Tensile Modulus (E) is the slope of stress strain curve of a specific material sample under direct tensile loading.

While flexural modulus/ bending modulus is obtained from slope of moment-curvature diagram dividing it by moment of intertia of the beam specimen; (E = EI / I). However, in reality flexural modulus value used is determined from the slope of the stress strain diagram of a material which is the modulus of Elasticity . These may at times vary for some material in tension than in compression. Fact remains that "modulus" represents the Young's Modulus of Elasticity.  

When the sample is loaded in longitudinal mode the elastic modulus obtained will refer to the orientation of the sample’s length. When a material is flexed, there is both tension and compression. For homogeneous and isotropic materials, the elastic modulus obtained from a bending test coincides with the elastic modulus measured in an axial test (longitudinal direction). Nevertheless, it is known that when flexed, the surface is the region that it is submitted to the greatest values of stress. For this reason, if a sample presents the stiffness of the surface different from the center (for example, if there is a stiffness gradient along the thickness); or if the sample presents small flaws such as pores, cracks and micro-cracks on the surface, there will be a difference between the values obtained using flexural and longitudinal modes.

Tensile modulus (also know as Young's modulus) is a measure of a materials flexibility along an axis of strain, which is not normalized for thickness. It's essentially the relationship between stress and strain (Hooke's Law) in the materials' linear elastic region where the stress is defined as the force distributed over the samples cross-sectional area perpendicular to the applied force and the strain is the relative change in length. The stress strain data is collected from a sample place under tensile loading.

Flexural modulus (also know as the Bend modulus) is a measure of a materials stiffness/ resistance to bend when a force is applied perpendicular to the long edge of a sample - the sampling configuration is known as the three point bend test. When bending occurs, the surface where the force is applied experiences compression forces while the opposite surface experiences tensile forces making this measure most suited to isotropic materials. Similar to the tensile modulus, the flexural modulus is defined as the relationship between stress and strain (Hooke's Law again) in the materials' linear elastic region where the stress is defined as the force applied over the length of the sample and the strain is the deflection of the samples' cross sectional area (width x height).

Ideally, the Tensile and Flexural modulus would be the same since by definition they are both the materials ability to resist deformation under loads, even though the loads they are resisting are different.  However the results are usually different because measurements are not made in an ideal state.

A little clarification regarding flexural modulus may be added. It is the ratio of the bending stress and bending strain (caused in axial direction by bending) at a particular level from the neutral surface. Even the flexural modulus in in tension and in compression are not identical because of non-isotropic nature of materials.

Thank you all for the explinations

In some materials, as wood and composites, the young mouduli obtained by tensile and compressive tests are different. Therefore, the flexural modulus has an intermediate value. You can find the relation between them and more references in

Determination of tensile and compressive moduli by flexural tests

• September 2006
• Polymer Testing 25(6):766-771

In simple terms, both TENSILE MODULUS and FLEXURAL MODULUS are same for an ideal material.

But real materials show different strength during compression and tension resulting in different values for both tensile modulus and flexural modulus.

Qn: How to choose between tensile modulus and flexural modulus ?

Ans: Closely examine your application of the material or the loading pattern of your member

ASTM FOR TENSILE D-638

ASTM FOR FLEXURAL D-790