): 279-319.

3) Srinivasa, A. R. “On the use of the upper triangular (or QR) decomposition for developing constitutive equations for Green-elastic materials.” International Journal of Engineering Science 60 (2012): 1-12.

4) Babu, Anju R., Achu G. Byju, and Namrata Gundiah. “Biomechanical properties of human ascending thoracic aortic dissections.” Journal of biomechanical engineering 137.8 (2015): 081013.

5) Spencer, Anthony James Merrill. “Deformations of fibre-reinforced materials.” (1972): 128.

6) Erel, Veysel, and Alan D. Freed. “Stress/strain basis pairs for anisotropic materials.” Composites Part B: Engineering 120 (2017): 152-158.